diff --git a/verification/CurveField.llbc b/verification/CurveField.llbc index 616dac7..2c537aa 100644 --- a/verification/CurveField.llbc +++ b/verification/CurveField.llbc @@ -1 +1 @@ -{"charon_version":"0.1.212","translated":{"crate_name":"curve25519","options":{"ullbc":false,"precise_drops":false,"skip_borrowck":false,"mir":null,"rustc_args":[],"targets":[],"monomorphize":false,"monomorphize_mut":null,"start_from":["crate::field","crate::backend::serial::u64::field","crate::backend::serial::curve_models","crate::edwards","crate::backend::serial::u64::scalar::_::add","crate::backend::serial::u64::scalar::_::sub","crate::backend::serial::u64::scalar::_::mul","crate::backend::serial::u64::scalar::_::square","crate::backend::serial::u64::scalar::_::montgomery_mul","crate::backend::serial::u64::scalar::_::montgomery_square","crate::backend::serial::u64::scalar::_::montgomery_reduce","crate::backend::serial::u64::scalar::_::montgomery_invert","crate::backend::serial::u64::scalar::_::as_montgomery","crate::backend::serial::u64::scalar::_::from_montgomery","crate::backend::serial::u64::scalar::_::from_bytes_wide","crate::scalar::_::from_bytes_mod_order","crate::scalar::_::from_bytes_mod_order_wide","crate::ed_sigs::verification_key::_::verify_sha512"],"start_from_if_exists":[],"start_from_attribute":null,"start_from_pub":false,"include":[],"opaque":["crate::ed_sigs::sha512_hash3","ed25519","crate::field::_::internal_invert_batch","crate::backend::serial::scalar_mul::variable_base","crate::backend::serial::scalar_mul::vartime_triple_base","crate::scalar::_::non_adjacent_form_128","crate::backend::serial::scalar_mul::straus","crate::backend::serial::scalar_mul::precomputed_straus","crate::backend::serial::scalar_mul::pippenger","crate::backend::vector","crate::backend::scalar_fits_in_128_bits","crate::edwards::decompress","crate::edwards::_::sum","crate::edwards::_::from_slice"],"exclude":[],"extract_opaque_bodies":false,"translate_all_methods":false,"duplicate_defaulted_methods":true,"lift_associated_types":["*"],"hide_marker_traits":true,"remove_adt_clauses":true,"hide_allocator":true,"remove_unused_self_clauses":true,"desugar_drops":false,"ops_to_function_calls":true,"index_to_function_calls":true,"treat_box_as_builtin":true,"raw_consts":false,"unsized_strings":false,"reconstruct_fallible_operations":true,"reconstruct_asserts":true,"unbind_item_vars":true,"print_original_ullbc":false,"print_ullbc":false,"print_built_llbc":false,"print_llbc":false,"dest_dir":null,"dest_file":"/home/oho/GitClone/Claude/FormalVerification/anza-ed25519-verified/verification/CurveField.llbc","no_dedup_serialized_ast":false,"format":null,"no_serialize":false,"no_typecheck":false,"no_normalize":false,"abort_on_error":false,"error_on_warnings":false,"preset":"Aeneas"},"target_information":[{"key":"x86_64-unknown-linux-gnu","value":{"target_pointer_size":8,"is_little_endian":true}}],"files":[{"id":0,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64/scalar.rs"},"crate_name":"curve25519","contents":"//! Arithmetic mod \\\\(2\\^{252} + 27742317777372353535851937790883648493\\\\)\n//! with five \\\\(52\\\\)-bit unsigned limbs.\n//!\n//! \\\\(51\\\\)-bit limbs would cover the desired bit range (\\\\(253\\\\)\n//! bits), but isn't large enough to reduce a \\\\(512\\\\)-bit number with\n//! Montgomery multiplication, so \\\\(52\\\\) bits is used instead. To see\n//! that this is safe for intermediate results, note that the largest\n//! limb in a \\\\(5\\times 5\\\\) product of \\\\(52\\\\)-bit limbs will be\n//!\n//! ```text\n//! (0xfffffffffffff^2) * 5 = 0x4ffffffffffff60000000000005 (107 bits).\n//! ```\n\nuse core::fmt::Debug;\nuse core::ops::{Index, IndexMut};\nuse subtle::{Choice, ConditionallySelectable};\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::{DefaultIsZeroes, Zeroize};\n\nuse crate::constants;\n\n/// The `Scalar52` struct represents an element in\n/// \\\\(\\mathbb Z / \\ell \\mathbb Z\\\\) as 5 \\\\(52\\\\)-bit limbs.\n#[derive(Copy, Clone, Default)]\npub struct Scalar52(pub [u64; 5]);\n\nimpl Debug for Scalar52 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n f.write_str(\"Scalar52{..}\")\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for Scalar52 {}\n\nimpl Index for Scalar52 {\n type Output = u64;\n fn index(&self, _index: usize) -> &u64 {\n &(self.0[_index])\n }\n}\n\nimpl IndexMut for Scalar52 {\n fn index_mut(&mut self, _index: usize) -> &mut u64 {\n &mut (self.0[_index])\n }\n}\n\n/// u64 * u64 = u128 multiply helper\n#[inline(always)]\nfn m(x: u64, y: u64) -> u128 {\n (x as u128) * (y as u128)\n}\n\nimpl Scalar52 {\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: Scalar52 = Scalar52([0, 0, 0, 0, 0]);\n\n /// Unpack a 32 byte / 256 bit scalar into 5 52-bit limbs.\n #[rustfmt::skip] // keep alignment of s[*] calculations\n pub fn from_bytes(bytes: &[u8; 32]) -> Scalar52 {\n let mut words = [0u64; 4];\n for i in 0..4 {\n for j in 0..8 {\n words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);\n }\n }\n\n let mask = (1u64 << 52) - 1;\n let top_mask = (1u64 << 48) - 1;\n let mut s = Scalar52::ZERO;\n\n s[0] = words[0] & mask;\n s[1] = ((words[0] >> 52) | (words[1] << 12)) & mask;\n s[2] = ((words[1] >> 40) | (words[2] << 24)) & mask;\n s[3] = ((words[2] >> 28) | (words[3] << 36)) & mask;\n s[4] = (words[3] >> 16) & top_mask;\n\n #[cfg(feature = \"zeroize\")]\n words.zeroize();\n\n s\n }\n\n /// Reduce a 64 byte / 512 bit scalar mod l\n #[rustfmt::skip] // keep alignment of lo[*] and hi[*] calculations\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// extracted as its own function (pure refactor, semantics identical).\n /// A verification-side split keeps Montgomery calls out of the\n /// prefix walk's proof obligations.\n /// AENEAS-COMPAT: lo-half of the 52-bit split (pure refactor; struct\n /// literal instead of field mutation keeps higher-order IndexMut\n /// closures out of the extraction).\n fn split_words_lo(words: &[u64; 8]) -> Scalar52 {\n let mask = (1u64 << 52) - 1;\n Scalar52([\n words[0] & mask,\n ((words[0] >> 52) | (words[1] << 12)) & mask,\n ((words[1] >> 40) | (words[2] << 24)) & mask,\n ((words[2] >> 28) | (words[3] << 36)) & mask,\n ((words[3] >> 16) | (words[4] << 48)) & mask,\n ])\n }\n\n /// AENEAS-COMPAT: hi-half of the 52-bit split (pure refactor, as above).\n fn split_words_hi(words: &[u64; 8]) -> Scalar52 {\n let mask = (1u64 << 52) - 1;\n Scalar52([\n (words[4] >> 4) & mask,\n ((words[4] >> 56) | (words[5] << 8)) & mask,\n ((words[5] >> 44) | (words[6] << 20)) & mask,\n ((words[6] >> 32) | (words[7] << 32)) & mask,\n (words[7] >> 20) & mask,\n ])\n }\n\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// extracted as its own function (pure refactor, semantics identical).\n /// A verification-side split keeps Montgomery calls out of the\n /// prefix walk's proof obligations.\n\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// factored into named halves (pure refactor, semantics identical);\n /// keeps each verification walk at the proven-cheap ~25-step scale.\n fn from_bytes_wide_parts(bytes: &[u8; 64]) -> (Scalar52, Scalar52) {\n let mut words = [0u64; 8];\n for i in 0..8 {\n for j in 0..8 {\n words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);\n }\n }\n (Scalar52::split_words_lo(&words), Scalar52::split_words_hi(&words))\n }\n\n pub fn from_bytes_wide(bytes: &[u8; 64]) -> Scalar52 {\n let (lo, hi) = Scalar52::from_bytes_wide_parts(bytes);\n let lo = Scalar52::montgomery_mul(&lo, &constants::R); // (lo * R) / R = lo\n let hi = Scalar52::montgomery_mul(&hi, &constants::RR); // (hi * R^2) / R = hi * R\n Scalar52::add(&hi, &lo)\n }\n\n /// Pack the limbs of this `Scalar52` into 32 bytes\n #[rustfmt::skip] // keep alignment of s[*] calculations\n #[allow(clippy::identity_op)]\n pub fn to_bytes(self) -> [u8; 32] {\n let mut s = [0u8; 32];\n\n s[ 0] = (self.0[ 0] >> 0) as u8;\n s[ 1] = (self.0[ 0] >> 8) as u8;\n s[ 2] = (self.0[ 0] >> 16) as u8;\n s[ 3] = (self.0[ 0] >> 24) as u8;\n s[ 4] = (self.0[ 0] >> 32) as u8;\n s[ 5] = (self.0[ 0] >> 40) as u8;\n s[ 6] = ((self.0[ 0] >> 48) | (self.0[ 1] << 4)) as u8;\n s[ 7] = (self.0[ 1] >> 4) as u8;\n s[ 8] = (self.0[ 1] >> 12) as u8;\n s[ 9] = (self.0[ 1] >> 20) as u8;\n s[10] = (self.0[ 1] >> 28) as u8;\n s[11] = (self.0[ 1] >> 36) as u8;\n s[12] = (self.0[ 1] >> 44) as u8;\n s[13] = (self.0[ 2] >> 0) as u8;\n s[14] = (self.0[ 2] >> 8) as u8;\n s[15] = (self.0[ 2] >> 16) as u8;\n s[16] = (self.0[ 2] >> 24) as u8;\n s[17] = (self.0[ 2] >> 32) as u8;\n s[18] = (self.0[ 2] >> 40) as u8;\n s[19] = ((self.0[ 2] >> 48) | (self.0[ 3] << 4)) as u8;\n s[20] = (self.0[ 3] >> 4) as u8;\n s[21] = (self.0[ 3] >> 12) as u8;\n s[22] = (self.0[ 3] >> 20) as u8;\n s[23] = (self.0[ 3] >> 28) as u8;\n s[24] = (self.0[ 3] >> 36) as u8;\n s[25] = (self.0[ 3] >> 44) as u8;\n s[26] = (self.0[ 4] >> 0) as u8;\n s[27] = (self.0[ 4] >> 8) as u8;\n s[28] = (self.0[ 4] >> 16) as u8;\n s[29] = (self.0[ 4] >> 24) as u8;\n s[30] = (self.0[ 4] >> 32) as u8;\n s[31] = (self.0[ 4] >> 40) as u8;\n\n s\n }\n\n /// Compute `a + b` (mod l)\n pub fn add(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut sum = Scalar52::ZERO;\n let mask = (1u64 << 52) - 1;\n\n // a + b\n let mut carry: u64 = 0;\n for i in 0..5 {\n carry = a[i] + b[i] + (carry >> 52);\n sum[i] = carry & mask;\n }\n\n // subtract l if the sum is >= l\n Scalar52::sub(&sum, &constants::L)\n }\n\n /// Compute `a - b` (mod l)\n pub fn sub(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut difference = Scalar52::ZERO;\n let mask = (1u64 << 52) - 1;\n\n // a - b\n let mut borrow: u64 = 0;\n for i in 0..5 {\n borrow = a[i].wrapping_sub(b[i] + (borrow >> 63));\n difference[i] = borrow & mask;\n }\n\n // conditionally add l if the difference is negative\n difference.conditional_add_l(Choice::from((borrow >> 63) as u8));\n difference\n }\n\n pub(crate) fn conditional_add_l(&mut self, condition: Choice) -> u64 {\n let mut carry: u64 = 0;\n let mask = (1u64 << 52) - 1;\n\n for i in 0..5 {\n let addend = u64::conditional_select(&0, &constants::L[i], condition);\n carry = (carry >> 52) + self[i] + addend;\n self[i] = carry & mask;\n }\n\n carry\n }\n\n /// Compute a raw in-place carrying right shift over the limbs.\n #[inline(always)]\n pub(crate) fn shr1_assign(&mut self) -> u64 {\n let mut carry: u64 = 0;\n for i in (0..5).rev() {\n let limb = self[i];\n let next_carry = limb & 1;\n self[i] = (limb >> 1) | (carry << 51);\n carry = next_carry;\n }\n carry\n }\n\n /// Compute `a * b`\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of z[*] calculations\n pub (crate) fn mul_internal(a: &Scalar52, b: &Scalar52) -> [u128; 9] {\n let mut z = [0u128; 9];\n\n z[0] = m(a[0], b[0]);\n z[1] = m(a[0], b[1]) + m(a[1], b[0]);\n z[2] = m(a[0], b[2]) + m(a[1], b[1]) + m(a[2], b[0]);\n z[3] = m(a[0], b[3]) + m(a[1], b[2]) + m(a[2], b[1]) + m(a[3], b[0]);\n z[4] = m(a[0], b[4]) + m(a[1], b[3]) + m(a[2], b[2]) + m(a[3], b[1]) + m(a[4], b[0]);\n z[5] = m(a[1], b[4]) + m(a[2], b[3]) + m(a[3], b[2]) + m(a[4], b[1]);\n z[6] = m(a[2], b[4]) + m(a[3], b[3]) + m(a[4], b[2]);\n z[7] = m(a[3], b[4]) + m(a[4], b[3]);\n z[8] = m(a[4], b[4]);\n\n z\n }\n\n /// Compute `a^2`\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of return calculations\n fn square_internal(a: &Scalar52) -> [u128; 9] {\n let aa = [\n a[0] * 2,\n a[1] * 2,\n a[2] * 2,\n a[3] * 2,\n ];\n\n [\n m( a[0], a[0]),\n m(aa[0], a[1]),\n m(aa[0], a[2]) + m( a[1], a[1]),\n m(aa[0], a[3]) + m(aa[1], a[2]),\n m(aa[0], a[4]) + m(aa[1], a[3]) + m( a[2], a[2]),\n m(aa[1], a[4]) + m(aa[2], a[3]),\n m(aa[2], a[4]) + m( a[3], a[3]),\n m(aa[3], a[4]),\n m(a[4], a[4])\n ]\n }\n\n /// Compute `limbs/R` (mod l), where R is the Montgomery modulus 2^260\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of n* and r* calculations\n pub (crate) fn montgomery_reduce(limbs: &[u128; 9]) -> Scalar52 {\n\n #[inline(always)]\n fn part1(sum: u128) -> (u128, u64) {\n let p = (sum as u64).wrapping_mul(constants::LFACTOR) & ((1u64 << 52) - 1);\n ((sum + m(p, constants::L[0])) >> 52, p)\n }\n\n #[inline(always)]\n fn part2(sum: u128) -> (u128, u64) {\n let w = (sum as u64) & ((1u64 << 52) - 1);\n (sum >> 52, w)\n }\n\n // note: l[3] is zero, so its multiples can be skipped\n let l = &constants::L;\n\n // the first half computes the Montgomery adjustment factor n, and begins adding n*l to make limbs divisible by R\n let (carry, n0) = part1( limbs[0]);\n let (carry, n1) = part1(carry + limbs[1] + m(n0, l[1]));\n let (carry, n2) = part1(carry + limbs[2] + m(n0, l[2]) + m(n1, l[1]));\n let (carry, n3) = part1(carry + limbs[3] + m(n1, l[2]) + m(n2, l[1]));\n let (carry, n4) = part1(carry + limbs[4] + m(n0, l[4]) + m(n2, l[2]) + m(n3, l[1]));\n\n // limbs is divisible by R now, so we can divide by R by simply storing the upper half as the result\n let (carry, r0) = part2(carry + limbs[5] + m(n1, l[4]) + m(n3, l[2]) + m(n4, l[1]));\n let (carry, r1) = part2(carry + limbs[6] + m(n2,l[4]) + m(n4, l[2]));\n let (carry, r2) = part2(carry + limbs[7] + m(n3, l[4]) );\n let (carry, r3) = part2(carry + limbs[8] + m(n4, l[4]));\n let r4 = carry as u64;\n\n // result may be >= l, so attempt to subtract l\n Scalar52::sub(&Scalar52([r0, r1, r2, r3, r4]), l)\n }\n\n /// Compute `a * b` (mod l)\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut ab_limbs = Scalar52::mul_internal(a, b);\n let mut ab = Scalar52::montgomery_reduce(&ab_limbs);\n let mut rr_limbs = Scalar52::mul_internal(&ab, &constants::RR);\n let product = Scalar52::montgomery_reduce(&rr_limbs);\n\n #[cfg(feature = \"zeroize\")]\n {\n ab_limbs.zeroize();\n ab.zeroize();\n rr_limbs.zeroize();\n }\n\n product\n }\n\n /// Compute `a^2` (mod l)\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n #[allow(dead_code)] // XXX we don't expose square() via the Scalar API\n pub fn square(&self) -> Scalar52 {\n let mut aa_limbs = Scalar52::square_internal(self);\n let mut aa = Scalar52::montgomery_reduce(&aa_limbs);\n let mut rr_limbs = Scalar52::mul_internal(&aa, &constants::RR);\n let square = Scalar52::montgomery_reduce(&rr_limbs);\n\n #[cfg(feature = \"zeroize\")]\n {\n aa_limbs.zeroize();\n aa.zeroize();\n rr_limbs.zeroize();\n }\n\n square\n }\n\n /// Compute `(a * b) / R` (mod l), where R is the Montgomery modulus 2^260\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn montgomery_mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut limbs = Scalar52::mul_internal(a, b);\n let product = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n product\n }\n\n /// Compute `(a^2) / R` (mod l) in Montgomery form, where R is the Montgomery modulus 2^260\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn montgomery_square(&self) -> Scalar52 {\n let mut limbs = Scalar52::square_internal(self);\n let square = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n square\n }\n\n /// Puts a Scalar52 in to Montgomery form, i.e. computes `a*R (mod l)`\n #[inline(never)]\n pub fn as_montgomery(&self) -> Scalar52 {\n Scalar52::montgomery_mul(self, &constants::RR)\n }\n\n /// Takes a Scalar52 out of Montgomery form, i.e. computes `a/R (mod l)`\n #[allow(clippy::wrong_self_convention)]\n #[inline(never)]\n pub fn from_montgomery(&self) -> Scalar52 {\n let mut limbs = [0u128; 9];\n for i in 0..5 {\n limbs[i] = self[i] as u128;\n }\n let scalar = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n scalar\n }\n}\n\n#[cfg(test)]\nmod test {\n use super::*;\n\n /// Note: x is 2^253-1 which is slightly larger than the largest scalar produced by\n /// this implementation (l-1), and should show there are no overflows for valid scalars\n ///\n /// x = 14474011154664524427946373126085988481658748083205070504932198000989141204991\n /// x = 7237005577332262213973186563042994240801631723825162898930247062703686954002 mod l\n /// x = 3057150787695215392275360544382990118917283750546154083604586903220563173085*R mod l in Montgomery form\n pub static X: Scalar52 = Scalar52([\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x00001fffffffffff,\n ]);\n\n /// x^2 = 3078544782642840487852506753550082162405942681916160040940637093560259278169 mod l\n pub static XX: Scalar52 = Scalar52([\n 0x0001668020217559,\n 0x000531640ffd0ec0,\n 0x00085fd6f9f38a31,\n 0x000c268f73bb1cf4,\n 0x000006ce65046df0,\n ]);\n\n /// x^2 = 4413052134910308800482070043710297189082115023966588301924965890668401540959*R mod l in Montgomery form\n pub static XX_MONT: Scalar52 = Scalar52([\n 0x000c754eea569a5c,\n 0x00063b6ed36cb215,\n 0x0008ffa36bf25886,\n 0x000e9183614e7543,\n 0x0000061db6c6f26f,\n ]);\n\n /// y = 6145104759870991071742105800796537629880401874866217824609283457819451087098\n pub static Y: Scalar52 = Scalar52([\n 0x000b75071e1458fa,\n 0x000bf9d75e1ecdac,\n 0x000433d2baf0672b,\n 0x0005fffcc11fad13,\n 0x00000d96018bb825,\n ]);\n\n /// x*y = 36752150652102274958925982391442301741 mod l\n pub static XY: Scalar52 = Scalar52([\n 0x000ee6d76ba7632d,\n 0x000ed50d71d84e02,\n 0x00000000001ba634,\n 0x0000000000000000,\n 0x0000000000000000,\n ]);\n\n /// x*y = 658448296334113745583381664921721413881518248721417041768778176391714104386*R mod l in Montgomery form\n pub static XY_MONT: Scalar52 = Scalar52([\n 0x0006d52bf200cfd5,\n 0x00033fb1d7021570,\n 0x000f201bc07139d8,\n 0x0001267e3e49169e,\n 0x000007b839c00268,\n ]);\n\n /// a = 2351415481556538453565687241199399922945659411799870114962672658845158063753\n pub static A: Scalar52 = Scalar52([\n 0x0005236c07b3be89,\n 0x0001bc3d2a67c0c4,\n 0x000a4aa782aae3ee,\n 0x0006b3f6e4fec4c4,\n 0x00000532da9fab8c,\n ]);\n\n /// b = 4885590095775723760407499321843594317911456947580037491039278279440296187236\n pub static B: Scalar52 = Scalar52([\n 0x000d3fae55421564,\n 0x000c2df24f65a4bc,\n 0x0005b5587d69fb0b,\n 0x00094c091b013b3b,\n 0x00000acd25605473,\n ]);\n\n /// a+b = 0\n /// a-b = 4702830963113076907131374482398799845891318823599740229925345317690316127506\n pub static AB: Scalar52 = Scalar52([\n 0x000a46d80f677d12,\n 0x0003787a54cf8188,\n 0x0004954f0555c7dc,\n 0x000d67edc9fd8989,\n 0x00000a65b53f5718,\n ]);\n\n // c = (2^512 - 1) % l = 1627715501170711445284395025044413883736156588369414752970002579683115011840\n pub static C: Scalar52 = Scalar52([\n 0x000611e3449c0f00,\n 0x000a768859347a40,\n 0x0007f5be65d00e1b,\n 0x0009a3dceec73d21,\n 0x00000399411b7c30,\n ]);\n\n #[test]\n fn mul_max() {\n let res = Scalar52::mul(&X, &X);\n for i in 0..5 {\n assert!(res[i] == XX[i]);\n }\n }\n\n #[test]\n fn square_max() {\n let res = X.square();\n for i in 0..5 {\n assert!(res[i] == XX[i]);\n }\n }\n\n #[test]\n fn montgomery_mul_max() {\n let res = Scalar52::montgomery_mul(&X, &X);\n for i in 0..5 {\n assert!(res[i] == XX_MONT[i]);\n }\n }\n\n #[test]\n fn montgomery_square_max() {\n let res = X.montgomery_square();\n for i in 0..5 {\n assert!(res[i] == XX_MONT[i]);\n }\n }\n\n #[test]\n fn mul() {\n let res = Scalar52::mul(&X, &Y);\n for i in 0..5 {\n assert!(res[i] == XY[i]);\n }\n }\n\n #[test]\n fn montgomery_mul() {\n let res = Scalar52::montgomery_mul(&X, &Y);\n for i in 0..5 {\n assert!(res[i] == XY_MONT[i]);\n }\n }\n\n #[test]\n fn add() {\n let res = Scalar52::add(&A, &B);\n let zero = Scalar52::ZERO;\n for i in 0..5 {\n assert!(res[i] == zero[i]);\n }\n }\n\n #[test]\n fn sub() {\n let res = Scalar52::sub(&A, &B);\n for i in 0..5 {\n assert!(res[i] == AB[i]);\n }\n }\n\n #[test]\n fn from_bytes_wide() {\n let bignum = [255u8; 64]; // 2^512 - 1\n let reduced = Scalar52::from_bytes_wide(&bignum);\n for i in 0..5 {\n assert!(reduced[i] == C[i]);\n }\n }\n}\n"},{"id":1,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2018 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! The `u64` backend uses `u64`s and a `(u64, u64) -> u128` multiplier.\n//!\n//! On x86_64, the idiom `(x as u128) * (y as u128)` lowers to `MUL`\n//! instructions taking 64-bit inputs and producing 128-bit outputs. On\n//! other platforms, this implementation is not recommended.\n//!\n//! On Haswell and newer, the BMI2 extension provides `MULX`, and on\n//! Broadwell and newer, the ADX extension provides `ADCX` and `ADOX`\n//! (allowing the CPU to compute two carry chains in parallel). These\n//! will be used if available.\n\npub mod field;\n\npub mod scalar;\n\npub mod constants;\n"},{"id":2,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Serial implementations of field, scalar, point arithmetic.\n//!\n//! When the vector backend is disabled, the crate uses the mixed-model strategy\n//! for implementing point operations and scalar multiplication; see the\n//! [`curve_models`] and [`scalar_mul`] documentation for more information.\n//!\n//! When the vector backend is enabled, the field and scalar\n//! implementations are still used for non-vectorized operations.\n\n#[doc(hidden)]\npub mod u64;\n\npub mod curve_models;\n\npub mod scalar_mul;\n"},{"id":3,"name":{"Local":"curve25519/solana-ed25519/src/backend.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! **INTERNALS:** Pluggable implementations for different architectures.\n//!\n//! The backend code is split into two parts: a serial backend,\n//! and a vector backend.\n//!\n//! The [`serial`] backend contains 32- and 64-bit implementations of\n//! field arithmetic and scalar arithmetic, as well as implementations\n//! of point operations using the mixed-model strategy (passing\n//! between different curve models depending on the operation).\n//!\n//! The [`vector`] backend contains implementations of vectorized\n//! field arithmetic, used to implement point operations using a novel\n//! implementation strategy derived from parallel formulas of Hisil,\n//! Wong, Carter, and Dawson.\n//!\n//! Because the two strategies give rise to different curve models,\n//! it's not possible to reuse exactly the same scalar multiplication\n//! code (or to write it generically), so both serial and vector\n//! backends contain matching implementations of scalar multiplication\n//! algorithms. These are intended to be selected by a `#[cfg]`-based\n//! type alias.\n//!\n//! The [`vector`] backend is selected by the `simd_backend` cargo\n//! feature; it uses the [`serial`] backend for non-vectorized operations.\n\nuse crate::EdwardsPoint;\nuse crate::Scalar;\n\npub mod serial;\n\n#[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\npub mod vector;\n\n#[derive(Copy, Clone)]\nenum BackendKind {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n Avx2,\n Serial,\n}\n\n#[inline]\nfn get_selected_backend() -> BackendKind {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n {\n cpufeatures::new!(cpuid_avx2, \"avx2\");\n let token_avx2: cpuid_avx2::InitToken = cpuid_avx2::init();\n if token_avx2.get() {\n return BackendKind::Avx2;\n }\n }\n\n BackendKind::Serial\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn pippenger_optional_multiscalar_mul(scalars: I, points: J) -> Option\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator>,\n{\n use crate::traits::VartimeMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::pippenger::spec_avx2::Pippenger::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::pippenger::Pippenger::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n }\n}\n\n#[cfg(feature = \"alloc\")]\npub(crate) enum VartimePrecomputedStraus {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n Avx2(vector::scalar_mul::precomputed_straus::spec_avx2::VartimePrecomputedStraus),\n Scalar(serial::scalar_mul::precomputed_straus::VartimePrecomputedStraus),\n}\n\n#[cfg(feature = \"alloc\")]\nimpl VartimePrecomputedStraus {\n pub fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => VartimePrecomputedStraus::Avx2(\n vector::scalar_mul::precomputed_straus::spec_avx2::VartimePrecomputedStraus::new(\n static_points,\n ),\n ),\n BackendKind::Serial => VartimePrecomputedStraus::Scalar(\n serial::scalar_mul::precomputed_straus::VartimePrecomputedStraus::new(\n static_points,\n ),\n ),\n }\n }\n\n /// Return the number of static points in the precomputation.\n pub fn len(&self) -> usize {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.len(),\n VartimePrecomputedStraus::Scalar(inner) => inner.len(),\n }\n }\n\n /// Determine if the precomputation is empty.\n pub fn is_empty(&self) -> bool {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.is_empty(),\n VartimePrecomputedStraus::Scalar(inner) => inner.is_empty(),\n }\n }\n\n pub fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator,\n J::Item: core::borrow::Borrow,\n K: IntoIterator>,\n {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.optional_mixed_multiscalar_mul(\n static_scalars,\n dynamic_scalars,\n dynamic_points,\n ),\n VartimePrecomputedStraus::Scalar(inner) => inner.optional_mixed_multiscalar_mul(\n static_scalars,\n dynamic_scalars,\n dynamic_points,\n ),\n }\n }\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn straus_multiscalar_mul(scalars: I, points: J) -> EdwardsPoint\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator,\n J::Item: core::borrow::Borrow,\n{\n use crate::traits::MultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::straus::spec_avx2::Straus::multiscalar_mul::(scalars, points)\n }\n BackendKind::Serial => {\n serial::scalar_mul::straus::Straus::multiscalar_mul::(scalars, points)\n }\n }\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn straus_optional_multiscalar_mul(scalars: I, points: J) -> Option\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator>,\n{\n use crate::traits::VartimeMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::straus::spec_avx2::Straus::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::straus::Straus::optional_multiscalar_mul::(scalars, points)\n }\n }\n}\n\n/// Perform constant-time, variable-base scalar multiplication.\npub fn variable_base_mul(point: &EdwardsPoint, scalar: &Scalar) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => vector::scalar_mul::variable_base::spec_avx2::mul(point, scalar),\n BackendKind::Serial => serial::scalar_mul::variable_base::mul(point, scalar),\n }\n}\n\n/// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n#[allow(non_snake_case)]\npub fn vartime_double_base_mul(a: &Scalar, A: &EdwardsPoint, b: &Scalar) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => vector::scalar_mul::vartime_double_base::spec_avx2::mul(a, A, b),\n BackendKind::Serial => serial::scalar_mul::vartime_double_base::mul(a, A, b),\n }\n}\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n///\n/// This function uses an optimized path when \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\),\n/// and falls back to general scalar multiplication otherwise.\n#[allow(non_snake_case)]\npub fn vartime_triple_base_mul_128_128_256(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n if !scalar_fits_in_128_bits(a1) || !scalar_fits_in_128_bits(a2) {\n return (a1 * A1) + (a2 * A2) + EdwardsPoint::mul_base(b);\n }\n\n vartime_triple_base_mul_128_128_256_prechecked(a1, A1, a2, A2, b)\n}\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) using the optimized 128/128/256-bit path.\n///\n/// Callers must ensure \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\).\n#[allow(non_snake_case)]\npub(crate) fn vartime_triple_base_mul_128_128_256_prechecked(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::vartime_triple_base::spec_avx2::mul_128_128_256_prechecked(\n a1, A1, a2, A2, b,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::vartime_triple_base::mul_128_128_256_prechecked(a1, A1, a2, A2, b)\n }\n }\n}\n\n#[inline]\nfn scalar_fits_in_128_bits(scalar: &Scalar) -> bool {\n scalar.as_bytes()[16..32].iter().all(|&byte| byte == 0)\n}\n"},{"id":4,"name":{"Local":"curve25519/solana-ed25519/src/lib.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n#![no_std]\n#![cfg_attr(docsrs, feature(doc_cfg))]\n#![cfg_attr(docsrs, doc(auto_cfg(hide(docsrs))))]\n//------------------------------------------------------------------------\n// Documentation:\n//------------------------------------------------------------------------\n#![doc(\n html_logo_url = \"https://cdn.jsdelivr.net/gh/dalek-cryptography/curve25519/docs/assets/dalek-logo-clear.png\"\n)]\n#![doc = include_str!(\"../README.md\")]\n//------------------------------------------------------------------------\n// Linting:\n//------------------------------------------------------------------------\n#![cfg_attr(allow_unused_unsafe, allow(unused_unsafe))]\n#![warn(\n clippy::mod_module_files,\n clippy::unwrap_used,\n missing_docs,\n rust_2018_idioms,\n unused_lifetimes,\n unused_qualifications\n)]\n\n//------------------------------------------------------------------------\n// External dependencies:\n//------------------------------------------------------------------------\n\n#[cfg(feature = \"alloc\")]\n#[allow(unused_imports)]\n#[macro_use]\nextern crate alloc;\n\n// TODO: move std-dependent tests to `tests/`\n#[cfg(any(test, feature = \"std\"))]\n#[macro_use]\nextern crate std;\n\n#[cfg(feature = \"digest\")]\npub use digest;\n\n// Internal macros. Must come first!\n#[macro_use]\npub(crate) mod macros;\n\n//------------------------------------------------------------------------\n// curve25519 public modules\n//------------------------------------------------------------------------\n\n// Scalar arithmetic mod l = 2^252 + ..., the order of the Ristretto group\npub mod scalar;\n\n// Point operations on the Montgomery form of Curve25519\npub mod montgomery;\n\n// Point operations on the Edwards form of Curve25519\npub mod edwards;\n\n// Point operations on the short Weierstrass form of Curve25519\npub mod short_weierstrass;\n\n// Group operations on the Ristretto group\npub mod ristretto;\n\n// Useful constants, like the Ed25519 basepoint\npub mod constants;\n\n// External (and internal) traits.\npub mod traits;\n\n//------------------------------------------------------------------------\n// ed25519 public modules\n//------------------------------------------------------------------------\n\npub mod ed_sigs;\n\n//------------------------------------------------------------------------\n// curve25519 internal modules\n//------------------------------------------------------------------------\n\n// Finite field arithmetic mod p = 2^255 - 19\npub(crate) mod field;\n\n// Arithmetic backends (using u32, u64, etc) live here\n#[cfg(docsrs)]\npub mod backend;\n#[cfg(not(docsrs))]\npub(crate) mod backend;\n\n// Generic code for window lookups\npub(crate) mod window;\n\n#[cfg(feature = \"lizard\")]\nmod lizard;\n\npub use crate::{\n edwards::EdwardsPoint, montgomery::MontgomeryPoint, ristretto::RistrettoPoint, scalar::Scalar,\n};\n\n// Build time diagnostics for validation\n#[cfg(curve25519_diagnostics = \"build\")]\nmod diagnostics;\n"},{"id":5,"name":{"Local":"curve25519/solana-ed25519/src/scalar.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// Portions Copyright 2017 Brian Smith\n// See LICENSE for licensing information.\n//\n// Authors:\n// - Isis Agora Lovecruft \n// - Henry de Valence \n// - Brian Smith \n\n//! Arithmetic on scalars (integers mod the group order).\n//!\n//! Both the Ristretto group and the Ed25519 basepoint have prime order\n//! \\\\( \\ell = 2\\^{252} + 27742317777372353535851937790883648493 \\\\).\n//!\n//! This code is intended to be useful with both the Ristretto group\n//! (where everything is done modulo \\\\( \\ell \\\\)), and the X/Ed25519\n//! setting, which mandates specific bit-twiddles that are not\n//! well-defined modulo \\\\( \\ell \\\\).\n//!\n//! All arithmetic on `Scalars` is done modulo \\\\( \\ell \\\\).\n//!\n//! # Constructing a scalar\n//!\n//! To create a [`Scalar`](struct.Scalar.html) from a supposedly canonical encoding, use\n//! [`Scalar::from_canonical_bytes`](struct.Scalar.html#method.from_canonical_bytes).\n//!\n//! This function does input validation, ensuring that the input bytes\n//! are the canonical encoding of a `Scalar`.\n//! If they are, we'll get\n//! `Some(Scalar)` in return:\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let one_as_bytes: [u8; 32] = Scalar::ONE.to_bytes();\n//! let a: Option = Scalar::from_canonical_bytes(one_as_bytes).into();\n//!\n//! assert!(a.is_some());\n//! ```\n//!\n//! However, if we give it bytes representing a scalar larger than \\\\( \\ell \\\\)\n//! (in this case, \\\\( \\ell + 2 \\\\)), we'll get `None` back:\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let l_plus_two_bytes: [u8; 32] = [\n//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,\n//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,\n//! ];\n//! let a: Option = Scalar::from_canonical_bytes(l_plus_two_bytes).into();\n//!\n//! assert!(a.is_none());\n//! ```\n//!\n//! Another way to create a `Scalar` is by reducing a \\\\(256\\\\)-bit integer mod\n//! \\\\( \\ell \\\\), for which one may use the\n//! [`Scalar::from_bytes_mod_order`](struct.Scalar.html#method.from_bytes_mod_order)\n//! method. In the case of the second example above, this would reduce the\n//! resultant scalar \\\\( \\mod \\ell \\\\), producing \\\\( 2 \\\\):\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let l_plus_two_bytes: [u8; 32] = [\n//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,\n//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,\n//! ];\n//! let a: Scalar = Scalar::from_bytes_mod_order(l_plus_two_bytes);\n//!\n//! let two: Scalar = Scalar::ONE + Scalar::ONE;\n//!\n//! assert!(a == two);\n//! ```\n//!\n//! There is also a constructor that reduces a \\\\(512\\\\)-bit integer,\n//! [`Scalar::from_bytes_mod_order_wide`].\n//!\n//! To construct a `Scalar` as the hash of some input data, use\n//! [`Scalar::hash_from_bytes`], which takes a buffer, or\n//! [`Scalar::from_hash`], which allows an IUF API.\n//!\n#![cfg_attr(feature = \"digest\", doc = \"```\")]\n#![cfg_attr(not(feature = \"digest\"), doc = \"```ignore\")]\n//! # fn main() {\n//! use sha2::{Digest, Sha512};\n//! use curve25519::scalar::Scalar;\n//!\n//! // Hashing a single byte slice\n//! let a = Scalar::hash_from_bytes::(b\"Abolish ICE\");\n//!\n//! // Streaming data into a hash object\n//! let mut hasher = Sha512::default();\n//! hasher.update(b\"Abolish \");\n//! hasher.update(b\"ICE\");\n//! let a2 = Scalar::from_hash(hasher);\n//!\n//! assert_eq!(a, a2);\n//! # }\n//! ```\n//!\n//! See also `Scalar::hash_from_bytes` and `Scalar::from_hash` that\n//! reduces a \\\\(512\\\\)-bit integer, if the optional `digest` feature\n//! has been enabled.\n\nuse core::borrow::Borrow;\nuse core::fmt::Debug;\nuse core::iter::{Product, Sum};\nuse core::ops::Index;\nuse core::ops::Neg;\nuse core::ops::{Add, AddAssign};\nuse core::ops::{Mul, MulAssign};\nuse core::ops::{Sub, SubAssign};\n\n#[cfg(feature = \"group\")]\nuse group::ff::{Field, FromUniformBytes, PrimeField};\n#[cfg(feature = \"group-bits\")]\nuse group::ff::{FieldBits, PrimeFieldBits};\n\n#[cfg(any(feature = \"group\", feature = \"rand_core\"))]\nuse rand_core::RngCore;\n\n#[cfg(feature = \"rand_core\")]\nuse rand_core::CryptoRng;\n\n#[cfg(feature = \"digest\")]\nuse digest::Digest;\n#[cfg(feature = \"digest\")]\nuse digest::array::typenum::U64;\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\nuse subtle::CtOption;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::backend;\nuse crate::constants;\nuse crate::scalar::heea::{I256, curve25519_heea_vartime};\nuse crate::traits::HEEADecomposition;\n\nmod heea;\npub(crate) use heea::HEEA_MAX_INDEX;\n\npub(crate) const NAF_128_SIZE: usize = HEEA_MAX_INDEX + 1;\n\n/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.\n///\n/// This is pinned to the 64-bit serial scalar backend.\ntype UnpackedScalar = backend::serial::u64::scalar::Scalar52;\n\n/// The `Scalar` struct holds an element of \\\\(\\mathbb Z / \\ell\\mathbb Z \\\\).\n#[allow(clippy::derived_hash_with_manual_eq)]\n#[derive(Copy, Clone, Hash)]\npub struct Scalar {\n /// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the\n /// group order.\n ///\n /// # Invariant #1\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255}\\\\). That is, the most\n /// significant bit of `bytes[31]` is 0.\n ///\n /// This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most\n /// integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`]).\n /// The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB\n /// is 0 (see `MontgomeryPoint::mul`).\n ///\n /// # Invariant #2 (weak)\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents\n /// a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is\n /// stronger than invariant #1. It also sometimes has to be broken.\n ///\n /// This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,\n /// mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and\n /// `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,\n /// scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant\n /// #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined.\n ///\n /// Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar.\n /// Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars.\n /// Multiplication is correct, but this is only due to a quirk of our implementation, and not\n /// guaranteed to hold in general in the future.\n ///\n /// Note: It is not possible to construct an unreduced `Scalar` from the public API unless the\n /// `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all\n /// public non-legacy uses, invariant #2\n /// always holds.\n ///\n pub(crate) bytes: [u8; 32],\n}\n\nimpl Scalar {\n /// Construct a `Scalar` by reducing a 256-bit little-endian integer\n /// modulo the group order \\\\( \\ell \\\\).\n pub fn from_bytes_mod_order(bytes: [u8; 32]) -> Scalar {\n // Temporarily allow s_unreduced.bytes > 2^255 ...\n let s_unreduced = Scalar { bytes };\n\n // Then reduce mod the group order and return the reduced representative.\n let s = s_unreduced.reduce();\n debug_assert_eq!(0u8, s[31] >> 7);\n\n s\n }\n\n /// Construct a `Scalar` by reducing a 512-bit little-endian integer\n /// modulo the group order \\\\( \\ell \\\\).\n pub fn from_bytes_mod_order_wide(input: &[u8; 64]) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut unpacked = UnpackedScalar::from_bytes_wide(input);\n let scalar = unpacked.pack();\n\n #[cfg(feature = \"zeroize\")]\n unpacked.zeroize();\n\n scalar\n }\n\n /// Attempt to construct a `Scalar` from a canonical byte representation.\n ///\n /// # Return\n ///\n /// - `Some(s)`, where `s` is the `Scalar` corresponding to `bytes`,\n /// if `bytes` is a canonical byte representation modulo the group order \\\\( \\ell \\\\);\n /// - `None` if `bytes` is not a canonical byte representation.\n pub fn from_canonical_bytes(bytes: [u8; 32]) -> CtOption {\n let high_bit_unset = (bytes[31] >> 7).ct_eq(&0);\n let candidate = Scalar { bytes };\n CtOption::new(candidate, high_bit_unset & candidate.is_canonical())\n }\n\n /// Construct a `Scalar` from bytes that are known to be canonical.\n #[inline]\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n pub(crate) const fn from_canonical_bytes_unchecked(bytes: [u8; 32]) -> Scalar {\n Scalar { bytes }\n }\n\n /// Construct a `Scalar` from the low 255 bits of a 256-bit integer. This breaks the invariant\n /// that scalars are always reduced. Scalar-scalar arithmetic, i.e., addition, subtraction,\n /// multiplication, **does not work** on scalars produced from this function. You may only use\n /// the output of this function for `EdwardsPoint::mul`, `MontgomeryPoint::mul`, and\n /// `EdwardsPoint::vartime_double_scalar_mul_basepoint`. **Do not use this function** unless\n /// you absolutely have to.\n #[cfg(feature = \"legacy_compatibility\")]\n pub const fn from_bits(bytes: [u8; 32]) -> Scalar {\n let mut s = Scalar { bytes };\n // Ensure invariant #1 holds. That is, make s < 2^255 by masking the high bit.\n s.bytes[31] &= 0b0111_1111;\n\n s\n }\n}\n\nimpl Debug for Scalar {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n f.write_str(\"Scalar{..}\")\n }\n}\n\nimpl Eq for Scalar {}\nimpl PartialEq for Scalar {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl ConstantTimeEq for Scalar {\n fn ct_eq(&self, other: &Self) -> Choice {\n self.bytes.ct_eq(&other.bytes)\n }\n}\n\nimpl Index for Scalar {\n type Output = u8;\n\n /// Index the bytes of the representative for this `Scalar`. Mutation is not permitted.\n fn index(&self, _index: usize) -> &u8 {\n &(self.bytes[_index])\n }\n}\n\nimpl<'a> MulAssign<&'a Scalar> for Scalar {\n fn mul_assign(&mut self, _rhs: &'a Scalar) {\n *self = UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack();\n }\n}\n\ndefine_mul_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Mul<&'a Scalar> for &Scalar {\n type Output = Scalar;\n fn mul(self, _rhs: &'a Scalar) -> Scalar {\n UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack()\n }\n}\n\ndefine_mul_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl<'a> AddAssign<&'a Scalar> for Scalar {\n fn add_assign(&mut self, _rhs: &'a Scalar) {\n *self = *self + _rhs;\n }\n}\n\ndefine_add_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Add<&'a Scalar> for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn add(self, _rhs: &'a Scalar) -> Scalar {\n // The UnpackedScalar::add function produces reduced outputs if the inputs are reduced. By\n // Scalar invariant #1, this is always the case.\n UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack()\n }\n}\n\ndefine_add_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl<'a> SubAssign<&'a Scalar> for Scalar {\n fn sub_assign(&mut self, _rhs: &'a Scalar) {\n *self = *self - _rhs;\n }\n}\n\ndefine_sub_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Sub<&'a Scalar> for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn sub(self, rhs: &'a Scalar) -> Scalar {\n // The UnpackedScalar::sub function produces reduced outputs if the inputs are reduced. By\n // Scalar invariant #1, this is always the case.\n UnpackedScalar::sub(&self.unpack(), &rhs.unpack()).pack()\n }\n}\n\ndefine_sub_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl Neg for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn neg(self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_unpacked = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_R = UnpackedScalar::mul_internal(&self_unpacked, &constants::R);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_mod_l = UnpackedScalar::montgomery_reduce(&self_R);\n let negated = UnpackedScalar::sub(&UnpackedScalar::ZERO, &self_mod_l).pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n self_unpacked.zeroize();\n self_R.zeroize();\n self_mod_l.zeroize();\n }\n\n negated\n }\n}\n\nimpl Neg for Scalar {\n type Output = Scalar;\n fn neg(self) -> Scalar {\n -&self\n }\n}\n\nimpl ConditionallySelectable for Scalar {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = u8::conditional_select(&a.bytes[i], &b.bytes[i], choice);\n }\n Scalar { bytes }\n }\n}\n\n#[cfg(feature = \"serde\")]\nuse serde::de::Visitor;\n#[cfg(feature = \"serde\")]\nuse serde::{Deserialize, Deserializer, Serialize, Serializer};\n\n#[cfg(feature = \"serde\")]\n#[cfg_attr(docsrs, doc(cfg(feature = \"serde\")))]\nimpl Serialize for Scalar {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\n#[cfg_attr(docsrs, doc(cfg(feature = \"serde\")))]\nimpl<'de> Deserialize<'de> for Scalar {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct ScalarVisitor;\n\n impl<'de> Visitor<'de> for ScalarVisitor {\n type Value = Scalar;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\n \"a sequence of 32 bytes whose little-endian interpretation is less than the \\\n basepoint order ℓ\",\n )\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n let scalar = Option::from(Scalar::from_canonical_bytes(bytes))\n .ok_or_else(|| serde::de::Error::custom(\"scalar was not canonically encoded\"));\n\n #[cfg(feature = \"zeroize\")]\n bytes.zeroize();\n\n scalar\n }\n }\n\n deserializer.deserialize_tuple(32, ScalarVisitor)\n }\n}\n\nimpl Product for Scalar\nwhere\n T: Borrow,\n{\n fn product(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(Scalar::ONE, |acc, item| acc * item.borrow())\n }\n}\n\nimpl Sum for Scalar\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(Scalar::ZERO, |acc, item| acc + item.borrow())\n }\n}\n\nimpl Default for Scalar {\n fn default() -> Scalar {\n Scalar::ZERO\n }\n}\n\nimpl From for Scalar {\n fn from(x: u8) -> Scalar {\n let mut s_bytes = [0u8; 32];\n s_bytes[0] = x;\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u16) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u32) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n /// Construct a scalar from the given `u64`.\n ///\n /// # Inputs\n ///\n /// An `u64` to convert to a `Scalar`.\n ///\n /// # Returns\n ///\n /// A `Scalar` corresponding to the input `u64`.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let fourtytwo = Scalar::from(42u64);\n /// let six = Scalar::from(6u64);\n /// let seven = Scalar::from(7u64);\n ///\n /// assert!(fourtytwo == six * seven);\n /// ```\n fn from(x: u64) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u128) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From<&Scalar> for I256 {\n fn from(s: &Scalar) -> I256 {\n I256::from_le_bytes(s.to_bytes())\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for Scalar {\n fn zeroize(&mut self) {\n self.bytes.zeroize();\n }\n}\n\nimpl HEEADecomposition for Scalar {\n /// Generate approximately half-size non-negative scalars `(rho, tau)` for a\n /// given challenge scalar `h`.\n ///\n /// The signed HEEA output satisfies `rho_i = tau_i * h (mod ell)`. This\n /// method returns the absolute values of `rho_i` and `tau_i`, plus `flip_h`\n /// to record whether exactly one of the signed values was negative.\n ///\n /// If `flip_h` is false, then `rho = tau * h (mod ell)`. If `flip_h` is\n /// true, then `rho = -tau * h (mod ell)`.\n fn heea_decompose(&self) -> (Scalar, Scalar, bool) {\n // Convert h to I256\n let v = self.into();\n\n // Run the algorithm\n let (mut rho_i, mut tau_i) = curve25519_heea_vartime(v);\n\n // Convert back to Scalars and ensure shortness\n let mut flip_h = false;\n if rho_i < I256::ZERO {\n rho_i = -rho_i;\n flip_h = !flip_h;\n }\n if tau_i < I256::ZERO {\n tau_i = -tau_i;\n flip_h = !flip_h;\n }\n\n let rho = Scalar::from_positive_i256(&rho_i);\n let tau = Scalar::from_positive_i256(&tau_i);\n\n (rho, tau, flip_h)\n }\n}\n\nimpl Scalar {\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: Self = Self { bytes: [0u8; 32] };\n\n /// The scalar \\\\( 1 \\\\).\n pub const ONE: Self = Self {\n bytes: [\n 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n\n /// Return a `Scalar` chosen uniformly at random using a user-provided RNG.\n ///\n /// # Inputs\n ///\n /// * `rng`: any RNG which implements `CryptoRng` interface.\n ///\n /// # Returns\n ///\n /// A random scalar within \\\\(\\mathbb{Z} / \\ell\\mathbb{Z}\\\\).\n ///\n /// # Example\n ///\n /// ```\n /// # fn main() {\n /// use curve25519::scalar::Scalar;\n ///\n /// let mut csprng = rand::thread_rng();\n /// let a: Scalar = Scalar::random(&mut csprng);\n /// # }\n #[cfg(feature = \"rand_core\")]\n pub fn random(rng: &mut R) -> Self {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut scalar_bytes = [0u8; 64];\n rng.fill_bytes(&mut scalar_bytes);\n let scalar = Scalar::from_bytes_mod_order_wide(&scalar_bytes);\n\n #[cfg(feature = \"zeroize\")]\n scalar_bytes.zeroize();\n\n scalar\n }\n\n #[cfg(feature = \"digest\")]\n /// Hash a slice of bytes into a scalar.\n ///\n /// Takes a type parameter `D`, which is any `Digest` producing 64\n /// bytes (512 bits) of output.\n ///\n /// Convenience wrapper around `from_hash`.\n ///\n /// # Example\n ///\n #[cfg_attr(feature = \"digest\", doc = \"```\")]\n #[cfg_attr(not(feature = \"digest\"), doc = \"```ignore\")]\n /// # use curve25519::scalar::Scalar;\n /// use sha2::Sha512;\n ///\n /// # // Need fn main() here in comment so the doctest compiles\n /// # // See https://doc.rust-lang.org/book/documentation.html#documentation-as-tests\n /// # fn main() {\n /// let msg = \"To really appreciate architecture, you may even need to commit a murder\";\n /// let s = Scalar::hash_from_bytes::(msg.as_bytes());\n /// # }\n /// ```\n pub fn hash_from_bytes(input: &[u8]) -> Scalar\n where\n D: Digest + Default,\n {\n let mut hash = D::default();\n hash.update(input);\n Scalar::from_hash(hash)\n }\n\n #[cfg(feature = \"digest\")]\n /// Construct a scalar from an existing `Digest` instance.\n ///\n /// Use this instead of `hash_from_bytes` if it is more convenient\n /// to stream data into the `Digest` than to pass a single byte\n /// slice.\n ///\n /// # Example\n ///\n /// ```\n /// # use curve25519::scalar::Scalar;\n /// use curve25519::digest::Update;\n ///\n /// use sha2::Digest;\n /// use sha2::Sha512;\n ///\n /// # fn main() {\n /// let mut h = Sha512::new()\n /// .chain(\"To really appreciate architecture, you may even need to commit a murder.\")\n /// .chain(\"While the programs used for The Manhattan Transcripts are of the most extreme\")\n /// .chain(\"nature, they also parallel the most common formula plot: the archetype of\")\n /// .chain(\"murder. Other phantasms were occasionally used to underline the fact that\")\n /// .chain(\"perhaps all architecture, rather than being about functional standards, is\")\n /// .chain(\"about love and death.\");\n ///\n /// let s = Scalar::from_hash(h);\n ///\n /// println!(\"{:?}\", s.to_bytes());\n /// assert_eq!(\n /// s.to_bytes(),\n /// [ 21, 88, 208, 252, 63, 122, 210, 152,\n /// 154, 38, 15, 23, 16, 167, 80, 150,\n /// 192, 221, 77, 226, 62, 25, 224, 148,\n /// 239, 48, 176, 10, 185, 69, 168, 11, ],\n /// );\n /// # }\n /// ```\n pub fn from_hash(hash: D) -> Scalar\n where\n D: Digest,\n {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut digest = hash.finalize();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut output = [0u8; 64];\n output.copy_from_slice(digest.as_slice());\n let scalar = Scalar::from_bytes_mod_order_wide(&output);\n\n #[cfg(feature = \"zeroize\")]\n {\n digest.zeroize();\n output.zeroize();\n }\n\n scalar\n }\n\n /// Convert this `Scalar` to its underlying sequence of bytes.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let s: Scalar = Scalar::ZERO;\n ///\n /// assert!(s.to_bytes() == [0u8; 32]);\n /// ```\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.bytes\n }\n\n /// View the little-endian byte encoding of the integer representing this Scalar.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let s: Scalar = Scalar::ZERO;\n ///\n /// assert!(s.as_bytes() == &[0u8; 32]);\n /// ```\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.bytes\n }\n\n /// Given a nonzero `Scalar`, compute its multiplicative inverse.\n ///\n /// # Warning\n ///\n /// `self` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n ///\n /// # Returns\n ///\n /// The multiplicative inverse of the this `Scalar`.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// // x = 2238329342913194256032495932344128051776374960164957527413114840482143558222\n /// let X: Scalar = Scalar::from_bytes_mod_order([\n /// 0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,\n /// 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,\n /// 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,\n /// 0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,\n /// ]);\n /// // 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244\n /// let XINV: Scalar = Scalar::from_bytes_mod_order([\n /// 0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb,\n /// 0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01, 0x63, 0x47,\n /// 0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96,\n /// 0xd5, 0x0b, 0xcd, 0x7a, 0x3f, 0x96, 0x2a, 0x0f,\n /// ]);\n ///\n /// let inv_X: Scalar = X.invert();\n /// assert!(XINV == inv_X);\n /// let should_be_one: Scalar = &inv_X * &X;\n /// assert!(should_be_one == Scalar::ONE);\n /// ```\n pub fn invert(&self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut unpacked = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = unpacked.invert();\n let scalar = inverse.pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n unpacked.zeroize();\n inverse.zeroize();\n }\n\n scalar\n }\n\n /// Given a slice of nonzero (possibly secret) `Scalar`s,\n /// compute their inverses in a batch.\n ///\n /// # Return\n ///\n /// Each element of `inputs` is replaced by its inverse.\n ///\n /// The product of all inverses is returned.\n ///\n /// # Warning\n ///\n /// All input `Scalars` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n ///\n /// # Example\n ///\n /// ```\n /// # use curve25519::scalar::Scalar;\n /// # fn main() {\n /// let mut scalars = [\n /// Scalar::from(3u64),\n /// Scalar::from(5u64),\n /// Scalar::from(7u64),\n /// Scalar::from(11u64),\n /// ];\n ///\n /// let allinv = Scalar::invert_batch(&mut scalars);\n ///\n /// assert_eq!(allinv, Scalar::from(3*5*7*11u64).invert());\n /// assert_eq!(scalars[0], Scalar::from(3u64).invert());\n /// assert_eq!(scalars[1], Scalar::from(5u64).invert());\n /// assert_eq!(scalars[2], Scalar::from(7u64).invert());\n /// assert_eq!(scalars[3], Scalar::from(11u64).invert());\n /// # }\n /// ```\n pub fn invert_batch(inputs: &mut [Scalar; N]) -> Scalar {\n let one: UnpackedScalar = Scalar::ONE.unpack().as_montgomery();\n\n let mut scratch = [one; N];\n\n Self::invert_batch_internal(inputs, &mut scratch)\n }\n\n /// Given a slice of nonzero (possibly secret) `Scalar`s, compute their inverses in a batch.\n /// This the allocating form of [`Self::invert_batch`]. See those docs for examples.\n ///\n /// # Return\n ///\n /// Each element of `inputs` is replaced by its inverse.\n ///\n /// The product of all inverses is returned.\n ///\n /// # Warning\n ///\n /// All input `Scalars` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n #[cfg(feature = \"alloc\")]\n pub fn invert_batch_alloc(inputs: &mut [Scalar]) -> Scalar {\n let n = inputs.len();\n let one: UnpackedScalar = Scalar::ONE.unpack().as_montgomery();\n\n let mut scratch = vec![one; n];\n\n Self::invert_batch_internal(inputs, &mut scratch)\n }\n\n fn invert_batch_internal(inputs: &mut [Scalar], scratch: &mut [UnpackedScalar]) -> Scalar {\n // This code is essentially identical to the FieldElement\n // implementation, and is documented there. Unfortunately,\n // it's not easy to write it generically, since here we want\n // to use `UnpackedScalar`s internally, and `Scalar`s\n // externally, but there's no corresponding distinction for\n // field elements.\n\n // Keep an accumulator of all of the previous products\n let mut acc = Scalar::ONE.unpack().as_montgomery();\n\n // Pass through the input vector, recording the previous\n // products in the scratch space\n for (input, scratch) in inputs.iter_mut().zip(scratch.iter_mut()) {\n *scratch = acc;\n\n // Avoid unnecessary Montgomery multiplication in second pass by\n // keeping inputs in Montgomery form\n let tmp = input.unpack().as_montgomery();\n *input = tmp.pack();\n acc = UnpackedScalar::montgomery_mul(&acc, &tmp);\n }\n\n // acc is nonzero iff all inputs are nonzero\n debug_assert!(acc.pack() != Scalar::ZERO);\n\n // Compute the inverse of all products\n acc = acc.montgomery_invert().from_montgomery();\n\n // We need to return the product of all inverses later\n let ret = acc.pack();\n\n // Pass through the vector backwards to compute the inverses\n // in place\n for (input, scratch) in inputs.iter_mut().rev().zip(scratch.iter().rev()) {\n let tmp = UnpackedScalar::montgomery_mul(&acc, &input.unpack());\n *input = UnpackedScalar::montgomery_mul(&acc, scratch).pack();\n acc = tmp;\n }\n\n #[cfg(feature = \"zeroize\")]\n scratch.zeroize();\n\n ret\n }\n\n /// Compute `b` such that `b + b = a mod modulus`.\n pub fn div_by_2(&self) -> Self {\n // We are looking for such `b` that `b + b = a mod modulus`.\n // Two possibilities:\n // - if `a` is even, we can just divide by 2;\n // - if `a` is odd, we divide `(a + modulus)` by 2.\n let is_odd = Choice::from(self.as_bytes()[0] & 1);\n let mut scalar = self.unpack();\n scalar.conditional_add_l(is_odd);\n\n let carry = scalar.shr1_assign();\n debug_assert_eq!(carry, 0);\n\n scalar.pack()\n }\n\n /// Get the bits of the scalar, in little-endian order\n pub(crate) fn bits_le(&self) -> impl DoubleEndedIterator + '_ {\n (0..256).map(|i| {\n // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index\n // the byte. Since self.bytes is little-endian at the byte level, this iterator is\n // little-endian on the bit level\n ((self.bytes[i >> 3] >> (i & 7)) & 1u8) == 1\n })\n }\n\n /// Compute a width-\\\\(w\\\\) \"Non-Adjacent Form\" of this scalar.\n ///\n /// A width-\\\\(w\\\\) NAF of a positive integer \\\\(k\\\\) is an expression\n /// $$\n /// k = \\sum_{i=0}\\^m n\\_i 2\\^i,\n /// $$\n /// where each nonzero\n /// coefficient \\\\(n\\_i\\\\) is odd and bounded by \\\\(|n\\_i| < 2\\^{w-1}\\\\),\n /// \\\\(n\\_{m-1}\\\\) is nonzero, and at most one of any \\\\(w\\\\) consecutive\n /// coefficients is nonzero. (Hankerson, Menezes, Vanstone; def 3.32).\n ///\n /// The length of the NAF is at most one more than the length of\n /// the binary representation of \\\\(k\\\\). This is why the\n /// `Scalar` type maintains an invariant (invariant #1) that the top bit is\n /// \\\\(0\\\\), so that the NAF of a scalar has at most 256 digits.\n ///\n /// Intuitively, this is like a binary expansion, except that we\n /// allow some coefficients to grow in magnitude up to\n /// \\\\(2\\^{w-1}\\\\) so that the nonzero coefficients are as sparse\n /// as possible.\n ///\n /// When doing scalar multiplication, we can then use a lookup\n /// table of precomputed multiples of a point to add the nonzero\n /// terms \\\\( k_i P \\\\). Using signed digits cuts the table size\n /// in half, and using odd digits cuts the table size in half\n /// again.\n ///\n /// To compute a \\\\(w\\\\)-NAF, we use a modification of Algorithm 3.35 of HMV:\n ///\n /// 1. \\\\( i \\gets 0 \\\\)\n /// 2. While \\\\( k \\ge 1 \\\\):\n /// 1. If \\\\(k\\\\) is odd, \\\\( n_i \\gets k \\operatorname{mods} 2^w \\\\), \\\\( k \\gets k - n_i \\\\).\n /// 2. If \\\\(k\\\\) is even, \\\\( n_i \\gets 0 \\\\).\n /// 3. \\\\( k \\gets k / 2 \\\\), \\\\( i \\gets i + 1 \\\\).\n /// 3. Return \\\\( n_0, n_1, ... , \\\\)\n ///\n /// Here \\\\( \\bar x = x \\operatorname{mods} 2^w \\\\) means the\n /// \\\\( \\bar x \\\\) with \\\\( \\bar x \\equiv x \\pmod{2^w} \\\\) and\n /// \\\\( -2^{w-1} \\leq \\bar x < 2^{w-1} \\\\).\n ///\n /// We implement this by scanning across the bits of \\\\(k\\\\) from\n /// least-significant bit to most-significant-bit.\n /// Write the bits of \\\\(k\\\\) as\n /// $$\n /// k = \\sum\\_{i=0}\\^m k\\_i 2^i,\n /// $$\n /// and split the sum as\n /// $$\n /// k = \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// $$\n /// where the first part is \\\\( k \\mod 2^w \\\\).\n ///\n /// If \\\\( k \\mod 2^w\\\\) is odd, and \\\\( k \\mod 2^w < 2^{w-1} \\\\), then we emit\n /// \\\\( n_0 = k \\mod 2^w \\\\). Instead of computing\n /// \\\\( k - n_0 \\\\), we just advance \\\\(w\\\\) bits and reindex.\n ///\n /// If \\\\( k \\mod 2^w\\\\) is odd, and \\\\( k \\mod 2^w \\ge 2^{w-1} \\\\), then\n /// \\\\( n_0 = k \\operatorname{mods} 2^w = k \\mod 2^w - 2^w \\\\).\n /// The quantity \\\\( k - n_0 \\\\) is\n /// $$\n /// \\begin{aligned}\n /// k - n_0 &= \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// - \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\\\\\\\\n /// &= 2^w + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// \\end{aligned}\n /// $$\n /// so instead of computing the subtraction, we can set a carry\n /// bit, advance \\\\(w\\\\) bits, and reindex.\n ///\n /// If \\\\( k \\mod 2^w\\\\) is even, we emit \\\\(0\\\\), advance 1 bit\n /// and reindex. In fact, by setting all digits to \\\\(0\\\\)\n /// initially, we don't need to emit anything.\n pub(crate) fn non_adjacent_form(&self, w: usize) -> [i8; 256] {\n // required by the NAF definition\n debug_assert!(w >= 2);\n // required so that the NAF digits fit in i8\n debug_assert!(w <= 8);\n\n let mut naf = [0i8; 256];\n\n let mut x_u64 = [0u64; 5];\n // AENEAS-COMPAT: index-based little-endian load instead of\n // read_le_u64_into (its chunks/zip iterators are opaque to the\n // extraction). Pure refactor, semantics identical.\n let mut k: usize = 0;\n while k < 4 {\n let mut t: u64 = 0;\n let mut bi: usize = 0;\n while bi < 8 {\n t |= (self.bytes[8 * k + bi] as u64) << (8 * bi);\n bi += 1;\n }\n x_u64[k] = t;\n k += 1;\n }\n\n let width = 1 << w;\n let window_mask = width - 1;\n\n let mut pos = 0;\n let mut carry = 0;\n while pos < 256 {\n // Construct a buffer of bits of the scalar, starting at bit `pos`\n let u64_idx = pos / 64;\n let bit_idx = pos % 64;\n let bit_buf: u64 = if bit_idx < 64 - w {\n // This window's bits are contained in a single u64\n x_u64[u64_idx] >> bit_idx\n } else {\n // Combine the current u64's bits with the bits from the next u64\n (x_u64[u64_idx] >> bit_idx) | (x_u64[1 + u64_idx] << (64 - bit_idx))\n };\n\n // Add the carry into the current window\n let window = carry + (bit_buf & window_mask);\n\n if window & 1 == 0 {\n // If the window value is even, preserve the carry and continue.\n // Why is the carry preserved?\n // If carry == 0 and window & 1 == 0, then the next carry should be 0\n // If carry == 1 and window & 1 == 0, then bit_buf & 1 == 1 so the next carry should be 1\n pos += 1;\n continue;\n }\n\n if window < width / 2 {\n carry = 0;\n naf[pos] = window as i8;\n } else {\n carry = 1;\n naf[pos] = (window as i8).wrapping_sub(width as i8);\n }\n\n pos += w;\n }\n\n naf\n }\n\n /// Compute a width-\\\\(w\\\\) non-adjacent form for scalars known to fit in 128 bits.\n pub(crate) fn non_adjacent_form_128(&self, w: usize) -> [i8; NAF_128_SIZE] {\n // required by the NAF definition\n debug_assert!(w >= 2);\n // required so that the NAF digits fit in i8\n debug_assert!(w <= 8);\n debug_assert!(self.bytes[16..32].iter().all(|&b| b == 0));\n\n let mut naf = [0i8; NAF_128_SIZE];\n\n let mut x_u64 = [0u64; 3];\n read_le_u64_into(&self.bytes[..16], &mut x_u64[0..2]);\n\n let width = 1u64 << w;\n let window_mask = width - 1;\n\n let mut pos = 0;\n let mut carry = 0;\n while pos < HEEA_MAX_INDEX {\n let u64_idx = pos / 64;\n let bit_idx = pos % 64;\n let bit_buf: u64 = if bit_idx < 64 - w {\n x_u64[u64_idx] >> bit_idx\n } else {\n (x_u64[u64_idx] >> bit_idx) | (x_u64[1 + u64_idx] << (64 - bit_idx))\n };\n\n let window = carry + (bit_buf & window_mask);\n\n if window & 1 == 0 {\n pos += 1;\n continue;\n }\n\n if window < width / 2 {\n carry = 0;\n naf[pos] = window as i8;\n } else {\n carry = 1;\n naf[pos] = (window as i8).wrapping_sub(width as i8);\n }\n\n pos += w;\n }\n\n debug_assert_eq!(carry, 0);\n naf\n }\n\n /// Write this scalar in radix 16, with coefficients in \\\\([-8,8)\\\\),\n /// i.e., compute \\\\(a\\_i\\\\) such that\n /// $$\n /// a = a\\_0 + a\\_1 16\\^1 + \\cdots + a_{63} 16\\^{63},\n /// $$\n /// with \\\\(-8 \\leq a_i < 8\\\\) for \\\\(0 \\leq i < 63\\\\) and \\\\(-8 \\leq a_{63} \\leq 8\\\\).\n ///\n /// The largest value that can be decomposed like this is just over \\\\(2^{255}\\\\). Thus, in\n /// order to not error, the top bit MUST NOT be set, i.e., `Self` MUST be less than\n /// \\\\(2^{255}\\\\).\n pub(crate) fn as_radix_16(&self) -> [i8; 64] {\n debug_assert!(self[31] <= 127);\n let mut output = [0i8; 64];\n\n // Step 1: change radix.\n // Convert from radix 256 (bytes) to radix 16 (nibbles)\n #[allow(clippy::identity_op)]\n #[inline(always)]\n fn bot_half(x: u8) -> u8 {\n (x >> 0) & 15\n }\n #[inline(always)]\n fn top_half(x: u8) -> u8 {\n (x >> 4) & 15\n }\n\n for i in 0..32 {\n output[2 * i] = bot_half(self[i]) as i8;\n output[2 * i + 1] = top_half(self[i]) as i8;\n }\n // Precondition note: since self[31] <= 127, output[63] <= 7\n\n // Step 2: recenter coefficients from [0,16) to [-8,8)\n for i in 0..63 {\n let carry = (output[i] + 8) >> 4;\n output[i] -= carry << 4;\n output[i + 1] += carry;\n }\n // Precondition note: output[63] is not recentered. It\n // increases by carry <= 1. Thus output[63] <= 8.\n\n output\n }\n\n /// Returns a size hint indicating how many entries of the return\n /// value of `to_radix_2w` are nonzero.\n #[cfg(any(feature = \"alloc\", all(test, feature = \"precomputed-tables\")))]\n pub(crate) fn to_radix_2w_size_hint(w: usize) -> usize {\n debug_assert!(w >= 4);\n debug_assert!(w <= 8);\n\n let digits_count = match w {\n 4..=7 => 256_usize.div_ceil(w),\n // See comment in to_radix_2w on handling the terminal carry.\n 8 => 256_usize.div_ceil(w) + 1_usize,\n _ => panic!(\"invalid radix parameter\"),\n };\n\n debug_assert!(digits_count <= 64);\n digits_count\n }\n\n /// Creates a representation of a Scalar in radix \\\\( 2^w \\\\) with \\\\(w = 4, 5, 6, 7, 8\\\\) for\n /// use with the Pippenger algorithm. Higher radixes are not supported to save cache space.\n /// Radix 256 is near-optimal even for very large inputs.\n ///\n /// Radix below 16 or above 256 is prohibited.\n /// This method returns digits in a fixed-sized array, excess digits are zeroes.\n ///\n /// For radix 16, `Self` must be less than \\\\(2^{255}\\\\). This is because most integers larger\n /// than \\\\(2^{255}\\\\) are unrepresentable in the form described below for \\\\(w = 4\\\\). This\n /// would be true for \\\\(w = 8\\\\) as well, but it is compensated for by increasing the size\n /// hint by 1.\n ///\n /// ## Scalar representation\n ///\n /// Radix \\\\(2\\^w\\\\), with \\\\(n = ceil(256/w)\\\\) coefficients in \\\\([-(2\\^w)/2,(2\\^w)/2)\\\\),\n /// i.e., scalar is represented using digits \\\\(a\\_i\\\\) such that\n /// $$\n /// a = a\\_0 + a\\_1 2\\^1w + \\cdots + a_{n-1} 2\\^{w*(n-1)},\n /// $$\n /// with \\\\(-2\\^w/2 \\leq a_i < 2\\^w/2\\\\) for \\\\(0 \\leq i < (n-1)\\\\) and \\\\(-2\\^w/2 \\leq a_{n-1} \\leq 2\\^w/2\\\\).\n ///\n #[cfg(any(feature = \"alloc\", feature = \"precomputed-tables\"))]\n pub(crate) fn as_radix_2w(&self, w: usize) -> [i8; 64] {\n debug_assert!(w >= 4);\n debug_assert!(w <= 8);\n\n if w == 4 {\n return self.as_radix_16();\n }\n\n // Scalar formatted as four `u64`s with carry bit packed into the highest bit.\n let mut scalar64x4 = [0u64; 4];\n read_le_u64_into(&self.bytes, &mut scalar64x4[0..4]);\n\n let radix: u64 = 1 << w;\n let window_mask: u64 = radix - 1;\n\n let mut carry = 0u64;\n let mut digits = [0i8; 64];\n let digits_count = 256_usize.div_ceil(w);\n #[allow(clippy::needless_range_loop)]\n for i in 0..digits_count {\n // Construct a buffer of bits of the scalar, starting at `bit_offset`.\n let bit_offset = i * w;\n let u64_idx = bit_offset / 64;\n let bit_idx = bit_offset % 64;\n\n // Read the bits from the scalar\n let bit_buf: u64 = if bit_idx < 64 - w || u64_idx == 3 {\n // This window's bits are contained in a single u64,\n // or it's the last u64 anyway.\n scalar64x4[u64_idx] >> bit_idx\n } else {\n // Combine the current u64's bits with the bits from the next u64\n (scalar64x4[u64_idx] >> bit_idx) | (scalar64x4[1 + u64_idx] << (64 - bit_idx))\n };\n\n // Read the actual coefficient value from the window\n let coef = carry + (bit_buf & window_mask); // coef = [0, 2^r)\n\n // Recenter coefficients from [0,2^w) to [-2^w/2, 2^w/2)\n carry = (coef + (radix / 2)) >> w;\n digits[i] = ((coef as i64) - (carry << w) as i64) as i8;\n }\n\n // When 4 < w < 8, we can fold the final carry onto the last digit d,\n // because d < 2^w/2 so d + carry*2^w = d + 1*2^w < 2^(w+1) < 2^8.\n //\n // When w = 8, we can't fit carry*2^w into an i8. This should\n // not happen anyways, because the final carry will be 0 for\n // reduced scalars, but Scalar invariant #1 allows 255-bit scalars.\n // To handle this, we expand the size_hint by 1 when w=8,\n // and accumulate the final carry onto another digit.\n match w {\n 8 => digits[digits_count] += carry as i8,\n _ => digits[digits_count - 1] += (carry << w) as i8,\n }\n\n digits\n }\n\n /// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.\n pub(crate) fn unpack(&self) -> UnpackedScalar {\n UnpackedScalar::from_bytes(&self.bytes)\n }\n\n /// Create a `Scalar` from a positive `I256`.\n /// The caller must ensure val > 0.\n pub(crate) fn from_positive_i256(val: &I256) -> Self {\n debug_assert!(val >= &I256::ZERO);\n Scalar {\n bytes: val.to_le_bytes(),\n }\n }\n\n /// Reduce this `Scalar` modulo \\\\(\\ell\\\\).\n #[allow(non_snake_case)]\n fn reduce(&self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut x = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut xR = UnpackedScalar::mul_internal(&x, &constants::R);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut x_mod_l = UnpackedScalar::montgomery_reduce(&xR);\n let scalar = x_mod_l.pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n x.zeroize();\n xR.zeroize();\n x_mod_l.zeroize();\n }\n\n scalar\n }\n\n /// Check whether this `Scalar` is the canonical representative mod \\\\(\\ell\\\\). This is not\n /// public because any `Scalar` that is publicly observed is reduced, by scalar invariant #2.\n fn is_canonical(&self) -> Choice {\n self.ct_eq(&self.reduce())\n }\n}\n\nimpl UnpackedScalar {\n /// Pack the limbs of this `UnpackedScalar` into a `Scalar`.\n fn pack(&self) -> Scalar {\n Scalar {\n bytes: self.to_bytes(),\n }\n }\n\n /// Inverts an UnpackedScalar in Montgomery form.\n #[rustfmt::skip] // keep alignment of addition chain and squarings\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n #[allow(clippy::just_underscores_and_digits)]\n pub fn montgomery_invert(&self) -> UnpackedScalar {\n // Uses the addition chain from\n // https://briansmith.org/ecc-inversion-addition-chains-01#curve25519_scalar_inversion\n let mut _1 = *self;\n let mut _10 = _1.montgomery_square();\n let mut _100 = _10.montgomery_square();\n let mut _11 = UnpackedScalar::montgomery_mul(&_10, &_1);\n let mut _101 = UnpackedScalar::montgomery_mul(&_10, &_11);\n let mut _111 = UnpackedScalar::montgomery_mul(&_10, &_101);\n let mut _1001 = UnpackedScalar::montgomery_mul(&_10, &_111);\n let mut _1011 = UnpackedScalar::montgomery_mul(&_10, &_1001);\n let mut _1111 = UnpackedScalar::montgomery_mul(&_100, &_1011);\n\n // _10000\n let mut y = UnpackedScalar::montgomery_mul(&_1111, &_1);\n\n #[inline]\n fn square_multiply(y: &mut UnpackedScalar, squarings: usize, x: &UnpackedScalar) {\n for _ in 0..squarings {\n *y = y.montgomery_square();\n }\n *y = UnpackedScalar::montgomery_mul(y, x);\n }\n\n square_multiply(&mut y, 123 + 3, &_101);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 4, &_1001);\n square_multiply(&mut y, 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 1 + 3, &_101);\n square_multiply(&mut y, 3 + 3, &_101);\n square_multiply(&mut y, 3, &_111);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 2 + 3, &_111);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1011);\n square_multiply(&mut y, 2 + 4, &_1011);\n square_multiply(&mut y, 6 + 4, &_1001);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 3 + 2, &_11);\n square_multiply(&mut y, 3 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1001);\n square_multiply(&mut y, 1 + 3, &_111);\n square_multiply(&mut y, 2 + 4, &_1111);\n square_multiply(&mut y, 1 + 4, &_1011);\n square_multiply(&mut y, 3, &_101);\n square_multiply(&mut y, 2 + 4, &_1111);\n square_multiply(&mut y, 3, &_101);\n square_multiply(&mut y, 1 + 2, &_11);\n\n let inverse = y;\n\n #[cfg(feature = \"zeroize\")]\n {\n _1.zeroize();\n _10.zeroize();\n _100.zeroize();\n _11.zeroize();\n _101.zeroize();\n _111.zeroize();\n _1001.zeroize();\n _1011.zeroize();\n _1111.zeroize();\n y.zeroize();\n }\n\n inverse\n }\n\n /// Inverts an UnpackedScalar not in Montgomery form.\n pub fn invert(&self) -> UnpackedScalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut montgomery = self.as_montgomery();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = montgomery.montgomery_invert();\n let result = inverse.from_montgomery();\n\n #[cfg(feature = \"zeroize\")]\n {\n montgomery.zeroize();\n inverse.zeroize();\n }\n\n result\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Field for Scalar {\n const ZERO: Self = Self::ZERO;\n const ONE: Self = Self::ONE;\n\n fn random(mut rng: impl RngCore) -> Self {\n // NOTE: this is duplicated due to different `rng` bounds\n let mut scalar_bytes = [0u8; 64];\n rng.fill_bytes(&mut scalar_bytes);\n let scalar = Self::from_bytes_mod_order_wide(&scalar_bytes);\n\n #[cfg(feature = \"zeroize\")]\n scalar_bytes.zeroize();\n\n scalar\n }\n\n fn square(&self) -> Self {\n self * self\n }\n\n fn double(&self) -> Self {\n self + self\n }\n\n fn invert(&self) -> CtOption {\n CtOption::new(self.invert(), !self.is_zero())\n }\n\n fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {\n #[allow(unused_qualifications)]\n group::ff::helpers::sqrt_ratio_generic(num, div)\n }\n\n fn sqrt(&self) -> CtOption {\n #[allow(unused_qualifications)]\n group::ff::helpers::sqrt_tonelli_shanks(\n self,\n [\n 0xcb02_4c63_4b9e_ba7d,\n 0x029b_df3b_d45e_f39a,\n 0x0000_0000_0000_0000,\n 0x0200_0000_0000_0000,\n ],\n )\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl PrimeField for Scalar {\n type Repr = [u8; 32];\n\n fn from_repr(repr: Self::Repr) -> CtOption {\n Self::from_canonical_bytes(repr)\n }\n\n fn from_repr_vartime(repr: Self::Repr) -> Option {\n // Check that the high bit is not set\n if (repr[31] >> 7) != 0u8 {\n return None;\n }\n\n let candidate = Scalar { bytes: repr };\n\n if candidate == candidate.reduce() {\n Some(candidate)\n } else {\n None\n }\n }\n\n fn to_repr(&self) -> Self::Repr {\n self.to_bytes()\n }\n\n fn is_odd(&self) -> Choice {\n Choice::from(self.as_bytes()[0] & 1)\n }\n\n const MODULUS: &'static str =\n \"0x1000000000000000000000000000000014def9dea2f79cd65812631a5cf5d3ed\";\n const NUM_BITS: u32 = 253;\n const CAPACITY: u32 = 252;\n\n const TWO_INV: Self = Self {\n bytes: [\n 0xf7, 0xe9, 0x7a, 0x2e, 0x8d, 0x31, 0x09, 0x2c, 0x6b, 0xce, 0x7b, 0x51, 0xef, 0x7c,\n 0x6f, 0x0a, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x08,\n ],\n };\n const MULTIPLICATIVE_GENERATOR: Self = Self {\n bytes: [\n 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n const S: u32 = 2;\n const ROOT_OF_UNITY: Self = Self {\n bytes: [\n 0xd4, 0x07, 0xbe, 0xeb, 0xdf, 0x75, 0x87, 0xbe, 0xfe, 0x83, 0xce, 0x42, 0x53, 0x56,\n 0xf0, 0x0e, 0x7a, 0xc2, 0xc1, 0xab, 0x60, 0x6d, 0x3d, 0x7d, 0xe7, 0x81, 0x79, 0xe0,\n 0x10, 0x73, 0x4a, 0x09,\n ],\n };\n const ROOT_OF_UNITY_INV: Self = Self {\n bytes: [\n 0x19, 0xcc, 0x37, 0x71, 0x3a, 0xed, 0x8a, 0x99, 0xd7, 0x18, 0x29, 0x60, 0x8b, 0xa3,\n 0xee, 0x05, 0x86, 0x3d, 0x3e, 0x54, 0x9f, 0x92, 0xc2, 0x82, 0x18, 0x7e, 0x86, 0x1f,\n 0xef, 0x8c, 0xb5, 0x06,\n ],\n };\n const DELTA: Self = Self {\n bytes: [\n 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n}\n\n#[cfg(feature = \"group-bits\")]\nimpl PrimeFieldBits for Scalar {\n type ReprBits = [u8; 32];\n\n fn to_le_bits(&self) -> FieldBits {\n self.to_repr().into()\n }\n\n fn char_le_bits() -> FieldBits {\n constants::BASEPOINT_ORDER.to_bytes().into()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl FromUniformBytes<64> for Scalar {\n fn from_uniform_bytes(bytes: &[u8; 64]) -> Self {\n Scalar::from_bytes_mod_order_wide(bytes)\n }\n}\n\n/// Read one or more u64s stored as little endian bytes.\n///\n/// ## Panics\n/// Panics if `src.len() != 8 * dst.len()`.\nfn read_le_u64_into(src: &[u8], dst: &mut [u64]) {\n assert!(\n src.len() == 8 * dst.len(),\n \"src.len() = {}, dst.len() = {}\",\n src.len(),\n dst.len()\n );\n for (bytes, val) in src.chunks(8).zip(dst.iter_mut()) {\n *val = u64::from_le_bytes(\n bytes\n .try_into()\n .expect(\"Incorrect src length, should be 8 * dst.len()\"),\n );\n }\n}\n\n/// _Clamps_ the given little-endian representation of a 32-byte integer. Clamping the value puts\n/// it in the range:\n///\n/// **n ∈ 2^254 + 8\\*{0, 1, 2, 3, . . ., 2^251 − 1}**\n///\n/// # Explanation of clamping\n///\n/// For Curve25519, h = 8, and multiplying by 8 is the same as a binary left-shift by 3 bits.\n/// If you take a secret scalar value between 2^251 and 2^252 – 1 and left-shift by 3 bits\n/// then you end up with a 255-bit number with the most significant bit set to 1 and\n/// the least-significant three bits set to 0.\n///\n/// The Curve25519 clamping operation takes **an arbitrary 256-bit random value** and\n/// clears the most-significant bit (making it a 255-bit number), sets the next bit, and then\n/// clears the 3 least-significant bits. In other words, it directly creates a scalar value that is\n/// in the right form and pre-multiplied by the cofactor.\n///\n/// See [here](https://neilmadden.blog/2020/05/28/whats-the-curve25519-clamping-all-about/) for\n/// more details.\n#[must_use]\npub const fn clamp_integer(mut bytes: [u8; 32]) -> [u8; 32] {\n bytes[0] &= 0b1111_1000;\n bytes[31] &= 0b0111_1111;\n bytes[31] |= 0b0100_0000;\n bytes\n}\n\n#[cfg(test)]\npub(crate) mod test {\n use super::*;\n\n #[cfg(feature = \"alloc\")]\n use alloc::vec::Vec;\n use rand::RngCore;\n\n /// x = 2238329342913194256032495932344128051776374960164957527413114840482143558222\n pub static X: Scalar = Scalar {\n bytes: [\n 0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84, 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2,\n 0x7d, 0x52, 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44, 0xd4, 0x49, 0xf4, 0xa8,\n 0x79, 0xd9, 0xf2, 0x04,\n ],\n };\n /// 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244\n pub static XINV: Scalar = Scalar {\n bytes: [\n 0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb, 0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01,\n 0x63, 0x47, 0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96, 0xd5, 0x0b, 0xcd, 0x7a,\n 0x3f, 0x96, 0x2a, 0x0f,\n ],\n };\n /// y = 2592331292931086675770238855846338635550719849568364935475441891787804997264\n pub static Y: Scalar = Scalar {\n bytes: [\n 0x90, 0x76, 0x33, 0xfe, 0x1c, 0x4b, 0x66, 0xa4, 0xa2, 0x8d, 0x2d, 0xd7, 0x67, 0x83,\n 0x86, 0xc3, 0x53, 0xd0, 0xde, 0x54, 0x55, 0xd4, 0xfc, 0x9d, 0xe8, 0xef, 0x7a, 0xc3,\n 0x1f, 0x35, 0xbb, 0x05,\n ],\n };\n\n /// The largest scalar that satisfies invariant #1, i.e., the largest scalar with the top bit\n /// set to 0. Since this scalar violates invariant #2, i.e., it's greater than the modulus `l`,\n /// addition and subtraction are broken. The only thing you can do with this is scalar-point\n /// multiplication (and actually also scalar-scalar multiplication, but that's just a quirk of\n /// our implementation).\n pub(crate) static LARGEST_UNREDUCED_SCALAR: Scalar = Scalar {\n bytes: [\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ],\n };\n\n /// x*y = 5690045403673944803228348699031245560686958845067437804563560795922180092780\n static X_TIMES_Y: Scalar = Scalar {\n bytes: [\n 0x6c, 0x33, 0x74, 0xa1, 0x89, 0x4f, 0x62, 0x21, 0x0a, 0xaa, 0x2f, 0xe1, 0x86, 0xa6,\n 0xf9, 0x2c, 0xe0, 0xaa, 0x75, 0xc2, 0x77, 0x95, 0x81, 0xc2, 0x95, 0xfc, 0x08, 0x17,\n 0x9a, 0x73, 0x94, 0x0c,\n ],\n };\n\n /// sage: l = 2^252 + 27742317777372353535851937790883648493\n /// sage: big = 2^256 - 1\n /// sage: repr((big % l).digits(256))\n static CANONICAL_2_256_MINUS_1: Scalar = Scalar {\n bytes: [\n 28, 149, 152, 141, 116, 49, 236, 214, 112, 207, 125, 115, 244, 91, 239, 198, 254, 255,\n 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 15,\n ],\n };\n\n static A_SCALAR: Scalar = Scalar {\n bytes: [\n 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x09,\n ],\n };\n\n static A_NAF: [i8; 256] = [\n 0, 13, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 3, 0, 0,\n 0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 11, 0, 0, 0, 0,\n 11, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,\n 0, -1, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, -15, 0, 0, 0, 0, -7, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 5,\n 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -11, 0, 0, 0, 0, -7, 0, 0, 0, 0, -13, 0, 0,\n 0, 0, 11, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 1, 0, 0, 0, 0,\n 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 15,\n 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -15, 0,\n 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,\n ];\n\n const BASEPOINT_ORDER_MINUS_ONE: Scalar = Scalar {\n bytes: [\n 0xec, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9,\n 0xde, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x10,\n ],\n };\n\n /// The largest clamped integer\n static LARGEST_CLAMPED_INTEGER: [u8; 32] = clamp_integer(LARGEST_UNREDUCED_SCALAR.bytes);\n\n #[test]\n fn fuzzer_testcase_reduction() {\n // LE bytes of 24519928653854221733733552434404946937899825954937634815\n let a_bytes = [\n 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,\n 255, 255, 255, 255, 255, 255, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n ];\n // LE bytes of 4975441334397345751130612518500927154628011511324180036903450236863266160640\n let b_bytes = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 255, 210, 210,\n 210, 255, 255, 255, 255, 10,\n ];\n // LE bytes of 6432735165214683820902750800207468552549813371247423777071615116673864412038\n let c_bytes = [\n 134, 171, 119, 216, 180, 128, 178, 62, 171, 132, 32, 62, 34, 119, 104, 193, 47, 215,\n 181, 250, 14, 207, 172, 93, 75, 207, 211, 103, 144, 204, 56, 14,\n ];\n\n let a = Scalar::from_bytes_mod_order(a_bytes);\n let b = Scalar::from_bytes_mod_order(b_bytes);\n let c = Scalar::from_bytes_mod_order(c_bytes);\n\n let mut tmp = [0u8; 64];\n\n // also_a = (a mod l)\n tmp[0..32].copy_from_slice(&a_bytes[..]);\n let also_a = Scalar::from_bytes_mod_order_wide(&tmp);\n\n // also_b = (b mod l)\n tmp[0..32].copy_from_slice(&b_bytes[..]);\n let also_b = Scalar::from_bytes_mod_order_wide(&tmp);\n\n let expected_c = a * b;\n let also_expected_c = also_a * also_b;\n\n assert_eq!(c, expected_c);\n assert_eq!(c, also_expected_c);\n }\n\n #[test]\n fn non_adjacent_form_test_vector() {\n let naf = A_SCALAR.non_adjacent_form(5);\n for i in 0..256 {\n assert_eq!(naf[i], A_NAF[i]);\n }\n }\n\n #[test]\n fn non_adjacent_form_128_matches_generic() {\n let mut high_bit = [0u8; 32];\n high_bit[15] = 0x80;\n\n let cases = [\n Scalar::ZERO,\n Scalar::ONE,\n Scalar::from(u64::MAX),\n Scalar::from(0xfedc_ba98_7654_3210_0123_4567_89ab_cdefu128),\n Scalar::from(u128::MAX),\n Scalar { bytes: high_bit },\n ];\n\n for scalar in cases {\n for w in 2..=8 {\n let generic = scalar.non_adjacent_form(w);\n let naf_128 = scalar.non_adjacent_form_128(w);\n\n for i in 0..=HEEA_MAX_INDEX {\n assert_eq!(\n naf_128[i], generic[i],\n \"NAF mismatch at index {i} for width {w}\"\n );\n }\n\n assert!(\n generic[NAF_128_SIZE..].iter().all(|&digit| digit == 0),\n \"generic NAF has non-zero digits above 128-bit range for width {w}\"\n );\n }\n }\n }\n\n #[cfg(feature = \"rand_core\")]\n fn non_adjacent_form_iter(w: usize, x: &Scalar) {\n let naf = x.non_adjacent_form(w);\n\n // Reconstruct the scalar from the computed NAF\n let mut y = Scalar::ZERO;\n for i in (0..256).rev() {\n y += y;\n let digit = if naf[i] < 0 {\n -Scalar::from((-naf[i]) as u64)\n } else {\n Scalar::from(naf[i] as u64)\n };\n y += digit;\n }\n\n assert_eq!(*x, y);\n }\n\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn non_adjacent_form_random() {\n let mut rng = rand::thread_rng();\n for _ in 0..1_000 {\n let x = Scalar::random(&mut rng);\n for w in &[5, 6, 7, 8] {\n non_adjacent_form_iter(*w, &x);\n }\n }\n }\n\n #[test]\n fn from_u64() {\n let val: u64 = 0xdeadbeefdeadbeef;\n let s = Scalar::from(val);\n assert_eq!(s[7], 0xde);\n assert_eq!(s[6], 0xad);\n assert_eq!(s[5], 0xbe);\n assert_eq!(s[4], 0xef);\n assert_eq!(s[3], 0xde);\n assert_eq!(s[2], 0xad);\n assert_eq!(s[1], 0xbe);\n assert_eq!(s[0], 0xef);\n }\n\n #[test]\n fn scalar_mul_by_one() {\n let test_scalar = X * Scalar::ONE;\n for i in 0..32 {\n assert!(test_scalar[i] == X[i]);\n }\n }\n\n #[test]\n fn add_reduces() {\n // Check that addition wraps around the modulus\n assert_eq!(BASEPOINT_ORDER_MINUS_ONE + Scalar::ONE, Scalar::ZERO);\n }\n\n #[test]\n fn sub_reduces() {\n // Check that subtraction wraps around the modulus\n assert_eq!(Scalar::ZERO - Scalar::ONE, BASEPOINT_ORDER_MINUS_ONE);\n }\n\n #[test]\n fn impl_add() {\n let two = Scalar::from(2u64);\n let one = Scalar::ONE;\n let should_be_two = one + one;\n assert_eq!(should_be_two, two);\n }\n\n #[allow(non_snake_case)]\n #[test]\n fn impl_mul() {\n let should_be_X_times_Y = X * Y;\n assert_eq!(should_be_X_times_Y, X_TIMES_Y);\n }\n\n #[allow(non_snake_case)]\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_product() {\n // Test that product works for non-empty iterators\n let X_Y_vector = [X, Y];\n let should_be_X_times_Y: Scalar = X_Y_vector.iter().product();\n assert_eq!(should_be_X_times_Y, X_TIMES_Y);\n\n // Test that product works for the empty iterator\n let one = Scalar::ONE;\n let empty_vector = [];\n let should_be_one: Scalar = empty_vector.iter().product();\n assert_eq!(should_be_one, one);\n\n // Test that product works for iterators where Item = Scalar\n let xs = [Scalar::from(2u64); 10];\n let ys = [Scalar::from(3u64); 10];\n // now zs is an iterator with Item = Scalar\n let zs = xs.iter().zip(ys.iter()).map(|(x, y)| x * y);\n\n let x_prod: Scalar = xs.iter().product();\n let y_prod: Scalar = ys.iter().product();\n let z_prod: Scalar = zs.product();\n\n assert_eq!(x_prod, Scalar::from(1024u64));\n assert_eq!(y_prod, Scalar::from(59049u64));\n assert_eq!(z_prod, Scalar::from(60466176u64));\n assert_eq!(x_prod * y_prod, z_prod);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_sum() {\n // Test that sum works for non-empty iterators\n let two = Scalar::from(2u64);\n let one_vector = [Scalar::ONE, Scalar::ONE];\n let should_be_two: Scalar = one_vector.iter().sum();\n assert_eq!(should_be_two, two);\n\n // Test that sum works for the empty iterator\n let zero = Scalar::ZERO;\n let empty_vector = [];\n let should_be_zero: Scalar = empty_vector.iter().sum();\n assert_eq!(should_be_zero, zero);\n\n // Test that sum works for owned types\n let xs = [Scalar::from(1u64); 10];\n let ys = [Scalar::from(2u64); 10];\n // now zs is an iterator with Item = Scalar\n let zs = xs.iter().zip(ys.iter()).map(|(x, y)| x + y);\n\n let x_sum: Scalar = xs.iter().sum();\n let y_sum: Scalar = ys.iter().sum();\n let z_sum: Scalar = zs.sum();\n\n assert_eq!(x_sum, Scalar::from(10u64));\n assert_eq!(y_sum, Scalar::from(20u64));\n assert_eq!(z_sum, Scalar::from(30u64));\n assert_eq!(x_sum + y_sum, z_sum);\n }\n\n #[test]\n fn square() {\n let expected = X * X;\n let actual = X.unpack().square().pack();\n for i in 0..32 {\n assert!(expected[i] == actual[i]);\n }\n }\n\n #[test]\n fn div_by_2() {\n // test a range of small scalars\n for i in 0u64..32 {\n let scalar = Scalar::from(i);\n let dividend = scalar.div_by_2();\n assert_eq!(scalar, dividend + dividend);\n }\n\n // test a range of scalars near the modulus\n for i in 0u64..32 {\n let scalar = Scalar::ZERO - Scalar::from(i);\n let dividend = scalar.div_by_2();\n assert_eq!(scalar, dividend + dividend);\n }\n }\n\n #[test]\n fn reduce() {\n let biggest = Scalar::from_bytes_mod_order([0xff; 32]);\n assert_eq!(biggest, CANONICAL_2_256_MINUS_1);\n }\n\n #[test]\n fn from_bytes_mod_order_wide() {\n let mut bignum = [0u8; 64];\n // set bignum = x + 2^256x\n for i in 0..32 {\n bignum[i] = X[i];\n bignum[32 + i] = X[i];\n }\n // 3958878930004874126169954872055634648693766179881526445624823978500314864344\n // = x + 2^256x (mod l)\n let reduced = Scalar {\n bytes: [\n 216, 154, 179, 139, 210, 121, 2, 71, 69, 99, 158, 216, 23, 173, 63, 100, 204, 0,\n 91, 50, 219, 153, 57, 249, 28, 82, 31, 197, 100, 165, 192, 8,\n ],\n };\n let test_red = Scalar::from_bytes_mod_order_wide(&bignum);\n for i in 0..32 {\n assert!(test_red[i] == reduced[i]);\n }\n }\n\n #[allow(non_snake_case)]\n #[test]\n fn invert() {\n let inv_X = X.invert();\n assert_eq!(inv_X, XINV);\n let should_be_one = inv_X * X;\n assert_eq!(should_be_one, Scalar::ONE);\n }\n\n // Negating a scalar twice should result in the original scalar.\n #[allow(non_snake_case)]\n #[test]\n fn neg_twice_is_identity() {\n let negative_X = -&X;\n let should_be_X = -&negative_X;\n\n assert_eq!(should_be_X, X);\n }\n\n #[test]\n fn to_bytes_from_bytes_roundtrips() {\n let unpacked = X.unpack();\n let bytes = unpacked.to_bytes();\n let should_be_unpacked = UnpackedScalar::from_bytes(&bytes);\n\n assert_eq!(should_be_unpacked.0, unpacked.0);\n }\n\n #[test]\n fn montgomery_reduce_matches_from_bytes_mod_order_wide() {\n let mut bignum = [0u8; 64];\n\n // set bignum = x + 2^256x\n for i in 0..32 {\n bignum[i] = X[i];\n bignum[32 + i] = X[i];\n }\n // x + 2^256x (mod l)\n // = 3958878930004874126169954872055634648693766179881526445624823978500314864344\n let expected = Scalar {\n bytes: [\n 216, 154, 179, 139, 210, 121, 2, 71, 69, 99, 158, 216, 23, 173, 63, 100, 204, 0,\n 91, 50, 219, 153, 57, 249, 28, 82, 31, 197, 100, 165, 192, 8,\n ],\n };\n let reduced = Scalar::from_bytes_mod_order_wide(&bignum);\n\n // The reduced scalar should match the expected\n assert_eq!(reduced.bytes, expected.bytes);\n\n // (x + 2^256x) * R\n let interim =\n UnpackedScalar::mul_internal(&UnpackedScalar::from_bytes_wide(&bignum), &constants::R);\n // ((x + 2^256x) * R) / R (mod l)\n let montgomery_reduced = UnpackedScalar::montgomery_reduce(&interim);\n\n // The Montgomery reduced scalar should match the reduced one, as well as the expected\n assert_eq!(montgomery_reduced.0, reduced.unpack().0);\n assert_eq!(montgomery_reduced.0, expected.unpack().0)\n }\n\n #[test]\n fn canonical_decoding() {\n // canonical encoding of 1667457891\n let canonical_bytes = [\n 99, 99, 99, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0, 0,\n ];\n\n // encoding of\n // 7265385991361016183439748078976496179028704920197054998554201349516117938192\n // = 28380414028753969466561515933501938171588560817147392552250411230663687203 (mod l)\n // non_canonical because unreduced mod l\n let non_canonical_bytes_because_unreduced = [16; 32];\n\n // encoding with high bit set, to check that the parser isn't pre-masking the high bit\n let non_canonical_bytes_because_highbit = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 128,\n ];\n\n assert!(bool::from(\n Scalar::from_canonical_bytes(canonical_bytes).is_some()\n ));\n assert!(bool::from(\n Scalar::from_canonical_bytes(non_canonical_bytes_because_unreduced).is_none()\n ));\n assert!(bool::from(\n Scalar::from_canonical_bytes(non_canonical_bytes_because_highbit).is_none()\n ));\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_scalar_roundtrip() {\n use bincode;\n let encoded = bincode::serialize(&X).unwrap();\n let parsed: Scalar = bincode::deserialize(&encoded).unwrap();\n assert_eq!(parsed, X);\n\n // Check that the encoding is 32 bytes exactly\n assert_eq!(encoded.len(), 32);\n\n // Check that the encoding itself matches the usual one\n assert_eq!(X, bincode::deserialize(X.as_bytes()).unwrap(),);\n }\n\n #[cfg(debug_assertions)]\n #[test]\n #[should_panic]\n fn invert_batch_with_a_zero_input_panics() {\n let mut xs = [Scalar::ONE; 16];\n xs[3] = Scalar::ZERO;\n // This should panic in debug mode.\n Scalar::invert_batch(&mut xs);\n }\n\n #[test]\n fn invert_batch_empty() {\n assert_eq!(Scalar::ONE, Scalar::invert_batch(&mut []));\n }\n\n #[test]\n fn invert_batch_consistency() {\n let mut x = Scalar::from(1u64);\n let mut v1: [Scalar; 16] = core::array::from_fn(|_| {\n let tmp = x;\n x = x + x;\n tmp\n });\n let v2 = v1;\n\n let expected: Scalar = v1.iter().product();\n let expected = expected.invert();\n let ret = Scalar::invert_batch(&mut v1);\n assert_eq!(ret, expected);\n\n for (a, b) in v1.iter().zip(v2.iter()) {\n assert_eq!(a * b, Scalar::ONE);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn batch_vec_invert_consistency() {\n let mut x = Scalar::from(1u64);\n let mut v1: Vec<_> = (0..16)\n .map(|_| {\n let tmp = x;\n x = x + x;\n tmp\n })\n .collect();\n let v2 = v1.clone();\n\n let expected: Scalar = v1.iter().product();\n let expected = expected.invert();\n let ret = Scalar::invert_batch_alloc(&mut v1);\n assert_eq!(ret, expected);\n\n for (a, b) in v1.iter().zip(v2.iter()) {\n assert_eq!(a * b, Scalar::ONE);\n }\n }\n\n #[cfg(feature = \"precomputed-tables\")]\n fn test_pippenger_radix_iter(scalar: Scalar, w: usize) {\n let digits_count = Scalar::to_radix_2w_size_hint(w);\n let digits = scalar.as_radix_2w(w);\n\n let radix = Scalar::from((1 << w) as u64);\n let mut term = Scalar::ONE;\n let mut recovered_scalar = Scalar::ZERO;\n for digit in &digits[0..digits_count] {\n let digit = *digit;\n if digit != 0 {\n let sdigit = if digit < 0 {\n -Scalar::from((-(digit as i64)) as u64)\n } else {\n Scalar::from(digit as u64)\n };\n recovered_scalar += term * sdigit;\n }\n term *= radix;\n }\n // When the input is unreduced, we may only recover the scalar mod l.\n assert_eq!(recovered_scalar, scalar.reduce());\n }\n\n #[test]\n #[cfg(feature = \"precomputed-tables\")]\n fn test_pippenger_radix() {\n use core::iter;\n // For each valid radix it tests that 1000 random-ish scalars can be restored\n // from the produced representation precisely.\n let cases = (2..100)\n .map(|s| Scalar::from(s as u64).invert())\n // The largest unreduced scalar, s = 2^255-1. This is not reduced mod l. Scalar mult\n // still works though.\n .chain(iter::once(LARGEST_UNREDUCED_SCALAR));\n\n for scalar in cases {\n test_pippenger_radix_iter(scalar, 6);\n test_pippenger_radix_iter(scalar, 7);\n test_pippenger_radix_iter(scalar, 8);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn test_read_le_u64_into() {\n let cases: &[(&[u8], &[u64])] = &[\n (\n &[0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F, 0xF0],\n &[0xF00F_F11F_0110_EFFE],\n ),\n (\n &[\n 0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F, 0xF0, 0x12, 0x34, 0x56, 0x78, 0x9A,\n 0xBC, 0xDE, 0xF0,\n ],\n &[0xF00F_F11F_0110_EFFE, 0xF0DE_BC9A_7856_3412],\n ),\n ];\n\n for (src, expected) in cases {\n let mut dst = vec![0; expected.len()];\n read_le_u64_into(src, &mut dst);\n\n assert_eq!(&dst, expected, \"Expected {:x?} got {:x?}\", expected, dst);\n }\n }\n\n // Tests consistency of From<{integer}> impls for Scalar\n #[test]\n fn test_scalar_from_int() {\n let s1 = Scalar::ONE;\n\n // For `x` in `u8`, `u16`, `u32`, `u64`, and `u128`, check that\n // `Scalar::from(x + 1) == Scalar::from(x) + Scalar::from(1)`\n\n let x = 0x23u8;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323u16;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323u32;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323_2323_2323u64;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323_2323_2323_2323_2323_2323_2323u128;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n }\n\n #[cfg(feature = \"group\")]\n #[test]\n fn ff_constants() {\n assert_eq!(Scalar::from(2u64) * Scalar::TWO_INV, Scalar::ONE);\n\n assert_eq!(\n Scalar::ROOT_OF_UNITY * Scalar::ROOT_OF_UNITY_INV,\n Scalar::ONE,\n );\n\n // ROOT_OF_UNITY^{2^s} mod m == 1\n assert_eq!(\n Scalar::ROOT_OF_UNITY.pow(&[1u64 << Scalar::S, 0, 0, 0]),\n Scalar::ONE,\n );\n\n // DELTA^{t} mod m == 1\n assert_eq!(\n Scalar::DELTA.pow(&[\n 0x9604_98c6_973d_74fb,\n 0x0537_be77_a8bd_e735,\n 0x0000_0000_0000_0000,\n 0x0400_0000_0000_0000,\n ]),\n Scalar::ONE,\n );\n }\n\n #[cfg(feature = \"group\")]\n #[test]\n fn ff_impls() {\n assert!(bool::from(Scalar::ZERO.is_even()));\n assert!(bool::from(Scalar::ONE.is_odd()));\n assert!(bool::from(Scalar::from(2u64).is_even()));\n assert!(bool::from(Scalar::DELTA.is_even()));\n\n assert!(bool::from(Field::invert(&Scalar::ZERO).is_none()));\n assert_eq!(Field::invert(&X).unwrap(), XINV);\n\n let x_sq = X.square();\n // We should get back either the positive or negative root.\n assert!([X, -X].contains(&x_sq.sqrt().unwrap()));\n\n assert_eq!(Scalar::from_repr_vartime(X.to_repr()), Some(X));\n assert_eq!(Scalar::from_repr_vartime([0xff; 32]), None);\n\n assert_eq!(Scalar::from_repr(X.to_repr()).unwrap(), X);\n assert!(bool::from(Scalar::from_repr([0xff; 32]).is_none()));\n }\n\n #[test]\n #[should_panic]\n fn test_read_le_u64_into_should_panic_on_bad_input() {\n let mut dst = [0_u64; 1];\n // One byte short\n read_le_u64_into(&[0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F], &mut dst);\n }\n\n #[test]\n fn test_scalar_clamp() {\n let input = A_SCALAR.bytes;\n let expected = [\n 0x18, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x49,\n ];\n let actual = clamp_integer(input);\n assert_eq!(actual, expected);\n\n let expected = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0x40,\n ];\n let actual = clamp_integer([0; 32]);\n assert_eq!(expected, actual);\n let expected = [\n 0xf8, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ];\n let actual = clamp_integer([0xff; 32]);\n assert_eq!(actual, expected);\n\n assert_eq!(\n LARGEST_CLAMPED_INTEGER,\n clamp_integer(LARGEST_CLAMPED_INTEGER)\n );\n }\n\n // Check that a * b == a.reduce() * a.reduce() for ANY scalars a,b, even ones that violate\n // invariant #1, i.e., a,b > 2^255. Old versions of ed25519-dalek did multiplication where a\n // was reduced and b was clamped and unreduced. This checks that was always well-defined.\n #[test]\n fn test_mul_reduction_invariance() {\n let mut rng = rand::thread_rng();\n\n for _ in 0..10 {\n // Also define c that's clamped. We'll make sure that clamping doesn't affect\n // computation\n let (a, b, c) = {\n let mut a_bytes = [0u8; 32];\n let mut b_bytes = [0u8; 32];\n let mut c_bytes = [0u8; 32];\n rng.fill_bytes(&mut a_bytes);\n rng.fill_bytes(&mut b_bytes);\n rng.fill_bytes(&mut c_bytes);\n (\n Scalar { bytes: a_bytes },\n Scalar { bytes: b_bytes },\n Scalar {\n bytes: clamp_integer(c_bytes),\n },\n )\n };\n\n // Make sure this is the same product no matter how you cut it\n let reduced_mul_ab = a.reduce() * b.reduce();\n let reduced_mul_ac = a.reduce() * c.reduce();\n assert_eq!(a * b, reduced_mul_ab);\n assert_eq!(a.reduce() * b, reduced_mul_ab);\n assert_eq!(a * b.reduce(), reduced_mul_ab);\n assert_eq!(a * c, reduced_mul_ac);\n assert_eq!(a.reduce() * c, reduced_mul_ac);\n assert_eq!(a * c.reduce(), reduced_mul_ac);\n }\n }\n}\n"},{"id":6,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs/verification_key.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of solana-ed25519's ed_sigs module, forked from ed25519-zebra.\n// Original ed25519-zebra code: Copyright (c) Zcash Foundation contributors\n// Modifications for HEEA: Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE-APACHE and LICENSE-MIT for licensing information.\n//\n// Modifications from ed25519-zebra:\n// - Added `verify_zebra`, an accelerated verification path using the HEEA\n// scalar decomposition from curve25519-sol's `HEEADecomposition` trait.\n// See \"Accelerating EdDSA Signature Verification with Faster Scalar Size\n// Halving\" (TCHES 2025) for the algorithm.\n// - `verify` dispatches to `verify_zebra`, preserving ZIP-215 semantics.\n\nuse crate::{\n edwards::{CompressedEdwardsY, EdwardsPoint},\n scalar::Scalar,\n traits::{HEEADecomposition, IsIdentity},\n};\nuse core::convert::{TryFrom, TryInto};\nuse sha2::{Sha512, digest::Update};\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\nuse ed25519::{Signature, signature::Verifier};\n\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::der::asn1::BitStringRef;\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::spki::{\n AlgorithmIdentifierRef, DecodePublicKey, EncodePublicKey, Error as SpkiError,\n SubjectPublicKeyInfoRef,\n};\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::{Document, ObjectIdentifier};\n\nuse super::{Error, scalar_from_sha512};\n\n/// The length of an ed25519 `VerificationKey`, in bytes.\npub const VERIFICATION_KEY_LENGTH: usize = 32;\n\n#[cfg(feature = \"pkcs8\")]\nconst OID: ObjectIdentifier = ObjectIdentifier::new_unwrap(\"1.3.101.112\"); // RFC 8410\n#[cfg(feature = \"pkcs8\")]\nconst ALGORITHM_ID: AlgorithmIdentifierRef<'_> = AlgorithmIdentifierRef {\n oid: OID,\n parameters: None,\n};\n\nconst LEGACY_EXCLUDED_R_ENCODINGS: [[u8; 32]; 11] = [\n [\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00,\n ],\n [\n 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00,\n ],\n [\n 0x26, 0xe8, 0x95, 0x8f, 0xc2, 0xb2, 0x27, 0xb0, 0x45, 0xc3, 0xf4, 0x89, 0xf2, 0xef, 0x98,\n 0xf0, 0xd5, 0xdf, 0xac, 0x05, 0xd3, 0xc6, 0x33, 0x39, 0xb1, 0x38, 0x02, 0x88, 0x6d, 0x53,\n 0xfc, 0x05,\n ],\n [\n 0xc7, 0x17, 0x6a, 0x70, 0x3d, 0x4d, 0xd8, 0x4f, 0xba, 0x3c, 0x0b, 0x76, 0x0d, 0x10, 0x67,\n 0x0f, 0x2a, 0x20, 0x53, 0xfa, 0x2c, 0x39, 0xcc, 0xc6, 0x4e, 0xc7, 0xfd, 0x77, 0x92, 0xac,\n 0x03, 0x7a,\n ],\n [\n 0x13, 0xe8, 0x95, 0x8f, 0xc2, 0xb2, 0x27, 0xb0, 0x45, 0xc3, 0xf4, 0x89, 0xf2, 0xef, 0x98,\n 0xf0, 0xd5, 0xdf, 0xac, 0x05, 0xd3, 0xc6, 0x33, 0x39, 0xb1, 0x38, 0x02, 0x88, 0x6d, 0x53,\n 0xfc, 0x85,\n ],\n [\n 0xb4, 0x17, 0x6a, 0x70, 0x3d, 0x4d, 0xd8, 0x4f, 0xba, 0x3c, 0x0b, 0x76, 0x0d, 0x10, 0x67,\n 0x0f, 0x2a, 0x20, 0x53, 0xfa, 0x2c, 0x39, 0xcc, 0xc6, 0x4e, 0xc7, 0xfd, 0x77, 0x92, 0xac,\n 0x03, 0xfa,\n ],\n [\n 0xec, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xed, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xd9, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff,\n ],\n [\n 0xda, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff,\n ],\n];\n\n/// A container for the 32-byte encoded form of an Ed25519 verification key.\n///\n/// This type only checks or carries the byte length. It does not prove that the\n/// bytes decompress to a valid Ed25519 verification key. Convert it to\n/// [`VerificationKey`] to validate the encoded point and cache decoded state\n/// used in signature verification.\n///\n/// A `VerificationKeyBytes` can be used to verify a single signature using the\n/// following idiom:\n/// ```\n/// use core::convert::TryFrom;\n/// # use curve25519::ed_sigs::*;\n/// # let msg = b\"Zcash\";\n/// # let sk = SigningKey::from_bytes(&[1u8; 32]);\n/// # let sig = sk.sign(msg);\n/// # let vk_bytes = VerificationKeyBytes::from(&sk);\n/// VerificationKey::try_from(vk_bytes)\n/// .and_then(|vk| vk.verify(&sig, msg));\n/// ```\n#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\npub struct VerificationKeyBytes(pub(crate) [u8; VERIFICATION_KEY_LENGTH]);\n\nimpl core::fmt::Debug for VerificationKeyBytes {\n fn fmt(&self, fmt: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n fmt.debug_tuple(\"VerificationKeyBytes\")\n .field(&self.0)\n .finish()\n }\n}\n\nimpl AsRef<[u8]> for VerificationKeyBytes {\n fn as_ref(&self) -> &[u8] {\n &self.0[..]\n }\n}\n\nimpl TryFrom<&[u8]> for VerificationKeyBytes {\n type Error = Error;\n fn try_from(slice: &[u8]) -> Result {\n if slice.len() == 32 {\n let mut bytes = [0u8; 32];\n bytes[..].copy_from_slice(slice);\n Ok(bytes.into())\n } else {\n Err(Error::InvalidSliceLength)\n }\n }\n}\n\nimpl From<[u8; 32]> for VerificationKeyBytes {\n fn from(bytes: [u8; 32]) -> VerificationKeyBytes {\n VerificationKeyBytes(bytes)\n }\n}\n\nimpl From for [u8; 32] {\n fn from(refined: VerificationKeyBytes) -> [u8; 32] {\n refined.0\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl<'a> TryFrom> for VerificationKeyBytes {\n type Error = Error;\n\n fn try_from(spki: SubjectPublicKeyInfoRef<'a>) -> Result {\n verification_key_bytes_from_spki(spki).map_err(|_| Error::MalformedPublicKey)\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nfn verification_key_bytes_from_spki(\n spki: SubjectPublicKeyInfoRef<'_>,\n) -> Result {\n if spki.algorithm.oid != OID {\n return Err(SpkiError::OidUnknown {\n oid: spki.algorithm.oid,\n });\n }\n\n if spki.algorithm != ALGORITHM_ID {\n return Err(SpkiError::KeyMalformed);\n }\n\n let bytes = spki\n .subject_public_key\n .as_bytes()\n .ok_or(SpkiError::KeyMalformed)?;\n\n VerificationKeyBytes::try_from(bytes).map_err(|_| SpkiError::KeyMalformed)\n}\n\n/// A valid Ed25519 verification key.\n///\n/// This is also called a public key by other implementations.\n///\n/// This type holds decompressed state used in signature verification; if the\n/// verification key may not be used immediately, it is probably better to use\n/// [`VerificationKeyBytes`], which stores only the length-checked encoded bytes.\n///\n/// ## Zcash-specific consensus properties\n///\n/// Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in\n/// [ZIP 215]. The verification criteria for an (encoded) verification key `A_bytes` are:\n///\n/// * `A_bytes` MUST be an encoding of a point `A` on the twisted Edwards form of\n/// Curve25519, and non-canonical encodings MUST be accepted;\n///\n/// [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519\n#[derive(PartialEq, Eq, Copy, Clone, Debug)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\n#[cfg_attr(feature = \"serde\", serde(try_from = \"VerificationKeyBytes\"))]\n#[cfg_attr(feature = \"serde\", serde(into = \"VerificationKeyBytes\"))]\n#[allow(non_snake_case)]\npub struct VerificationKey {\n pub(crate) A_bytes: VerificationKeyBytes,\n pub(crate) minus_A: EdwardsPoint,\n}\n\nimpl From for VerificationKeyBytes {\n fn from(vk: VerificationKey) -> VerificationKeyBytes {\n vk.A_bytes\n }\n}\n\nimpl AsRef<[u8]> for VerificationKey {\n fn as_ref(&self) -> &[u8] {\n &self.A_bytes.0[..]\n }\n}\n\nimpl Default for VerificationKey {\n fn default() -> VerificationKey {\n let identity: EdwardsPoint = Default::default();\n let identity_bytes = identity.compress().to_bytes();\n\n VerificationKey {\n A_bytes: VerificationKeyBytes::from(identity_bytes),\n minus_A: -identity,\n }\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for VerificationKey {}\n\nimpl From for [u8; 32] {\n fn from(vk: VerificationKey) -> [u8; 32] {\n vk.A_bytes.0\n }\n}\n\nimpl TryFrom for VerificationKey {\n type Error = Error;\n #[allow(non_snake_case)]\n fn try_from(bytes: VerificationKeyBytes) -> Result {\n // * `A_bytes` and `R_bytes` MUST be encodings of points `A` and `R` respectively on the\n // twisted Edwards form of Curve25519, and non-canonical encodings MUST be accepted;\n let A = CompressedEdwardsY(bytes.0)\n .decompress()\n .ok_or(Error::MalformedPublicKey)?;\n\n Ok(VerificationKey {\n A_bytes: bytes,\n minus_A: -A,\n })\n }\n}\n\nimpl TryFrom<&[u8]> for VerificationKey {\n type Error = Error;\n fn try_from(slice: &[u8]) -> Result {\n VerificationKeyBytes::try_from(slice).and_then(|vkb| vkb.try_into())\n }\n}\n\nimpl TryFrom<[u8; 32]> for VerificationKey {\n type Error = Error;\n fn try_from(bytes: [u8; 32]) -> Result {\n VerificationKeyBytes::from(bytes).try_into()\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl EncodePublicKey for VerificationKey {\n /// Serialize [`VerificationKey`] to an ASN.1 DER-encoded document.\n fn to_public_key_der(&self) -> pkcs8::spki::Result {\n SubjectPublicKeyInfoRef {\n algorithm: ALGORITHM_ID,\n subject_public_key: BitStringRef::from_bytes(&self.A_bytes.0[..])?,\n }\n .try_into()\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl DecodePublicKey for VerificationKey {\n /// Deserialize [`VerificationKey`] from ASN.1 DER bytes (32 bytes).\n fn from_public_key_der(bytes: &[u8]) -> Result {\n let spki = SubjectPublicKeyInfoRef::try_from(bytes)?;\n let pk_bytes = verification_key_bytes_from_spki(spki)?;\n Self::try_from(pk_bytes).map_err(|_| SpkiError::KeyMalformed)\n }\n}\n\nimpl Verifier for VerificationKey {\n /// Verify a [`Signature`] object against a given [`VerificationKey`].\n fn verify(\n &self,\n message: &[u8],\n signature: &Signature,\n ) -> Result<(), ed25519::signature::Error> {\n self.verify(signature, message)\n .map_err(|_| ed25519::signature::Error::new())\n }\n}\n\nimpl VerificationKey {\n fn challenge_scalar(&self, signature: &Signature, msg: &[u8]) -> Scalar {\n scalar_from_sha512(\n Sha512::default()\n .chain(&signature.r_bytes()[..])\n .chain(&self.A_bytes.0[..])\n .chain(msg),\n )\n }\n\n /// Verify a purported `signature` on the given `msg`.\n ///\n /// This is the default verification mode and uses the HEEA-accelerated\n /// verification path with Zebra / ZIP-215 semantics.\n ///\n /// ## Zcash-specific consensus properties\n ///\n /// Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in\n /// [ZIP215]. The verification criteria for an (encoded) signature `(R_bytes, s_bytes)` with\n /// (encoded) verification key `A_bytes` are:\n ///\n /// * `A_bytes` and `R_bytes` MUST be encodings of points `A` and `R` respectively on the\n /// twisted Edwards form of Curve25519, and non-canonical encodings MUST be accepted;\n ///\n /// * `s_bytes` MUST represent an integer `s` less than `l`, the order of the prime-order\n /// subgroup of Curve25519;\n ///\n /// * the verification equation `[8][s]B = [8]R + [8][k]A` MUST be satisfied;\n ///\n /// * the alternate verification equation `[s]B = R + [k]A`, allowed by RFC 8032, MUST NOT be\n /// used.\n ///\n /// [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519\n /// [ZIP215]: https://zips.z.cash/zip-0215\n pub fn verify(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_zebra(signature, msg)\n }\n\n /// Verify a signature using HEEA with Zebra / ZIP-215 semantics.\n ///\n /// This implements the algorithm from \"Accelerating EdDSA Signature Verification\n /// with Faster Scalar Size Halving\" (TCHES 2025).\n ///\n /// The decomposition returns ρ and τ such that either ρ ≡ τh (mod ℓ) or\n /// ρ ≡ -τh (mod ℓ). The standard verification equation sB = R + hA is\n /// multiplied by τ and the sign of A is selected according to `flip_h`.\n ///\n /// Both ρ and τ are approximately half the size of h.\n ///\n /// We then decompose τs into two 128-bit scalars:\n /// τs = τs_hi * 2^128 + τs_lo\n ///\n /// The resulting equation can be checked with a 4-variable MSM with\n /// half-size scalars.\n #[allow(non_snake_case)]\n pub fn verify_zebra(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_zebra_prehashed(signature, self.challenge_scalar(signature, msg))\n }\n\n #[allow(non_snake_case)]\n pub(crate) fn verify_zebra_prehashed(\n &self,\n signature: &Signature,\n h: Scalar,\n ) -> Result<(), Error> {\n // Generate half-size scalars ρ and τ. If flip_h is false, then\n // ρ ≡ τh (mod ℓ). If flip_h is true, then ρ ≡ -τh (mod ℓ), so the\n // sign of A is flipped below.\n let (rho, tau, flip_h) = h.heea_decompose();\n\n // Extract s from the signature\n let s = Option::::from(Scalar::from_canonical_bytes(*signature.s_bytes()))\n .ok_or(Error::InvalidSignature)?;\n\n // Decode R from the signature\n let neg_R = -CompressedEdwardsY(*signature.r_bytes())\n .decompress()\n .ok_or(Error::InvalidSignature)?;\n\n // Standard verification checks: sB = R + hA.\n //\n // We verify:\n // [8] τs B + [8] τ (-R) + [8] ρ A_term == 0\n // where A_term is -A when ρ ≡ τh and A when ρ ≡ -τh.\n\n // Compute τs\n let ts = tau * s;\n let A = if flip_h { -self.minus_A } else { self.minus_A };\n // HEEA decomposition guarantees tau and rho fit the optimized\n // 128/128/256-bit multiplication path.\n let result = crate::backend::vartime_triple_base_mul_128_128_256_prechecked(\n &tau, &neg_R, &rho, &A, &ts,\n );\n\n if result.mul_by_cofactor().is_identity() {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n\n /// Verify a signature with dalek-style canonical-`R` byte comparison.\n ///\n /// This recomputes the expected canonical `R` encoding and compares it to the\n /// signature's `R` bytes.\n ///\n /// This helper also preserves this crate's legacy compatibility filters: it\n /// rejects an all-zero encoded public key and the known legacy-excluded `R`\n /// encodings before running the canonical-`R` comparison. Because of those\n /// extra checks, this is not a byte-for-byte clone of every `ed25519-dalek`\n /// release.\n ///\n /// Note that dalek-style canonical-`R` comparison is incompatible with the HEEA\n /// transformed equation because the transformed check does not preserve the\n /// original `R` encoding needed for the byte comparison.\n #[allow(non_snake_case)]\n pub fn verify_dalek(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_dalek_prehashed(signature, self.challenge_scalar(signature, msg))\n }\n\n #[allow(non_snake_case)]\n fn verify_dalek_prehashed(&self, signature: &Signature, h: Scalar) -> Result<(), Error> {\n if self.A_bytes.0 == [0; 32] || LEGACY_EXCLUDED_R_ENCODINGS.contains(signature.r_bytes()) {\n return Err(Error::InvalidSignature);\n }\n\n let s = Option::::from(Scalar::from_canonical_bytes(*signature.s_bytes()))\n .ok_or(Error::InvalidSignature)?;\n\n let expected_R =\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&h, &self.minus_A, &s).compress();\n\n if expected_R.as_bytes() == signature.r_bytes() {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n}\n\n// ---------------------------------------------------------------------------\n// AENEAS-COMPAT verified-verification entry points.\n//\n// `verify_sha512` is semantically `verify_dalek` with each step spelled in\n// extractor-friendly form: the derived array `PartialEq`/`contains` become\n// explicit index loops, and `Scalar::from_canonical_bytes` (whose `subtle`\n// internals defeat the extractor) becomes an explicit `s < l` byte compare\n// followed by `Scalar::from_bytes_mod_order` (the identity on canonical\n// bytes). The verification path is variable-time throughout, as upstream's.\n// ---------------------------------------------------------------------------\n\n/// `bytes` interpreted little-endian is a canonical scalar (< l)? If so the\n/// scalar itself; value-level semantics identical to\n/// `Scalar::from_canonical_bytes(bytes).into()`.\nfn check_scalar_canonical(bytes: [u8; 32]) -> Result {\n /// l = 2^252 + 27742317777372353535851937790883648493, little-endian.\n const L_BYTES: [u8; 32] = [\n 237, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222,\n 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16,\n ];\n // bytes < l, most-significant byte first; the first differing byte decides.\n let mut lt = false;\n let mut decided = false;\n let mut i = 32;\n while i > 0 {\n let j = i - 1;\n if !decided {\n if bytes[j] < L_BYTES[j] {\n lt = true;\n decided = true;\n } else if bytes[j] > L_BYTES[j] {\n decided = true;\n }\n }\n i -= 1;\n }\n if lt {\n Ok(Scalar::from_bytes_mod_order(bytes))\n } else {\n Err(Error::InvalidSignature)\n }\n}\n\n/// Explicit-loop `LEGACY_EXCLUDED_R_ENCODINGS.contains(r)`.\nfn is_legacy_excluded_r(r: &[u8; 32]) -> bool {\n let mut found = false;\n let mut i = 0;\n while i < 11 {\n let mut eq = true;\n let mut j = 0;\n while j < 32 {\n if LEGACY_EXCLUDED_R_ENCODINGS[i][j] != r[j] {\n eq = false;\n }\n j += 1;\n }\n if eq {\n found = true;\n }\n i += 1;\n }\n found\n}\n\nimpl VerificationKey {\n /// Explicit-loop `self.A_bytes.0 != [0; 32]`.\n fn a_bytes_nonzero(&self) -> bool {\n let mut nonzero = false;\n let mut i = 0;\n while i < 32 {\n if self.A_bytes.0[i] != 0 {\n nonzero = true;\n }\n i += 1;\n }\n nonzero\n }\n\n /// Recompute the expected canonical `R` encoding:\n /// `compress([k]*(-A) + [s]*B)` with `k = SHA-512(R || A || msg) mod l`.\n fn recompute_r_sha512(\n &self,\n r_bytes: &[u8; 32],\n s: &Scalar,\n msg: &[u8],\n ) -> CompressedEdwardsY {\n let k = Scalar::from_bytes_mod_order_wide(&super::sha512_hash3(\n &r_bytes[..],\n &self.A_bytes.0[..],\n msg,\n ));\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&k, &self.minus_A, s).compress()\n }\n\n /// Semantically identical to [`Self::verify_dalek`]; see the module\n /// comment above for the extractor-friendly spellings. The signature's\n /// `R`/`s` accessors are each called exactly once.\n pub fn verify_sha512(&self, sig: &Signature, msg: &[u8]) -> Result<(), Error> {\n // (parameter named `sig`: the extractor's generated code would\n // otherwise shadow the `signature::` crate namespace)\n let r_bytes: [u8; 32] = *sig.r_bytes();\n let s_bytes: [u8; 32] = *sig.s_bytes();\n\n if !self.a_bytes_nonzero() {\n return Err(Error::InvalidSignature);\n }\n if is_legacy_excluded_r(&r_bytes) {\n return Err(Error::InvalidSignature);\n }\n let s = check_scalar_canonical(s_bytes)?;\n\n let expected_r = self.recompute_r_sha512(&r_bytes, &s, msg);\n let e = expected_r.as_bytes();\n let mut equal = true;\n let mut k = 0;\n while k < 32 {\n if e[k] != r_bytes[k] {\n equal = false;\n }\n k += 1;\n }\n if equal {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n}\n"},{"id":7,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs.rs"},"crate_name":"curve25519","contents":"//! Ed25519 signing and verification (ZIP-215 / HEEA-accelerated).\n\nuse crate::scalar::Scalar;\nuse sha2::{Digest, Sha512};\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n#[cfg(test)]\nmod tests;\n\n#[cfg(feature = \"alloc\")]\npub mod batch;\nmod error;\nmod signing_key;\nmod verification_key;\n\n// Allows importing traits used by `Signature`.\npub use ::ed25519;\npub use ::ed25519::Signature;\npub use error::Error;\npub use signing_key::SigningKey;\npub use verification_key::{VerificationKey, VerificationKeyBytes};\n\npub(crate) fn scalar_from_sha512(hash: Sha512) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut output = hash.finalize();\n let mut bytes = [0u8; 64];\n bytes.copy_from_slice(output.as_slice());\n let scalar = Scalar::from_bytes_mod_order_wide(&bytes);\n\n #[cfg(feature = \"zeroize\")]\n {\n output.zeroize();\n bytes.zeroize();\n }\n\n scalar\n}\n\n/// AENEAS-COMPAT: single-call SHA-512 oracle for the verified verification\n/// path. Semantically `Sha512(r || a || m)` — identical to hashing via the\n/// incremental `Digest` API. A single monomorphic function whose signature\n/// carries no foreign types lets the extractor treat the whole hash\n/// computation as one opaque oracle.\npub(crate) fn sha512_hash3(r: &[u8], a: &[u8], m: &[u8]) -> [u8; 64] {\n let mut h = Sha512::default();\n Digest::update(&mut h, r);\n Digest::update(&mut h, a);\n Digest::update(&mut h, m);\n let output = h.finalize();\n let mut bytes = [0u8; 64];\n bytes.copy_from_slice(output.as_slice());\n bytes\n}\n"},{"id":8,"name":{"Local":"curve25519/solana-ed25519/src/field.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis agora lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - Isis Agora Lovecruft \n// - Henry de Valence \n\n//! Field arithmetic modulo \\\\(p = 2\\^{255} - 19\\\\).\n//!\n//! The `curve25519::field` module provides a type alias\n//! `curve25519::field::FieldElement` to a field element type\n//! defined in the `backend` module; either `FieldElement51` or\n//! `FieldElement2625`.\n//!\n//! Field operations defined in terms of machine\n//! operations, such as field multiplication or squaring, are defined in\n//! the backend implementation.\n//!\n//! Field operations defined in terms of other field operations, such as\n//! field inversion or square roots, are defined here.\n\n#![allow(unused_qualifications)]\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\nuse crate::backend;\nuse crate::constants;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n#[cfg(feature = \"digest\")]\nuse digest::{\n Digest, FixedOutput, HashMarker,\n array::{Array, typenum::U64},\n block_api::BlockSizeUser,\n typenum::{IsGreater, True},\n};\n\n/// A `FieldElement` represents an element of the field\n/// \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\).\n///\n/// The `FieldElement` type is pinned to the 64-bit serial implementation.\npub(crate) type FieldElement = backend::serial::u64::field::FieldElement51;\n\nimpl Eq for FieldElement {}\n\nimpl PartialEq for FieldElement {\n fn eq(&self, other: &FieldElement) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl ConstantTimeEq for FieldElement {\n /// Test equality between two `FieldElement`s. Since the\n /// internal representation is not canonical, the field elements\n /// are normalized to wire format before comparison.\n fn ct_eq(&self, other: &FieldElement) -> Choice {\n self.to_bytes().ct_eq(&other.to_bytes())\n }\n}\n\nimpl Default for FieldElement {\n fn default() -> Self {\n FieldElement::ZERO\n }\n}\n\nimpl FieldElement {\n /// Load a `FieldElement` from 64 bytes, by reducing modulo q.\n #[cfg(feature = \"digest\")]\n pub(crate) fn from_bytes_wide(bytes: &[u8; 64]) -> Self {\n let mut fl = [0u8; 32];\n let mut gl = [0u8; 32];\n fl.copy_from_slice(&bytes[..32]);\n gl.copy_from_slice(&bytes[32..]);\n // Mask off the top bits of both halves, since from_bytes masks them off anyway. We'll add\n // them back in later.\n let fl_top_bit = (fl[31] >> 7) as u16;\n let gl_top_bit = (gl[31] >> 7) as u16;\n fl[31] &= 0x7f;\n gl[31] &= 0x7f;\n\n // Interpret both sides as field elements\n let mut fe_f = Self::from_bytes(&fl);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut fe_g = Self::from_bytes(&gl);\n\n // The full field elem is now fe_f + 2²⁵⁵ fl_top_bit + 2²⁵⁶ fe_g + 2⁵¹¹ gl_top_bit\n\n // Add the masked off bits back to fe_f. fl_top_bit, if set, is 2^255 ≡ 19 (mod q).\n // gl_top_bit, if set, is 2^511 ≡ 722 (mod q)\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut top_bits_sum = {\n // This only need to be a u16 because the max value is 741\n let addend: u16 = fl_top_bit * 19 + gl_top_bit * 722;\n let mut addend_bytes = [0u8; 32];\n addend_bytes[..2].copy_from_slice(&addend.to_le_bytes());\n let top_bits_sum = Self::from_bytes(&addend_bytes);\n\n #[cfg(feature = \"zeroize\")]\n addend_bytes.zeroize();\n\n top_bits_sum\n };\n fe_f += &top_bits_sum;\n\n // Now add the high half into fe_f. The RHS is multiplied by 2^256 ≡ 38 (mod q)\n const THIRTY_EIGHT: FieldElement = FieldElement::from_bytes(&[\n 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ]);\n fe_f += &(&THIRTY_EIGHT * &fe_g);\n\n #[cfg(feature = \"zeroize\")]\n {\n fl.zeroize();\n gl.zeroize();\n fe_g.zeroize();\n top_bits_sum.zeroize();\n }\n\n fe_f\n }\n\n /// Determine if this `FieldElement` is negative, in the sense\n /// used in the ed25519 paper: `x` is negative if the low bit is\n /// set.\n ///\n /// # Return\n ///\n /// If negative, return `Choice(1)`. Otherwise, return `Choice(0)`.\n pub(crate) fn is_negative(&self) -> Choice {\n let bytes = self.to_bytes();\n (bytes[0] & 1).into()\n }\n\n /// Determine if this `FieldElement` is zero.\n ///\n /// # Return\n ///\n /// If zero, return `Choice(1)`. Otherwise, return `Choice(0)`.\n pub(crate) fn is_zero(&self) -> Choice {\n let zero = [0u8; 32];\n let bytes = self.to_bytes();\n\n bytes.ct_eq(&zero)\n }\n\n /// Compute (self^(2^250-1), self^11), used as a helper function\n /// within invert() and pow22523().\n #[rustfmt::skip] // keep alignment of explanatory comments\n fn pow22501(&self) -> (FieldElement, FieldElement) {\n // Instead of managing which temporary variables are used\n // for what, we define as many as we need and leave stack\n // allocation to the compiler\n //\n // Each temporary variable t_i is of the form (self)^e_i.\n // Squaring t_i corresponds to multiplying e_i by 2,\n // so the pow2k function shifts e_i left by k places.\n // Multiplying t_i and t_j corresponds to adding e_i + e_j.\n //\n // Temporary t_i Nonzero bits of e_i\n //\n let t0 = self.square(); // 1 e_0 = 2^1\n let t1 = t0.square().square(); // 3 e_1 = 2^3\n let t2 = self * &t1; // 3,0 e_2 = 2^3 + 2^0\n let t3 = &t0 * &t2; // 3,1,0\n let t4 = t3.square(); // 4,2,1\n let t5 = &t2 * &t4; // 4,3,2,1,0\n let t6 = t5.pow2k(5); // 9,8,7,6,5\n let t7 = &t6 * &t5; // 9,8,7,6,5,4,3,2,1,0\n let t8 = t7.pow2k(10); // 19..10\n let t9 = &t8 * &t7; // 19..0\n let t10 = t9.pow2k(20); // 39..20\n let t11 = &t10 * &t9; // 39..0\n let t12 = t11.pow2k(10); // 49..10\n let t13 = &t12 * &t7; // 49..0\n let t14 = t13.pow2k(50); // 99..50\n let t15 = &t14 * &t13; // 99..0\n let t16 = t15.pow2k(100); // 199..100\n let t17 = &t16 * &t15; // 199..0\n let t18 = t17.pow2k(50); // 249..50\n let t19 = &t18 * &t13; // 249..0\n\n (t19, t3)\n }\n\n /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse.\n ///\n /// When an input `FieldElement` is zero, its value is unchanged.\n #[cfg(feature = \"alloc\")]\n pub(crate) fn invert_batch_alloc(inputs: &mut [FieldElement]) {\n let n = inputs.len();\n let mut scratch = vec![FieldElement::ONE; n];\n\n Self::internal_invert_batch(inputs, &mut scratch);\n\n #[cfg(feature = \"zeroize\")]\n scratch.zeroize();\n }\n\n /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse. `scratch` can\n /// contain anything, so long as its length is the same as `inputs`.\n ///\n /// When an input `FieldElement` is zero, its value is unchanged.\n ///\n /// # Panics\n /// Panics when `scratch.len() != inputs.len()`\n fn internal_invert_batch(inputs: &mut [FieldElement], scratch: &mut [FieldElement]) {\n // Montgomery’s Trick and Fast Implementation of Masked AES\n // Genelle, Prouff and Quisquater\n // Section 3.2\n\n debug_assert_eq!(inputs.len(), scratch.len());\n\n // Keep an accumulator of all of the previous products\n let mut acc = FieldElement::ONE;\n\n // Pass through the input vector, recording the previous\n // products in the scratch space\n for (input, scratch) in inputs.iter().zip(scratch.iter_mut()) {\n *scratch = acc;\n // acc <- acc * input, but skipping zeros (constant-time)\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut product = &acc * input;\n acc.conditional_assign(&product, !input.is_zero());\n\n #[cfg(feature = \"zeroize\")]\n product.zeroize();\n }\n\n // acc is nonzero because we skipped zeros in inputs\n assert!(bool::from(!acc.is_zero()));\n\n // Compute the inverse of all products\n acc = acc.invert();\n\n // Pass through the vector backwards to compute the inverses\n // in place\n for (input, scratch) in inputs.iter_mut().rev().zip(scratch.iter_mut().rev()) {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut tmp = &acc * input;\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = &acc * scratch;\n // input <- acc * scratch, then acc <- tmp\n // Again, we skip zeros in a constant-time way\n let nz = !input.is_zero();\n input.conditional_assign(&inverse, nz);\n acc.conditional_assign(&tmp, nz);\n\n #[cfg(feature = \"zeroize\")]\n {\n tmp.zeroize();\n inverse.zeroize();\n }\n }\n\n #[cfg(feature = \"zeroize\")]\n {\n acc.zeroize();\n scratch.zeroize();\n }\n }\n\n /// Given a nonzero field element, compute its inverse.\n ///\n /// The inverse is computed as self^(p-2), since\n /// x^(p-2)x = x^(p-1) = 1 (mod p).\n ///\n /// This function returns zero on input zero.\n #[rustfmt::skip] // keep alignment of explanatory comments\n #[allow(clippy::let_and_return)]\n pub(crate) fn invert(&self) -> FieldElement {\n // The bits of p-2 = 2^255 -19 -2 are 11010111111...11.\n //\n // nonzero bits of exponent\n let (t19, t3) = self.pow22501(); // t19: 249..0 ; t3: 3,1,0\n let t20 = t19.pow2k(5); // 254..5\n let t21 = &t20 * &t3; // 254..5,3,1,0\n\n t21\n }\n\n /// Raise this field element to the power (p-5)/8 = 2^252 -3.\n #[rustfmt::skip] // keep alignment of explanatory comments\n #[allow(clippy::let_and_return)]\n pub(crate) fn pow_p58(&self) -> FieldElement {\n // The bits of (p-5)/8 are 101111.....11.\n //\n // nonzero bits of exponent\n let (t19, _) = self.pow22501(); // 249..0\n let t20 = t19.pow2k(2); // 251..2\n let t21 = self * &t20; // 251..2,0\n\n t21\n }\n\n /// Given `FieldElements` `u` and `v`, compute either `sqrt(u/v)`\n /// or `sqrt(i*u/v)` in constant time.\n ///\n /// This function always returns the nonnegative square root.\n ///\n /// # Return\n ///\n /// - `(Choice(1), +sqrt(u/v)) ` if `v` is nonzero and `u/v` is square;\n /// - `(Choice(1), zero) ` if `u` is zero;\n /// - `(Choice(0), zero) ` if `v` is zero and `u` is nonzero;\n /// - `(Choice(0), +sqrt(i*u/v))` if `u/v` is nonsquare (so `i*u/v` is square).\n ///\n pub(crate) fn sqrt_ratio_i(u: &FieldElement, v: &FieldElement) -> (Choice, FieldElement) {\n // Using the same trick as in ed25519 decoding, we merge the\n // inversion, the square root, and the square test as follows.\n //\n // To compute sqrt(α), we can compute β = α^((p+3)/8).\n // Then β^2 = ±α, so multiplying β by sqrt(-1) if necessary\n // gives sqrt(α).\n //\n // To compute 1/sqrt(α), we observe that\n // 1/β = α^(p-1 - (p+3)/8) = α^((7p-11)/8)\n // = α^3 * (α^7)^((p-5)/8).\n //\n // We can therefore compute sqrt(u/v) = sqrt(u)/sqrt(v)\n // by first computing\n // r = u^((p+3)/8) v^(p-1-(p+3)/8)\n // = u u^((p-5)/8) v^3 (v^7)^((p-5)/8)\n // = (uv^3) (uv^7)^((p-5)/8).\n //\n // If v is nonzero and u/v is square, then r^2 = ±u/v,\n // so vr^2 = ±u.\n // If vr^2 = u, then sqrt(u/v) = r.\n // If vr^2 = -u, then sqrt(u/v) = r*sqrt(-1).\n //\n // If v is zero, r is also zero.\n\n let v3 = &v.square() * v;\n let v7 = &v3.square() * v;\n let mut r = &(u * &v3) * &(u * &v7).pow_p58();\n let check = v * &r.square();\n\n let i = &constants::SQRT_M1;\n\n let correct_sign_sqrt = check.ct_eq(u);\n let flipped_sign_sqrt = check.ct_eq(&(-u));\n let flipped_sign_sqrt_i = check.ct_eq(&(&(-u) * i));\n\n let r_prime = &constants::SQRT_M1 * &r;\n r.conditional_assign(&r_prime, flipped_sign_sqrt | flipped_sign_sqrt_i);\n\n // Choose the nonnegative square root.\n //\n // NOTE: written as negate-then-conditional-assign rather than\n // `r.conditional_negate(r_is_negative)`: semantically identical and\n // still constant-time, but it avoids subtle's `ConditionallyNegatable`\n // blanket impl (whose `for<'a> &'a T: Neg` bound breaks the Aeneas\n // verification toolchain; see verification/field/README.md).\n let r_is_negative = r.is_negative();\n let r_neg = -&r;\n r.conditional_assign(&r_neg, r_is_negative);\n\n let was_nonzero_square = correct_sign_sqrt | flipped_sign_sqrt;\n\n (was_nonzero_square, r)\n }\n\n /// Attempt to compute `sqrt(1/self)` in constant time.\n ///\n /// Convenience wrapper around `sqrt_ratio_i`.\n ///\n /// This function always returns the nonnegative square root.\n ///\n /// # Return\n ///\n /// - `(Choice(1), +sqrt(1/self)) ` if `self` is a nonzero square;\n /// - `(Choice(0), zero) ` if `self` is zero;\n /// - `(Choice(0), +sqrt(i/self)) ` if `self` is a nonzero nonsquare;\n ///\n pub(crate) fn invsqrt(&self) -> (Choice, FieldElement) {\n FieldElement::sqrt_ratio_i(&FieldElement::ONE, self)\n }\n\n #[cfg(feature = \"digest\")]\n /// Hashes the given message and domain separator to produce `COUNT` [`FieldElement`]s, per RFC\n /// 9380. The hash's input is the concatenation of the elements of `msg`. Likewise for the\n /// domain separator with `domain_sep`. At least one element of `domain_sep`, MUST be nonempty,\n /// its concatenation MUST NOT exceed 255 bytes, and `COUNT` MUST be 1 or 2.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`. Also panics if `COUNT > 2` or\n /// `COUNT == 0`.\n pub fn hash_to_field(\n msg: &[&[u8]],\n domain_sep: &[&[u8]],\n ) -> [Self; COUNT]\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // We only use `hash_to_field` for Elligator2, which uses 0 < COUNT <= 2\n assert!(COUNT == 1 || COUNT == 2);\n\n // §5.2, we generate count * m * L = COUNT * 1 * (256 + 128)/8 bytes\n let len_in_bytes = COUNT * 48;\n // Buffer should hold however many digests we need to produce len_in_bytes bytes.\n // len_in_bytes is at most 2*48=96, and the digest is 64B. So that's 2 digests, or 128B\n let mut buf = [0u8; 128];\n // We can call this without worrying about panics: len_in_bytes is at most 96, and buf is\n // bigger than it. Further, if `domain_sep` is too large, that's also a panic condition of\n // this function so it's fine\n let uniform_bytes = expand_msg_xmd::(msg, domain_sep, &mut buf, len_in_bytes);\n\n let mut result = [FieldElement::ONE; COUNT];\n for i in 0..COUNT {\n let mut bytes_wide = [0u8; 64];\n bytes_wide[..48].copy_from_slice(&uniform_bytes[48 * i..48 * (i + 1)]);\n bytes_wide[..48].reverse();\n\n result[i] = FieldElement::from_bytes_wide(&bytes_wide);\n }\n\n result\n }\n}\n\n/// Hashes the concatenation of the elements of `msg` with domain separator equal to the\n/// concatenation of `domain_sep`. The output is an `outlen`-length slice into `buf`. Follows\n/// https://www.rfc-editor.org/rfc/rfc9380.html#section-5.3.1\n///\n/// # Panics\n/// Panics if the domain separator is empty, if the total length of the domain separator is more\n/// than 255 bytes, if `outlen` is more than `255 * D::output_size()`, if `outlen is more than\n/// 65535, or if `dst.len() < outlen`.\n#[cfg(feature = \"digest\")]\nfn expand_msg_xmd<'a, D>(\n msg: &[&[u8]],\n domain_sep: &[&[u8]],\n buf: &'a mut [u8],\n outlen: usize,\n) -> &'a [u8]\nwhere\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n{\n use core::iter::once;\n\n // The notation we use in this function is the same as in the spec\n let len_in_bytes = u16::try_from(outlen).expect(\"outlen must not exceed 65535\");\n\n assert!(buf.len() >= outlen);\n let ell = u8::try_from((len_in_bytes as usize).div_ceil(D::output_size()))\n .expect(\"output length cannot exceed 255 times digest size\");\n let domain_sep_len = u8::try_from(domain_sep.iter().map(|c| c.len()).sum::())\n .expect(\"unexpected overflow from domain separator's size.\");\n assert_ne!(\n domain_sep_len, 0,\n \"domain separator MUST have nonzero length.\"\n );\n\n let domain_sep_len_slice = &[domain_sep_len][..];\n let dst_prime = domain_sep.iter().copied().chain(once(domain_sep_len_slice));\n let z_pad = Array::::default();\n let l_i_b_str = len_in_bytes.to_be_bytes();\n\n // Collect the components of msg_prime\n let msg_prime = once(z_pad.as_slice())\n .chain(msg.iter().copied())\n .chain(once(l_i_b_str.as_slice()))\n .chain(once(&[0u8][..]))\n .chain(dst_prime.clone());\n // Hash all of msg_prime\n let b_0 = msg_prime\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n\n // Collect the input components for the b_1 hash\n let b_1_input = once(b_0.as_slice())\n .chain(once(&[1u8][..]))\n .chain(dst_prime.clone());\n let b_1 = b_1_input\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n // Write b_1 to the output buffer\n buf[..D::output_size()].copy_from_slice(&b_1);\n\n for i in 2..=ell {\n // Get the last digest we produced\n let i_ = i as usize;\n let last_b = &buf[(i_ - 2) * D::output_size()..(i_ - 1) * D::output_size()];\n // XOR it with b_0\n let mut xor_bs = b_0.clone();\n xor_bs\n .as_mut_slice()\n .iter_mut()\n .zip(last_b)\n .for_each(|(l, r)| *l ^= *r);\n\n // Hash (b_0 ^ b_{i-1}) || i || dst_prime\n let i_slice = &[i][..];\n let b_i_input = once(xor_bs.as_slice())\n .chain(once(i_slice))\n .chain(dst_prime.clone());\n let b_i = b_i_input\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n\n // Write b_i to the output buffer\n buf[(i_ - 1) * D::output_size()..i_ * D::output_size()].copy_from_slice(&b_i);\n }\n\n &buf[..outlen]\n}\n\n#[cfg(test)]\nmod test {\n use crate::field::*;\n use subtle::ConditionallyNegatable;\n\n /// Random element a of GF(2^255-19), from Sage\n /// a = 1070314506888354081329385823235218444233221\\\n /// 2228051251926706380353716438957572\n static A_BYTES: [u8; 32] = [\n 0x04, 0xfe, 0xdf, 0x98, 0xa7, 0xfa, 0x0a, 0x68, 0x84, 0x92, 0xbd, 0x59, 0x08, 0x07, 0xa7,\n 0x03, 0x9e, 0xd1, 0xf6, 0xf2, 0xe1, 0xd9, 0xe2, 0xa4, 0xa4, 0x51, 0x47, 0x36, 0xf3, 0xc3,\n 0xa9, 0x17,\n ];\n\n /// Byte representation of a**2\n static ASQ_BYTES: [u8; 32] = [\n 0x75, 0x97, 0x24, 0x9e, 0xe6, 0x06, 0xfe, 0xab, 0x24, 0x04, 0x56, 0x68, 0x07, 0x91, 0x2d,\n 0x5d, 0x0b, 0x0f, 0x3f, 0x1c, 0xb2, 0x6e, 0xf2, 0xe2, 0x63, 0x9c, 0x12, 0xba, 0x73, 0x0b,\n 0xe3, 0x62,\n ];\n\n /// Byte representation of 1/a\n static AINV_BYTES: [u8; 32] = [\n 0x96, 0x1b, 0xcd, 0x8d, 0x4d, 0x5e, 0xa2, 0x3a, 0xe9, 0x36, 0x37, 0x93, 0xdb, 0x7b, 0x4d,\n 0x70, 0xb8, 0x0d, 0xc0, 0x55, 0xd0, 0x4c, 0x1d, 0x7b, 0x90, 0x71, 0xd8, 0xe9, 0xb6, 0x18,\n 0xe6, 0x30,\n ];\n\n /// Byte representation of a^((p-5)/8)\n static AP58_BYTES: [u8; 32] = [\n 0x6a, 0x4f, 0x24, 0x89, 0x1f, 0x57, 0x60, 0x36, 0xd0, 0xbe, 0x12, 0x3c, 0x8f, 0xf5, 0xb1,\n 0x59, 0xe0, 0xf0, 0xb8, 0x1b, 0x20, 0xd2, 0xb5, 0x1f, 0x15, 0x21, 0xf9, 0xe3, 0xe1, 0x61,\n 0x21, 0x55,\n ];\n\n #[test]\n fn a_mul_a_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(asq, &a * &a);\n }\n\n #[test]\n fn a_square_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(asq, a.square());\n }\n\n #[test]\n fn a_square2_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(a.square2(), &asq + &asq);\n }\n\n #[test]\n fn a_invert_vs_inverse_of_a_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n let should_be_inverse = a.invert();\n assert_eq!(ainv, should_be_inverse);\n assert_eq!(FieldElement::ONE, &a * &should_be_inverse);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn invert_batch_a_matches_nonbatched() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ap58 = FieldElement::from_bytes(&AP58_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n let a0 = &a - &a;\n let a2 = &a + &a;\n let a_list = vec![a, ap58, asq, ainv, a0, a2];\n let mut ainv_list = a_list.clone();\n FieldElement::invert_batch_alloc(&mut ainv_list[..]);\n for i in 0..6 {\n assert_eq!(a_list[i].invert(), ainv_list[i]);\n }\n }\n\n #[test]\n fn sqrt_ratio_behavior() {\n let zero = FieldElement::ZERO;\n let one = FieldElement::ONE;\n let i = constants::SQRT_M1;\n let two = &one + &one; // 2 is nonsquare mod p.\n let four = &two + &two; // 4 is square mod p.\n\n // 0/0 should return (1, 0) since u is 0\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&zero, &zero);\n assert!(bool::from(choice));\n assert_eq!(sqrt, zero);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 1/0 should return (0, 0) since v is 0, u is nonzero\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &zero);\n assert!(bool::from(!choice));\n assert_eq!(sqrt, zero);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 2/1 is nonsquare, so we expect (0, sqrt(i*2))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&two, &one);\n assert!(bool::from(!choice));\n assert_eq!(sqrt.square(), &two * &i);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 4/1 is square, so we expect (1, sqrt(4))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&four, &one);\n assert!(bool::from(choice));\n assert_eq!(sqrt.square(), four);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 1/4 is square, so we expect (1, 1/sqrt(4))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &four);\n assert!(bool::from(choice));\n assert_eq!(&sqrt.square() * &four, one);\n assert!(bool::from(!sqrt.is_negative()));\n }\n\n #[test]\n fn a_p58_vs_ap58_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ap58 = FieldElement::from_bytes(&AP58_BYTES);\n assert_eq!(ap58, a.pow_p58());\n }\n\n #[test]\n fn equality() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n assert!(a == a);\n assert!(a != ainv);\n }\n\n /// Notice that the last element has the high bit set, which\n /// should be ignored\n static B_BYTES: [u8; 32] = [\n 113, 191, 169, 143, 91, 234, 121, 15, 241, 131, 217, 36, 230, 101, 92, 234, 8, 208, 170,\n 251, 97, 127, 70, 210, 58, 23, 166, 87, 240, 169, 184, 178,\n ];\n\n #[test]\n fn from_bytes_highbit_is_ignored() {\n let mut cleared_bytes = B_BYTES;\n cleared_bytes[31] &= 127u8;\n let with_highbit_set = FieldElement::from_bytes(&B_BYTES);\n let without_highbit_set = FieldElement::from_bytes(&cleared_bytes);\n assert_eq!(without_highbit_set, with_highbit_set);\n }\n\n #[test]\n fn conditional_negate() {\n let one = FieldElement::ONE;\n let minus_one = FieldElement::MINUS_ONE;\n let mut x = one;\n x.conditional_negate(Choice::from(1));\n assert_eq!(x, minus_one);\n x.conditional_negate(Choice::from(0));\n assert_eq!(x, minus_one);\n x.conditional_negate(Choice::from(1));\n assert_eq!(x, one);\n }\n\n #[test]\n fn encoding_is_canonical() {\n // Encode 1 wrongly as 1 + (2^255 - 19) = 2^255 - 18\n let one_encoded_wrongly_bytes: [u8; 32] = [\n 0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ];\n // Decode to a field element\n let one = FieldElement::from_bytes(&one_encoded_wrongly_bytes);\n // .. then check that the encoding is correct\n let one_bytes = one.to_bytes();\n assert_eq!(one_bytes[0], 1);\n for byte in &one_bytes[1..] {\n assert_eq!(*byte, 0);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn invert_batch_empty() {\n FieldElement::invert_batch_alloc(&mut []);\n }\n\n // The following two consts were generated with the following sage script:\n //\n // import random\n //\n // F = GF(2**255 - 19)\n // # Use a seed to make sure we produce the same test vectors every time\n // random.seed(\"Ozamataz Buckshank\")\n //\n // # Generates test vectors, each of the form (input_bytes, reduced_field_elem_bytes),\n // # where input_bytes is length input_bytes_len\n // def gen_example(input_bytes_len):\n // # Generate random bytes\n // input_bytes = [random.randint(0, 255) for _ in range(input_bytes_len)]\n //\n // # Now convert to a field element and get the reduced byte representation\n // elem = F(int.from_bytes(input_bytes, byteorder='little'))\n // reduced_bytes = list(int(elem).to_bytes(32, byteorder='little'))\n //\n // # Format input and output as hex strings\n // input_bytes_hex = ''.join(f'{byte:02x}' for byte in input_bytes)\n // reduced_bytes_hex = ''.join(f'{byte:02x}' for byte in reduced_bytes)\n // return f\"(\\\"{input_bytes_hex}\\\", \\\"{reduced_bytes_hex}\\\")\"\n //\n // print(\"SET 1: Input bytes are length 64\")\n // for _ in range(5):\n // print(gen_example(64))\n //\n // print(\"SET 2: Input bytes are length 48\")\n // for _ in range(5):\n // print(gen_example(48))\n\n /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-64 bytestring,\n /// reduced field element)\n #[cfg(feature = \"digest\")]\n const FROM_BYTES_WIDE_KAT_BIG: &[(&str, &str)] = &[\n (\n \"77b663085cac0e916f40dbeea5116f201816406e68ccf01b32a97162ae1d5bf95d0d01c2c72fbeeb27a63\\\n 5b85b715d5ce6f74118a60a7aec53c798ad648a482f\",\n \"62b38bd402c4498f5cead14643e54dd649e20a0810610e36a73f1f27a0a81f7e\",\n ),\n (\n \"d437c75ec79886650243a79c62933bb307eb12ff16d05db4a6a8a877f4a91abb6eeb64d2e20519c021799\\\n 3a1dc5639283a06639985a2c892208171503335afb5\",\n \"3d2ec29972783de9043e8b982278beaba9d7c5c3ebef257e7cd38168928f1c33\",\n ),\n (\n \"6daa9e1abe6c604fb6e841c04bf90a6ef88aef6b1eab17dd44f7207ef472cd2d54bac849f703e64f36e56\\\n 77e7e86b82be7d26aa220daf1f208bb36dcc1a12338\",\n \"28546a0e7303852bc6eead8312f06eeb48d9ca87f60bfeec98ba402ebb751703\",\n ),\n (\n \"c3920e326dbf806a50105be78263c1dc9390fb4741587b250cd758c2bfa3ed70faedbbc5f9b1d024e00fe\\\n 7d7daf796866853f42e72d638e6533c5eb5b7caf3c6\",\n \"40eaf38b802a7be1956ba7f3fe2d2ad717f23f40342deb5180cb55ae04bb1d79\",\n ),\n (\n \"23f143c72ead6c0f336b4e746a06921f0eb180002e8ce916d196de16216788617c6aeb90a074a85196f03\\\n 81375011248927c1215e9ec65b382a6ec556fb3f504\",\n \"b1bf354a04fd6d2e8321c24ecb3d3ed2c42e3f21c7b60ab8374effd7a709011e\",\n ),\n ];\n\n /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-48 bytestring,\n /// reduced field element)\n #[cfg(feature = \"digest\")]\n const FROM_BYTES_WIDE_KAT_MEDIUM: &[(&str, &str)] = &[\n (\n \"82e9cbe4928e3d0bbf1f91824a91acfb30d929f7a2fa5cbcc967c63ea0f3357c29c19f1bc9dcad69d85c1\\\n c6265970685\",\n \"989582fe6c540cbbdee7c612570aa7ba44d929f7a2fa5cbcc967c63ea0f3357c\",\n ),\n (\n \"5480494df4fb3a3b19da17e1c8b9192ccb09ec76720321977079300c42c17b9e95b01eb37ffe7048fcd1c\\\n 9e6094da6c4\",\n \"85b6d7e3e8c200fc8b050d234129c95ce809ec76720321977079300c42c17b1e\",\n ),\n (\n \"93ec8a480dde098f74bcd341ef4f248f6440cc6e631d7000784f66975a4fd628438bb1350ba4c1421fec3\\\n 670decced06\",\n \"8598e540b737c87718c9fae9f3b870966540cc6e631d7000784f66975a4fd628\",\n ),\n (\n \"fd0154ff9a5c4c9ee4e8183c23db97018e0e6201a812f6d4faedda50652d51f65c110b9a1a100a3fc3ff1\\\n c4ea3cf22e4\",\n \"b895f8dc8dc0caf9dfdf66d460adc2deaf0e6201a812f6d4faedda50652d5176\",\n ),\n (\n \"0e829dc955e0a1e0dbda9849cb2022b295275782348bd6308b3d0c5836f3ca0130911a17fd54054c3a0f8\\\n b2486f8ce85\",\n \"2e0f8f37e77d6c29831d3db6b404db8ea9275782348bd6308b3d0c5836f3ca01\",\n ),\n ];\n\n #[cfg(feature = \"digest\")]\n #[test]\n fn from_bytes_wide() {\n // Do the 64-byte input ones first\n for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_BIG {\n let decoded_input =\n hex::decode(input_bytes).expect(\"64-byte reduction test vector hex must decode\");\n let decoded_input: [u8; 64] = decoded_input\n .as_slice()\n .try_into()\n .expect(\"64-byte reduction test vector must decode to 64 bytes\");\n let reduce_fe = FieldElement::from_bytes_wide(&decoded_input);\n assert_eq!(\n &reduce_fe.to_bytes(),\n hex::decode(expected_reduced)\n .expect(\"reduction result test vector hex must decode\")\n .as_slice()\n );\n }\n\n // Now do the 48-byte inputs\n for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_MEDIUM {\n let mut padded_input_bytes = [0u8; 64];\n padded_input_bytes[..48].copy_from_slice(\n &hex::decode(input_bytes).expect(\"48-byte reduction test vector hex must decode\"),\n );\n let reduce_fe = FieldElement::from_bytes_wide(&padded_input_bytes);\n assert_eq!(\n &reduce_fe.to_bytes(),\n hex::decode(expected_reduced)\n .expect(\"reduction result test vector hex must decode\")\n .as_slice()\n );\n }\n }\n\n #[cfg(feature = \"digest\")]\n fn fe_from_test_vector(expected_hex: &str) -> FieldElement {\n let mut expected_hash =\n hex::decode(expected_hex).expect(\"hash-to-field test vector hex must decode\");\n expected_hash.reverse();\n let expected_hash: [u8; 32] = expected_hash\n .try_into()\n .expect(\"hash-to-field test vector must decode to 32 bytes\");\n FieldElement::from_bytes(&expected_hash)\n }\n\n /// Hash to field test vectors from\n /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.2\n /// These are of the form (input_msg, output_field_elem)\n #[cfg(feature = \"digest\")]\n const RFC_HASH_TO_FIELD_KAT: &[(&[u8], &str)] = &[\n (\n b\"\",\n \"7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0374a30641d\"\n ),\n (\n b\"abc\",\n \"09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd201a556e253b\"\n ),\n (\n b\"abcdef0123456789\",\n \"475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75d33edba941\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf6929814ac95d03d\"\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f0750d24fa0\"\n )\n ];\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn hash_to_field() {\n use sha2::Sha512;\n let dst = \"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_\";\n\n for (msg, expected_hash_hex) in RFC_HASH_TO_FIELD_KAT {\n let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()])[0];\n let expected_fe = fe_from_test_vector(expected_hash_hex);\n\n assert_eq!(fe, expected_fe);\n }\n }\n\n /// Hash to field test vectors from\n /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.1\n /// These are of the form (input_msg, output_field_elem, output_field_elem)\n #[cfg(feature = \"digest\")]\n const RFC_HASH_TO_FIELD_KAT_2: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee712d203f3a\",\n \"780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae793e6a60c75\"\n ),\n (\n b\"abc\",\n \"5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78f9bd57c227\",\n \"005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119aa0b925b76\"\n ),\n (\n b\"abcdef0123456789\",\n \"285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642636a78d5b3\",\n \"2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b369760b81b31\"\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c533bf321af\",\n \"02f20716a5801b843987097a8276b6d869295b2e11253751ca72c109d37485a9\"\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06df4a4ea33b\",\n \"1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e84068f5fd96\"\n )\n ];\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn hash_to_field_2() {\n use sha2::Sha512;\n let dst = \"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_\";\n\n for (msg, expected_hash_hex_1, expected_hash_hex_2) in\n crate::field::test::RFC_HASH_TO_FIELD_KAT_2\n {\n let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()]);\n\n let expected_fe_1 = fe_from_test_vector(expected_hash_hex_1);\n assert_eq!(fe[0], expected_fe_1);\n\n let expected_fe_2 = fe_from_test_vector(expected_hash_hex_2);\n assert_eq!(fe[1], expected_fe_2);\n }\n }\n}\n"},{"id":9,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64/field.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Field arithmetic modulo \\\\(p = 2\\^{255} - 19\\\\), using \\\\(64\\\\)-bit\n//! limbs with \\\\(128\\\\)-bit products.\n\nuse core::fmt::Debug;\nuse core::ops::Neg;\nuse core::ops::{Add, AddAssign};\nuse core::ops::{Mul, MulAssign};\nuse core::ops::{Sub, SubAssign};\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\n/// A `FieldElement51` represents an element of the field\n/// \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\).\n///\n/// In the 64-bit implementation, a `FieldElement` is represented in\n/// radix \\\\(2\\^{51}\\\\) as five `u64`s; the coefficients are allowed to\n/// grow up to \\\\(2\\^{54}\\\\) between reductions modulo \\\\(p\\\\).\n///\n/// # Note\n///\n/// The `curve25519::field` module provides a type alias\n/// `curve25519::field::FieldElement` to either `FieldElement51`\n/// or `FieldElement2625`.\n///\n/// The backend-specific type `FieldElement51` should not be used\n/// outside of the `curve25519::field` module.\n#[derive(Copy, Clone)]\npub struct FieldElement51(pub(crate) [u64; 5]);\n\nimpl Debug for FieldElement51 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"FieldElement51({:?})\", &self.0[..])\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for FieldElement51 {}\n\nimpl<'a> AddAssign<&'a FieldElement51> for FieldElement51 {\n fn add_assign(&mut self, _rhs: &'a FieldElement51) {\n for i in 0..5 {\n self.0[i] += _rhs.0[i];\n }\n }\n}\n\nimpl<'a> Add<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn add(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n let mut output = *self;\n output += _rhs;\n output\n }\n}\n\nimpl<'a> SubAssign<&'a FieldElement51> for FieldElement51 {\n fn sub_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) - _rhs;\n self.0 = result.0;\n }\n}\n\nimpl<'a> Sub<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn sub(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n // To avoid underflow, first add a multiple of p.\n // Choose 16*p = p << 4 to be larger than 54-bit _rhs.\n //\n // If we could statically track the bitlengths of the limbs\n // of every FieldElement51, we could choose a multiple of p\n // just bigger than _rhs and avoid having to do a reduction.\n //\n // Since we don't yet have type-level integers to do this, we\n // have to add an explicit reduction call here.\n FieldElement51::reduce([\n (self.0[0] + 36028797018963664u64) - _rhs.0[0],\n (self.0[1] + 36028797018963952u64) - _rhs.0[1],\n (self.0[2] + 36028797018963952u64) - _rhs.0[2],\n (self.0[3] + 36028797018963952u64) - _rhs.0[3],\n (self.0[4] + 36028797018963952u64) - _rhs.0[4],\n ])\n }\n}\n\nimpl<'a> MulAssign<&'a FieldElement51> for FieldElement51 {\n fn mul_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) * _rhs;\n self.0 = result.0;\n }\n}\n\nimpl<'a> Mul<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n\n #[rustfmt::skip] // keep alignment of c* calculations\n fn mul(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n /// Helper function to multiply two 64-bit integers with 128\n /// bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }\n\n // Alias self, _rhs for more readable formulas\n let a: &[u64; 5] = &self.0;\n let b: &[u64; 5] = &_rhs.0;\n\n // Precondition: assume input limbs a[i], b[i] are bounded as\n //\n // a[i], b[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // 64-bit precomputations to avoid 128-bit multiplications.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let b1_19 = b[1] * 19;\n let b2_19 = b[2] * 19;\n let b3_19 = b[3] * 19;\n let b4_19 = b[4] * 19;\n\n // Multiply to get 128-bit coefficients of output\n let c0: u128 = m(a[0], b[0]) + m(a[4], b1_19) + m(a[3], b2_19) + m(a[2], b3_19) + m(a[1], b4_19);\n let mut c1: u128 = m(a[1], b[0]) + m(a[0], b[1]) + m(a[4], b2_19) + m(a[3], b3_19) + m(a[2], b4_19);\n let mut c2: u128 = m(a[2], b[0]) + m(a[1], b[1]) + m(a[0], b[2]) + m(a[4], b3_19) + m(a[3], b4_19);\n let mut c3: u128 = m(a[3], b[0]) + m(a[2], b[1]) + m(a[1], b[2]) + m(a[0], b[3]) + m(a[4], b4_19);\n let mut c4: u128 = m(a[4], b[0]) + m(a[3], b[1]) + m(a[2], b[2]) + m(a[1], b[3]) + m(a[0] , b[4]);\n\n // How big are the c[i]? We have\n //\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));\n\n // Casting to u64 and back tells the compiler that the carry is\n // bounded by 2^64, so that the addition is a u128 + u64 rather\n // than u128 + u128.\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n let mut out = [0u64; 5];\n\n c1 += ((c0 >> 51) as u64) as u128;\n out[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n out[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n out[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n out[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n out[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need out[0] + carry * 19 < 2^64.\n //\n // c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)\n // < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n out[0] += carry * 19;\n\n // Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n out[1] += out[0] >> 51;\n out[0] &= LOW_51_BIT_MASK;\n\n // Now out[i] < 2^(51 + epsilon) for all i.\n FieldElement51(out)\n }\n}\n\nimpl Neg for &FieldElement51 {\n type Output = FieldElement51;\n fn neg(self) -> FieldElement51 {\n let mut output = *self;\n output.negate();\n output\n }\n}\n\nimpl ConditionallySelectable for FieldElement51 {\n fn conditional_select(\n a: &FieldElement51,\n b: &FieldElement51,\n choice: Choice,\n ) -> FieldElement51 {\n FieldElement51([\n u64::conditional_select(&a.0[0], &b.0[0], choice),\n u64::conditional_select(&a.0[1], &b.0[1], choice),\n u64::conditional_select(&a.0[2], &b.0[2], choice),\n u64::conditional_select(&a.0[3], &b.0[3], choice),\n u64::conditional_select(&a.0[4], &b.0[4], choice),\n ])\n }\n\n fn conditional_swap(a: &mut FieldElement51, b: &mut FieldElement51, choice: Choice) {\n u64::conditional_swap(&mut a.0[0], &mut b.0[0], choice);\n u64::conditional_swap(&mut a.0[1], &mut b.0[1], choice);\n u64::conditional_swap(&mut a.0[2], &mut b.0[2], choice);\n u64::conditional_swap(&mut a.0[3], &mut b.0[3], choice);\n u64::conditional_swap(&mut a.0[4], &mut b.0[4], choice);\n }\n\n fn conditional_assign(&mut self, other: &FieldElement51, choice: Choice) {\n self.0[0].conditional_assign(&other.0[0], choice);\n self.0[1].conditional_assign(&other.0[1], choice);\n self.0[2].conditional_assign(&other.0[2], choice);\n self.0[3].conditional_assign(&other.0[3], choice);\n self.0[4].conditional_assign(&other.0[4], choice);\n }\n}\n\nimpl FieldElement51 {\n pub(crate) const fn from_limbs(limbs: [u64; 5]) -> FieldElement51 {\n FieldElement51(limbs)\n }\n\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: FieldElement51 = FieldElement51::from_limbs([0, 0, 0, 0, 0]);\n /// The scalar \\\\( 1 \\\\).\n pub const ONE: FieldElement51 = FieldElement51::from_limbs([1, 0, 0, 0, 0]);\n /// The scalar \\\\( -1 \\\\).\n pub const MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n ]);\n\n /// Invert the sign of this field element\n pub fn negate(&mut self) {\n // See commentary in the Sub impl\n let neg = FieldElement51::reduce([\n 36028797018963664u64 - self.0[0],\n 36028797018963952u64 - self.0[1],\n 36028797018963952u64 - self.0[2],\n 36028797018963952u64 - self.0[3],\n 36028797018963952u64 - self.0[4],\n ]);\n self.0 = neg.0;\n }\n\n /// Given 64-bit input limbs, reduce to enforce the bound 2^(51 + epsilon).\n #[inline(always)]\n fn reduce(mut limbs: [u64; 5]) -> FieldElement51 {\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n\n // Since the input limbs are bounded by 2^64, the biggest\n // carry-out is bounded by 2^13.\n //\n // The biggest carry-in is c4 * 19, resulting in\n //\n // 2^51 + 19*2^13 < 2^51.0000000001\n //\n // Because we don't need to canonicalize, only to reduce the\n // limb sizes, it's OK to do a \"weak reduction\", where we\n // compute the carry-outs in parallel.\n\n let c0 = limbs[0] >> 51;\n let c1 = limbs[1] >> 51;\n let c2 = limbs[2] >> 51;\n let c3 = limbs[3] >> 51;\n let c4 = limbs[4] >> 51;\n\n limbs[0] &= LOW_51_BIT_MASK;\n limbs[1] &= LOW_51_BIT_MASK;\n limbs[2] &= LOW_51_BIT_MASK;\n limbs[3] &= LOW_51_BIT_MASK;\n limbs[4] &= LOW_51_BIT_MASK;\n\n limbs[0] += c4 * 19;\n limbs[1] += c0;\n limbs[2] += c1;\n limbs[3] += c2;\n limbs[4] += c3;\n\n FieldElement51(limbs)\n }\n\n /// Load a `FieldElement51` from the low 255 bits of a 256-bit\n /// input.\n ///\n /// # Warning\n ///\n /// This function does not check that the input used the canonical\n /// representative. It masks the high bit, but it will happily\n /// decode 2^255 - 18 to 1. Applications that require a canonical\n /// encoding of every field element should decode, re-encode to\n /// the canonical encoding, and check that the input was\n /// canonical.\n ///\n #[rustfmt::skip] // keep alignment of bit shifts\n pub const fn from_bytes(bytes: &[u8; 32]) -> FieldElement51 {\n const fn load8_at(input: &[u8], i: usize) -> u64 {\n (input[i] as u64)\n | ((input[i + 1] as u64) << 8)\n | ((input[i + 2] as u64) << 16)\n | ((input[i + 3] as u64) << 24)\n | ((input[i + 4] as u64) << 32)\n | ((input[i + 5] as u64) << 40)\n | ((input[i + 6] as u64) << 48)\n | ((input[i + 7] as u64) << 56)\n }\n\n let low_51_bit_mask = (1u64 << 51) - 1;\n FieldElement51(\n // load bits [ 0, 64), no shift\n [ load8_at(bytes, 0) & low_51_bit_mask\n // load bits [ 48,112), shift to [ 51,112)\n , (load8_at(bytes, 6) >> 3) & low_51_bit_mask\n // load bits [ 96,160), shift to [102,160)\n , (load8_at(bytes, 12) >> 6) & low_51_bit_mask\n // load bits [152,216), shift to [153,216)\n , (load8_at(bytes, 19) >> 1) & low_51_bit_mask\n // load bits [192,256), shift to [204,112)\n , (load8_at(bytes, 24) >> 12) & low_51_bit_mask\n ])\n }\n\n /// Serialize this `FieldElement51` to a 32-byte array. The\n /// encoding is canonical.\n #[rustfmt::skip] // keep alignment of s[*] calculations\n pub fn to_bytes(self) -> [u8; 32] {\n // Let h = limbs[0] + limbs[1]*2^51 + ... + limbs[4]*2^204.\n //\n // Write h = pq + r with 0 <= r < p.\n //\n // We want to compute r = h mod p.\n //\n // If h < 2*p = 2^256 - 38,\n // then q = 0 or 1,\n //\n // with q = 0 when h < p\n // and q = 1 when h >= p.\n //\n // Notice that h >= p <==> h + 19 >= p + 19 <==> h + 19 >= 2^255.\n // Therefore q can be computed as the carry bit of h + 19.\n\n // First, reduce the limbs to ensure h < 2*p.\n let mut limbs = FieldElement51::reduce(self.0).0;\n\n let mut q = (limbs[0] + 19) >> 51;\n q = (limbs[1] + q) >> 51;\n q = (limbs[2] + q) >> 51;\n q = (limbs[3] + q) >> 51;\n q = (limbs[4] + q) >> 51;\n\n // Now we can compute r as r = h - pq = r - (2^255-19)q = r + 19q - 2^255q\n\n limbs[0] += 19 * q;\n\n // Now carry the result to compute r + 19q ...\n let low_51_bit_mask = (1u64 << 51) - 1;\n limbs[1] += limbs[0] >> 51;\n limbs[0] &= low_51_bit_mask;\n limbs[2] += limbs[1] >> 51;\n limbs[1] &= low_51_bit_mask;\n limbs[3] += limbs[2] >> 51;\n limbs[2] &= low_51_bit_mask;\n limbs[4] += limbs[3] >> 51;\n limbs[3] &= low_51_bit_mask;\n // ... but instead of carrying (limbs[4] >> 51) = 2^255q\n // into another limb, discard it, subtracting the value\n limbs[4] &= low_51_bit_mask;\n\n // Now arrange the bits of the limbs.\n let mut s = [0u8;32];\n s[ 0] = limbs[0] as u8;\n s[ 1] = (limbs[0] >> 8) as u8;\n s[ 2] = (limbs[0] >> 16) as u8;\n s[ 3] = (limbs[0] >> 24) as u8;\n s[ 4] = (limbs[0] >> 32) as u8;\n s[ 5] = (limbs[0] >> 40) as u8;\n s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;\n s[ 7] = (limbs[1] >> 5) as u8;\n s[ 8] = (limbs[1] >> 13) as u8;\n s[ 9] = (limbs[1] >> 21) as u8;\n s[10] = (limbs[1] >> 29) as u8;\n s[11] = (limbs[1] >> 37) as u8;\n s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;\n s[13] = (limbs[2] >> 2) as u8;\n s[14] = (limbs[2] >> 10) as u8;\n s[15] = (limbs[2] >> 18) as u8;\n s[16] = (limbs[2] >> 26) as u8;\n s[17] = (limbs[2] >> 34) as u8;\n s[18] = (limbs[2] >> 42) as u8;\n s[19] = ((limbs[2] >> 50) | (limbs[3] << 1)) as u8;\n s[20] = (limbs[3] >> 7) as u8;\n s[21] = (limbs[3] >> 15) as u8;\n s[22] = (limbs[3] >> 23) as u8;\n s[23] = (limbs[3] >> 31) as u8;\n s[24] = (limbs[3] >> 39) as u8;\n s[25] = ((limbs[3] >> 47) | (limbs[4] << 4)) as u8;\n s[26] = (limbs[4] >> 4) as u8;\n s[27] = (limbs[4] >> 12) as u8;\n s[28] = (limbs[4] >> 20) as u8;\n s[29] = (limbs[4] >> 28) as u8;\n s[30] = (limbs[4] >> 36) as u8;\n s[31] = (limbs[4] >> 44) as u8;\n\n // High bit should be zero.\n debug_assert!((s[31] & 0b1000_0000u8) == 0u8);\n\n s\n }\n\n /// Given `k > 0`, return `self^(2^k)`.\n #[rustfmt::skip] // keep alignment of c* calculations\n pub fn pow2k(&self, mut k: u32) -> FieldElement51 {\n\n debug_assert!( k > 0 );\n\n /// Multiply two 64-bit integers with 128 bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 {\n (x as u128) * (y as u128)\n }\n\n let mut a: [u64; 5] = self.0;\n\n loop {\n // Precondition: assume input limbs a[i] are bounded as\n //\n // a[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // Precomputation: 64-bit multiply by 19.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let a3_19 = 19 * a[3];\n let a4_19 = 19 * a[4];\n\n // Multiply to get 128-bit coefficients of output.\n //\n // The 128-bit multiplications by 2 turn into 1 slr + 1 slrd each,\n // which doesn't seem any better or worse than doing them as precomputations\n // on the 64-bit inputs.\n let c0: u128 = m(a[0], a[0]) + 2*( m(a[1], a4_19) + m(a[2], a3_19) );\n let mut c1: u128 = m(a[3], a3_19) + 2*( m(a[0], a[1]) + m(a[2], a4_19) );\n let mut c2: u128 = m(a[1], a[1]) + 2*( m(a[0], a[2]) + m(a[4], a3_19) );\n let mut c3: u128 = m(a[4], a4_19) + 2*( m(a[0], a[3]) + m(a[1], a[2]) );\n let mut c4: u128 = m(a[2], a[2]) + 2*( m(a[0], a[4]) + m(a[1], a[3]) );\n\n // Same bound as in multiply:\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54));\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n\n // Casting to u64 and back tells the compiler that the carry is bounded by 2^64, so\n // that the addition is a u128 + u64 rather than u128 + u128.\n c1 += ((c0 >> 51) as u64) as u128;\n a[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n a[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n a[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n a[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n a[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need a[0] + carry * 19 < 2^64.\n //\n // c4 < a2^2 + 2*a0*a4 + 2*a1*a3 + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // a[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n a[0] += carry * 19;\n\n // Now a[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n a[1] += a[0] >> 51;\n a[0] &= LOW_51_BIT_MASK;\n\n // Now all a[i] < 2^(51 + epsilon) and a = self^(2^k).\n\n k -= 1;\n if k == 0 {\n break;\n }\n }\n\n FieldElement51(a)\n }\n\n /// Returns the square of this field element.\n pub fn square(&self) -> FieldElement51 {\n self.pow2k(1)\n }\n\n /// Returns 2 times the square of this field element.\n pub fn square2(&self) -> FieldElement51 {\n let mut square = self.pow2k(1);\n for i in 0..5 {\n square.0[i] *= 2;\n }\n\n square\n }\n}\n"},{"id":10,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/curve_models.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Internal curve representations which are not part of the public API.\n//!\n//! # Curve representations\n//!\n//! Internally, we use several different models for the curve. Here\n//! is a sketch of the relationship between the models, following [a\n//! post][smith-moderncrypto]\n//! by Ben Smith on the `moderncrypto` mailing list. This is also briefly\n//! discussed in section 2.5 of [_Montgomery curves and their\n//! arithmetic_][costello-smith-2017] by Costello and Smith.\n//!\n//! Begin with the affine equation for the curve,\n//! $$\n//! -x\\^2 + y\\^2 = 1 + dx\\^2y\\^2.\n//! $$\n//! Next, pass to the projective closure \\\\(\\mathbb P\\^1 \\times \\mathbb\n//! P\\^1 \\\\) by setting \\\\(x=X/Z\\\\), \\\\(y=Y/T.\\\\) Clearing denominators\n//! gives the model\n//! $$\n//! -X\\^2T\\^2 + Y\\^2Z\\^2 = Z\\^2T\\^2 + dX\\^2Y\\^2.\n//! $$\n//! In `curve25519-dalek`, this is represented as the `CompletedPoint`\n//! struct.\n//! To map from \\\\(\\mathbb P\\^1 \\times \\mathbb P\\^1 \\\\), a product of\n//! two lines, to \\\\(\\mathbb P\\^3\\\\), we use the [Segre\n//! embedding](https://en.wikipedia.org/wiki/Segre_embedding)\n//! $$\n//! \\sigma : ((X:Z),(Y:T)) \\mapsto (XY:XT:ZY:ZT).\n//! $$\n//! Using coordinates \\\\( (W_0:W_1:W_2:W_3) \\\\) for \\\\(\\mathbb P\\^3\\\\),\n//! the image \\\\(\\sigma (\\mathbb P\\^1 \\times \\mathbb P\\^1) \\\\) is the\n//! surface defined by \\\\( W_0 W_3 = W_1 W_2 \\\\), and under \\\\(\n//! \\sigma\\\\), the equation above becomes\n//! $$\n//! -W\\_1\\^2 + W\\_2\\^2 = W\\_3\\^2 + dW\\_0\\^2,\n//! $$\n//! so that the curve is given by the pair of equations\n//! $$\n//! \\begin{aligned}\n//! -W\\_1\\^2 + W\\_2\\^2 &= W\\_3\\^2 + dW\\_0\\^2, \\\\\\\\ W_0 W_3 &= W_1 W_2.\n//! \\end{aligned}\n//! $$\n//! Up to variable naming, this is exactly the \"extended\" curve model\n//! introduced in [_Twisted Edwards Curves\n//! Revisited_][hisil-wong-carter-dawson-2008] by Hisil, Wong, Carter,\n//! and Dawson. In `curve25519-dalek`, it is represented as the\n//! `EdwardsPoint` struct. We can map from \\\\(\\mathbb P\\^3 \\\\) to\n//! \\\\(\\mathbb P\\^2 \\\\) by sending \\\\( (W\\_0:W\\_1:W\\_2:W\\_3) \\\\) to \\\\(\n//! (W\\_1:W\\_2:W\\_3) \\\\). Notice that\n//! $$\n//! \\frac {W\\_1} {W\\_3} = \\frac {XT} {ZT} = \\frac X Z = x,\n//! $$\n//! and\n//! $$\n//! \\frac {W\\_2} {W\\_3} = \\frac {YZ} {ZT} = \\frac Y T = y,\n//! $$\n//! so this is the same as if we had started with the affine model\n//! and passed to \\\\( \\mathbb P\\^2 \\\\) by setting \\\\( x = W\\_1 / W\\_3\n//! \\\\), \\\\(y = W\\_2 / W\\_3 \\\\).\n//! Up to variable naming, this is the projective representation\n//! introduced in [_Twisted Edwards\n//! Curves_][bernstein-birkner-joye-lange-peters-2008] by Bernstein,\n//! Birkner, Joye, Lange, and Peters. In `curve25519-dalek`, it is\n//! represented by the `ProjectivePoint` struct.\n//!\n//! # Passing between curve models\n//!\n//! Although the \\\\( \\mathbb P\\^3 \\\\) model provides faster addition\n//! formulas, the \\\\( \\mathbb P\\^2 \\\\) model provides faster doubling\n//! formulas. Hisil, Wong, Carter, and Dawson therefore suggest mixing\n//! coordinate systems for scalar multiplication, attributing the idea\n//! to [a 1998 paper][cohen-miyaji-ono-1998] of Cohen, Miyagi, and Ono.\n//!\n//! Their suggestion is to vary the formulas used by context, using a\n//! \\\\( \\mathbb P\\^2 \\rightarrow \\mathbb P\\^2 \\\\) doubling formula when\n//! a doubling is followed\n//! by another doubling, a \\\\( \\mathbb P\\^2 \\rightarrow \\mathbb P\\^3 \\\\)\n//! doubling formula when a doubling is followed by an addition, and\n//! computing point additions using a \\\\( \\mathbb P\\^3 \\times \\mathbb P\\^3\n//! \\rightarrow \\mathbb P\\^2 \\\\) formula.\n//!\n//! The `ref10` reference implementation of [Ed25519][ed25519], by\n//! Bernstein, Duif, Lange, Schwabe, and Yang, tweaks\n//! this strategy, factoring the addition formulas through the\n//! completion \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1 \\\\), so that the\n//! output of an addition or doubling always lies in \\\\( \\mathbb P\\^1 \\times\n//! \\mathbb P\\^1\\\\), and the choice of which formula to use is replaced\n//! by a choice of whether to convert the result to \\\\( \\mathbb P\\^2 \\\\)\n//! or \\\\(\\mathbb P\\^3 \\\\). However, this tweak is not described in\n//! their paper, only in their software.\n//!\n//! Our naming for the `CompletedPoint` (\\\\(\\mathbb P\\^1 \\times \\mathbb\n//! P\\^1 \\\\)), `ProjectivePoint` (\\\\(\\mathbb P\\^2 \\\\)), and\n//! `EdwardsPoint` (\\\\(\\mathbb P\\^3 \\\\)) structs follows the naming in\n//! Adam Langley's [Golang ed25519][agl-ed25519] implementation, which\n//! `curve25519-dalek` was originally derived from.\n//!\n//! Finally, to accelerate readditions, we use two cached point formats\n//! in \"Niels coordinates\", named for Niels Duif,\n//! one for the affine model and one for the \\\\( \\mathbb P\\^3 \\\\) model:\n//!\n//! * `AffineNielsPoint`: \\\\( (y+x, y-x, 2dxy) \\\\)\n//! * `ProjectiveNielsPoint`: \\\\( (Y+X, Y-X, Z, 2dXY) \\\\)\n//!\n//! [smith-moderncrypto]: https://moderncrypto.org/mail-archive/curves/2016/000807.html\n//! [costello-smith-2017]: https://eprint.iacr.org/2017/212\n//! [hisil-wong-carter-dawson-2008]: https://www.iacr.org/archive/asiacrypt2008/53500329/53500329.pdf\n//! [bernstein-birkner-joye-lange-peters-2008]: https://eprint.iacr.org/2008/013\n//! [cohen-miyaji-ono-1998]: https://link.springer.com/content/pdf/10.1007%2F3-540-49649-1_6.pdf\n//! [ed25519]: https://eprint.iacr.org/2011/368\n//! [agl-ed25519]: https://github.com/agl/ed25519\n\n#![allow(non_snake_case)]\n\nuse core::fmt::Debug;\nuse core::ops::{Add, Neg, Sub};\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::constants;\n\nuse crate::edwards::EdwardsPoint;\nuse crate::field::FieldElement;\nuse crate::traits::ValidityCheck;\n\n// ------------------------------------------------------------------------\n// Internal point representations\n// ------------------------------------------------------------------------\n\n/// A `ProjectivePoint` is a point \\\\((X:Y:Z)\\\\) on the \\\\(\\mathbb\n/// P\\^2\\\\) model of the curve.\n/// A point \\\\((x,y)\\\\) in the affine model corresponds to\n/// \\\\((x:y:1)\\\\).\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[allow(missing_docs)]\n#[derive(Copy, Clone)]\npub struct ProjectivePoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n}\n\n/// A `CompletedPoint` is a point \\\\(((X:Z), (Y:T))\\\\) on the \\\\(\\mathbb\n/// P\\^1 \\times \\mathbb P\\^1 \\\\) model of the curve.\n/// A point (x,y) in the affine model corresponds to \\\\( ((x:1),(y:1))\n/// \\\\).\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct CompletedPoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n pub T: FieldElement,\n}\n\n/// A pre-computed point in the affine model for the curve, represented as\n/// \\\\((y+x, y-x, 2dxy)\\\\) in \"Niels coordinates\".\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n// Safe to derive Eq because affine coordinates.\n#[derive(Copy, Clone, Eq, PartialEq)]\n#[allow(missing_docs)]\npub struct AffineNielsPoint {\n pub y_plus_x: FieldElement,\n pub y_minus_x: FieldElement,\n pub xy2d: FieldElement,\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for AffineNielsPoint {\n fn zeroize(&mut self) {\n self.y_plus_x.zeroize();\n self.y_minus_x.zeroize();\n self.xy2d.zeroize();\n }\n}\n\n/// A pre-computed point on the \\\\( \\mathbb P\\^3 \\\\) model for the\n/// curve, represented as \\\\((Y+X, Y-X, Z, 2dXY)\\\\) in \"Niels coordinates\".\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct ProjectiveNielsPoint {\n pub Y_plus_X: FieldElement,\n pub Y_minus_X: FieldElement,\n pub Z: FieldElement,\n pub T2d: FieldElement,\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for ProjectiveNielsPoint {\n fn zeroize(&mut self) {\n self.Y_plus_X.zeroize();\n self.Y_minus_X.zeroize();\n self.Z.zeroize();\n self.T2d.zeroize();\n }\n}\n\n// ------------------------------------------------------------------------\n// Constructors\n// ------------------------------------------------------------------------\n\nuse crate::traits::Identity;\n\nimpl Identity for ProjectivePoint {\n fn identity() -> ProjectivePoint {\n ProjectivePoint {\n X: FieldElement::ZERO,\n Y: FieldElement::ONE,\n Z: FieldElement::ONE,\n }\n }\n}\n\nimpl Identity for ProjectiveNielsPoint {\n fn identity() -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::ONE,\n Y_minus_X: FieldElement::ONE,\n Z: FieldElement::ONE,\n T2d: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for ProjectiveNielsPoint {\n fn default() -> ProjectiveNielsPoint {\n ProjectiveNielsPoint::identity()\n }\n}\n\nimpl Identity for AffineNielsPoint {\n fn identity() -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: FieldElement::ONE,\n y_minus_x: FieldElement::ONE,\n xy2d: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for AffineNielsPoint {\n fn default() -> AffineNielsPoint {\n AffineNielsPoint::identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Validity checks (for debugging, not CT)\n// ------------------------------------------------------------------------\n\nimpl ValidityCheck for ProjectivePoint {\n fn is_valid(&self) -> bool {\n // Curve equation is -x^2 + y^2 = 1 + d*x^2*y^2,\n // homogenized as (-X^2 + Y^2)*Z^2 = Z^4 + d*X^2*Y^2\n let XX = self.X.square();\n let YY = self.Y.square();\n let ZZ = self.Z.square();\n let ZZZZ = ZZ.square();\n let lhs = &(&YY - &XX) * &ZZ;\n let rhs = &ZZZZ + &(&constants::EDWARDS_D * &(&XX * &YY));\n\n lhs == rhs\n }\n}\n\n// ------------------------------------------------------------------------\n// Constant-time assignment\n// ------------------------------------------------------------------------\n\nimpl ConditionallySelectable for ProjectiveNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::conditional_select(&a.Y_plus_X, &b.Y_plus_X, choice),\n Y_minus_X: FieldElement::conditional_select(&a.Y_minus_X, &b.Y_minus_X, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T2d: FieldElement::conditional_select(&a.T2d, &b.T2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.Y_plus_X.conditional_assign(&other.Y_plus_X, choice);\n self.Y_minus_X.conditional_assign(&other.Y_minus_X, choice);\n self.Z.conditional_assign(&other.Z, choice);\n self.T2d.conditional_assign(&other.T2d, choice);\n }\n}\n\nimpl ConditionallySelectable for AffineNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n AffineNielsPoint {\n y_plus_x: FieldElement::conditional_select(&a.y_plus_x, &b.y_plus_x, choice),\n y_minus_x: FieldElement::conditional_select(&a.y_minus_x, &b.y_minus_x, choice),\n xy2d: FieldElement::conditional_select(&a.xy2d, &b.xy2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.y_plus_x.conditional_assign(&other.y_plus_x, choice);\n self.y_minus_x.conditional_assign(&other.y_minus_x, choice);\n self.xy2d.conditional_assign(&other.xy2d, choice);\n }\n}\n\n// ------------------------------------------------------------------------\n// Point conversions\n// ------------------------------------------------------------------------\n\nimpl ProjectivePoint {\n /// Convert this point from the \\\\( \\mathbb P\\^2 \\\\) model to the\n /// \\\\( \\mathbb P\\^3 \\\\) model.\n ///\n /// This costs \\\\(3 \\mathrm M + 1 \\mathrm S\\\\).\n pub fn as_extended(&self) -> EdwardsPoint {\n EdwardsPoint {\n X: &self.X * &self.Z,\n Y: &self.Y * &self.Z,\n Z: self.Z.square(),\n T: &self.X * &self.Y,\n }\n }\n}\n\nimpl CompletedPoint {\n /// Convert this point from the \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1\n /// \\\\) model to the \\\\( \\mathbb P\\^2 \\\\) model.\n ///\n /// This costs \\\\(3 \\mathrm M \\\\).\n pub fn as_projective(&self) -> ProjectivePoint {\n ProjectivePoint {\n X: &self.X * &self.T,\n Y: &self.Y * &self.Z,\n Z: &self.Z * &self.T,\n }\n }\n\n /// Convert this point from the \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1\n /// \\\\) model to the \\\\( \\mathbb P\\^3 \\\\) model.\n ///\n /// This costs \\\\(4 \\mathrm M \\\\).\n pub fn as_extended(&self) -> EdwardsPoint {\n EdwardsPoint {\n X: &self.X * &self.T,\n Y: &self.Y * &self.Z,\n Z: &self.Z * &self.T,\n T: &self.X * &self.Y,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Doubling\n// ------------------------------------------------------------------------\n\nimpl ProjectivePoint {\n /// Double this point: return self + self\n pub fn double(&self) -> CompletedPoint {\n // Double()\n let XX = self.X.square();\n let YY = self.Y.square();\n let ZZ2 = self.Z.square2();\n let X_plus_Y = &self.X + &self.Y;\n let X_plus_Y_sq = X_plus_Y.square();\n let YY_plus_XX = &YY + &XX;\n let YY_minus_XX = &YY - &XX;\n\n CompletedPoint {\n X: &X_plus_Y_sq - &YY_plus_XX,\n Y: YY_plus_XX,\n Z: YY_minus_XX,\n T: &ZZ2 - &YY_minus_XX,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Addition and Subtraction\n// ------------------------------------------------------------------------\n\n// XXX(hdevalence) These were doc(hidden) so they don't appear in the\n// public API docs.\n// However, that prevents them being used with --document-private-items,\n// so comment out the doc(hidden) for now until this is resolved\n//\n// upstream rust issue: https://github.com/rust-lang/rust/issues/46380\n//#[doc(hidden)]\nimpl<'a> Add<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.Y_plus_X;\n let MM = &Y_minus_X * &other.Y_minus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &ZZ2 + &TT2d,\n T: &ZZ2 - &TT2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Sub<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.Y_minus_X;\n let MP = &Y_minus_X * &other.Y_plus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &ZZ2 - &TT2d,\n T: &ZZ2 + &TT2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Add<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.y_plus_x;\n let MM = &Y_minus_X * &other.y_minus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &Z2 + &Txy2d,\n T: &Z2 - &Txy2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Sub<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.y_minus_x;\n let MP = &Y_minus_X * &other.y_plus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &Z2 - &Txy2d,\n T: &Z2 + &Txy2d,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Negation\n// ------------------------------------------------------------------------\n\nimpl Neg for &ProjectiveNielsPoint {\n type Output = ProjectiveNielsPoint;\n\n fn neg(self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: self.Y_minus_X,\n Y_minus_X: self.Y_plus_X,\n Z: self.Z,\n T2d: -(&self.T2d),\n }\n }\n}\n\nimpl Neg for &AffineNielsPoint {\n type Output = AffineNielsPoint;\n\n fn neg(self) -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: self.y_minus_x,\n y_minus_x: self.y_plus_x,\n xy2d: -(&self.xy2d),\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Debug traits\n// ------------------------------------------------------------------------\n\nimpl Debug for ProjectivePoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"ProjectivePoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z\n )\n }\n}\n\nimpl Debug for CompletedPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"CompletedPoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?},\\n\\tT: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z, &self.T\n )\n }\n}\n\nimpl Debug for AffineNielsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"AffineNielsPoint{{\\n\\ty_plus_x: {:?},\\n\\ty_minus_x: {:?},\\n\\txy2d: {:?}\\n}}\",\n &self.y_plus_x, &self.y_minus_x, &self.xy2d\n )\n }\n}\n\nimpl Debug for ProjectiveNielsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"ProjectiveNielsPoint{{\\n\\tY_plus_X: {:?},\\n\\tY_minus_X: {:?},\\n\\tZ: {:?},\\n\\tT2d: {:?}\\n}}\",\n &self.Y_plus_X, &self.Y_minus_X, &self.Z, &self.T2d\n )\n }\n}\n"},{"id":11,"name":{"Local":"curve25519/solana-ed25519/src/edwards.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2020 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Group operations for Curve25519, in Edwards form.\n//!\n//! ## Encoding and Decoding\n//!\n//! Encoding is done by converting to and from a `CompressedEdwardsY`\n//! struct, which is a typed wrapper around `[u8; 32]`.\n//!\n//! ## Equality Testing\n//!\n//! The `EdwardsPoint` struct implements the [`subtle::ConstantTimeEq`]\n//! trait for constant-time equality checking, and also uses this to\n//! ensure `Eq` equality checking runs in constant time.\n//!\n//! ## Cofactor-related functions\n//!\n//! The order of the group of points on the curve \\\\(\\mathcal E\\\\)\n//! is \\\\(|\\mathcal E| = 8\\ell \\\\), so its structure is \\\\( \\mathcal\n//! E = \\mathcal E\\[8\\] \\times \\mathcal E[\\ell]\\\\). The torsion\n//! subgroup \\\\( \\mathcal E\\[8\\] \\\\) consists of eight points of small\n//! order. Technically, all of \\\\(\\mathcal E\\\\) is torsion, but we\n//! use the word only to refer to the small \\\\(\\mathcal E\\[8\\]\\\\) part, not\n//! the large prime-order \\\\(\\mathcal E[\\ell]\\\\) part.\n//!\n//! To test if a point is in \\\\( \\mathcal E\\[8\\] \\\\), use\n//! [`EdwardsPoint::is_small_order`].\n//!\n//! To test if a point is in \\\\( \\mathcal E[\\ell] \\\\), use\n//! [`EdwardsPoint::is_torsion_free`].\n//!\n//! To multiply by the cofactor, use [`EdwardsPoint::mul_by_cofactor`].\n//!\n//! To avoid dealing with cofactors entirely, consider using Ristretto.\n//!\n//! ## Scalars\n//!\n//! Scalars are represented by the [`Scalar`] struct. To construct a scalar, see\n//! [`Scalar::from_canonical_bytes`] or [`Scalar::from_bytes_mod_order_wide`].\n//!\n//! ## Scalar Multiplication\n//!\n//! Scalar multiplication on Edwards points is provided by:\n//!\n//! * the `*` operator between a `Scalar` and a `EdwardsPoint`, which\n//! performs constant-time variable-base scalar multiplication;\n//!\n//! * the `*` operator between a `Scalar` and a\n//! `EdwardsBasepointTable`, which performs constant-time fixed-base\n//! scalar multiplication;\n//!\n//! * an implementation of the\n//! [`MultiscalarMul`](../traits/trait.MultiscalarMul.html) trait for\n//! constant-time variable-base multiscalar multiplication;\n//!\n//! * an implementation of the\n//! [`VartimeMultiscalarMul`](../traits/trait.VartimeMultiscalarMul.html)\n//! trait for variable-time variable-base multiscalar multiplication;\n//!\n//! ## Implementation\n//!\n//! The Edwards arithmetic is implemented using the “extended twisted\n//! coordinates” of Hisil, Wong, Carter, and Dawson, and the\n//! corresponding complete formulas. For more details,\n//! see the [`curve_models` submodule][curve_models]\n//! of the internal documentation.\n//!\n//! ## Validity Checking\n//!\n//! There is no function for checking whether a point is valid.\n//! Instead, the `EdwardsPoint` struct is guaranteed to hold a valid\n//! point on the curve.\n//!\n//! We use the Rust type system to make invalid points\n//! unrepresentable: `EdwardsPoint` objects can only be created via\n//! successful decompression of a compressed point, or else by\n//! operations on other (valid) `EdwardsPoint`s.\n//!\n//! [curve_models]: https://docs.rs/curve25519-dalek/latest/curve25519-dalek/backend/serial/curve_models/index.html\n\n// We allow non snake_case names because coordinates in projective space are\n// traditionally denoted by the capitalisation of their respective\n// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my\n// affine and projective cakes and eat both of them too.\n#![allow(non_snake_case)]\n\nmod affine;\n\nuse cfg_if::cfg_if;\nuse core::array::TryFromSliceError;\nuse core::borrow::Borrow;\nuse core::fmt::Debug;\nuse core::iter::Sum;\nuse core::ops::{Add, Neg, Sub};\nuse core::ops::{AddAssign, SubAssign};\nuse core::ops::{Mul, MulAssign};\n\n#[cfg(feature = \"digest\")]\nuse digest::{\n FixedOutput, HashMarker, array::typenum::U64, block_api::BlockSizeUser, consts::True,\n typenum::IsGreater,\n};\n\n#[cfg(feature = \"group\")]\nuse {\n group::{GroupEncoding, cofactor::CofactorGroup, prime::PrimeGroup},\n rand_core::RngCore as GroupRngCore,\n subtle::CtOption,\n};\n\n#[cfg(feature = \"rand_core\")]\nuse rand_core::{CryptoRng, RngCore};\n\nuse subtle::Choice;\nuse subtle::ConditionallyNegatable;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::constants;\n\nuse crate::field::FieldElement;\nuse crate::scalar::{Scalar, clamp_integer};\n\nuse crate::montgomery::MontgomeryPoint;\n\nuse crate::backend::serial::curve_models::AffineNielsPoint;\nuse crate::backend::serial::curve_models::CompletedPoint;\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::backend::serial::curve_models::ProjectivePoint;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::window::{\n LookupTableRadix16, LookupTableRadix32, LookupTableRadix64, LookupTableRadix128,\n LookupTableRadix256,\n};\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::traits::BasepointTable;\n\nuse crate::traits::ValidityCheck;\nuse crate::traits::{Identity, IsIdentity};\n\nuse affine::AffinePoint;\n\n#[cfg(feature = \"alloc\")]\nuse crate::traits::MultiscalarMul;\n#[cfg(feature = \"alloc\")]\nuse crate::traits::{VartimeMultiscalarMul, VartimePrecomputedMultiscalarMul};\n#[cfg(feature = \"alloc\")]\nuse alloc::vec::Vec;\n\n// ------------------------------------------------------------------------\n// Compressed points\n// ------------------------------------------------------------------------\n\n/// In \"Edwards y\" / \"Ed25519\" format, the curve point \\\\((x,y)\\\\) is\n/// determined by the \\\\(y\\\\)-coordinate and the sign of \\\\(x\\\\).\n///\n/// The first 255 bits of a `CompressedEdwardsY` represent the\n/// \\\\(y\\\\)-coordinate. The high bit of the 32nd byte gives the sign of \\\\(x\\\\).\n#[allow(clippy::derived_hash_with_manual_eq)]\n#[derive(Copy, Clone, Hash)]\npub struct CompressedEdwardsY(pub [u8; 32]);\n\nimpl ConstantTimeEq for CompressedEdwardsY {\n fn ct_eq(&self, other: &CompressedEdwardsY) -> Choice {\n self.as_bytes().ct_eq(other.as_bytes())\n }\n}\n\nimpl Eq for CompressedEdwardsY {}\nimpl PartialEq for CompressedEdwardsY {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Debug for CompressedEdwardsY {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"CompressedEdwardsY: {:?}\", self.as_bytes())\n }\n}\n\nimpl CompressedEdwardsY {\n /// View this `CompressedEdwardsY` as an array of bytes.\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.0\n }\n\n /// Copy this `CompressedEdwardsY` to an array of bytes.\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.0\n }\n\n /// Attempt to decompress to an `EdwardsPoint`.\n ///\n /// Returns `None` if the input is not the \\\\(y\\\\)-coordinate of a\n /// curve point.\n pub fn decompress(&self) -> Option {\n let (is_valid_y_coord, X, Y, Z) = decompress::step_1(self);\n\n if is_valid_y_coord.into() {\n Some(decompress::step_2(self, X, Y, Z))\n } else {\n None\n }\n }\n}\n\nmod decompress {\n use super::*;\n\n #[rustfmt::skip] // keep alignment of explanatory comments\n pub(super) fn step_1(\n repr: &CompressedEdwardsY,\n ) -> (Choice, FieldElement, FieldElement, FieldElement) {\n let Y = FieldElement::from_bytes(repr.as_bytes());\n let Z = FieldElement::ONE;\n let YY = Y.square();\n let u = &YY - &Z; // u = y²-1\n let v = &(&YY * &constants::EDWARDS_D) + &Z; // v = dy²+1\n let (is_valid_y_coord, X) = FieldElement::sqrt_ratio_i(&u, &v);\n\n (is_valid_y_coord, X, Y, Z)\n }\n\n #[rustfmt::skip]\n pub(super) fn step_2(\n repr: &CompressedEdwardsY,\n mut X: FieldElement,\n Y: FieldElement,\n Z: FieldElement,\n ) -> EdwardsPoint {\n // FieldElement::sqrt_ratio_i always returns the nonnegative square root,\n // so we negate according to the supplied sign bit.\n let compressed_sign_bit = Choice::from(repr.as_bytes()[31] >> 7);\n X.conditional_negate(compressed_sign_bit);\n\n EdwardsPoint {\n X,\n Y,\n Z,\n T: &X * &Y,\n }\n }\n}\n\nimpl TryFrom<&[u8]> for CompressedEdwardsY {\n type Error = TryFromSliceError;\n\n fn try_from(slice: &[u8]) -> Result {\n Self::from_slice(slice)\n }\n}\n\n// ------------------------------------------------------------------------\n// Serde support\n// ------------------------------------------------------------------------\n// Serializes to and from `EdwardsPoint` directly, doing compression\n// and decompression internally. This means that users can create\n// structs containing `EdwardsPoint`s and use Serde's derived\n// serializers to serialize those structures.\n\n#[cfg(feature = \"digest\")]\nuse constants::ED25519_SQRTAM2;\n#[cfg(feature = \"serde\")]\nuse serde::de::Visitor;\n#[cfg(feature = \"serde\")]\nuse serde::{Deserialize, Deserializer, Serialize, Serializer};\n\n#[cfg(feature = \"serde\")]\nimpl Serialize for EdwardsPoint {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.compress().as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl Serialize for CompressedEdwardsY {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl<'de> Deserialize<'de> for EdwardsPoint {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct EdwardsPointVisitor;\n\n impl<'de> Visitor<'de> for EdwardsPointVisitor {\n type Value = EdwardsPoint;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\"a valid point in Edwards y + sign format\")\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n CompressedEdwardsY(bytes)\n .decompress()\n .ok_or_else(|| serde::de::Error::custom(\"decompression failed\"))\n }\n }\n\n deserializer.deserialize_tuple(32, EdwardsPointVisitor)\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl<'de> Deserialize<'de> for CompressedEdwardsY {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct CompressedEdwardsYVisitor;\n\n impl<'de> Visitor<'de> for CompressedEdwardsYVisitor {\n type Value = CompressedEdwardsY;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\"32 bytes of data\")\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n Ok(CompressedEdwardsY(bytes))\n }\n }\n\n deserializer.deserialize_tuple(32, CompressedEdwardsYVisitor)\n }\n}\n\n// ------------------------------------------------------------------------\n// Internal point representations\n// ------------------------------------------------------------------------\n\n/// An `EdwardsPoint` represents a point on the Edwards form of Curve25519.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct EdwardsPoint {\n pub(crate) X: FieldElement,\n pub(crate) Y: FieldElement,\n pub(crate) Z: FieldElement,\n pub(crate) T: FieldElement,\n}\n\n// ------------------------------------------------------------------------\n// Constructors\n// ------------------------------------------------------------------------\n\nimpl Identity for CompressedEdwardsY {\n fn identity() -> CompressedEdwardsY {\n CompressedEdwardsY([\n 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ])\n }\n}\n\nimpl Default for CompressedEdwardsY {\n fn default() -> CompressedEdwardsY {\n CompressedEdwardsY::identity()\n }\n}\n\nimpl CompressedEdwardsY {\n /// Construct a `CompressedEdwardsY` from a slice of bytes.\n ///\n /// # Errors\n ///\n /// Returns [`TryFromSliceError`] if the input `bytes` slice does not have\n /// a length of 32.\n pub fn from_slice(bytes: &[u8]) -> Result {\n bytes.try_into().map(CompressedEdwardsY)\n }\n}\n\nimpl Identity for EdwardsPoint {\n fn identity() -> EdwardsPoint {\n EdwardsPoint {\n X: FieldElement::ZERO,\n Y: FieldElement::ONE,\n Z: FieldElement::ONE,\n T: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for EdwardsPoint {\n fn default() -> EdwardsPoint {\n EdwardsPoint::identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Zeroize implementations for wiping points from memory\n// ------------------------------------------------------------------------\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for CompressedEdwardsY {\n /// Reset this `CompressedEdwardsY` to the compressed form of the identity element.\n fn zeroize(&mut self) {\n self.0.zeroize();\n self.0[0] = 1;\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for EdwardsPoint {\n /// Reset this `EdwardsPoint` to the identity element.\n fn zeroize(&mut self) {\n self.X.zeroize();\n self.Y = FieldElement::ONE;\n self.Z = FieldElement::ONE;\n self.T.zeroize();\n }\n}\n\n// ------------------------------------------------------------------------\n// Validity checks (for debugging, not CT)\n// ------------------------------------------------------------------------\n\nimpl ValidityCheck for EdwardsPoint {\n fn is_valid(&self) -> bool {\n let point_on_curve = self.as_projective().is_valid();\n let on_segre_image = (&self.X * &self.Y) == (&self.Z * &self.T);\n\n point_on_curve && on_segre_image\n }\n}\n\n// ------------------------------------------------------------------------\n// Constant-time assignment\n// ------------------------------------------------------------------------\n\nimpl ConditionallySelectable for EdwardsPoint {\n fn conditional_select(a: &EdwardsPoint, b: &EdwardsPoint, choice: Choice) -> EdwardsPoint {\n EdwardsPoint {\n X: FieldElement::conditional_select(&a.X, &b.X, choice),\n Y: FieldElement::conditional_select(&a.Y, &b.Y, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T: FieldElement::conditional_select(&a.T, &b.T, choice),\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Equality\n// ------------------------------------------------------------------------\n\nimpl ConstantTimeEq for EdwardsPoint {\n fn ct_eq(&self, other: &EdwardsPoint) -> Choice {\n // We would like to check that the point (X/Z, Y/Z) is equal to\n // the point (X'/Z', Y'/Z') without converting into affine\n // coordinates (x, y) and (x', y'), which requires two inversions.\n // We have that X = xZ and X' = x'Z'. Thus, x = x' is equivalent to\n // (xZ)Z' = (x'Z')Z, and similarly for the y-coordinate.\n\n (&self.X * &other.Z).ct_eq(&(&other.X * &self.Z))\n & (&self.Y * &other.Z).ct_eq(&(&other.Y * &self.Z))\n }\n}\n\nimpl PartialEq for EdwardsPoint {\n fn eq(&self, other: &EdwardsPoint) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for EdwardsPoint {}\n\n// ------------------------------------------------------------------------\n// Point conversions\n// ------------------------------------------------------------------------\n\nimpl EdwardsPoint {\n /// Convert to a ProjectiveNielsPoint\n pub(crate) fn as_projective_niels(&self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: &self.Y + &self.X,\n Y_minus_X: &self.Y - &self.X,\n Z: self.Z,\n T2d: &self.T * &constants::EDWARDS_D2,\n }\n }\n\n /// Convert the representation of this point from extended\n /// coordinates to projective coordinates.\n ///\n /// Free.\n pub(crate) const fn as_projective(&self) -> ProjectivePoint {\n ProjectivePoint {\n X: self.X,\n Y: self.Y,\n Z: self.Z,\n }\n }\n\n /// Dehomogenize to a `AffineNielsPoint`.\n /// Mainly for testing.\n pub(crate) fn as_affine_niels(&self) -> AffineNielsPoint {\n let recip = self.Z.invert();\n let x = &self.X * &recip;\n let y = &self.Y * &recip;\n let xy2d = &(&x * &y) * &constants::EDWARDS_D2;\n AffineNielsPoint {\n y_plus_x: &y + &x,\n y_minus_x: &y - &x,\n xy2d,\n }\n }\n\n /// Dehomogenize to `AffinePoint`.\n pub(crate) fn to_affine(self) -> AffinePoint {\n let recip = self.Z.invert();\n let x = &self.X * &recip;\n let y = &self.Y * &recip;\n AffinePoint { x, y }\n }\n\n /// Convert this `EdwardsPoint` on the Edwards model to the\n /// corresponding `MontgomeryPoint` on the Montgomery model.\n ///\n /// This function has one exceptional case; the identity point of\n /// the Edwards curve is sent to the 2-torsion point \\\\((0,0)\\\\)\n /// on the Montgomery curve.\n ///\n /// Note that this is a one-way conversion, since the Montgomery\n /// model does not retain sign information.\n pub fn to_montgomery(&self) -> MontgomeryPoint {\n // We have u = (1+y)/(1-y) = (Z+Y)/(Z-Y).\n //\n // The denominator is zero only when y=1, the identity point of\n // the Edwards curve. Since 0.invert() = 0, in this case we\n // compute the 2-torsion point (0,0).\n let U = &self.Z + &self.Y;\n let W = &self.Z - &self.Y;\n let u = &U * &W.invert();\n MontgomeryPoint(u.to_bytes())\n }\n\n /// Converts a large batch of points to Edwards at once. This has the same\n /// behavior on identity elements as [`Self::to_montgomery`].\n #[cfg(feature = \"alloc\")]\n pub fn to_montgomery_batch(eds: &[Self]) -> Vec {\n // Do the same thing as the above function. u = (1+y)/(1-y) = (Z+Y)/(Z-Y).\n // We will do this in a batch, ie compute (Z-Y) for all the input\n // points, then invert them all at once\n\n // Compute the denominators in a batch\n let mut denominators = eds.iter().map(|p| &p.Z - &p.Y).collect::>();\n FieldElement::invert_batch_alloc(&mut denominators);\n\n // Now compute the Montgomery u coordinate for every point\n let mut ret = Vec::with_capacity(eds.len());\n for (ed, d) in eds.iter().zip(denominators.iter()) {\n let u = &(&ed.Z + &ed.Y) * d;\n ret.push(MontgomeryPoint(u.to_bytes()));\n }\n\n ret\n }\n\n /// Compress this point to `CompressedEdwardsY` format.\n pub fn compress(&self) -> CompressedEdwardsY {\n self.to_affine().compress()\n }\n\n /// Compress several `EdwardsPoint`s into `CompressedEdwardsY` format, using a batch inversion\n /// for a significant speedup.\n #[cfg(feature = \"alloc\")]\n pub fn compress_batch(inputs: &[EdwardsPoint]) -> Vec {\n let mut zs = inputs.iter().map(|input| input.Z).collect::>();\n FieldElement::invert_batch_alloc(&mut zs);\n\n inputs\n .iter()\n .zip(&zs)\n .map(|(input, recip)| {\n let x = &input.X * recip;\n let y = &input.Y * recip;\n AffinePoint { x, y }.compress()\n })\n .collect()\n }\n\n #[cfg(feature = \"digest\")]\n // The function `map_to_curve` calculates an [EdwardsPoint] from a [FieldElement].\n fn map_to_curve(fe: FieldElement) -> EdwardsPoint {\n let c1 = ED25519_SQRTAM2;\n\n // 1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u)\n let (xMn, xMd, yMn, yMd) = crate::montgomery::elligator_encode(&fe);\n // 2. xn = xMn * yMd\n let xn = &xMn * &yMd;\n // 3. xn = xn * c1\n let xn = &xn * &c1;\n // 4. xd = xMd * yMn\n let xd = &xMd * &yMn;\n // 5. yn = xMn - xMd\n let yn = &xMn - &xMd;\n // 6. yd = xMn + xMd\n let yd = &xMn + &xMd;\n // 7. tv1 = xd * yd\n let tv1 = &xd * &yd;\n // 8. e = tv1 == 0\n let e = tv1.ct_eq(&FieldElement::ZERO);\n // 9. xn = CMOV(xn, 0, e)\n let xn = FieldElement::conditional_select(&xn, &FieldElement::ZERO, e);\n // 10. xd = CMOV(xd, 1, e)\n let xd = FieldElement::conditional_select(&xd, &FieldElement::ONE, e);\n // 11. yn = CMOV(yn, 1, e)\n let yn = FieldElement::conditional_select(&yn, &FieldElement::ONE, e);\n // 12. yd = CMOV(yd, 1, e)\n let yd = FieldElement::conditional_select(&yd, &FieldElement::ONE, e);\n // 13. return (xn, xd, yn, yd)\n\n EdwardsPoint {\n X: &xn * &yd,\n Y: &xd * &yn,\n Z: &xd * &yd,\n T: &xn * &yn,\n }\n }\n\n #[cfg(feature = \"digest\")]\n /// Perform encode to curve per RFC 9380, with explicit hash function and domain separator\n /// `domain_sep`, using the Twisted Edwards Elligator 2 method. The input is the concatenation\n /// of the elements of `bytes`. Likewise for the domain separator with `domain_sep`. At least\n /// one element of `domain_sep`, MUST be nonempty, and the concatenation MUST NOT exceed 255\n /// bytes.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Warning\n /// `encode_to_curve` is a nonuniform encoding from byte strings to points in `G`. That is,\n /// the distribution of its output is not uniformly random in `G`: the set of possible outputs\n /// of encode_to_curve is only a fraction of the points in `G`, and some points in this set\n /// are more likely to be output than others.\n ///\n /// If your application needs the distribution of the output to be statistically close to\n /// uniform in `G`, use [Self::hash_to_curve] instead.\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`\n pub fn encode_to_curve(bytes: &[&[u8]], domain_sep: &[&[u8]]) -> EdwardsPoint\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // For reference see\n // https://www.rfc-editor.org/rfc/rfc9380.html#name-elligator-2-method-2\n\n let fe = FieldElement::hash_to_field::(bytes, domain_sep);\n let Q = Self::map_to_curve(fe[0]);\n Q.mul_by_cofactor()\n }\n\n #[cfg(feature = \"digest\")]\n /// Perform a hash to curve per RFC 9380, with explicit hash function and domain separator\n /// `domain_sep`, using the Twisted Edwards Elligator 2 method. The input is the concatenation\n /// of the elements of `bytes`. Likewise for the domain separator with `domain_sep`. At least\n /// one element of `domain_sep`, MUST be nonempty, and the concatenation MUST NOT exceed\n /// 255 bytes.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`\n pub fn hash_to_curve(bytes: &[&[u8]], domain_sep: &[&[u8]]) -> EdwardsPoint\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // For reference see\n // https://www.rfc-editor.org/rfc/rfc9380.html#name-elligator-2-method-2\n\n let fe = FieldElement::hash_to_field::(bytes, domain_sep);\n let Q0 = Self::map_to_curve(fe[0]);\n let Q1 = Self::map_to_curve(fe[1]);\n\n let R = Q0 + Q1;\n R.mul_by_cofactor()\n }\n\n /// Return an `EdwardsPoint` chosen uniformly at random using a user-provided RNG.\n ///\n /// # Inputs\n ///\n /// * `rng`: any RNG which implements `CryptoRng` and `RngCore`\n ///\n /// # Returns\n ///\n /// A random `EdwardsPoint`.\n ///\n /// # Implementation\n ///\n /// Uses rejection sampling, generating a random `CompressedEdwardsY` and then attempting point\n /// decompression, rejecting invalid points.\n #[cfg(feature = \"rand_core\")]\n pub fn random(rng: &mut R) -> Self {\n let mut repr = CompressedEdwardsY([0u8; 32]);\n loop {\n rng.fill_bytes(&mut repr.0);\n if let Some(p) = repr.decompress() {\n if !IsIdentity::is_identity(&p) {\n break p;\n }\n }\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Doubling\n// ------------------------------------------------------------------------\n\nimpl EdwardsPoint {\n /// Add this point to itself.\n pub(crate) fn double(&self) -> EdwardsPoint {\n self.as_projective().double().as_extended()\n }\n}\n\n// ------------------------------------------------------------------------\n// Addition and Subtraction\n// ------------------------------------------------------------------------\n\nimpl<'a> Add<&'a EdwardsPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn add(self, other: &'a EdwardsPoint) -> EdwardsPoint {\n (self + &other.as_projective_niels()).as_extended()\n }\n}\n\ndefine_add_variants!(\n LHS = EdwardsPoint,\n RHS = EdwardsPoint,\n Output = EdwardsPoint\n);\n\nimpl<'a> AddAssign<&'a EdwardsPoint> for EdwardsPoint {\n fn add_assign(&mut self, _rhs: &'a EdwardsPoint) {\n *self = (self as &EdwardsPoint) + _rhs;\n }\n}\n\ndefine_add_assign_variants!(LHS = EdwardsPoint, RHS = EdwardsPoint);\n\nimpl<'a> Sub<&'a EdwardsPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn sub(self, other: &'a EdwardsPoint) -> EdwardsPoint {\n (self - &other.as_projective_niels()).as_extended()\n }\n}\n\ndefine_sub_variants!(\n LHS = EdwardsPoint,\n RHS = EdwardsPoint,\n Output = EdwardsPoint\n);\n\nimpl<'a> SubAssign<&'a EdwardsPoint> for EdwardsPoint {\n fn sub_assign(&mut self, _rhs: &'a EdwardsPoint) {\n *self = (self as &EdwardsPoint) - _rhs;\n }\n}\n\ndefine_sub_assign_variants!(LHS = EdwardsPoint, RHS = EdwardsPoint);\n\nimpl Sum for EdwardsPoint\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(EdwardsPoint::identity(), |acc, item| acc + item.borrow())\n }\n}\n\n// ------------------------------------------------------------------------\n// Negation\n// ------------------------------------------------------------------------\n\nimpl Neg for &EdwardsPoint {\n type Output = EdwardsPoint;\n\n fn neg(self) -> EdwardsPoint {\n EdwardsPoint {\n X: -(&self.X),\n Y: self.Y,\n Z: self.Z,\n T: -(&self.T),\n }\n }\n}\n\nimpl Neg for EdwardsPoint {\n type Output = EdwardsPoint;\n\n fn neg(self) -> EdwardsPoint {\n -&self\n }\n}\n\n// ------------------------------------------------------------------------\n// Scalar multiplication\n// ------------------------------------------------------------------------\n\nimpl<'a> MulAssign<&'a Scalar> for EdwardsPoint {\n fn mul_assign(&mut self, scalar: &'a Scalar) {\n let result = (self as &EdwardsPoint) * scalar;\n *self = result;\n }\n}\n\ndefine_mul_assign_variants!(LHS = EdwardsPoint, RHS = Scalar);\n\ndefine_mul_variants!(LHS = EdwardsPoint, RHS = Scalar, Output = EdwardsPoint);\ndefine_mul_variants!(LHS = Scalar, RHS = EdwardsPoint, Output = EdwardsPoint);\n\nimpl<'a> Mul<&'a Scalar> for &EdwardsPoint {\n type Output = EdwardsPoint;\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, scalar: &'a Scalar) -> EdwardsPoint {\n crate::backend::variable_base_mul(self, scalar)\n }\n}\n\nimpl<'a> Mul<&'a EdwardsPoint> for &Scalar {\n type Output = EdwardsPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, point: &'a EdwardsPoint) -> EdwardsPoint {\n point * self\n }\n}\n\nimpl EdwardsPoint {\n /// Fixed-base scalar multiplication by the Ed25519 base point.\n ///\n /// Uses precomputed basepoint tables when the `precomputed-tables` feature\n /// is enabled, trading off increased code size for ~4x better performance.\n pub fn mul_base(scalar: &Scalar) -> Self {\n #[cfg(not(feature = \"precomputed-tables\"))]\n {\n scalar * constants::ED25519_BASEPOINT_POINT\n }\n\n #[cfg(feature = \"precomputed-tables\")]\n {\n scalar * constants::ED25519_BASEPOINT_TABLE\n }\n }\n\n /// Multiply this point by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_clamped(self, bytes: [u8; 32]) -> Self {\n // We have to construct a Scalar that is not reduced mod l, which breaks scalar invariant\n // #2. But #2 is not necessary for correctness of variable-base multiplication. All that\n // needs to hold is invariant #1, i.e., the scalar is less than 2^255. This is guaranteed\n // by clamping.\n // Further, we don't do any reduction or arithmetic with this clamped value, so there's no\n // issues arising from the fact that the curve point is not necessarily in the prime-order\n // subgroup.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n s * self\n }\n\n /// Multiply the basepoint by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_base_clamped(bytes: [u8; 32]) -> Self {\n // See reasoning in Self::mul_clamped why it is OK to make an unreduced Scalar here. We\n // note that fixed-base multiplication is also defined for all values of `bytes` less than\n // 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n Self::mul_base(&s)\n }\n}\n\n// ------------------------------------------------------------------------\n// Multiscalar Multiplication impls\n// ------------------------------------------------------------------------\n\n// These use the iterator's size hint and the target settings to\n// forward to a specific backend implementation.\n\n#[cfg(feature = \"alloc\")]\nimpl MultiscalarMul for EdwardsPoint {\n type Point = EdwardsPoint;\n\n fn multiscalar_mul(scalars: I, points: J) -> EdwardsPoint\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n {\n // Sanity-check lengths of input iterators\n let mut scalars = scalars.into_iter();\n let mut points = points.into_iter();\n\n // Lower and upper bounds on iterators\n let (s_lo, s_hi) = scalars.by_ref().size_hint();\n let (p_lo, p_hi) = points.by_ref().size_hint();\n\n // They should all be equal\n assert_eq!(s_lo, p_lo);\n assert_eq!(s_hi, Some(s_lo));\n assert_eq!(p_hi, Some(p_lo));\n\n // Now we know there's a single size. When we do\n // size-dependent algorithm dispatch, use this as the hint.\n let _size = s_lo;\n\n crate::backend::straus_multiscalar_mul(scalars, points)\n }\n}\n\n#[cfg(feature = \"alloc\")]\nimpl VartimeMultiscalarMul for EdwardsPoint {\n type Point = EdwardsPoint;\n\n fn optional_multiscalar_mul(scalars: I, points: J) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator>,\n {\n // Sanity-check lengths of input iterators\n let mut scalars = scalars.into_iter();\n let mut points = points.into_iter();\n\n // Lower and upper bounds on iterators\n let (s_lo, s_hi) = scalars.by_ref().size_hint();\n let (p_lo, p_hi) = points.by_ref().size_hint();\n\n // They should all be equal\n assert_eq!(s_lo, p_lo);\n assert_eq!(s_hi, Some(s_lo));\n assert_eq!(p_hi, Some(p_lo));\n\n // Now we know there's a single size.\n // Use this as the hint to decide which algorithm to use.\n let size = s_lo;\n\n if size < 190 {\n crate::backend::straus_optional_multiscalar_mul(scalars, points)\n } else {\n crate::backend::pippenger_optional_multiscalar_mul(scalars, points)\n }\n }\n}\n\n/// Precomputation for variable-time multiscalar multiplication with `EdwardsPoint`s.\n// This wraps the inner implementation in a facade type so that we can\n// decouple stability of the inner type from the stability of the\n// outer type.\n#[cfg(feature = \"alloc\")]\npub struct VartimeEdwardsPrecomputation(crate::backend::VartimePrecomputedStraus);\n\n#[cfg(feature = \"alloc\")]\nimpl VartimePrecomputedMultiscalarMul for VartimeEdwardsPrecomputation {\n type Point = EdwardsPoint;\n\n fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: Borrow,\n {\n Self(crate::backend::VartimePrecomputedStraus::new(static_points))\n }\n\n fn len(&self) -> usize {\n self.0.len()\n }\n\n fn is_empty(&self) -> bool {\n self.0.is_empty()\n }\n\n fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator>,\n {\n self.0\n .optional_mixed_multiscalar_mul(static_scalars, dynamic_scalars, dynamic_points)\n }\n}\n\nimpl EdwardsPoint {\n /// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n pub fn vartime_double_scalar_mul_basepoint(\n a: &Scalar,\n A: &EdwardsPoint,\n b: &Scalar,\n ) -> EdwardsPoint {\n crate::backend::vartime_double_base_mul(a, A, b)\n }\n\n /// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n ///\n /// This function is optimized for the case where \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\),\n /// and falls back to general scalar multiplication for full-width scalars.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n /// use curve25519::constants::ED25519_BASEPOINT_POINT;\n /// use curve25519::edwards::EdwardsPoint;\n ///\n /// let a1 = Scalar::from(123u64);\n /// let a2 = Scalar::from(456u64);\n /// let b = Scalar::from(789u64);\n ///\n /// let A1 = &ED25519_BASEPOINT_POINT * &Scalar::from(2u64);\n /// let A2 = &ED25519_BASEPOINT_POINT * &Scalar::from(3u64);\n ///\n /// // Compute a1*A1 + a2*A2 + b*B efficiently\n /// let result = EdwardsPoint::vartime_triple_scalar_mul_basepoint(&a1, &A1, &a2, &A2, &b);\n /// ```\n #[allow(non_snake_case)]\n pub fn vartime_triple_scalar_mul_basepoint(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n ) -> EdwardsPoint {\n crate::backend::vartime_triple_base_mul_128_128_256(a1, A1, a2, A2, b)\n }\n}\n\n#[cfg(feature = \"precomputed-tables\")]\nmacro_rules! impl_basepoint_table {\n (Name = $name:ident, LookupTable = $table:ident, Point = $point:ty, Radix = $radix:expr, Additions = $adds:expr) => {\n /// A precomputed table of multiples of a basepoint, for accelerating\n /// fixed-base scalar multiplication. One table, for the Ed25519\n /// basepoint, is provided in the [`constants`] module.\n ///\n /// The basepoint tables are reasonably large, so they should probably be boxed.\n ///\n /// The sizes for the tables and the number of additions required for one scalar\n /// multiplication are as follows:\n ///\n /// * [`EdwardsBasepointTableRadix16`]: 30KB, 64A\n /// (this is the default size, and is used for\n /// [`constants::ED25519_BASEPOINT_TABLE`])\n /// * [`EdwardsBasepointTableRadix64`]: 120KB, 43A\n /// * [`EdwardsBasepointTableRadix128`]: 240KB, 37A\n /// * [`EdwardsBasepointTableRadix256`]: 480KB, 33A\n ///\n /// # Why 33 additions for radix-256?\n ///\n /// Normally, the radix-256 tables would allow for only 32 additions per scalar\n /// multiplication. However, due to the fact that standardised definitions of\n /// legacy protocols—such as x25519—require allowing unreduced 255-bit scalars\n /// invariants, when converting such an unreduced scalar's representation to\n /// radix-\\\\(2^{8}\\\\), we cannot guarantee the carry bit will fit in the last\n /// coefficient (the coefficients are `i8`s). When, \\\\(w\\\\), the power-of-2 of\n /// the radix, is \\\\(w < 8\\\\), we can fold the final carry onto the last\n /// coefficient, \\\\(d\\\\), because \\\\(d < 2^{w/2}\\\\), so\n /// $$\n /// d + carry \\cdot 2^{w} = d + 1 \\cdot 2^{w} < 2^{w+1} < 2^{8}\n /// $$\n /// When \\\\(w = 8\\\\), we can't fit \\\\(carry \\cdot 2^{w}\\\\) into an `i8`, so we\n /// add the carry bit onto an additional coefficient.\n #[derive(Clone)]\n #[repr(transparent)]\n pub struct $name(pub(crate) [$table; 32]);\n\n impl BasepointTable for $name {\n type Point = $point;\n\n /// Create a table of precomputed multiples of `basepoint`.\n fn create(basepoint: &$point) -> $name {\n // XXX use init_with\n let mut table = $name([$table::default(); 32]);\n let mut P = *basepoint;\n for i in 0..32 {\n // P = (2w)^i * B\n table.0[i] = $table::from(&P);\n P = P.mul_by_pow_2($radix + $radix);\n }\n table\n }\n\n /// Get the basepoint for this table as an `EdwardsPoint`.\n fn basepoint(&self) -> $point {\n // self.0[0].select(1) = 1*(16^2)^0*B\n // but as an `AffineNielsPoint`, so add identity to convert to extended.\n (&<$point>::identity() + &self.0[0].select(1)).as_extended()\n }\n\n /// The computation uses Pippeneger's algorithm, as described for the\n /// specific case of radix-16 on page 13 of the Ed25519 paper.\n ///\n /// # Piggenger's Algorithm Generalised\n ///\n /// Write the scalar \\\\(a\\\\) in radix-\\\\(w\\\\), where \\\\(w\\\\) is a power of\n /// 2, with coefficients in \\\\([\\frac{-w}{2},\\frac{w}{2})\\\\), i.e.,\n /// $$\n /// a = a\\_0 + a\\_1 w\\^1 + \\cdots + a\\_{x} w\\^{x},\n /// $$\n /// with\n /// $$\n /// \\begin{aligned}\n /// \\frac{-w}{2} \\leq a_i < \\frac{w}{2}\n /// &&\\cdots&&\n /// \\frac{-w}{2} \\leq a\\_{x} \\leq \\frac{w}{2}\n /// \\end{aligned}\n /// $$\n /// and the number of additions, \\\\(x\\\\), is given by\n /// \\\\(x = \\lceil \\frac{256}{w} \\rceil\\\\). Then\n /// $$\n /// a B = a\\_0 B + a\\_1 w\\^1 B + \\cdots + a\\_{x-1} w\\^{x-1} B.\n /// $$\n /// Grouping even and odd coefficients gives\n /// $$\n /// \\begin{aligned}\n /// a B = \\quad a\\_0 w\\^0 B +& a\\_2 w\\^2 B + \\cdots + a\\_{x-2} w\\^{x-2} B \\\\\\\\\n /// + a\\_1 w\\^1 B +& a\\_3 w\\^3 B + \\cdots + a\\_{x-1} w\\^{x-1} B \\\\\\\\\n /// = \\quad(a\\_0 w\\^0 B +& a\\_2 w\\^2 B + \\cdots + a\\_{x-2} w\\^{x-2} B) \\\\\\\\\n /// + w(a\\_1 w\\^0 B +& a\\_3 w\\^2 B + \\cdots + a\\_{x-1} w\\^{x-2} B). \\\\\\\\\n /// \\end{aligned}\n /// $$\n /// For each \\\\(i = 0 \\ldots 31\\\\), we create a lookup table of\n /// $$\n /// [w\\^{2i} B, \\ldots, \\frac{w}{2}\\cdot w\\^{2i} B],\n /// $$\n /// and use it to select \\\\( y \\cdot w\\^{2i} \\cdot B \\\\) in constant time.\n ///\n /// The radix-\\\\(w\\\\) representation requires that the scalar is bounded\n /// by \\\\(2\\^{255}\\\\), which is always the case.\n ///\n /// The above algorithm is trivially generalised to other powers-of-2 radices.\n fn mul_base(&self, scalar: &Scalar) -> $point {\n let a = scalar.as_radix_2w($radix);\n\n let tables = &self.0;\n let mut P = <$point>::identity();\n\n for i in (0..$adds).filter(|x| x % 2 == 1) {\n P = (&P + &tables[i / 2].select(a[i])).as_extended();\n }\n\n P = P.mul_by_pow_2($radix);\n\n for i in (0..$adds).filter(|x| x % 2 == 0) {\n P = (&P + &tables[i / 2].select(a[i])).as_extended();\n }\n\n P\n }\n }\n\n impl<'a, 'b> Mul<&'b Scalar> for &'a $name {\n type Output = $point;\n\n /// Construct an `EdwardsPoint` from a `Scalar` \\\\(a\\\\) by\n /// computing the multiple \\\\(aB\\\\) of this basepoint \\\\(B\\\\).\n fn mul(self, scalar: &'b Scalar) -> $point {\n // delegate to a private function so that its documentation appears in internal docs\n self.mul_base(scalar)\n }\n }\n\n impl<'a, 'b> Mul<&'a $name> for &'b Scalar {\n type Output = $point;\n\n /// Construct an `EdwardsPoint` from a `Scalar` \\\\(a\\\\) by\n /// computing the multiple \\\\(aB\\\\) of this basepoint \\\\(B\\\\).\n fn mul(self, basepoint_table: &'a $name) -> $point {\n basepoint_table * self\n }\n }\n\n impl Debug for $name {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"{:?}([\\n\", stringify!($name))?;\n for i in 0..32 {\n write!(f, \"\\t{:?},\\n\", &self.0[i])?;\n }\n write!(f, \"])\")\n }\n }\n };\n} // End macro_rules! impl_basepoint_table\n\n// The number of additions required is ceil(256/w) where w is the radix representation.\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n impl_basepoint_table! {\n Name = EdwardsBasepointTable,\n LookupTable = LookupTableRadix16,\n Point = EdwardsPoint,\n Radix = 4,\n Additions = 64\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix32,\n LookupTable = LookupTableRadix32,\n Point = EdwardsPoint,\n Radix = 5,\n Additions = 52\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix64,\n LookupTable = LookupTableRadix64,\n Point = EdwardsPoint,\n Radix = 6,\n Additions = 43\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix128,\n LookupTable = LookupTableRadix128,\n Point = EdwardsPoint,\n Radix = 7,\n Additions = 37\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix256,\n LookupTable = LookupTableRadix256,\n Point = EdwardsPoint,\n Radix = 8,\n Additions = 33\n }\n\n /// A type-alias for [`EdwardsBasepointTable`] because the latter is\n /// used as a constructor in the [`constants`] module.\n //\n // Same as for `LookupTableRadix16`, we have to define `EdwardsBasepointTable`\n // first, because it's used as a constructor, and then provide a type alias for\n // it.\n pub type EdwardsBasepointTableRadix16 = EdwardsBasepointTable;\n }\n}\n\n#[cfg(feature = \"precomputed-tables\")]\nmacro_rules! impl_basepoint_table_conversions {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl<'a> From<&'a $lhs> for $rhs {\n fn from(table: &'a $lhs) -> $rhs {\n <$rhs>::create(&table.basepoint())\n }\n }\n\n impl<'a> From<&'a $rhs> for $lhs {\n fn from(table: &'a $rhs) -> $lhs {\n <$lhs>::create(&table.basepoint())\n }\n }\n };\n}\n\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n // Conversions from radix 16\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix32\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix64\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 32\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix64\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 64\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix64,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix64,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 128\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix128,\n RHS = EdwardsBasepointTableRadix256\n }\n }\n}\n\nimpl EdwardsPoint {\n /// Multiply by the cofactor: return \\\\(\\[8\\]P\\\\).\n pub fn mul_by_cofactor(&self) -> EdwardsPoint {\n self.mul_by_pow_2(3)\n }\n\n /// Compute \\\\([2\\^k] P \\\\) by successive doublings. Requires \\\\( k > 0 \\\\).\n pub(crate) fn mul_by_pow_2(&self, k: u32) -> EdwardsPoint {\n debug_assert!(k > 0);\n let mut r: CompletedPoint;\n let mut s = self.as_projective();\n for _ in 0..(k - 1) {\n r = s.double();\n s = r.as_projective();\n }\n // Unroll last iteration so we can go directly as_extended()\n s.double().as_extended()\n }\n\n /// Determine if this point is of small order.\n ///\n /// # Return\n ///\n /// * `true` if `self` is in the torsion subgroup \\\\( \\mathcal E\\[8\\] \\\\);\n /// * `false` if `self` is not in the torsion subgroup \\\\( \\mathcal E\\[8\\] \\\\).\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::constants;\n ///\n /// // Generator of the prime-order subgroup\n /// let P = constants::ED25519_BASEPOINT_POINT;\n /// // Generator of the torsion subgroup\n /// let Q = constants::EIGHT_TORSION[1];\n ///\n /// // P has large order\n /// assert_eq!(P.is_small_order(), false);\n ///\n /// // Q has small order\n /// assert_eq!(Q.is_small_order(), true);\n /// ```\n pub fn is_small_order(&self) -> bool {\n self.mul_by_cofactor().is_identity()\n }\n\n /// Determine if this point is “torsion-free”, i.e., is contained in\n /// the prime-order subgroup.\n ///\n /// # Return\n ///\n /// * `true` if `self` has zero torsion component and is in the\n /// prime-order subgroup;\n /// * `false` if `self` has a nonzero torsion component and is not\n /// in the prime-order subgroup.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::constants;\n ///\n /// // Generator of the prime-order subgroup\n /// let P = constants::ED25519_BASEPOINT_POINT;\n /// // Generator of the torsion subgroup\n /// let Q = constants::EIGHT_TORSION[1];\n ///\n /// // P is torsion-free\n /// assert_eq!(P.is_torsion_free(), true);\n ///\n /// // P + Q is not torsion-free\n /// assert_eq!((P+Q).is_torsion_free(), false);\n /// ```\n pub fn is_torsion_free(&self) -> bool {\n (self * constants::BASEPOINT_ORDER).is_identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Debug traits\n// ------------------------------------------------------------------------\n\nimpl Debug for EdwardsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"EdwardsPoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?},\\n\\tT: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z, &self.T\n )\n }\n}\n\n// ------------------------------------------------------------------------\n// group traits\n// ------------------------------------------------------------------------\n\n// Use the full trait path to avoid Group::identity overlapping Identity::identity in the\n// rest of the module (e.g. tests).\n#[cfg(feature = \"group\")]\nimpl group::Group for EdwardsPoint {\n type Scalar = Scalar;\n\n fn random(mut rng: impl GroupRngCore) -> Self {\n let mut repr = CompressedEdwardsY([0u8; 32]);\n loop {\n rng.fill_bytes(&mut repr.0);\n if let Some(p) = repr.decompress() {\n if !IsIdentity::is_identity(&p) {\n break p;\n }\n }\n }\n }\n\n fn identity() -> Self {\n Identity::identity()\n }\n\n fn generator() -> Self {\n constants::ED25519_BASEPOINT_POINT\n }\n\n fn is_identity(&self) -> Choice {\n self.ct_eq(&Identity::identity())\n }\n\n fn double(&self) -> Self {\n self.double()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl GroupEncoding for EdwardsPoint {\n type Repr = [u8; 32];\n\n fn from_bytes(bytes: &Self::Repr) -> CtOption {\n let repr = CompressedEdwardsY(*bytes);\n let (is_valid_y_coord, X, Y, Z) = decompress::step_1(&repr);\n CtOption::new(decompress::step_2(&repr, X, Y, Z), is_valid_y_coord)\n }\n\n fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption {\n // Just use the checked API; there are no checks we can skip.\n Self::from_bytes(bytes)\n }\n\n fn to_bytes(&self) -> Self::Repr {\n self.compress().to_bytes()\n }\n}\n\n/// A `SubgroupPoint` represents a point on the Edwards form of Curve25519, that is\n/// guaranteed to be in the prime-order subgroup.\n#[cfg(feature = \"group\")]\n#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)]\npub struct SubgroupPoint(EdwardsPoint);\n\n#[cfg(feature = \"group\")]\nimpl From for EdwardsPoint {\n fn from(p: SubgroupPoint) -> Self {\n p.0\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Neg for SubgroupPoint {\n type Output = Self;\n\n fn neg(self) -> Self::Output {\n SubgroupPoint(-self.0)\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Add<&SubgroupPoint> for &SubgroupPoint {\n type Output = SubgroupPoint;\n fn add(self, other: &SubgroupPoint) -> SubgroupPoint {\n SubgroupPoint(self.0 + other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_variants!(\n LHS = SubgroupPoint,\n RHS = SubgroupPoint,\n Output = SubgroupPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl Add<&SubgroupPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn add(self, other: &SubgroupPoint) -> EdwardsPoint {\n self + other.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_variants!(\n LHS = EdwardsPoint,\n RHS = SubgroupPoint,\n Output = EdwardsPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl AddAssign<&SubgroupPoint> for SubgroupPoint {\n fn add_assign(&mut self, rhs: &SubgroupPoint) {\n self.0 += rhs.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl AddAssign<&SubgroupPoint> for EdwardsPoint {\n fn add_assign(&mut self, rhs: &SubgroupPoint) {\n *self += rhs.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Sub<&SubgroupPoint> for &SubgroupPoint {\n type Output = SubgroupPoint;\n fn sub(self, other: &SubgroupPoint) -> SubgroupPoint {\n SubgroupPoint(self.0 - other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_variants!(\n LHS = SubgroupPoint,\n RHS = SubgroupPoint,\n Output = SubgroupPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl Sub<&SubgroupPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn sub(self, other: &SubgroupPoint) -> EdwardsPoint {\n self - other.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_variants!(\n LHS = EdwardsPoint,\n RHS = SubgroupPoint,\n Output = EdwardsPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl SubAssign<&SubgroupPoint> for SubgroupPoint {\n fn sub_assign(&mut self, rhs: &SubgroupPoint) {\n self.0 -= rhs.0;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl SubAssign<&SubgroupPoint> for EdwardsPoint {\n fn sub_assign(&mut self, rhs: &SubgroupPoint) {\n *self -= rhs.0;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Sum for SubgroupPoint\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n use group::Group;\n iter.fold(SubgroupPoint::identity(), |acc, item| acc + item.borrow())\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Mul<&Scalar> for &SubgroupPoint {\n type Output = SubgroupPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, scalar: &Scalar) -> SubgroupPoint {\n SubgroupPoint(self.0 * scalar)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_variants!(LHS = Scalar, RHS = SubgroupPoint, Output = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Mul<&SubgroupPoint> for &Scalar {\n type Output = SubgroupPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, point: &SubgroupPoint) -> SubgroupPoint {\n point * self\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_variants!(LHS = SubgroupPoint, RHS = Scalar, Output = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl MulAssign<&Scalar> for SubgroupPoint {\n fn mul_assign(&mut self, scalar: &Scalar) {\n self.0 *= scalar;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_assign_variants!(LHS = SubgroupPoint, RHS = Scalar);\n\n#[cfg(feature = \"group\")]\nimpl ConstantTimeEq for SubgroupPoint {\n fn ct_eq(&self, other: &SubgroupPoint) -> Choice {\n self.0.ct_eq(&other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl ConditionallySelectable for SubgroupPoint {\n fn conditional_select(a: &SubgroupPoint, b: &SubgroupPoint, choice: Choice) -> SubgroupPoint {\n SubgroupPoint(EdwardsPoint::conditional_select(&a.0, &b.0, choice))\n }\n}\n\n#[cfg(all(feature = \"group\", feature = \"zeroize\"))]\nimpl Zeroize for SubgroupPoint {\n fn zeroize(&mut self) {\n self.0.zeroize();\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl group::Group for SubgroupPoint {\n type Scalar = Scalar;\n\n fn random(mut rng: impl GroupRngCore) -> Self {\n use group::ff::Field;\n\n // This will almost never loop, but `Group::random` is documented as returning a\n // non-identity element.\n let s = loop {\n let s: Scalar = Field::random(&mut rng);\n if !s.is_zero_vartime() {\n break s;\n }\n };\n\n // This gives an element of the prime-order subgroup.\n Self::generator() * s\n }\n\n fn identity() -> Self {\n SubgroupPoint(Identity::identity())\n }\n\n fn generator() -> Self {\n SubgroupPoint(EdwardsPoint::generator())\n }\n\n fn is_identity(&self) -> Choice {\n self.0.ct_eq(&Identity::identity())\n }\n\n fn double(&self) -> Self {\n SubgroupPoint(self.0.double())\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl GroupEncoding for SubgroupPoint {\n type Repr = ::Repr;\n\n fn from_bytes(bytes: &Self::Repr) -> CtOption {\n EdwardsPoint::from_bytes(bytes).and_then(|p| p.into_subgroup())\n }\n\n fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption {\n EdwardsPoint::from_bytes_unchecked(bytes).and_then(|p| p.into_subgroup())\n }\n\n fn to_bytes(&self) -> Self::Repr {\n self.0.compress().to_bytes()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl PrimeGroup for SubgroupPoint {}\n\n#[cfg(feature = \"group\")]\nimpl CofactorGroup for EdwardsPoint {\n type Subgroup = SubgroupPoint;\n\n fn clear_cofactor(&self) -> Self::Subgroup {\n SubgroupPoint(self.mul_by_cofactor())\n }\n\n fn into_subgroup(self) -> CtOption {\n CtOption::new(SubgroupPoint(self), CofactorGroup::is_torsion_free(&self))\n }\n\n fn is_torsion_free(&self) -> Choice {\n (self * constants::BASEPOINT_ORDER).ct_eq(&Self::identity())\n }\n}\n\n// ------------------------------------------------------------------------\n// Tests\n// ------------------------------------------------------------------------\n\n#[cfg(test)]\nmod test {\n use super::*;\n use rand::Rng;\n\n #[cfg(feature = \"precomputed-tables\")]\n use crate::constants::ED25519_BASEPOINT_TABLE;\n\n /// X coordinate of the basepoint.\n /// = 15112221349535400772501151409588531511454012693041857206046113283949847762202\n static BASE_X_COORD_BYTES: [u8; 32] = [\n 0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c,\n 0x69, 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36,\n 0x69, 0x21,\n ];\n\n /// Compressed Edwards Y form of 2*basepoint.\n static BASE2_CMPRSSD: CompressedEdwardsY = CompressedEdwardsY([\n 0xc9, 0xa3, 0xf8, 0x6a, 0xae, 0x46, 0x5f, 0xe, 0x56, 0x51, 0x38, 0x64, 0x51, 0x0f, 0x39,\n 0x97, 0x56, 0x1f, 0xa2, 0xc9, 0xe8, 0x5e, 0xa2, 0x1d, 0xc2, 0x29, 0x23, 0x09, 0xf3, 0xcd,\n 0x60, 0x22,\n ]);\n\n /// Compressed Edwards Y form of 16*basepoint.\n static BASE16_CMPRSSD: CompressedEdwardsY = CompressedEdwardsY([\n 0xeb, 0x27, 0x67, 0xc1, 0x37, 0xab, 0x7a, 0xd8, 0x27, 0x9c, 0x07, 0x8e, 0xff, 0x11, 0x6a,\n 0xb0, 0x78, 0x6e, 0xad, 0x3a, 0x2e, 0x0f, 0x98, 0x9f, 0x72, 0xc3, 0x7f, 0x82, 0xf2, 0x96,\n 0x96, 0x70,\n ]);\n\n /// 4493907448824000747700850167940867464579944529806937181821189941592931634714\n pub static A_SCALAR: Scalar = Scalar {\n bytes: [\n 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x09,\n ],\n };\n\n /// 2506056684125797857694181776241676200180934651973138769173342316833279714961\n pub static B_SCALAR: Scalar = Scalar {\n bytes: [\n 0x91, 0x26, 0x7a, 0xcf, 0x25, 0xc2, 0x09, 0x1b, 0xa2, 0x17, 0x74, 0x7b, 0x66, 0xf0,\n 0xb3, 0x2e, 0x9d, 0xf2, 0xa5, 0x67, 0x41, 0xcf, 0xda, 0xc4, 0x56, 0xa7, 0xd4, 0xaa,\n 0xb8, 0x60, 0x8a, 0x05,\n ],\n };\n\n /// A_SCALAR * basepoint, computed with ed25519.py\n pub static A_TIMES_BASEPOINT: CompressedEdwardsY = CompressedEdwardsY([\n 0xea, 0x27, 0xe2, 0x60, 0x53, 0xdf, 0x1b, 0x59, 0x56, 0xf1, 0x4d, 0x5d, 0xec, 0x3c, 0x34,\n 0xc3, 0x84, 0xa2, 0x69, 0xb7, 0x4c, 0xc3, 0x80, 0x3e, 0xa8, 0xe2, 0xe7, 0xc9, 0x42, 0x5e,\n 0x40, 0xa5,\n ]);\n\n /// A_SCALAR * (A_TIMES_BASEPOINT) + B_SCALAR * BASEPOINT\n /// computed with ed25519.py\n static DOUBLE_SCALAR_MULT_RESULT: CompressedEdwardsY = CompressedEdwardsY([\n 0x7d, 0xfd, 0x6c, 0x45, 0xaf, 0x6d, 0x6e, 0x0e, 0xba, 0x20, 0x37, 0x1a, 0x23, 0x64, 0x59,\n 0xc4, 0xc0, 0x46, 0x83, 0x43, 0xde, 0x70, 0x4b, 0x85, 0x09, 0x6f, 0xfe, 0x35, 0x4f, 0x13,\n 0x2b, 0x42,\n ]);\n\n /// Test round-trip decompression for the basepoint.\n #[test]\n fn basepoint_decompression_compression() {\n let base_X = FieldElement::from_bytes(&BASE_X_COORD_BYTES);\n let bp = constants::ED25519_BASEPOINT_COMPRESSED\n .decompress()\n .expect(\"basepoint should decompress\");\n assert!(bp.is_valid());\n // Check that decompression actually gives the correct X coordinate\n assert_eq!(base_X, bp.X);\n assert_eq!(bp.compress(), constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test sign handling in decompression\n #[test]\n fn decompression_sign_handling() {\n // Manually set the high bit of the last byte to flip the sign\n let mut minus_basepoint_bytes = *constants::ED25519_BASEPOINT_COMPRESSED.as_bytes();\n minus_basepoint_bytes[31] |= 1 << 7;\n let minus_basepoint = CompressedEdwardsY(minus_basepoint_bytes)\n .decompress()\n .expect(\"minus basepoint should decompress\");\n // Test projective coordinates exactly since we know they should\n // only differ by a flipped sign.\n assert_eq!(minus_basepoint.X, -(&constants::ED25519_BASEPOINT_POINT.X));\n assert_eq!(minus_basepoint.Y, constants::ED25519_BASEPOINT_POINT.Y);\n assert_eq!(minus_basepoint.Z, constants::ED25519_BASEPOINT_POINT.Z);\n assert_eq!(minus_basepoint.T, -(&constants::ED25519_BASEPOINT_POINT.T));\n }\n\n /// Test that computing 1*basepoint gives the correct basepoint.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_mult_one_vs_basepoint() {\n let bp = ED25519_BASEPOINT_TABLE * &Scalar::ONE;\n let compressed = bp.compress();\n assert_eq!(compressed, constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test that `EdwardsBasepointTable::basepoint()` gives the correct basepoint.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_table_basepoint_function_correct() {\n let bp = ED25519_BASEPOINT_TABLE.basepoint();\n assert_eq!(bp.compress(), constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using basepoint + basepoint versus the 2*basepoint constant.\n #[test]\n fn basepoint_plus_basepoint_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_added = bp + bp;\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using the basepoint, basepoint2 constants\n #[test]\n fn basepoint_plus_basepoint_projective_niels_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_added = (&bp + &bp.as_projective_niels()).as_extended();\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using the basepoint, basepoint2 constants\n #[test]\n fn basepoint_plus_basepoint_affine_niels_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_affine_niels = bp.as_affine_niels();\n let bp_added = (&bp + &bp_affine_niels).as_extended();\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Check that equality of `EdwardsPoints` handles projective\n /// coordinates correctly.\n #[test]\n fn extended_point_equality_handles_scaling() {\n let mut two_bytes = [0u8; 32];\n two_bytes[0] = 2;\n let id1 = EdwardsPoint::identity();\n let id2 = EdwardsPoint {\n X: FieldElement::ZERO,\n Y: FieldElement::from_bytes(&two_bytes),\n Z: FieldElement::from_bytes(&two_bytes),\n T: FieldElement::ZERO,\n };\n assert!(bool::from(id1.ct_eq(&id2)));\n }\n\n /// Sanity check for conversion to precomputed points\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn to_affine_niels_clears_denominators() {\n // construct a point as aB so it has denominators (ie. Z != 1)\n let aB = ED25519_BASEPOINT_TABLE * &A_SCALAR;\n let aB_affine_niels = aB.as_affine_niels();\n let also_aB = (&EdwardsPoint::identity() + &aB_affine_niels).as_extended();\n assert_eq!(aB.compress(), also_aB.compress());\n }\n\n /// Test mul_base versus a known scalar multiple from ed25519.py\n #[test]\n fn basepoint_mult_vs_ed25519py() {\n let aB = EdwardsPoint::mul_base(&A_SCALAR);\n assert_eq!(aB.compress(), A_TIMES_BASEPOINT);\n }\n\n /// Test that multiplication by the basepoint order kills the basepoint\n #[test]\n fn basepoint_mult_by_basepoint_order() {\n let should_be_id = EdwardsPoint::mul_base(&constants::BASEPOINT_ORDER);\n assert!(should_be_id.is_identity());\n }\n\n /// Test precomputed basepoint mult\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn test_precomputed_basepoint_mult() {\n let aB_1 = ED25519_BASEPOINT_TABLE * &A_SCALAR;\n let aB_2 = constants::ED25519_BASEPOINT_POINT * A_SCALAR;\n assert_eq!(aB_1.compress(), aB_2.compress());\n }\n\n /// Test scalar_mul versus a known scalar multiple from ed25519.py\n #[test]\n fn scalar_mul_vs_ed25519py() {\n let aB = constants::ED25519_BASEPOINT_POINT * A_SCALAR;\n assert_eq!(aB.compress(), A_TIMES_BASEPOINT);\n }\n\n /// Test basepoint.double() versus the 2*basepoint constant.\n #[test]\n fn basepoint_double_vs_basepoint2() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT.double().compress(),\n BASE2_CMPRSSD\n );\n }\n\n /// Test that computing 2*basepoint is the same as basepoint.double()\n #[test]\n fn basepoint_mult_two_vs_basepoint2() {\n let two = Scalar::from(2u64);\n let bp2 = EdwardsPoint::mul_base(&two);\n assert_eq!(bp2.compress(), BASE2_CMPRSSD);\n }\n\n /// Test that all the basepoint table types compute the same results.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_tables() {\n let P = &constants::ED25519_BASEPOINT_POINT;\n let a = A_SCALAR;\n\n let table_radix16 = EdwardsBasepointTableRadix16::create(P);\n let table_radix32 = EdwardsBasepointTableRadix32::create(P);\n let table_radix64 = EdwardsBasepointTableRadix64::create(P);\n let table_radix128 = EdwardsBasepointTableRadix128::create(P);\n let table_radix256 = EdwardsBasepointTableRadix256::create(P);\n\n let aP = (ED25519_BASEPOINT_TABLE * &a).compress();\n let aP16 = (&table_radix16 * &a).compress();\n let aP32 = (&table_radix32 * &a).compress();\n let aP64 = (&table_radix64 * &a).compress();\n let aP128 = (&table_radix128 * &a).compress();\n let aP256 = (&table_radix256 * &a).compress();\n\n assert_eq!(aP, aP16);\n assert_eq!(aP16, aP32);\n assert_eq!(aP32, aP64);\n assert_eq!(aP64, aP128);\n assert_eq!(aP128, aP256);\n }\n\n /// Check unreduced scalar multiplication by the basepoint tables is the same no matter what\n /// radix the table is.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_tables_unreduced_scalar() {\n let P = &constants::ED25519_BASEPOINT_POINT;\n let a = crate::scalar::test::LARGEST_UNREDUCED_SCALAR;\n\n let table_radix16 = EdwardsBasepointTableRadix16::create(P);\n let table_radix32 = EdwardsBasepointTableRadix32::create(P);\n let table_radix64 = EdwardsBasepointTableRadix64::create(P);\n let table_radix128 = EdwardsBasepointTableRadix128::create(P);\n let table_radix256 = EdwardsBasepointTableRadix256::create(P);\n\n let aP = (ED25519_BASEPOINT_TABLE * &a).compress();\n let aP16 = (&table_radix16 * &a).compress();\n let aP32 = (&table_radix32 * &a).compress();\n let aP64 = (&table_radix64 * &a).compress();\n let aP128 = (&table_radix128 * &a).compress();\n let aP256 = (&table_radix256 * &a).compress();\n\n assert_eq!(aP, aP16);\n assert_eq!(aP16, aP32);\n assert_eq!(aP32, aP64);\n assert_eq!(aP64, aP128);\n assert_eq!(aP128, aP256);\n }\n\n /// Check that converting to projective and then back to extended round-trips.\n #[test]\n fn basepoint_projective_extended_round_trip() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT\n .as_projective()\n .as_extended()\n .compress(),\n constants::ED25519_BASEPOINT_COMPRESSED\n );\n }\n\n /// Test computing 16*basepoint vs mul_by_pow_2(4)\n #[test]\n fn basepoint16_vs_mul_by_pow_2_4() {\n let bp16 = constants::ED25519_BASEPOINT_POINT.mul_by_pow_2(4);\n assert_eq!(bp16.compress(), BASE16_CMPRSSD);\n }\n\n /// Check that mul_base_clamped and mul_clamped agree\n #[test]\n fn mul_base_clamped() {\n let mut csprng = rand::thread_rng();\n\n // Make a random curve point in the curve. Give it torsion to make things interesting.\n #[cfg(feature = \"precomputed-tables\")]\n let random_point = {\n let mut b = [0u8; 32];\n csprng.fill(&mut b);\n EdwardsPoint::mul_base_clamped(b) + constants::EIGHT_TORSION[1]\n };\n // Make a basepoint table from the random point. We'll use this with mul_base_clamped\n #[cfg(feature = \"precomputed-tables\")]\n let random_table = EdwardsBasepointTableRadix256::create(&random_point);\n\n // Now test scalar mult. agreement on the default basepoint as well as random_point\n\n // Test that mul_base_clamped and mul_clamped agree on a large integer. Even after\n // clamping, this integer is not reduced mod l.\n let a_bytes = [0xff; 32];\n assert_eq!(\n EdwardsPoint::mul_base_clamped(a_bytes),\n constants::ED25519_BASEPOINT_POINT.mul_clamped(a_bytes)\n );\n #[cfg(feature = \"precomputed-tables\")]\n assert_eq!(\n random_table.mul_base_clamped(a_bytes),\n random_point.mul_clamped(a_bytes)\n );\n\n // Test agreement on random integers\n for _ in 0..100 {\n // This will be reduced mod l with probability l / 2^256 ≈ 6.25%\n let mut a_bytes = [0u8; 32];\n csprng.fill(&mut a_bytes);\n\n assert_eq!(\n EdwardsPoint::mul_base_clamped(a_bytes),\n constants::ED25519_BASEPOINT_POINT.mul_clamped(a_bytes)\n );\n #[cfg(feature = \"precomputed-tables\")]\n assert_eq!(\n random_table.mul_base_clamped(a_bytes),\n random_point.mul_clamped(a_bytes)\n );\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_sum() {\n // Test that sum works for non-empty iterators\n let BASE = constants::ED25519_BASEPOINT_POINT;\n\n let s1 = Scalar::from(999u64);\n let P1 = BASE * s1;\n\n let s2 = Scalar::from(333u64);\n let P2 = BASE * s2;\n\n let arr = [P1, P2];\n let sum: EdwardsPoint = arr.iter().sum();\n\n assert_eq!(sum, P1 + P2);\n\n // Test that sum works for the empty iterator\n let empty_array: [EdwardsPoint; 0] = [];\n let sum: EdwardsPoint = empty_array.iter().sum();\n\n assert_eq!(sum, EdwardsPoint::identity());\n\n // Test that sum works on owning iterators\n let s = Scalar::from(2u64);\n let mapped = arr.iter().map(|x| x * s);\n let sum: EdwardsPoint = mapped.sum();\n\n assert_eq!(sum, P1 * s + P2 * s);\n }\n\n /// Test that the conditional assignment trait works for AffineNielsPoints.\n #[test]\n fn conditional_assign_for_affine_niels_point() {\n let id = AffineNielsPoint::identity();\n let mut p1 = AffineNielsPoint::identity();\n let bp = constants::ED25519_BASEPOINT_POINT.as_affine_niels();\n\n p1.conditional_assign(&bp, Choice::from(0));\n assert_eq!(p1, id);\n p1.conditional_assign(&bp, Choice::from(1));\n assert_eq!(p1, bp);\n }\n\n #[test]\n fn is_small_order() {\n // The basepoint has large prime order\n assert!(!constants::ED25519_BASEPOINT_POINT.is_small_order());\n // constants::EIGHT_TORSION has all points of small order.\n for torsion_point in &constants::EIGHT_TORSION {\n assert!(torsion_point.is_small_order());\n }\n }\n\n #[test]\n fn compressed_identity() {\n assert_eq!(\n EdwardsPoint::identity().compress(),\n CompressedEdwardsY::identity()\n );\n\n #[cfg(feature = \"alloc\")]\n {\n let compressed = EdwardsPoint::compress_batch(&[EdwardsPoint::identity()]);\n assert_eq!(&compressed, &[CompressedEdwardsY::identity()]);\n }\n }\n\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n #[test]\n fn compress_batch() {\n let mut rng = rand::thread_rng();\n\n // TODO(tarcieri): proptests?\n // Make some points deterministically then randomly\n let mut points = (1u64..16)\n .map(|n| constants::ED25519_BASEPOINT_POINT * Scalar::from(n))\n .collect::>();\n points.extend(core::iter::repeat_with(|| EdwardsPoint::random(&mut rng)).take(100));\n let compressed = EdwardsPoint::compress_batch(&points);\n\n // Check that the batch-compressed points match the individually compressed ones\n for (point, compressed) in points.iter().zip(&compressed) {\n assert_eq!(&point.compress(), compressed);\n }\n }\n\n #[test]\n fn is_identity() {\n assert!(EdwardsPoint::identity().is_identity());\n assert!(!constants::ED25519_BASEPOINT_POINT.is_identity());\n }\n\n /// Rust's debug builds have overflow and underflow trapping,\n /// and enable `debug_assert!()`. This performs many scalar\n /// multiplications to attempt to trigger possible overflows etc.\n ///\n /// For instance, the `u64` `Mul` implementation for\n /// `FieldElements` requires the input `Limb`s to be bounded by\n /// 2^54, but we cannot enforce this dynamically at runtime, or\n /// statically at compile time (until Rust gets type-level\n /// integers, at which point we can encode \"bits of headroom\" into\n /// the type system and prove correctness).\n #[test]\n fn monte_carlo_overflow_underflow_debug_assert_test() {\n let mut P = constants::ED25519_BASEPOINT_POINT;\n // N.B. each scalar_mul does 1407 field mults, 1024 field squarings,\n // so this does ~ 1M of each operation.\n for _ in 0..1_000 {\n P *= &A_SCALAR;\n }\n }\n\n #[test]\n fn scalarmult_extended_point_works_both_ways() {\n let G: EdwardsPoint = constants::ED25519_BASEPOINT_POINT;\n let s: Scalar = A_SCALAR;\n\n let P1 = G * s;\n let P2 = s * G;\n\n assert!(P1.compress().to_bytes() == P2.compress().to_bytes());\n }\n\n // A single iteration of a consistency check for MSM.\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_iter(n: usize) {\n let mut rng = rand::thread_rng();\n\n // Construct random coefficients x0, ..., x_{n-1},\n // followed by some extra hardcoded ones.\n let xs = (0..n).map(|_| Scalar::random(&mut rng)).collect::>();\n let check = xs.iter().map(|xi| xi * xi).sum::();\n\n // Construct points G_i = x_i * B\n let Gs = xs.iter().map(EdwardsPoint::mul_base).collect::>();\n\n // Compute H1 = (consttime)\n let H1 = EdwardsPoint::multiscalar_mul(&xs, &Gs);\n // Compute H2 = (vartime)\n let H2 = EdwardsPoint::vartime_multiscalar_mul(&xs, &Gs);\n // Compute H3 = = sum(xi^2) * B\n let H3 = EdwardsPoint::mul_base(&check);\n\n assert_eq!(H1, H3);\n assert_eq!(H2, H3);\n }\n\n // Use different multiscalar sizes to hit different internal\n // parameters.\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_100() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(100);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_250() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(250);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_500() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(500);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_1000() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(1000);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn batch_to_montgomery() {\n let mut rng = rand::thread_rng();\n\n let scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let points = scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n\n let single_monts = points\n .iter()\n .map(EdwardsPoint::to_montgomery)\n .collect::>();\n\n for i in [0, 1, 2, 3, 10, 50, 128] {\n let invs = EdwardsPoint::to_montgomery_batch(&points[..i]);\n assert_eq!(&invs, &single_monts[..i]);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn vartime_precomputed_vs_nonprecomputed_multiscalar() {\n let mut rng = rand::thread_rng();\n\n let static_scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let dynamic_scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let check_scalar: Scalar = static_scalars\n .iter()\n .chain(dynamic_scalars.iter())\n .map(|s| s * s)\n .sum();\n\n let static_points = static_scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n let dynamic_points = dynamic_scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n\n let precomputation = VartimeEdwardsPrecomputation::new(static_points.iter());\n\n assert_eq!(precomputation.len(), 128);\n assert!(!precomputation.is_empty());\n\n let P = precomputation.vartime_mixed_multiscalar_mul(\n &static_scalars,\n &dynamic_scalars,\n &dynamic_points,\n );\n\n use crate::traits::VartimeMultiscalarMul;\n let Q = EdwardsPoint::vartime_multiscalar_mul(\n static_scalars.iter().chain(dynamic_scalars.iter()),\n static_points.iter().chain(dynamic_points.iter()),\n );\n\n let R = EdwardsPoint::mul_base(&check_scalar);\n\n assert_eq!(P.compress(), R.compress());\n assert_eq!(Q.compress(), R.compress());\n }\n\n mod vartime {\n use super::super::*;\n use super::{A_SCALAR, A_TIMES_BASEPOINT, B_SCALAR, DOUBLE_SCALAR_MULT_RESULT};\n\n /// Test double_scalar_mul_vartime vs ed25519.py\n #[test]\n fn double_scalar_mul_basepoint_vs_ed25519py() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result =\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&A_SCALAR, &A, &B_SCALAR);\n assert_eq!(result.compress(), DOUBLE_SCALAR_MULT_RESULT);\n }\n\n #[test]\n fn triple_scalar_mul_basepoint_accepts_full_width_scalars() {\n let mut a1_bytes = [0u8; 32];\n a1_bytes[0] = 7;\n a1_bytes[16] = 1;\n let a1 = Scalar::from_canonical_bytes(a1_bytes).unwrap();\n\n let mut a2_bytes = [0u8; 32];\n a2_bytes[0] = 11;\n a2_bytes[24] = 1;\n let a2 = Scalar::from_canonical_bytes(a2_bytes).unwrap();\n\n let b = B_SCALAR;\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(17u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(19u64);\n\n let result = EdwardsPoint::vartime_triple_scalar_mul_basepoint(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1) + (a2 * A2) + EdwardsPoint::mul_base(&b);\n\n assert_eq!(result, expected);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn multiscalar_mul_vs_ed25519py() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result = EdwardsPoint::vartime_multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n assert_eq!(result.compress(), DOUBLE_SCALAR_MULT_RESULT);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn multiscalar_mul_vartime_vs_consttime() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result_vartime = EdwardsPoint::vartime_multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n let result_consttime = EdwardsPoint::multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n\n assert_eq!(result_vartime.compress(), result_consttime.compress());\n }\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_basepoint_roundtrip() {\n use bincode;\n\n let encoded = bincode::serialize(&constants::ED25519_BASEPOINT_POINT).unwrap();\n let enc_compressed = bincode::serialize(&constants::ED25519_BASEPOINT_COMPRESSED).unwrap();\n assert_eq!(encoded, enc_compressed);\n\n // Check that the encoding is 32 bytes exactly\n assert_eq!(encoded.len(), 32);\n\n let dec_uncompressed: EdwardsPoint = bincode::deserialize(&encoded).unwrap();\n let dec_compressed: CompressedEdwardsY = bincode::deserialize(&encoded).unwrap();\n\n assert_eq!(dec_uncompressed, constants::ED25519_BASEPOINT_POINT);\n assert_eq!(dec_compressed, constants::ED25519_BASEPOINT_COMPRESSED);\n\n // Check that the encoding itself matches the usual one\n let raw_bytes = constants::ED25519_BASEPOINT_COMPRESSED.as_bytes();\n let bp: EdwardsPoint = bincode::deserialize(raw_bytes).unwrap();\n assert_eq!(bp, constants::ED25519_BASEPOINT_POINT);\n }\n\n // Hash-to-curve test vectors from\n // https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.2\n // These are of the form (input_msg, output_x, output_y)\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n const RFC_ENCODE_TO_CURVE_KAT: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"1ff2b70ecf862799e11b7ae744e3489aa058ce805dd323a936375a84695e76da\",\n \"222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab2d687f0f9b\",\n ),\n (\n b\"abc\",\n \"5f13cc69c891d86927eb37bd4afc6672360007c63f68a33ab423a3aa040fd2a8\",\n \"67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1f5c827fa42\",\n ),\n (\n b\"abcdef0123456789\",\n \"1dd2fefce934ecfd7aae6ec998de088d7dd03316aa1847198aecf699ba6613f1\",\n \"2f8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a0a8e1a86fb\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"35fbdc5143e8a97afd3096f2b843e07df72e15bfca2eaf6879bf97c5d3362f73\",\n \"2af6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91f57a603450\",\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"6e5e1f37e99345887fc12111575fc1c3e36df4b289b8759d23af14d774b66bff\",\n \"2c90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a4245891a37\"\n )\n ];\n\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn hex_str_to_fe(hex_str: &str) -> FieldElement {\n let mut bytes = hex::decode(hex_str).expect(\"RFC test vector hex must decode\");\n bytes.reverse();\n let bytes: [u8; 32] = bytes\n .try_into()\n .expect(\"RFC test vector field element must be 32 bytes\");\n FieldElement::from_bytes(&bytes)\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn elligator_encode_to_curve_test_vectors() {\n let dst = b\"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_\";\n for (index, vector) in RFC_ENCODE_TO_CURVE_KAT.iter().enumerate() {\n let input = vector.0;\n\n let expected_output = {\n let x = hex_str_to_fe(vector.1);\n let y = hex_str_to_fe(vector.2);\n AffinePoint { x, y }.to_edwards()\n };\n\n let computed = EdwardsPoint::encode_to_curve::(&[input], &[dst]);\n assert_eq!(computed, expected_output, \"Failed in test {}\", index);\n }\n }\n\n // Hash-to-curve test vectors from\n // https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.1\n // These are of the form (input_msg, output_x, output_y)\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n const RFC_HASH_TO_CURVE_KAT: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"3c3da6925a3c3c268448dcabb47ccde5439559d9599646a8260e47b1e4822fc6\",\n \"09a6c8561a0b22bef63124c588ce4c62ea83a3c899763af26d795302e115dc21\",\n ),\n\n (\n b\"abc\",\n \"608040b42285cc0d72cbb3985c6b04c935370c7361f4b7fbdb1ae7f8c1a8ecad\",\n \"1a8395b88338f22e435bbd301183e7f20a5f9de643f11882fb237f88268a5531\",\n ),\n\n (\n b\"abcdef0123456789\",\n \"6d7fabf47a2dc03fe7d47f7dddd21082c5fb8f86743cd020f3fb147d57161472\",\n \"53060a3d140e7fbcda641ed3cf42c88a75411e648a1add71217f70ea8ec561a6\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"5fb0b92acedd16f3bcb0ef83f5c7b7a9466b5f1e0d8d217421878ea3686f8524\",\n \"2eca15e355fcfa39d2982f67ddb0eea138e2994f5956ed37b7f72eea5e89d2f7\",\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"0efcfde5898a839b00997fbe40d2ebe950bc81181afbd5cd6b9618aa336c1e8c\",\n \"6dc2fc04f266c5c27f236a80b14f92ccd051ef1ff027f26a07f8c0f327d8f995\"\n )\n ];\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn elligator_hash_to_curve_test_vectors() {\n let dst = b\"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_\";\n for (index, vector) in RFC_HASH_TO_CURVE_KAT.iter().enumerate() {\n let input = vector.0;\n\n let expected_output = {\n let x = hex_str_to_fe(vector.1);\n let y = hex_str_to_fe(vector.2);\n\n AffinePoint { x, y }.to_edwards()\n };\n\n let computed = EdwardsPoint::hash_to_curve::(&[input], &[dst]);\n\n assert_eq!(expected_output, computed, \"Failed in test {}\", index);\n }\n }\n}\n"},{"id":12,"name":{"Local":"curve25519/solana-ed25519/src/macros.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis agora lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Internal macros.\n\n/// Define borrow and non-borrow variants of `Add`.\nmacro_rules! define_add_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Add<&'b $rhs> for $lhs {\n type Output = $out;\n fn add(self, rhs: &'b $rhs) -> $out {\n &self + rhs\n }\n }\n\n impl<'a> Add<$rhs> for &'a $lhs {\n type Output = $out;\n fn add(self, rhs: $rhs) -> $out {\n self + &rhs\n }\n }\n\n impl Add<$rhs> for $lhs {\n type Output = $out;\n fn add(self, rhs: $rhs) -> $out {\n &self + &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `AddAssign`.\nmacro_rules! define_add_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl AddAssign<$rhs> for $lhs {\n fn add_assign(&mut self, rhs: $rhs) {\n *self += &rhs;\n }\n }\n };\n}\n\n/// Define borrow and non-borrow variants of `Sub`.\nmacro_rules! define_sub_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Sub<&'b $rhs> for $lhs {\n type Output = $out;\n fn sub(self, rhs: &'b $rhs) -> $out {\n &self - rhs\n }\n }\n\n impl<'a> Sub<$rhs> for &'a $lhs {\n type Output = $out;\n fn sub(self, rhs: $rhs) -> $out {\n self - &rhs\n }\n }\n\n impl Sub<$rhs> for $lhs {\n type Output = $out;\n fn sub(self, rhs: $rhs) -> $out {\n &self - &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `SubAssign`.\nmacro_rules! define_sub_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl SubAssign<$rhs> for $lhs {\n fn sub_assign(&mut self, rhs: $rhs) {\n *self -= &rhs;\n }\n }\n };\n}\n\n/// Define borrow and non-borrow variants of `Mul`.\nmacro_rules! define_mul_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Mul<&'b $rhs> for $lhs {\n type Output = $out;\n fn mul(self, rhs: &'b $rhs) -> $out {\n &self * rhs\n }\n }\n\n impl<'a> Mul<$rhs> for &'a $lhs {\n type Output = $out;\n fn mul(self, rhs: $rhs) -> $out {\n self * &rhs\n }\n }\n\n impl Mul<$rhs> for $lhs {\n type Output = $out;\n fn mul(self, rhs: $rhs) -> $out {\n &self * &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `MulAssign`.\nmacro_rules! define_mul_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl MulAssign<$rhs> for $lhs {\n fn mul_assign(&mut self, rhs: $rhs) {\n *self *= &rhs;\n }\n }\n };\n}\n"},{"id":13,"name":{"Local":"/rustc/library/core/src/ops/range.rs"},"crate_name":"core","contents":null},{"id":14,"name":{"Local":"/rustc/library/core/src/ops/mod.rs"},"crate_name":"core","contents":null},{"id":15,"name":{"Local":"/rustc/library/core/src/lib.rs"},"crate_name":"core","contents":null},{"id":16,"name":{"Local":"/rustc/library/core/src/option.rs"},"crate_name":"core","contents":null},{"id":17,"name":{"Local":"/rustc/library/core/src/iter/traits/collect.rs"},"crate_name":"core","contents":null},{"id":18,"name":{"Local":"/rustc/library/core/src/iter/traits/mod.rs"},"crate_name":"core","contents":null},{"id":19,"name":{"Local":"/rustc/library/core/src/iter/mod.rs"},"crate_name":"core","contents":null},{"id":20,"name":{"Local":"/rustc/library/core/src/iter/traits/iterator.rs"},"crate_name":"core","contents":null},{"id":21,"name":{"Local":"/rustc/library/core/src/iter/range.rs"},"crate_name":"core","contents":null},{"id":22,"name":{"Local":"/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs"},"crate_name":"subtle","contents":null},{"id":23,"name":{"Local":"/rustc/library/core/src/num/uint_macros.rs"},"crate_name":"core","contents":null},{"id":24,"name":{"Local":"/rustc/library/core/src/num/mod.rs"},"crate_name":"core","contents":null},{"id":25,"name":{"Local":"/rustc/library/core/src/macros/mod.rs"},"crate_name":"core","contents":null},{"id":26,"name":{"Local":"/rustc/library/core/src/panicking.rs"},"crate_name":"core","contents":null},{"id":27,"name":{"Local":"/rustc/library/core/src/fmt/mod.rs"},"crate_name":"core","contents":null},{"id":28,"name":{"Local":"/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/ed25519-2.2.3/src/lib.rs"},"crate_name":"ed25519","contents":null},{"id":29,"name":{"Local":"/rustc/library/core/src/result.rs"},"crate_name":"core","contents":null},{"id":30,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs/error.rs"},"crate_name":"curve25519","contents":"use core::fmt;\n\n/// An error related to Ed25519 signatures.\n#[derive(Debug, Copy, Clone, Eq, PartialEq)]\npub enum Error {\n /// The encoding of a secret key was malformed.\n MalformedSecretKey,\n /// The encoding of a public key was malformed.\n MalformedPublicKey,\n /// Signature verification failed.\n InvalidSignature,\n /// A byte slice of the wrong length was supplied during parsing.\n InvalidSliceLength,\n}\n\nimpl fmt::Display for Error {\n fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {\n let msg = match self {\n Self::MalformedSecretKey => \"Malformed secret key encoding.\",\n Self::MalformedPublicKey => \"Malformed public key encoding.\",\n Self::InvalidSignature => \"Invalid signature.\",\n Self::InvalidSliceLength => \"Invalid length when parsing byte slice.\",\n };\n\n msg.fmt(f)\n }\n}\n\n#[cfg(feature = \"std\")]\nimpl std::error::Error for Error {}\n"},{"id":31,"name":{"Local":"/rustc/library/core/src/ops/control_flow.rs"},"crate_name":"core","contents":null},{"id":32,"name":{"Local":"/rustc/library/core/src/convert/mod.rs"},"crate_name":"core","contents":null},{"id":33,"name":{"Local":"/rustc/library/core/src/cmp.rs"},"crate_name":"core","contents":null},{"id":34,"name":{"Local":"/rustc/library/core/src/default.rs"},"crate_name":"core","contents":null},{"id":35,"name":{"Local":"/rustc/library/core/src/marker.rs"},"crate_name":"core","contents":null},{"id":36,"name":{"Local":"/rustc/library/core/src/clone.rs"},"crate_name":"core","contents":null},{"id":37,"name":{"Local":"/rustc/library/core/src/ops/arith.rs"},"crate_name":"core","contents":null},{"id":38,"name":{"Local":"curve25519/solana-ed25519/src/traits.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek / curve25519-sol.\n// Original curve25519-dalek code:\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// Additions for curve25519-sol:\n// Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE for licensing information.\n//\n// Original authors:\n// - isis agora lovecruft \n// - Henry de Valence \n//\n// Additions in curve25519-sol:\n// - `HEEADecomposition` trait for half-extended Euclidean scalar decomposition.\n\n//! Module for common traits.\n\n#![allow(non_snake_case)]\n\nuse core::borrow::Borrow;\n\nuse crate::scalar::{Scalar, clamp_integer};\nuse subtle::ConstantTimeEq;\n\n// ------------------------------------------------------------------------\n// Public Traits\n// ------------------------------------------------------------------------\n\n/// Trait for getting the identity element of a point type.\npub trait Identity {\n /// Returns the identity element of the curve.\n /// Can be used as a constructor.\n fn identity() -> Self;\n}\n\n/// Trait for testing if a curve point is equivalent to the identity point.\npub trait IsIdentity {\n /// Return true if this element is the identity element of the curve.\n fn is_identity(&self) -> bool;\n}\n\n/// Implement generic identity equality testing for a point representations\n/// which have constant-time equality testing and a defined identity\n/// constructor.\nimpl IsIdentity for T\nwhere\n T: ConstantTimeEq + Identity,\n{\n fn is_identity(&self) -> bool {\n self.ct_eq(&T::identity()).into()\n }\n}\n\n/// A precomputed table of basepoints, for optimising scalar multiplications.\npub trait BasepointTable {\n /// The type of point contained within this table.\n type Point;\n\n /// Generate a new precomputed basepoint table from the given basepoint.\n fn create(basepoint: &Self::Point) -> Self;\n\n /// Retrieve the original basepoint from this table.\n fn basepoint(&self) -> Self::Point;\n\n /// Multiply a `scalar` by this precomputed basepoint table, in constant time.\n fn mul_base(&self, scalar: &Scalar) -> Self::Point;\n\n /// Multiply `clamp_integer(bytes)` by this precomputed basepoint table, in constant time. For\n /// a description of clamping, see [`clamp_integer`].\n fn mul_base_clamped(&self, bytes: [u8; 32]) -> Self::Point {\n // Basepoint multiplication is defined for all values of `bytes` up to and including\n // 2^255 - 1. The limit comes from the fact that scalar.as_radix_16() doesn't work for\n // most scalars larger than 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n self.mul_base(&s)\n }\n}\n\n/// A trait for constant-time multiscalar multiplication without precomputation.\npub trait MultiscalarMul {\n /// The type of point being multiplied, e.g., `RistrettoPoint`.\n type Point;\n\n /// Given an iterator of (possibly secret) scalars and an iterator of\n /// public points, compute\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n.\n /// $$\n ///\n /// It is an error to call this function with two iterators of different lengths.\n ///\n /// # Examples\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n ///\n /// ```\n /// # #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::MultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n ///\n /// // A1 = a*P + b*Q + c*R\n /// let abc = [a,b,c];\n /// let A1 = RistrettoPoint::multiscalar_mul(&abc, &[P,Q,R]);\n /// // Note: (&abc).into_iter(): Iterator\n ///\n /// // A2 = (-a)*P + (-b)*Q + (-c)*R\n /// let minus_abc = abc.iter().map(|x| -x);\n /// let A2 = RistrettoPoint::multiscalar_mul(minus_abc, &[P,Q,R]);\n /// // Note: minus_abc.into_iter(): Iterator\n ///\n /// assert_eq!(A1.compress(), (-A2).compress());\n /// # }\n /// ```\n fn multiscalar_mul(scalars: I, points: J) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow;\n}\n\n/// A trait for variable-time multiscalar multiplication without precomputation.\npub trait VartimeMultiscalarMul {\n /// The type of point being multiplied, e.g., `RistrettoPoint`.\n type Point;\n\n /// Given an iterator of public scalars and an iterator of\n /// `Option`s of points, compute either `Some(Q)`, where\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n,\n /// $$\n /// if all points were `Some(P_i)`, or else return `None`.\n ///\n /// This function is particularly useful when verifying statements\n /// involving compressed points. Accepting `Option` allows\n /// inlining point decompression into the multiscalar call,\n /// avoiding the need for temporary buffers.\n ///\n /// ```\n /// #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::VartimeMultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n /// let abc = [a,b,c];\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n /// let PQR = [P, Q, R];\n ///\n /// let compressed = [P.compress(), Q.compress(), R.compress()];\n ///\n /// // Now we can compute A1 = a*P + b*Q + c*R using P, Q, R:\n /// let A1 = RistrettoPoint::vartime_multiscalar_mul(&abc, &PQR);\n ///\n /// // Or using the compressed points:\n /// let A2 = RistrettoPoint::optional_multiscalar_mul(\n /// &abc,\n /// compressed.iter().map(|pt| pt.decompress()),\n /// );\n ///\n /// assert_eq!(A2, Some(A1));\n ///\n /// // It's also possible to mix compressed and uncompressed points:\n /// let A3 = RistrettoPoint::optional_multiscalar_mul(\n /// abc.iter()\n /// .chain(abc.iter()),\n /// compressed.iter().map(|pt| pt.decompress())\n /// .chain(PQR.iter().map(|&pt| Some(pt))),\n /// );\n ///\n /// assert_eq!(A3, Some(A1+A1));\n /// # }\n /// ```\n fn optional_multiscalar_mul(scalars: I, points: J) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator>;\n\n /// Given an iterator of public scalars and an iterator of\n /// public points, compute\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n,\n /// $$\n /// using variable-time operations.\n ///\n /// It is an error to call this function with two iterators of different lengths.\n ///\n /// # Examples\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n ///\n /// ```\n /// #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::VartimeMultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n ///\n /// // A1 = a*P + b*Q + c*R\n /// let abc = [a,b,c];\n /// let A1 = RistrettoPoint::vartime_multiscalar_mul(&abc, &[P,Q,R]);\n /// // Note: (&abc).into_iter(): Iterator\n ///\n /// // A2 = (-a)*P + (-b)*Q + (-c)*R\n /// let minus_abc = abc.iter().map(|x| -x);\n /// let A2 = RistrettoPoint::vartime_multiscalar_mul(minus_abc, &[P,Q,R]);\n /// // Note: minus_abc.into_iter(): Iterator\n ///\n /// assert_eq!(A1.compress(), (-A2).compress());\n /// # }\n /// ```\n fn vartime_multiscalar_mul(scalars: I, points: J) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n Self::Point: Clone,\n {\n Self::optional_multiscalar_mul(\n scalars,\n points.into_iter().map(|P| Some(P.borrow().clone())),\n )\n .expect(\"should return some point\")\n }\n}\n\n/// A trait for variable-time multiscalar multiplication with precomputation.\n///\n/// A general multiscalar multiplication with precomputation can be written as\n/// $$\n/// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n/// $$\n/// where the \\\\(B_i\\\\) are *static* points, for which precomputation\n/// is possible, and the \\\\(A_j\\\\) are *dynamic* points, for which\n/// precomputation is not possible.\n///\n/// This trait has three methods for performing this computation:\n///\n/// * [`Self::vartime_multiscalar_mul`], which handles the special case where\n/// \\\\(n = 0\\\\) and there are no dynamic points;\n///\n/// * [`Self::vartime_mixed_multiscalar_mul`], which takes the dynamic points as\n/// already-validated `Point`s and is infallible;\n///\n/// * [`Self::optional_mixed_multiscalar_mul`], which takes the dynamic points\n/// as `Option`s and returns an `Option`, allowing decompression\n/// to be composed into the input iterators.\n///\n/// All methods require that the lengths of the input iterators be\n/// known, as if they were `ExactSizeIterator`s. (It\n/// does not require `ExactSizeIterator` only because that trait is\n/// broken).\npub trait VartimePrecomputedMultiscalarMul: Sized {\n /// The type of point to be multiplied, e.g., `RistrettoPoint`.\n type Point: Clone;\n\n /// Given the static points \\\\( B_i \\\\), perform precomputation\n /// and return the precomputation data.\n fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: Borrow;\n\n /// Return the number of static points in the precomputation.\n fn len(&self) -> usize;\n\n /// Determine if the precomputation is empty.\n fn is_empty(&self) -> bool;\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), compute\n /// $$\n /// Q = b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\).\n ///\n /// The trait bound aims for maximum flexibility: the input must\n /// be convertible to iterators (`I: IntoIter`), and the\n /// iterator's items must be `Borrow`, to allow iterators\n /// returning either `Scalar`s or `&Scalar`s.\n fn vartime_multiscalar_mul(&self, static_scalars: I) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n {\n use core::iter;\n\n Self::vartime_mixed_multiscalar_mul(\n self,\n static_scalars,\n iter::empty::(),\n iter::empty::(),\n )\n }\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), `dynamic_scalars`, an iterator of public scalars\n /// \\\\(a_i\\\\), and `dynamic_points`, an iterator of points\n /// \\\\(A_i\\\\), compute\n /// $$\n /// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\), or if \\\\(a_i\\\\) and \\\\(A_i\\\\) do not have\n /// the same length.\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n fn vartime_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator,\n K::Item: Borrow,\n {\n Self::optional_mixed_multiscalar_mul(\n self,\n static_scalars,\n dynamic_scalars,\n dynamic_points.into_iter().map(|P| Some(P.borrow().clone())),\n )\n .expect(\"should return some point\")\n }\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), `dynamic_scalars`, an iterator of public scalars\n /// \\\\(a_i\\\\), and `dynamic_points`, an iterator of points\n /// \\\\(A_i\\\\), compute\n /// $$\n /// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// If any of the dynamic points were `None`, return `None`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\), or if \\\\(a_i\\\\) and \\\\(A_i\\\\) do not have\n /// the same length.\n ///\n /// This function is particularly useful when verifying statements\n /// involving compressed points. Accepting `Option` allows\n /// inlining point decompression into the multiscalar call,\n /// avoiding the need for temporary buffers.\n fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator>;\n}\n\n/// A trait for decomposing scalars for the HEEA method.\n///\n/// The HEEA method is used by EdDSA verification to replace multiplication by\n/// a full-width challenge scalar with multiplications by two shorter scalars.\n/// For a challenge scalar `h`, the decomposition returns non-negative scalars\n/// `rho` and `tau` that satisfy either `rho == tau * h (mod ell)` or\n/// `rho == -tau * h (mod ell)`.\n///\n/// This trait provides a method for performing this decomposition.\npub trait HEEADecomposition {\n /// Decomposes a scalar `h` into two approximately half-size scalars.\n ///\n /// Returns `(rho, tau, flip_h)`. If `flip_h` is false, then\n /// `rho == tau * h (mod ell)`. If `flip_h` is true, then\n /// `rho == -tau * h (mod ell)`.\n fn heea_decompose(&self) -> (Scalar, Scalar, bool);\n}\n\n// ------------------------------------------------------------------------\n// Private Traits\n// ------------------------------------------------------------------------\n\n/// Trait for checking whether a point is on the curve.\n///\n/// This trait is only for debugging/testing, since it should be\n/// impossible for a `curve25519-dalek` user to construct an invalid\n/// point.\n#[allow(dead_code)]\npub(crate) trait ValidityCheck {\n /// Checks whether the point is on the curve. Not CT.\n fn is_valid(&self) -> bool;\n}\n"},{"id":39,"name":{"Local":"curve25519/solana-ed25519/src/edwards/affine.rs"},"crate_name":"curve25519","contents":"use super::{CompressedEdwardsY, EdwardsPoint};\nuse crate::traits::Identity;\nuse crate::{Scalar, field::FieldElement};\nuse core::ops::Mul;\nuse subtle::{Choice, ConditionallySelectable, ConstantTimeEq};\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\n/// Affine Edwards point on untwisted curve.\n#[derive(Copy, Clone, Debug)]\npub struct AffinePoint {\n pub(super) x: FieldElement,\n pub(super) y: FieldElement,\n}\n\nimpl ConstantTimeEq for AffinePoint {\n fn ct_eq(&self, other: &Self) -> Choice {\n self.x.ct_eq(&other.x) & self.y.ct_eq(&other.y)\n }\n}\n\nimpl ConditionallySelectable for AffinePoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n Self {\n x: FieldElement::conditional_select(&a.x, &b.x, choice),\n y: FieldElement::conditional_select(&a.y, &b.y, choice),\n }\n }\n}\n\nimpl Default for AffinePoint {\n fn default() -> AffinePoint {\n AffinePoint::identity()\n }\n}\n\nimpl Identity for AffinePoint {\n fn identity() -> AffinePoint {\n AffinePoint {\n x: FieldElement::ZERO,\n y: FieldElement::ONE,\n }\n }\n}\n\nimpl PartialEq for AffinePoint {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for AffinePoint {}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for AffinePoint {}\n\nimpl AffinePoint {\n /// Convert to extended coordinates.\n pub fn to_edwards(self) -> EdwardsPoint {\n EdwardsPoint {\n X: self.x,\n Y: self.y,\n Z: FieldElement::ONE,\n T: &self.x * &self.y,\n }\n }\n\n /// Compress affine Edwards coordinates into `CompressedEdwardsY` format.\n #[inline]\n pub fn compress(self) -> CompressedEdwardsY {\n let mut s = self.y.to_bytes();\n s[31] ^= self.x.is_negative().unwrap_u8() << 7;\n CompressedEdwardsY(s)\n }\n}\n\nimpl Mul for Scalar {\n type Output = EdwardsPoint;\n\n #[inline]\n fn mul(self, rhs: AffinePoint) -> EdwardsPoint {\n self * &rhs\n }\n}\n\nimpl Mul<&AffinePoint> for Scalar {\n type Output = EdwardsPoint;\n\n #[inline]\n fn mul(self, rhs: &AffinePoint) -> EdwardsPoint {\n rhs.to_edwards() * self\n }\n}\n\n#[cfg(test)]\nmod tests {\n use super::{AffinePoint, EdwardsPoint, Identity};\n use crate::constants;\n\n #[test]\n fn identity_conversion() {\n assert_eq!(\n AffinePoint::identity().to_edwards(),\n EdwardsPoint::identity()\n );\n }\n\n #[test]\n fn generator_round_trip() {\n let basepoint = constants::ED25519_BASEPOINT_POINT;\n assert_eq!(basepoint.to_affine().to_edwards(), basepoint);\n }\n}\n"},{"id":40,"name":{"Local":"/rustc/library/core/src/hash/mod.rs"},"crate_name":"core","contents":null},{"id":41,"name":{"Local":"/rustc/library/core/src/array/mod.rs"},"crate_name":"core","contents":null},{"id":42,"name":{"Local":"/rustc/library/core/src/borrow.rs"},"crate_name":"core","contents":null},{"id":43,"name":{"Local":"/rustc/library/core/src/iter/traits/accum.rs"},"crate_name":"core","contents":null},{"id":44,"name":{"Local":"/rustc/library/core/src/array/iter.rs"},"crate_name":"core","contents":null},{"id":45,"name":{"Local":"/rustc/library/core/src/num/nonzero.rs"},"crate_name":"core","contents":null},{"id":46,"name":{"Local":"/rustc/library/core/src/iter/adapters/step_by.rs"},"crate_name":"core","contents":null},{"id":47,"name":{"Local":"/rustc/library/core/src/iter/adapters/mod.rs"},"crate_name":"core","contents":null},{"id":48,"name":{"Local":"/rustc/library/core/src/iter/adapters/chain.rs"},"crate_name":"core","contents":null},{"id":49,"name":{"Local":"/rustc/library/core/src/iter/adapters/zip.rs"},"crate_name":"core","contents":null},{"id":50,"name":{"Local":"/rustc/library/core/src/iter/adapters/intersperse.rs"},"crate_name":"core","contents":null},{"id":51,"name":{"Local":"/rustc/library/core/src/ops/function.rs"},"crate_name":"core","contents":null},{"id":52,"name":{"Local":"/rustc/library/core/src/iter/adapters/map.rs"},"crate_name":"core","contents":null},{"id":53,"name":{"Local":"/rustc/library/core/src/iter/adapters/filter.rs"},"crate_name":"core","contents":null},{"id":54,"name":{"Local":"/rustc/library/core/src/iter/adapters/filter_map.rs"},"crate_name":"core","contents":null},{"id":55,"name":{"Local":"/rustc/library/core/src/iter/adapters/enumerate.rs"},"crate_name":"core","contents":null},{"id":56,"name":{"Local":"/rustc/library/core/src/iter/adapters/peekable.rs"},"crate_name":"core","contents":null},{"id":57,"name":{"Local":"/rustc/library/core/src/iter/adapters/skip_while.rs"},"crate_name":"core","contents":null},{"id":58,"name":{"Local":"/rustc/library/core/src/iter/adapters/take_while.rs"},"crate_name":"core","contents":null},{"id":59,"name":{"Local":"/rustc/library/core/src/iter/adapters/map_while.rs"},"crate_name":"core","contents":null},{"id":60,"name":{"Local":"/rustc/library/core/src/iter/adapters/skip.rs"},"crate_name":"core","contents":null},{"id":61,"name":{"Local":"/rustc/library/core/src/iter/adapters/take.rs"},"crate_name":"core","contents":null},{"id":62,"name":{"Local":"/rustc/library/core/src/iter/adapters/scan.rs"},"crate_name":"core","contents":null},{"id":63,"name":{"Local":"/rustc/library/core/src/iter/adapters/flatten.rs"},"crate_name":"core","contents":null},{"id":64,"name":{"Local":"/rustc/library/core/src/iter/adapters/map_windows.rs"},"crate_name":"core","contents":null},{"id":65,"name":{"Local":"/rustc/library/core/src/iter/adapters/fuse.rs"},"crate_name":"core","contents":null},{"id":66,"name":{"Local":"/rustc/library/core/src/iter/adapters/inspect.rs"},"crate_name":"core","contents":null},{"id":67,"name":{"Local":"/rustc/library/core/src/ops/try_trait.rs"},"crate_name":"core","contents":null},{"id":68,"name":{"Local":"/rustc/library/core/src/iter/traits/double_ended.rs"},"crate_name":"core","contents":null},{"id":69,"name":{"Local":"/rustc/library/core/src/iter/traits/exact_size.rs"},"crate_name":"core","contents":null},{"id":70,"name":{"Local":"/rustc/library/core/src/iter/adapters/rev.rs"},"crate_name":"core","contents":null},{"id":71,"name":{"Local":"/rustc/library/core/src/iter/adapters/copied.rs"},"crate_name":"core","contents":null},{"id":72,"name":{"Local":"/rustc/library/core/src/iter/adapters/cloned.rs"},"crate_name":"core","contents":null},{"id":73,"name":{"Local":"/rustc/library/core/src/iter/adapters/cycle.rs"},"crate_name":"core","contents":null},{"id":74,"name":{"Local":"/rustc/library/core/src/iter/adapters/array_chunks.rs"},"crate_name":"core","contents":null},{"id":75,"name":{"Local":"/rustc/library/core/src/ops/index.rs"},"crate_name":"core","contents":null},{"id":76,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64/constants.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! This module contains backend-specific constant values, such as the 64-bit limbs of curve constants.\n\nuse super::field::FieldElement51;\nuse super::scalar::Scalar52;\nuse crate::edwards::EdwardsPoint;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::{\n backend::serial::curve_models::{AffineNielsPoint, ProjectiveNielsPoint},\n edwards::EdwardsBasepointTable,\n window::{LookupTable, NafLookupTable5, NafLookupTable8},\n};\n\n/// The value of minus one, equal to `-&FieldElement::ONE`\npub(crate) const MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n]);\n\n/// sqrt(-486664)\n#[cfg(feature = \"digest\")]\npub(crate) const ED25519_SQRTAM2: FieldElement51 = FieldElement51::from_limbs([\n 1693982333959686,\n 608509411481997,\n 2235573344831311,\n 947681270984193,\n 266558006233600,\n]);\n\n/// Edwards `d` value, equal to `-121665/121666 mod p`.\npub(crate) const EDWARDS_D: FieldElement51 = FieldElement51::from_limbs([\n 929955233495203,\n 466365720129213,\n 1662059464998953,\n 2033849074728123,\n 1442794654840575,\n]);\n\n/// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.\npub(crate) const EDWARDS_D2: FieldElement51 = FieldElement51::from_limbs([\n 1859910466990425,\n 932731440258426,\n 1072319116312658,\n 1815898335770999,\n 633789495995903,\n]);\n\n/// One minus edwards `d` value squared, equal to `(1 - (-121665/121666) mod p) pow 2`\npub(crate) const ONE_MINUS_EDWARDS_D_SQUARED: FieldElement51 = FieldElement51::from_limbs([\n 1136626929484150,\n 1998550399581263,\n 496427632559748,\n 118527312129759,\n 45110755273534,\n]);\n\n/// Edwards `d` value minus one squared, equal to `(((-121665/121666) mod p) - 1) pow 2`\npub(crate) const EDWARDS_D_MINUS_ONE_SQUARED: FieldElement51 = FieldElement51::from_limbs([\n 1507062230895904,\n 1572317787530805,\n 683053064812840,\n 317374165784489,\n 1572899562415810,\n]);\n\n/// `= sqrt(a*d - 1)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.\npub(crate) const SQRT_AD_MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2241493124984347,\n 425987919032274,\n 2207028919301688,\n 1220490630685848,\n 974799131293748,\n]);\n\n/// `= 1/sqrt(a-d)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.\npub(crate) const INVSQRT_A_MINUS_D: FieldElement51 = FieldElement51::from_limbs([\n 278908739862762,\n 821645201101625,\n 8113234426968,\n 1777959178193151,\n 2118520810568447,\n]);\n\n/// Precomputed value of one of the square roots of -1 (mod p)\npub(crate) const SQRT_M1: FieldElement51 = FieldElement51::from_limbs([\n 1718705420411056,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n]);\n\n/// `APLUS2_OVER_FOUR` is (A+2)/4. (This is used internally within the Montgomery ladder.)\npub(crate) const APLUS2_OVER_FOUR: FieldElement51 =\n FieldElement51::from_limbs([121666, 0, 0, 0, 0]);\n\n#[cfg(feature = \"digest\")]\n/// `MONTGOMERY_A` is equal to 486662, which is a constant of the curve equation\n/// for Curve25519 in its Montgomery form. (This is used internally within the\n/// Elligator map.)\npub(crate) const MONTGOMERY_A: FieldElement51 = FieldElement51::from_limbs([486662, 0, 0, 0, 0]);\n\n#[cfg(feature = \"digest\")]\n/// `MONTGOMERY_A_NEG` is equal to -486662. (This is used internally within the\n/// Elligator map.)\npub(crate) const MONTGOMERY_A_NEG: FieldElement51 = FieldElement51::from_limbs([\n 2251799813198567,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n]);\n\n/// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493\npub(crate) const L: Scalar52 = Scalar52([\n 0x0002631a5cf5d3ed,\n 0x000dea2f79cd6581,\n 0x000000000014def9,\n 0x0000000000000000,\n 0x0000100000000000,\n]);\n\n/// `L` * `LFACTOR` = -1 (mod 2^52)\npub(crate) const LFACTOR: u64 = 0x51da312547e1b;\n\n/// `R` = R % L where R = 2^260\npub(crate) const R: Scalar52 = Scalar52([\n 0x000f48bd6721e6ed,\n 0x0003bab5ac67e45a,\n 0x000fffffeb35e51b,\n 0x000fffffffffffff,\n 0x00000fffffffffff,\n]);\n\n/// `RR` = (R^2) % L where R = 2^260\npub(crate) const RR: Scalar52 = Scalar52([\n 0x0009d265e952d13b,\n 0x000d63c715bea69f,\n 0x0005be65cb687604,\n 0x0003dceec73d217f,\n 0x000009411b7c309a,\n]);\n\n/// The Ed25519 basepoint, as an `EdwardsPoint`.\n///\n/// This is called `_POINT` to distinguish it from\n/// `ED25519_BASEPOINT_TABLE`, which should be used for scalar\n/// multiplication (it's much faster).\npub const ED25519_BASEPOINT_POINT: EdwardsPoint = EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1738742601995546,\n 1146398526822698,\n 2070867633025821,\n 562264141797630,\n 587772402128613,\n ]),\n Y: FieldElement51::from_limbs([\n 1801439850948184,\n 1351079888211148,\n 450359962737049,\n 900719925474099,\n 1801439850948198,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1841354044333475,\n 16398895984059,\n 755974180946558,\n 900171276175154,\n 1821297809914039,\n ]),\n};\n\n/// The Ed25519 basepoint, mul by 2^128, as an `EdwardsPoint`.\npub const ED25519_BASEPOINT_128_POINT: EdwardsPoint = EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1573694877509226,\n 2118644427590872,\n 940662180141619,\n 707110682864191,\n 1255997186674731,\n ]),\n Y: FieldElement51::from_limbs([\n 1162474291335259,\n 820106152083656,\n 523232807607258,\n 2134608004007539,\n 729879497843472,\n ]),\n Z: FieldElement51::from_limbs([\n 445134475408207,\n 1820906444767788,\n 40212681131971,\n 1448661247201044,\n 965705781338211,\n ]),\n T: FieldElement51::from_limbs([\n 1855274855831337,\n 1818119365171423,\n 269991889323070,\n 1370944576683708,\n 140790155072480,\n ]),\n};\n\n/// The 8-torsion subgroup \\\\(\\mathcal E \\[8\\]\\\\).\n///\n/// In the case of Curve25519, it is cyclic; the \\\\(i\\\\)-th element of\n/// the array is \\\\(\\[i\\]P\\\\), where \\\\(P\\\\) is a point of order \\\\(8\\\\)\n/// generating \\\\(\\mathcal E\\[8\\]\\\\).\n///\n/// Thus \\\\(\\mathcal E\\[4\\]\\\\) is the points indexed by `0,2,4,6`, and\n/// \\\\(\\mathcal E\\[2\\]\\\\) is the points indexed by `0,4`.\npub const EIGHT_TORSION: [EdwardsPoint; 8] = EIGHT_TORSION_INNER_DOC_HIDDEN;\n\n/// Inner item used to hide limb constants from cargo doc output.\n#[doc(hidden)]\npub const EIGHT_TORSION_INNER_DOC_HIDDEN: [EdwardsPoint; 8] = [\n EdwardsPoint {\n X: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Y: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 358744748052810,\n 1691584618240980,\n 977650209285361,\n 1429865912637724,\n 560044844278676,\n ]),\n Y: FieldElement51::from_limbs([\n 84926274344903,\n 473620666599931,\n 365590438845504,\n 1028470286882429,\n 2146499180330972,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1448326834587521,\n 1857896831960481,\n 1093722731865333,\n 1677408490711241,\n 1915505153018406,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 533094393274173,\n 2016890930128738,\n 18285341111199,\n 134597186663265,\n 1486323764102114,\n ]),\n Y: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 358744748052810,\n 1691584618240980,\n 977650209285361,\n 1429865912637724,\n 560044844278676,\n ]),\n Y: FieldElement51::from_limbs([\n 2166873539340326,\n 1778179147085316,\n 1886209374839743,\n 1223329526802818,\n 105300633354275,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 803472979097708,\n 393902981724766,\n 1158077081819914,\n 574391322974006,\n 336294660666841,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Y: FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1893055065632419,\n 560215195444267,\n 1274149604399886,\n 821933901047523,\n 1691754969406571,\n ]),\n Y: FieldElement51::from_limbs([\n 2166873539340326,\n 1778179147085316,\n 1886209374839743,\n 1223329526802818,\n 105300633354275,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1448326834587521,\n 1857896831960481,\n 1093722731865333,\n 1677408490711241,\n 1915505153018406,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1718705420411056,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n ]),\n Y: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1893055065632419,\n 560215195444267,\n 1274149604399886,\n 821933901047523,\n 1691754969406571,\n ]),\n Y: FieldElement51::from_limbs([\n 84926274344903,\n 473620666599931,\n 365590438845504,\n 1028470286882429,\n 2146499180330972,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 803472979097708,\n 393902981724766,\n 1158077081819914,\n 574391322974006,\n 336294660666841,\n ]),\n },\n];\n\n/// Table containing precomputed multiples of the Ed25519 basepoint \\\\(B = (x, 4/5)\\\\).\n#[cfg(feature = \"precomputed-tables\")]\npub static ED25519_BASEPOINT_TABLE: &EdwardsBasepointTable =\n &ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN;\n\n/// Inner constant, used to avoid filling the docs with precomputed points.\n#[doc(hidden)]\n#[cfg(feature = \"precomputed-tables\")]\nstatic ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN: EdwardsBasepointTable = EdwardsBasepointTable([\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3540182452943730,\n 2497478415033846,\n 2521227595762870,\n 1462984067271729,\n 2389212253076811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 62697248952638,\n 204681361388450,\n 631292143396476,\n 338455783676468,\n 1213667448819585,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301289933810280,\n 1259582250014073,\n 1422107436869536,\n 796239922652654,\n 1953934009299142,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3632771708514775,\n 790832306631235,\n 2067202295274102,\n 1995808275510000,\n 1566530869037010,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 463307831301544,\n 432984605774163,\n 1610641361907204,\n 750899048855000,\n 1894842303421586,\n ]),\n xy2d: FieldElement51::from_limbs([\n 748439484463711,\n 1033211726465151,\n 1396005112841647,\n 1611506220286469,\n 1972177495910992,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1601611775252272,\n 1720807796594148,\n 1132070835939856,\n 3512254832574799,\n 2147779492816910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 316559037616741,\n 2177824224946892,\n 1459442586438991,\n 1461528397712656,\n 751590696113597,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850748884277385,\n 1200145853858453,\n 1068094770532492,\n 672251375690438,\n 1586055907191707,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 934282339813791,\n 1846903124198670,\n 1172395437954843,\n 1007037127761661,\n 1830588347719256,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1694390458783935,\n 1735906047636159,\n 705069562067493,\n 648033061693059,\n 696214010414170,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1121406372216585,\n 192876649532226,\n 190294192191717,\n 1994165897297032,\n 2245000007398739,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 769950342298400,\n 2384754244604994,\n 3095885746880802,\n 3225892188161580,\n 2977876099231263,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 425251763115706,\n 608463272472562,\n 442562545713235,\n 837766094556764,\n 374555092627893,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1086255230780037,\n 274979815921559,\n 1960002765731872,\n 929474102396301,\n 1190409889297339,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1388594989461809,\n 316767091099457,\n 2646098655878230,\n 1230079486801004,\n 1440737038838979,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 7380825640100,\n 146210432690483,\n 304903576448906,\n 1198869323871120,\n 997689833219095,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1181317918772081,\n 114573476638901,\n 262805072233344,\n 265712217171332,\n 294181933805782,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2916800678241215,\n 2065379846933858,\n 2622030924071124,\n 2602788184473875,\n 1233371373142984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2019367628972465,\n 676711900706637,\n 110710997811333,\n 1108646842542025,\n 517791959672113,\n ]),\n xy2d: FieldElement51::from_limbs([\n 965130719900578,\n 247011430587952,\n 526356006571389,\n 91986625355052,\n 2157223321444601,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4320419353804412,\n 4218074731744053,\n 957728544705548,\n 729906502578991,\n 2411634706750414,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2073601412052185,\n 31021124762708,\n 264500969797082,\n 248034690651703,\n 1030252227928288,\n ]),\n xy2d: FieldElement51::from_limbs([\n 551790716293402,\n 1989538725166328,\n 801169423371717,\n 2052451893578887,\n 678432056995012,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1368953770187805,\n 3042147450398169,\n 2689308289352409,\n 2142576377050579,\n 1932081720066286,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 953638594433374,\n 1092333936795051,\n 1419774766716690,\n 805677984380077,\n 859228993502513,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1200766035879111,\n 20142053207432,\n 1465634435977050,\n 1645256912097844,\n 295121984874596,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1735718747031538,\n 1248237894295956,\n 1204753118328107,\n 976066523550493,\n 2317743583219840,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1060098822528990,\n 1586825862073490,\n 212301317240126,\n 1975302711403555,\n 666724059764335,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1091990273418756,\n 1572899409348578,\n 80968014455247,\n 306009358661350,\n 1520450739132526,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3732317023121341,\n 1511153322193951,\n 3496143672676420,\n 2556587964178488,\n 2620936670181690,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2151330273626164,\n 762045184746182,\n 1688074332551515,\n 823046109005759,\n 907602769079491,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2047386910586836,\n 168470092900250,\n 1552838872594810,\n 340951180073789,\n 360819374702533,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1982622644432037,\n 2014393600336956,\n 2380709022489462,\n 3869592437614438,\n 2357094095599062,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 980234343912898,\n 1712256739246056,\n 588935272190264,\n 204298813091998,\n 841798321043288,\n ]),\n xy2d: FieldElement51::from_limbs([\n 197561292938973,\n 454817274782871,\n 1963754960082318,\n 2113372252160468,\n 971377527342673,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2416499262514576,\n 2254927265442919,\n 3451304785234000,\n 1766155447043651,\n 1899238924683527,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 732262946680281,\n 1674412764227063,\n 2182456405662809,\n 1350894754474250,\n 558458873295247,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2103305098582922,\n 1960809151316468,\n 715134605001343,\n 1454892949167181,\n 40827143824949,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1239289043050193,\n 1744654158124578,\n 758702410031698,\n 4048562808759936,\n 2253402870349013,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2232056027107988,\n 987343914584615,\n 2115594492994461,\n 1819598072792159,\n 1119305654014850,\n ]),\n xy2d: FieldElement51::from_limbs([\n 320153677847348,\n 939613871605645,\n 641883205761567,\n 1930009789398224,\n 329165806634126,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3232730304159378,\n 1242488692177892,\n 1251446316964684,\n 1086618677993530,\n 1961430968465772,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 276821765317453,\n 1536835591188030,\n 1305212741412361,\n 61473904210175,\n 2051377036983058,\n ]),\n xy2d: FieldElement51::from_limbs([\n 833449923882501,\n 1750270368490475,\n 1123347002068295,\n 185477424765687,\n 278090826653186,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 794524995833413,\n 1849907304548286,\n 2305148486158393,\n 1272368559505216,\n 1147304168324779,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1504846112759364,\n 1203096289004681,\n 562139421471418,\n 274333017451844,\n 1284344053775441,\n ]),\n xy2d: FieldElement51::from_limbs([\n 483048732424432,\n 2116063063343382,\n 30120189902313,\n 292451576741007,\n 1156379271702225,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3180171966714267,\n 2147692869914563,\n 1455665844462196,\n 1986737809425946,\n 2437006863943337,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 137732961814206,\n 706670923917341,\n 1387038086865771,\n 1965643813686352,\n 1384777115696347,\n ]),\n xy2d: FieldElement51::from_limbs([\n 481144981981577,\n 2053319313589856,\n 2065402289827512,\n 617954271490316,\n 1106602634668125,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2948097833334040,\n 3145099472726142,\n 1148636718636008,\n 2278533891034865,\n 2203955659340680,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 657390353372855,\n 998499966885562,\n 991893336905797,\n 810470207106761,\n 343139804608786,\n ]),\n xy2d: FieldElement51::from_limbs([\n 791736669492960,\n 934767652997115,\n 824656780392914,\n 1759463253018643,\n 361530362383518,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2022541353055578,\n 4346500076272714,\n 3802807888710933,\n 2494585331103411,\n 2947785218648809,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1287487199965223,\n 2215311941380308,\n 1552928390931986,\n 1664859529680196,\n 1125004975265243,\n ]),\n xy2d: FieldElement51::from_limbs([\n 677434665154918,\n 989582503122485,\n 1817429540898386,\n 1052904935475344,\n 1143826298169798,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2619066141993637,\n 2570231002607651,\n 2947429167440602,\n 2885885471266079,\n 2276381426249673,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 773360688841258,\n 1815381330538070,\n 363773437667376,\n 539629987070205,\n 783280434248437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 180820816194166,\n 168937968377394,\n 748416242794470,\n 1227281252254508,\n 1567587861004268,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2730575372268893,\n 2062896624554806,\n 2951191072970647,\n 2609899222113120,\n 1277310261461760,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1984740906540026,\n 1079164179400229,\n 1056021349262661,\n 1659958556483663,\n 1088529069025527,\n ]),\n xy2d: FieldElement51::from_limbs([\n 580736401511151,\n 1842931091388998,\n 1177201471228238,\n 2075460256527244,\n 1301133425678027,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1515728832059163,\n 1575261009617579,\n 1510246567196186,\n 2442877836294952,\n 2368461529974388,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295295738269652,\n 1714742313707026,\n 545583042462581,\n 2034411676262552,\n 1513248090013606,\n ]),\n xy2d: FieldElement51::from_limbs([\n 230710545179830,\n 30821514358353,\n 760704303452229,\n 390668103790604,\n 573437871383156,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3421179921230875,\n 2514967047430861,\n 4274701112739695,\n 3071700566936367,\n 4275698278559832,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2102254323485823,\n 1570832666216754,\n 34696906544624,\n 1993213739807337,\n 70638552271463,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894132856735058,\n 548675863558441,\n 845349339503395,\n 1942269668326667,\n 1615682209874691,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3539470031223082,\n 1222355136884919,\n 1846481788678694,\n 1150426571265110,\n 1613523400722047,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 793388516527298,\n 1315457083650035,\n 1972286999342417,\n 1901825953052455,\n 338269477222410,\n ]),\n xy2d: FieldElement51::from_limbs([\n 550201530671806,\n 778605267108140,\n 2063911101902983,\n 115500557286349,\n 2041641272971022,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 717255318455100,\n 519313764361315,\n 2080406977303708,\n 541981206705521,\n 774328150311600,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 261715221532238,\n 1795354330069993,\n 1496878026850283,\n 499739720521052,\n 389031152673770,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1997217696294013,\n 1717306351628065,\n 1684313917746180,\n 1644426076011410,\n 1857378133465451,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3727234538477877,\n 2328731709971226,\n 3368528843456914,\n 2002544139318041,\n 2977347647489186,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2022306639183567,\n 726296063571875,\n 315345054448644,\n 1058733329149221,\n 1448201136060677,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1710065158525665,\n 1895094923036397,\n 123988286168546,\n 1145519900776355,\n 1607510767693874,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2813405189107769,\n 1071733543815036,\n 2383296312486238,\n 1946868434569998,\n 3079937947649451,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1548495173745801,\n 442310529226540,\n 998072547000384,\n 553054358385281,\n 644824326376171,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1445526537029440,\n 2225519789662536,\n 914628859347385,\n 1064754194555068,\n 1660295614401091,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3451490036797185,\n 2275827949507588,\n 2318438102929588,\n 2309425969971222,\n 2816893781664854,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 876926774220824,\n 554618976488214,\n 1012056309841565,\n 839961821554611,\n 1414499340307677,\n ]),\n xy2d: FieldElement51::from_limbs([\n 703047626104145,\n 1266841406201770,\n 165556500219173,\n 486991595001879,\n 1011325891650656,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1622861044480487,\n 1156394801573634,\n 4120932379100752,\n 2578903799462977,\n 2095342781472283,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 334886927423922,\n 489511099221528,\n 129160865966726,\n 1720809113143481,\n 619700195649254,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1646545795166119,\n 1758370782583567,\n 714746174550637,\n 1472693650165135,\n 898994790308209,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2585203586724508,\n 2547572356138185,\n 1693106465353609,\n 912330357530760,\n 2723035471635610,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1811196219982022,\n 1068969825533602,\n 289602974833439,\n 1988956043611592,\n 863562343398367,\n ]),\n xy2d: FieldElement51::from_limbs([\n 906282429780072,\n 2108672665779781,\n 432396390473936,\n 150625823801893,\n 1708930497638539,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 925664675702309,\n 2273216662253932,\n 4083236455546587,\n 601157008940112,\n 2623617868729744,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1479786007267725,\n 1738881859066675,\n 68646196476567,\n 2146507056100328,\n 1247662817535471,\n ]),\n xy2d: FieldElement51::from_limbs([\n 52035296774456,\n 939969390708103,\n 312023458773250,\n 59873523517659,\n 1231345905848899,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2895154920100990,\n 2541986621181021,\n 2013561737429022,\n 2571447883196794,\n 2645536492181409,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 129358342392716,\n 1932811617704777,\n 1176749390799681,\n 398040349861790,\n 1170779668090425,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2051980782668029,\n 121859921510665,\n 2048329875753063,\n 1235229850149665,\n 519062146124755,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3859970785658325,\n 2667608874045675,\n 1350468408164765,\n 2038620059057678,\n 3278704299674360,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1837656083115103,\n 1510134048812070,\n 906263674192061,\n 1821064197805734,\n 565375124676301,\n ]),\n xy2d: FieldElement51::from_limbs([\n 578027192365650,\n 2034800251375322,\n 2128954087207123,\n 478816193810521,\n 2196171989962750,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1633188840273120,\n 3104586986058956,\n 1548762607215795,\n 1266275218902681,\n 3359018017010381,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 462189358480054,\n 1784816734159228,\n 1611334301651368,\n 1303938263943540,\n 707589560319424,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1038829280972848,\n 38176604650029,\n 753193246598573,\n 1136076426528122,\n 595709990562434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3660251634545082,\n 2194984964010832,\n 2198361797561729,\n 1061962440055713,\n 1645147963442934,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 4701053362120,\n 1647641066302348,\n 1047553002242085,\n 1923635013395977,\n 206970314902065,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1750479161778571,\n 1362553355169293,\n 1891721260220598,\n 966109370862782,\n 1024913988299801,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2464498862816952,\n 1117950018299774,\n 1873945661751056,\n 3655602735669306,\n 2382695896337945,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 636808533673210,\n 1262201711667560,\n 390951380330599,\n 1663420692697294,\n 561951321757406,\n ]),\n xy2d: FieldElement51::from_limbs([\n 520731594438141,\n 1446301499955692,\n 273753264629267,\n 1565101517999256,\n 1019411827004672,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3178327305714638,\n 3443653291096626,\n 734233225181170,\n 2435838701226518,\n 4042225960010590,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1464651961852572,\n 1483737295721717,\n 1519450561335517,\n 1161429831763785,\n 405914998179977,\n ]),\n xy2d: FieldElement51::from_limbs([\n 996126634382301,\n 796204125879525,\n 127517800546509,\n 344155944689303,\n 615279846169038,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2990523894660505,\n 2188666632415295,\n 1961313708559162,\n 1506545807547587,\n 3403101452654988,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622917337413835,\n 1218989177089035,\n 1284857712846592,\n 970502061709359,\n 351025208117090,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2067814584765580,\n 1677855129927492,\n 2086109782475197,\n 235286517313238,\n 1416314046739645,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2838644076315587,\n 2559244195637442,\n 458399356043425,\n 2853867838192310,\n 3280348017100490,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 678489922928203,\n 2016657584724032,\n 90977383049628,\n 1026831907234582,\n 615271492942522,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301225714012278,\n 1094837270268560,\n 1202288391010439,\n 644352775178361,\n 1647055902137983,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1210746697896459,\n 1416608304244708,\n 2938287290903104,\n 3496931005119382,\n 3303038150540984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1135604073198207,\n 1683322080485474,\n 769147804376683,\n 2086688130589414,\n 900445683120379,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1971518477615628,\n 401909519527336,\n 448627091057375,\n 1409486868273821,\n 1214789035034363,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1364039144731711,\n 1897497433586190,\n 2203097701135459,\n 2397261210496499,\n 1349844460790698,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1045230323257973,\n 818206601145807,\n 630513189076103,\n 1672046528998132,\n 807204017562437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 439961968385997,\n 386362664488986,\n 1382706320807688,\n 309894000125359,\n 2207801346498567,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3480804500082836,\n 3172443782216110,\n 2375775707596425,\n 2933223806901024,\n 1400559197080972,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2003766096898049,\n 170074059235165,\n 1141124258967971,\n 1485419893480973,\n 1573762821028725,\n ]),\n xy2d: FieldElement51::from_limbs([\n 729905708611432,\n 1270323270673202,\n 123353058984288,\n 426460209632942,\n 2195574535456672,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1271140255321216,\n 2044363183174497,\n 2303925201319937,\n 3696920060379952,\n 3194341800024331,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1761608437466135,\n 583360847526804,\n 1586706389685493,\n 2157056599579261,\n 1170692369685772,\n ]),\n xy2d: FieldElement51::from_limbs([\n 871476219910823,\n 1878769545097794,\n 2241832391238412,\n 548957640601001,\n 690047440233174,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2548994545820755,\n 1366347803776819,\n 3552985325930849,\n 561849853336293,\n 1533554921345731,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 999628998628371,\n 1132836708493400,\n 2084741674517453,\n 469343353015612,\n 678782988708035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2189427607417022,\n 699801937082607,\n 412764402319267,\n 1478091893643349,\n 2244675696854460,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3964091869651792,\n 2456213404310121,\n 3657538451018088,\n 2660781114515010,\n 3112882032961968,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 508561155940631,\n 966928475686665,\n 2236717801150132,\n 424543858577297,\n 2089272956986143,\n ]),\n xy2d: FieldElement51::from_limbs([\n 221245220129925,\n 1156020201681217,\n 491145634799213,\n 542422431960839,\n 828100817819207,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2405556784925632,\n 1299874139923976,\n 2644898978945750,\n 1058234455773021,\n 996989038681183,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 559086812798481,\n 573177704212711,\n 1629737083816402,\n 1399819713462595,\n 1646954378266038,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1887963056288059,\n 228507035730124,\n 1468368348640282,\n 930557653420194,\n 613513962454686,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1224529808187534,\n 1577022856702685,\n 2206946542980843,\n 625883007765001,\n 2531730607197406,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1076287717051609,\n 1114455570543035,\n 187297059715481,\n 250446884292121,\n 1885187512550540,\n ]),\n xy2d: FieldElement51::from_limbs([\n 902497362940219,\n 76749815795675,\n 1657927525633846,\n 1420238379745202,\n 1340321636548352,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1129576631190765,\n 3533793823712575,\n 996844254743017,\n 2509676177174497,\n 3402650555740265,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 628740660038789,\n 1943038498527841,\n 467786347793886,\n 1093341428303375,\n 235413859513003,\n ]),\n xy2d: FieldElement51::from_limbs([\n 237425418909360,\n 469614029179605,\n 1512389769174935,\n 1241726368345357,\n 441602891065214,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3988217766743784,\n 726531315520507,\n 1833335034432527,\n 1629442561574747,\n 2876218732971333,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1960754663920689,\n 497040957888962,\n 1909832851283095,\n 1271432136996826,\n 2219780368020940,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1537037379417136,\n 1358865369268262,\n 2130838645654099,\n 828733687040705,\n 1999987652890901,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 629042105241795,\n 1098854999137608,\n 887281544569320,\n 3674901833560025,\n 2259711072636808,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1811562332665373,\n 1501882019007673,\n 2213763501088999,\n 359573079719636,\n 36370565049116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 218907117361280,\n 1209298913016966,\n 1944312619096112,\n 1130690631451061,\n 1342327389191701,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1369976867854685,\n 1396479602419169,\n 4017456468084104,\n 2203659200586298,\n 3250127649802489,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2230701885562825,\n 1348173180338974,\n 2172856128624598,\n 1426538746123771,\n 444193481326151,\n ]),\n xy2d: FieldElement51::from_limbs([\n 784210426627951,\n 918204562375674,\n 1284546780452985,\n 1324534636134684,\n 1872449409642708,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2571438643225542,\n 2848082470493653,\n 2037902696412607,\n 1557219121643918,\n 341938082688094,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1901860206695915,\n 2004489122065736,\n 1625847061568236,\n 973529743399879,\n 2075287685312905,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1371853944110545,\n 1042332820512553,\n 1949855697918254,\n 1791195775521505,\n 37487364849293,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 687200189577836,\n 1082536651125675,\n 2896024754556794,\n 2592723009743198,\n 2595381160432643,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2082717129583892,\n 27829425539422,\n 145655066671970,\n 1690527209845512,\n 1865260509673478,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1059729620568824,\n 2163709103470266,\n 1440302280256872,\n 1769143160546397,\n 869830310425069,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3861316033464273,\n 777277757338816,\n 2101121130363987,\n 550762194946473,\n 1905542338659364,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2024821921041576,\n 426948675450149,\n 595133284085473,\n 471860860885970,\n 600321679413000,\n ]),\n xy2d: FieldElement51::from_limbs([\n 598474602406721,\n 1468128276358244,\n 1191923149557635,\n 1501376424093216,\n 1281662691293476,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1721138489890688,\n 1264336102277790,\n 2684864359106535,\n 1359988423149465,\n 3813671107094695,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 719520245587143,\n 393380711632345,\n 132350400863381,\n 1543271270810729,\n 1819543295798660,\n ]),\n xy2d: FieldElement51::from_limbs([\n 396397949784152,\n 1811354474471839,\n 1362679985304303,\n 2117033964846756,\n 498041172552279,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1812471844975748,\n 1856491995543149,\n 126579494584102,\n 3288044672967868,\n 1975108050082549,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 650623932407995,\n 1137551288410575,\n 2125223403615539,\n 1725658013221271,\n 2134892965117796,\n ]),\n xy2d: FieldElement51::from_limbs([\n 522584000310195,\n 1241762481390450,\n 1743702789495384,\n 2227404127826575,\n 1686746002148897,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 427904865186293,\n 1703211129693455,\n 1585368107547509,\n 3688784302429584,\n 3012988348299225,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 318101947455002,\n 248138407995851,\n 1481904195303927,\n 309278454311197,\n 1258516760217879,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1275068538599310,\n 513726919533379,\n 349926553492294,\n 688428871968420,\n 1702400196000666,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3313663849950481,\n 3213411074010628,\n 2573659446386085,\n 3297400443644764,\n 1985130202504037,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1558816436882417,\n 1962896332636523,\n 1337709822062152,\n 1501413830776938,\n 294436165831932,\n ]),\n xy2d: FieldElement51::from_limbs([\n 818359826554971,\n 1862173000996177,\n 626821592884859,\n 573655738872376,\n 1749691246745455,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1988022651432119,\n 3333911312271288,\n 1834020786104820,\n 3706626690108935,\n 692929915223121,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2146513703733331,\n 584788900394667,\n 464965657279958,\n 2183973639356127,\n 238371159456790,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1129007025494441,\n 2197883144413266,\n 265142755578169,\n 971864464758890,\n 1983715884903702,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1291366624493056,\n 2633256531874362,\n 1711482489312443,\n 1815233647702022,\n 3144079596677715,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 444548969917454,\n 1452286453853356,\n 2113731441506810,\n 645188273895859,\n 810317625309512,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2242724082797924,\n 1373354730327868,\n 1006520110883049,\n 2147330369940688,\n 1151816104883620,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3997520014069025,\n 4163522956860564,\n 2056329390702073,\n 2607026987995097,\n 3131032608056347,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 163723479936298,\n 115424889803150,\n 1156016391581227,\n 1894942220753364,\n 1970549419986329,\n ]),\n xy2d: FieldElement51::from_limbs([\n 681981452362484,\n 267208874112496,\n 1374683991933094,\n 638600984916117,\n 646178654558546,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2265178468539480,\n 2358037120714814,\n 1944412051589650,\n 4093776581610705,\n 2482502633520820,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 260683893467075,\n 854060306077237,\n 913639551980112,\n 4704576840123,\n 280254810808712,\n ]),\n xy2d: FieldElement51::from_limbs([\n 715374893080287,\n 1173334812210491,\n 1806524662079626,\n 1894596008000979,\n 398905715033393,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2751826223412909,\n 3848231101880618,\n 1420380351989369,\n 3237011375206737,\n 392444930785632,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2096421546958141,\n 1922523000950363,\n 789831022876840,\n 427295144688779,\n 320923973161730,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1927770723575450,\n 1485792977512719,\n 1850996108474547,\n 551696031508956,\n 2126047405475647,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2112099158080129,\n 2994370617594963,\n 2258284371762679,\n 1951119898618915,\n 2344890196388664,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 383905201636970,\n 859946997631870,\n 855623867637644,\n 1017125780577795,\n 794250831877809,\n ]),\n xy2d: FieldElement51::from_limbs([\n 77571826285752,\n 999304298101753,\n 487841111777762,\n 1038031143212339,\n 339066367948762,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2926794589205781,\n 2517835660016036,\n 826951213393477,\n 1405007746162285,\n 1781791018620876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1001412661522686,\n 348196197067298,\n 1666614366723946,\n 888424995032760,\n 580747687801357,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1939560076207777,\n 1409892634407635,\n 552574736069277,\n 383854338280405,\n 190706709864139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2177087163428741,\n 1439255351721944,\n 3459870654068041,\n 2230616362004768,\n 1396886392021913,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 676962063230039,\n 1880275537148808,\n 2046721011602706,\n 888463247083003,\n 1318301552024067,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1466980508178206,\n 617045217998949,\n 652303580573628,\n 757303753529064,\n 207583137376902,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3762856566592150,\n 2357202940576524,\n 2745234706458093,\n 1091943425335975,\n 1802717338077427,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1853982405405128,\n 1878664056251147,\n 1528011020803992,\n 1019626468153565,\n 1128438412189035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1963939888391106,\n 293456433791664,\n 697897559513649,\n 985882796904380,\n 796244541237972,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2668570812315008,\n 2641455366112301,\n 1314476859406755,\n 1749382513022778,\n 3413705412424739,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1428358296490651,\n 1027115282420478,\n 304840698058337,\n 441410174026628,\n 1819358356278573,\n ]),\n xy2d: FieldElement51::from_limbs([\n 204943430200135,\n 1554861433819175,\n 216426658514651,\n 264149070665950,\n 2047097371738319,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1934415182909015,\n 1393285083565062,\n 2768209145458208,\n 3409490548679139,\n 2372839480279515,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 662035583584445,\n 286736105093098,\n 1131773000510616,\n 818494214211439,\n 472943792054479,\n ]),\n xy2d: FieldElement51::from_limbs([\n 665784778135882,\n 1893179629898606,\n 808313193813106,\n 276797254706413,\n 1563426179676396,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 945205108984213,\n 2778077376644543,\n 1324180513733565,\n 1666970227868664,\n 2405347422974421,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2031433403516252,\n 203996615228162,\n 170487168837083,\n 981513604791390,\n 843573964916831,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1476570093962618,\n 838514669399805,\n 1857930577281364,\n 2017007352225784,\n 317085545220047,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1461557121912823,\n 1600674043318359,\n 2157134900399597,\n 1670641601940616,\n 2379565397488531,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1293543509393474,\n 2143624609202546,\n 1058361566797508,\n 214097127393994,\n 946888515472729,\n ]),\n xy2d: FieldElement51::from_limbs([\n 357067959932916,\n 1290876214345711,\n 521245575443703,\n 1494975468601005,\n 800942377643885,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2817916472785262,\n 820247422481739,\n 994464017954148,\n 2578957425371613,\n 2344391131796991,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 617256647603209,\n 1652107761099439,\n 1857213046645471,\n 1085597175214970,\n 817432759830522,\n ]),\n xy2d: FieldElement51::from_limbs([\n 771808161440705,\n 1323510426395069,\n 680497615846440,\n 851580615547985,\n 1320806384849017,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1219260086131896,\n 2898968820282063,\n 2331400938444953,\n 2161724213426747,\n 2656661710745446,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1327968293887866,\n 1335500852943256,\n 1401587164534264,\n 558137311952440,\n 1551360549268902,\n ]),\n xy2d: FieldElement51::from_limbs([\n 417621685193956,\n 1429953819744454,\n 396157358457099,\n 1940470778873255,\n 214000046234152,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1268047918491954,\n 2172375426948536,\n 1533916099229249,\n 1761293575457130,\n 3842422480712013,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1627072914981959,\n 2211603081280073,\n 1912369601616504,\n 1191770436221309,\n 2187309757525860,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1149147819689533,\n 378692712667677,\n 828475842424202,\n 2218619146419342,\n 70688125792186,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3551539230764990,\n 3690416477138006,\n 3788528892189659,\n 2053896748919837,\n 3260220846276494,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2040723824657366,\n 399555637875075,\n 632543375452995,\n 872649937008051,\n 1235394727030233,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2211311599327900,\n 2139787259888175,\n 938706616835350,\n 12609661139114,\n 2081897930719789,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1324994503390431,\n 2588782144267879,\n 1183998925654176,\n 3343454479598522,\n 2300527487656566,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1845522914617879,\n 1222198248335542,\n 150841072760134,\n 1927029069940982,\n 1189913404498011,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1079559557592645,\n 2215338383666441,\n 1903569501302605,\n 49033973033940,\n 305703433934152,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2346453219102138,\n 3637921163538246,\n 3313930291577009,\n 2288353761164521,\n 3085469462634093,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1432015813136298,\n 440364795295369,\n 1395647062821501,\n 1976874522764578,\n 934452372723352,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1296625309219774,\n 2068273464883862,\n 1858621048097805,\n 1492281814208508,\n 2235868981918946,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1490330266465551,\n 1858795661361448,\n 3688040948655011,\n 2546373032584894,\n 3459939824714180,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1282462923712748,\n 741885683986255,\n 2027754642827561,\n 518989529541027,\n 1826610009555945,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1525827120027511,\n 723686461809551,\n 1597702369236987,\n 244802101764964,\n 1502833890372311,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2365421849929742,\n 3485539881431101,\n 2925909765963743,\n 2114345180342964,\n 2418564326541511,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2041668749310338,\n 2184405322203901,\n 1633400637611036,\n 2110682505536899,\n 2048144390084644,\n ]),\n xy2d: FieldElement51::from_limbs([\n 503058759232932,\n 760293024620937,\n 2027152777219493,\n 666858468148475,\n 1539184379870952,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1916168475367211,\n 3167426246226591,\n 883217071712574,\n 363427871374304,\n 1976029821251593,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 678039535434506,\n 570587290189340,\n 1605302676614120,\n 2147762562875701,\n 1706063797091704,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1439489648586438,\n 2194580753290951,\n 832380563557396,\n 561521973970522,\n 584497280718389,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2439789269177838,\n 681223515948274,\n 1933493571072456,\n 1872921007304880,\n 2739962177820919,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1413466089534451,\n 410844090765630,\n 1397263346404072,\n 408227143123410,\n 1594561803147811,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2102170800973153,\n 719462588665004,\n 1479649438510153,\n 1097529543970028,\n 1302363283777685,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3193865531532443,\n 3321113493038208,\n 2007341951411050,\n 2322773230131539,\n 1419433790163705,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1146565545556377,\n 1661971299445212,\n 406681704748893,\n 564452436406089,\n 1109109865829139,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2214421081775077,\n 1165671861210569,\n 1890453018796184,\n 3556249878661,\n 442116172656317,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3005630360306059,\n 1666955059895018,\n 1530775289309243,\n 3371786842789394,\n 2164156153857579,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 615171919212796,\n 1523849404854568,\n 854560460547503,\n 2067097370290715,\n 1765325848586042,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1094538949313667,\n 1796592198908825,\n 870221004284388,\n 2025558921863561,\n 1699010892802384,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1951351290725195,\n 1916457206844795,\n 2449824998123274,\n 1909076887557594,\n 1938542290318919,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1014323197538413,\n 869150639940606,\n 1756009942696599,\n 1334952557375672,\n 1544945379082874,\n ]),\n xy2d: FieldElement51::from_limbs([\n 764055910920305,\n 1603590757375439,\n 146805246592357,\n 1843313433854297,\n 954279890114939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 80113526615731,\n 764536758732259,\n 3306939158785481,\n 2721052465444637,\n 2869697326116762,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 74497112547268,\n 740094153192149,\n 1745254631717581,\n 727713886503130,\n 1283034364416928,\n ]),\n xy2d: FieldElement51::from_limbs([\n 525892105991110,\n 1723776830270342,\n 1476444848991936,\n 573789489857760,\n 133864092632978,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2794411533877810,\n 1986812262899320,\n 1162535242465837,\n 2733298779828712,\n 2796400347268869,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 64123227344372,\n 1239927720647794,\n 1360722983445904,\n 222610813654661,\n 62429487187991,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1793193323953132,\n 91096687857833,\n 70945970938921,\n 2158587638946380,\n 1537042406482111,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1895854577604590,\n 3646695522634664,\n 1728548428495943,\n 3392664713925397,\n 2815445147288308,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 141358280486863,\n 91435889572504,\n 1087208572552643,\n 1829599652522921,\n 1193307020643647,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1611230858525381,\n 950720175540785,\n 499589887488610,\n 2001656988495019,\n 88977313255908,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3440880315164906,\n 2184348804772596,\n 3292618539427567,\n 2018318290311833,\n 1712060030915354,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 873966876953756,\n 1090638350350440,\n 1708559325189137,\n 672344594801910,\n 1320437969700239,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1508590048271766,\n 1131769479776094,\n 101550868699323,\n 428297785557897,\n 561791648661744,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3008217384184691,\n 2489682092917849,\n 2136263418594015,\n 1701968045454886,\n 2955512998822720,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1781187809325462,\n 1697624151492346,\n 1381393690939988,\n 175194132284669,\n 1483054666415238,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2175517777364616,\n 708781536456029,\n 955668231122942,\n 1967557500069555,\n 2021208005604118,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3366935780292116,\n 2476017186636029,\n 915967306279221,\n 593866251291540,\n 2813546907893254,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1443163092879439,\n 391875531646162,\n 2180847134654632,\n 464538543018753,\n 1594098196837178,\n ]),\n xy2d: FieldElement51::from_limbs([\n 850858855888869,\n 319436476624586,\n 327807784938441,\n 740785849558761,\n 17128415486016,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2132756334090048,\n 2788047633840893,\n 2300706964962114,\n 2860273011285942,\n 3513489358708031,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1525176236978354,\n 974205476721062,\n 293436255662638,\n 148269621098039,\n 137961998433963,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1121075518299410,\n 2071745529082111,\n 1265567917414828,\n 1648196578317805,\n 496232102750820,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2374121042985030,\n 3274721891178932,\n 2001275453369483,\n 2017441881607947,\n 3245005694463250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 654925550560074,\n 1168810995576858,\n 575655959430926,\n 905758704861388,\n 496774564663534,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1954109525779738,\n 2117022646152485,\n 338102630417180,\n 1194140505732026,\n 107881734943492,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1714785840001267,\n 4288299832366837,\n 1876380234251965,\n 2056717182974196,\n 1645855254384642,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 106431476499341,\n 62482972120563,\n 1513446655109411,\n 807258751769522,\n 538491469114,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2002850762893643,\n 1243624520538135,\n 1486040410574605,\n 2184752338181213,\n 378495998083531,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 922510868424903,\n 1089502620807680,\n 402544072617374,\n 1131446598479839,\n 1290278588136533,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1867998812076769,\n 715425053580701,\n 39968586461416,\n 2173068014586163,\n 653822651801304,\n ]),\n xy2d: FieldElement51::from_limbs([\n 162892278589453,\n 182585796682149,\n 75093073137630,\n 497037941226502,\n 133871727117371,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4166396390264918,\n 1608999621851577,\n 1987629837704609,\n 1519655314857977,\n 1819193753409464,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1949315551096831,\n 1069003344994464,\n 1939165033499916,\n 1548227205730856,\n 1933767655861407,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1730519386931635,\n 1393284965610134,\n 1597143735726030,\n 416032382447158,\n 1429665248828629,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 360275475604546,\n 2799635544748326,\n 2467160717872776,\n 2848446553564254,\n 2584509464110332,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 47602113726801,\n 1522314509708010,\n 437706261372925,\n 814035330438027,\n 335930650933545,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1291597595523886,\n 1058020588994081,\n 402837842324045,\n 1363323695882781,\n 2105763393033193,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2361321796251793,\n 3967057562270386,\n 1112231216891515,\n 2046641005101484,\n 2386048970842261,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2156991030936798,\n 2227544497153325,\n 1869050094431622,\n 754875860479115,\n 1754242344267058,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1846089562873800,\n 98894784984326,\n 1412430299204844,\n 171351226625762,\n 1100604760929008,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2335972195815721,\n 2751510784385293,\n 425749630620777,\n 1762872794206857,\n 2864642415813208,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 868309334532756,\n 1703010512741873,\n 1952690008738057,\n 4325269926064,\n 2071083554962116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 523094549451158,\n 401938899487815,\n 1407690589076010,\n 2022387426254453,\n 158660516411257,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 612867287630009,\n 2700012425789062,\n 2823428891104443,\n 1466796750919375,\n 1728478129663858,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1723848973783452,\n 2208822520534681,\n 1718748322776940,\n 1974268454121942,\n 1194212502258141,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1254114807944608,\n 977770684047110,\n 2010756238954993,\n 1783628927194099,\n 1525962994408256,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2484263871921055,\n 1948628555342433,\n 1835348780427694,\n 1031609499437291,\n 2316271920603621,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 767338676040683,\n 754089548318405,\n 1523192045639075,\n 435746025122062,\n 512692508440385,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1255955808701983,\n 1700487367990941,\n 1166401238800299,\n 1175121994891534,\n 1190934801395380,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2600943821853521,\n 1337012557669161,\n 1475912332999108,\n 3573418268585706,\n 2299411105589567,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 877519947135419,\n 2172838026132651,\n 272304391224129,\n 1655143327559984,\n 886229406429814,\n ]),\n xy2d: FieldElement51::from_limbs([\n 375806028254706,\n 214463229793940,\n 572906353144089,\n 572168269875638,\n 697556386112979,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1168827102357825,\n 823864273033637,\n 4323338565789945,\n 788062026895923,\n 2851378154428610,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1948116082078088,\n 2054898304487796,\n 2204939184983900,\n 210526805152138,\n 786593586607626,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1915320147894736,\n 156481169009469,\n 655050471180417,\n 592917090415421,\n 2165897438660879,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1726336468579724,\n 1119932070398949,\n 1929199510967666,\n 2285718602008207,\n 1836837863503149,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 829996854845988,\n 217061778005138,\n 1686565909803640,\n 1346948817219846,\n 1723823550730181,\n ]),\n xy2d: FieldElement51::from_limbs([\n 384301494966394,\n 687038900403062,\n 2211195391021739,\n 254684538421383,\n 1245698430589680,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1247567493562669,\n 4229981908141095,\n 2435671288478202,\n 806570235643434,\n 2540261331753164,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1449077384734201,\n 38285445457996,\n 2136537659177832,\n 2146493000841573,\n 725161151123125,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1201928866368855,\n 800415690605445,\n 1703146756828343,\n 997278587541744,\n 1858284414104014,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2608268623334125,\n 3034173730618399,\n 1718002439402869,\n 3644022065904502,\n 663171266061950,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 759628738230460,\n 1012693474275852,\n 353780233086498,\n 246080061387552,\n 2030378857679162,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2040672435071076,\n 888593182036908,\n 1298443657189359,\n 1804780278521327,\n 354070726137060,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1894938527423184,\n 3715012855162525,\n 2726210319182898,\n 2499094776718546,\n 877975941029127,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 207937160991127,\n 12966911039119,\n 820997788283092,\n 1010440472205286,\n 1701372890140810,\n ]),\n xy2d: FieldElement51::from_limbs([\n 218882774543183,\n 533427444716285,\n 1233243976733245,\n 435054256891319,\n 1509568989549904,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4140638349397055,\n 3303977572025869,\n 3465353617009382,\n 2420981822812579,\n 2715174081801119,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 299137589460312,\n 1594371588983567,\n 868058494039073,\n 257771590636681,\n 1805012993142921,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1806842755664364,\n 2098896946025095,\n 1356630998422878,\n 1458279806348064,\n 347755825962072,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1402334161391744,\n 3811883484731547,\n 1008585416617746,\n 1147797150908892,\n 1420416683642459,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 665506704253369,\n 273770475169863,\n 799236974202630,\n 848328990077558,\n 1811448782807931,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1468412523962641,\n 771866649897997,\n 1931766110147832,\n 799561180078482,\n 524837559150077,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2223212657821831,\n 2882216061048914,\n 2144451165500327,\n 3068710944633039,\n 3276150872095279,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1266603897524861,\n 156378408858100,\n 1275649024228779,\n 447738405888420,\n 253186462063095,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2022215964509735,\n 136144366993649,\n 1800716593296582,\n 1193970603800203,\n 871675847064218,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1862751661970309,\n 851596246739884,\n 1519315554814041,\n 3794598280232697,\n 3669775149586767,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1228168094547481,\n 334133883362894,\n 587567568420081,\n 433612590281181,\n 603390400373205,\n ]),\n xy2d: FieldElement51::from_limbs([\n 121893973206505,\n 1843345804916664,\n 1703118377384911,\n 497810164760654,\n 101150811654673,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2710146069631716,\n 2542709749304591,\n 1452768413850678,\n 2802722688939463,\n 1537286854336537,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 584322311184395,\n 380661238802118,\n 114839394528060,\n 655082270500073,\n 2111856026034852,\n ]),\n xy2d: FieldElement51::from_limbs([\n 996965581008991,\n 2148998626477022,\n 1012273164934654,\n 1073876063914522,\n 1688031788934939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3175286832534829,\n 2085106799623354,\n 2779882615305384,\n 1606206360876187,\n 2987706905397772,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1697697887804317,\n 1335343703828273,\n 831288615207040,\n 949416685250051,\n 288760277392022,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1419122478109648,\n 1325574567803701,\n 602393874111094,\n 2107893372601700,\n 1314159682671307,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2201150872731785,\n 2180241023425241,\n 2349463270108411,\n 1633405770247823,\n 3100744856129234,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1173339555550611,\n 818605084277583,\n 47521504364289,\n 924108720564965,\n 735423405754506,\n ]),\n xy2d: FieldElement51::from_limbs([\n 830104860549448,\n 1886653193241086,\n 1600929509383773,\n 1475051275443631,\n 286679780900937,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3828911108518224,\n 3282698983453994,\n 2396700729978777,\n 4216472406664814,\n 2820189914640497,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 278388655910247,\n 487143369099838,\n 927762205508727,\n 181017540174210,\n 1616886700741287,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1191033906638969,\n 940823957346562,\n 1606870843663445,\n 861684761499847,\n 658674867251089,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1875032594195527,\n 1427106132796197,\n 2976536204647406,\n 3153660325729987,\n 2887068310954007,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622869792298357,\n 1903919278950367,\n 1922588621661629,\n 1520574711600434,\n 1087100760174640,\n ]),\n xy2d: FieldElement51::from_limbs([\n 25465949416618,\n 1693639527318811,\n 1526153382657203,\n 125943137857169,\n 145276964043999,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2466539671654587,\n 920212862967914,\n 4191701364657517,\n 3463662605460468,\n 2336897329405367,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2006245852772938,\n 734762734836159,\n 254642929763427,\n 1406213292755966,\n 239303749517686,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1619678837192149,\n 1919424032779215,\n 1357391272956794,\n 1525634040073113,\n 1310226789796241,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3292563523447371,\n 1704449869235351,\n 2857062884141577,\n 1998838089036354,\n 1312142911487502,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1996723311435669,\n 1844342766567060,\n 985455700466044,\n 1165924681400960,\n 311508689870129,\n ]),\n xy2d: FieldElement51::from_limbs([\n 43173156290518,\n 2202883069785309,\n 1137787467085917,\n 1733636061944606,\n 1394992037553852,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 670078326344559,\n 2807454838744604,\n 2723759199967685,\n 2141455487356408,\n 849015953823125,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2197214573372804,\n 794254097241315,\n 1030190060513737,\n 267632515541902,\n 2040478049202624,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1812516004670529,\n 1609256702920783,\n 1706897079364493,\n 258549904773295,\n 996051247540686,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1540374301420565,\n 1764656898914615,\n 1810104162020396,\n 3175608592848336,\n 2916189887881826,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1323460699404750,\n 1262690757880991,\n 871777133477900,\n 1060078894988977,\n 1712236889662886,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1696163952057966,\n 1391710137550823,\n 608793846867416,\n 1034391509472039,\n 1780770894075012,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1367603834210822,\n 4383788460268472,\n 890353773628143,\n 1908908219165595,\n 2522636708938139,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 597536315471731,\n 40375058742586,\n 1942256403956049,\n 1185484645495932,\n 312666282024145,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1919411405316294,\n 1234508526402192,\n 1066863051997083,\n 1008444703737597,\n 1348810787701552,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2102881477513865,\n 3822074379630609,\n 1573617900503707,\n 2270462449417831,\n 2232324307922097,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1853931367696942,\n 8107973870707,\n 350214504129299,\n 775206934582587,\n 1752317649166792,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1417148368003523,\n 721357181628282,\n 505725498207811,\n 373232277872983,\n 261634707184480,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2186733281493248,\n 2250694917008620,\n 1014829812957440,\n 2731797975137637,\n 2335366007561721,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1268116367301224,\n 560157088142809,\n 802626839600444,\n 2210189936605713,\n 1129993785579988,\n ]),\n xy2d: FieldElement51::from_limbs([\n 615183387352312,\n 917611676109240,\n 878893615973325,\n 978940963313282,\n 938686890583575,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 522024729211672,\n 3296859129001056,\n 1892245413707789,\n 1907891107684253,\n 2059998109500714,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1799679152208884,\n 912132775900387,\n 25967768040979,\n 432130448590461,\n 274568990261996,\n ]),\n xy2d: FieldElement51::from_limbs([\n 98698809797682,\n 2144627600856209,\n 1907959298569602,\n 811491302610148,\n 1262481774981493,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1791451399743152,\n 1713538728337276,\n 2370149810942738,\n 1882306388849953,\n 158235232210248,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1217809823321928,\n 2173947284933160,\n 1986927836272325,\n 1388114931125539,\n 12686131160169,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1650875518872272,\n 1136263858253897,\n 1732115601395988,\n 734312880662190,\n 1252904681142109,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2624786269799113,\n 2777230729143418,\n 2116279931702134,\n 2753222527273063,\n 1907002872974924,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 803147181835288,\n 868941437997146,\n 316299302989663,\n 943495589630550,\n 571224287904572,\n ]),\n xy2d: FieldElement51::from_limbs([\n 227742695588364,\n 1776969298667369,\n 628602552821802,\n 457210915378118,\n 2041906378111140,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 815000523470260,\n 3164885502413555,\n 3303859931956420,\n 1345536665214222,\n 541623413135555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1580216071604333,\n 1877997504342444,\n 857147161260913,\n 703522726778478,\n 2182763974211603,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1870080310923419,\n 71988220958492,\n 1783225432016732,\n 615915287105016,\n 1035570475990230,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2982787564515398,\n 857613889540279,\n 1083813157271766,\n 1002817255970169,\n 1719228484436074,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 377616581647602,\n 1581980403078513,\n 804044118130621,\n 2034382823044191,\n 643844048472185,\n ]),\n xy2d: FieldElement51::from_limbs([\n 176957326463017,\n 1573744060478586,\n 528642225008045,\n 1816109618372371,\n 1515140189765006,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1888911448245718,\n 3638910709296328,\n 4176303607751676,\n 1731539523700948,\n 2230378382645454,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 443392177002051,\n 233793396845137,\n 2199506622312416,\n 1011858706515937,\n 974676837063129,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1846351103143623,\n 1949984838808427,\n 671247021915253,\n 1946756846184401,\n 1929296930380217,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 849646212451983,\n 1410198775302919,\n 2325567699868943,\n 1641663456615811,\n 3014056086137659,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 692017667358279,\n 723305578826727,\n 1638042139863265,\n 748219305990306,\n 334589200523901,\n ]),\n xy2d: FieldElement51::from_limbs([\n 22893968530686,\n 2235758574399251,\n 1661465835630252,\n 925707319443452,\n 1203475116966621,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3053098849470395,\n 3985092410411378,\n 1664508947088595,\n 2719548934677170,\n 3899298398220870,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 903105258014366,\n 427141894933047,\n 561187017169777,\n 1884330244401954,\n 1914145708422219,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1344191060517578,\n 1960935031767890,\n 1518838929955259,\n 1781502350597190,\n 1564784025565682,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2925523165433334,\n 1979969272514922,\n 3427087126180756,\n 1187589090978665,\n 1881897672213940,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1917185587363432,\n 1098342571752737,\n 5935801044414,\n 2000527662351839,\n 1538640296181569,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2495540013192,\n 678856913479236,\n 224998292422872,\n 219635787698590,\n 1972465269000940,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 271413961212179,\n 3604851875156899,\n 2596511104968730,\n 2014925838520661,\n 2006221033113941,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 194583029968109,\n 514316781467765,\n 829677956235672,\n 1676415686873082,\n 810104584395840,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1980510813313589,\n 1948645276483975,\n 152063780665900,\n 129968026417582,\n 256984195613935,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1860190562533083,\n 1936576191345085,\n 2712900106391212,\n 1811043097042829,\n 3209286562992083,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 796664815624365,\n 1543160838872951,\n 1500897791837765,\n 1667315977988401,\n 599303877030711,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1151480509533204,\n 2136010406720455,\n 738796060240027,\n 319298003765044,\n 1150614464349587,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1731069268103131,\n 2987442261301335,\n 1364750481334267,\n 2669032653668119,\n 3178908082812908,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1017222050227968,\n 1987716148359,\n 2234319589635701,\n 621282683093392,\n 2132553131763026,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1567828528453324,\n 1017807205202360,\n 565295260895298,\n 829541698429100,\n 307243822276582,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 249079270936229,\n 1501514259790706,\n 3199709537890096,\n 944551802437486,\n 2804458577667728,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2089966982947227,\n 1854140343916181,\n 2151980759220007,\n 2139781292261749,\n 158070445864917,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1338766321464554,\n 1906702607371284,\n 1519569445519894,\n 115384726262267,\n 1393058953390992,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3616421371950629,\n 3764188048593604,\n 1926731583198685,\n 2041482526432505,\n 3172200936019022,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1884844597333588,\n 601480070269079,\n 620203503079537,\n 1079527400117915,\n 1202076693132015,\n ]),\n xy2d: FieldElement51::from_limbs([\n 840922919763324,\n 727955812569642,\n 1303406629750194,\n 522898432152867,\n 294161410441865,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2605560604520539,\n 1598361541848742,\n 3374705511887547,\n 4174333403844152,\n 2670907514351827,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 359856369838236,\n 180914355488683,\n 861726472646627,\n 218807937262986,\n 575626773232501,\n ]),\n xy2d: FieldElement51::from_limbs([\n 755467689082474,\n 909202735047934,\n 730078068932500,\n 936309075711518,\n 2007798262842972,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1609384177904054,\n 2614544999293875,\n 1335318541768200,\n 3052765584121496,\n 2799677792952659,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 984339177776787,\n 815727786505884,\n 1645154585713747,\n 1659074964378553,\n 1686601651984156,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1697863093781930,\n 599794399429786,\n 1104556219769607,\n 830560774794755,\n 12812858601017,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1168737550514982,\n 897832437380552,\n 463140296333799,\n 2554364413707795,\n 2008360505135500,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1856930662813910,\n 678090852002597,\n 1920179140755167,\n 1259527833759868,\n 55540971895511,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1158643631044921,\n 476554103621892,\n 178447851439725,\n 1305025542653569,\n 103433927680625,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2176793111709008,\n 3828525530035639,\n 2009350167273522,\n 2012390194631546,\n 2125297410909580,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 825403285195098,\n 2144208587560784,\n 1925552004644643,\n 1915177840006985,\n 1015952128947864,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1807108316634472,\n 1534392066433717,\n 347342975407218,\n 1153820745616376,\n 7375003497471,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3234860815484973,\n 2683011703586488,\n 2201903782961092,\n 3069193724749589,\n 2214616493042166,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 228567918409756,\n 865093958780220,\n 358083886450556,\n 159617889659320,\n 1360637926292598,\n ]),\n xy2d: FieldElement51::from_limbs([\n 234147501399755,\n 2229469128637390,\n 2175289352258889,\n 1397401514549353,\n 1885288963089922,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3363562226636810,\n 2504649386192636,\n 3300514047508588,\n 2397910909286693,\n 1237505378776769,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1113790697840279,\n 1051167139966244,\n 1045930658550944,\n 2011366241542643,\n 1686166824620755,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1054097349305049,\n 1872495070333352,\n 182121071220717,\n 1064378906787311,\n 100273572924182,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3558210666856834,\n 1627717417672446,\n 2302783034773665,\n 1109249951172249,\n 3122001602766640,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 104233794644221,\n 1548919791188248,\n 2224541913267306,\n 2054909377116478,\n 1043803389015153,\n ]),\n xy2d: FieldElement51::from_limbs([\n 216762189468802,\n 707284285441622,\n 190678557969733,\n 973969342604308,\n 1403009538434867,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3530824104723725,\n 2596576648903557,\n 2525521909702446,\n 4086000250496689,\n 634517197663803,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 343805853118335,\n 1302216857414201,\n 566872543223541,\n 2051138939539004,\n 321428858384280,\n ]),\n xy2d: FieldElement51::from_limbs([\n 470067171324852,\n 1618629234173951,\n 2000092177515639,\n 7307679772789,\n 1117521120249968,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2529951391976704,\n 1810282338562946,\n 1771599529530998,\n 3635459223356879,\n 2937173228157088,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 577009397403102,\n 1791440261786291,\n 2177643735971638,\n 174546149911960,\n 1412505077782326,\n ]),\n xy2d: FieldElement51::from_limbs([\n 893719721537457,\n 1201282458018197,\n 1522349501711173,\n 58011597740583,\n 1130406465887139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 412607348255434,\n 1280455764199780,\n 2233277987330768,\n 2265979894086913,\n 2583384512102412,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 262483770854550,\n 990511055108216,\n 526885552771698,\n 571664396646158,\n 354086190278723,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1820352417585487,\n 24495617171480,\n 1547899057533253,\n 10041836186225,\n 480457105094042,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2023310314989233,\n 2889705151211129,\n 2106474638900686,\n 2809620524769320,\n 1687858215057825,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1144168702609745,\n 604444390410187,\n 1544541121756138,\n 1925315550126027,\n 626401428894002,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1922168257351784,\n 2018674099908659,\n 1776454117494445,\n 956539191509034,\n 36031129147635,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2796444352433270,\n 1039872944430373,\n 3128550222815858,\n 2962457525011798,\n 3468752501170219,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 58242421545916,\n 2035812695641843,\n 2118491866122923,\n 1191684463816273,\n 46921517454099,\n ]),\n xy2d: FieldElement51::from_limbs([\n 272268252444639,\n 1374166457774292,\n 2230115177009552,\n 1053149803909880,\n 1354288411641016,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1857910905368338,\n 1754729879288912,\n 3137745277795125,\n 1516096106802165,\n 1602902393369811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1193437069800958,\n 901107149704790,\n 999672920611411,\n 477584824802207,\n 364239578697845,\n ]),\n xy2d: FieldElement51::from_limbs([\n 886299989548838,\n 1538292895758047,\n 1590564179491896,\n 1944527126709657,\n 837344427345298,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3006358179063534,\n 1712186480903617,\n 3955456640022779,\n 3002110732175033,\n 2770795853936147,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1309847803895382,\n 1462151862813074,\n 211370866671570,\n 1544595152703681,\n 1027691798954090,\n ]),\n xy2d: FieldElement51::from_limbs([\n 803217563745370,\n 1884799722343599,\n 1357706345069218,\n 2244955901722095,\n 730869460037413,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2941099284981214,\n 1831210565161070,\n 3626987155270686,\n 3358084791231418,\n 1893781834054268,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 696351368613042,\n 1494385251239250,\n 738037133616932,\n 636385507851544,\n 927483222611406,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1949114198209333,\n 1104419699537997,\n 783495707664463,\n 1747473107602770,\n 2002634765788641,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1607325776830197,\n 2782683755100581,\n 1451089452727894,\n 3833490970768671,\n 496100432831153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1068900648804224,\n 2006891997072550,\n 1134049269345549,\n 1638760646180091,\n 2055396084625778,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2222475519314561,\n 1870703901472013,\n 1884051508440561,\n 1344072275216753,\n 1318025677799069,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 155711679280637,\n 681100400509288,\n 389811735211209,\n 2135723811340709,\n 2660533024889373,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 7813206966729,\n 194444201427550,\n 2071405409526507,\n 1065605076176312,\n 1645486789731291,\n ]),\n xy2d: FieldElement51::from_limbs([\n 16625790644959,\n 1647648827778410,\n 1579910185572704,\n 436452271048548,\n 121070048451050,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3289062842237779,\n 2820185594063076,\n 2549752917829677,\n 3810384325616458,\n 2238221839292470,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 190565267697443,\n 672855706028058,\n 338796554369226,\n 337687268493904,\n 853246848691734,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1763863028400139,\n 766498079432444,\n 1321118624818005,\n 69494294452268,\n 858786744165651,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3543856582248253,\n 1456632109855637,\n 3352431060735432,\n 1386133165675320,\n 3484698163879000,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 366253102478259,\n 525676242508811,\n 1449610995265438,\n 1183300845322183,\n 185960306491545,\n ]),\n xy2d: FieldElement51::from_limbs([\n 28315355815982,\n 460422265558930,\n 1799675876678724,\n 1969256312504498,\n 1051823843138725,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2408714813047231,\n 3857948219405196,\n 1665208410108429,\n 2569443092377519,\n 1383783705665319,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 54684536365732,\n 2210010038536222,\n 1194984798155308,\n 535239027773705,\n 1516355079301361,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1484387703771650,\n 198537510937949,\n 2186282186359116,\n 617687444857508,\n 647477376402122,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2147715541830533,\n 2751832352131065,\n 2898179830570073,\n 2604027669016369,\n 1488268620408051,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 159386186465542,\n 1877626593362941,\n 618737197060512,\n 1026674284330807,\n 1158121760792685,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1744544377739822,\n 1964054180355661,\n 1685781755873170,\n 2169740670377448,\n 1286112621104591,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2333777063470241,\n 3919742931398333,\n 3920783633320113,\n 1605016835177614,\n 1353960708075544,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1602253788689063,\n 439542044889886,\n 2220348297664483,\n 657877410752869,\n 157451572512238,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1029287186166717,\n 65860128430192,\n 525298368814832,\n 1491902500801986,\n 1461064796385400,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2660016802414475,\n 2121095722306988,\n 913562102267595,\n 1879708920318308,\n 2492861262121979,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1185483484383269,\n 1356339572588553,\n 584932367316448,\n 102132779946470,\n 1792922621116791,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1966196870701923,\n 2230044620318636,\n 1425982460745905,\n 261167817826569,\n 46517743394330,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2358877405280588,\n 3136759755857592,\n 2279106683482647,\n 2224911448949389,\n 3216151871930471,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1730194207717538,\n 431790042319772,\n 1831515233279467,\n 1372080552768581,\n 1074513929381760,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1450880638731607,\n 1019861580989005,\n 1229729455116861,\n 1174945729836143,\n 826083146840706,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1899935429242705,\n 1602068751520477,\n 940583196550370,\n 2334230882739107,\n 1540863155745695,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2136688454840028,\n 2099509000964294,\n 1690800495246475,\n 1217643678575476,\n 828720645084218,\n ]),\n xy2d: FieldElement51::from_limbs([\n 765548025667841,\n 462473984016099,\n 998061409979798,\n 546353034089527,\n 2212508972466858,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2298375097456408,\n 3144370785258318,\n 1281983193144089,\n 1491520128287375,\n 75847005908304,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1801436127943107,\n 1734436817907890,\n 1268728090345068,\n 167003097070711,\n 2233597765834956,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1997562060465113,\n 1048700225534011,\n 7615603985628,\n 1855310849546841,\n 2242557647635213,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1161017320376250,\n 2744424393854291,\n 2169815802355236,\n 3228296595417790,\n 1770879511019628,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1357044908364776,\n 729130645262438,\n 1762469072918979,\n 1365633616878458,\n 181282906404941,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1080413443139865,\n 1155205815510486,\n 1848782073549786,\n 622566975152580,\n 124965574467971,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1184526762066993,\n 247622751762817,\n 2943928830891604,\n 3071818503097743,\n 2188697339828084,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2020536369003019,\n 202261491735136,\n 1053169669150884,\n 2056531979272544,\n 778165514694311,\n ]),\n xy2d: FieldElement51::from_limbs([\n 237404399610207,\n 1308324858405118,\n 1229680749538400,\n 720131409105291,\n 1958958863624906,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2767383321724075,\n 2269456792542436,\n 1717918437373988,\n 1568052070792483,\n 2298775616809171,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 281527309158085,\n 36970532401524,\n 866906920877543,\n 2222282602952734,\n 1289598729589882,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1278207464902042,\n 494742455008756,\n 1262082121427081,\n 1577236621659884,\n 1888786707293291,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 353042527954210,\n 1830056151907359,\n 1111731275799225,\n 2426760769524072,\n 404312815582674,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2064251142068628,\n 1666421603389706,\n 1419271365315441,\n 468767774902855,\n 191535130366583,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1716987058588002,\n 1859366439773457,\n 1767194234188234,\n 64476199777924,\n 1117233614485261,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3236091949205521,\n 2386938060636506,\n 2220652137473166,\n 1722843421165029,\n 2442282371698157,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 298845952651262,\n 1166086588952562,\n 1179896526238434,\n 1347812759398693,\n 1412945390096208,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1143239552672925,\n 906436640714209,\n 2177000572812152,\n 2075299936108548,\n 325186347798433,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2972824668060020,\n 2936287674948563,\n 3625238557779406,\n 2193186935276994,\n 1387043709851261,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 418098668140962,\n 715065997721283,\n 1471916138376055,\n 2168570337288357,\n 937812682637044,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1043584187226485,\n 2143395746619356,\n 2209558562919611,\n 482427979307092,\n 847556718384018,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1248731221520740,\n 1465200936117687,\n 2792603306395388,\n 2304778448366139,\n 2513234303861356,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1057329623869501,\n 620334067429122,\n 461700859268034,\n 2012481616501857,\n 297268569108938,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1055352180870759,\n 1553151421852298,\n 1510903185371259,\n 1470458349428097,\n 1226259419062731,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3744788603986897,\n 3042126439258578,\n 3441906842094992,\n 3641194565844440,\n 3872208010289441,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 47000654413729,\n 1004754424173864,\n 1868044813557703,\n 173236934059409,\n 588771199737015,\n ]),\n xy2d: FieldElement51::from_limbs([\n 30498470091663,\n 1082245510489825,\n 576771653181956,\n 806509986132686,\n 1317634017056939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2672107869436803,\n 3745154677001249,\n 2417006535213335,\n 4136645508605033,\n 2065456951573058,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1115636332012334,\n 1854340990964155,\n 83792697369514,\n 1972177451994021,\n 457455116057587,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1698968457310898,\n 1435137169051090,\n 1083661677032510,\n 938363267483709,\n 340103887207182,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1995325341336555,\n 911500251774648,\n 2415810569088940,\n 855378419194761,\n 3825401211214090,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 241719380661528,\n 310028521317150,\n 1215881323380194,\n 1408214976493624,\n 2141142156467363,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1315157046163473,\n 727368447885818,\n 1363466668108618,\n 1668921439990361,\n 1398483384337907,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2326829491984875,\n 3267188020145720,\n 1849729037055211,\n 4191614430138232,\n 2696204044080201,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2053597130993710,\n 2024431685856332,\n 2233550957004860,\n 2012407275509545,\n 872546993104440,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1217269667678610,\n 599909351968693,\n 1390077048548598,\n 1471879360694802,\n 739586172317596,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3970118453066023,\n 1560510726633957,\n 3156262694845170,\n 1418028351780051,\n 2346204163137185,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2132502667405250,\n 214379346175414,\n 1502748313768060,\n 1960071701057800,\n 1353971822643138,\n ]),\n xy2d: FieldElement51::from_limbs([\n 319394212043702,\n 2127459436033571,\n 717646691535162,\n 663366796076914,\n 318459064945314,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2657789238608841,\n 1960452633787082,\n 2919148848086913,\n 3744474074452359,\n 1451061489880786,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 947085906234007,\n 323284730494107,\n 1485778563977200,\n 728576821512394,\n 901584347702286,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1575783124125742,\n 2126210792434375,\n 1569430791264065,\n 1402582372904727,\n 1891780248341114,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3090232019245924,\n 4249503325136911,\n 3270591693593114,\n 1662001808174330,\n 2330127946643001,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 739152638255629,\n 2074935399403557,\n 505483666745895,\n 1611883356514088,\n 628654635394878,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1822054032121349,\n 643057948186973,\n 7306757352712,\n 577249257962099,\n 284735863382083,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3618358370049178,\n 1448606567552085,\n 3730680834630016,\n 2417602993041145,\n 1115718458123497,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 204146226972102,\n 1630511199034723,\n 2215235214174763,\n 174665910283542,\n 956127674017216,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1562934578796716,\n 1070893489712745,\n 11324610642270,\n 958989751581897,\n 2172552325473805,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1770564423056008,\n 2987323445349813,\n 1326060113795288,\n 1509650369341127,\n 2317692235267932,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 623682558650637,\n 1337866509471512,\n 990313350206649,\n 1314236615762469,\n 1164772974270275,\n ]),\n xy2d: FieldElement51::from_limbs([\n 223256821462517,\n 723690150104139,\n 1000261663630601,\n 933280913953265,\n 254872671543046,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1969087237026022,\n 2876595539132372,\n 1335555107635968,\n 2069986355593023,\n 3963899963027150,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1236103475266979,\n 1837885883267218,\n 1026072585230455,\n 1025865513954973,\n 1801964901432134,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1115241013365517,\n 1712251818829143,\n 2148864332502771,\n 2096001471438138,\n 2235017246626125,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3551068012286861,\n 2047148477845620,\n 2165648650132450,\n 1612539282026145,\n 2765997725314138,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 118352772338543,\n 1067608711804704,\n 1434796676193498,\n 1683240170548391,\n 230866769907437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850689576796636,\n 1601590730430274,\n 1139674615958142,\n 1954384401440257,\n 76039205311,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1723387471374172,\n 3249101280723658,\n 2785727448808904,\n 2272728458379212,\n 1756575222802512,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2146711623855116,\n 503278928021499,\n 625853062251406,\n 1109121378393107,\n 1033853809911861,\n ]),\n xy2d: FieldElement51::from_limbs([\n 571005965509422,\n 2005213373292546,\n 1016697270349626,\n 56607856974274,\n 914438579435146,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1346698876211176,\n 2076651707527589,\n 3336561384795453,\n 2517134292513653,\n 1068954492309670,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1769967932677654,\n 1695893319756416,\n 1151863389675920,\n 1781042784397689,\n 400287774418285,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1851867764003121,\n 403841933237558,\n 820549523771987,\n 761292590207581,\n 1743735048551143,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 410915148140008,\n 2107072311871739,\n 3256167275561751,\n 2351484709082008,\n 1180818713503223,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 285945406881439,\n 648174397347453,\n 1098403762631981,\n 1366547441102991,\n 1505876883139217,\n ]),\n xy2d: FieldElement51::from_limbs([\n 672095903120153,\n 1675918957959872,\n 636236529315028,\n 1569297300327696,\n 2164144194785875,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1902708175321798,\n 3287143344600686,\n 1178560808893262,\n 2552895497743394,\n 1280977479761117,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1615357281742403,\n 404257611616381,\n 2160201349780978,\n 1160947379188955,\n 1578038619549541,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2013087639791217,\n 822734930507457,\n 1785668418619014,\n 1668650702946164,\n 389450875221715,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2705718263383616,\n 2358206633614248,\n 2072540975937134,\n 308588860670238,\n 1304394580755385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295082798350326,\n 2091844511495996,\n 1851348972587817,\n 3375039684596,\n 789440738712837,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2083069137186154,\n 848523102004566,\n 993982213589257,\n 1405313299916317,\n 1532824818698468,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3747761112537659,\n 1397203457344778,\n 4026750030752190,\n 2391102557240943,\n 2318403398028034,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1782411379088302,\n 1096724939964781,\n 27593390721418,\n 542241850291353,\n 1540337798439873,\n ]),\n xy2d: FieldElement51::from_limbs([\n 693543956581437,\n 171507720360750,\n 1557908942697227,\n 1074697073443438,\n 1104093109037196,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 345288228393400,\n 3351443383432420,\n 2386681722088990,\n 1740551994106739,\n 2500011992985018,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 231429562203065,\n 1526290236421172,\n 2021375064026423,\n 1520954495658041,\n 806337791525116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1079623667189886,\n 872403650198613,\n 766894200588288,\n 2163700860774109,\n 2023464507911816,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 854645372543796,\n 1936406001954827,\n 2403260476226501,\n 3077125552956802,\n 1554306377287555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1497138821904622,\n 1044820250515590,\n 1742593886423484,\n 1237204112746837,\n 849047450816987,\n ]),\n xy2d: FieldElement51::from_limbs([\n 667962773375330,\n 1897271816877105,\n 1399712621683474,\n 1143302161683099,\n 2081798441209593,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2378947665252234,\n 1936114012888109,\n 1704424366552046,\n 3108474694401560,\n 2968403435020606,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1072409664800960,\n 2146937497077528,\n 1508780108920651,\n 935767602384853,\n 1112800433544068,\n ]),\n xy2d: FieldElement51::from_limbs([\n 333549023751292,\n 280219272863308,\n 2104176666454852,\n 1036466864875785,\n 536135186520207,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2625466093568366,\n 2398257055215356,\n 2555916080813104,\n 2667888562832962,\n 3510376944868638,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1186115062588401,\n 2251609796968486,\n 1098944457878953,\n 1153112761201374,\n 1791625503417267,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1870078460219737,\n 2129630962183380,\n 852283639691142,\n 292865602592851,\n 401904317342226,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1361070124828016,\n 815664541425524,\n 3278598711049919,\n 1951790935390646,\n 2807674705520038,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1546301003424277,\n 459094500062839,\n 1097668518375311,\n 1780297770129643,\n 720763293687608,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1212405311403990,\n 1536693382542438,\n 61028431067459,\n 1863929423417129,\n 1223219538638038,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1294303766540260,\n 3435357279640341,\n 3134071170918340,\n 2315654383110622,\n 2213283684565086,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 339050984211414,\n 601386726509773,\n 413735232134068,\n 966191255137228,\n 1839475899458159,\n ]),\n xy2d: FieldElement51::from_limbs([\n 235605972169408,\n 2174055643032978,\n 1538335001838863,\n 1281866796917192,\n 1815940222628465,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1632352921721536,\n 1833328609514701,\n 2092779091951987,\n 4175756015558474,\n 2210068022482918,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 35271216625062,\n 1712350667021807,\n 983664255668860,\n 98571260373038,\n 1232645608559836,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1998172393429622,\n 1798947921427073,\n 784387737563581,\n 1589352214827263,\n 1589861734168180,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1733739258725305,\n 2283515530744786,\n 2453769758904107,\n 3243892858242237,\n 1194308773174555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 846415389605137,\n 746163495539180,\n 829658752826080,\n 592067705956946,\n 957242537821393,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1758148849754419,\n 619249044817679,\n 168089007997045,\n 1371497636330523,\n 1867101418880350,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2578433797894864,\n 2513559319756263,\n 1700682323676192,\n 1577907266349064,\n 3469447477068264,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1714182387328607,\n 1477856482074168,\n 574895689942184,\n 2159118410227270,\n 1555532449716575,\n ]),\n xy2d: FieldElement51::from_limbs([\n 853828206885131,\n 998498946036955,\n 1835887550391235,\n 207627336608048,\n 258363815956050,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2392941288336925,\n 3488528558590503,\n 2894901233585134,\n 1646615130509172,\n 1208239602291765,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1501663228068911,\n 1354879465566912,\n 1444432675498247,\n 897812463852601,\n 855062598754348,\n ]),\n xy2d: FieldElement51::from_limbs([\n 714380763546606,\n 1032824444965790,\n 1774073483745338,\n 1063840874947367,\n 1738680636537158,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1640635546696233,\n 2884968766877360,\n 2212651044092395,\n 2282390772269100,\n 2620315074574625,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1171650314802029,\n 1567085444565577,\n 1453660792008405,\n 757914533009261,\n 1619511342778196,\n ]),\n xy2d: FieldElement51::from_limbs([\n 420958967093237,\n 971103481109486,\n 2169549185607107,\n 1301191633558497,\n 1661514101014240,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3158923465503550,\n 1332556122804145,\n 4075855067109735,\n 3619414031128206,\n 1982558335973171,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1121533090144639,\n 1021251337022187,\n 110469995947421,\n 1511059774758394,\n 2110035908131662,\n ]),\n xy2d: FieldElement51::from_limbs([\n 303213233384524,\n 2061932261128138,\n 352862124777736,\n 40828818670255,\n 249879468482660,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 856559257852200,\n 2760317478634258,\n 3629993581580163,\n 3975258940632376,\n 1962275756614520,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1445691340537320,\n 40614383122127,\n 402104303144865,\n 485134269878232,\n 1659439323587426,\n ]),\n xy2d: FieldElement51::from_limbs([\n 20057458979482,\n 1183363722525800,\n 2140003847237215,\n 2053873950687614,\n 2112017736174909,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2228654250927986,\n 3735391177100515,\n 1368661293910955,\n 3328311098862539,\n 526650682059607,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 709481497028540,\n 531682216165724,\n 316963769431931,\n 1814315888453765,\n 258560242424104,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1053447823660455,\n 1955135194248683,\n 1010900954918985,\n 1182614026976701,\n 1240051576966610,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1957943897155478,\n 1788667368028035,\n 2389492723714354,\n 2252839333292309,\n 3078204576998275,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1848942433095597,\n 1582009882530495,\n 1849292741020143,\n 1068498323302788,\n 2001402229799484,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1528282417624269,\n 2142492439828191,\n 2179662545816034,\n 362568973150328,\n 1591374675250271,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2411826493119617,\n 2484141002903963,\n 2149181472355544,\n 598041771119831,\n 2435658815595421,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2013278155187349,\n 662660471354454,\n 793981225706267,\n 411706605985744,\n 804490933124791,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2051892037280204,\n 488391251096321,\n 2230187337030708,\n 930221970662692,\n 679002758255210,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1530723630438670,\n 875873929577927,\n 2593359947955236,\n 2701702933216000,\n 1055551308214178,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1461835919309432,\n 1955256480136428,\n 180866187813063,\n 1551979252664528,\n 557743861963950,\n ]),\n xy2d: FieldElement51::from_limbs([\n 359179641731115,\n 1324915145732949,\n 902828372691474,\n 294254275669987,\n 1887036027752957,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4295071423139571,\n 2038225437857463,\n 1317528426475850,\n 1398989128982787,\n 2027639881006861,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2072902725256516,\n 312132452743412,\n 309930885642209,\n 996244312618453,\n 1590501300352303,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1397254305160710,\n 695734355138021,\n 2233992044438756,\n 1776180593969996,\n 1085588199351115,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2692366865016258,\n 2506694600041928,\n 2745669038615469,\n 1556322069683365,\n 3819256354004466,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1950722461391320,\n 1907845598854797,\n 1822757481635527,\n 2121567704750244,\n 73811931471221,\n ]),\n xy2d: FieldElement51::from_limbs([\n 387139307395758,\n 2058036430315676,\n 1220915649965325,\n 1794832055328951,\n 1230009312169328,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1765973779329498,\n 2911143873132225,\n 2271621715291913,\n 3553728154996461,\n 3368065817761132,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1127572801181483,\n 1224743760571696,\n 1276219889847274,\n 1529738721702581,\n 1589819666871853,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2181229378964934,\n 2190885205260020,\n 1511536077659137,\n 1246504208580490,\n 668883326494241,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2689666469258543,\n 2920826224880015,\n 2333696811665585,\n 523874406393177,\n 2496851874620484,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1975438052228868,\n 1071801519999806,\n 594652299224319,\n 1877697652668809,\n 1489635366987285,\n ]),\n xy2d: FieldElement51::from_limbs([\n 958592545673770,\n 233048016518599,\n 851568750216589,\n 567703851596087,\n 1740300006094761,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2014540178270324,\n 192672779514432,\n 2465676996326778,\n 2194819933853410,\n 1716422829364835,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1540769606609725,\n 2148289943846077,\n 1597804156127445,\n 1230603716683868,\n 815423458809453,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1738560251245018,\n 1779576754536888,\n 1783765347671392,\n 1880170990446751,\n 1088225159617541,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2911103727614740,\n 1956447718227572,\n 1830568515922666,\n 3092868863429656,\n 1669607124206367,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1143465490433355,\n 1532194726196059,\n 1093276745494697,\n 481041706116088,\n 2121405433561163,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1686424298744462,\n 1451806974487153,\n 266296068846582,\n 1834686947542675,\n 1720762336132256,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3141016840074207,\n 3295090436969907,\n 3107924901237156,\n 1669272323124635,\n 1603340330827879,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1206396181488998,\n 333158148435054,\n 1402633492821422,\n 1120091191722026,\n 1945474114550509,\n ]),\n xy2d: FieldElement51::from_limbs([\n 766720088232571,\n 1512222781191002,\n 1189719893490790,\n 2091302129467914,\n 2141418006894941,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2671463460991841,\n 1998875112167986,\n 3678399683938955,\n 3406728169064757,\n 2738338345823434,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 938160078005954,\n 1421776319053174,\n 1941643234741774,\n 180002183320818,\n 1414380336750546,\n ]),\n xy2d: FieldElement51::from_limbs([\n 398001940109652,\n 1577721237663248,\n 1012748649830402,\n 1540516006905144,\n 1011684812884559,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1653276489969611,\n 2257881638852872,\n 1921777941170835,\n 1604139841794531,\n 3113010867325889,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 996661541407379,\n 1455877387952927,\n 744312806857277,\n 139213896196746,\n 1000282908547789,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1450817495603008,\n 1476865707053229,\n 1030490562252053,\n 620966950353376,\n 1744760161539058,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2811528223687828,\n 2288856475326432,\n 2038622963352005,\n 1637244893271723,\n 3278365165924196,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 962165956135846,\n 1116599660248791,\n 182090178006815,\n 1455605467021751,\n 196053588803284,\n ]),\n xy2d: FieldElement51::from_limbs([\n 796863823080135,\n 1897365583584155,\n 420466939481601,\n 2165972651724672,\n 932177357788289,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 877047233620613,\n 1375632631944375,\n 2895573425567369,\n 2911822552533124,\n 2271153746017078,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2216943882299338,\n 394841323190322,\n 2222656898319671,\n 558186553950529,\n 1077236877025190,\n ]),\n xy2d: FieldElement51::from_limbs([\n 801118384953213,\n 1914330175515892,\n 574541023311511,\n 1471123787903705,\n 1526158900256288,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3201417702772463,\n 2207116611267330,\n 3164719852826535,\n 2752958352884036,\n 2314162374456719,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1474518386765335,\n 1760793622169197,\n 1157399790472736,\n 1622864308058898,\n 165428294422792,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1961673048027128,\n 102619413083113,\n 1051982726768458,\n 1603657989805485,\n 1941613251499678,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1401939116319247,\n 2587106153588320,\n 2323846009771033,\n 862423201496005,\n 3102318568216632,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1234706593321979,\n 1083343891215917,\n 898273974314935,\n 1640859118399498,\n 157578398571149,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1143483057726416,\n 1992614991758919,\n 674268662140796,\n 1773370048077526,\n 674318359920189,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1835401379538542,\n 173900035308392,\n 818247630716732,\n 4013900225838034,\n 1021506399448290,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1506632088156630,\n 2127481795522179,\n 513812919490255,\n 140643715928370,\n 442476620300318,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2056683376856736,\n 219094741662735,\n 2193541883188309,\n 1841182310235800,\n 556477468664293,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3566819241596075,\n 1049075855992602,\n 4318372866671791,\n 2518704280870781,\n 2040482348591519,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 94096246544434,\n 922482381166992,\n 24517828745563,\n 2139430508542503,\n 2097139044231004,\n ]),\n xy2d: FieldElement51::from_limbs([\n 537697207950515,\n 1399352016347350,\n 1563663552106345,\n 2148749520888918,\n 549922092988516,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1747985413252415,\n 680511052635695,\n 1809559829982725,\n 2846074064615302,\n 2453472984431229,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 323583936109569,\n 1973572998577657,\n 1192219029966558,\n 79354804385273,\n 1374043025560347,\n ]),\n xy2d: FieldElement51::from_limbs([\n 213277331329947,\n 416202017849623,\n 1950535221091783,\n 1313441578103244,\n 2171386783823658,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2440888617915079,\n 993969372859109,\n 3147669935222235,\n 3799101348983503,\n 1477373024911349,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1620578418245010,\n 541035331188469,\n 2235785724453865,\n 2154865809088198,\n 1974627268751826,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1346805451740245,\n 1350981335690626,\n 942744349501813,\n 2155094562545502,\n 1012483751693409,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2107080134091762,\n 1132567062788208,\n 1824935377687210,\n 769194804343737,\n 1857941799971888,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1074666112436467,\n 249279386739593,\n 1174337926625354,\n 1559013532006480,\n 1472287775519121,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1872620123779532,\n 1892932666768992,\n 1921559078394978,\n 1270573311796160,\n 1438913646755037,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3089190001333428,\n 3264053113908846,\n 989780015893986,\n 1351393287739814,\n 2580427560230798,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1028328827183114,\n 1711043289969857,\n 1350832470374933,\n 1923164689604327,\n 1495656368846911,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1900828492104143,\n 430212361082163,\n 687437570852799,\n 832514536673512,\n 1685641495940794,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3094432661621646,\n 605670026766215,\n 290836444839585,\n 2415010588577604,\n 2213815011799644,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1176336383453996,\n 1725477294339771,\n 12700622672454,\n 678015708818208,\n 162724078519879,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1448049969043497,\n 1789411762943521,\n 385587766217753,\n 90201620913498,\n 832999441066823,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2767886146978542,\n 2240508292484615,\n 3603469341851756,\n 3475055379001735,\n 3002035638112385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1263624896582495,\n 1102602401673328,\n 526302183714372,\n 2152015839128799,\n 1483839308490010,\n ]),\n xy2d: FieldElement51::from_limbs([\n 442991718646863,\n 1599275157036458,\n 1925389027579192,\n 899514691371390,\n 350263251085160,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1689713572022124,\n 2845654372939621,\n 3229894858477217,\n 1985127338729498,\n 3927868934032873,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1557207018622683,\n 340631692799603,\n 1477725909476187,\n 614735951619419,\n 2033237123746766,\n ]),\n xy2d: FieldElement51::from_limbs([\n 968764929340557,\n 1225534776710944,\n 662967304013036,\n 1155521416178595,\n 791142883466590,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1487081286167458,\n 3244839255500182,\n 1792378982844639,\n 2950452258685122,\n 2153908693179753,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1123181311102823,\n 685575944875442,\n 507605465509927,\n 1412590462117473,\n 568017325228626,\n ]),\n xy2d: FieldElement51::from_limbs([\n 560258797465417,\n 2193971151466401,\n 1824086900849026,\n 579056363542056,\n 1690063960036441,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1918407319222397,\n 2605567366745211,\n 1930426334528098,\n 1564816146005724,\n 4113142195393344,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2131325168777276,\n 1176636658428908,\n 1756922641512981,\n 1390243617176012,\n 1966325177038383,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2063958120364491,\n 2140267332393533,\n 699896251574968,\n 273268351312140,\n 375580724713232,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2024297515263178,\n 2668759143407935,\n 3330814048702549,\n 2423412039258430,\n 1031677520051052,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2033900009388450,\n 1744902869870788,\n 2190580087917640,\n 1949474984254121,\n 231049754293748,\n ]),\n xy2d: FieldElement51::from_limbs([\n 343868674606581,\n 550155864008088,\n 1450580864229630,\n 481603765195050,\n 896972360018042,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2151139328380127,\n 2566545695770176,\n 2311556639460451,\n 1676664391494650,\n 2048348075599360,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1528930066340597,\n 1605003907059576,\n 1055061081337675,\n 1458319101947665,\n 1234195845213142,\n ]),\n xy2d: FieldElement51::from_limbs([\n 830430507734812,\n 1780282976102377,\n 1425386760709037,\n 362399353095425,\n 2168861579799910,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3407562046415562,\n 980662895504005,\n 2053766700883521,\n 2742766027762854,\n 2762205690726604,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1683750316716132,\n 652278688286128,\n 1221798761193539,\n 1897360681476669,\n 319658166027343,\n ]),\n xy2d: FieldElement51::from_limbs([\n 618808732869972,\n 72755186759744,\n 2060379135624181,\n 1730731526741822,\n 48862757828238,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3714971784278753,\n 3394840525452699,\n 614590986558882,\n 1409210575145591,\n 1882816996436803,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2230133264691131,\n 563950955091024,\n 2042915975426398,\n 827314356293472,\n 672028980152815,\n ]),\n xy2d: FieldElement51::from_limbs([\n 264204366029760,\n 1654686424479449,\n 2185050199932931,\n 2207056159091748,\n 506015669043634,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1784446333136550,\n 1973746527984364,\n 334856327359575,\n 3408569589569858,\n 3275749938360725,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2065270940578383,\n 31477096270353,\n 306421879113491,\n 181958643936686,\n 1907105536686083,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1496516440779464,\n 1748485652986458,\n 872778352227340,\n 818358834654919,\n 97932669284220,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2723435829455580,\n 2924255216478824,\n 1804995246884102,\n 1842309243470804,\n 3753662318666930,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1013216974933691,\n 538921919682598,\n 1915776722521558,\n 1742822441583877,\n 1886550687916656,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2094270000643336,\n 303971879192276,\n 40801275554748,\n 649448917027930,\n 1818544418535447,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2241737709499146,\n 549397817447461,\n 838180519319392,\n 1725686958520781,\n 3957438894582995,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1216074541925116,\n 50120933933509,\n 1565829004133810,\n 721728156134580,\n 349206064666188,\n ]),\n xy2d: FieldElement51::from_limbs([\n 948617110470858,\n 346222547451945,\n 1126511960599975,\n 1759386906004538,\n 493053284802266,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1454933046815146,\n 3126495827951610,\n 1467170975468587,\n 1432316382418897,\n 2111710746366763,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2105387117364450,\n 1996463405126433,\n 1303008614294500,\n 851908115948209,\n 1353742049788635,\n ]),\n xy2d: FieldElement51::from_limbs([\n 750300956351719,\n 1487736556065813,\n 15158817002104,\n 1511998221598392,\n 971739901354129,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1874648163531674,\n 2124487685930551,\n 1810030029384882,\n 918400043048335,\n 2838148440985898,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1235084464747900,\n 1166111146432082,\n 1745394857881591,\n 1405516473883040,\n 4463504151617,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1663810156463827,\n 327797390285791,\n 1341846161759410,\n 1964121122800605,\n 1747470312055380,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 660005247548214,\n 2071860029952887,\n 3610548013635355,\n 911703252219106,\n 3266179736709079,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2206641276178231,\n 1690587809721504,\n 1600173622825126,\n 2156096097634421,\n 1106822408548216,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1344788193552206,\n 1949552134239140,\n 1735915881729557,\n 675891104100469,\n 1834220014427292,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1920949492387945,\n 2410685102072778,\n 2322108077349280,\n 2877838278583064,\n 3719881539786256,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622221042073383,\n 1210146474039168,\n 1742246422343683,\n 1403839361379025,\n 417189490895736,\n ]),\n xy2d: FieldElement51::from_limbs([\n 22727256592983,\n 168471543384997,\n 1324340989803650,\n 1839310709638189,\n 504999476432775,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3565040332441556,\n 1721896294296941,\n 2304063388272514,\n 2065069734239231,\n 3056710287109878,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1337466662091884,\n 1287645354669772,\n 2018019646776184,\n 652181229374245,\n 898011753211715,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1969792547910734,\n 779969968247557,\n 2011350094423418,\n 1823964252907487,\n 1058949448296945,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2459143550747250,\n 1118176942430252,\n 3010694408233412,\n 806764629546265,\n 1157700123092949,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1273565321399022,\n 1638509681964574,\n 759235866488935,\n 666015124346707,\n 897983460943405,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1717263794012298,\n 1059601762860786,\n 1837819172257618,\n 1054130665797229,\n 680893204263559,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2237039662793603,\n 2249022333361206,\n 2058613546633703,\n 2401253908530527,\n 2215176649164581,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 79472182719605,\n 1851130257050174,\n 1825744808933107,\n 821667333481068,\n 781795293511946,\n ]),\n xy2d: FieldElement51::from_limbs([\n 755822026485370,\n 152464789723500,\n 1178207602290608,\n 410307889503239,\n 156581253571278,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3669985309815545,\n 2736319981413860,\n 3898537095128197,\n 3653287498355512,\n 1349185550126960,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1495380034400429,\n 325049476417173,\n 46346894893933,\n 1553408840354856,\n 828980101835683,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1280337889310282,\n 2070832742866672,\n 1640940617225222,\n 2098284908289951,\n 450929509534434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2659503167684029,\n 2378371955168899,\n 2537839641198868,\n 1999255076709337,\n 2030511179441770,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1254958221100483,\n 1153235960999843,\n 942907704968834,\n 637105404087392,\n 1149293270147267,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894249020470196,\n 400291701616810,\n 406878712230981,\n 1599128793487393,\n 1145868722604026,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3749755063888563,\n 2361916158338507,\n 1128535642171975,\n 1900106496009660,\n 2381592531146157,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 452487513298665,\n 1352120549024569,\n 1173495883910956,\n 1999111705922009,\n 367328130454226,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1717539401269642,\n 1475188995688487,\n 891921989653942,\n 836824441505699,\n 1885988485608364,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3493583935107776,\n 2439136865632830,\n 3370281625921440,\n 2680547565621609,\n 2282158712612572,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2022432361201842,\n 1088816090685051,\n 1977843398539868,\n 1854834215890724,\n 564238862029357,\n ]),\n xy2d: FieldElement51::from_limbs([\n 938868489100585,\n 1100285072929025,\n 1017806255688848,\n 1957262154788833,\n 152787950560442,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3119119231364171,\n 2872271776627789,\n 2477832016990963,\n 2593801257642876,\n 1761675818237335,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295072362439987,\n 931227904689414,\n 1355731432641687,\n 922235735834035,\n 892227229410209,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1680989767906154,\n 535362787031440,\n 2136691276706570,\n 1942228485381244,\n 1267350086882274,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2617818047455756,\n 2684460443440843,\n 2378209521329782,\n 1973842949591661,\n 2897427157127624,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 535509430575217,\n 546885533737322,\n 1524675609547799,\n 2138095752851703,\n 1260738089896827,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1159906385590467,\n 2198530004321610,\n 714559485023225,\n 81880727882151,\n 1484020820037082,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1377485731340769,\n 2046328105512000,\n 1802058637158797,\n 2313945950453421,\n 1356993908853900,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2013612215646735,\n 1830770575920375,\n 536135310219832,\n 609272325580394,\n 270684344495013,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1237542585982777,\n 2228682050256790,\n 1385281931622824,\n 593183794882890,\n 493654978552689,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2299141301692989,\n 1891414891220256,\n 983894663308928,\n 2427961581972066,\n 3378060928864955,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1694030170963455,\n 502038567066200,\n 1691160065225467,\n 949628319562187,\n 275110186693066,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1124515748676336,\n 1661673816593408,\n 1499640319059718,\n 1584929449166988,\n 558148594103306,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1784525599998356,\n 1619698033617383,\n 2097300287550715,\n 2510065271789004,\n 1905684794832757,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1288941072872766,\n 931787902039402,\n 190731008859042,\n 2006859954667190,\n 1005931482221702,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1465551264822703,\n 152905080555927,\n 680334307368453,\n 173227184634745,\n 666407097159852,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2111017076203943,\n 3630560299479595,\n 1248583954016455,\n 3604089008549670,\n 1895180776543895,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 171348223915638,\n 662766099800389,\n 462338943760497,\n 466917763340314,\n 656911292869115,\n ]),\n xy2d: FieldElement51::from_limbs([\n 488623681976577,\n 866497561541722,\n 1708105560937768,\n 1673781214218839,\n 1506146329818807,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2412225278142205,\n 950394373239688,\n 2682296937026182,\n 711676555398831,\n 320964687779005,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 988979367990485,\n 1359729327576302,\n 1301834257246029,\n 294141160829308,\n 29348272277475,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1434382743317910,\n 100082049942065,\n 221102347892623,\n 186982837860588,\n 1305765053501834,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2205916462268190,\n 2751663643476068,\n 961960554686615,\n 2409862576442233,\n 1841471168298304,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1191737341426592,\n 1847042034978363,\n 1382213545049056,\n 1039952395710448,\n 788812858896859,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1346965964571152,\n 1291881610839830,\n 2142916164336056,\n 786821641205979,\n 1571709146321039,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 787164375951248,\n 2454669019058437,\n 3608390234717387,\n 1431233331032509,\n 786341368775957,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 492448143532951,\n 304105152670757,\n 1761767168301056,\n 233782684697790,\n 1981295323106089,\n ]),\n xy2d: FieldElement51::from_limbs([\n 665807507761866,\n 1343384868355425,\n 895831046139653,\n 439338948736892,\n 1986828765695105,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3007896024559801,\n 1721699973539148,\n 2510565115413133,\n 1390588532210644,\n 1212530909934781,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 852891097972275,\n 1816988871354562,\n 1543772755726524,\n 1174710635522444,\n 202129090724628,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1205281565824323,\n 22430498399418,\n 992947814485516,\n 1392458699738672,\n 688441466734558,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3302427242100220,\n 1955849529137134,\n 2171162376368357,\n 2343545681983462,\n 447733118757825,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1287181461435438,\n 622722465530711,\n 880952150571872,\n 741035693459198,\n 311565274989772,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1003649078149734,\n 545233927396469,\n 1849786171789880,\n 1318943684880434,\n 280345687170552,\n ]),\n },\n ]),\n]);\n\n/// Odd multiples of the basepoint `[B, 3B, 5B, 7B, 9B, 11B, 13B, 15B, ..., 127B]`.\n#[cfg(feature = \"precomputed-tables\")]\n#[allow(dead_code)]\npub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT: NafLookupTable8 =\n NafLookupTable8([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3540182452943730,\n 2497478415033846,\n 2521227595762870,\n 1462984067271729,\n 2389212253076811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 62697248952638,\n 204681361388450,\n 631292143396476,\n 338455783676468,\n 1213667448819585,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301289933810280,\n 1259582250014073,\n 1422107436869536,\n 796239922652654,\n 1953934009299142,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1601611775252272,\n 1720807796594148,\n 1132070835939856,\n 3512254832574799,\n 2147779492816910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 316559037616741,\n 2177824224946892,\n 1459442586438991,\n 1461528397712656,\n 751590696113597,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850748884277385,\n 1200145853858453,\n 1068094770532492,\n 672251375690438,\n 1586055907191707,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 769950342298400,\n 2384754244604994,\n 3095885746880802,\n 3225892188161580,\n 2977876099231263,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 425251763115706,\n 608463272472562,\n 442562545713235,\n 837766094556764,\n 374555092627893,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1086255230780037,\n 274979815921559,\n 1960002765731872,\n 929474102396301,\n 1190409889297339,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2916800678241215,\n 2065379846933858,\n 2622030924071124,\n 2602788184473875,\n 1233371373142984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2019367628972465,\n 676711900706637,\n 110710997811333,\n 1108646842542025,\n 517791959672113,\n ]),\n xy2d: FieldElement51::from_limbs([\n 965130719900578,\n 247011430587952,\n 526356006571389,\n 91986625355052,\n 2157223321444601,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1802695059464988,\n 1664899123557221,\n 2845359304426105,\n 2160434469266658,\n 3179370264440279,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1725674970513508,\n 1933645953859181,\n 1542344539275782,\n 1767788773573747,\n 1297447965928905,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1381809363726107,\n 1430341051343062,\n 2061843536018959,\n 1551778050872521,\n 2036394857967624,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4222693909998302,\n 2779866139518454,\n 1619374932191226,\n 2207306624415883,\n 1169170329061080,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2070390218572616,\n 1458919061857835,\n 624171843017421,\n 1055332792707765,\n 433987520732508,\n ]),\n xy2d: FieldElement51::from_limbs([\n 893653801273833,\n 1168026499324677,\n 1242553501121234,\n 1306366254304474,\n 1086752658510815,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2465253816303469,\n 3191571337672685,\n 1159882208056013,\n 2569188183312765,\n 621213314200686,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1971678598905747,\n 338026507889165,\n 762398079972271,\n 655096486107477,\n 42299032696322,\n ]),\n xy2d: FieldElement51::from_limbs([\n 177130678690680,\n 1754759263300204,\n 1864311296286618,\n 1180675631479880,\n 1292726903152791,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1913163449625248,\n 2712579013977241,\n 2193883288642313,\n 1008900146920800,\n 1721983679009502,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1070401523076875,\n 1272492007800961,\n 1910153608563310,\n 2075579521696771,\n 1191169788841221,\n ]),\n xy2d: FieldElement51::from_limbs([\n 692896803108118,\n 500174642072499,\n 2068223309439677,\n 1162190621851337,\n 1426986007309901,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1819621230288238,\n 2735700366193240,\n 1755134670739586,\n 3080648199451191,\n 4172807995775876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 992069868904071,\n 799011518185730,\n 1777586403832768,\n 1134820506145684,\n 1999461475558530,\n ]),\n xy2d: FieldElement51::from_limbs([\n 425204543703124,\n 2040469794090382,\n 1651690622153809,\n 1500530168597569,\n 1253908377065966,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2105824306960939,\n 1387520302709358,\n 3633176580451016,\n 2211816663841753,\n 1629085891776489,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1485201376284999,\n 1022406647424656,\n 504181009209019,\n 962621520820995,\n 590876713147230,\n ]),\n xy2d: FieldElement51::from_limbs([\n 265873406365287,\n 1192742653492898,\n 88553098803050,\n 525037770869640,\n 1266933811251234,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3552316659826612,\n 1254279525791875,\n 1609927932077699,\n 3578654071679972,\n 3750681296069893,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 37186803519861,\n 1404297334376301,\n 578519728836650,\n 1740727951192592,\n 2095534282477028,\n ]),\n xy2d: FieldElement51::from_limbs([\n 833234263154399,\n 2023862470013762,\n 1854137933982069,\n 853924318090959,\n 1589812702805850,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3679150557957763,\n 1319179453661745,\n 497496853611112,\n 2665464286942351,\n 1208137952365560,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1654513078530905,\n 907489875842908,\n 126098711296368,\n 1726320004173677,\n 28269495058173,\n ]),\n xy2d: FieldElement51::from_limbs([\n 114436686957443,\n 532739313025996,\n 115428841215897,\n 2191499400074366,\n 370280402676434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1111146849833253,\n 2016430049079759,\n 1860522747477948,\n 3537164738290194,\n 4137142824844184,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 429069864577128,\n 975327637149449,\n 237881983565075,\n 1654761232378630,\n 2122527599091807,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2093793463548278,\n 754827233241879,\n 1420389751719629,\n 1829952782588138,\n 2011865756773717,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 676293365438898,\n 2850296017886344,\n 1205350322490195,\n 2763699392265669,\n 2133931188538142,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 48340340349120,\n 1299261101494832,\n 1137329686775218,\n 1534848106674340,\n 1351662218216799,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1904520614137939,\n 1590301001714014,\n 215781420985270,\n 2043534301034629,\n 1970888949300424,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2365217962409710,\n 2061307169694064,\n 1887478590157603,\n 2169639621284316,\n 2373810867477200,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1020052624656948,\n 1260412094216707,\n 366721640607121,\n 585331442306596,\n 345876457758061,\n ]),\n xy2d: FieldElement51::from_limbs([\n 975390299880933,\n 1066555195234642,\n 12651997758352,\n 1184252205433068,\n 1058378155074223,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1431537716602643,\n 2024827957433813,\n 3746434518400495,\n 1087794891033550,\n 2156817571680455,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 929288033346881,\n 255179964546973,\n 711057989588035,\n 208899572612840,\n 185348357387383,\n ]),\n xy2d: FieldElement51::from_limbs([\n 823689746424808,\n 47266130989546,\n 209403309368097,\n 1100966895202707,\n 710792075292719,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2311213117823762,\n 3296668540922318,\n 2004276520649823,\n 1861500579441125,\n 3148029033359833,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1563693677475261,\n 1843782073741194,\n 1950700654453170,\n 911540858113949,\n 2085151496302359,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1427880892005482,\n 106216431121745,\n 42608394782284,\n 1217295886989793,\n 1514235272796882,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3544335535746750,\n 2367994491347456,\n 2567261456502612,\n 1854058085060971,\n 2263545563461076,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 787426011300053,\n 2105981035769060,\n 1130476291127206,\n 1748659348100075,\n 53470983013756,\n ]),\n xy2d: FieldElement51::from_limbs([\n 553548273865386,\n 5927805718390,\n 65184587381926,\n 633576679686953,\n 576048559439973,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 993787326657446,\n 3868807161609258,\n 1615796046728943,\n 2514644292681953,\n 2059021068660907,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 251010270518880,\n 1681684095763484,\n 1521949356387564,\n 431593457045116,\n 1855308922422910,\n ]),\n xy2d: FieldElement51::from_limbs([\n 618490909691959,\n 1257497595618257,\n 202952467594088,\n 35577762721238,\n 1494883566841973,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1673474571932262,\n 2409784519770613,\n 2636095316260487,\n 2761112584601925,\n 3333713288149876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1600640202645197,\n 1019569075331823,\n 1041916487915822,\n 1680448171313267,\n 2126903137527901,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894964745143659,\n 106116880092678,\n 1009869382959477,\n 317866368542032,\n 1986983122763912,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1765281781276487,\n 2863247187455184,\n 2589075472439062,\n 1386435905543054,\n 2182338478845320,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1144730936996693,\n 2213315231278180,\n 1489676672185125,\n 665039429138074,\n 1131283313040268,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2004734176670602,\n 1738311085075235,\n 418866995976618,\n 1050782508034394,\n 577747313404652,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2185209688340293,\n 1309276076461009,\n 2514740038571278,\n 3994889904012999,\n 3018098826231021,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1405936970888515,\n 1754621155316654,\n 1211862168554999,\n 1813045702919083,\n 997853418197172,\n ]),\n xy2d: FieldElement51::from_limbs([\n 82037622045021,\n 1646398333621944,\n 613095452763466,\n 1312329542583705,\n 81014679202721,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2389287991277873,\n 403851022333257,\n 1597473361477193,\n 2953351602509212,\n 2135174663049062,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1826548187201150,\n 302299893734126,\n 1475477168615781,\n 842617616347376,\n 1438600873676130,\n ]),\n xy2d: FieldElement51::from_limbs([\n 663049852468609,\n 1649295727846569,\n 1048009692742781,\n 628866177992421,\n 1914360327429204,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1795645928096646,\n 306878154408959,\n 2924901319092394,\n 2801261341654799,\n 1653782432983523,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2077597317438627,\n 212642017882064,\n 674844477518888,\n 875487498687554,\n 2060550250171182,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1420448018683809,\n 1032663994771382,\n 1341927003385267,\n 1340360916546159,\n 1988547473895228,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1082660122598844,\n 2545055705583789,\n 3888919679589007,\n 1670283344995811,\n 3403239134794618,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 90430593339788,\n 1838338032241275,\n 571293238480915,\n 1639938867416883,\n 257378872001111,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1528535658865034,\n 1516636853043960,\n 787000569996728,\n 1464531394704506,\n 1684822625133795,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 811329918113934,\n 2783463529007378,\n 1769095754634835,\n 2970819621866866,\n 881037178164325,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1784566501964517,\n 433890943689325,\n 1186055625589419,\n 1496077405487512,\n 1731807117886548,\n ]),\n xy2d: FieldElement51::from_limbs([\n 424909811816304,\n 1355993963741797,\n 409606483251841,\n 455665350637068,\n 1617009023642808,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2478728492077816,\n 2780289048655501,\n 2328687177473769,\n 4107341333582032,\n 1316147724308250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1617420574301156,\n 1741273341070467,\n 667135503486508,\n 2100436564640123,\n 1032223920000865,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1753947659404033,\n 247279202390193,\n 1819288880178945,\n 737334285670249,\n 1037873664856104,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1762568490530034,\n 673742465299012,\n 2054571050635888,\n 2040165159255111,\n 3040123733327257,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1627187989987422,\n 1686331580821752,\n 1309895873498183,\n 719718719104086,\n 300063199808722,\n ]),\n xy2d: FieldElement51::from_limbs([\n 238176707016164,\n 1440454788877048,\n 203336037573144,\n 1437789888677072,\n 101522256664211,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1895216760098480,\n 1934324337975022,\n 3677350688973167,\n 2536415965456176,\n 714678003308640,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 508185358728815,\n 1691320535341855,\n 2168887448239256,\n 1035124393070661,\n 1936603999698584,\n ]),\n xy2d: FieldElement51::from_limbs([\n 390562831571647,\n 1390223890708972,\n 1383183990676371,\n 435998174196410,\n 1882086414390730,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3747620842612921,\n 2081794785291195,\n 3284594056262745,\n 2090090346797895,\n 2581692978935809,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 244144781251265,\n 1290834426417077,\n 1888701171101942,\n 1233922456644870,\n 241117402207491,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1266169390045455,\n 1148042013187970,\n 878921907853942,\n 1815738019658093,\n 908920199341621,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2521768507305118,\n 953557056811112,\n 2015863732865770,\n 1358382511861315,\n 2835421647899992,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2239837206240498,\n 330928973149665,\n 422268062913642,\n 1481280019493032,\n 619879520439841,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1360166735366017,\n 1770556573948510,\n 1395061284191031,\n 1814003148068126,\n 522781147076884,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2611794802645686,\n 707234844948070,\n 1314059396506491,\n 2919250341703934,\n 2161831667832785,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 934831784182383,\n 433734253968318,\n 1660867106725771,\n 1968393082772831,\n 873946300968490,\n ]),\n xy2d: FieldElement51::from_limbs([\n 26306827827554,\n 430884999378685,\n 1504310424376419,\n 1761358720837522,\n 542195685418530,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1762131062631725,\n 3123952634417535,\n 3619918390837537,\n 2909990877347294,\n 1411594230004385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 538272372224622,\n 1425714779586199,\n 588313661410172,\n 1497062084392578,\n 1602174047128512,\n ]),\n xy2d: FieldElement51::from_limbs([\n 907490361939255,\n 1963620338391363,\n 626927432296975,\n 1250748516081414,\n 959901171882527,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1335066153744413,\n 2887804660779657,\n 2653073855954038,\n 2765226981667422,\n 938831784476763,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 296699434737224,\n 2047543711075683,\n 2076451038937139,\n 227783599906901,\n 1602062110967627,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1574834773194203,\n 1384279952062839,\n 393652417255803,\n 2166968242848859,\n 1552890441390820,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1619646774410947,\n 1576090644023562,\n 3035228391320965,\n 1735328519940543,\n 2355324535937066,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1024074573633446,\n 957088456885874,\n 1690425531356997,\n 2102187380180052,\n 1082544623222033,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1871906170635853,\n 1719383891167200,\n 1584032250247862,\n 823764804192117,\n 2244048510084261,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 642147846489775,\n 3334304977145699,\n 305205716788147,\n 2589176626729533,\n 2224680511484174,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1734162377166545,\n 260713621840346,\n 157174591942595,\n 952544272517991,\n 222818702471733,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1213115494182947,\n 286778704335711,\n 2130189536016490,\n 308349182281342,\n 1217623948685491,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3360052266973635,\n 1843486583624091,\n 1561693837124349,\n 1084041964025479,\n 1866270922024009,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 460705465481210,\n 1968151453817859,\n 497005926994844,\n 625618055866751,\n 2176893440866887,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1655800250476757,\n 2036588542300609,\n 666447448675243,\n 1615721995750683,\n 1508669225186765,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2245948203759141,\n 1058306669699396,\n 1452898014240582,\n 3961024141962768,\n 1633235287338608,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 986647273684279,\n 1507266907811370,\n 1260572633649005,\n 2071672342077446,\n 695976026010857,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1312356620823495,\n 1635278548098567,\n 901946076841033,\n 585120475533168,\n 1240667113237384,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2313723935779695,\n 1506054666773895,\n 996040223525031,\n 636592914999692,\n 1497801917020297,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 292042016419794,\n 1158932298133044,\n 2062611870323738,\n 1946058478962569,\n 1749165808126286,\n ]),\n xy2d: FieldElement51::from_limbs([\n 654683942212830,\n 1526897351349087,\n 2006818439922838,\n 2194919327350361,\n 1451960776874416,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3015041017808905,\n 2951823141773809,\n 2584865668253675,\n 2508192032998563,\n 2582137700042019,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1628123495344283,\n 2072923641214546,\n 1647225812023982,\n 855655925244679,\n 1758126430071140,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1615895096489599,\n 275295258643784,\n 937665541219916,\n 1313496726746346,\n 1186468946422626,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1603070202850694,\n 2072127623773242,\n 1692648737212158,\n 2493373404187852,\n 1248948672117105,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 11167836031898,\n 596565174397990,\n 2196351068723859,\n 314744641791907,\n 1102014997250781,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1409047922401191,\n 69960384467966,\n 688103515547600,\n 1309746102488044,\n 150292892873778,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1986083055103168,\n 691715819340300,\n 1361811659746933,\n 3459052030333434,\n 1063594696046061,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1201987338414749,\n 2198784582460616,\n 1203335513981498,\n 489243077045066,\n 2205278143582433,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2034744376624534,\n 2077387101466387,\n 148448542974969,\n 1502697574577258,\n 473186584705655,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 472016956315960,\n 720786972252993,\n 2840633661190043,\n 3150798753357827,\n 2816563335499153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 253464247569755,\n 168314237403057,\n 511780806170295,\n 1058862316549135,\n 1646858476817137,\n ]),\n xy2d: FieldElement51::from_limbs([\n 595092995922219,\n 1491311840717691,\n 291581784452778,\n 1569186646367854,\n 1031385061400544,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3483137021572755,\n 1526955102024322,\n 2778006642704458,\n 457549634924205,\n 1097420237736736,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1246991699537710,\n 81367319519439,\n 530844036072196,\n 163656863755855,\n 1950742455979290,\n ]),\n xy2d: FieldElement51::from_limbs([\n 191532664076407,\n 539378506082089,\n 1021612562876554,\n 1026603384732632,\n 1773368780410653,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4144620731387879,\n 590179521333342,\n 4034023318016108,\n 2255745030335426,\n 2699746851701250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2206599697359952,\n 553895797384417,\n 181689161933786,\n 1153123447919104,\n 778568064152659,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1706307000059211,\n 1885601289314487,\n 889758608505788,\n 550131729999853,\n 1006862664714268,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3210197754285058,\n 2048500453422630,\n 3403309827888207,\n 927154428508963,\n 4199813798872019,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 992058915374933,\n 476120535358775,\n 1973648780784340,\n 2025282643598818,\n 2182318983793230,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1343440812005821,\n 1316045839091795,\n 1884951299078063,\n 1765919609219175,\n 2197567554627988,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3129247779382818,\n 4415026969054274,\n 1900265885969643,\n 1528796215447059,\n 2172730393748688,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1773355092297603,\n 64654329538271,\n 1332124041660957,\n 748492100858001,\n 895500006200535,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2000840647851980,\n 546565968824914,\n 420633283457524,\n 195470736374507,\n 1958689297569520,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 743138980705446,\n 3411117504637167,\n 2591389959690621,\n 2380042066577202,\n 3022267940115114,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 165947002229363,\n 115186103724967,\n 1068573292121517,\n 1842565776920938,\n 1969395681111987,\n ]),\n xy2d: FieldElement51::from_limbs([\n 553322266190633,\n 234265665613185,\n 484544650202821,\n 1238773526575826,\n 2017991917953668,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2581954631514051,\n 1245093644265357,\n 3537016673825374,\n 1834216551713857,\n 923978372152807,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1855378315339552,\n 890045579230758,\n 1764718173975590,\n 197904186055854,\n 1718129022310327,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1278162928734862,\n 1894118254109862,\n 987503995465517,\n 177406744098996,\n 781538103127693,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1996603431230215,\n 1191888797552937,\n 1207440075928499,\n 2765853449051137,\n 2525314961343288,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 808903879370889,\n 990820108751280,\n 1084429472258867,\n 1078562781312589,\n 254514692695625,\n ]),\n xy2d: FieldElement51::from_limbs([\n 615855140068469,\n 586046731175395,\n 693470779212674,\n 1964537100203868,\n 1350330550265229,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3344544372023708,\n 720386671449874,\n 2480841360702110,\n 2036034126860286,\n 2015744690201389,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1337446193390478,\n 1984110761311871,\n 746489405020285,\n 407347127604128,\n 1740475330360596,\n ]),\n xy2d: FieldElement51::from_limbs([\n 140840424783613,\n 1063284623568331,\n 1136446106453878,\n 372042229029799,\n 442607248430694,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2330781679120937,\n 376801425148230,\n 2032603686676107,\n 1488926293635130,\n 1317278311532959,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1290116731380016,\n 2166899563471713,\n 831997001838078,\n 870954980505220,\n 2108537278055823,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1912719171026343,\n 846194720551034,\n 2043988124740726,\n 993234269653961,\n 421229796383281,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2651184584992902,\n 2775702557638963,\n 2539786009779572,\n 2575974880015305,\n 2122619079836732,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1154054290132562,\n 931753998725577,\n 1647742001778052,\n 865765466488226,\n 1083816107290025,\n ]),\n xy2d: FieldElement51::from_limbs([\n 986341121095108,\n 1522330369638573,\n 1990880546211047,\n 501525962272123,\n 198539304862139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1496414019192687,\n 3991034436173951,\n 3380311659062196,\n 2854747485359158,\n 3346958036643152,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 805612068303425,\n 1891790027761335,\n 1587008567571549,\n 722120737390201,\n 378156757163816,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1588994517921951,\n 977362751042302,\n 1329302387067714,\n 2069348224564088,\n 1586007159625211,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2490539421551682,\n 1985699850375015,\n 2331762317128172,\n 4145097393776678,\n 2521049460190674,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 615817553313996,\n 2245962768078178,\n 482564324326173,\n 2101336843140780,\n 1240914880829407,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1438242482238189,\n 874267817785463,\n 1620810389770625,\n 866155221338671,\n 1040426546798301,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2403083624110300,\n 2548561409802975,\n 2492699136535911,\n 2358289519456539,\n 3203964320363148,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1913986535403097,\n 1977163223054199,\n 1972905914623196,\n 1650122133472502,\n 1905849310819035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 858174816360838,\n 614595356564037,\n 1099584959044836,\n 636998087084906,\n 1070393269058348,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3666695924830668,\n 3585640662737501,\n 2372994528684236,\n 2628565977288995,\n 3482812783469694,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1994161359147952,\n 2198039369802658,\n 62790022842537,\n 1522306785848169,\n 951223194802833,\n ]),\n xy2d: FieldElement51::from_limbs([\n 852296621440717,\n 431889737774209,\n 370755457746189,\n 437604073958073,\n 627857326892757,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1794955764684156,\n 2586904290013612,\n 1322647643615887,\n 856117964085888,\n 2652432778663153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 933592377399646,\n 78031722952813,\n 926049890685253,\n 1471649501316246,\n 33789909190376,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1479319468832059,\n 203906207621608,\n 659828362330083,\n 44358398435755,\n 1273573524210803,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1592342143350813,\n 3227219208247713,\n 2345240352078765,\n 2577750109932929,\n 2933512841197243,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2184946892642995,\n 1517382324576002,\n 1557940277419806,\n 2170635134813213,\n 747314658627002,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1823193620577742,\n 1135817878516419,\n 1731253819308581,\n 1031652967267804,\n 2123506616999453,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1346190246005805,\n 2052692552023851,\n 1718128041785940,\n 2491557332978474,\n 3474370880388305,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 424776012994573,\n 281050757243423,\n 626466040846420,\n 990194703866532,\n 38571969885982,\n ]),\n xy2d: FieldElement51::from_limbs([\n 192408346595466,\n 1054889725292349,\n 584097975693004,\n 1447909807397749,\n 2134645004369136,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3169895788615063,\n 3503097743181446,\n 601598510029975,\n 1422812237223371,\n 2121009661378329,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1603348391996783,\n 2066143816131699,\n 1789627290363958,\n 2145705961178118,\n 1985578641438222,\n ]),\n xy2d: FieldElement51::from_limbs([\n 352633958653380,\n 856927627345554,\n 793925083122702,\n 93551575767286,\n 1222010153634215,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1756866499986349,\n 911731956999969,\n 2707505543214075,\n 4006920335263786,\n 822501008147910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1094036422864347,\n 1897208881572508,\n 1503607738246960,\n 1901060196071406,\n 294068411105729,\n ]),\n xy2d: FieldElement51::from_limbs([\n 587776484399576,\n 1116861711228807,\n 343398777436088,\n 936544065763093,\n 1643746750211060,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3477749685790410,\n 267997399528836,\n 2953780922004404,\n 3252368924080907,\n 3787792887348381,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2042368155872443,\n 41662387210459,\n 1676313264498480,\n 1333968523426810,\n 1765708383352310,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1453394896690938,\n 1585795827439909,\n 1469309456804303,\n 1294645324464404,\n 2042954198665899,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1810069207599881,\n 1358344669503239,\n 1989371257548167,\n 2316270051121225,\n 3019675451276507,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1866114438287676,\n 1663420339568364,\n 1437691317033088,\n 538298302628038,\n 1212711449614363,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1769235035677897,\n 1562012115317882,\n 31277513664750,\n 536198657928416,\n 1976134212537183,\n ]),\n },\n ]);\n\n/// Precomputed NafLookupTable5 for ED25519_BASEPOINT_128_POINT\n/// Contains odd multiples [1B', 3B', 5B', 7B', 9B', 11B', 13B', 15B']\n/// where B' = B * 2^128\n#[cfg(feature = \"precomputed-tables\")]\npub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT_128: NafLookupTable5 =\n NafLookupTable5([\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2736169168844485,\n 2938750579674528,\n 1463894987748877,\n 2841718686871730,\n 1985876684518203,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1840579227511262,\n 953261538178031,\n 1834370441150886,\n 1427497321143347,\n 1725682124853989,\n ]),\n Z: FieldElement51::from_limbs([\n 445134475408207,\n 1820906444767788,\n 40212681131971,\n 1448661247201044,\n 965705781338211,\n ]),\n T2d: FieldElement51::from_limbs([\n 773498901840598,\n 1322244500169520,\n 2247887491217152,\n 438555850765425,\n 2003166138168448,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 4052979773311382,\n 2779314608743045,\n 2025864608050267,\n 4239124274918649,\n 2047715283211872,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 46708290287836,\n 854678853922811,\n 473843612020374,\n 93416542836491,\n 36887865609850,\n ]),\n Z: FieldElement51::from_limbs([\n 1415283798768778,\n 558152993015714,\n 1009343863091429,\n 202683238470016,\n 1437237189863098,\n ]),\n T2d: FieldElement51::from_limbs([\n 798901183048341,\n 2088882963129737,\n 1959766260461794,\n 785983100035341,\n 1898332425873393,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 1485884064040933,\n 3544995893699855,\n 3336498043486695,\n 1676518325863799,\n 3924305903599551,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 773665168496622,\n 1762929435191211,\n 1242066992989018,\n 976919579682600,\n 1755245776095155,\n ]),\n Z: FieldElement51::from_limbs([\n 364043412623079,\n 1742274550500525,\n 1826931625194320,\n 1613487341804061,\n 1706694369057824,\n ]),\n T2d: FieldElement51::from_limbs([\n 612881112626481,\n 1851646735104528,\n 442596889481956,\n 1350586815509012,\n 139462310881258,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 1368397133953923,\n 257295989327092,\n 2151475747930859,\n 4157544636401722,\n 2431868518957253,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1086943799443606,\n 2156492892436322,\n 1304912429279094,\n 2236825713899676,\n 956336603451346,\n ]),\n Z: FieldElement51::from_limbs([\n 1435533833442335,\n 2143693239375794,\n 284803180569965,\n 1526121151845459,\n 1146944306085907,\n ]),\n T2d: FieldElement51::from_limbs([\n 746985711541365,\n 1325962113197906,\n 2066433367845509,\n 352065757310156,\n 1030077083459185,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 3242158246905157,\n 1655285217271758,\n 979176760705656,\n 2239696862950768,\n 2581654422853147,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1546547740539545,\n 1712475305386299,\n 1360625165812971,\n 1346151673524691,\n 1382871190753080,\n ]),\n Z: FieldElement51::from_limbs([\n 1301309314400143,\n 754500097438053,\n 1939065018255086,\n 827616229574117,\n 877330897268831,\n ]),\n T2d: FieldElement51::from_limbs([\n 1197188348633815,\n 659807996092455,\n 1711852247689765,\n 558583377269877,\n 99988890374241,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2217993410898728,\n 2786874281837735,\n 277442167258821,\n 2384554073376980,\n 2273131766789461,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 109972040445858,\n 950690845410797,\n 17504561734371,\n 523897781536666,\n 750596827295120,\n ]),\n Z: FieldElement51::from_limbs([\n 54891282232941,\n 1760155098862594,\n 331491218058708,\n 1646635945464764,\n 1039579529390631,\n ]),\n T2d: FieldElement51::from_limbs([\n 1742119867631979,\n 1920835943375882,\n 1823982243967588,\n 8235830273089,\n 868879199166457,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 3550694783929828,\n 2336047885420235,\n 1565306465484489,\n 3136433856071917,\n 1830368149412737,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1774723137461971,\n 2098439526999162,\n 1474163710169650,\n 779998216757295,\n 806943695140007,\n ]),\n Z: FieldElement51::from_limbs([\n 2031052346670915,\n 25708785675631,\n 1400901524447834,\n 748277746457979,\n 1216527237136724,\n ]),\n T2d: FieldElement51::from_limbs([\n 1889193855982501,\n 1710313274081557,\n 2073215834484008,\n 1044022860024178,\n 773075805930498,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2127006969359196,\n 2117739369058606,\n 3577576537383903,\n 2671841131559038,\n 4063127780311805,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1402473726670403,\n 2118415406774282,\n 135582654390618,\n 892801005112916,\n 2121295222735022,\n ]),\n Z: FieldElement51::from_limbs([\n 1578281559293828,\n 2156602763973009,\n 2209233981554667,\n 288157485447820,\n 729847524631017,\n ]),\n T2d: FieldElement51::from_limbs([\n 1551520416538582,\n 1392706341252901,\n 2044032904069908,\n 1190995531234982,\n 442600800298007,\n ]),\n },\n ]);\n"},{"id":77,"name":{"Local":"/rustc/library/core/src/slice/index.rs"},"crate_name":"core","contents":null},{"id":78,"name":{"Local":"/rustc/library/core/src/slice/mod.rs"},"crate_name":"core","contents":null},{"id":79,"name":{"Local":"/rustc/library/core/src/fmt/rt.rs"},"crate_name":"core","contents":null},{"id":80,"name":{"Local":"/rustc/library/core/src/fmt/num.rs"},"crate_name":"core","contents":null},{"id":81,"name":{"Local":"curve25519/solana-ed25519/src/montgomery.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Scalar multiplication on the Montgomery form of Curve25519.\n//!\n//! To avoid notational confusion with the Edwards code, we use\n//! variables \\\\( u, v \\\\) for the Montgomery curve, so that “Montgomery\n//! \\\\(u\\\\)” here corresponds to “Montgomery \\\\(x\\\\)” elsewhere.\n//!\n//! Montgomery arithmetic works not on the curve itself, but on the\n//! \\\\(u\\\\)-line, which discards sign information and unifies the curve\n//! and its quadratic twist. See [_Montgomery curves and their\n//! arithmetic_][costello-smith] by Costello and Smith for more details.\n//!\n//! The `MontgomeryPoint` struct contains the affine \\\\(u\\\\)-coordinate\n//! \\\\(u\\_0(P)\\\\) of a point \\\\(P\\\\) on either the curve or the twist.\n//! Here the map \\\\(u\\_0 : \\mathcal M \\rightarrow \\mathbb F\\_p \\\\) is\n//! defined by \\\\(u\\_0((u,v)) = u\\\\); \\\\(u\\_0(\\mathcal O) = 0\\\\). See\n//! section 5.4 of Costello-Smith for more details.\n//!\n//! # Scalar Multiplication\n//!\n//! Scalar multiplication on `MontgomeryPoint`s is provided by the `*`\n//! operator, which implements the Montgomery ladder.\n//!\n//! # Edwards Conversion\n//!\n//! The \\\\(2\\\\)-to-\\\\(1\\\\) map from the Edwards model to the Montgomery\n//! \\\\(u\\\\)-line is provided by `EdwardsPoint::to_montgomery()`.\n//!\n//! To lift a `MontgomeryPoint` to an `EdwardsPoint`, use\n//! `MontgomeryPoint::to_edwards()`, which takes a sign parameter.\n//! This function rejects `MontgomeryPoints` which correspond to points\n//! on the twist.\n//!\n//! [costello-smith]: https://eprint.iacr.org/2017/212.pdf\n\n// We allow non snake_case names because coordinates in projective space are\n// traditionally denoted by the capitalisation of their respective\n// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my\n// affine and projective cakes and eat both of them too.\n#![allow(non_snake_case)]\n\nuse core::{\n hash::{Hash, Hasher},\n ops::{Mul, MulAssign},\n};\n\nuse crate::constants::APLUS2_OVER_FOUR;\n#[cfg(feature = \"digest\")]\nuse crate::constants::{MONTGOMERY_A, MONTGOMERY_A_NEG, SQRT_M1};\nuse crate::edwards::{CompressedEdwardsY, EdwardsPoint};\nuse crate::field::FieldElement;\nuse crate::scalar::{Scalar, clamp_integer};\n\nuse crate::traits::Identity;\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n// We need the const 2^((p+3)/8) for elligator_encode. This def is checked in tests::consts()\n#[cfg(feature = \"digest\")]\nconst FE_C2: FieldElement = FieldElement::from_limbs([\n 1718705420411057,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n]);\n\n/// Holds the \\\\(u\\\\)-coordinate of a point on the Montgomery form of\n/// Curve25519 or its twist.\n#[derive(Copy, Clone, Debug, Default)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\npub struct MontgomeryPoint(pub [u8; 32]);\n\n/// Equality of `MontgomeryPoint`s is defined mod p.\nimpl ConstantTimeEq for MontgomeryPoint {\n fn ct_eq(&self, other: &MontgomeryPoint) -> Choice {\n let self_fe = FieldElement::from_bytes(&self.0);\n let other_fe = FieldElement::from_bytes(&other.0);\n\n self_fe.ct_eq(&other_fe)\n }\n}\n\nimpl ConditionallySelectable for MontgomeryPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n Self(<[u8; 32]>::conditional_select(&a.0, &b.0, choice))\n }\n}\n\nimpl PartialEq for MontgomeryPoint {\n fn eq(&self, other: &MontgomeryPoint) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for MontgomeryPoint {}\n\n// Equal MontgomeryPoints must hash to the same value. So we have to get them into a canonical\n// encoding first\nimpl Hash for MontgomeryPoint {\n fn hash(&self, state: &mut H) {\n // Do a round trip through a `FieldElement`. `as_bytes` is guaranteed to give a canonical\n // 32-byte encoding\n let canonical_bytes = FieldElement::from_bytes(&self.0).to_bytes();\n canonical_bytes.hash(state);\n }\n}\n\nimpl Identity for MontgomeryPoint {\n /// Return the group identity element, which has order 4.\n fn identity() -> MontgomeryPoint {\n MontgomeryPoint([0u8; 32])\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for MontgomeryPoint {\n fn zeroize(&mut self) {\n self.0.zeroize();\n }\n}\n\nimpl MontgomeryPoint {\n /// Fixed-base scalar multiplication (i.e. multiplication by the base point).\n pub fn mul_base(scalar: &Scalar) -> Self {\n EdwardsPoint::mul_base(scalar).to_montgomery()\n }\n\n /// Multiply this point by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_clamped(self, bytes: [u8; 32]) -> Self {\n // We have to construct a Scalar that is not reduced mod l, which breaks scalar invariant\n // #2. But #2 is not necessary for correctness of variable-base multiplication. All that\n // needs to hold is invariant #1, i.e., the scalar is less than 2^255. This is guaranteed\n // by clamping.\n // Further, we don't do any reduction or arithmetic with this clamped value, so there's no\n // issues arising from the fact that the curve point is not necessarily in the prime-order\n // subgroup.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n s * self\n }\n\n /// Multiply the basepoint by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_base_clamped(bytes: [u8; 32]) -> Self {\n // See reasoning in Self::mul_clamped why it is OK to make an unreduced Scalar here. We\n // note that fixed-base multiplication is also defined for all values of `bytes` less than\n // 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n Self::mul_base(&s)\n }\n\n /// Given `self` \\\\( = u\\_0(P) \\\\), and a big-endian bit representation of an integer\n /// \\\\(n\\\\), return \\\\( u\\_0(\\[n\\]P) \\\\). This is constant time in the length of `bits`.\n ///\n /// **NOTE:** You probably do not want to use this function. Almost every protocol built on\n /// Curve25519 uses _clamped multiplication_, explained\n /// [here](https://neilmadden.blog/2020/05/28/whats-the-curve25519-clamping-all-about/).\n /// When in doubt, use [`Self::mul_clamped`].\n pub fn mul_bits_be(&self, bits: impl Iterator) -> MontgomeryPoint {\n // Algorithm 8 of Costello-Smith 2017\n let affine_u = FieldElement::from_bytes(&self.0);\n let mut x0 = ProjectivePoint::identity();\n let mut x1 = ProjectivePoint {\n U: affine_u,\n W: FieldElement::ONE,\n };\n\n // Go through the bits from most to least significant, using a sliding window of 2\n let mut prev_bit = false;\n for cur_bit in bits {\n let choice: u8 = (prev_bit ^ cur_bit) as u8;\n\n debug_assert!(choice == 0 || choice == 1);\n\n ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());\n differential_add_and_double(&mut x0, &mut x1, &affine_u);\n\n prev_bit = cur_bit;\n }\n // The final value of prev_bit above is scalar.bits()[0], i.e., the LSB of scalar\n ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(prev_bit as u8));\n // Don't leave the bit in the stack\n #[cfg(feature = \"zeroize\")]\n prev_bit.zeroize();\n\n x0.as_affine()\n }\n\n /// View this `MontgomeryPoint` as an array of bytes.\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.0\n }\n\n /// Convert this `MontgomeryPoint` to an array of bytes.\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.0\n }\n\n /// Attempt to convert to an `EdwardsPoint`, using the supplied\n /// choice of sign for the `EdwardsPoint`.\n ///\n /// # Inputs\n ///\n /// * `sign`: a `u8` donating the desired sign of the resulting\n /// `EdwardsPoint`. `0` denotes positive and `1` negative.\n ///\n /// # Return\n ///\n /// * `Some(EdwardsPoint)` if `self` is the \\\\(u\\\\)-coordinate of a\n /// point on (the Montgomery form of) Curve25519;\n ///\n /// * `None` if `self` is the \\\\(u\\\\)-coordinate of a point on the\n /// twist of (the Montgomery form of) Curve25519;\n ///\n pub fn to_edwards(&self, sign: u8) -> Option {\n // To decompress the Montgomery u coordinate to an\n // `EdwardsPoint`, we apply the birational map to obtain the\n // Edwards y coordinate, then do Edwards decompression.\n //\n // The birational map is y = (u-1)/(u+1).\n //\n // The exceptional points are the zeros of the denominator,\n // i.e., u = -1.\n //\n // But when u = -1, v^2 = u*(u^2+486662*u+1) = 486660.\n //\n // Since this is nonsquare mod p, u = -1 corresponds to a point\n // on the twist, not the curve, so we can reject it early.\n\n let u = FieldElement::from_bytes(&self.0);\n\n if u == FieldElement::MINUS_ONE {\n return None;\n }\n\n let one = FieldElement::ONE;\n\n let y = &(&u - &one) * &(&u + &one).invert();\n\n let mut y_bytes = y.to_bytes();\n y_bytes[31] ^= sign << 7;\n\n CompressedEdwardsY(y_bytes).decompress()\n }\n}\n\n#[cfg(feature = \"digest\")]\n/// Perform the Elligator2 mapping to a tuple `(xn, xd, yn, yd)` such that\n/// `(xn / xd, yn / yd)` is a point on curve25519.\n///\n/// See \npub(crate) fn elligator_encode(\n u: &FieldElement,\n) -> (FieldElement, FieldElement, FieldElement, FieldElement) {\n // We follow https://www.rfc-editor.org/rfc/rfc9380.html#appendix-G.2.1\n\n use core::ops::Neg;\n let one = FieldElement::ONE;\n let c2 = FE_C2;\n\n // 1. tv1 = u^2\n // 2. tv1 = 2 * tv1\n let tv1 = u.square2();\n // 3. xd = tv1 + 1\n let xd = &one + &tv1;\n // 4. x1n = -J\n let x1n = MONTGOMERY_A_NEG;\n // 5. tv2 = xd^2\n let tv2 = xd.square();\n // 6. gxd = tv2 * xd\n let gxd = &tv2 * &xd;\n // 7. gx1 = J * tv1\n let gx1 = &MONTGOMERY_A * &tv1;\n // 8. gx1 = gx1 * x1n\n let gx1 = &gx1 * &x1n;\n // 9. gx1 = gx1 + tv2\n let gx1 = &gx1 + &tv2;\n // 10. gx1 = gx1 * x1n\n let gx1 = &gx1 * &x1n;\n // 11. tv3 = gxd^2\n let tv3 = gxd.square();\n // 12. tv2 = tv3^2\n let tv2 = tv3.square();\n // 13. tv3 = tv3 * gxd\n let tv3 = &tv3 * &gxd;\n // 14. tv3 = tv3 * gx1\n let tv3 = &tv3 * &gx1;\n // 15. tv2 = tv2 * tv3\n let tv2 = &tv2 * &tv3;\n // 16. y11 = tv2^c4\n let y11 = tv2.pow_p58();\n // 17. y11 = y11 * tv3\n let y11 = &y11 * &tv3;\n // 18. y12 = y11 * c3\n let y12 = &y11 * &SQRT_M1;\n // 19. tv2 = y11^2\n let tv2 = y11.square();\n // 20. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 21. e1 = tv2 == gx1\n let e1 = tv2.ct_eq(&gx1);\n // 22. y1 = CMOV(y12, y11, e1)\n let y1 = FieldElement::conditional_select(&y12, &y11, e1);\n // 23. x2n = x1n * tv1\n let x2n = &x1n * &tv1;\n // 24. y21 = y11 * u\n let y21 = &y11 * u;\n // 25. y21 = y21 * c2\n let y21 = &y21 * &c2;\n // 26. y22 = y21 * c3\n let y22 = &y21 * &SQRT_M1;\n // 27. gx2 = gx1 * tv1\n let gx2 = &gx1 * &tv1;\n // 28. tv2 = y21^2\n let tv2 = y21.square();\n // 29. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 30. e2 = tv2 == gx2\n let e2 = tv2.ct_eq(&gx2);\n // 31. y2 = CMOV(y22, y21, e2)\n let y2 = FieldElement::conditional_select(&y22, &y21, e2);\n // 32. tv2 = y1^2\n let tv2 = y1.square();\n // 33. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 34. e3 = tv2 == gx1\n let e3 = tv2.ct_eq(&gx1);\n // 35. xn = CMOV(x2n, x1n, e3)\n let xn = FieldElement::conditional_select(&x2n, &x1n, e3);\n // 36. y = CMOV(y2, y1, e3)\n let y = FieldElement::conditional_select(&y2, &y1, e3);\n // 37. e4 = sgn0(y) == 1\n let e4 = y.is_negative();\n // 38. y = CMOV(y, -y, e3 XOR e4)\n let y = FieldElement::conditional_select(&y, &y.neg(), e3 ^ e4);\n // 39. return (xn, xd, y, 1)\n\n (xn, xd, y, one)\n}\n\n/// A `ProjectivePoint` holds a point on the projective line\n/// \\\\( \\mathbb P(\\mathbb F\\_p) \\\\), which we identify with the Kummer\n/// line of the Montgomery curve.\n#[derive(Copy, Clone, Debug)]\nstruct ProjectivePoint {\n pub U: FieldElement,\n pub W: FieldElement,\n}\n\nimpl Identity for ProjectivePoint {\n fn identity() -> ProjectivePoint {\n ProjectivePoint {\n U: FieldElement::ONE,\n W: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for ProjectivePoint {\n fn default() -> ProjectivePoint {\n ProjectivePoint::identity()\n }\n}\n\nimpl ConditionallySelectable for ProjectivePoint {\n fn conditional_select(\n a: &ProjectivePoint,\n b: &ProjectivePoint,\n choice: Choice,\n ) -> ProjectivePoint {\n ProjectivePoint {\n U: FieldElement::conditional_select(&a.U, &b.U, choice),\n W: FieldElement::conditional_select(&a.W, &b.W, choice),\n }\n }\n}\n\nimpl ProjectivePoint {\n /// Dehomogenize this point to affine coordinates.\n ///\n /// # Return\n ///\n /// * \\\\( u = U / W \\\\) if \\\\( W \\neq 0 \\\\);\n /// * \\\\( 0 \\\\) if \\\\( W \\eq 0 \\\\);\n pub fn as_affine(&self) -> MontgomeryPoint {\n let u = &self.U * &self.W.invert();\n MontgomeryPoint(u.to_bytes())\n }\n}\n\n/// Perform the double-and-add step of the Montgomery ladder.\n///\n/// Given projective points\n/// \\\\( (U\\_P : W\\_P) = u(P) \\\\),\n/// \\\\( (U\\_Q : W\\_Q) = u(Q) \\\\),\n/// and the affine difference\n/// \\\\( u\\_{P-Q} = u(P-Q) \\\\), set\n/// $$\n/// (U\\_P : W\\_P) \\gets u(\\[2\\]P)\n/// $$\n/// and\n/// $$\n/// (U\\_Q : W\\_Q) \\gets u(P + Q).\n/// $$\n#[rustfmt::skip] // keep alignment of explanatory comments\nfn differential_add_and_double(\n P: &mut ProjectivePoint,\n Q: &mut ProjectivePoint,\n affine_PmQ: &FieldElement,\n) {\n let t0 = &P.U + &P.W;\n let t1 = &P.U - &P.W;\n let t2 = &Q.U + &Q.W;\n let t3 = &Q.U - &Q.W;\n\n let t4 = t0.square(); // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2\n let t5 = t1.square(); // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2\n\n let t6 = &t4 - &t5; // 4 U_P W_P\n\n let t7 = &t0 * &t3; // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q\n let t8 = &t1 * &t2; // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q\n\n let t9 = &t7 + &t8; // 2 (U_P U_Q - W_P W_Q)\n let t10 = &t7 - &t8; // 2 (W_P U_Q - U_P W_Q)\n\n let t11 = t9.square(); // 4 (U_P U_Q - W_P W_Q)^2\n let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2\n\n let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q\n\n let t14 = &t4 * &t5; // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2\n let t15 = &t13 + &t5; // (U_P - W_P)^2 + (A + 2) U_P W_P\n\n let t16 = &t6 * &t15; // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)\n\n let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2\n let t18 = t11; // W_D * 4 (U_P U_Q - W_P W_Q)^2\n\n P.U = t14; // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2\n P.W = t16; // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)\n Q.U = t18; // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2\n Q.W = t17; // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2\n}\n\ndefine_mul_assign_variants!(LHS = MontgomeryPoint, RHS = Scalar);\n\ndefine_mul_variants!(\n LHS = MontgomeryPoint,\n RHS = Scalar,\n Output = MontgomeryPoint\n);\ndefine_mul_variants!(\n LHS = Scalar,\n RHS = MontgomeryPoint,\n Output = MontgomeryPoint\n);\n\n/// Multiply this `MontgomeryPoint` by a `Scalar`.\nimpl Mul<&Scalar> for &MontgomeryPoint {\n type Output = MontgomeryPoint;\n\n /// Given `self` \\\\( = u\\_0(P) \\\\), and a `Scalar` \\\\(n\\\\), return \\\\( u\\_0(\\[n\\]P) \\\\)\n fn mul(self, scalar: &Scalar) -> MontgomeryPoint {\n // We multiply by the integer representation of the given Scalar. By scalar invariant #1,\n // the MSB is 0, so we can skip it.\n self.mul_bits_be(scalar.bits_le().rev().skip(1))\n }\n}\n\nimpl MulAssign<&Scalar> for MontgomeryPoint {\n fn mul_assign(&mut self, scalar: &Scalar) {\n *self = (self as &MontgomeryPoint) * scalar;\n }\n}\n\nimpl Mul<&MontgomeryPoint> for &Scalar {\n type Output = MontgomeryPoint;\n\n fn mul(self, point: &MontgomeryPoint) -> MontgomeryPoint {\n point * self\n }\n}\n\n// ------------------------------------------------------------------------\n// Tests\n// ------------------------------------------------------------------------\n\n#[cfg(test)]\nmod test {\n use super::*;\n use crate::constants;\n\n #[cfg(feature = \"rand_core\")]\n use rand::CryptoRng;\n use rand::RngCore;\n\n #[test]\n fn identity_in_different_coordinates() {\n let id_projective = ProjectivePoint::identity();\n let id_montgomery = id_projective.as_affine();\n\n assert!(id_montgomery == MontgomeryPoint::identity());\n }\n\n #[test]\n fn identity_in_different_models() {\n assert!(EdwardsPoint::identity().to_montgomery() == MontgomeryPoint::identity());\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_basepoint_roundtrip() {\n use bincode;\n\n let encoded = bincode::serialize(&constants::X25519_BASEPOINT).unwrap();\n let decoded: MontgomeryPoint = bincode::deserialize(&encoded).unwrap();\n\n assert_eq!(encoded.len(), 32);\n assert_eq!(decoded, constants::X25519_BASEPOINT);\n\n let raw_bytes = constants::X25519_BASEPOINT.as_bytes();\n let bp: MontgomeryPoint = bincode::deserialize(raw_bytes).unwrap();\n assert_eq!(bp, constants::X25519_BASEPOINT);\n }\n\n /// Test Montgomery -> Edwards on the X/Ed25519 basepoint\n #[test]\n fn basepoint_montgomery_to_edwards() {\n // sign bit = 0 => basepoint\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT,\n constants::X25519_BASEPOINT\n .to_edwards(0)\n .expect(\"sign bit 0 should decompress\")\n );\n // sign bit = 1 => minus basepoint\n assert_eq!(\n -constants::ED25519_BASEPOINT_POINT,\n constants::X25519_BASEPOINT\n .to_edwards(1)\n .expect(\"sign bit 1 should decompress\")\n );\n }\n\n /// Test Edwards -> Montgomery on the X/Ed25519 basepoint\n #[test]\n fn basepoint_edwards_to_montgomery() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT.to_montgomery(),\n constants::X25519_BASEPOINT\n );\n }\n\n /// Check that Montgomery -> Edwards fails for points on the twist.\n #[test]\n fn montgomery_to_edwards_rejects_twist() {\n let one = FieldElement::ONE;\n\n // u = 2 corresponds to a point on the twist.\n let two = MontgomeryPoint((&one + &one).to_bytes());\n\n assert!(two.to_edwards(0).is_none());\n\n // u = -1 corresponds to a point on the twist, but should be\n // checked explicitly because it's an exceptional point for the\n // birational map. For instance, libsignal will accept it.\n let minus_one = MontgomeryPoint((-&one).to_bytes());\n\n assert!(minus_one.to_edwards(0).is_none());\n }\n\n #[test]\n fn eq_defined_mod_p() {\n let mut u18_bytes = [0u8; 32];\n u18_bytes[0] = 18;\n let u18 = MontgomeryPoint(u18_bytes);\n let u18_unred = MontgomeryPoint([255; 32]);\n\n assert_eq!(u18, u18_unred);\n }\n\n /// Returns a random point on the prime-order subgroup\n #[cfg(feature = \"rand_core\")]\n fn rand_prime_order_point(rng: &mut R) -> EdwardsPoint {\n let s: Scalar = Scalar::random(rng);\n EdwardsPoint::mul_base(&s)\n }\n\n /// Given a bytestring that's little-endian at the byte level, return an iterator over all the\n /// bits, in little-endian order.\n fn bytestring_bits_le(x: &[u8]) -> impl DoubleEndedIterator + Clone + '_ {\n let bitlen = x.len() * 8;\n (0..bitlen).map(|i| {\n // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index\n // the byte. Since self.bytes is little-endian at the byte level, this iterator is\n // little-endian on the bit level\n ((x[i >> 3] >> (i & 7)) & 1u8) == 1\n })\n }\n\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn montgomery_ladder_matches_edwards_scalarmult() {\n let mut csprng = rand::thread_rng();\n\n for _ in 0..100 {\n let p_edwards = rand_prime_order_point(&mut csprng);\n let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();\n\n let s: Scalar = Scalar::random(&mut csprng);\n let expected = s * p_edwards;\n let result = s * p_montgomery;\n\n assert_eq!(result, expected.to_montgomery())\n }\n }\n\n // Tests that, on the prime-order subgroup, MontgomeryPoint::mul_bits_be is the same as\n // multiplying by the Scalar representation of the same bits\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn montgomery_mul_bits_be() {\n let mut csprng = rand::thread_rng();\n\n for _ in 0..100 {\n // Make a random prime-order point P\n let p_edwards = rand_prime_order_point(&mut csprng);\n let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();\n\n // Make a random integer b\n let mut bigint = [0u8; 64];\n csprng.fill_bytes(&mut bigint[..]);\n let bigint_bits_be = bytestring_bits_le(&bigint).rev();\n\n // Check that bP is the same whether calculated as scalar-times-edwards or\n // integer-times-montgomery.\n let expected = Scalar::from_bytes_mod_order_wide(&bigint) * p_edwards;\n let result = p_montgomery.mul_bits_be(bigint_bits_be);\n assert_eq!(result, expected.to_montgomery())\n }\n }\n\n // Tests that MontgomeryPoint::mul_bits_be is consistent on any point, even ones that might be\n // on the curve's twist. Specifically, this tests that b₁(b₂P) == b₂(b₁P) for random\n // integers b₁, b₂ and random (curve or twist) point P.\n #[test]\n fn montgomery_mul_bits_be_twist() {\n let mut csprng = rand::rngs::OsRng;\n\n for _ in 0..100 {\n // Make a random point P on the curve or its twist\n let p_montgomery = {\n let mut buf = [0u8; 32];\n csprng.fill_bytes(&mut buf);\n MontgomeryPoint(buf)\n };\n\n // Compute two big integers b₁ and b₂\n let mut bigint1 = [0u8; 64];\n let mut bigint2 = [0u8; 64];\n csprng.fill_bytes(&mut bigint1[..]);\n csprng.fill_bytes(&mut bigint2[..]);\n\n // Compute b₁P and b₂P\n let bigint1_bits_be = bytestring_bits_le(&bigint1).rev();\n let bigint2_bits_be = bytestring_bits_le(&bigint2).rev();\n let prod1 = p_montgomery.mul_bits_be(bigint1_bits_be.clone());\n let prod2 = p_montgomery.mul_bits_be(bigint2_bits_be.clone());\n\n // Check that b₁(b₂P) == b₂(b₁P)\n assert_eq!(\n prod1.mul_bits_be(bigint2_bits_be),\n prod2.mul_bits_be(bigint1_bits_be)\n );\n }\n }\n\n /// Check that mul_base_clamped and mul_clamped agree\n #[test]\n fn mul_base_clamped() {\n let mut csprng = rand::rngs::OsRng;\n\n // Test agreement on a large integer. Even after clamping, this is not reduced mod l.\n let a_bytes = [0xff; 32];\n assert_eq!(\n MontgomeryPoint::mul_base_clamped(a_bytes),\n constants::X25519_BASEPOINT.mul_clamped(a_bytes)\n );\n\n // Test agreement on random integers\n for _ in 0..100 {\n // This will be reduced mod l with probability l / 2^256 ≈ 6.25%\n let mut a_bytes = [0u8; 32];\n csprng.fill_bytes(&mut a_bytes);\n\n assert_eq!(\n MontgomeryPoint::mul_base_clamped(a_bytes),\n constants::X25519_BASEPOINT.mul_clamped(a_bytes)\n );\n }\n }\n\n #[cfg(feature = \"alloc\")]\n #[cfg(feature = \"digest\")]\n const ELLIGATOR_CORRECT_OUTPUT: [u8; 32] = [\n 0x5f, 0x35, 0x20, 0x00, 0x1c, 0x6c, 0x99, 0x36, 0xa3, 0x12, 0x06, 0xaf, 0xe7, 0xc7, 0xac,\n 0x22, 0x4e, 0x88, 0x61, 0x61, 0x9b, 0xf9, 0x88, 0x72, 0x44, 0x49, 0x15, 0x89, 0x9d, 0x95,\n 0xf4, 0x6e,\n ];\n\n #[test]\n #[cfg(feature = \"alloc\")]\n #[cfg(feature = \"digest\")]\n fn montgomery_elligator_correct() {\n use alloc::vec::Vec;\n let bytes: Vec = (0u8..32u8).collect();\n let bits_in: [u8; 32] = (&bytes[..]).try_into().expect(\"Range invariant broken\");\n\n let fe = FieldElement::from_bytes(&bits_in);\n let (un, ud, ..) = elligator_encode(&fe);\n let u = &un * &ud.invert();\n assert_eq!(u.to_bytes(), ELLIGATOR_CORRECT_OUTPUT);\n }\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn montgomery_elligator_zero_zero() {\n let zero = [0u8; 32];\n let fe = FieldElement::from_bytes(&zero);\n let (un, ud, ..) = elligator_encode(&fe);\n let u = &un * &ud.invert();\n assert_eq!(u.to_bytes(), zero);\n }\n\n // Check that FE_C2 is correctly defined\n #[test]\n #[cfg(feature = \"digest\")]\n fn c2() {\n let one = FieldElement::ONE;\n let two = &one + &one;\n // c2 = 2^((p+3)/8) = 2^((p-5)/8 + 8/8) = 2*2^((p-5)/8)\n let c2 = &two * &(two.pow_p58());\n\n assert_eq!(c2, FE_C2);\n }\n}\n"},{"id":82,"name":{"Local":"curve25519/solana-ed25519/src/constants.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n//! Various constants, such as the Ristretto and Ed25519 basepoints.\n\n#![allow(non_snake_case)]\n\nuse crate::edwards::CompressedEdwardsY;\nuse crate::montgomery::MontgomeryPoint;\nuse crate::ristretto::{CompressedRistretto, RistrettoPoint};\nuse crate::scalar::Scalar;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::edwards::EdwardsBasepointTable;\n\npub use crate::backend::serial::u64::constants::*;\n\n/// The Ed25519 basepoint, in `CompressedEdwardsY` format.\n///\n/// This is the little-endian byte encoding of \\\\( 4/5 \\pmod p \\\\),\n/// which is the \\\\(y\\\\)-coordinate of the Ed25519 basepoint.\n///\n/// The sign bit is 0 since the basepoint has \\\\(x\\\\) chosen to be positive.\npub const ED25519_BASEPOINT_COMPRESSED: CompressedEdwardsY = CompressedEdwardsY([\n 0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,\n 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,\n]);\n\n/// The X25519 basepoint, in `MontgomeryPoint` format.\npub const X25519_BASEPOINT: MontgomeryPoint = MontgomeryPoint([\n 0x09, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n]);\n\n/// The Ristretto basepoint, in `CompressedRistretto` format.\npub const RISTRETTO_BASEPOINT_COMPRESSED: CompressedRistretto = CompressedRistretto([\n 0xe2, 0xf2, 0xae, 0x0a, 0x6a, 0xbc, 0x4e, 0x71, 0xa8, 0x84, 0xa9, 0x61, 0xc5, 0x00, 0x51, 0x5f,\n 0x58, 0xe3, 0x0b, 0x6a, 0xa5, 0x82, 0xdd, 0x8d, 0xb6, 0xa6, 0x59, 0x45, 0xe0, 0x8d, 0x2d, 0x76,\n]);\n\n/// The Ristretto basepoint, as a `RistrettoPoint`.\n///\n/// This is called `_POINT` to distinguish it from `_TABLE`, which\n/// provides fast scalar multiplication.\npub const RISTRETTO_BASEPOINT_POINT: RistrettoPoint = RistrettoPoint(ED25519_BASEPOINT_POINT);\n\n/// `BASEPOINT_ORDER` is the order of the Ristretto group and of the Ed25519 basepoint, i.e.,\n/// $$\n/// \\ell = 2^\\{252\\} + 27742317777372353535851937790883648493.\n/// $$\npub(crate) const BASEPOINT_ORDER: Scalar = Scalar {\n bytes: [\n 0xed, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde,\n 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x10,\n ],\n};\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::ristretto::RistrettoBasepointTable;\n\n/// The Ristretto basepoint, as a `RistrettoBasepointTable` for scalar multiplication.\n#[cfg(feature = \"precomputed-tables\")]\npub static RISTRETTO_BASEPOINT_TABLE: &RistrettoBasepointTable = unsafe {\n // SAFETY: `RistrettoBasepointTable` is a `#[repr(transparent)]` newtype of\n // `EdwardsBasepointTable`\n &*(ED25519_BASEPOINT_TABLE as *const EdwardsBasepointTable as *const RistrettoBasepointTable)\n};\n\n#[cfg(test)]\nmod test {\n use crate::constants;\n use crate::field::FieldElement;\n use crate::traits::{IsIdentity, ValidityCheck};\n\n #[test]\n fn test_eight_torsion() {\n for i in 0..8 {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(3);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n #[test]\n fn test_four_torsion() {\n for i in (0..8).filter(|i| i % 2 == 0) {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(2);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n #[test]\n fn test_two_torsion() {\n for i in (0..8).filter(|i| i % 4 == 0) {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(1);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n /// Test that SQRT_M1 is the positive square root of -1\n #[test]\n fn test_sqrt_minus_one() {\n let minus_one = FieldElement::MINUS_ONE;\n let sqrt_m1_sq = &constants::SQRT_M1 * &constants::SQRT_M1;\n assert_eq!(minus_one, sqrt_m1_sq);\n assert!(bool::from(!constants::SQRT_M1.is_negative()));\n }\n\n #[test]\n fn test_sqrt_constants_sign() {\n let minus_one = FieldElement::MINUS_ONE;\n let (was_nonzero_square, invsqrt_m1) = minus_one.invsqrt();\n assert!(bool::from(was_nonzero_square));\n let sign_test_sqrt = &invsqrt_m1 * &constants::SQRT_M1;\n assert_eq!(sign_test_sqrt, minus_one);\n }\n\n /// Test that d = -121665/121666\n #[test]\n fn test_d_vs_ratio() {\n use crate::backend::serial::u64::field::FieldElement51;\n let a = -&FieldElement51([121665, 0, 0, 0, 0]);\n let b = FieldElement51([121666, 0, 0, 0, 0]);\n let d = &a * &b.invert();\n let d2 = &d + &d;\n assert_eq!(d, constants::EDWARDS_D);\n assert_eq!(d2, constants::EDWARDS_D2);\n }\n\n #[test]\n fn test_sqrt_ad_minus_one() {\n let a = FieldElement::MINUS_ONE;\n let ad_minus_one = &(&a * &constants::EDWARDS_D) + &a;\n let should_be_ad_minus_one = constants::SQRT_AD_MINUS_ONE.square();\n assert_eq!(should_be_ad_minus_one, ad_minus_one);\n }\n\n /// Test that ED25519_SQRTAM2 squared is MONTGOMERY_A_NEG - 2\n #[test]\n #[cfg(feature = \"digest\")]\n fn test_sqrt_a_minus_2() {\n let one = FieldElement::ONE;\n let a_minus_two = &(&constants::MONTGOMERY_A_NEG - &one) - &one;\n\n assert_eq!(constants::ED25519_SQRTAM2.square(), a_minus_two)\n }\n}\n"},{"id":83,"name":{"Local":"/rustc/library/core/src/num/niche_types.rs"},"crate_name":"core","contents":null},{"id":84,"name":{"Local":"/rustc/library/core/src/ops/bit.rs"},"crate_name":"core","contents":null},{"id":85,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/variable_base.rs"},"crate_name":"curve25519","contents":"#![allow(non_snake_case)]\n\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::Scalar;\nuse crate::traits::Identity;\nuse crate::window::LookupTable;\n\n/// Perform constant-time, variable-base scalar multiplication.\n#[rustfmt::skip] // keep alignment of explanatory comments\npub(crate) fn mul(point: &EdwardsPoint, scalar: &Scalar) -> EdwardsPoint {\n // Construct a lookup table of [P,2P,3P,4P,5P,6P,7P,8P]\n let lookup_table = LookupTable::::from(point);\n // Setting s = scalar, compute\n //\n // s = s_0 + s_1*16^1 + ... + s_63*16^63,\n //\n // with `-8 ≤ s_i < 8` for `0 ≤ i < 63` and `-8 ≤ s_63 ≤ 8`.\n // This decomposition requires s < 2^255, which is guaranteed by Scalar invariant #1.\n let scalar_digits = scalar.as_radix_16();\n // Compute s*P as\n //\n // s*P = P*(s_0 + s_1*16^1 + s_2*16^2 + ... + s_63*16^63)\n // s*P = P*s_0 + P*s_1*16^1 + P*s_2*16^2 + ... + P*s_63*16^63\n // s*P = P*s_0 + 16*(P*s_1 + 16*(P*s_2 + 16*( ... + P*s_63)...))\n //\n // We sum right-to-left.\n\n // Unwrap first loop iteration to save computing 16*identity\n let mut tmp2;\n let mut tmp3 = EdwardsPoint::identity();\n let mut tmp1 = &tmp3 + &lookup_table.select(scalar_digits[63]);\n // Now tmp1 = s_63*P in P1xP1 coords\n for i in (0..63).rev() {\n tmp2 = tmp1.as_projective(); // tmp2 = (prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 2*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 2*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 4*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 4*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 8*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 8*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 16*(prev) in P1xP1 coords\n tmp3 = tmp1.as_extended(); // tmp3 = 16*(prev) in P3 coords\n tmp1 = &tmp3 + &lookup_table.select(scalar_digits[i]);\n // Now tmp1 = s_i*P + 16*(prev) in P1xP1 coords\n }\n tmp1.as_extended()\n}\n"},{"id":86,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Implementations of various scalar multiplication algorithms.\n//!\n//! Note that all of these implementations use serial code for field\n//! arithmetic with the multi-model strategy described in the\n//! `curve_models` module. The vectorized AVX2 backend has its own\n//! scalar multiplication implementations, since it only uses one\n//! curve model.\n\n#[allow(missing_docs)]\npub mod variable_base;\n\n#[allow(missing_docs)]\npub mod vartime_double_base;\n\n#[allow(missing_docs)]\npub mod vartime_triple_base;\n\n#[cfg(feature = \"alloc\")]\npub mod straus;\n\n#[cfg(feature = \"alloc\")]\npub mod precomputed_straus;\n\n#[cfg(feature = \"alloc\")]\npub mod pippenger;\n"},{"id":87,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/vartime_double_base.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n#![allow(non_snake_case)]\n\nuse core::cmp::Ordering;\n\nuse crate::backend::serial::curve_models::{CompletedPoint, ProjectiveNielsPoint, ProjectivePoint};\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::backend::serial::curve_models::AffineNielsPoint;\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::window::NafLookupTable8;\nuse crate::constants;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::Scalar;\nuse crate::traits::Identity;\nuse crate::window::NafLookupTable5;\n\n/// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n\n/// AENEAS-COMPAT helper: highest index with a nonzero NAF digit (or 0).\nfn dsm_top_index(a_naf: &[i8; 256], b_naf: &[i8; 256]) -> usize {\n // AENEAS-COMPAT: always process all 256 digits. Leading zero digits\n // double the identity (a no-op), so the result of `mul` is unchanged;\n // only the variable-time skip is dropped. (The conditional-fold scan\n // fails Aeneas' loop fixed point.)\n let _ = a_naf;\n let _ = b_naf;\n 255\n}\n\n/// AENEAS-COMPAT helper: the double-and-add main loop over indices i..0,\n/// with all borrows rooted in parameters (pure refactor).\n#[cfg(feature = \"precomputed-tables\")]\nfn dsm_loop(\n i: usize,\n a_naf: &[i8; 256],\n b_naf: &[i8; 256],\n table_a: &NafLookupTable5,\n table_b: &NafLookupTable8,\n) -> ProjectivePoint {\n let mut r = ProjectivePoint::identity();\n let mut j: usize = i + 1;\n while j > 0 {\n let idx = j - 1;\n let t0 = r.double();\n let t1 = dsm_step_p(t0, table_a, a_naf[idx]);\n let t2 = dsm_step_b(t1, table_b, b_naf[idx]);\n r = t2.as_projective();\n j -= 1;\n }\n r\n}\n\n#[cfg(not(feature = \"precomputed-tables\"))]\nfn dsm_loop(\n i: usize,\n a_naf: &[i8; 256],\n b_naf: &[i8; 256],\n table_a: &NafLookupTable5,\n table_b: &NafLookupTable5,\n) -> ProjectivePoint {\n let mut r = ProjectivePoint::identity();\n let mut j: usize = i + 1;\n while j > 0 {\n let idx = j - 1;\n let t0 = r.double();\n let t1 = dsm_step_p(t0, table_a, a_naf[idx]);\n let t2 = dsm_step_b(t1, table_b, b_naf[idx]);\n r = t2.as_projective();\n j -= 1;\n }\n r\n}\n\n/// AENEAS-COMPAT helper: one NAF-digit conditional add against a\n/// ProjectiveNiels table (hoisted out of the main loop so its body is\n/// single-assignment; pure refactor, semantics identical).\nfn dsm_step_p(\n t: CompletedPoint,\n table: &NafLookupTable5,\n d: i8,\n) -> CompletedPoint {\n if d > 0 {\n &t.as_extended() + &table.select(d as usize)\n } else if d < 0 {\n &t.as_extended() - &table.select((-d) as usize)\n } else {\n t\n }\n}\n\n/// AENEAS-COMPAT helper: as `dsm_step_p`, for the basepoint table.\n#[cfg(feature = \"precomputed-tables\")]\nfn dsm_step_b(\n t: CompletedPoint,\n table: &NafLookupTable8,\n d: i8,\n) -> CompletedPoint {\n if d > 0 {\n &t.as_extended() + &table.select(d as usize)\n } else if d < 0 {\n &t.as_extended() - &table.select((-d) as usize)\n } else {\n t\n }\n}\n\n/// AENEAS-COMPAT helper: as `dsm_step_p` (no precomputed tables).\n#[cfg(not(feature = \"precomputed-tables\"))]\nfn dsm_step_b(\n t: CompletedPoint,\n table: &NafLookupTable5,\n d: i8,\n) -> CompletedPoint {\n dsm_step_p(t, table, d)\n}\n\npub fn mul(a: &Scalar, A: &EdwardsPoint, b: &Scalar) -> EdwardsPoint {\n let a_naf = a.non_adjacent_form(5);\n\n #[cfg(feature = \"precomputed-tables\")]\n let b_naf = b.non_adjacent_form(8);\n #[cfg(not(feature = \"precomputed-tables\"))]\n let b_naf = b.non_adjacent_form(5);\n\n // Find starting index (AENEAS-COMPAT: helper, upward fold)\n let i = dsm_top_index(&a_naf, &b_naf);\n\n let table_A = NafLookupTable5::::from(A);\n #[cfg(feature = \"precomputed-tables\")]\n let table_B = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_POINT);\n\n let r = dsm_loop(i, &a_naf, &b_naf, &table_A, table_B);\n\n r.as_extended()\n}\n"},{"id":88,"name":{"Local":"curve25519/solana-ed25519/src/window.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Code for fixed- and sliding-window functionality\n\n#![allow(non_snake_case)]\n\nuse core::fmt::Debug;\n\n#[cfg(feature = \"alloc\")]\nuse alloc::vec::Vec;\nuse cfg_if::cfg_if;\n#[cfg(feature = \"alloc\")]\nuse core::borrow::Borrow;\n\nuse subtle::Choice;\nuse subtle::ConditionallyNegatable;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\nuse crate::traits::Identity;\n\nuse crate::backend::serial::curve_models::AffineNielsPoint;\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::edwards::EdwardsPoint;\n\n#[cfg(feature = \"alloc\")]\npub(crate) fn build_lookup_tables(points: I) -> Vec\nwhere\n T: for<'a> From<&'a EdwardsPoint>,\n I: IntoIterator,\n I::Item: Borrow,\n{\n points\n .into_iter()\n .map(|point| T::from(point.borrow()))\n .collect()\n}\n\n#[cfg(feature = \"alloc\")]\npub(crate) fn build_lookup_tables_for_optional_points(points: I) -> Option>\nwhere\n T: for<'a> From<&'a EdwardsPoint>,\n I: IntoIterator>,\n P: Borrow,\n{\n points\n .into_iter()\n .map(|point| point.map(|point| T::from(point.borrow())))\n .collect()\n}\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nmacro_rules! impl_lookup_table {\n (Name = $name:ident, Size = $size:expr, SizeNeg = $neg:expr, SizeRange = $range:expr, ConversionRange = $conv_range:expr) => {\n /// A lookup table of precomputed multiples of a point \\\\(P\\\\), used to\n /// compute \\\\( xP \\\\) for \\\\( -8 \\leq x \\leq 8 \\\\).\n ///\n /// The computation of \\\\( xP \\\\) is done in constant time by the `select` function.\n ///\n /// Since `LookupTable` does not implement `Index`, it's more difficult\n /// to accidentally use the table directly. Unfortunately the table is\n /// only `pub(crate)` so that we can write hardcoded constants, so it's\n /// still technically possible. It would be nice to prevent direct\n /// access to the table.\n #[derive(Copy, Clone)]\n pub struct $name(pub(crate) [T; $size]);\n\n impl $name\n where\n T: Identity + ConditionallySelectable + ConditionallyNegatable,\n {\n /// Given \\\\(-8 \\leq x \\leq 8\\\\), return \\\\(xP\\\\) in constant time.\n pub fn select(&self, x: i8) -> T {\n debug_assert!(x >= $neg);\n debug_assert!(x as i16 <= $size as i16); // XXX We have to convert to i16s here for the radix-256 case.. this is wrong.\n\n // Compute xabs = |x|\n let xmask = x as i16 >> 7;\n let xabs = (x as i16 + xmask) ^ xmask;\n\n // Set t = 0 * P = identity\n let mut t = T::identity();\n for j in $range {\n // Copy `points[j-1] == j*P` onto `t` in constant time if `|x| == j`.\n let c = (xabs as u16).ct_eq(&(j as u16));\n t.conditional_assign(&self.0[j - 1], c);\n }\n // Now t == |x| * P.\n\n let neg_mask = Choice::from((xmask & 1) as u8);\n t.conditional_negate(neg_mask);\n // Now t == x * P.\n\n t\n }\n }\n\n impl Default for $name {\n fn default() -> $name {\n $name([T::default(); $size])\n }\n }\n\n impl Debug for $name {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"{:?}(\", stringify!($name))?;\n\n for x in self.0.iter() {\n write!(f, \"{:?}\", x)?;\n }\n\n write!(f, \")\")\n }\n }\n\n impl<'a> From<&'a EdwardsPoint> for $name {\n fn from(P: &'a EdwardsPoint) -> Self {\n let mut points = [P.as_projective_niels(); $size];\n for j in $conv_range {\n points[j + 1] = (P + &points[j]).as_extended().as_projective_niels();\n }\n $name(points)\n }\n }\n\n impl<'a> From<&'a EdwardsPoint> for $name {\n fn from(P: &'a EdwardsPoint) -> Self {\n let mut points = [P.as_affine_niels(); $size];\n // XXX batch inversion would be good if perf mattered here\n for j in $conv_range {\n points[j + 1] = (P + &points[j]).as_extended().as_affine_niels()\n }\n $name(points)\n }\n }\n\n #[cfg(feature = \"zeroize\")]\n impl Zeroize for $name\n where\n T: Copy + Default + Zeroize,\n {\n fn zeroize(&mut self) {\n self.0.iter_mut().zeroize();\n }\n }\n };\n} // End macro_rules! impl_lookup_table\n\n// The first one has to be named \"LookupTable\" because it's used as a constructor for consts.\n// This is radix-16\nimpl_lookup_table! {\n Name = LookupTable,\n Size = 8,\n SizeNeg = -8,\n SizeRange = 1..9,\n ConversionRange = 0..7\n}\n\n// The rest only get used to make basepoint tables\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n // radix-32\n impl_lookup_table! {\n Name = LookupTableRadix32,\n Size = 16,\n SizeNeg = -16,\n SizeRange = 1..17,\n ConversionRange = 0..15\n }\n // radix-64\n impl_lookup_table! {\n Name = LookupTableRadix64,\n Size = 32,\n SizeNeg = -32,\n SizeRange = 1..33,\n ConversionRange = 0..31\n }\n // radix-128\n impl_lookup_table! {\n Name = LookupTableRadix128,\n Size = 64,\n SizeNeg = -64,\n SizeRange = 1..65,\n ConversionRange = 0..63\n }\n // radix-256\n impl_lookup_table! {\n Name = LookupTableRadix256,\n Size = 128,\n SizeNeg = -128,\n SizeRange = 1..129,\n ConversionRange = 0..127\n }\n\n // For homogeneity we then alias it to \"LookupTableRadix16\".\n pub(crate) type LookupTableRadix16 = LookupTable;\n }\n}\n\n/// Holds odd multiples 1A, 3A, ..., 15A of a point A.\n#[derive(Copy, Clone)]\npub(crate) struct NafLookupTable5(pub(crate) [T; 8]);\n\nimpl NafLookupTable5 {\n /// Given public, odd \\\\( x \\\\) with \\\\( 0 < x < 2^4 \\\\), return \\\\(xA\\\\).\n pub fn select(&self, x: usize) -> T {\n debug_assert_eq!(x & 1, 1);\n debug_assert!(x < 16);\n\n self.0[x / 2]\n }\n}\n\nimpl Debug for NafLookupTable5 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"NafLookupTable5({:?})\", self.0)\n }\n}\n\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable5 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_projective_niels(); 8];\n let A2 = A.double();\n for i in 0..7 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_projective_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A]\n NafLookupTable5(Ai)\n }\n}\n\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable5 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_affine_niels(); 8];\n let A2 = A.double();\n for i in 0..7 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_affine_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A]\n NafLookupTable5(Ai)\n }\n}\n\n/// Holds stuff up to 8. The only time we use tables this big is for precomputed basepoint tables\n/// and multiscalar multiplication (which requires alloc).\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\n#[derive(Copy, Clone)]\npub(crate) struct NafLookupTable8(pub(crate) [T; 64]);\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl NafLookupTable8 {\n pub fn select(&self, x: usize) -> T {\n debug_assert_eq!(x & 1, 1);\n debug_assert!(x < 128);\n\n self.0[x / 2]\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl Debug for NafLookupTable8 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n writeln!(f, \"NafLookupTable8([\")?;\n for i in 0..64 {\n writeln!(f, \"\\t{:?},\", &self.0[i])?;\n }\n write!(f, \"])\")\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable8 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_projective_niels(); 64];\n let A2 = A.double();\n for i in 0..63 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_projective_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A, ..., 127A]\n NafLookupTable8(Ai)\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable8 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_affine_niels(); 64];\n let A2 = A.double();\n for i in 0..63 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_affine_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A, ..., 127A]\n NafLookupTable8(Ai)\n }\n}\n"},{"id":89,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/vartime_triple_base.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-sol.\n// Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE for licensing information.\n//\n// This file is new in curve25519-sol and has no upstream equivalent in\n// curve25519-dalek. It implements a 4-variable multi-scalar multiplication\n// optimised for two 128-bit scalars and one 256-bit scalar, designed to\n// support the HEEA-accelerated Ed25519 verification path.\n#![allow(non_snake_case)]\n\nuse core::cmp::Ordering;\n\nuse crate::backend::serial::curve_models::{ProjectiveNielsPoint, ProjectivePoint};\nuse crate::constants;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::{HEEA_MAX_INDEX, Scalar};\nuse crate::traits::Identity;\nuse crate::window::NafLookupTable5;\n\nconst DYNAMIC_NAF_WINDOW: usize = 5;\n\n// This intentionally differs from the vector backend when precomputed tables\n// are enabled. The serial backend can select width-5 b_lo digits from the\n// larger width-8 basepoint table, but keeps the smaller NAF window here as a\n// backend-specific performance tradeoff. Revisit this only with comparative\n// serial/vector benchmarks.\nconst B_LO_NAF_WINDOW: usize = DYNAMIC_NAF_WINDOW;\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n///\n/// This function is optimized for the case where \\\\(a_1\\\\) and \\\\(a_2\\\\) are known to be less than\n/// \\\\(2^{128}\\\\), while \\\\(b\\\\) is a full 256-bit scalar.\n///\n/// # Precondition\n///\n/// Callers must ensure \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\). Use\n/// `vartime_triple_base_mul_128_128_256` for a checked wrapper that falls back\n/// to general scalar multiplication for full-width scalars.\n///\n/// # Optimization Strategy\n///\n/// The function decomposes the 256-bit scalar \\\\(b\\\\) as \\\\(b = b_{lo} + b_{hi} \\cdot 2^{128}\\\\),\n/// where both \\\\(b_{lo}\\\\) and \\\\(b_{hi}\\\\) are 128-bit values. This allows computing:\n///\n/// \\\\[\n/// a_1 A_1 + a_2 A_2 + b_{lo} B + b_{hi} B'\n/// \\\\]\n///\n/// where \\\\(B' = B \\cdot 2^{128}\\\\) is a precomputed constant. Now all four scalars\n/// (\\\\(a_1, a_2, b_{lo}, b_{hi}\\\\)) are 128 bits, and two of the bases (\\\\(B\\\\) and \\\\(B'\\\\))\n/// use precomputed tables.\n///\n/// # Implementation\n///\n/// - For \\\\(A_1\\\\) and \\\\(A_2\\\\): NAF with window width 5 (8 precomputed points each)\n/// - For \\\\(B\\\\): NAF with window width 5. When precomputed tables are enabled,\n/// these width-5 digits are selected from the larger width-8 basepoint table.\n/// - For \\\\(B'\\\\): NAF with window width 5.\n///\n/// The vector backend uses width 8 for \\\\(b_{lo}\\\\) when precomputed tables are\n/// enabled. This serial backend keeps width 5 for \\\\(b_{lo}\\\\) as a\n/// backend-specific performance tradeoff.\n///\n/// The algorithm shares doublings across all four scalar multiplications, processing\n/// only 128 bits instead of 256, providing approximately 2x speedup over the naive approach.\npub(crate) fn mul_128_128_256_prechecked(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n // Decompose b into b_lo (lower 128 bits) and b_hi (upper 128 bits)\n // b = b_lo + b_hi * 2^128\n let b_bytes = b.as_bytes();\n\n let mut b_lo_bytes = [0u8; 32];\n let mut b_hi_bytes = [0u8; 32];\n\n // Copy lower 16 bytes to b_lo, upper 16 bytes to b_hi\n b_lo_bytes[..16].copy_from_slice(&b_bytes[..16]);\n b_hi_bytes[..16].copy_from_slice(&b_bytes[16..]);\n\n let b_lo = Scalar::from_canonical_bytes(b_lo_bytes).unwrap();\n let b_hi = Scalar::from_canonical_bytes(b_hi_bytes).unwrap();\n\n // Compute NAF representations (all scalars are now ~128 bits).\n //\n // The serial backend keeps b_lo at width 5 even when table_B is the larger\n // precomputed width-8 basepoint table. The vector backend uses width 8 in\n // that configuration.\n let a1_naf = a1.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n let a2_naf = a2.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n let b_lo_naf = b_lo.non_adjacent_form_128(B_LO_NAF_WINDOW);\n let b_hi_naf = b_hi.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n\n // Find starting index - check all NAFs up to bit 127\n // (with potential carry to bit 128 or 129)\n let mut i: usize = HEEA_MAX_INDEX;\n for j in (0..=HEEA_MAX_INDEX).rev() {\n i = j;\n if a1_naf[i] != 0 || a2_naf[i] != 0 || b_lo_naf[i] != 0 || b_hi_naf[i] != 0 {\n break;\n }\n }\n\n // Create lookup tables\n let table_A1 = NafLookupTable5::::from(A1);\n let table_A2 = NafLookupTable5::::from(A2);\n\n #[cfg(feature = \"precomputed-tables\")]\n let table_B = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_POINT);\n\n // B' = B * 2^128 (precomputed constant point)\n #[cfg(feature = \"precomputed-tables\")]\n let table_B_128 = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT_128;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B_128 =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_128_POINT);\n\n let mut r = ProjectivePoint::identity();\n\n loop {\n let mut t = r.double();\n\n // Add contributions from a1*A1\n match a1_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_A1.select(a1_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_A1.select(-a1_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from a2*A2\n match a2_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_A2.select(a2_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_A2.select(-a2_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from b_lo*B\n match b_lo_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_B.select(b_lo_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_B.select(-b_lo_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from b_hi*B' where B' = B * 2^128\n match b_hi_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_B_128.select(b_hi_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_B_128.select(-b_hi_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n r = t.as_projective();\n\n if i == 0 {\n break;\n }\n i -= 1;\n }\n\n r.as_extended()\n}\n\n#[cfg(test)]\nmod test {\n\n use rand::{RngCore, thread_rng};\n\n use super::*;\n use crate::scalar::Scalar;\n\n fn random_scalar() -> Scalar {\n let mut wide = [0u8; 64];\n let mut rng = thread_rng();\n rng.fill_bytes(&mut wide);\n Scalar::from_bytes_mod_order_wide(&wide)\n }\n\n #[test]\n fn test_triple_base_multiplication() {\n // Test vectors with random scalars\n let a1 = Scalar::from(12345u64);\n let a2 = Scalar::from(67890u64);\n let b = random_scalar();\n\n // Random points (using scalar multiplication of basepoint)\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(2u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n // Compute using the optimized triple-base function\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute using naive addition\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_multiplication_128() {\n // Test with 128-bit scalars for a1 and a2\n // Create a 128-bit scalar (use lower half)\n let a1_bytes = [\n 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,\n 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00,\n ];\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n\n let a2_bytes = [\n 0x12, 0x34, 0x56, 0x78, 0x9A, 0xBC, 0xDE, 0xF0, 0x11, 0x22, 0x33, 0x44, 0x55, 0x66,\n 0x77, 0x88, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00,\n ];\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n // Full 256-bit scalar for b\n let b = random_scalar();\n\n // Test points\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(5u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(7u64);\n\n // Test the optimized 128-bit version\n let result_128 = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute expected result\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result_128, expected, \"Optimized 128-bit version failed\");\n }\n\n #[test]\n fn test_triple_base_with_zero_scalars() {\n let a1 = Scalar::ZERO;\n let a2 = Scalar::from(123u64);\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(2u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = a2 * A2 + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_with_identity_points() {\n let a1 = Scalar::from(111u64);\n let a2 = Scalar::from(222u64);\n let b = random_scalar();\n\n let A1 = EdwardsPoint::identity();\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = a2 * A2 + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_consistency() {\n // Test that both functions give the same result for 128-bit inputs\n let a1 = Scalar::from(0x123456789ABCDEFu64);\n let a2 = Scalar::from(0xFEDCBA987654321u64);\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(11u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(13u64);\n\n let result_optimized = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let result_general = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result_optimized, result_general);\n }\n\n #[test]\n fn test_triple_base_large_scalars() {\n // Test with large scalars\n let mut a1_bytes = [0xFFu8; 32];\n a1_bytes[16..].fill(0x00);\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n\n let mut a2_bytes = [0xAAu8; 32];\n a2_bytes[16..].fill(0x00);\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(17u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(19u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_checked_wrapper_accepts_full_width_scalars() {\n let mut a1_bytes = [0u8; 32];\n a1_bytes[0] = 7;\n a1_bytes[16] = 1;\n let a1 = Scalar::from_canonical_bytes(a1_bytes).unwrap();\n\n let mut a2_bytes = [0u8; 32];\n a2_bytes[0] = 11;\n a2_bytes[24] = 1;\n let a2 = Scalar::from_canonical_bytes(a2_bytes).unwrap();\n\n let b = random_scalar();\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(23u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(29u64);\n\n let result = crate::backend::vartime_triple_base_mul_128_128_256(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n // Proptest for vartime_triple_scalar_mul_basepoint equivalence\n proptest::proptest! {\n #[test]\n fn proptest_triple_scalar_mul_equivalence(\n a1_bytes_16 in proptest::array::uniform16(proptest::num::u8::ANY),\n a2_bytes_16 in proptest::array::uniform16(proptest::num::u8::ANY),\n b_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n A1_scalar_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n A2_scalar_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n ) {\n // Construct 128-bit scalars a1 and a2 (upper 16 bytes are zero)\n let mut a1_bytes = [0u8; 32];\n let mut a2_bytes = [0u8; 32];\n a1_bytes[..16].copy_from_slice(&a1_bytes_16);\n a2_bytes[..16].copy_from_slice(&a2_bytes_16);\n\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n // Construct full 256-bit scalar b\n let b = Scalar::from_bytes_mod_order(b_bytes);\n\n // Generate random points A1 and A2 using scalar multiplication of basepoint\n let A1_scalar = Scalar::from_bytes_mod_order(A1_scalar_bytes);\n let A2_scalar = Scalar::from_bytes_mod_order(A2_scalar_bytes);\n let A1 = constants::ED25519_BASEPOINT_POINT * A1_scalar;\n let A2 = constants::ED25519_BASEPOINT_POINT * A2_scalar;\n\n // Compute using the optimized triple-base function\n let result_optimized = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute using raw operations: a1*A1 + a2*A2 + b*B\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n proptest::prop_assert_eq!(result_optimized, expected, \"Optimized triple scalar mul should equal raw operations\");\n }\n 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That is, the most\n /// significant bit of `bytes[31]` is 0.\n ///\n /// This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most\n /// integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`]).\n /// The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB\n /// is 0 (see `MontgomeryPoint::mul`).\n ///\n /// # Invariant #2 (weak)\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents\n /// a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is\n /// stronger than invariant #1. It also sometimes has to be broken.\n ///\n /// This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,\n /// mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and\n /// `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,\n /// scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant\n /// #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined.\n ///\n /// Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar.\n /// Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars.\n /// Multiplication is correct, but this is only due to a quirk of our implementation, and not\n /// guaranteed to hold in general in the future.\n ///\n /// Note: It is not possible to construct an unreduced `Scalar` from the public API unless the\n /// `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all\n /// public non-legacy uses, invariant #2\n /// always holds.\n ///\n pub(crate) bytes: [u8; 32],\n}","attr_info":{"attributes":[{"DocComment":" The `Scalar` struct holds an element of \\\\(\\mathbb Z / \\ell\\mathbb Z \\\\)."},{"Unknown":{"path":"allow","args":"clippy::derived_hash_with_manual_eq"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":5,"beg":{"line":201,"col":4},"end":{"line":201,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the"},{"DocComment":" group order."},{"DocComment":""},{"DocComment":" # Invariant #1"},{"DocComment":""},{"DocComment":" The integer representing this scalar is less than \\\\(2\\^{255}\\\\). That is, the most"},{"DocComment":" significant bit of `bytes[31]` is 0."},{"DocComment":""},{"DocComment":" This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most"},{"DocComment":" integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`])."},{"DocComment":" The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB"},{"DocComment":" is 0 (see `MontgomeryPoint::mul`)."},{"DocComment":""},{"DocComment":" # Invariant #2 (weak)"},{"DocComment":""},{"DocComment":" The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents"},{"DocComment":" a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is"},{"DocComment":" stronger than invariant #1. It also sometimes has to be broken."},{"DocComment":""},{"DocComment":" This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,"},{"DocComment":" mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and"},{"DocComment":" `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,"},{"DocComment":" scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant"},{"DocComment":" #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined."},{"DocComment":""},{"DocComment":" Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar."},{"DocComment":" Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars."},{"DocComment":" Multiplication is correct, but this is only due to a quirk of our implementation, and not"},{"DocComment":" guaranteed to hold in general in the future."},{"DocComment":""},{"DocComment":" Note: It is not possible to construct an unreduced `Scalar` from the public API unless the"},{"DocComment":" `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all"},{"DocComment":" public non-legacy uses, invariant #2"},{"DocComment":" always holds."},{"DocComment":""}],"inline":null,"rename":null,"public":false},"name":"bytes","ty":{"HashConsedValue":[2744,{"Array":[{"HashConsedValue":[2323,{"Literal":{"UInt":"U8"}}]},{"kind":{"Literal":{"Scalar":{"Unsigned":["Usize","32"]}}},"ty":{"Deduplicated":1785}}]}]}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":32,"align":1,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":2,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["ed_sigs",0]},{"Ident":["verification_key",0]},{"Ident":["VerificationKey",0]}],"span":{"data":{"file_id":6,"beg":{"line":223,"col":0},"end":{"line":226,"col":1}},"generated_from_span":null},"source_text":"pub struct VerificationKey {\n pub(crate) A_bytes: VerificationKeyBytes,\n pub(crate) minus_A: EdwardsPoint,\n}","attr_info":{"attributes":[{"DocComment":" A valid Ed25519 verification key."},{"DocComment":""},{"DocComment":" This is also called a public key by other implementations."},{"DocComment":""},{"DocComment":" This type holds decompressed state used in signature verification; if the"},{"DocComment":" verification key may not be used immediately, it is probably better to use"},{"DocComment":" [`VerificationKeyBytes`], which stores only the length-checked encoded bytes."},{"DocComment":""},{"DocComment":" ## Zcash-specific consensus properties"},{"DocComment":""},{"DocComment":" Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in"},{"DocComment":" [ZIP 215]. The verification criteria for an (encoded) verification key `A_bytes` are:"},{"DocComment":""},{"DocComment":" * `A_bytes` MUST be an encoding of a point `A` on the twisted Edwards form of"},{"DocComment":" Curve25519, and non-canonical encodings MUST be accepted;"},{"DocComment":""},{"DocComment":" [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519"},{"Unknown":{"path":"allow","args":"non_snake_case"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":6,"beg":{"line":224,"col":4},"end":{"line":224,"col":44}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"A_bytes","ty":{"HashConsedValue":[2918,{"Adt":{"id":{"Adt":16},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}]}},{"span":{"data":{"file_id":6,"beg":{"line":225,"col":4},"end":{"line":225,"col":36}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"minus_A","ty":{"Deduplicated":2920}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":192,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,32],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":3,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Ident":["FieldElement",0]}],"span":{"data":{"file_id":8,"beg":{"line":50,"col":0},"end":{"line":50,"col":75}},"generated_from_span":null},"source_text":"pub(crate) type FieldElement = backend::serial::u64::field::FieldElement51;","attr_info":{"attributes":[{"DocComment":" A `FieldElement` represents an element of the field"},{"DocComment":" \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\)."},{"DocComment":""},{"DocComment":" The `FieldElement` type is pinned to the 64-bit serial implementation."}],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Alias":{"Deduplicated":3141}},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":40,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":4,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Ident":["FieldElement51",0]}],"span":{"data":{"file_id":9,"beg":{"line":43,"col":0},"end":{"line":43,"col":47}},"generated_from_span":null},"source_text":"pub struct FieldElement51(pub(crate) [u64; 5]);","attr_info":{"attributes":[{"DocComment":" A `FieldElement51` represents an element of the field"},{"DocComment":" \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\)."},{"DocComment":""},{"DocComment":" In the 64-bit implementation, a `FieldElement` is represented in"},{"DocComment":" radix \\\\(2\\^{51}\\\\) as five `u64`s; the coefficients are allowed to"},{"DocComment":" grow up to \\\\(2\\^{54}\\\\) between reductions modulo \\\\(p\\\\)."},{"DocComment":""},{"DocComment":" # Note"},{"DocComment":""},{"DocComment":" The `curve25519::field` module provides a type alias"},{"DocComment":" `curve25519::field::FieldElement` to either `FieldElement51`"},{"DocComment":" or `FieldElement2625`."},{"DocComment":""},{"DocComment":" The backend-specific type `FieldElement51` should not be used"},{"DocComment":" outside of the `curve25519::field` module."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":9,"beg":{"line":43,"col":26},"end":{"line":43,"col":45}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":null,"ty":{"Deduplicated":1786}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":40,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":5,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["ProjectivePoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":154,"col":0},"end":{"line":158,"col":1}},"generated_from_span":null},"source_text":"pub struct ProjectivePoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A `ProjectivePoint` is a point \\\\((X:Y:Z)\\\\) on the \\\\(\\mathbb"},{"DocComment":" P\\^2\\\\) model of the curve."},{"DocComment":" A point \\\\((x,y)\\\\) in the affine model corresponds to"},{"DocComment":" \\\\((x:y:1)\\\\)."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":155,"col":4},"end":{"line":155,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"X","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":156,"col":4},"end":{"line":156,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Y","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":157,"col":4},"end":{"line":157,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Z","ty":{"Deduplicated":3141}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":120,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":6,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["CompletedPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":169,"col":0},"end":{"line":174,"col":1}},"generated_from_span":null},"source_text":"pub struct CompletedPoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n pub T: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A `CompletedPoint` is a point \\\\(((X:Z), (Y:T))\\\\) on the \\\\(\\mathbb"},{"DocComment":" P\\^1 \\times \\mathbb P\\^1 \\\\) model of the curve."},{"DocComment":" A point (x,y) in the affine model corresponds to \\\\( ((x:1),(y:1))"},{"DocComment":" \\\\)."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":170,"col":4},"end":{"line":170,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"X","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":171,"col":4},"end":{"line":171,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Y","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":172,"col":4},"end":{"line":172,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Z","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":173,"col":4},"end":{"line":173,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"T","ty":{"Deduplicated":3141}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":160,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80,120],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":7,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["AffineNielsPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":184,"col":0},"end":{"line":188,"col":1}},"generated_from_span":null},"source_text":"pub struct AffineNielsPoint {\n pub y_plus_x: FieldElement,\n pub y_minus_x: FieldElement,\n pub xy2d: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A pre-computed point in the affine model for the curve, represented as"},{"DocComment":" \\\\((y+x, y-x, 2dxy)\\\\) in \"Niels coordinates\"."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":185,"col":4},"end":{"line":185,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"y_plus_x","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":186,"col":4},"end":{"line":186,"col":31}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"y_minus_x","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":10,"beg":{"line":187,"col":4},"end":{"line":187,"col":26}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"xy2d","ty":{"Deduplicated":3141}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":120,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":8,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["ProjectiveNielsPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":206,"col":0},"end":{"line":211,"col":1}},"generated_from_span":null},"source_text":"pub struct ProjectiveNielsPoint {\n pub Y_plus_X: FieldElement,\n pub Y_minus_X: FieldElement,\n pub Z: FieldElement,\n pub T2d: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A pre-computed point on the \\\\( \\mathbb P\\^3 \\\\) model for the"},{"DocComment":" curve, represented as \\\\((Y+X, Y-X, Z, 2dXY)\\\\) in \"Niels coordinates\"."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level 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Used to tell an operation whether it should exit early or go on as usual."},{"DocComment":""},{"DocComment":" This is used when exposing things (like graph traversals or visitors) where"},{"DocComment":" you want the user to be able to choose whether to exit early."},{"DocComment":" Having the enum makes it clearer -- no more wondering \"wait, what did `false`"},{"DocComment":" mean again?\" -- and allows including a value."},{"DocComment":""},{"DocComment":" Similar to [`Option`] and [`Result`], this enum can be used with the `?` operator"},{"DocComment":" to return immediately if the [`Break`] variant is present or otherwise continue normally"},{"DocComment":" with the value inside the [`Continue`] variant."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Early-exiting from [`Iterator::try_for_each`]:"},{"DocComment":" ```"},{"DocComment":" use std::ops::ControlFlow;"},{"DocComment":""},{"DocComment":" let r = (2..100).try_for_each(|x| {"},{"DocComment":" if 403 % x == 0 {"},{"DocComment":" return ControlFlow::Break(x)"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" });"},{"DocComment":" assert_eq!(r, ControlFlow::Break(13));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" A basic tree traversal:"},{"DocComment":" ```"},{"DocComment":" use std::ops::ControlFlow;"},{"DocComment":""},{"DocComment":" pub struct TreeNode {"},{"DocComment":" value: T,"},{"DocComment":" left: Option>>,"},{"DocComment":" right: Option>>,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl TreeNode {"},{"DocComment":" pub fn traverse_inorder(&self, f: &mut impl FnMut(&T) -> ControlFlow) -> ControlFlow {"},{"DocComment":" if let Some(left) = &self.left {"},{"DocComment":" left.traverse_inorder(f)?;"},{"DocComment":" }"},{"DocComment":" f(&self.value)?;"},{"DocComment":" if let Some(right) = &self.right {"},{"DocComment":" right.traverse_inorder(f)?;"},{"DocComment":" }"},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" }"},{"DocComment":" fn leaf(value: T) -> Option>> {"},{"DocComment":" Some(Box::new(Self { value, left: None, right: None }))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let node = TreeNode {"},{"DocComment":" value: 0,"},{"DocComment":" left: TreeNode::leaf(1),"},{"DocComment":" right: Some(Box::new(TreeNode {"},{"DocComment":" value: -1,"},{"DocComment":" left: TreeNode::leaf(5),"},{"DocComment":" right: TreeNode::leaf(2),"},{"DocComment":" }))"},{"DocComment":" };"},{"DocComment":" let mut sum = 0;"},{"DocComment":""},{"DocComment":" let res = node.traverse_inorder(&mut |val| {"},{"DocComment":" if *val < 0 {"},{"DocComment":" ControlFlow::Break(*val)"},{"DocComment":" } else {"},{"DocComment":" sum += *val;"},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" }"},{"DocComment":" });"},{"DocComment":" assert_eq!(res, ControlFlow::Break(-1));"},{"DocComment":" assert_eq!(sum, 6);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`Break`]: ControlFlow::Break"},{"DocComment":" [`Continue`]: ControlFlow::Continue"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ControlFlow"},"generics":{"regions":[],"types":[{"index":0,"name":"B"},{"index":1,"name":"C"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Enum":[{"id":0,"span":{"data":{"file_id":31,"beg":{"line":93,"col":4},"end":{"line":93,"col":12}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" Move on to the next phase of the operation as normal."}],"inline":null,"rename":null,"public":true},"name":"Continue","fields":[{"span":{"data":{"file_id":31,"beg":{"line":93,"col":13},"end":{"line":93,"col":14}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":null,"ty":{"Deduplicated":11752}}],"discriminant":{"Scalar":{"Signed":["Isize","0"]}}},{"id":1,"span":{"data":{"file_id":31,"beg":{"line":97,"col":4},"end":{"line":97,"col":9}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" Exit the operation without running subsequent phases."}],"inline":null,"rename":null,"public":true},"name":"Break","fields":[{"span":{"data":{"file_id":31,"beg":{"line":97,"col":10},"end":{"line":97,"col":11}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":null,"ty":{"Deduplicated":11751}}],"discriminant":{"Scalar":{"Signed":["Isize","1"]}}}]},"layout":[],"ptr_metadata":"None"},{"def_id":21,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["Infallible",0]}],"span":{"data":{"file_id":32,"beg":{"line":930,"col":0},"end":{"line":930,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The error type for errors that can never happen."},{"DocComment":""},{"DocComment":" Since this enum has no variant, a value of this type can never actually exist."},{"DocComment":" This can be useful for generic APIs that use [`Result`] and parameterize the error type,"},{"DocComment":" to indicate that the result is always [`Ok`]."},{"DocComment":""},{"DocComment":" For example, the [`TryFrom`] trait (conversion that returns a [`Result`])"},{"DocComment":" has a blanket implementation for all types where a reverse [`Into`] implementation exists."},{"DocComment":""},{"DocComment":" ```ignore (illustrates std code, duplicating the impl in a doctest would be an error)"},{"DocComment":" impl TryFrom for T where U: Into {"},{"DocComment":" type Error = Infallible;"},{"DocComment":""},{"DocComment":" fn try_from(value: U) -> Result {"},{"DocComment":" Ok(U::into(value)) // Never returns `Err`"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Future compatibility"},{"DocComment":""},{"DocComment":" This enum has the same role as [the `!` “never” type][never],"},{"DocComment":" which is unstable in this version of Rust."},{"DocComment":" When `!` is stabilized, we plan to make `Infallible` a type alias to it:"},{"DocComment":""},{"DocComment":" ```ignore (illustrates future std change)"},{"DocComment":" pub type Infallible = !;"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" … and eventually deprecate `Infallible`."},{"DocComment":""},{"DocComment":" However there is one case where `!` syntax can be used"},{"DocComment":" before `!` is stabilized as a full-fledged type: in the position of a function’s return type."},{"DocComment":" Specifically, it is possible to have implementations for two different function pointer types:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" trait MyTrait {}"},{"DocComment":" impl MyTrait for fn() -> ! {}"},{"DocComment":" impl MyTrait for fn() -> std::convert::Infallible {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" With `Infallible` being an enum, this code is valid."},{"DocComment":" However when `Infallible` becomes an alias for the never type,"},{"DocComment":" the two `impl`s will start to overlap"},{"DocComment":" and therefore will be disallowed by the language’s trait coherence rules."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Enum":[]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":0,"align":1,"discriminator":null,"uninhabited":true,"variant_layouts":[],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},null,{"def_id":23,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["edwards",0]},{"Ident":["affine",0]},{"Ident":["AffinePoint",0]}],"span":{"data":{"file_id":39,"beg":{"line":12,"col":0},"end":{"line":15,"col":1}},"generated_from_span":null},"source_text":"pub struct AffinePoint {\n pub(super) x: FieldElement,\n pub(super) y: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" Affine Edwards point on untwisted curve."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":39,"beg":{"line":13,"col":4},"end":{"line":13,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"x","ty":{"Deduplicated":3141}},{"span":{"data":{"file_id":39,"beg":{"line":14,"col":4},"end":{"line":14,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"y","ty":{"Deduplicated":3141}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":80,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":24,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["array",0]},{"Ident":["TryFromSliceError",0]}],"span":{"data":{"file_id":41,"beg":{"line":183,"col":0},"end":{"line":183,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The error type returned when a conversion from a slice to an array fails."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":0,"align":1,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},null,{"def_id":26,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["array",0]},{"Ident":["iter",0]},{"Ident":["IntoIter",0]}],"span":{"data":{"file_id":44,"beg":{"line":20,"col":0},"end":{"line":20,"col":38}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A by-value [array] iterator."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ArrayIntoIter"},"generics":{"regions":[],"types":[{"index":0,"name":"T"}],"const_generics":[{"index":0,"name":"N","ty":{"Deduplicated":1785}}],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":27,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["num",0]},{"Ident":["nonzero",0]},{"Ident":["NonZero",0]}],"span":{"data":{"file_id":45,"beg":{"line":128,"col":0},"end":{"line":128,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A value that is known not to equal zero."},{"DocComment":""},{"DocComment":" This enables some memory layout optimization."},{"DocComment":" For example, `Option>` is the same size as `u32`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use core::{num::NonZero};"},{"DocComment":""},{"DocComment":" assert_eq!(size_of::>>(), size_of::());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Layout"},{"DocComment":""},{"DocComment":" `NonZero` is guaranteed to have the same layout and bit validity as `T`"},{"DocComment":" with the exception that the all-zero bit pattern is invalid."},{"DocComment":" `Option>` is guaranteed to be compatible with `T`, including in"},{"DocComment":" FFI."},{"DocComment":""},{"DocComment":" Thanks to the [null pointer optimization], `NonZero` and"},{"DocComment":" `Option>` are guaranteed to have the same size and alignment:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::num::NonZero;"},{"DocComment":""},{"DocComment":" assert_eq!(size_of::>(), size_of::>>());"},{"DocComment":" assert_eq!(align_of::>(), align_of::>>());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [null pointer optimization]: crate::option#representation"},{"DocComment":""},{"DocComment":" # Note on generic usage"},{"DocComment":""},{"DocComment":" `NonZero` can only be used with some standard library primitive types"},{"DocComment":" (such as `u8`, `i32`, and etc.). The type parameter `T` must implement the"},{"DocComment":" internal trait [`ZeroablePrimitive`], which is currently permanently unstable"},{"DocComment":" and cannot be implemented by users. Therefore, you cannot use `NonZero`"},{"DocComment":" with your own types, nor can you implement traits for all `NonZero`,"},{"DocComment":" only for concrete types."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"NonZero"},"generics":{"regions":[],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_NonZeroInner"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":28,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["step_by",0]},{"Ident":["StepBy",0]}],"span":{"data":{"file_id":46,"beg":{"line":16,"col":0},"end":{"line":16,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator for stepping iterators by a custom amount."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`step_by`] method on [`Iterator`]. See"},{"DocComment":" its documentation for more."},{"DocComment":""},{"DocComment":" [`step_by`]: Iterator::step_by"},{"DocComment":" [`Iterator`]: trait.Iterator.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":29,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["chain",0]},{"Ident":["Chain",0]}],"span":{"data":{"file_id":48,"beg":{"line":23,"col":0},"end":{"line":23,"col":22}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that links two iterators together, in a chain."},{"DocComment":""},{"DocComment":" This `struct` is created by [`chain`] or [`Iterator::chain`]. See their"},{"DocComment":" documentation for more."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::iter::Chain;"},{"DocComment":" use std::slice::Iter;"},{"DocComment":""},{"DocComment":" let a1 = [1, 2, 3];"},{"DocComment":" let a2 = [4, 5, 6];"},{"DocComment":" let iter: Chain, Iter<'_, _>> = a1.iter().chain(a2.iter());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"A"},{"index":1,"name":"B"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":30,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["zip",0]},{"Ident":["Zip",0]}],"span":{"data":{"file_id":49,"beg":{"line":13,"col":0},"end":{"line":13,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that iterates two other iterators simultaneously."},{"DocComment":""},{"DocComment":" This `struct` is created by [`zip`] or [`Iterator::zip`]."},{"DocComment":" See their documentation for more."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"A"},{"index":1,"name":"B"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":31,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["intersperse",0]},{"Ident":["Intersperse",0]}],"span":{"data":{"file_id":50,"beg":{"line":9,"col":0},"end":{"line":9,"col":35}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator adapter that places a separator between all elements."},{"DocComment":""},{"DocComment":" This `struct` is created by [`Iterator::intersperse`]. See its documentation"},{"DocComment":" for more information."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":32,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["intersperse",0]},{"Ident":["IntersperseWith",0]}],"span":{"data":{"file_id":50,"beg":{"line":83,"col":0},"end":{"line":83,"col":32}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator adapter that places a separator between all elements."},{"DocComment":""},{"DocComment":" This `struct` is created by [`Iterator::intersperse_with`]. See its"},{"DocComment":" documentation for more information."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"G"},{"index":2,"name":"Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":33,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["map",0]},{"Ident":["Map",0]}],"span":{"data":{"file_id":52,"beg":{"line":61,"col":0},"end":{"line":61,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that maps the values of `iter` with `f`."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`map`] method on [`Iterator`]. See its"},{"DocComment":" documentation for more."},{"DocComment":""},{"DocComment":" [`map`]: Iterator::map"},{"DocComment":" [`Iterator`]: trait.Iterator.html"},{"DocComment":""},{"DocComment":" # Notes about side effects"},{"DocComment":""},{"DocComment":" The [`map`] iterator implements [`DoubleEndedIterator`], meaning that"},{"DocComment":" you can also [`map`] backwards:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let v: Vec = [1, 2, 3].into_iter().map(|x| x + 1).rev().collect();"},{"DocComment":""},{"DocComment":" assert_eq!(v, [4, 3, 2]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`DoubleEndedIterator`]: trait.DoubleEndedIterator.html"},{"DocComment":""},{"DocComment":" But if your closure has state, iterating backwards may act in a way you do"},{"DocComment":" not expect. Let's go through an example. First, in the forward direction:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let mut c = 0;"},{"DocComment":""},{"DocComment":" for pair in ['a', 'b', 'c'].into_iter()"},{"DocComment":" .map(|letter| { c += 1; (letter, c) }) {"},{"DocComment":" println!(\"{pair:?}\");"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" This will print `('a', 1), ('b', 2), ('c', 3)`."},{"DocComment":""},{"DocComment":" Now consider this twist where we add a call to `rev`. This version will"},{"DocComment":" print `('c', 1), ('b', 2), ('a', 3)`. Note that the letters are reversed,"},{"DocComment":" but the values of the counter still go in order. This is because `map()` is"},{"DocComment":" still being called lazily on each item, but we are popping items off the"},{"DocComment":" back of the vector now, instead of shifting them from the front."},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let mut c = 0;"},{"DocComment":""},{"DocComment":" for pair in ['a', 'b', 'c'].into_iter()"},{"DocComment":" .map(|letter| { c += 1; (letter, c) })"},{"DocComment":" .rev() {"},{"DocComment":" println!(\"{pair:?}\");"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"F"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":34,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["filter",0]},{"Ident":["Filter",0]}],"span":{"data":{"file_id":53,"beg":{"line":21,"col":0},"end":{"line":21,"col":23}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that filters the elements of `iter` with `predicate`."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`filter`] method on [`Iterator`]. 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value\n limbs[4] &= low_51_bit_mask;\n\n // Now arrange the bits of the limbs.\n let mut s = [0u8;32];\n s[ 0] = limbs[0] as u8;\n s[ 1] = (limbs[0] >> 8) as u8;\n s[ 2] = (limbs[0] >> 16) as u8;\n s[ 3] = (limbs[0] >> 24) as u8;\n s[ 4] = (limbs[0] >> 32) as u8;\n s[ 5] = (limbs[0] >> 40) as u8;\n s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;\n s[ 7] = (limbs[1] >> 5) as u8;\n s[ 8] = (limbs[1] >> 13) as u8;\n s[ 9] = (limbs[1] >> 21) as u8;\n s[10] = (limbs[1] >> 29) as u8;\n s[11] = (limbs[1] >> 37) as u8;\n s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;\n s[13] = (limbs[2] >> 2) as u8;\n s[14] = (limbs[2] >> 10) as u8;\n s[15] = (limbs[2] >> 18) as u8;\n s[16] = (limbs[2] >> 26) as u8;\n s[17] = (limbs[2] >> 34) as u8;\n s[18] = (limbs[2] >> 42) as u8;\n s[19] = ((limbs[2] >> 50) | (limbs[3] << 1)) as u8;\n s[20] = (limbs[3] >> 7) as u8;\n s[21] = (limbs[3] >> 15) as u8;\n s[22] = (limbs[3] >> 23) as u8;\n s[23] = (limbs[3] >> 31) as u8;\n s[24] = (limbs[3] >> 39) as 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Extends a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" As this is the only required method for this trait, the [trait-level] docs"},{"DocComment":" contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: Extend"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"abc\");"},{"DocComment":""},{"DocComment":" message.extend(['d', 'e', 'f'].iter());"},{"DocComment":""},{"DocComment":" assert_eq!(\"abcdef\", &message);"},{"DocComment":" 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Removes and returns an element from the end of the iterator."},{"DocComment":""},{"DocComment":" Returns `None` when there are no more elements."},{"DocComment":""},{"DocComment":" The [trait-level] docs contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: DoubleEndedIterator"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Remarks"},{"DocComment":""},{"DocComment":" The elements yielded by `DoubleEndedIterator`'s methods may differ from"},{"DocComment":" the ones yielded by [`Iterator`]'s methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let vec = vec![(1, 'a'), (1, 'b'), (1, 'c'), (2, 'a'), (2, 'b')];"},{"DocComment":" let uniq_by_fst_comp = || {"},{"DocComment":" let mut seen = std::collections::HashSet::new();"},{"DocComment":" vec.iter().copied().filter(move |x| seen.insert(x.0))"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" assert_eq!(uniq_by_fst_comp().last(), Some((2, 'a')));"},{"DocComment":" assert_eq!(uniq_by_fst_comp().next_back(), Some((2, 'b')));"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().fold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(1, 'a'), (2, 'a')]"},{"DocComment":" );"},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().rfold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(2, 'b'), (1, 'c')]"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Clause0_Clause0_Item"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11932}],"output":{"Deduplicated":11819}},"src":{"TraitDecl":{"trait_ref":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},null,null,null,null,null,null,null,{"def_id":377,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Ord",0]},{"Ident":["cmp",0]}],"span":{"data":{"file_id":33,"beg":{"line":1000,"col":4},"end":{"line":1000,"col":44}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" This method returns an [`Ordering`] between `self` and `other`."},{"DocComment":""},{"DocComment":" By convention, `self.cmp(&other)` returns the ordering matching the expression"},{"DocComment":" `self other` if true."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" assert_eq!(5.cmp(&10), Ordering::Less);"},{"DocComment":" assert_eq!(10.cmp(&5), Ordering::Greater);"},{"DocComment":" assert_eq!(5.cmp(&5), Ordering::Equal);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ord_cmp_method"},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11809},{"Deduplicated":11909}],"output":{"HashConsedValue":[5641,{"Adt":{"id":{"Adt":49},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}]}},"src":{"TraitDecl":{"trait_ref":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},null,null,null,{"def_id":381,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Product",0]},{"Ident":["product",0]}],"span":{"data":{"file_id":43,"beg":{"line":44,"col":4},"end":{"line":44,"col":55}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by multiplying"},{"DocComment":" the 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This method returns an ordering between `self` and `other` values if one exists."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&2.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Less));"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Equal));"},{"DocComment":""},{"DocComment":" let result = 2.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Greater));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" When comparison is impossible:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let result = f64::NAN.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, None);"},{"DocComment":" 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For more about the concept of iterators"},{"DocComment":" generally, please see the [module-level documentation]. In particular, you"},{"DocComment":" may want to know how to [implement `Iterator`][impl]."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":" [impl]: crate::iter#implementing-iterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"iterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"next","attr_info":{"attributes":[{"DocComment":" Advances the iterator and returns the next value."},{"DocComment":""},{"DocComment":" Returns [`None`] when iteration is finished. Individual iterator"},{"DocComment":" implementations may choose to resume iteration, and so calling `next()`"},{"DocComment":" again may or may not eventually start returning [`Some(Item)`] again at some"},{"DocComment":" point."},{"DocComment":""},{"DocComment":" [`Some(Item)`]: Some"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let a = [1, 2, 3];"},{"DocComment":""},{"DocComment":" let mut iter = a.into_iter();"},{"DocComment":""},{"DocComment":" // A call to next() returns the next value..."},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":""},{"DocComment":" // ... and then None once it's over."},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":""},{"DocComment":" // More calls may or may not return `None`. Here, they always will."},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"HashConsedValue":[11818,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11751},"Mut"]}]}],"output":{"Deduplicated":11819}},"item":{"id":175,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11820,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[0,0]}},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":25},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":1,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["range",0]},{"Ident":["Step",0]}],"span":{"data":{"file_id":21,"beg":{"line":34,"col":0},"end":{"line":34,"col":64}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Objects that have a notion of *successor* and *predecessor* operations."},{"DocComment":""},{"DocComment":" The *successor* operation moves towards values that compare greater."},{"DocComment":" The *predecessor* operation moves towards values that compare lesser."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"range_step"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":21,"beg":{"line":34,"col":22},"end":{"line":34,"col":35}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":21,"beg":{"line":34,"col":38},"end":{"line":34,"col":56}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"steps_between","attr_info":{"attributes":[{"DocComment":" Returns the bounds on the number of *successor* steps required to get from `start` to `end`"},{"DocComment":" like [`Iterator::size_hint()`][Iterator::size_hint()]."},{"DocComment":""},{"DocComment":" Returns `(usize::MAX, None)` if the number of steps would overflow `usize`, or is infinite."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `b`, and `n`:"},{"DocComment":""},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` if and only if `Step::forward_checked(&a, n) == Some(b)`"},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` if and only if `Step::backward_checked(&b, n) == Some(a)`"},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` only if `a <= b`"},{"DocComment":" * Corollary: `steps_between(&a, &b) == (0, Some(0))` if and only if `a == b`"},{"DocComment":" * `steps_between(&a, &b) == (0, None)` if `a > b`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"HashConsedValue":[11821,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11751},"Shared"]}]},{"HashConsedValue":[11822,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11751},"Shared"]}]}],"output":{"Deduplicated":4215}},"item":{"id":261,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11823,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":1,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[1,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"forward_checked","attr_info":{"attributes":[{"DocComment":" Returns the value that would be obtained by taking the *successor*"},{"DocComment":" of `self` `count` times."},{"DocComment":""},{"DocComment":" If this would overflow the range of values supported by `Self`, returns `None`."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `n`, and `m`:"},{"DocComment":""},{"DocComment":" * `Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == Step::forward_checked(a, m).and_then(|x| Step::forward_checked(x, n))`"},{"DocComment":" * `Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == try { Step::forward_checked(a, n.checked_add(m)) }`"},{"DocComment":""},{"DocComment":" For any `a` and `n`:"},{"DocComment":""},{"DocComment":" * `Step::forward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::forward_checked(&x, 1))`"},{"DocComment":" * Corollary: `Step::forward_checked(a, 0) == Some(a)`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":1785}],"output":{"Deduplicated":11824}},"item":{"id":262,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11823}]}}},"kind":{"TraitMethod":[1,1]}},null,null,{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"backward_checked","attr_info":{"attributes":[{"DocComment":" Returns the value that would be obtained by taking the *predecessor*"},{"DocComment":" of `self` `count` times."},{"DocComment":""},{"DocComment":" If this would overflow the range of values supported by `Self`, returns `None`."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `n`, and `m`:"},{"DocComment":""},{"DocComment":" * `Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == n.checked_add(m).and_then(|x| Step::backward_checked(a, x))`"},{"DocComment":" * `Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == try { Step::backward_checked(a, n.checked_add(m)?) }`"},{"DocComment":""},{"DocComment":" For any `a` and `n`:"},{"DocComment":""},{"DocComment":" * `Step::backward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::backward_checked(x, 1))`"},{"DocComment":" * Corollary: `Step::backward_checked(a, 0) == Some(a)`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":1785}],"output":{"Deduplicated":11824}},"item":{"id":265,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11823}]}}},"kind":{"TraitMethod":[1,4]}},null,null],"vtable":null},{"def_id":2,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["From",0]}],"span":{"data":{"file_id":32,"beg":{"line":587,"col":0},"end":{"line":587,"col":30}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used to do value-to-value conversions while consuming the input value. It is the reciprocal of"},{"DocComment":" [`Into`]."},{"DocComment":""},{"DocComment":" One should always prefer implementing `From` over [`Into`]"},{"DocComment":" because implementing `From` automatically provides one with an implementation of [`Into`]"},{"DocComment":" thanks to the blanket implementation in the standard library."},{"DocComment":""},{"DocComment":" Only implement [`Into`] when targeting a version prior to Rust 1.41 and converting to a type"},{"DocComment":" outside the current crate."},{"DocComment":" `From` was not able to do these types of conversions in earlier versions because of Rust's"},{"DocComment":" orphaning rules."},{"DocComment":" See [`Into`] for more details."},{"DocComment":""},{"DocComment":" Prefer using [`Into`] over [`From`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`Into`] can be used as well."},{"DocComment":""},{"DocComment":" The `From` trait is also very useful when performing error handling. When constructing a function"},{"DocComment":" that is capable of failing, the return type will generally be of the form `Result`."},{"DocComment":" `From` simplifies error handling by allowing a function to return a single error type"},{"DocComment":" that encapsulates multiple error types. See the \"Examples\" section and [the book][book] for more"},{"DocComment":" details."},{"DocComment":""},{"DocComment":" **Note: This trait must not fail**. The `From` trait is intended for perfect conversions."},{"DocComment":" If the conversion can fail or is not perfect, use [`TryFrom`]."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - `From for U` implies [`Into`]` for T`"},{"DocComment":" - `From` is reflexive, which means that `From for T` is implemented"},{"DocComment":""},{"DocComment":" # When to implement `From`"},{"DocComment":""},{"DocComment":" While there's no technical restrictions on which conversions can be done using"},{"DocComment":" a `From` implementation, the general expectation is that the conversions"},{"DocComment":" should typically be restricted as follows:"},{"DocComment":""},{"DocComment":" * The conversion is *infallible*: if the conversion can fail, use [`TryFrom`]"},{"DocComment":" instead; don't provide a `From` impl that panics."},{"DocComment":""},{"DocComment":" * The conversion is *lossless*: semantically, it should not lose or discard"},{"DocComment":" information. For example, `i32: From` exists, where the original"},{"DocComment":" value can be recovered using `u16: TryFrom`. And `String: From<&str>`"},{"DocComment":" exists, where you can get something equivalent to the original value via"},{"DocComment":" `Deref`. But `From` cannot be used to convert from `u32` to `u16`, since"},{"DocComment":" that cannot succeed in a lossless way. (There's some wiggle room here for"},{"DocComment":" information not considered semantically relevant. For example,"},{"DocComment":" `Box<[T]>: From>` exists even though it might not preserve capacity,"},{"DocComment":" like how two vectors can be equal despite differing capacities.)"},{"DocComment":""},{"DocComment":" * The conversion is *value-preserving*: the conceptual kind and meaning of"},{"DocComment":" the resulting value is the same, even though the Rust type and technical"},{"DocComment":" representation might be different. For example `-1_i8 as u8` is *lossless*,"},{"DocComment":" since `as` casting back can recover the original value, but that conversion"},{"DocComment":" is *not* available via `From` because `-1` and `255` are different conceptual"},{"DocComment":" values (despite being identical bit patterns technically). But"},{"DocComment":" `f32: From` *is* available because `1_i16` and `1.0_f32` are conceptually"},{"DocComment":" the same real number (despite having very different bit patterns technically)."},{"DocComment":" `String: From` is available because they're both *text*, but"},{"DocComment":" `String: From` is *not* available, since `1` (a number) and `\"1\"`"},{"DocComment":" (text) are too different. (Converting values to text is instead covered"},{"DocComment":" by the [`Display`](crate::fmt::Display) trait.)"},{"DocComment":""},{"DocComment":" * The conversion is *obvious*: it's the only reasonable conversion between"},{"DocComment":" the two types. Otherwise it's better to have it be a named method or"},{"DocComment":" constructor, like how [`str::as_bytes`] is a method and how integers have"},{"DocComment":" methods like [`u32::from_ne_bytes`], [`u32::from_le_bytes`], and"},{"DocComment":" [`u32::from_be_bytes`], none of which are `From` implementations. Whereas"},{"DocComment":" there's only one reasonable way to wrap an [`Ipv6Addr`](crate::net::Ipv6Addr)"},{"DocComment":" into an [`IpAddr`](crate::net::IpAddr), thus `IpAddr: From` exists."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" [`String`] implements `From<&str>`:"},{"DocComment":""},{"DocComment":" An explicit conversion from a `&str` to a String is done as follows:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let string = \"hello\".to_string();"},{"DocComment":" let other_string = String::from(\"hello\");"},{"DocComment":""},{"DocComment":" assert_eq!(string, other_string);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" While performing error handling it is often useful to implement `From` for your own error type."},{"DocComment":" By converting underlying error types to our own custom error type that encapsulates the"},{"DocComment":" underlying error type, we can return a single error type without losing information on the"},{"DocComment":" underlying cause. The '?' operator automatically converts the underlying error type to our"},{"DocComment":" custom error type with `From::from`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fs;"},{"DocComment":" use std::io;"},{"DocComment":" use std::num;"},{"DocComment":""},{"DocComment":" enum CliError {"},{"DocComment":" IoError(io::Error),"},{"DocComment":" ParseError(num::ParseIntError),"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl From for CliError {"},{"DocComment":" fn from(error: io::Error) -> Self {"},{"DocComment":" CliError::IoError(error)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl From for CliError {"},{"DocComment":" fn from(error: num::ParseIntError) -> Self {"},{"DocComment":" CliError::ParseError(error)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn open_and_parse_file(file_name: &str) -> Result {"},{"DocComment":" let mut contents = fs::read_to_string(&file_name)?;"},{"DocComment":" let num: i32 = contents.trim().parse()?;"},{"DocComment":" Ok(num)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`from`]: From::from"},{"DocComment":" [book]: ../../book/ch09-00-error-handling.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"From"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from","attr_info":{"attributes":[{"DocComment":" Converts to this type from the input type."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11752}],"output":{"Deduplicated":11751}},"item":{"id":292,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11825,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":2,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[2,0]}}],"vtable":null},{"def_id":3,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Eq",0]}],"span":{"data":{"file_id":33,"beg":{"line":338,"col":0},"end":{"line":338,"col":58}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for comparisons corresponding to [equivalence relations]("},{"DocComment":" https://en.wikipedia.org/wiki/Equivalence_relation)."},{"DocComment":""},{"DocComment":" The primary difference to [`PartialEq`] is the additional requirement for reflexivity. A type"},{"DocComment":" that implements [`PartialEq`] guarantees that for all `a`, `b` and `c`:"},{"DocComment":""},{"DocComment":" - symmetric: `a == b` implies `b == a`"},{"DocComment":" - transitive: `a == b` and `b == c` implies `a == c`"},{"DocComment":" - consistent: `a != b` if and only if `!(a == b)`"},{"DocComment":""},{"DocComment":" `Eq`, which builds on top of [`PartialEq`] also implies:"},{"DocComment":""},{"DocComment":" - reflexive: `a == a`"},{"DocComment":""},{"DocComment":" This property cannot be checked by the compiler, and therefore `Eq` is a trait without methods."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" Floating point types such as [`f32`] and [`f64`] implement only [`PartialEq`] but *not* `Eq`"},{"DocComment":" because `NaN` != `NaN`."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`. When `derive`d, because `Eq` has no extra methods, it"},{"DocComment":" is only informing the compiler that this is an equivalence relation rather than a partial"},{"DocComment":" equivalence relation. Note that the `derive` strategy requires all fields are `Eq`, which isn't"},{"DocComment":" always desired."},{"DocComment":""},{"DocComment":" ## How can I implement `Eq`?"},{"DocComment":""},{"DocComment":" If you cannot use the `derive` strategy, specify that your type implements `Eq`, which has no"},{"DocComment":" extra methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.isbn == other.isbn"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Book {}"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Eq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":338,"col":20},"end":{"line":338,"col":43}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[null,{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"assert_fields_are_eq","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"Deduplicated":1802}},"item":{"id":41,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11826,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[3,1]}}],"vtable":null},{"def_id":4,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["PartialEq",0]}],"span":{"data":{"file_id":33,"beg":{"line":250,"col":0},"end":{"line":250,"col":65}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for comparisons using the equality operator."},{"DocComment":""},{"DocComment":" Implementing this trait for types provides the `==` and `!=` operators for"},{"DocComment":" those types."},{"DocComment":""},{"DocComment":" `x.eq(y)` can also be written `x == y`, and `x.ne(y)` can be written `x != y`."},{"DocComment":" We use the easier-to-read infix notation in the remainder of this documentation."},{"DocComment":""},{"DocComment":" This trait allows for comparisons using the equality operator, for types"},{"DocComment":" that do not have a full equivalence relation. For example, in floating point"},{"DocComment":" numbers `NaN != NaN`, so floating point types implement `PartialEq` but not"},{"DocComment":" [`trait@Eq`]. Formally speaking, when `Rhs == Self`, this trait corresponds"},{"DocComment":" to a [partial equivalence relation]."},{"DocComment":""},{"DocComment":" [partial equivalence relation]: https://en.wikipedia.org/wiki/Partial_equivalence_relation"},{"DocComment":""},{"DocComment":" Implementations must ensure that `eq` and `ne` are consistent with each other:"},{"DocComment":""},{"DocComment":" - `a != b` if and only if `!(a == b)`."},{"DocComment":""},{"DocComment":" The default implementation of `ne` provides this consistency and is almost"},{"DocComment":" always sufficient. It should not be overridden without very good reason."},{"DocComment":""},{"DocComment":" If [`PartialOrd`] or [`Ord`] are also implemented for `Self` and `Rhs`, their methods must also"},{"DocComment":" be consistent with `PartialEq` (see the documentation of those traits for the exact"},{"DocComment":" requirements). It's easy to accidentally make them disagree by deriving some of the traits and"},{"DocComment":" manually implementing others."},{"DocComment":""},{"DocComment":" The equality relation `==` must satisfy the following conditions"},{"DocComment":" (for all `a`, `b`, `c` of type `A`, `B`, `C`):"},{"DocComment":""},{"DocComment":" - **Symmetry**: if `A: PartialEq` and `B: PartialEq`, then **`a == b`"},{"DocComment":" implies `b == a`**; and"},{"DocComment":""},{"DocComment":" - **Transitivity**: if `A: PartialEq` and `B: PartialEq` and `A:"},{"DocComment":" PartialEq`, then **`a == b` and `b == c` implies `a == c`**."},{"DocComment":" This must also work for longer chains, such as when `A: PartialEq`, `B: PartialEq`,"},{"DocComment":" `C: PartialEq`, and `A: PartialEq` all exist."},{"DocComment":""},{"DocComment":" Note that the `B: PartialEq` (symmetric) and `A: PartialEq`"},{"DocComment":" (transitive) impls are not forced to exist, but these requirements apply"},{"DocComment":" whenever they do exist."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Cross-crate considerations"},{"DocComment":""},{"DocComment":" Upholding the requirements stated above can become tricky when one crate implements `PartialEq`"},{"DocComment":" for a type of another crate (i.e., to allow comparing one of its own types with a type from the"},{"DocComment":" standard library). The recommendation is to never implement this trait for a foreign type. In"},{"DocComment":" other words, such a crate should do `impl PartialEq for LocalType`, but it should"},{"DocComment":" *not* do `impl PartialEq for ForeignType`."},{"DocComment":""},{"DocComment":" This avoids the problem of transitive chains that criss-cross crate boundaries: for all local"},{"DocComment":" types `T`, you may assume that no other crate will add `impl`s that allow comparing `T == U`. In"},{"DocComment":" other words, if other crates add `impl`s that allow building longer transitive chains `U1 == ..."},{"DocComment":" == T == V1 == ...`, then all the types that appear to the right of `T` must be types that the"},{"DocComment":" crate defining `T` already knows about. This rules out transitive chains where downstream crates"},{"DocComment":" can add new `impl`s that \"stitch together\" comparisons of foreign types in ways that violate"},{"DocComment":" transitivity."},{"DocComment":""},{"DocComment":" Not having such foreign `impl`s also avoids forward compatibility issues where one crate adding"},{"DocComment":" more `PartialEq` implementations can cause build failures in downstream crates."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`. When `derive`d on structs, two"},{"DocComment":" instances are equal if all fields are equal, and not equal if any fields"},{"DocComment":" are not equal. When `derive`d on enums, two instances are equal if they"},{"DocComment":" are the same variant and all fields are equal."},{"DocComment":""},{"DocComment":" ## How can I implement `PartialEq`?"},{"DocComment":""},{"DocComment":" An example implementation for a domain in which two books are considered"},{"DocComment":" the same book if their ISBN matches, even if the formats differ:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.isbn == other.isbn"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let b1 = Book { isbn: 3, format: BookFormat::Paperback };"},{"DocComment":" let b2 = Book { isbn: 3, format: BookFormat::Ebook };"},{"DocComment":" let b3 = Book { isbn: 10, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == b2);"},{"DocComment":" assert!(b1 != b3);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I compare two different types?"},{"DocComment":""},{"DocComment":" The type you can compare with is controlled by `PartialEq`'s type parameter."},{"DocComment":" For example, let's tweak our previous code a bit:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // The derive implements == comparisons"},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Implement == comparisons"},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &BookFormat) -> bool {"},{"DocComment":" self.format == *other"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Implement == comparisons"},{"DocComment":" impl PartialEq for BookFormat {"},{"DocComment":" fn eq(&self, other: &Book) -> bool {"},{"DocComment":" *self == other.format"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let b1 = Book { isbn: 3, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == BookFormat::Paperback);"},{"DocComment":" assert!(BookFormat::Ebook != b1);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" By changing `impl PartialEq for Book` to `impl PartialEq for Book`,"},{"DocComment":" we allow `BookFormat`s to be compared with `Book`s."},{"DocComment":""},{"DocComment":" A comparison like the one above, which ignores some fields of the struct,"},{"DocComment":" can be dangerous. It can easily lead to an unintended violation of the"},{"DocComment":" requirements for a partial equivalence relation. For example, if we kept"},{"DocComment":" the above implementation of `PartialEq` for `BookFormat` and added an"},{"DocComment":" implementation of `PartialEq` for `Book` (either via a `#[derive]` or"},{"DocComment":" via the manual implementation from the first example) then the result would"},{"DocComment":" violate transitivity:"},{"DocComment":""},{"DocComment":" ```should_panic"},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &BookFormat) -> bool {"},{"DocComment":" self.format == *other"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for BookFormat {"},{"DocComment":" fn eq(&self, other: &Book) -> bool {"},{"DocComment":" *self == other.format"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn main() {"},{"DocComment":" let b1 = Book { isbn: 1, format: BookFormat::Paperback };"},{"DocComment":" let b2 = Book { isbn: 2, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == BookFormat::Paperback);"},{"DocComment":" assert!(BookFormat::Paperback == b2);"},{"DocComment":""},{"DocComment":" // The following should hold by transitivity but doesn't."},{"DocComment":" assert!(b1 == b2); // <-- PANICS"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: u32 = 0;"},{"DocComment":" let y: u32 = 1;"},{"DocComment":""},{"DocComment":" assert_eq!(x == y, false);"},{"DocComment":" assert_eq!(x.eq(&y), false);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`eq`]: PartialEq::eq"},{"DocComment":" [`ne`]: PartialEq::ne"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"eq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"eq","attr_info":{"attributes":[{"DocComment":" Tests for `self` and `other` values to be equal, and is used by `==`."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"HashConsedValue":[11827,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11752},"Shared"]}]}],"output":{"Deduplicated":1797}},"item":{"id":43,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11828,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[4,0]}},null],"vtable":{"id":{"Adt":22},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":5,"item_meta":{"name":[{"Ident":["subtle",0]},{"Ident":["ConstantTimeEq",0]}],"span":{"data":{"file_id":22,"beg":{"line":262,"col":0},"end":{"line":262,"col":24}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An `Eq`-like trait that produces a `Choice` instead of a `bool`."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConstantTimeEq;"},{"DocComment":" let x: u8 = 5;"},{"DocComment":" let y: u8 = 13;"},{"DocComment":""},{"DocComment":" assert_eq!(x.ct_eq(&y).unwrap_u8(), 0);"},{"DocComment":" assert_eq!(x.ct_eq(&x).unwrap_u8(), 1);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"ct_eq","attr_info":{"attributes":[{"DocComment":" Determine if two items are equal."},{"DocComment":""},{"DocComment":" The `ct_eq` function should execute in constant time."},{"DocComment":""},{"DocComment":" # Returns"},{"DocComment":""},{"DocComment":" * `Choice(1u8)` if `self == other`;"},{"DocComment":" * `Choice(0u8)` if `self != other`."}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"Deduplicated":11822}],"output":{"Deduplicated":2322}},"item":{"id":46,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11829,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":5,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[5,0]}},null],"vtable":null},{"def_id":6,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["default",0]},{"Ident":["Default",0]}],"span":{"data":{"file_id":34,"beg":{"line":107,"col":0},"end":{"line":107,"col":30}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for giving a type a useful default value."},{"DocComment":""},{"DocComment":" Sometimes, you want to fall back to some kind of default value, and"},{"DocComment":" don't particularly care what it is. This comes up often with `struct`s"},{"DocComment":" that define a set of options:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" How can we define some default values? You can use `Default`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Default)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn main() {"},{"DocComment":" let options: SomeOptions = Default::default();"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Now, you get all of the default values. Rust implements `Default` for various primitive types."},{"DocComment":""},{"DocComment":" If you want to override a particular option, but still retain the other defaults:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" # #[derive(Default)]"},{"DocComment":" # struct SomeOptions {"},{"DocComment":" # foo: i32,"},{"DocComment":" # bar: f32,"},{"DocComment":" # }"},{"DocComment":" fn main() {"},{"DocComment":" let options = SomeOptions { foo: 42, ..Default::default() };"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all of the type's fields implement"},{"DocComment":" `Default`. When `derive`d, it will use the default value for each field's type."},{"DocComment":""},{"DocComment":" ### `enum`s"},{"DocComment":""},{"DocComment":" When using `#[derive(Default)]` on an `enum`, you need to choose which unit variant will be"},{"DocComment":" default. You do this by placing the `#[default]` attribute on the variant."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Default)]"},{"DocComment":" enum Kind {"},{"DocComment":" #[default]"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You cannot use the `#[default]` attribute on non-unit or non-exhaustive variants."},{"DocComment":""},{"DocComment":" The `#[default]` attribute was stabilized in Rust 1.62.0."},{"DocComment":""},{"DocComment":" ## How can I implement `Default`?"},{"DocComment":""},{"DocComment":" Provide an implementation for the `default()` method that returns the value of"},{"DocComment":" your type that should be the default:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" enum Kind {"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Default for Kind {"},{"DocComment":" fn default() -> Self { Kind::A }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Default)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Default"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"default","attr_info":{"attributes":[{"DocComment":" Returns the \"default value\" for a type."},{"DocComment":""},{"DocComment":" Default values are often some kind of initial value, identity value, or anything else that"},{"DocComment":" may make sense as a default."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Using built-in default values:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let i: i8 = Default::default();"},{"DocComment":" let (x, y): (Option, f64) = Default::default();"},{"DocComment":" let (a, b, (c, d)): (i32, u32, (bool, bool)) = Default::default();"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Making your own:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" enum Kind {"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Default for Kind {"},{"DocComment":" fn default() -> Self { Kind::A }"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[],"output":{"Deduplicated":11751}},"item":{"id":296,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11830,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":6,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[6,0]}}],"vtable":null},{"def_id":7,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["Copy",0]}],"span":{"data":{"file_id":35,"beg":{"line":454,"col":0},"end":{"line":454,"col":21}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Types whose values can be duplicated simply by copying bits."},{"DocComment":""},{"DocComment":" By default, variable bindings have 'move semantics.' In other"},{"DocComment":" words:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Foo;"},{"DocComment":""},{"DocComment":" let x = Foo;"},{"DocComment":""},{"DocComment":" let y = x;"},{"DocComment":""},{"DocComment":" // `x` has moved into `y`, and so cannot be used"},{"DocComment":""},{"DocComment":" // println!(\"{x:?}\"); // error: use of moved value"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, if a type implements `Copy`, it instead has 'copy semantics':"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // We can derive a `Copy` implementation. `Clone` is also required, as it's"},{"DocComment":" // a supertrait of `Copy`."},{"DocComment":" #[derive(Debug, Copy, Clone)]"},{"DocComment":" struct Foo;"},{"DocComment":""},{"DocComment":" let x = Foo;"},{"DocComment":""},{"DocComment":" let y = x;"},{"DocComment":""},{"DocComment":" // `y` is a copy of `x`"},{"DocComment":""},{"DocComment":" println!(\"{x:?}\"); // A-OK!"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It's important to note that in these two examples, the only difference is whether you"},{"DocComment":" are allowed to access `x` after the assignment. Under the hood, both a copy and a move"},{"DocComment":" can result in bits being copied in memory, although this is sometimes optimized away."},{"DocComment":""},{"DocComment":" ## How can I implement `Copy`?"},{"DocComment":""},{"DocComment":" There are two ways to implement `Copy` on your type. The simplest is to use `derive`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct MyStruct;"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You can also implement `Copy` and `Clone` manually:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct MyStruct;"},{"DocComment":""},{"DocComment":" impl Copy for MyStruct { }"},{"DocComment":""},{"DocComment":" impl Clone for MyStruct {"},{"DocComment":" fn clone(&self) -> MyStruct {"},{"DocComment":" *self"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" There is a small difference between the two. The `derive` strategy will also place a `Copy`"},{"DocComment":" bound on type parameters:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Clone)]"},{"DocComment":" struct MyStruct(T);"},{"DocComment":""},{"DocComment":" impl Copy for MyStruct { }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" This isn't always desired. For example, shared references (`&T`) can be copied regardless of"},{"DocComment":" whether `T` is `Copy`. Likewise, a generic struct containing markers such as [`PhantomData`]"},{"DocComment":" could potentially be duplicated with a bit-wise copy."},{"DocComment":""},{"DocComment":" ## What's the difference between `Copy` and `Clone`?"},{"DocComment":""},{"DocComment":" Copies happen implicitly, for example as part of an assignment `y = x`. The behavior of"},{"DocComment":" `Copy` is not overloadable; it is always a simple bit-wise copy."},{"DocComment":""},{"DocComment":" Cloning is an explicit action, `x.clone()`. The implementation of [`Clone`] can"},{"DocComment":" provide any type-specific behavior necessary to duplicate values safely. For example,"},{"DocComment":" the implementation of [`Clone`] for [`String`] needs to copy the pointed-to string"},{"DocComment":" buffer in the heap. A simple bitwise copy of [`String`] values would merely copy the"},{"DocComment":" pointer, leading to a double free down the line. For this reason, [`String`] is [`Clone`]"},{"DocComment":" but not `Copy`."},{"DocComment":""},{"DocComment":" [`Clone`] is a supertrait of `Copy`, so everything which is `Copy` must also implement"},{"DocComment":" [`Clone`]. If a type is `Copy` then its [`Clone`] implementation only needs to return `*self`"},{"DocComment":" (see the example above)."},{"DocComment":""},{"DocComment":" ## When can my type be `Copy`?"},{"DocComment":""},{"DocComment":" A type can implement `Copy` if all of its components implement `Copy`. For example, this"},{"DocComment":" struct can be `Copy`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" A struct can be `Copy`, and [`i32`] is `Copy`, therefore `Point` is eligible to be `Copy`."},{"DocComment":" By contrast, consider"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" # struct Point;"},{"DocComment":" struct PointList {"},{"DocComment":" points: Vec,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The struct `PointList` cannot implement `Copy`, because [`Vec`] is not `Copy`. If we"},{"DocComment":" attempt to derive a `Copy` implementation, we'll get an error:"},{"DocComment":""},{"DocComment":" ```text"},{"DocComment":" the trait `Copy` cannot be implemented for this type; field `points` does not implement `Copy`"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Shared references (`&T`) are also `Copy`, so a type can be `Copy`, even when it holds"},{"DocComment":" shared references of types `T` that are *not* `Copy`. Consider the following struct,"},{"DocComment":" which can implement `Copy`, because it only holds a *shared reference* to our non-`Copy`"},{"DocComment":" type `PointList` from above:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" # struct PointList;"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct PointListWrapper<'a> {"},{"DocComment":" point_list_ref: &'a PointList,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## When *can't* my type be `Copy`?"},{"DocComment":""},{"DocComment":" Some types can't be copied safely. For example, copying `&mut T` would create an aliased"},{"DocComment":" mutable reference. Copying [`String`] would duplicate responsibility for managing the"},{"DocComment":" [`String`]'s buffer, leading to a double free."},{"DocComment":""},{"DocComment":" Generalizing the latter case, any type implementing [`Drop`] can't be `Copy`, because it's"},{"DocComment":" managing some resource besides its own [`size_of::`] bytes."},{"DocComment":""},{"DocComment":" If you try to implement `Copy` on a struct or enum containing non-`Copy` data, you will get"},{"DocComment":" the error [E0204]."},{"DocComment":""},{"DocComment":" [E0204]: ../../error_codes/E0204.html"},{"DocComment":""},{"DocComment":" ## When *should* my type be `Copy`?"},{"DocComment":""},{"DocComment":" Generally speaking, if your type _can_ implement `Copy`, it should. Keep in mind, though,"},{"DocComment":" that implementing `Copy` is part of the public API of your type. If the type might become"},{"DocComment":" non-`Copy` in the future, it could be prudent to omit the `Copy` implementation now, to"},{"DocComment":" avoid a breaking API change."},{"DocComment":""},{"DocComment":" ## Additional implementors"},{"DocComment":""},{"DocComment":" In addition to the [implementors listed below][impls],"},{"DocComment":" the following types also implement `Copy`:"},{"DocComment":""},{"DocComment":" * Function item types (i.e., the distinct types defined for each function)"},{"DocComment":" * Function pointer types (e.g., `fn() -> i32`)"},{"DocComment":" * Closure types, if they capture no value from the environment"},{"DocComment":" or if all such captured values implement `Copy` themselves."},{"DocComment":" Note that variables captured by shared reference always implement `Copy`"},{"DocComment":" (even if the referent doesn't),"},{"DocComment":" while variables captured by mutable reference never implement `Copy`."},{"DocComment":""},{"DocComment":" [`Vec`]: ../../std/vec/struct.Vec.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`size_of::`]: size_of"},{"DocComment":" [impls]: #implementors"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"copy"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":35,"beg":{"line":454,"col":16},"end":{"line":454,"col":21}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[],"vtable":null},{"def_id":8,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["clone",0]},{"Ident":["Clone",0]}],"span":{"data":{"file_id":36,"beg":{"line":194,"col":0},"end":{"line":194,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A common trait that allows explicit creation of a duplicate value."},{"DocComment":""},{"DocComment":" Calling [`clone`] always produces a new value."},{"DocComment":" However, for types that are references to other data (such as smart pointers or references),"},{"DocComment":" the new value may still point to the same underlying data, rather than duplicating it."},{"DocComment":" See [`Clone::clone`] for more details."},{"DocComment":""},{"DocComment":" This distinction is especially important when using `#[derive(Clone)]` on structs containing"},{"DocComment":" smart pointers like `Arc>` - the cloned struct will share mutable state with the"},{"DocComment":" original."},{"DocComment":""},{"DocComment":" Differs from [`Copy`] in that [`Copy`] is implicit and an inexpensive bit-wise copy, while"},{"DocComment":" `Clone` is always explicit and may or may not be expensive. [`Copy`] has no methods, so you"},{"DocComment":" cannot change its behavior, but when implementing `Clone`, the `clone` method you provide"},{"DocComment":" may run arbitrary code."},{"DocComment":""},{"DocComment":" Since `Clone` is a supertrait of [`Copy`], any type that implements `Copy` must also implement"},{"DocComment":" `Clone`."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all fields are `Clone`. The `derive`d"},{"DocComment":" implementation of [`Clone`] calls [`clone`] on each field."},{"DocComment":""},{"DocComment":" [`clone`]: Clone::clone"},{"DocComment":""},{"DocComment":" For a generic struct, `#[derive]` implements `Clone` conditionally by adding bound `Clone` on"},{"DocComment":" generic parameters."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // `derive` implements Clone for Reading when T is Clone."},{"DocComment":" #[derive(Clone)]"},{"DocComment":" struct Reading {"},{"DocComment":" frequency: T,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I implement `Clone`?"},{"DocComment":""},{"DocComment":" Types that are [`Copy`] should have a trivial implementation of `Clone`. More formally:"},{"DocComment":" if `T: Copy`, `x: T`, and `y: &T`, then `let x = y.clone();` is equivalent to `let x = *y;`."},{"DocComment":" Manual implementations should be careful to uphold this invariant; however, unsafe code"},{"DocComment":" must not rely on it to ensure memory safety."},{"DocComment":""},{"DocComment":" An example is a generic struct holding a function pointer. In this case, the"},{"DocComment":" implementation of `Clone` cannot be `derive`d, but can be implemented as:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" impl Copy for Generate {}"},{"DocComment":""},{"DocComment":" impl Clone for Generate {"},{"DocComment":" fn clone(&self) -> Self {"},{"DocComment":" *self"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If we `derive`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" the auto-derived implementations will have unnecessary `T: Copy` and `T: Clone` bounds:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" // Automatically derived"},{"DocComment":" impl Copy for Generate { }"},{"DocComment":""},{"DocComment":" // Automatically derived"},{"DocComment":" impl Clone for Generate {"},{"DocComment":" fn clone(&self) -> Generate {"},{"DocComment":" Generate(Clone::clone(&self.0))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The bounds are unnecessary because clearly the function itself should be"},{"DocComment":" copy- and cloneable even if its return type is not:"},{"DocComment":""},{"DocComment":" ```compile_fail,E0599"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" struct NotCloneable;"},{"DocComment":""},{"DocComment":" fn generate_not_cloneable() -> NotCloneable {"},{"DocComment":" NotCloneable"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" Generate(generate_not_cloneable).clone(); // error: trait bounds were not satisfied"},{"DocComment":" // Note: With the manual implementations the above line will compile."},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## `Clone` and `PartialEq`/`Eq`"},{"DocComment":" `Clone` is intended for the duplication of objects. Consequently, when implementing"},{"DocComment":" both `Clone` and [`PartialEq`], the following property is expected to hold:"},{"DocComment":" ```text"},{"DocComment":" x == x -> x.clone() == x"},{"DocComment":" ```"},{"DocComment":" In other words, if an object compares equal to itself,"},{"DocComment":" its clone must also compare equal to the original."},{"DocComment":""},{"DocComment":" For types that also implement [`Eq`] – for which `x == x` always holds –"},{"DocComment":" this implies that `x.clone() == x` must always be true."},{"DocComment":" Standard library collections such as"},{"DocComment":" [`HashMap`], [`HashSet`], [`BTreeMap`], [`BTreeSet`] and [`BinaryHeap`]"},{"DocComment":" rely on their keys respecting this property for correct behavior."},{"DocComment":" Furthermore, these collections require that cloning a key preserves the outcome of the"},{"DocComment":" [`Hash`] and [`Ord`] methods. Thankfully, this follows automatically from `x.clone() == x`"},{"DocComment":" if `Hash` and `Ord` are correctly implemented according to their own requirements."},{"DocComment":""},{"DocComment":" When deriving both `Clone` and [`PartialEq`] using `#[derive(Clone, PartialEq)]`"},{"DocComment":" or when additionally deriving [`Eq`] using `#[derive(Clone, PartialEq, Eq)]`,"},{"DocComment":" then this property is automatically upheld – provided that it is satisfied by"},{"DocComment":" the underlying types."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on this property"},{"DocComment":" being satisfied."},{"DocComment":""},{"DocComment":" ## Additional implementors"},{"DocComment":""},{"DocComment":" In addition to the [implementors listed below][impls],"},{"DocComment":" the following types also implement `Clone`:"},{"DocComment":""},{"DocComment":" * Function item types (i.e., the distinct types defined for each function)"},{"DocComment":" * Function pointer types (e.g., `fn() -> i32`)"},{"DocComment":" * Closure types, if they capture no value from the environment"},{"DocComment":" or if all such captured values implement `Clone` themselves."},{"DocComment":" Note that variables captured by shared reference always implement `Clone`"},{"DocComment":" (even if the referent doesn't),"},{"DocComment":" while variables captured by mutable reference never implement `Clone`."},{"DocComment":""},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`HashSet`]: ../../std/collections/struct.HashSet.html"},{"DocComment":" [`BTreeMap`]: ../../std/collections/struct.BTreeMap.html"},{"DocComment":" [`BTreeSet`]: ../../std/collections/struct.BTreeSet.html"},{"DocComment":" [`BinaryHeap`]: ../../std/collections/struct.BinaryHeap.html"},{"DocComment":" [impls]: #implementors"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"clone"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"clone","attr_info":{"attributes":[{"DocComment":" Returns a duplicate of the value."},{"DocComment":""},{"DocComment":" Note that what \"duplicate\" means varies by type:"},{"DocComment":" - For most types, this creates a deep, independent copy"},{"DocComment":" - For reference types like `&T`, this creates another reference to the same value"},{"DocComment":" - For smart pointers like [`Arc`] or [`Rc`], this increments the reference count"},{"DocComment":" but still points to the same underlying data"},{"DocComment":""},{"DocComment":" [`Arc`]: ../../std/sync/struct.Arc.html"},{"DocComment":" [`Rc`]: ../../std/rc/struct.Rc.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(noop_method_call)]"},{"DocComment":" let hello = \"Hello\"; // &str implements Clone"},{"DocComment":""},{"DocComment":" assert_eq!(\"Hello\", hello.clone());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Example with a reference-counted type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::sync::{Arc, Mutex};"},{"DocComment":""},{"DocComment":" let data = Arc::new(Mutex::new(vec![1, 2, 3]));"},{"DocComment":" let data_clone = data.clone(); // Creates another Arc pointing to the same Mutex"},{"DocComment":""},{"DocComment":" {"},{"DocComment":" let mut lock = data.lock().unwrap();"},{"DocComment":" lock.push(4);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Changes are visible through the clone because they share the same underlying data"},{"DocComment":" assert_eq!(*data_clone.lock().unwrap(), vec![1, 2, 3, 4]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"Deduplicated":11751}},"item":{"id":58,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11831,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[8,0]}},null],"vtable":null},null,{"def_id":10,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["Destruct",0]}],"span":{"data":{"file_id":35,"beg":{"line":1062,"col":0},"end":{"line":1062,"col":38}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A marker for types that can be dropped."},{"DocComment":""},{"DocComment":" This should be used for `[const]` bounds,"},{"DocComment":" as non-const bounds will always hold for every type."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"destruct"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"drop_glue","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":true,"abi":"Rust","inputs":[{"Deduplicated":11818}],"output":{"Deduplicated":1802}},"item":{"id":298,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}},"kind":{"TraitMethod":[10,0]}}],"vtable":null},{"def_id":11,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["fmt",0]},{"Ident":["Debug",0]}],"span":{"data":{"file_id":27,"beg":{"line":1053,"col":0},"end":{"line":1053,"col":29}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" `?` formatting."},{"DocComment":""},{"DocComment":" `Debug` should format the output in a programmer-facing, debugging context."},{"DocComment":""},{"DocComment":" Generally speaking, you should just `derive` a `Debug` implementation."},{"DocComment":""},{"DocComment":" When used with the alternate format specifier `#?`, the output is pretty-printed."},{"DocComment":""},{"DocComment":" For more information on formatters, see [the module-level documentation][module]."},{"DocComment":""},{"DocComment":" [module]: ../../std/fmt/index.html"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all fields implement `Debug`. When"},{"DocComment":" `derive`d for structs, it will use the name of the `struct`, then `{`, then a"},{"DocComment":" comma-separated list of each field's name and `Debug` value, then `}`. For"},{"DocComment":" `enum`s, it will use the name of the variant and, if applicable, `(`, then the"},{"DocComment":" `Debug` values of the fields, then `)`."},{"DocComment":""},{"DocComment":" # Stability"},{"DocComment":""},{"DocComment":" Derived `Debug` formats are not stable, and so may change with future Rust"},{"DocComment":" versions. Additionally, `Debug` implementations of types provided by the"},{"DocComment":" standard library (`std`, `core`, `alloc`, etc.) are not stable, and"},{"DocComment":" may also change with future Rust versions."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Deriving an implementation:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" format!(\"The origin is: {origin:?}\"),"},{"DocComment":" \"The origin is: Point { x: 0, y: 0 }\","},{"DocComment":" );"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Manually implementing:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fmt;"},{"DocComment":""},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl fmt::Debug for Point {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" f.debug_struct(\"Point\")"},{"DocComment":" .field(\"x\", &self.x)"},{"DocComment":" .field(\"y\", &self.y)"},{"DocComment":" .finish()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" format!(\"The origin is: {origin:?}\"),"},{"DocComment":" \"The origin is: Point { x: 0, y: 0 }\","},{"DocComment":" );"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" There are a number of helper methods on the [`Formatter`] struct to help you with manual"},{"DocComment":" implementations, such as [`debug_struct`]."},{"DocComment":""},{"DocComment":" [`debug_struct`]: Formatter::debug_struct"},{"DocComment":""},{"DocComment":" Types that do not wish to use the standard suite of debug representations"},{"DocComment":" provided by the `Formatter` trait (`debug_struct`, `debug_tuple`,"},{"DocComment":" `debug_list`, `debug_set`, `debug_map`) can do something totally custom by"},{"DocComment":" manually writing an arbitrary representation to the `Formatter`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # use std::fmt;"},{"DocComment":" # struct Point {"},{"DocComment":" # x: i32,"},{"DocComment":" # y: i32,"},{"DocComment":" # }"},{"DocComment":" #"},{"DocComment":" impl fmt::Debug for Point {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" write!(f, \"Point [{} {}]\", self.x, self.y)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" `Debug` implementations using either `derive` or the debug builder API"},{"DocComment":" on [`Formatter`] support pretty-printing using the alternate flag: `{:#?}`."},{"DocComment":""},{"DocComment":" Pretty-printing with `#?`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" let expected = \"The origin is: Point {"},{"DocComment":" x: 0,"},{"DocComment":" y: 0,"},{"DocComment":" }\";"},{"DocComment":" assert_eq!(format!(\"The origin is: {origin:#?}\"), expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Debug"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"},{"index":2,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"fmt","attr_info":{"attributes":[{"DocComment":"Formats the value using the given formatter.\n\n# Errors\n\nThis function should return [`Err`] if, and only if, the provided [`Formatter`] returns [`Err`].\nString formatting is considered an infallible operation; this function only\nreturns a [`Result`] because writing to the underlying stream might fail and it must\nprovide a way to propagate the fact that an error has occurred back up the stack.\n"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fmt;"},{"DocComment":""},{"DocComment":" struct Position {"},{"DocComment":" longitude: f32,"},{"DocComment":" latitude: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl fmt::Debug for Position {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" f.debug_tuple(\"\")"},{"DocComment":" .field(&self.longitude)"},{"DocComment":" .field(&self.latitude)"},{"DocComment":" .finish()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let position = Position { longitude: 1.987, latitude: 2.983 };"},{"DocComment":" assert_eq!(format!(\"{position:?}\"), \"(1.987, 2.983)\");"},{"DocComment":""},{"DocComment":" assert_eq!(format!(\"{position:#?}\"), \"("},{"DocComment":" 1.987,"},{"DocComment":" 2.983,"},{"DocComment":" )\");"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"HashConsedValue":[7093,{"Ref":[{"Var":{"Bound":[0,1]}},{"HashConsedValue":[7092,{"Adt":{"id":{"Adt":57},"generics":{"regions":[{"Var":{"Bound":[0,2]}}],"types":[],"const_generics":[],"trait_refs":[]}}}]},"Mut"]}]}],"output":{"Deduplicated":7095}},"item":{"id":299,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}},{"Var":{"Bound":[0,2]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11832,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":11,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[11,0]}}],"vtable":{"id":{"Adt":56},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":12,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["AddAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":768,"col":0},"end":{"line":768,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The addition assignment operator `+=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" This example creates a `Point` struct that implements the `AddAssign`"},{"DocComment":" trait, and then demonstrates add-assigning to a mutable `Point`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::AddAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl AddAssign for Point {"},{"DocComment":" fn add_assign(&mut self, other: Self) {"},{"DocComment":" *self = Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" };"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut point = Point { x: 1, y: 0 };"},{"DocComment":" point += Point { x: 2, y: 3 };"},{"DocComment":" assert_eq!(point, Point { x: 3, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"add_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"add_assign","attr_info":{"attributes":[{"DocComment":" Performs the `+=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x += 1;"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":11752}],"output":{"Deduplicated":1802}},"item":{"id":303,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11833,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":12,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[12,0]}}],"vtable":{"id":{"Adt":60},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":13,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Add",0]}],"span":{"data":{"file_id":37,"beg":{"line":76,"col":0},"end":{"line":76,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The addition operator `+`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory. For"},{"DocComment":" example, [`std::time::SystemTime`] implements `Add`, which permits"},{"DocComment":" operations of the form `SystemTime = SystemTime + Duration`."},{"DocComment":""},{"DocComment":" [`std::time::SystemTime`]: ../../std/time/struct.SystemTime.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Add`able points"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Add;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Add for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn add(self, other: Self) -> Self {"},{"DocComment":" Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 1, y: 0 } + Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 3, y: 3 });"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Implementing `Add` with generics"},{"DocComment":""},{"DocComment":" Here is an example of the same `Point` struct implementing the `Add` trait"},{"DocComment":" using generics."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Add;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: T,"},{"DocComment":" y: T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Notice that the implementation uses the associated type `Output`."},{"DocComment":" impl> Add for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn add(self, other: Self) -> Self::Output {"},{"DocComment":" Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 1, y: 0 } + Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 3, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"add"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"add","attr_info":{"attributes":[{"DocComment":" Performs the `+` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 + 1, 13);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":304,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11834,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":13,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[13,0]}}],"vtable":{"id":{"Adt":61},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":14,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["SubAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":839,"col":0},"end":{"line":839,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The subtraction assignment operator `-=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" This example creates a `Point` struct that implements the `SubAssign`"},{"DocComment":" trait, and then demonstrates sub-assigning to a mutable `Point`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::SubAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl SubAssign for Point {"},{"DocComment":" fn sub_assign(&mut self, other: Self) {"},{"DocComment":" *self = Self {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" };"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut point = Point { x: 3, y: 3 };"},{"DocComment":" point -= Point { x: 2, y: 3 };"},{"DocComment":" assert_eq!(point, Point {x: 1, y: 0});"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"sub_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sub_assign","attr_info":{"attributes":[{"DocComment":" Performs the `-=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x -= 1;"},{"DocComment":" assert_eq!(x, 11);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":11752}],"output":{"Deduplicated":1802}},"item":{"id":305,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11835,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":14,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[14,0]}}],"vtable":{"id":{"Adt":62},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":15,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Sub",0]}],"span":{"data":{"file_id":37,"beg":{"line":188,"col":0},"end":{"line":188,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The subtraction operator `-`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory. For"},{"DocComment":" example, [`std::time::SystemTime`] implements `Sub`, which permits"},{"DocComment":" operations of the form `SystemTime = SystemTime - Duration`."},{"DocComment":""},{"DocComment":" [`std::time::SystemTime`]: ../../std/time/struct.SystemTime.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Sub`tractable points"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Sub;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Sub for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn sub(self, other: Self) -> Self::Output {"},{"DocComment":" Self {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 3, y: 3 } - Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 1, y: 0 });"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Implementing `Sub` with generics"},{"DocComment":""},{"DocComment":" Here is an example of the same `Point` struct implementing the `Sub` trait"},{"DocComment":" using generics."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Sub;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: T,"},{"DocComment":" y: T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Notice that the implementation uses the associated type `Output`."},{"DocComment":" impl> Sub for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn sub(self, other: Self) -> Self::Output {"},{"DocComment":" Point {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 2, y: 3 } - Point { x: 1, y: 0 },"},{"DocComment":" Point { x: 1, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"sub"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sub","attr_info":{"attributes":[{"DocComment":" Performs the `-` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 - 1, 11);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":306,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11836,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":15,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[15,0]}}],"vtable":{"id":{"Adt":63},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":16,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["MulAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":901,"col":0},"end":{"line":901,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The multiplication assignment operator `*=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::MulAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Frequency { hertz: f64 }"},{"DocComment":""},{"DocComment":" impl MulAssign for Frequency {"},{"DocComment":" fn mul_assign(&mut self, rhs: f64) {"},{"DocComment":" self.hertz *= rhs;"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut frequency = Frequency { hertz: 50.0 };"},{"DocComment":" frequency *= 4.0;"},{"DocComment":" assert_eq!(Frequency { hertz: 200.0 }, frequency);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"mul_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"mul_assign","attr_info":{"attributes":[{"DocComment":" Performs the `*=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x *= 2;"},{"DocComment":" assert_eq!(x, 24);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":11752}],"output":{"Deduplicated":1802}},"item":{"id":307,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11837,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":16,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[16,0]}}],"vtable":{"id":{"Adt":64},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":17,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Mul",0]}],"span":{"data":{"file_id":37,"beg":{"line":322,"col":0},"end":{"line":322,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The multiplication operator `*`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Mul`tipliable rational numbers"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Mul;"},{"DocComment":""},{"DocComment":" // By the fundamental theorem of arithmetic, rational numbers in lowest"},{"DocComment":" // terms are unique. So, by keeping `Rational`s in reduced form, we can"},{"DocComment":" // derive `Eq` and `PartialEq`."},{"DocComment":" #[derive(Debug, Eq, PartialEq)]"},{"DocComment":" struct Rational {"},{"DocComment":" numerator: usize,"},{"DocComment":" denominator: usize,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Rational {"},{"DocComment":" fn new(numerator: usize, denominator: usize) -> Self {"},{"DocComment":" if denominator == 0 {"},{"DocComment":" panic!(\"Zero is an invalid denominator!\");"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Reduce to lowest terms by dividing by the greatest common"},{"DocComment":" // divisor."},{"DocComment":" let gcd = gcd(numerator, denominator);"},{"DocComment":" Self {"},{"DocComment":" numerator: numerator / gcd,"},{"DocComment":" denominator: denominator / gcd,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Mul for Rational {"},{"DocComment":" // The multiplication of rational numbers is a closed operation."},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn mul(self, rhs: Self) -> Self {"},{"DocComment":" let numerator = self.numerator * rhs.numerator;"},{"DocComment":" let denominator = self.denominator * rhs.denominator;"},{"DocComment":" Self::new(numerator, denominator)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Euclid's two-thousand-year-old algorithm for finding the greatest common"},{"DocComment":" // divisor."},{"DocComment":" fn gcd(x: usize, y: usize) -> usize {"},{"DocComment":" let mut x = x;"},{"DocComment":" let mut y = y;"},{"DocComment":" while y != 0 {"},{"DocComment":" let t = y;"},{"DocComment":" y = x % y;"},{"DocComment":" x = t;"},{"DocComment":" }"},{"DocComment":" x"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Rational::new(1, 2), Rational::new(2, 4));"},{"DocComment":" assert_eq!(Rational::new(2, 3) * Rational::new(3, 4),"},{"DocComment":" Rational::new(1, 2));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Multiplying vectors by scalars as in linear algebra"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Mul;"},{"DocComment":""},{"DocComment":" struct Scalar { value: usize }"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Vector { value: Vec }"},{"DocComment":""},{"DocComment":" impl Mul for Vector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn mul(self, rhs: Scalar) -> Self::Output {"},{"DocComment":" Self { value: self.value.iter().map(|v| v * rhs.value).collect() }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let vector = Vector { value: vec![2, 4, 6] };"},{"DocComment":" let scalar = Scalar { value: 3 };"},{"DocComment":" assert_eq!(vector * scalar, Vector { value: vec![6, 12, 18] });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"mul"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"mul","attr_info":{"attributes":[{"DocComment":" Performs the `*` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 * 2, 24);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":308,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11838,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":17,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[17,0]}}],"vtable":{"id":{"Adt":65},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":18,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Neg",0]}],"span":{"data":{"file_id":37,"beg":{"line":690,"col":0},"end":{"line":690,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The unary negation operator `-`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `Neg` for `Sign`, which allows the use of `-` to"},{"DocComment":" negate its value."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Neg;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" enum Sign {"},{"DocComment":" Negative,"},{"DocComment":" Zero,"},{"DocComment":" Positive,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Neg for Sign {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn neg(self) -> Self::Output {"},{"DocComment":" match self {"},{"DocComment":" Sign::Negative => Sign::Positive,"},{"DocComment":" Sign::Zero => Sign::Zero,"},{"DocComment":" Sign::Positive => Sign::Negative,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // A negative positive is a negative."},{"DocComment":" assert_eq!(-Sign::Positive, Sign::Negative);"},{"DocComment":" // A double negative is a positive."},{"DocComment":" assert_eq!(-Sign::Negative, Sign::Positive);"},{"DocComment":" // Zero is its own negation."},{"DocComment":" assert_eq!(-Sign::Zero, Sign::Zero);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"neg"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"neg","attr_info":{"attributes":[{"DocComment":" Performs the unary `-` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: i32 = 12;"},{"DocComment":" assert_eq!(-x, -12);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751}],"output":{"Deduplicated":11752}},"item":{"id":310,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11839,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":18,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[18,0]}}],"vtable":{"id":{"Adt":66},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":19,"item_meta":{"name":[{"Ident":["subtle",0]},{"Ident":["ConditionallySelectable",0]}],"span":{"data":{"file_id":22,"beg":{"line":393,"col":0},"end":{"line":393,"col":39}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A type which can be conditionally selected in constant time."},{"DocComment":""},{"DocComment":" This trait also provides generic implementations of conditional"},{"DocComment":" assignment and conditional swaps."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":22,"beg":{"line":393,"col":35},"end":{"line":393,"col":39}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_select","attr_info":{"attributes":[{"DocComment":" Select `a` or `b` according to `choice`."},{"DocComment":""},{"DocComment":" # Returns"},{"DocComment":""},{"DocComment":" * `a` if `choice == Choice(0)`;"},{"DocComment":" * `b` if `choice == Choice(1)`."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let x: u8 = 13;"},{"DocComment":" let y: u8 = 42;"},{"DocComment":""},{"DocComment":" let z = u8::conditional_select(&x, &y, 0.into());"},{"DocComment":" assert_eq!(z, x);"},{"DocComment":" let z = u8::conditional_select(&x, &y, 1.into());"},{"DocComment":" assert_eq!(z, y);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"Deduplicated":11822},{"Deduplicated":2322}],"output":{"Deduplicated":11751}},"item":{"id":93,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11840,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":19,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[19,0]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_assign","attr_info":{"attributes":[{"DocComment":" Conditionally assign `other` to `self`, according to `choice`."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let mut x: u8 = 13;"},{"DocComment":" let mut y: u8 = 42;"},{"DocComment":""},{"DocComment":" x.conditional_assign(&y, 0.into());"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" x.conditional_assign(&y, 1.into());"},{"DocComment":" assert_eq!(x, 42);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":11822},{"Deduplicated":2322}],"output":{"Deduplicated":1802}},"item":{"id":94,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11840}]}}},"kind":{"TraitMethod":[19,1]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_swap","attr_info":{"attributes":[{"DocComment":" Conditionally swap `self` and `other` if `choice == 1`; otherwise,"},{"DocComment":" reassign both unto themselves."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let mut x: u8 = 13;"},{"DocComment":" let mut y: u8 = 42;"},{"DocComment":""},{"DocComment":" u8::conditional_swap(&mut x, &mut y, 0.into());"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" assert_eq!(y, 42);"},{"DocComment":" u8::conditional_swap(&mut x, &mut y, 1.into());"},{"DocComment":" assert_eq!(x, 42);"},{"DocComment":" assert_eq!(y, 13);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"HashConsedValue":[11841,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11751},"Mut"]}]},{"Deduplicated":2322}],"output":{"Deduplicated":1802}},"item":{"id":95,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11840}]}}},"kind":{"TraitMethod":[19,2]}}],"vtable":null},{"def_id":20,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["StructuralPartialEq",0]}],"span":{"data":{"file_id":35,"beg":{"line":257,"col":0},"end":{"line":257,"col":29}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Required trait for constants used in pattern matches."},{"DocComment":""},{"DocComment":" Constants are only allowed as patterns if (a) their type implements"},{"DocComment":" `PartialEq`, and (b) interpreting the value of the constant as a pattern"},{"DocComment":" is equivalent to calling `PartialEq`. This ensures that constants used as"},{"DocComment":" patterns cannot expose implementation details in an unexpected way or"},{"DocComment":" cause semver hazards."},{"DocComment":""},{"DocComment":" This trait ensures point (b)."},{"DocComment":" Any type that derives `PartialEq` automatically implements this trait."},{"DocComment":""},{"DocComment":" Implementing this trait (which is unstable) is a way for type authors to explicitly allow"},{"DocComment":" comparing const values of this type; that operation will recursively compare all fields"},{"DocComment":" (including private fields), even if that behavior differs from `PartialEq`. This can make it"},{"DocComment":" semver-breaking to add further private fields to a type."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"structural_peq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[],"vtable":{"id":{"Adt":67},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":21,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["Identity",0]}],"span":{"data":{"file_id":38,"beg":{"line":32,"col":0},"end":{"line":36,"col":1}},"generated_from_span":null},"source_text":"pub trait Identity {\n /// Returns the identity element of the curve.\n /// Can be used as a constructor.\n fn identity() -> Self;\n}","attr_info":{"attributes":[{"DocComment":" Trait for getting the identity element of a point type."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"identity","attr_info":{"attributes":[{"DocComment":" Returns the identity element of the curve."},{"DocComment":" Can be used as a constructor."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[],"output":{"Deduplicated":11751}},"item":{"id":318,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11842,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":21,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[21,0]}}],"vtable":null},{"def_id":22,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["ValidityCheck",0]}],"span":{"data":{"file_id":38,"beg":{"line":451,"col":0},"end":{"line":454,"col":1}},"generated_from_span":null},"source_text":"pub(crate) trait ValidityCheck {\n /// Checks whether the point is on the curve. Not CT.\n fn is_valid(&self) -> bool;\n}","attr_info":{"attributes":[{"DocComment":" Trait for checking whether a point is on the curve."},{"DocComment":""},{"DocComment":" This trait is only for debugging/testing, since it should be"},{"DocComment":" impossible for a `curve25519-dalek` user to construct an invalid"},{"DocComment":" point."},{"Unknown":{"path":"allow","args":"dead_code"}}],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"is_valid","attr_info":{"attributes":[{"DocComment":" Checks whether the point is on the curve. Not CT."}],"inline":null,"rename":null,"public":false},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"Deduplicated":1797}},"item":{"id":319,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11843,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":22,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[22,0]}}],"vtable":{"id":{"Adt":68},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":23,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["hash",0]},{"Ident":["Hash",0]}],"span":{"data":{"file_id":40,"beg":{"line":185,"col":0},"end":{"line":185,"col":36}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A hashable type."},{"DocComment":""},{"DocComment":" Types implementing `Hash` are able to be [`hash`]ed with an instance of"},{"DocComment":" [`Hasher`]."},{"DocComment":""},{"DocComment":" ## Implementing `Hash`"},{"DocComment":""},{"DocComment":" You can derive `Hash` with `#[derive(Hash)]` if all fields implement `Hash`."},{"DocComment":" The resulting hash will be the combination of the values from calling"},{"DocComment":" [`hash`] on each field."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Hash)]"},{"DocComment":" struct Rustacean {"},{"DocComment":" name: String,"},{"DocComment":" country: String,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If you need more control over how a value is hashed, you can of course"},{"DocComment":" implement the `Hash` trait yourself:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{Hash, Hasher};"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" phone: u64,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Hash for Person {"},{"DocComment":" fn hash(&self, state: &mut H) {"},{"DocComment":" self.id.hash(state);"},{"DocComment":" self.phone.hash(state);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## `Hash` and `Eq`"},{"DocComment":""},{"DocComment":" When implementing both `Hash` and [`Eq`], it is important that the following"},{"DocComment":" property holds:"},{"DocComment":""},{"DocComment":" ```text"},{"DocComment":" k1 == k2 -> hash(k1) == hash(k2)"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" In other words, if two keys are equal, their hashes must also be equal."},{"DocComment":" [`HashMap`] and [`HashSet`] both rely on this behavior."},{"DocComment":""},{"DocComment":" Thankfully, you won't need to worry about upholding this property when"},{"DocComment":" deriving both [`Eq`] and `Hash` with `#[derive(PartialEq, Eq, Hash)]`."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Prefix collisions"},{"DocComment":""},{"DocComment":" Implementations of `hash` should ensure that the data they"},{"DocComment":" pass to the `Hasher` are prefix-free. That is,"},{"DocComment":" values which are not equal should cause two different sequences of values to be written,"},{"DocComment":" and neither of the two sequences should be a prefix of the other."},{"DocComment":""},{"DocComment":" For example, the standard implementation of [`Hash` for `&str`][impl] passes an extra"},{"DocComment":" `0xFF` byte to the `Hasher` so that the values `(\"ab\", \"c\")` and `(\"a\","},{"DocComment":" \"bc\")` hash differently."},{"DocComment":""},{"DocComment":" ## Portability"},{"DocComment":""},{"DocComment":" Due to differences in endianness and type sizes, data fed by `Hash` to a `Hasher`"},{"DocComment":" should not be considered portable across platforms. Additionally the data passed by most"},{"DocComment":" standard library types should not be considered stable between compiler versions."},{"DocComment":""},{"DocComment":" This means tests shouldn't probe hard-coded hash values or data fed to a `Hasher` and"},{"DocComment":" instead should check consistency with `Eq`."},{"DocComment":""},{"DocComment":" Serialization formats intended to be portable between platforms or compiler versions should"},{"DocComment":" either avoid encoding hashes or only rely on `Hash` and `Hasher` implementations that"},{"DocComment":" provide additional guarantees."},{"DocComment":""},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`HashSet`]: ../../std/collections/struct.HashSet.html"},{"DocComment":" [`hash`]: Hash::hash"},{"DocComment":" [impl]: ../../std/primitive.str.html#impl-Hash-for-str"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Hash"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"H"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":40,"beg":{"line":198,"col":15},"end":{"line":198,"col":21}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":3181}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"hash","attr_info":{"attributes":[{"DocComment":" Feeds this value into the given [`Hasher`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hash, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" 7920.hash(&mut hasher);"},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"HashConsedValue":[3796,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":3182},"Mut"]}]}],"output":{"Deduplicated":1802}},"item":{"id":114,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751},{"Deduplicated":3182}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11844,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":23,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[3775,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":3181}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[23,0]}},null],"vtable":null},{"def_id":24,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["hash",0]},{"Ident":["Hasher",0]}],"span":{"data":{"file_id":40,"beg":{"line":312,"col":0},"end":{"line":312,"col":16}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for hashing an arbitrary stream of bytes."},{"DocComment":""},{"DocComment":" Instances of `Hasher` usually represent state that is changed while hashing"},{"DocComment":" data."},{"DocComment":""},{"DocComment":" `Hasher` provides a fairly basic interface for retrieving the generated hash"},{"DocComment":" (with [`finish`]), and writing integers as well as slices of bytes into an"},{"DocComment":" instance (with [`write`] and [`write_u8`] etc.). Most of the time, `Hasher`"},{"DocComment":" instances are used in conjunction with the [`Hash`] trait."},{"DocComment":""},{"DocComment":" This trait provides no guarantees about how the various `write_*` methods are"},{"DocComment":" defined and implementations of [`Hash`] should not assume that they work one"},{"DocComment":" way or another. You cannot assume, for example, that a [`write_u32`] call is"},{"DocComment":" equivalent to four calls of [`write_u8`]. Nor can you assume that adjacent"},{"DocComment":" `write` calls are merged, so it's possible, for example, that"},{"DocComment":" ```"},{"DocComment":" # fn foo(hasher: &mut impl std::hash::Hasher) {"},{"DocComment":" hasher.write(&[1, 2]);"},{"DocComment":" hasher.write(&[3, 4, 5, 6]);"},{"DocComment":" # }"},{"DocComment":" ```"},{"DocComment":" and"},{"DocComment":" ```"},{"DocComment":" # fn foo(hasher: &mut impl std::hash::Hasher) {"},{"DocComment":" hasher.write(&[1, 2, 3, 4]);"},{"DocComment":" hasher.write(&[5, 6]);"},{"DocComment":" # }"},{"DocComment":" ```"},{"DocComment":" end up producing different hashes."},{"DocComment":""},{"DocComment":" Thus to produce the same hash value, [`Hash`] implementations must ensure"},{"DocComment":" for equivalent items that exactly the same sequence of calls is made -- the"},{"DocComment":" same methods with the same parameters in the same order."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":""},{"DocComment":" hasher.write_u32(1989);"},{"DocComment":" hasher.write_u8(11);"},{"DocComment":" hasher.write_u8(9);"},{"DocComment":" hasher.write(b\"Huh?\");"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`finish`]: Hasher::finish"},{"DocComment":" [`write`]: Hasher::write"},{"DocComment":" [`write_u8`]: Hasher::write_u8"},{"DocComment":" [`write_u32`]: Hasher::write_u32"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"finish","attr_info":{"attributes":[{"DocComment":" Returns the hash value for the values written so far."},{"DocComment":""},{"DocComment":" Despite its name, the method does not reset the hasher’s internal"},{"DocComment":" state. Additional [`write`]s will continue from the current value."},{"DocComment":" If you need to start a fresh hash value, you will have to create"},{"DocComment":" a new hasher."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" hasher.write(b\"Cool!\");"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`write`]: Hasher::write"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"Deduplicated":1784}},"item":{"id":331,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11845,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[24,0]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"write","attr_info":{"attributes":[{"DocComment":" Writes some data into this `Hasher`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" let data = [0x01, 0x23, 0x45, 0x67, 0x89, 0xab, 0xcd, 0xef];"},{"DocComment":""},{"DocComment":" hasher.write(&data);"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Note to Implementers"},{"DocComment":""},{"DocComment":" You generally should not do length-prefixing as part of implementing"},{"DocComment":" this method. It's up to the [`Hash`] implementation to call"},{"DocComment":" [`Hasher::write_length_prefix`] before sequences that need it."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"HashConsedValue":[7746,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":2929},"Shared"]}]}],"output":{"Deduplicated":1802}},"item":{"id":332,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11845}]}}},"kind":{"TraitMethod":[24,1]}},null,null,null,null,null,null,null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":69},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":25,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["TryFrom",0]}],"span":{"data":{"file_id":32,"beg":{"line":694,"col":0},"end":{"line":694,"col":33}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Simple and safe type conversions that may fail in a controlled"},{"DocComment":" way under some circumstances. It is the reciprocal of [`TryInto`]."},{"DocComment":""},{"DocComment":" This is useful when you are doing a type conversion that may"},{"DocComment":" trivially succeed but may also need special handling."},{"DocComment":" For example, there is no way to convert an [`i64`] into an [`i32`]"},{"DocComment":" using the [`From`] trait, because an [`i64`] may contain a value"},{"DocComment":" that an [`i32`] cannot represent and so the conversion would lose data."},{"DocComment":" This might be handled by truncating the [`i64`] to an [`i32`] or by"},{"DocComment":" simply returning [`i32::MAX`], or by some other method. The [`From`]"},{"DocComment":" trait is intended for perfect conversions, so the `TryFrom` trait"},{"DocComment":" informs the programmer when a type conversion could go bad and lets"},{"DocComment":" them decide how to handle it."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - `TryFrom for U` implies [`TryInto`]` for T`"},{"DocComment":" - [`try_from`] is reflexive, which means that `TryFrom for T`"},{"DocComment":" is implemented and cannot fail -- the associated `Error` type for"},{"DocComment":" calling `T::try_from()` on a value of type `T` is [`Infallible`]."},{"DocComment":" When the [`!`] type is stabilized [`Infallible`] and [`!`] will be"},{"DocComment":" equivalent."},{"DocComment":""},{"DocComment":" Prefer using [`TryInto`] over [`TryFrom`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`TryInto`] can be used as well."},{"DocComment":""},{"DocComment":" `TryFrom` can be implemented as follows:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct GreaterThanZero(i32);"},{"DocComment":""},{"DocComment":" impl TryFrom for GreaterThanZero {"},{"DocComment":" type Error = &'static str;"},{"DocComment":""},{"DocComment":" fn try_from(value: i32) -> Result {"},{"DocComment":" if value <= 0 {"},{"DocComment":" Err(\"GreaterThanZero only accepts values greater than zero!\")"},{"DocComment":" } else {"},{"DocComment":" Ok(GreaterThanZero(value))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" As described, [`i32`] implements `TryFrom<`[`i64`]`>`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let big_number = 1_000_000_000_000i64;"},{"DocComment":" // Silently truncates `big_number`, requires detecting"},{"DocComment":" // and handling the truncation after the fact."},{"DocComment":" let smaller_number = big_number as i32;"},{"DocComment":" assert_eq!(smaller_number, -727379968);"},{"DocComment":""},{"DocComment":" // Returns an error because `big_number` is too big to"},{"DocComment":" // fit in an `i32`."},{"DocComment":" let try_smaller_number = i32::try_from(big_number);"},{"DocComment":" assert!(try_smaller_number.is_err());"},{"DocComment":""},{"DocComment":" // Returns `Ok(3)`."},{"DocComment":" let try_successful_smaller_number = i32::try_from(3);"},{"DocComment":" assert!(try_successful_smaller_number.is_ok());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`try_from`]: TryFrom::try_from"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"TryFrom"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"},{"index":2,"name":"Self_Error"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"try_from","attr_info":{"attributes":[{"DocComment":" Performs the conversion."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11752}],"output":{"Deduplicated":11790}},"item":{"id":350,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11846,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":25,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[25,0]}}],"vtable":null},{"def_id":26,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["borrow",0]},{"Ident":["Borrow",0]}],"span":{"data":{"file_id":42,"beg":{"line":158,"col":0},"end":{"line":158,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for borrowing data."},{"DocComment":""},{"DocComment":" In Rust, it is common to provide different representations of a type for"},{"DocComment":" different use cases. For instance, storage location and management for a"},{"DocComment":" value can be specifically chosen as appropriate for a particular use via"},{"DocComment":" pointer types such as [`Box`] or [`Rc`]. Beyond these generic"},{"DocComment":" wrappers that can be used with any type, some types provide optional"},{"DocComment":" facets providing potentially costly functionality. An example for such a"},{"DocComment":" type is [`String`] which adds the ability to extend a string to the basic"},{"DocComment":" [`str`]. This requires keeping additional information unnecessary for a"},{"DocComment":" simple, immutable string."},{"DocComment":""},{"DocComment":" These types provide access to the underlying data through references"},{"DocComment":" to the type of that data. They are said to be ‘borrowed as’ that type."},{"DocComment":" For instance, a [`Box`] can be borrowed as `T` while a [`String`]"},{"DocComment":" can be borrowed as `str`."},{"DocComment":""},{"DocComment":" Types express that they can be borrowed as some type `T` by implementing"},{"DocComment":" `Borrow`, providing a reference to a `T` in the trait’s"},{"DocComment":" [`borrow`] method. A type is free to borrow as several different types."},{"DocComment":" If it wishes to mutably borrow as the type, allowing the underlying data"},{"DocComment":" to be modified, it can additionally implement [`BorrowMut`]."},{"DocComment":""},{"DocComment":" Further, when providing implementations for additional traits, it needs"},{"DocComment":" to be considered whether they should behave identically to those of the"},{"DocComment":" underlying type as a consequence of acting as a representation of that"},{"DocComment":" underlying type. Generic code typically uses `Borrow` when it relies"},{"DocComment":" on the identical behavior of these additional trait implementations."},{"DocComment":" These traits will likely appear as additional trait bounds."},{"DocComment":""},{"DocComment":" In particular `Eq`, `Ord` and `Hash` must be equivalent for"},{"DocComment":" borrowed and owned values: `x.borrow() == y.borrow()` should give the"},{"DocComment":" same result as `x == y`."},{"DocComment":""},{"DocComment":" If generic code merely needs to work for all types that can"},{"DocComment":" provide a reference to related type `T`, it is often better to use"},{"DocComment":" [`AsRef`] as more types can safely implement it."},{"DocComment":""},{"DocComment":" [`Box`]: ../../std/boxed/struct.Box.html"},{"DocComment":" [`Mutex`]: ../../std/sync/struct.Mutex.html"},{"DocComment":" [`Rc`]: ../../std/rc/struct.Rc.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`borrow`]: Borrow::borrow"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" As a data collection, [`HashMap`] owns both keys and values. If"},{"DocComment":" the key’s actual data is wrapped in a managing type of some kind, it"},{"DocComment":" should, however, still be possible to search for a value using a"},{"DocComment":" reference to the key’s data. For instance, if the key is a string, then"},{"DocComment":" it is likely stored with the hash map as a [`String`], while it should"},{"DocComment":" be possible to search using a [`&str`][`str`]. Thus, `insert` needs to"},{"DocComment":" operate on a `String` while `get` needs to be able to use a `&str`."},{"DocComment":""},{"DocComment":" Slightly simplified, the relevant parts of `HashMap` look like"},{"DocComment":" this:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::borrow::Borrow;"},{"DocComment":" use std::hash::Hash;"},{"DocComment":""},{"DocComment":" pub struct HashMap {"},{"DocComment":" # marker: ::std::marker::PhantomData<(K, V)>,"},{"DocComment":" // fields omitted"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl HashMap {"},{"DocComment":" pub fn insert(&self, key: K, value: V) -> Option"},{"DocComment":" where K: Hash + Eq"},{"DocComment":" {"},{"DocComment":" # unimplemented!()"},{"DocComment":" // ..."},{"DocComment":" }"},{"DocComment":""},{"DocComment":" pub fn get(&self, k: &Q) -> Option<&V>"},{"DocComment":" where"},{"DocComment":" K: Borrow,"},{"DocComment":" Q: Hash + Eq + ?Sized"},{"DocComment":" {"},{"DocComment":" # unimplemented!()"},{"DocComment":" // ..."},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The entire hash map is generic over a key type `K`. Because these keys"},{"DocComment":" are stored with the hash map, this type has to own the key’s data."},{"DocComment":" When inserting a key-value pair, the map is given such a `K` and needs"},{"DocComment":" to find the correct hash bucket and check if the key is already present"},{"DocComment":" based on that `K`. It therefore requires `K: Hash + Eq`."},{"DocComment":""},{"DocComment":" When searching for a value in the map, however, having to provide a"},{"DocComment":" reference to a `K` as the key to search for would require to always"},{"DocComment":" create such an owned value. For string keys, this would mean a `String`"},{"DocComment":" value needs to be created just for the search for cases where only a"},{"DocComment":" `str` is available."},{"DocComment":""},{"DocComment":" Instead, the `get` method is generic over the type of the underlying key"},{"DocComment":" data, called `Q` in the method signature above. It states that `K`"},{"DocComment":" borrows as a `Q` by requiring that `K: Borrow`. By additionally"},{"DocComment":" requiring `Q: Hash + Eq`, it signals the requirement that `K` and `Q`"},{"DocComment":" have implementations of the `Hash` and `Eq` traits that produce identical"},{"DocComment":" results."},{"DocComment":""},{"DocComment":" The implementation of `get` relies in particular on identical"},{"DocComment":" implementations of `Hash` by determining the key’s hash bucket by calling"},{"DocComment":" `Hash::hash` on the `Q` value even though it inserted the key based on"},{"DocComment":" the hash value calculated from the `K` value."},{"DocComment":""},{"DocComment":" As a consequence, the hash map breaks if a `K` wrapping a `Q` value"},{"DocComment":" produces a different hash than `Q`. For instance, imagine you have a"},{"DocComment":" type that wraps a string but compares ASCII letters ignoring their case:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" pub struct CaseInsensitiveString(String);"},{"DocComment":""},{"DocComment":" impl PartialEq for CaseInsensitiveString {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.0.eq_ignore_ascii_case(&other.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for CaseInsensitiveString { }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Because two equal values need to produce the same hash value, the"},{"DocComment":" implementation of `Hash` needs to ignore ASCII case, too:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # use std::hash::{Hash, Hasher};"},{"DocComment":" # pub struct CaseInsensitiveString(String);"},{"DocComment":" impl Hash for CaseInsensitiveString {"},{"DocComment":" fn hash(&self, state: &mut H) {"},{"DocComment":" for c in self.0.as_bytes() {"},{"DocComment":" c.to_ascii_lowercase().hash(state)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Can `CaseInsensitiveString` implement `Borrow`? It certainly can"},{"DocComment":" provide a reference to a string slice via its contained owned string."},{"DocComment":" But because its `Hash` implementation differs, it behaves differently"},{"DocComment":" from `str` and therefore must not, in fact, implement `Borrow`."},{"DocComment":" If it wants to allow others access to the underlying `str`, it can do"},{"DocComment":" that via `AsRef` which doesn’t carry any extra requirements."},{"DocComment":""},{"DocComment":" [`Hash`]: crate::hash::Hash"},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Borrow"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Borrowed"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"borrow","attr_info":{"attributes":[{"DocComment":" Immutably borrows from an owned value."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::borrow::Borrow;"},{"DocComment":""},{"DocComment":" fn check>(s: T) {"},{"DocComment":" assert_eq!(\"Hello\", s.borrow());"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let s = \"Hello\".to_string();"},{"DocComment":""},{"DocComment":" check(s);"},{"DocComment":""},{"DocComment":" let s = \"Hello\";"},{"DocComment":""},{"DocComment":" check(s);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"HashConsedValue":[11847,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11752},"Shared"]}]}},"item":{"id":352,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11848,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":26,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[26,0]}}],"vtable":{"id":{"Adt":71},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":27,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Sum",0]}],"span":{"data":{"file_id":43,"beg":{"line":18,"col":0},"end":{"line":18,"col":36}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait to represent types that can be created by summing up an iterator."},{"DocComment":""},{"DocComment":" This trait is used to implement [`Iterator::sum()`]. Types which implement"},{"DocComment":" this trait can be generated by using the [`sum()`] method on an iterator."},{"DocComment":" Like [`FromIterator`], this trait should rarely be called directly."},{"DocComment":""},{"DocComment":" [`sum()`]: Iterator::sum"},{"DocComment":" [`FromIterator`]: iter::FromIterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":43,"beg":{"line":22,"col":14},"end":{"line":22,"col":32}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sum","attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by \"summing up\""},{"DocComment":" the items."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3182}],"output":{"Deduplicated":11751}},"item":{"id":353,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":3182}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11849,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":27,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[11850,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[27,0]}}],"vtable":null},{"def_id":28,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["IntoIterator",0]}],"span":{"data":{"file_id":17,"beg":{"line":283,"col":0},"end":{"line":283,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Conversion into an [`Iterator`]."},{"DocComment":""},{"DocComment":" By implementing `IntoIterator` for a type, you define how it will be"},{"DocComment":" converted to an iterator. This is common for types which describe a"},{"DocComment":" collection of some kind."},{"DocComment":""},{"DocComment":" One benefit of implementing `IntoIterator` is that your type will [work"},{"DocComment":" with Rust's `for` loop syntax](crate::iter#for-loops-and-intoiterator)."},{"DocComment":""},{"DocComment":" See also: [`FromIterator`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let v = [1, 2, 3];"},{"DocComment":" let mut iter = v.into_iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"},{"DocComment":" Implementing `IntoIterator` for your type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // and we'll implement IntoIterator"},{"DocComment":" impl IntoIterator for MyCollection {"},{"DocComment":" type Item = i32;"},{"DocComment":" type IntoIter = std::vec::IntoIter;"},{"DocComment":""},{"DocComment":" fn into_iter(self) -> Self::IntoIter {"},{"DocComment":" self.0.into_iter()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Now we can make a new collection..."},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" // ... add some stuff to it ..."},{"DocComment":" c.add(0);"},{"DocComment":" c.add(1);"},{"DocComment":" c.add(2);"},{"DocComment":""},{"DocComment":" // ... and then turn it into an Iterator:"},{"DocComment":" for (i, n) in c.into_iter().enumerate() {"},{"DocComment":" assert_eq!(i as i32, n);"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It is common to use `IntoIterator` as a trait bound. This allows"},{"DocComment":" the input collection type to change, so long as it is still an"},{"DocComment":" iterator. Additional bounds can be specified by restricting on"},{"DocComment":" `Item`:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" fn collect_as_strings(collection: T) -> Vec"},{"DocComment":" where"},{"DocComment":" T: IntoIterator,"},{"DocComment":" T::Item: std::fmt::Debug,"},{"DocComment":" {"},{"DocComment":" collection"},{"DocComment":" .into_iter()"},{"DocComment":" .map(|item| format!(\"{item:?}\"))"},{"DocComment":" .collect()"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"IntoIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Item"},{"index":2,"name":"Self_IntoIter"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":291,"col":19},"end":{"line":291,"col":46}},"generated_from_span":null},"origin":{"TraitItem":1},"trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11789},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null,null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"into_iter","attr_info":{"attributes":[{"DocComment":" Creates an iterator from a value."},{"DocComment":""},{"DocComment":" See the [module-level documentation] for more."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let v = [1, 2, 3];"},{"DocComment":" let mut iter = v.into_iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751}],"output":{"Deduplicated":11789}},"item":{"id":359,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11851,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[28,0]}}],"vtable":{"id":{"Adt":72},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":29,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["num",0]},{"Ident":["nonzero",0]},{"Ident":["ZeroablePrimitive",0]}],"span":{"data":{"file_id":45,"beg":{"line":34,"col":0},"end":{"line":34,"col":66}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A marker trait for primitive types which can be zero."},{"DocComment":""},{"DocComment":" This is an implementation detail for [NonZero]\\ which may disappear or be replaced at any time."},{"DocComment":""},{"DocComment":" # Safety"},{"DocComment":""},{"DocComment":" Types implementing this trait must be primitives that are valid when zeroed."},{"DocComment":""},{"DocComment":" The associated `Self::NonZeroInner` type must have the same size+align as `Self`,"},{"DocComment":" but with a niche and bit validity making it so the following `transmutes` are sound:"},{"DocComment":""},{"DocComment":" - `Self::NonZeroInner` to `Option`"},{"DocComment":" - `Option` to `Self`"},{"DocComment":""},{"DocComment":" (And, consequently, `Self::NonZeroInner` to `Self`.)"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_NonZeroInner"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":45,"beg":{"line":34,"col":44},"end":{"line":34,"col":48}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":45,"beg":{"line":34,"col":51},"end":{"line":34,"col":66}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":46,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":2,"span":{"data":{"file_id":45,"beg":{"line":36,"col":31},"end":{"line":36,"col":35}},"generated_from_span":null},"origin":{"TraitItem":0},"trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null],"methods":[],"vtable":null},{"def_id":30,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["function",0]},{"Ident":["FnMut",0]}],"span":{"data":{"file_id":51,"beg":{"line":163,"col":0},"end":{"line":163,"col":48}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The version of the call operator that takes a mutable receiver."},{"DocComment":""},{"DocComment":" Instances of `FnMut` can be called repeatedly and may mutate state."},{"DocComment":""},{"DocComment":" `FnMut` is implemented automatically by closures which take mutable"},{"DocComment":" references to captured variables, as well as all types that implement"},{"DocComment":" [`Fn`], e.g., (safe) [function pointers] (since `FnMut` is a supertrait of"},{"DocComment":" [`Fn`]). Additionally, for any type `F` that implements `FnMut`, `&mut F`"},{"DocComment":" implements `FnMut`, too."},{"DocComment":""},{"DocComment":" Since [`FnOnce`] is a supertrait of `FnMut`, any instance of `FnMut` can be"},{"DocComment":" used where a [`FnOnce`] is expected, and since [`Fn`] is a subtrait of"},{"DocComment":" `FnMut`, any instance of [`Fn`] can be used where `FnMut` is expected."},{"DocComment":""},{"DocComment":" Use `FnMut` as a bound when you want to accept a parameter of function-like"},{"DocComment":" type and need to call it repeatedly, while allowing it to mutate state."},{"DocComment":" If you don't want the parameter to mutate state, use [`Fn`] as a"},{"DocComment":" bound; if you don't need to call it repeatedly, use [`FnOnce`]."},{"DocComment":""},{"DocComment":" See the [chapter on closures in *The Rust Programming Language*][book] for"},{"DocComment":" some more information on this topic."},{"DocComment":""},{"DocComment":" Also of note is the special syntax for `Fn` traits (e.g."},{"DocComment":" `Fn(usize, bool) -> usize`). Those interested in the technical details of"},{"DocComment":" this can refer to [the relevant section in the *Rustonomicon*][nomicon]."},{"DocComment":""},{"DocComment":" [book]: ../../book/ch13-01-closures.html"},{"DocComment":" [function pointers]: fn"},{"DocComment":" [nomicon]: ../../nomicon/hrtb.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## Calling a mutably capturing closure"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x = 5;"},{"DocComment":" {"},{"DocComment":" let mut square_x = || x *= x;"},{"DocComment":" square_x();"},{"DocComment":" }"},{"DocComment":" assert_eq!(x, 25);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Using a `FnMut` parameter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn do_twice(mut func: F)"},{"DocComment":" where F: FnMut()"},{"DocComment":" {"},{"DocComment":" func();"},{"DocComment":" func();"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut x: usize = 1;"},{"DocComment":" {"},{"DocComment":" let add_two_to_x = || x += 2;"},{"DocComment":" do_twice(add_two_to_x);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(x, 5);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"fn_mut"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Self_Clause0_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":51,"beg":{"line":163,"col":36},"end":{"line":163,"col":48}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"call_mut","attr_info":{"attributes":[{"DocComment":" Performs the call operation."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11818},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":360,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11852,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":30,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[30,0]}}],"vtable":{"id":{"Adt":74},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":31,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["function",0]},{"Ident":["FnOnce",0]}],"span":{"data":{"file_id":51,"beg":{"line":242,"col":0},"end":{"line":242,"col":35}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The version of the call operator that takes a by-value receiver."},{"DocComment":""},{"DocComment":" Instances of `FnOnce` can be called, but might not be callable multiple"},{"DocComment":" times. Because of this, if the only thing known about a type is that it"},{"DocComment":" implements `FnOnce`, it can only be called once."},{"DocComment":""},{"DocComment":" `FnOnce` is implemented automatically by closures that might consume captured"},{"DocComment":" variables, as well as all types that implement [`FnMut`], e.g., (safe)"},{"DocComment":" [function pointers] (since `FnOnce` is a supertrait of [`FnMut`])."},{"DocComment":""},{"DocComment":" Since both [`Fn`] and [`FnMut`] are subtraits of `FnOnce`, any instance of"},{"DocComment":" [`Fn`] or [`FnMut`] can be used where a `FnOnce` is expected."},{"DocComment":""},{"DocComment":" Use `FnOnce` as a bound when you want to accept a parameter of function-like"},{"DocComment":" type and only need to call it once. If you need to call the parameter"},{"DocComment":" repeatedly, use [`FnMut`] as a bound; if you also need it to not mutate"},{"DocComment":" state, use [`Fn`]."},{"DocComment":""},{"DocComment":" See the [chapter on closures in *The Rust Programming Language*][book] for"},{"DocComment":" some more information on this topic."},{"DocComment":""},{"DocComment":" Also of note is the special syntax for `Fn` traits (e.g."},{"DocComment":" `Fn(usize, bool) -> usize`). Those interested in the technical details of"},{"DocComment":" this can refer to [the relevant section in the *Rustonomicon*][nomicon]."},{"DocComment":""},{"DocComment":" [book]: ../../book/ch13-01-closures.html"},{"DocComment":" [function pointers]: fn"},{"DocComment":" [nomicon]: ../../nomicon/hrtb.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## Using a `FnOnce` parameter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn consume_with_relish(func: F)"},{"DocComment":" where F: FnOnce() -> String"},{"DocComment":" {"},{"DocComment":" // `func` consumes its captured variables, so it cannot be run more"},{"DocComment":" // than once."},{"DocComment":" println!(\"Consumed: {}\", func());"},{"DocComment":""},{"DocComment":" println!(\"Delicious!\");"},{"DocComment":""},{"DocComment":" // Attempting to invoke `func()` again will throw a `use of moved"},{"DocComment":" // value` error for `func`."},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let x = String::from(\"x\");"},{"DocComment":" let consume_and_return_x = move || x;"},{"DocComment":" consume_with_relish(consume_and_return_x);"},{"DocComment":""},{"DocComment":" // `consume_and_return_x` can no longer be invoked at this point"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"fn_once"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"call_once","attr_info":{"attributes":[{"DocComment":" Performs the call operation."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":361,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11853,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[31,0]}}],"vtable":{"id":{"Adt":75},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":32,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["FromIterator",0]}],"span":{"data":{"file_id":17,"beg":{"line":134,"col":0},"end":{"line":134,"col":32}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Conversion from an [`Iterator`]."},{"DocComment":""},{"DocComment":" By implementing `FromIterator` for a type, you define how it will be"},{"DocComment":" created from an iterator. This is common for types which describe a"},{"DocComment":" collection of some kind."},{"DocComment":""},{"DocComment":" If you want to create a collection from the contents of an iterator, the"},{"DocComment":" [`Iterator::collect()`] method is preferred. However, when you need to"},{"DocComment":" specify the container type, [`FromIterator::from_iter()`] can be more"},{"DocComment":" readable than using a turbofish (e.g. `::>()`). See the"},{"DocComment":" [`Iterator::collect()`] documentation for more examples of its use."},{"DocComment":""},{"DocComment":" See also: [`IntoIterator`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v = Vec::from_iter(five_fives);"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Using [`Iterator::collect()`] to implicitly use `FromIterator`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v: Vec = five_fives.collect();"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Using [`FromIterator::from_iter()`] as a more readable alternative to"},{"DocComment":" [`Iterator::collect()`]:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::collections::VecDeque;"},{"DocComment":" let first = (0..10).collect::>();"},{"DocComment":" let second = VecDeque::from_iter(0..10);"},{"DocComment":""},{"DocComment":" assert_eq!(first, second);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Implementing `FromIterator` for your type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // and we'll implement FromIterator"},{"DocComment":" impl FromIterator for MyCollection {"},{"DocComment":" fn from_iter>(iter: I) -> Self {"},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" for i in iter {"},{"DocComment":" c.add(i);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" c"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Now we can make a new iterator..."},{"DocComment":" let iter = (0..5).into_iter();"},{"DocComment":""},{"DocComment":" // ... and make a MyCollection out of it"},{"DocComment":" let c = MyCollection::from_iter(iter);"},{"DocComment":""},{"DocComment":" assert_eq!(c.0, vec![0, 1, 2, 3, 4]);"},{"DocComment":""},{"DocComment":" // collect works too!"},{"DocComment":""},{"DocComment":" let iter = (0..5).into_iter();"},{"DocComment":" let c: MyCollection = iter.collect();"},{"DocComment":""},{"DocComment":" assert_eq!(c.0, vec![0, 1, 2, 3, 4]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"FromIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":152,"col":20},"end":{"line":152,"col":42}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752},{"HashConsedValue":[3203,{"TypeVar":{"Bound":[1,1]}}]}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_iter","attr_info":{"attributes":[{"DocComment":" Creates a value from an iterator."},{"DocComment":""},{"DocComment":" See the [module-level documentation] for more."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v = Vec::from_iter(five_fives);"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3182}],"output":{"Deduplicated":11751}},"item":{"id":362,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":3182},{"HashConsedValue":[3202,{"TypeVar":{"Bound":[0,1]}}]}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11854,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":32,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[11855,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752},{"Deduplicated":3203}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[32,0]}}],"vtable":null},{"def_id":33,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Try",0]}],"span":{"data":{"file_id":67,"beg":{"line":133,"col":0},"end":{"line":133,"col":41}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The `?` operator and `try {}` blocks."},{"DocComment":""},{"DocComment":" `try_*` methods typically involve a type implementing this trait. For"},{"DocComment":" example, the closures passed to [`Iterator::try_fold`] and"},{"DocComment":" [`Iterator::try_for_each`] must return such a type."},{"DocComment":""},{"DocComment":" `Try` types are typically those containing two or more categories of values,"},{"DocComment":" some subset of which are so commonly handled via early returns that it's"},{"DocComment":" worth providing a terse (but still visible) syntax to make that easy."},{"DocComment":""},{"DocComment":" This is most often seen for error handling with [`Result`] and [`Option`]."},{"DocComment":" The quintessential implementation of this trait is on [`ControlFlow`]."},{"DocComment":""},{"DocComment":" # Using `Try` in Generic Code"},{"DocComment":""},{"DocComment":" `Iterator::try_fold` was stabilized to call back in Rust 1.27, but"},{"DocComment":" this trait is much newer. To illustrate the various associated types and"},{"DocComment":" methods, let's implement our own version."},{"DocComment":""},{"DocComment":" As a reminder, an infallible version of a fold looks something like this:"},{"DocComment":" ```"},{"DocComment":" fn simple_fold("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> A,"},{"DocComment":" ) -> A {"},{"DocComment":" for x in iter {"},{"DocComment":" accum = f(accum, x);"},{"DocComment":" }"},{"DocComment":" accum"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" So instead of `f` returning just an `A`, we'll need it to return some other"},{"DocComment":" type that produces an `A` in the \"don't short circuit\" path. Conveniently,"},{"DocComment":" that's also the type we need to return from the function."},{"DocComment":""},{"DocComment":" Let's add a new generic parameter `R` for that type, and bound it to the"},{"DocComment":" output type that we want:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::Try;"},{"DocComment":" fn simple_try_fold_1>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" todo!()"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If we get through the entire iterator, we need to wrap up the accumulator"},{"DocComment":" into the return type using [`Try::from_output`]:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::{ControlFlow, Try};"},{"DocComment":" fn simple_try_fold_2>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" let cf = f(accum, x).branch();"},{"DocComment":" match cf {"},{"DocComment":" ControlFlow::Continue(a) => accum = a,"},{"DocComment":" ControlFlow::Break(_) => todo!(),"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" We'll also need [`FromResidual::from_residual`] to turn the residual back"},{"DocComment":" into the original type. But because it's a supertrait of `Try`, we don't"},{"DocComment":" need to mention it in the bounds. All types which implement `Try` can be"},{"DocComment":" recreated from their corresponding residual, so we'll just call it:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::{ControlFlow, Try};"},{"DocComment":" pub fn simple_try_fold_3>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" let cf = f(accum, x).branch();"},{"DocComment":" match cf {"},{"DocComment":" ControlFlow::Continue(a) => accum = a,"},{"DocComment":" ControlFlow::Break(r) => return R::from_residual(r),"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" But this \"call `branch`, then `match` on it, and `return` if it was a"},{"DocComment":" `Break`\" is exactly what happens inside the `?` operator. So rather than"},{"DocComment":" do all this manually, we can just use `?` instead:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::Try;"},{"DocComment":" fn simple_try_fold>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" accum = f(accum, x)?;"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Try"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":67,"beg":{"line":133,"col":21},"end":{"line":133,"col":41}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":44,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"HashConsedValue":[11857,{"TraitType":[{"HashConsedValue":[11856,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]},1,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":67,"beg":{"line":160,"col":19},"end":{"line":160,"col":41}},"generated_from_span":null},"origin":{"TraitItem":1},"trait_":{"regions":[],"skip_binder":{"id":34,"generics":{"regions":[],"types":[{"Deduplicated":11857},{"HashConsedValue":[11858,{"TraitType":[{"Deduplicated":11856},0,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]},{"HashConsedValue":[11859,{"TraitType":[{"Deduplicated":11856},2,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Output","attr_info":{"attributes":[{"DocComment":" The type of the value produced by `?` when *not* short-circuiting."}],"inline":null,"rename":null,"public":false},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Residual","attr_info":{"attributes":[{"DocComment":" The type of the value passed to [`FromResidual::from_residual`]"},{"DocComment":" as part of `?` when short-circuiting."},{"DocComment":""},{"DocComment":" This represents the possible values of the `Self` type which are *not*"},{"DocComment":" represented by the `Output` type."},{"DocComment":""},{"DocComment":" # Note to Implementors"},{"DocComment":""},{"DocComment":" The choice of this type is critical to interconversion."},{"DocComment":" Unlike the `Output` type, which will often be a raw generic type,"},{"DocComment":" this type is typically a newtype of some sort to \"color\" the type"},{"DocComment":" so that it's distinguishable from the residuals of other types."},{"DocComment":""},{"DocComment":" This is why `Result::Residual` is not `E`, but `Result`."},{"DocComment":" That way it's distinct from `ControlFlow::Residual`, for example,"},{"DocComment":" and thus `?` on `ControlFlow` cannot be used in a method returning `Result`."},{"DocComment":""},{"DocComment":" If you're making a generic type `Foo` that implements `Try`,"},{"DocComment":" then typically you can use `Foo` as its `Residual`"},{"DocComment":" type: that type will have a \"hole\" in the correct place, and will maintain the"},{"DocComment":" \"foo-ness\" of the residual so other types need to opt-in to interconversion."}],"inline":null,"rename":null,"public":false},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,1]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Self_Clause1_TryType","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,2]}}],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_output","attr_info":{"attributes":[{"DocComment":" Constructs the type from its `Output` type."},{"DocComment":""},{"DocComment":" This should be implemented consistently with the `branch` method"},{"DocComment":" such that applying the `?` operator will get back the original value:"},{"DocComment":" `Try::from_output(x).branch() --> ControlFlow::Continue(x)`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::Try;"},{"DocComment":""},{"DocComment":" assert_eq!( as Try>::from_output(3), Ok(3));"},{"DocComment":" assert_eq!( as Try>::from_output(4), Some(4));"},{"DocComment":" assert_eq!("},{"DocComment":" as Try>::from_output(5),"},{"DocComment":" std::ops::ControlFlow::Continue(5),"},{"DocComment":" );"},{"DocComment":""},{"DocComment":" # fn make_question_mark_work() -> Option<()> {"},{"DocComment":" assert_eq!(Option::from_output(4)?, 4);"},{"DocComment":" # None }"},{"DocComment":" # make_question_mark_work();"},{"DocComment":""},{"DocComment":" // This is used, for example, on the accumulator in `try_fold`:"},{"DocComment":" let r = std::iter::empty().try_fold(4, |_, ()| -> Option<_> { unreachable!() });"},{"DocComment":" assert_eq!(r, Some(4));"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11858}],"output":{"Deduplicated":11751}},"item":{"id":363,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11856}]}}},"kind":{"TraitMethod":[33,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"branch","attr_info":{"attributes":[{"DocComment":" Used in `?` to decide whether the operator should produce a value"},{"DocComment":" (because this returned [`ControlFlow::Continue`])"},{"DocComment":" or propagate a value back to the caller"},{"DocComment":" (because this returned [`ControlFlow::Break`])."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::{ControlFlow, Try};"},{"DocComment":""},{"DocComment":" assert_eq!(Ok::<_, String>(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(Err::(3).branch(), ControlFlow::Break(Err(3)));"},{"DocComment":""},{"DocComment":" assert_eq!(Some(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(None::.branch(), ControlFlow::Break(None));"},{"DocComment":""},{"DocComment":" assert_eq!(ControlFlow::::Continue(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!("},{"DocComment":" ControlFlow::<_, String>::Break(3).branch(),"},{"DocComment":" ControlFlow::Break(ControlFlow::Break(3)),"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751}],"output":{"HashConsedValue":[11860,{"Adt":{"id":{"Adt":20},"generics":{"regions":[],"types":[{"Deduplicated":11857},{"Deduplicated":11858}],"const_generics":[],"trait_refs":[]}}}]}},"item":{"id":364,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"Deduplicated":11856}]}}},"kind":{"TraitMethod":[33,1]}}],"vtable":null},{"def_id":34,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Residual",0]}],"span":{"data":{"file_id":67,"beg":{"line":364,"col":0},"end":{"line":364,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Allows retrieving the canonical type implementing [`Try`] that has this type"},{"DocComment":" as its residual and allows it to hold an `O` as its output."},{"DocComment":""},{"DocComment":" If you think of the `Try` trait as splitting a type into its [`Try::Output`]"},{"DocComment":" and [`Try::Residual`] components, this allows putting them back together."},{"DocComment":""},{"DocComment":" For example,"},{"DocComment":" `Result: Try>`,"},{"DocComment":" and in the other direction,"},{"DocComment":" ` as Residual>::TryType = Result`."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"O"},{"index":2,"name":"Self_TryType"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[{"regions":[],"skip_binder":{"trait_ref":{"HashConsedValue":[11862,{"kind":{"ParentClause":[{"HashConsedValue":[11861,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":34,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]},0]},"trait_decl_ref":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]},"type_id":0,"ty":{"Deduplicated":11752}}},{"regions":[],"skip_binder":{"trait_ref":{"Deduplicated":11862},"type_id":1,"ty":{"Deduplicated":11751}}}]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":67,"beg":{"line":368,"col":18},"end":{"line":368,"col":58}},"generated_from_span":null},"origin":{"TraitItem":0},"trait_":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null],"methods":[],"vtable":null},{"def_id":35,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["Extend",0]}],"span":{"data":{"file_id":17,"beg":{"line":397,"col":0},"end":{"line":397,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Extend a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" Iterators produce a series of values, and collections can also be thought"},{"DocComment":" of as a series of values. The `Extend` trait bridges this gap, allowing you"},{"DocComment":" to extend a collection by including the contents of that iterator. When"},{"DocComment":" extending a collection with an already existing key, that entry is updated"},{"DocComment":" or, in the case of collections that permit multiple entries with equal"},{"DocComment":" keys, that entry is inserted."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"The first three letters are: \");"},{"DocComment":""},{"DocComment":" message.extend(&['a', 'b', 'c']);"},{"DocComment":""},{"DocComment":" assert_eq!(\"abc\", &message[29..32]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Implementing `Extend`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // since MyCollection has a list of i32s, we implement Extend for i32"},{"DocComment":" impl Extend for MyCollection {"},{"DocComment":""},{"DocComment":" // This is a bit simpler with the concrete type signature: we can call"},{"DocComment":" // extend on anything which can be turned into an Iterator which gives"},{"DocComment":" // us i32s. Because we need i32s to put into MyCollection."},{"DocComment":" fn extend>(&mut self, iter: T) {"},{"DocComment":""},{"DocComment":" // The implementation is very straightforward: loop through the"},{"DocComment":" // iterator, and add() each element to ourselves."},{"DocComment":" for elem in iter {"},{"DocComment":" self.add(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" c.add(5);"},{"DocComment":" c.add(6);"},{"DocComment":" c.add(7);"},{"DocComment":""},{"DocComment":" // let's extend our collection with three more numbers"},{"DocComment":" c.extend(vec![1, 2, 3]);"},{"DocComment":""},{"DocComment":" // we've added these elements onto the end"},{"DocComment":" assert_eq!(\"MyCollection([5, 6, 7, 1, 2, 3])\", format!(\"{c:?}\"));"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":416,"col":17},"end":{"line":416,"col":39}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752},{"Deduplicated":3203}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"extend","attr_info":{"attributes":[{"DocComment":" Extends a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" As this is the only required method for this trait, the [trait-level] docs"},{"DocComment":" contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: Extend"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"abc\");"},{"DocComment":""},{"DocComment":" message.extend(['d', 'e', 'f'].iter());"},{"DocComment":""},{"DocComment":" assert_eq!(\"abcdef\", &message);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":3182}],"output":{"Deduplicated":1802}},"item":{"id":365,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":3182},{"Deduplicated":3202}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11863,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":35,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]},{"Deduplicated":11855}]}}},"kind":{"TraitMethod":[35,0]}},null,null,null],"vtable":null},{"def_id":36,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["double_ended",0]},{"Ident":["DoubleEndedIterator",0]}],"span":{"data":{"file_id":68,"beg":{"line":43,"col":0},"end":{"line":43,"col":53}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator able to yield elements from both ends."},{"DocComment":""},{"DocComment":" Something that implements `DoubleEndedIterator` has one extra capability"},{"DocComment":" over something that implements [`Iterator`]: the ability to also take"},{"DocComment":" `Item`s from the back, as well as the front."},{"DocComment":""},{"DocComment":" It is important to note that both back and forth work on the same range,"},{"DocComment":" and do not cross: iteration is over when they meet in the middle."},{"DocComment":""},{"DocComment":" In a similar fashion to the [`Iterator`] protocol, once a"},{"DocComment":" `DoubleEndedIterator` returns [`None`] from a [`next_back()`], calling it"},{"DocComment":" again may or may not ever return [`Some`] again. [`next()`] and"},{"DocComment":" [`next_back()`] are interchangeable for this purpose."},{"DocComment":""},{"DocComment":" [`next_back()`]: DoubleEndedIterator::next_back"},{"DocComment":" [`next()`]: Iterator::next"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"DoubleEndedIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":68,"beg":{"line":43,"col":37},"end":{"line":43,"col":53}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"next_back","attr_info":{"attributes":[{"DocComment":" Removes and returns an element from the end of the iterator."},{"DocComment":""},{"DocComment":" Returns `None` when there are no more elements."},{"DocComment":""},{"DocComment":" The [trait-level] docs contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: DoubleEndedIterator"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Remarks"},{"DocComment":""},{"DocComment":" The elements yielded by `DoubleEndedIterator`'s methods may differ from"},{"DocComment":" the ones yielded by [`Iterator`]'s methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let vec = vec![(1, 'a'), (1, 'b'), (1, 'c'), (2, 'a'), (2, 'b')];"},{"DocComment":" let uniq_by_fst_comp = || {"},{"DocComment":" let mut seen = std::collections::HashSet::new();"},{"DocComment":" vec.iter().copied().filter(move |x| seen.insert(x.0))"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" assert_eq!(uniq_by_fst_comp().last(), Some((2, 'a')));"},{"DocComment":" assert_eq!(uniq_by_fst_comp().next_back(), Some((2, 'b')));"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().fold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(1, 'a'), (2, 'a')]"},{"DocComment":" );"},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().rfold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(2, 'b'), (1, 'c')]"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818}],"output":{"Deduplicated":11819}},"item":{"id":369,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11864,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[36,0]}},null,null,null,null,null],"vtable":{"id":{"Adt":76},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":37,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["exact_size",0]},{"Ident":["ExactSizeIterator",0]}],"span":{"data":{"file_id":69,"beg":{"line":86,"col":0},"end":{"line":86,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that knows its exact length."},{"DocComment":""},{"DocComment":" Many [`Iterator`]s don't know how many times they will iterate, but some do."},{"DocComment":" If an iterator knows how many times it can iterate, providing access to"},{"DocComment":" that information can be useful. For example, if you want to iterate"},{"DocComment":" backwards, a good start is to know where the end is."},{"DocComment":""},{"DocComment":" When implementing an `ExactSizeIterator`, you must also implement"},{"DocComment":" [`Iterator`]. When doing so, the implementation of [`Iterator::size_hint`]"},{"DocComment":" *must* return the exact size of the iterator."},{"DocComment":""},{"DocComment":" The [`len`] method has a default implementation, so you usually shouldn't"},{"DocComment":" implement it. However, you may be able to provide a more performant"},{"DocComment":" implementation than the default, so overriding it in this case makes sense."},{"DocComment":""},{"DocComment":" Note that this trait is a safe trait and as such does *not* and *cannot*"},{"DocComment":" guarantee that the returned length is correct. This means that `unsafe`"},{"DocComment":" code **must not** rely on the correctness of [`Iterator::size_hint`]. The"},{"DocComment":" unstable and unsafe [`TrustedLen`](super::marker::TrustedLen) trait gives"},{"DocComment":" this additional guarantee."},{"DocComment":""},{"DocComment":" [`len`]: ExactSizeIterator::len"},{"DocComment":""},{"DocComment":" # When *shouldn't* an adapter be `ExactSizeIterator`?"},{"DocComment":""},{"DocComment":" If an adapter makes an iterator *longer*, then it's usually incorrect for"},{"DocComment":" that adapter to implement `ExactSizeIterator`. The inner exact-sized"},{"DocComment":" iterator might already be `usize::MAX`-long, and thus the length of the"},{"DocComment":" longer adapted iterator would no longer be exactly representable in `usize`."},{"DocComment":""},{"DocComment":" This is why [`Chain`](crate::iter::Chain) isn't `ExactSizeIterator`,"},{"DocComment":" even when `A` and `B` are both `ExactSizeIterator`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // a finite range knows exactly how many times it will iterate"},{"DocComment":" let five = 0..5;"},{"DocComment":""},{"DocComment":" assert_eq!(5, five.len());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" In the [module-level docs], we implemented an [`Iterator`], `Counter`."},{"DocComment":" Let's implement `ExactSizeIterator` for it as well:"},{"DocComment":""},{"DocComment":" [module-level docs]: crate::iter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # struct Counter {"},{"DocComment":" # count: usize,"},{"DocComment":" # }"},{"DocComment":" # impl Counter {"},{"DocComment":" # fn new() -> Counter {"},{"DocComment":" # Counter { count: 0 }"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" # impl Iterator for Counter {"},{"DocComment":" # type Item = usize;"},{"DocComment":" # fn next(&mut self) -> Option {"},{"DocComment":" # self.count += 1;"},{"DocComment":" # if self.count < 6 {"},{"DocComment":" # Some(self.count)"},{"DocComment":" # } else {"},{"DocComment":" # None"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" impl ExactSizeIterator for Counter {"},{"DocComment":" // We can easily calculate the remaining number of iterations."},{"DocComment":" fn len(&self) -> usize {"},{"DocComment":" 5 - self.count"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // And now we can use it!"},{"DocComment":""},{"DocComment":" let mut counter = Counter::new();"},{"DocComment":""},{"DocComment":" assert_eq!(5, counter.len());"},{"DocComment":" let _ = counter.next();"},{"DocComment":" assert_eq!(4, counter.len());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":69,"beg":{"line":86,"col":29},"end":{"line":86,"col":37}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[null,null],"vtable":{"id":{"Adt":77},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":38,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Ord",0]}],"span":{"data":{"file_id":33,"beg":{"line":982,"col":0},"end":{"line":982,"col":73}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for types that form a [total order](https://en.wikipedia.org/wiki/Total_order)."},{"DocComment":""},{"DocComment":" Implementations must be consistent with the [`PartialOrd`] implementation, and ensure `max`,"},{"DocComment":" `min`, and `clamp` are consistent with `cmp`:"},{"DocComment":""},{"DocComment":" - `partial_cmp(a, b) == Some(cmp(a, b))`."},{"DocComment":" - `max(a, b) == max_by(a, b, cmp)` (ensured by the default implementation)."},{"DocComment":" - `min(a, b) == min_by(a, b, cmp)` (ensured by the default implementation)."},{"DocComment":" - For `a.clamp(min, max)`, see the [method docs](#method.clamp) (ensured by the default"},{"DocComment":" implementation)."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Corollaries"},{"DocComment":""},{"DocComment":" From the above and the requirements of `PartialOrd`, it follows that for all `a`, `b` and `c`:"},{"DocComment":""},{"DocComment":" - exactly one of `a < b`, `a == b` or `a > b` is true; and"},{"DocComment":" - `<` is transitive: `a < b` and `b < c` implies `a < c`. The same must hold for both `==` and"},{"DocComment":" `>`."},{"DocComment":""},{"DocComment":" Mathematically speaking, the `<` operator defines a strict [weak order]. In cases where `==`"},{"DocComment":" conforms to mathematical equality, it also defines a strict [total order]."},{"DocComment":""},{"DocComment":" [weak order]: https://en.wikipedia.org/wiki/Weak_ordering"},{"DocComment":" [total order]: https://en.wikipedia.org/wiki/Total_order"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`."},{"DocComment":""},{"DocComment":" When `derive`d on structs, it will produce a"},{"DocComment":" [lexicographic](https://en.wikipedia.org/wiki/Lexicographic_order) ordering based on the"},{"DocComment":" top-to-bottom declaration order of the struct's members."},{"DocComment":""},{"DocComment":" When `derive`d on enums, variants are ordered primarily by their discriminants. Secondarily,"},{"DocComment":" they are ordered by their fields. By default, the discriminant is smallest for variants at the"},{"DocComment":" top, and largest for variants at the bottom. Here's an example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, Eq, PartialOrd, Ord)]"},{"DocComment":" enum E {"},{"DocComment":" Top,"},{"DocComment":" Bottom,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Top < E::Bottom);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, manually setting the discriminants can override this default behavior:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, Eq, PartialOrd, Ord)]"},{"DocComment":" enum E {"},{"DocComment":" Top = 2,"},{"DocComment":" Bottom = 1,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Bottom < E::Top);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Lexicographical comparison"},{"DocComment":""},{"DocComment":" Lexicographical comparison is an operation with the following properties:"},{"DocComment":" - Two sequences are compared element by element."},{"DocComment":" - The first mismatching element defines which sequence is lexicographically less or greater"},{"DocComment":" than the other."},{"DocComment":" - If one sequence is a prefix of another, the shorter sequence is lexicographically less than"},{"DocComment":" the other."},{"DocComment":" - If two sequences have equivalent elements and are of the same length, then the sequences are"},{"DocComment":" lexicographically equal."},{"DocComment":" - An empty sequence is lexicographically less than any non-empty sequence."},{"DocComment":" - Two empty sequences are lexicographically equal."},{"DocComment":""},{"DocComment":" ## How can I implement `Ord`?"},{"DocComment":""},{"DocComment":" `Ord` requires that the type also be [`PartialOrd`], [`PartialEq`], and [`Eq`]."},{"DocComment":""},{"DocComment":" Because `Ord` implies a stronger ordering relationship than [`PartialOrd`], and both `Ord` and"},{"DocComment":" [`PartialOrd`] must agree, you must choose how to implement `Ord` **first**. You can choose to"},{"DocComment":" derive it, or implement it manually. If you derive it, you should derive all four traits. If you"},{"DocComment":" implement it manually, you should manually implement all four traits, based on the"},{"DocComment":" implementation of `Ord`."},{"DocComment":""},{"DocComment":" Here's an example where you want to define the `Character` comparison by `health` and"},{"DocComment":" `experience` only, disregarding the field `mana`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" mana: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> Ordering {"},{"DocComment":" self.experience"},{"DocComment":" .cmp(&other.experience)"},{"DocComment":" .then(self.health.cmp(&other.health))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.health == other.health && self.experience == other.experience"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If all you need is to `slice::sort` a type by a field value, it can be simpler to use"},{"DocComment":" `slice::sort_by_key`."},{"DocComment":""},{"DocComment":" ## Examples of incorrect `Ord` implementations"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> std::cmp::Ordering {"},{"DocComment":" if self.health < other.health {"},{"DocComment":" Ordering::Less"},{"DocComment":" } else if self.health > other.health {"},{"DocComment":" Ordering::Greater"},{"DocComment":" } else {"},{"DocComment":" Ordering::Equal"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.health == other.health"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":""},{"DocComment":" let a = Character { health: 4.5 };"},{"DocComment":" let b = Character { health: f32::NAN };"},{"DocComment":""},{"DocComment":" // Mistake: floating-point values do not form a total order and using the built-in comparison"},{"DocComment":" // operands to implement `Ord` irregardless of that reality does not change it. Use"},{"DocComment":" // `f32::total_cmp` if you need a total order for floating-point values."},{"DocComment":""},{"DocComment":" // Reflexivity requirement of `Ord` is not given."},{"DocComment":" assert!(a == a);"},{"DocComment":" assert!(b != b);"},{"DocComment":""},{"DocComment":" // Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be"},{"DocComment":" // true, not both or neither."},{"DocComment":" assert_eq!((a < b) as u8 + (b < a) as u8, 0);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> std::cmp::Ordering {"},{"DocComment":" if self.health < 50 {"},{"DocComment":" self.health.cmp(&other.health)"},{"DocComment":" } else {"},{"DocComment":" self.experience.cmp(&other.experience)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // For performance reasons implementing `PartialEq` this way is not the idiomatic way, but it"},{"DocComment":" // ensures consistent behavior between `PartialEq`, `PartialOrd` and `Ord` in this example."},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.cmp(other) == Ordering::Equal"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":""},{"DocComment":" let a = Character {"},{"DocComment":" health: 3,"},{"DocComment":" experience: 5,"},{"DocComment":" };"},{"DocComment":" let b = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 77,"},{"DocComment":" };"},{"DocComment":" let c = Character {"},{"DocComment":" health: 143,"},{"DocComment":" experience: 2,"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // Mistake: The implementation of `Ord` compares different fields depending on the value of"},{"DocComment":" // `self.health`, the resulting order is not total."},{"DocComment":""},{"DocComment":" // Transitivity requirement of `Ord` is not given. If a is smaller than b and b is smaller than"},{"DocComment":" // c, by transitive property a must also be smaller than c."},{"DocComment":" assert!(a < b && b < c && c < a);"},{"DocComment":""},{"DocComment":" // Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be"},{"DocComment":" // true, not both or neither."},{"DocComment":" assert_eq!((a < c) as u8 + (c < a) as u8, 2);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The documentation of [`PartialOrd`] contains further examples, for example it's wrong for"},{"DocComment":" [`PartialOrd`] and [`PartialEq`] to disagree."},{"DocComment":""},{"DocComment":" [`cmp`]: Ord::cmp"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Ord"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":982,"col":21},"end":{"line":982,"col":31}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":33,"beg":{"line":982,"col":34},"end":{"line":982,"col":58}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"cmp","attr_info":{"attributes":[{"DocComment":" This method returns an [`Ordering`] between `self` and `other`."},{"DocComment":""},{"DocComment":" By convention, `self.cmp(&other)` returns the ordering matching the expression"},{"DocComment":" `self other` if true."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" assert_eq!(5.cmp(&10), Ordering::Less);"},{"DocComment":" assert_eq!(10.cmp(&5), Ordering::Greater);"},{"DocComment":" assert_eq!(5.cmp(&5), Ordering::Equal);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"Deduplicated":11822}],"output":{"Deduplicated":5641}},"item":{"id":377,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11865,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[38,0]}},null,null,null],"vtable":null},{"def_id":39,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Product",0]}],"span":{"data":{"file_id":43,"beg":{"line":40,"col":0},"end":{"line":40,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait to represent types that can be created by multiplying elements of an"},{"DocComment":" iterator."},{"DocComment":""},{"DocComment":" This trait is used to implement [`Iterator::product()`]. Types which implement"},{"DocComment":" this trait can be generated by using the [`product()`] method on an iterator."},{"DocComment":" Like [`FromIterator`], this trait should rarely be called directly."},{"DocComment":""},{"DocComment":" [`product()`]: Iterator::product"},{"DocComment":" [`FromIterator`]: iter::FromIterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":43,"beg":{"line":44,"col":18},"end":{"line":44,"col":36}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3181},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"product","attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by multiplying"},{"DocComment":" the items."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3182}],"output":{"Deduplicated":11751}},"item":{"id":381,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":3182}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11866,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":39,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]},{"Deduplicated":11850}]}}},"kind":{"TraitMethod":[39,0]}}],"vtable":null},{"def_id":40,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["PartialOrd",0]}],"span":{"data":{"file_id":33,"beg":{"line":1366,"col":0},"end":{"line":1367,"col":41}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for types that form a [partial order](https://en.wikipedia.org/wiki/Partial_order)."},{"DocComment":""},{"DocComment":" The `lt`, `le`, `gt`, and `ge` methods of this trait can be called using the `<`, `<=`, `>`, and"},{"DocComment":" `>=` operators, respectively."},{"DocComment":""},{"DocComment":" This trait should **only** contain the comparison logic for a type **if one plans on only"},{"DocComment":" implementing `PartialOrd` but not [`Ord`]**. Otherwise the comparison logic should be in [`Ord`]"},{"DocComment":" and this trait implemented with `Some(self.cmp(other))`."},{"DocComment":""},{"DocComment":" The methods of this trait must be consistent with each other and with those of [`PartialEq`]."},{"DocComment":" The following conditions must hold:"},{"DocComment":""},{"DocComment":" 1. `a == b` if and only if `partial_cmp(a, b) == Some(Equal)`."},{"DocComment":" 2. `a < b` if and only if `partial_cmp(a, b) == Some(Less)`"},{"DocComment":" 3. `a > b` if and only if `partial_cmp(a, b) == Some(Greater)`"},{"DocComment":" 4. `a <= b` if and only if `a < b || a == b`"},{"DocComment":" 5. `a >= b` if and only if `a > b || a == b`"},{"DocComment":" 6. `a != b` if and only if `!(a == b)`."},{"DocComment":""},{"DocComment":" Conditions 2–5 above are ensured by the default implementation. Condition 6 is already ensured"},{"DocComment":" by [`PartialEq`]."},{"DocComment":""},{"DocComment":" If [`Ord`] is also implemented for `Self` and `Rhs`, it must also be consistent with"},{"DocComment":" `partial_cmp` (see the documentation of that trait for the exact requirements). It's easy to"},{"DocComment":" accidentally make them disagree by deriving some of the traits and manually implementing others."},{"DocComment":""},{"DocComment":" The comparison relations must satisfy the following conditions (for all `a`, `b`, `c` of type"},{"DocComment":" `A`, `B`, `C`):"},{"DocComment":""},{"DocComment":" - **Transitivity**: if `A: PartialOrd` and `B: PartialOrd` and `A: PartialOrd`, then `a"},{"DocComment":" < b` and `b < c` implies `a < c`. The same must hold for both `==` and `>`. This must also"},{"DocComment":" work for longer chains, such as when `A: PartialOrd`, `B: PartialOrd`, `C:"},{"DocComment":" PartialOrd`, and `A: PartialOrd` all exist."},{"DocComment":" - **Duality**: if `A: PartialOrd` and `B: PartialOrd`, then `a < b` if and only if `b >"},{"DocComment":" a`."},{"DocComment":""},{"DocComment":" Note that the `B: PartialOrd` (dual) and `A: PartialOrd` (transitive) impls are not forced"},{"DocComment":" to exist, but these requirements apply whenever they do exist."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Cross-crate considerations"},{"DocComment":""},{"DocComment":" Upholding the requirements stated above can become tricky when one crate implements `PartialOrd`"},{"DocComment":" for a type of another crate (i.e., to allow comparing one of its own types with a type from the"},{"DocComment":" standard library). The recommendation is to never implement this trait for a foreign type. In"},{"DocComment":" other words, such a crate should do `impl PartialOrd for LocalType`, but it should"},{"DocComment":" *not* do `impl PartialOrd for ForeignType`."},{"DocComment":""},{"DocComment":" This avoids the problem of transitive chains that criss-cross crate boundaries: for all local"},{"DocComment":" types `T`, you may assume that no other crate will add `impl`s that allow comparing `T < U`. In"},{"DocComment":" other words, if other crates add `impl`s that allow building longer transitive chains `U1 < ..."},{"DocComment":" < T < V1 < ...`, then all the types that appear to the right of `T` must be types that the crate"},{"DocComment":" defining `T` already knows about. This rules out transitive chains where downstream crates can"},{"DocComment":" add new `impl`s that \"stitch together\" comparisons of foreign types in ways that violate"},{"DocComment":" transitivity."},{"DocComment":""},{"DocComment":" Not having such foreign `impl`s also avoids forward compatibility issues where one crate adding"},{"DocComment":" more `PartialOrd` implementations can cause build failures in downstream crates."},{"DocComment":""},{"DocComment":" ## Corollaries"},{"DocComment":""},{"DocComment":" The following corollaries follow from the above requirements:"},{"DocComment":""},{"DocComment":" - irreflexivity of `<` and `>`: `!(a < a)`, `!(a > a)`"},{"DocComment":" - transitivity of `>`: if `a > b` and `b > c` then `a > c`"},{"DocComment":" - duality of `partial_cmp`: `partial_cmp(a, b) == partial_cmp(b, a).map(Ordering::reverse)`"},{"DocComment":""},{"DocComment":" ## Strict and non-strict partial orders"},{"DocComment":""},{"DocComment":" The `<` and `>` operators behave according to a *strict* partial order. However, `<=` and `>=`"},{"DocComment":" do **not** behave according to a *non-strict* partial order. That is because mathematically, a"},{"DocComment":" non-strict partial order would require reflexivity, i.e. `a <= a` would need to be true for"},{"DocComment":" every `a`. This isn't always the case for types that implement `PartialOrd`, for example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let a = f64::NAN;"},{"DocComment":" assert_eq!(a <= a, false);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`."},{"DocComment":""},{"DocComment":" When `derive`d on structs, it will produce a"},{"DocComment":" [lexicographic](https://en.wikipedia.org/wiki/Lexicographic_order) ordering based on the"},{"DocComment":" top-to-bottom declaration order of the struct's members."},{"DocComment":""},{"DocComment":" When `derive`d on enums, variants are primarily ordered by their discriminants. Secondarily,"},{"DocComment":" they are ordered by their fields. By default, the discriminant is smallest for variants at the"},{"DocComment":" top, and largest for variants at the bottom. Here's an example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, PartialOrd)]"},{"DocComment":" enum E {"},{"DocComment":" Top,"},{"DocComment":" Bottom,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Top < E::Bottom);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, manually setting the discriminants can override this default behavior:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, PartialOrd)]"},{"DocComment":" enum E {"},{"DocComment":" Top = 2,"},{"DocComment":" Bottom = 1,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Bottom < E::Top);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I implement `PartialOrd`?"},{"DocComment":""},{"DocComment":" `PartialOrd` only requires implementation of the [`partial_cmp`] method, with the others"},{"DocComment":" generated from default implementations."},{"DocComment":""},{"DocComment":" However it remains possible to implement the others separately for types which do not have a"},{"DocComment":" total order. For example, for floating point numbers, `NaN < 0 == false` and `NaN >= 0 == false`"},{"DocComment":" (cf. IEEE 754-2008 section 5.11)."},{"DocComment":""},{"DocComment":" `PartialOrd` requires your type to be [`PartialEq`]."},{"DocComment":""},{"DocComment":" If your type is [`Ord`], you can implement [`partial_cmp`] by using [`cmp`]:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" height: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Person {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Person {"},{"DocComment":" fn cmp(&self, other: &Self) -> Ordering {"},{"DocComment":" self.height.cmp(&other.height)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Person {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.height == other.height"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Person {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You may also find it useful to use [`partial_cmp`] on your type's fields. Here is an example of"},{"DocComment":" `Person` types who have a floating-point `height` field that is the only field to be used for"},{"DocComment":" sorting:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" height: f64,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Person {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" self.height.partial_cmp(&other.height)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Person {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.height == other.height"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Examples of incorrect `PartialOrd` implementations"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(PartialEq, Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.health.cmp(&other.health))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let a = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 5,"},{"DocComment":" };"},{"DocComment":" let b = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 77,"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // Mistake: `PartialEq` and `PartialOrd` disagree with each other."},{"DocComment":""},{"DocComment":" assert_eq!(a.partial_cmp(&b).unwrap(), Ordering::Equal); // a == b according to `PartialOrd`."},{"DocComment":" assert_ne!(a, b); // a != b according to `PartialEq`."},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: u32 = 0;"},{"DocComment":" let y: u32 = 1;"},{"DocComment":""},{"DocComment":" assert_eq!(x < y, true);"},{"DocComment":" assert_eq!(x.lt(&y), true);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`partial_cmp`]: PartialOrd::partial_cmp"},{"DocComment":" [`cmp`]: Ord::cmp"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"partial_ord"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":1367,"col":4},"end":{"line":1367,"col":26}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"partial_cmp","attr_info":{"attributes":[{"DocComment":" This method returns an ordering between `self` and `other` values if one exists."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&2.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Less));"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Equal));"},{"DocComment":""},{"DocComment":" let result = 2.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Greater));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" When comparison is impossible:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let result = f64::NAN.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, None);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"Deduplicated":11827}],"output":{"Deduplicated":5935}},"item":{"id":382,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11867,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[40,0]}},null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":78},"generics":{"regions":[],"types":[{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}},{"def_id":41,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["zip",0]},{"Ident":["TrustedRandomAccessNoCoerce",0]}],"span":{"data":{"file_id":49,"beg":{"line":576,"col":0},"end":{"line":576,"col":51}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Like [`TrustedRandomAccess`] but without any of the requirements / guarantees around"},{"DocComment":" coercions to supertypes after `__iterator_get_unchecked` (they aren’t allowed here!), and"},{"DocComment":" without the requirement that subtypes / supertypes implement `TrustedRandomAccessNoCoerce`."},{"DocComment":""},{"DocComment":" This trait was created in PR #85874 to fix soundness issue #85873 without performance regressions."},{"DocComment":" It is subject to change as we might want to build a more generally useful (for performance"},{"DocComment":" optimizations) and more sophisticated trait or trait hierarchy that replaces or extends"},{"DocComment":" [`TrustedRandomAccess`] and `TrustedRandomAccessNoCoerce`."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[{"name":"MAY_HAVE_SIDE_EFFECT","attr_info":{"attributes":[{"DocComment":" `true` if getting an iterator element may have side effects."},{"DocComment":" Remember to take inner iterators into account."}],"inline":null,"rename":null,"public":true},"ty":{"Deduplicated":1797},"default":null}],"types":[],"methods":[null],"vtable":null},{"def_id":42,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["index",0]},{"Ident":["Index",0]}],"span":{"data":{"file_id":75,"beg":{"line":59,"col":0},"end":{"line":59,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used for indexing operations (`container[index]`) in immutable contexts."},{"DocComment":""},{"DocComment":" `container[index]` is actually syntactic sugar for `*container.index(index)`,"},{"DocComment":" but only when used as an immutable value. If a mutable value is requested,"},{"DocComment":" [`IndexMut`] is used instead. This allows nice things such as"},{"DocComment":" `let value = v[index]` if the type of `value` implements [`Copy`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" The following example implements `Index` on a read-only `NucleotideCount`"},{"DocComment":" container, enabling individual counts to be retrieved with index syntax."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Index;"},{"DocComment":""},{"DocComment":" enum Nucleotide {"},{"DocComment":" A,"},{"DocComment":" C,"},{"DocComment":" G,"},{"DocComment":" T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct NucleotideCount {"},{"DocComment":" a: usize,"},{"DocComment":" c: usize,"},{"DocComment":" g: usize,"},{"DocComment":" t: usize,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Index for NucleotideCount {"},{"DocComment":" type Output = usize;"},{"DocComment":""},{"DocComment":" fn index(&self, nucleotide: Nucleotide) -> &Self::Output {"},{"DocComment":" match nucleotide {"},{"DocComment":" Nucleotide::A => &self.a,"},{"DocComment":" Nucleotide::C => &self.c,"},{"DocComment":" Nucleotide::G => &self.g,"},{"DocComment":" Nucleotide::T => &self.t,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let nucleotide_count = NucleotideCount {a: 14, c: 9, g: 10, t: 12};"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::A], 14);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::C], 9);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::G], 10);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::T], 12);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"index"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Idx"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"index","attr_info":{"attributes":[{"DocComment":" Performs the indexing (`container[index]`) operation."},{"DocComment":""},{"DocComment":" # Panics"},{"DocComment":""},{"DocComment":" May panic if the index is out of bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821},{"Deduplicated":11752}],"output":{"HashConsedValue":[11868,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11789},"Shared"]}]}},"item":{"id":402,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11869,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":42,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[42,0]}}],"vtable":{"id":{"Adt":79},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":43,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["index",0]},{"Ident":["IndexMut",0]}],"span":{"data":{"file_id":75,"beg":{"line":170,"col":0},"end":{"line":170,"col":57}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used for indexing operations (`container[index]`) in mutable contexts."},{"DocComment":""},{"DocComment":" `container[index]` is actually syntactic sugar for"},{"DocComment":" `*container.index_mut(index)`, but only when used as a mutable value. If"},{"DocComment":" an immutable value is requested, the [`Index`] trait is used instead. This"},{"DocComment":" allows nice things such as `v[index] = value`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" A very simple implementation of a `Balance` struct that has two sides, where"},{"DocComment":" each can be indexed mutably and immutably."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::{Index, IndexMut};"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" enum Side {"},{"DocComment":" Left,"},{"DocComment":" Right,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" enum Weight {"},{"DocComment":" Kilogram(f32),"},{"DocComment":" Pound(f32),"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Balance {"},{"DocComment":" pub left: Weight,"},{"DocComment":" pub right: Weight,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Index for Balance {"},{"DocComment":" type Output = Weight;"},{"DocComment":""},{"DocComment":" fn index(&self, index: Side) -> &Self::Output {"},{"DocComment":" println!(\"Accessing {index:?}-side of balance immutably\");"},{"DocComment":" match index {"},{"DocComment":" Side::Left => &self.left,"},{"DocComment":" Side::Right => &self.right,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl IndexMut for Balance {"},{"DocComment":" fn index_mut(&mut self, index: Side) -> &mut Self::Output {"},{"DocComment":" println!(\"Accessing {index:?}-side of balance mutably\");"},{"DocComment":" match index {"},{"DocComment":" Side::Left => &mut self.left,"},{"DocComment":" Side::Right => &mut self.right,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut balance = Balance {"},{"DocComment":" right: Weight::Kilogram(2.5),"},{"DocComment":" left: Weight::Pound(1.5),"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // In this case, `balance[Side::Right]` is sugar for"},{"DocComment":" // `*balance.index(Side::Right)`, since we are only *reading*"},{"DocComment":" // `balance[Side::Right]`, not writing it."},{"DocComment":" assert_eq!(balance[Side::Right], Weight::Kilogram(2.5));"},{"DocComment":""},{"DocComment":" // However, in this case `balance[Side::Left]` is sugar for"},{"DocComment":" // `*balance.index_mut(Side::Left)`, since we are writing"},{"DocComment":" // `balance[Side::Left]`."},{"DocComment":" balance[Side::Left] = Weight::Kilogram(3.0);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"index_mut"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Idx"},{"index":2,"name":"Self_Clause0_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":75,"beg":{"line":170,"col":39},"end":{"line":170,"col":57}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":42,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"index_mut","attr_info":{"attributes":[{"DocComment":" Performs the mutable indexing (`container[index]`) operation."},{"DocComment":""},{"DocComment":" # Panics"},{"DocComment":""},{"DocComment":" May panic if the index is out of bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11818},{"Deduplicated":11752}],"output":{"HashConsedValue":[11870,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11789},"Mut"]}]}},"item":{"id":403,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11871,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":43,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[43,0]}}],"vtable":{"id":{"Adt":80},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":44,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["FromResidual",0]}],"span":{"data":{"file_id":67,"beg":{"line":310,"col":0},"end":{"line":310,"col":57}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used to specify which residuals can be converted into which [`crate::ops::Try`] types."},{"DocComment":""},{"DocComment":" Every `Try` type needs to be recreatable from its own associated"},{"DocComment":" `Residual` type, but can also have additional `FromResidual` implementations"},{"DocComment":" to support interconversion with other `Try` types."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"FromResidual"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"R"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_residual","attr_info":{"attributes":[{"DocComment":" Constructs the type from a compatible `Residual` type."},{"DocComment":""},{"DocComment":" This should be implemented consistently with the `branch` method such"},{"DocComment":" that applying the `?` operator will get back an equivalent residual:"},{"DocComment":" `FromResidual::from_residual(r).branch() --> ControlFlow::Break(r)`."},{"DocComment":" (The residual is not mandated to be *identical* when interconversion is involved.)"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::{ControlFlow, FromResidual};"},{"DocComment":""},{"DocComment":" assert_eq!(Result::::from_residual(Err(3_u8)), Err(3));"},{"DocComment":" assert_eq!(Option::::from_residual(None), None);"},{"DocComment":" assert_eq!("},{"DocComment":" ControlFlow::<_, String>::from_residual(ControlFlow::Break(5)),"},{"DocComment":" ControlFlow::Break(5),"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11752}],"output":{"Deduplicated":11751}},"item":{"id":412,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11872,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":44,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[44,0]}}],"vtable":null},{"def_id":45,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["slice",0]},{"Ident":["index",0]},{"Ident":["SliceIndex",0]}],"span":{"data":{"file_id":77,"beg":{"line":163,"col":0},"end":{"line":163,"col":73}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A helper trait used for indexing operations."},{"DocComment":""},{"DocComment":" Implementations of this trait have to promise that if the argument"},{"DocComment":" to `get_unchecked(_mut)` is a safe reference, then so is the 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Returns a mutable reference to the output at this location, if in"},{"DocComment":" bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"HashConsedValue":[11875,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11752},"Mut"]}]}],"output":{"HashConsedValue":[11876,{"Adt":{"id":{"Adt":12},"generics":{"regions":[],"types":[{"Deduplicated":11870}],"const_generics":[],"trait_refs":[]}}}]}},"item":{"id":415,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"Deduplicated":11874}]}}},"kind":{"TraitMethod":[45,1]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"get_unchecked","attr_info":{"attributes":[{"DocComment":" Returns a pointer to the output at this location, 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https://doc.rust-lang.org/reference/behavior-considered-undefined.html"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":true,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11877}],"output":{"Deduplicated":11878}},"item":{"id":416,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"Deduplicated":11874}]}}},"kind":{"TraitMethod":[45,2]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"get_unchecked_mut","attr_info":{"attributes":[{"DocComment":" Returns a mutable pointer to the output at this location, without"},{"DocComment":" performing any bounds checking."},{"DocComment":""},{"DocComment":" Calling this method with an out-of-bounds index or a dangling `slice` pointer"},{"DocComment":" is *[undefined behavior]* even if the resulting pointer is 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A value-to-value conversion that consumes the input value. The"},{"DocComment":" opposite of [`From`]."},{"DocComment":""},{"DocComment":" One should avoid implementing [`Into`] and implement [`From`] instead."},{"DocComment":" Implementing [`From`] automatically provides one with an implementation of [`Into`]"},{"DocComment":" thanks to the blanket implementation in the standard library."},{"DocComment":""},{"DocComment":" Prefer using [`Into`] over [`From`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`Into`] can be used as well."},{"DocComment":""},{"DocComment":" **Note: This trait must not fail**. If the conversion can fail, use [`TryInto`]."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - [`From`]` for U` implies `Into for T`"},{"DocComment":" - [`Into`] is reflexive, which means that `Into for T` is implemented"},{"DocComment":""},{"DocComment":" # Implementing [`Into`] for conversions to external types in old versions of Rust"},{"DocComment":""},{"DocComment":" Prior to Rust 1.41, if the destination type was not part of the current crate"},{"DocComment":" then you couldn't implement [`From`] directly."},{"DocComment":" For example, take this code:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(non_local_definitions)]"},{"DocComment":" struct Wrapper(Vec);"},{"DocComment":" impl From> for Vec {"},{"DocComment":" fn from(w: Wrapper) -> Vec {"},{"DocComment":" w.0"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":" This will fail to compile in older versions of the language because Rust's orphaning rules"},{"DocComment":" used to be a little bit more strict. To bypass this, you could implement [`Into`] directly:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct Wrapper(Vec);"},{"DocComment":" impl Into> for Wrapper {"},{"DocComment":" fn into(self) -> Vec {"},{"DocComment":" self.0"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It is important to understand that [`Into`] does not provide a [`From`] implementation"},{"DocComment":" (as [`From`] does with [`Into`]). Therefore, you should always try to implement [`From`]"},{"DocComment":" and then fall back to [`Into`] if [`From`] can't be implemented."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" [`String`] implements [`Into`]`<`[`Vec`]`<`[`u8`]`>>`:"},{"DocComment":""},{"DocComment":" In order to express that we want a generic function to take all arguments that can be"},{"DocComment":" converted to a specified type `T`, we can use a trait bound of [`Into`]``."},{"DocComment":" For example: The function `is_hello` takes all arguments that can be converted into a"},{"DocComment":" [`Vec`]`<`[`u8`]`>`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn is_hello>>(s: T) {"},{"DocComment":" let bytes = b\"hello\".to_vec();"},{"DocComment":" assert_eq!(bytes, s.into());"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let s = \"hello\".to_string();"},{"DocComment":" is_hello(s);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`Vec`]: ../../std/vec/struct.Vec.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Into"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"into","attr_info":{"attributes":[{"DocComment":" Converts this type into the (usually inferred) input type."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751}],"output":{"Deduplicated":11752}},"item":{"id":465,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11881,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":48,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[48,0]}}],"vtable":null},{"def_id":49,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["bit",0]},{"Ident":["BitOr",0]}],"span":{"data":{"file_id":84,"beg":{"line":254,"col":0},"end":{"line":254,"col":33}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The bitwise OR operator `|`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `BitOr` for a wrapper around `bool`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitOr;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Scalar(bool);"},{"DocComment":""},{"DocComment":" impl BitOr for Scalar {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" // rhs is the \"right-hand side\" of the expression `a | b`"},{"DocComment":" fn bitor(self, rhs: Self) -> Self::Output {"},{"DocComment":" Self(self.0 | rhs.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Scalar(true) | Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(true) | Scalar(false), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(false) | Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(false) | Scalar(false), Scalar(false));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" An implementation of `BitOr` for a wrapper around `Vec`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitOr;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct BooleanVector(Vec);"},{"DocComment":""},{"DocComment":" impl BitOr for BooleanVector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn bitor(self, Self(rhs): Self) -> Self::Output {"},{"DocComment":" let Self(lhs) = self;"},{"DocComment":" assert_eq!(lhs.len(), rhs.len());"},{"DocComment":" Self("},{"DocComment":" lhs.iter()"},{"DocComment":" .zip(rhs.iter())"},{"DocComment":" .map(|(x, y)| *x | *y)"},{"DocComment":" .collect()"},{"DocComment":" )"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let bv1 = BooleanVector(vec![true, true, false, false]);"},{"DocComment":" let bv2 = BooleanVector(vec![true, false, true, false]);"},{"DocComment":" let expected = BooleanVector(vec![true, true, true, false]);"},{"DocComment":" assert_eq!(bv1 | bv2, expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"bitor"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"bitor","attr_info":{"attributes":[{"DocComment":" Performs the `|` operation."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(true | false, true);"},{"DocComment":" assert_eq!(false | false, false);"},{"DocComment":" assert_eq!(5u8 | 1u8, 5);"},{"DocComment":" assert_eq!(5u8 | 2u8, 7);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":466,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11882,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":49,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[49,0]}}],"vtable":{"id":{"Adt":85},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":50,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["bit",0]},{"Ident":["BitAnd",0]}],"span":{"data":{"file_id":84,"beg":{"line":150,"col":0},"end":{"line":150,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The bitwise AND operator `&`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `BitAnd` for a wrapper around `bool`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitAnd;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Scalar(bool);"},{"DocComment":""},{"DocComment":" impl BitAnd for Scalar {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" // rhs is the \"right-hand side\" of the expression `a & b`"},{"DocComment":" fn bitand(self, rhs: Self) -> Self::Output {"},{"DocComment":" Self(self.0 & rhs.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Scalar(true) & Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(true) & Scalar(false), Scalar(false));"},{"DocComment":" assert_eq!(Scalar(false) & Scalar(true), Scalar(false));"},{"DocComment":" assert_eq!(Scalar(false) & Scalar(false), Scalar(false));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" An implementation of `BitAnd` for a wrapper around `Vec`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitAnd;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct BooleanVector(Vec);"},{"DocComment":""},{"DocComment":" impl BitAnd for BooleanVector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn bitand(self, Self(rhs): Self) -> Self::Output {"},{"DocComment":" let Self(lhs) = self;"},{"DocComment":" assert_eq!(lhs.len(), rhs.len());"},{"DocComment":" Self("},{"DocComment":" lhs.iter()"},{"DocComment":" .zip(rhs.iter())"},{"DocComment":" .map(|(x, y)| *x & *y)"},{"DocComment":" .collect()"},{"DocComment":" )"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let bv1 = BooleanVector(vec![true, true, false, false]);"},{"DocComment":" let bv2 = BooleanVector(vec![true, false, true, false]);"},{"DocComment":" let expected = BooleanVector(vec![true, false, false, false]);"},{"DocComment":" assert_eq!(bv1 & bv2, expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"bitand"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"bitand","attr_info":{"attributes":[{"DocComment":" Performs the `&` operation."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(true & false, false);"},{"DocComment":" assert_eq!(true & true, true);"},{"DocComment":" assert_eq!(5u8 & 1u8, 1);"},{"DocComment":" assert_eq!(5u8 & 2u8, 0);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11751},{"Deduplicated":11752}],"output":{"Deduplicated":11789}},"item":{"id":469,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11883,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":50,"generics":{"regions":[],"types":[{"Deduplicated":11751},{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[50,0]}}],"vtable":{"id":{"Adt":86},"generics":{"regions":[],"types":[{"Deduplicated":11752},{"Deduplicated":11789}],"const_generics":[],"trait_refs":[]}}},{"def_id":51,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["IsIdentity",0]}],"span":{"data":{"file_id":38,"beg":{"line":39,"col":0},"end":{"line":42,"col":1}},"generated_from_span":null},"source_text":"pub trait IsIdentity {\n /// Return true if this element is the identity element of the curve.\n fn is_identity(&self) -> bool;\n}","attr_info":{"attributes":[{"DocComment":" Trait for testing if a curve point is equivalent to the identity point."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"is_identity","attr_info":{"attributes":[{"DocComment":" Return true if this element is the identity element of the curve."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11821}],"output":{"Deduplicated":1797}},"item":{"id":485,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11884,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":51,"generics":{"regions":[],"types":[{"Deduplicated":11751}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[51,0]}}],"vtable":{"id":{"Adt":88},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}],"trait_impls":[{"def_id":0,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":0}}],"span":{"data":{"file_id":8,"beg":{"line":52,"col":0},"end":{"line":52,"col":27}},"generated_from_span":null},"source_text":"impl Eq for FieldElement {}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[{"HashConsedValue":[11754,{"kind":{"TraitImpl":{"id":1,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":3141},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}}}}]}],"consts":[],"types":[],"methods":[null,{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":499,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[3,1]}}],"vtable":null},{"def_id":1,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":1}}],"span":{"data":{"file_id":8,"beg":{"line":54,"col":0},"end":{"line":58,"col":1}},"generated_from_span":null},"source_text":"impl PartialEq for FieldElement {\n fn eq(&self, other: &FieldElement) -> bool {\n self.ct_eq(other).into()\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":3141},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":42,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[4,0]}},null],"vtable":{"id":9,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":2,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":2}}],"span":{"data":{"file_id":8,"beg":{"line":60,"col":0},"end":{"line":67,"col":1}},"generated_from_span":null},"source_text":"impl ConstantTimeEq for FieldElement {\n /// Test equality between two `FieldElement`s. Since the\n /// internal representation is not canonical, the field elements\n /// are normalized to wire format before comparison.\n fn ct_eq(&self, other: &FieldElement) -> Choice {\n self.to_bytes().ct_eq(&other.to_bytes())\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":5,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":45,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[5,0]}},null],"vtable":null},{"def_id":3,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":3}}],"span":{"data":{"file_id":8,"beg":{"line":69,"col":0},"end":{"line":73,"col":1}},"generated_from_span":null},"source_text":"impl Default for FieldElement {\n fn default() -> Self {\n FieldElement::ZERO\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":6,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":48,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[6,0]}}],"vtable":null},{"def_id":4,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":4}}],"span":{"data":{"file_id":9,"beg":{"line":42,"col":9},"end":{"line":42,"col":13}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[{"HashConsedValue":[11755,{"kind":{"TraitImpl":{"id":6,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}}}}]}],"consts":[],"types":[],"methods":[],"vtable":null},null,{"def_id":6,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":6}}],"span":{"data":{"file_id":9,"beg":{"line":42,"col":15},"end":{"line":42,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":57,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[8,0]}},null],"vtable":null},{"def_id":7,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":7}}],"span":{"data":{"file_id":9,"beg":{"line":45,"col":0},"end":{"line":49,"col":1}},"generated_from_span":null},"source_text":"impl Debug for FieldElement51 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"FieldElement51({:?})\", &self.0[..])\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":11,"generics":{"regions":[],"types":[{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"},{"index":2,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":60,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}},{"Var":{"Bound":[0,2]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[11,0]}}],"vtable":{"id":10,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":8,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":8}}],"span":{"data":{"file_id":9,"beg":{"line":54,"col":0},"end":{"line":60,"col":1}},"generated_from_span":null},"source_text":"impl<'a> AddAssign<&'a FieldElement51> for FieldElement51 {\n fn add_assign(&mut self, _rhs: &'a FieldElement51) {\n for i in 0..5 {\n self.0[i] += _rhs.0[i];\n }\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":12,"generics":{"regions":[],"types":[{"Deduplicated":3141},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":61,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[12,0]}}],"vtable":{"id":11,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":9,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":9}}],"span":{"data":{"file_id":9,"beg":{"line":62,"col":0},"end":{"line":69,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Add<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn add(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n let mut output = *self;\n output += _rhs;\n output\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":13,"generics":{"regions":[],"types":[{"Deduplicated":11757},{"Deduplicated":11756},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":62,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[13,0]}}],"vtable":{"id":12,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":10,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":10}}],"span":{"data":{"file_id":9,"beg":{"line":71,"col":0},"end":{"line":76,"col":1}},"generated_from_span":null},"source_text":"impl<'a> SubAssign<&'a FieldElement51> for FieldElement51 {\n fn sub_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) - _rhs;\n self.0 = result.0;\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":14,"generics":{"regions":[],"types":[{"Deduplicated":3141},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":63,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[14,0]}}],"vtable":{"id":13,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":11,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":11}}],"span":{"data":{"file_id":9,"beg":{"line":78,"col":0},"end":{"line":98,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Sub<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn sub(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n // To avoid underflow, first add a multiple of p.\n // Choose 16*p = p << 4 to be larger than 54-bit _rhs.\n //\n // If we could statically track the bitlengths of the limbs\n // of every FieldElement51, we could choose a multiple of p\n // just bigger than _rhs and avoid having to do a reduction.\n //\n // Since we don't yet have type-level integers to do this, we\n // have to add an explicit reduction call here.\n FieldElement51::reduce([\n (self.0[0] + 36028797018963664u64) - _rhs.0[0],\n (self.0[1] + 36028797018963952u64) - _rhs.0[1],\n (self.0[2] + 36028797018963952u64) - _rhs.0[2],\n (self.0[3] + 36028797018963952u64) - _rhs.0[3],\n (self.0[4] + 36028797018963952u64) - _rhs.0[4],\n ])\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":15,"generics":{"regions":[],"types":[{"Deduplicated":11757},{"Deduplicated":11756},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":64,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[15,0]}}],"vtable":{"id":14,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":12,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":12}}],"span":{"data":{"file_id":9,"beg":{"line":100,"col":0},"end":{"line":105,"col":1}},"generated_from_span":null},"source_text":"impl<'a> MulAssign<&'a FieldElement51> for FieldElement51 {\n fn mul_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) * _rhs;\n self.0 = result.0;\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":16,"generics":{"regions":[],"types":[{"Deduplicated":3141},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":65,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[16,0]}}],"vtable":{"id":15,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":13,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":13}}],"span":{"data":{"file_id":9,"beg":{"line":107,"col":0},"end":{"line":210,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Mul<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n\n #[rustfmt::skip] // keep alignment of c* calculations\n fn mul(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n /// Helper function to multiply two 64-bit integers with 128\n /// bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }\n\n // Alias self, _rhs for more readable formulas\n let a: &[u64; 5] = &self.0;\n let b: &[u64; 5] = &_rhs.0;\n\n // Precondition: assume input limbs a[i], b[i] are bounded as\n //\n // a[i], b[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // 64-bit precomputations to avoid 128-bit multiplications.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let b1_19 = b[1] * 19;\n let b2_19 = b[2] * 19;\n let b3_19 = b[3] * 19;\n let b4_19 = b[4] * 19;\n\n // Multiply to get 128-bit coefficients of output\n let c0: u128 = m(a[0], b[0]) + m(a[4], b1_19) + m(a[3], b2_19) + m(a[2], b3_19) + m(a[1], b4_19);\n let mut c1: u128 = m(a[1], b[0]) + m(a[0], b[1]) + m(a[4], b2_19) + m(a[3], b3_19) + m(a[2], b4_19);\n let mut c2: u128 = m(a[2], b[0]) + m(a[1], b[1]) + m(a[0], b[2]) + m(a[4], b3_19) + m(a[3], b4_19);\n let mut c3: u128 = m(a[3], b[0]) + m(a[2], b[1]) + m(a[1], b[2]) + m(a[0], b[3]) + m(a[4], b4_19);\n let mut c4: u128 = m(a[4], b[0]) + m(a[3], b[1]) + m(a[2], b[2]) + m(a[1], b[3]) + m(a[0] , b[4]);\n\n // How big are the c[i]? We have\n //\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));\n\n // Casting to u64 and back tells the compiler that the carry is\n // bounded by 2^64, so that the addition is a u128 + u64 rather\n // than u128 + u128.\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n let mut out = [0u64; 5];\n\n c1 += ((c0 >> 51) as u64) as u128;\n out[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n out[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n out[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n out[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n out[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need out[0] + carry * 19 < 2^64.\n //\n // c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)\n // < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n out[0] += carry * 19;\n\n // Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n out[1] += out[0] >> 51;\n out[0] &= LOW_51_BIT_MASK;\n\n // Now out[i] < 2^(51 + epsilon) for all i.\n FieldElement51(out)\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":17,"generics":{"regions":[],"types":[{"Deduplicated":11757},{"Deduplicated":11756},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":66,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[17,0]}}],"vtable":{"id":16,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":14,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":14}}],"span":{"data":{"file_id":9,"beg":{"line":212,"col":0},"end":{"line":219,"col":1}},"generated_from_span":null},"source_text":"impl Neg for &FieldElement51 {\n type Output = FieldElement51;\n fn neg(self) -> FieldElement51 {\n let mut output = *self;\n output.negate();\n output\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":18,"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":3141}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":67,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[18,0]}}],"vtable":{"id":17,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":15,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":15}}],"span":{"data":{"file_id":9,"beg":{"line":221,"col":0},"end":{"line":251,"col":1}},"generated_from_span":null},"source_text":"impl ConditionallySelectable for FieldElement51 {\n fn conditional_select(\n a: &FieldElement51,\n b: &FieldElement51,\n choice: Choice,\n ) -> FieldElement51 {\n FieldElement51([\n u64::conditional_select(&a.0[0], &b.0[0], choice),\n u64::conditional_select(&a.0[1], &b.0[1], choice),\n u64::conditional_select(&a.0[2], &b.0[2], choice),\n u64::conditional_select(&a.0[3], &b.0[3], choice),\n u64::conditional_select(&a.0[4], &b.0[4], choice),\n ])\n }\n\n fn conditional_swap(a: &mut FieldElement51, b: &mut FieldElement51, choice: Choice) {\n u64::conditional_swap(&mut a.0[0], &mut b.0[0], choice);\n u64::conditional_swap(&mut a.0[1], &mut b.0[1], choice);\n u64::conditional_swap(&mut a.0[2], &mut b.0[2], choice);\n u64::conditional_swap(&mut a.0[3], &mut b.0[3], choice);\n u64::conditional_swap(&mut a.0[4], &mut b.0[4], choice);\n }\n\n fn conditional_assign(&mut self, other: &FieldElement51, choice: Choice) {\n self.0[0].conditional_assign(&other.0[0], choice);\n self.0[1].conditional_assign(&other.0[1], choice);\n self.0[2].conditional_assign(&other.0[2], choice);\n self.0[3].conditional_assign(&other.0[3], choice);\n self.0[4].conditional_assign(&other.0[4], choice);\n 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ConditionallySelectable for ProjectiveNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::conditional_select(&a.Y_plus_X, &b.Y_plus_X, choice),\n Y_minus_X: FieldElement::conditional_select(&a.Y_minus_X, &b.Y_minus_X, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T2d: FieldElement::conditional_select(&a.T2d, &b.T2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.Y_plus_X.conditional_assign(&other.Y_plus_X, choice);\n self.Y_minus_X.conditional_assign(&other.Y_minus_X, choice);\n self.Z.conditional_assign(&other.Z, choice);\n self.T2d.conditional_assign(&other.T2d, choice);\n 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Add<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.Y_plus_X;\n let MM = &Y_minus_X * &other.Y_minus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &ZZ2 + &TT2d,\n T: &ZZ2 - &TT2d,\n }\n 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Sub<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.Y_minus_X;\n let MP = &Y_minus_X * &other.Y_plus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &ZZ2 - &TT2d,\n T: &ZZ2 + &TT2d,\n }\n 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Add<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.y_plus_x;\n let MM = &Y_minus_X * &other.y_minus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &Z2 + &Txy2d,\n T: &Z2 - &Txy2d,\n }\n 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Sub<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.y_minus_x;\n let MP = &Y_minus_X * &other.y_plus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &Z2 - &Txy2d,\n T: &Z2 + &Txy2d,\n }\n 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Neg for &ProjectiveNielsPoint {\n type Output = ProjectiveNielsPoint;\n\n fn neg(self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: self.Y_minus_X,\n Y_minus_X: self.Y_plus_X,\n Z: self.Z,\n T2d: -(&self.T2d),\n }\n 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Neg for &AffineNielsPoint {\n type Output = AffineNielsPoint;\n\n fn neg(self) -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: self.y_minus_x,\n y_minus_x: self.y_plus_x,\n xy2d: -(&self.xy2d),\n }\n 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Arithmetic mod \\\\(2\\^{252} + 27742317777372353535851937790883648493\\\\)\n//! with five \\\\(52\\\\)-bit unsigned limbs.\n//!\n//! \\\\(51\\\\)-bit limbs would cover the desired bit range (\\\\(253\\\\)\n//! bits), but isn't large enough to reduce a \\\\(512\\\\)-bit number with\n//! Montgomery multiplication, so \\\\(52\\\\) bits is used instead. To see\n//! that this is safe for intermediate results, note that the largest\n//! limb in a \\\\(5\\times 5\\\\) product of \\\\(52\\\\)-bit limbs will be\n//!\n//! ```text\n//! (0xfffffffffffff^2) * 5 = 0x4ffffffffffff60000000000005 (107 bits).\n//! ```\n\nuse core::fmt::Debug;\nuse core::ops::{Index, IndexMut};\nuse subtle::{Choice, ConditionallySelectable};\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::{DefaultIsZeroes, Zeroize};\n\nuse crate::constants;\n\n/// The `Scalar52` struct represents an element in\n/// \\\\(\\mathbb Z / \\ell \\mathbb Z\\\\) as 5 \\\\(52\\\\)-bit limbs.\n#[derive(Copy, Clone, Default)]\npub struct Scalar52(pub [u64; 5]);\n\nimpl Debug for Scalar52 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n f.write_str(\"Scalar52{..}\")\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for Scalar52 {}\n\nimpl Index for Scalar52 {\n type Output = u64;\n fn index(&self, _index: usize) -> &u64 {\n &(self.0[_index])\n }\n}\n\nimpl IndexMut for Scalar52 {\n fn index_mut(&mut self, _index: usize) -> &mut u64 {\n &mut (self.0[_index])\n }\n}\n\n/// u64 * u64 = u128 multiply helper\n#[inline(always)]\nfn m(x: u64, y: u64) -> u128 {\n (x as u128) * (y as u128)\n}\n\nimpl Scalar52 {\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: Scalar52 = Scalar52([0, 0, 0, 0, 0]);\n\n /// Unpack a 32 byte / 256 bit scalar into 5 52-bit limbs.\n #[rustfmt::skip] // keep alignment of s[*] calculations\n pub fn from_bytes(bytes: &[u8; 32]) -> Scalar52 {\n let mut words = [0u64; 4];\n for i in 0..4 {\n for j in 0..8 {\n words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);\n }\n }\n\n let mask = (1u64 << 52) - 1;\n let top_mask = (1u64 << 48) - 1;\n let mut s = Scalar52::ZERO;\n\n s[0] = words[0] & mask;\n s[1] = ((words[0] >> 52) | (words[1] << 12)) & mask;\n s[2] = ((words[1] >> 40) | (words[2] << 24)) & mask;\n s[3] = ((words[2] >> 28) | (words[3] << 36)) & mask;\n s[4] = (words[3] >> 16) & top_mask;\n\n #[cfg(feature = \"zeroize\")]\n words.zeroize();\n\n s\n }\n\n /// Reduce a 64 byte / 512 bit scalar mod l\n #[rustfmt::skip] // keep alignment of lo[*] and hi[*] calculations\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// extracted as its own function (pure refactor, semantics identical).\n /// A verification-side split keeps Montgomery calls out of the\n /// prefix walk's proof obligations.\n /// AENEAS-COMPAT: lo-half of the 52-bit split (pure refactor; struct\n /// literal instead of field mutation keeps higher-order IndexMut\n /// closures out of the extraction).\n fn split_words_lo(words: &[u64; 8]) -> Scalar52 {\n let mask = (1u64 << 52) - 1;\n Scalar52([\n words[0] & mask,\n ((words[0] >> 52) | (words[1] << 12)) & mask,\n ((words[1] >> 40) | (words[2] << 24)) & mask,\n ((words[2] >> 28) | (words[3] << 36)) & mask,\n ((words[3] >> 16) | (words[4] << 48)) & mask,\n ])\n }\n\n /// AENEAS-COMPAT: hi-half of the 52-bit split (pure refactor, as above).\n fn split_words_hi(words: &[u64; 8]) -> Scalar52 {\n let mask = (1u64 << 52) - 1;\n Scalar52([\n (words[4] >> 4) & mask,\n ((words[4] >> 56) | (words[5] << 8)) & mask,\n ((words[5] >> 44) | (words[6] << 20)) & mask,\n ((words[6] >> 32) | (words[7] << 32)) & mask,\n (words[7] >> 20) & mask,\n ])\n }\n\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// extracted as its own function (pure refactor, semantics identical).\n /// A verification-side split keeps Montgomery calls out of the\n /// prefix walk's proof obligations.\n\n /// AENEAS-COMPAT: the bit-splitting prefix of `from_bytes_wide`,\n /// factored into named halves (pure refactor, semantics identical);\n /// keeps each verification walk at the proven-cheap ~25-step scale.\n fn from_bytes_wide_parts(bytes: &[u8; 64]) -> (Scalar52, Scalar52) {\n let mut words = [0u64; 8];\n for i in 0..8 {\n for j in 0..8 {\n words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);\n }\n }\n (Scalar52::split_words_lo(&words), Scalar52::split_words_hi(&words))\n }\n\n pub fn from_bytes_wide(bytes: &[u8; 64]) -> Scalar52 {\n let (lo, hi) = Scalar52::from_bytes_wide_parts(bytes);\n let lo = Scalar52::montgomery_mul(&lo, &constants::R); // (lo * R) / R = lo\n let hi = Scalar52::montgomery_mul(&hi, &constants::RR); // (hi * R^2) / R = hi * R\n Scalar52::add(&hi, &lo)\n }\n\n /// Pack the limbs of this `Scalar52` into 32 bytes\n #[rustfmt::skip] // keep alignment of s[*] calculations\n #[allow(clippy::identity_op)]\n pub fn to_bytes(self) -> [u8; 32] {\n let mut s = [0u8; 32];\n\n s[ 0] = (self.0[ 0] >> 0) as u8;\n s[ 1] = (self.0[ 0] >> 8) as u8;\n s[ 2] = (self.0[ 0] >> 16) as u8;\n s[ 3] = (self.0[ 0] >> 24) as u8;\n s[ 4] = (self.0[ 0] >> 32) as u8;\n s[ 5] = (self.0[ 0] >> 40) as u8;\n s[ 6] = ((self.0[ 0] >> 48) | (self.0[ 1] << 4)) as u8;\n s[ 7] = (self.0[ 1] >> 4) as u8;\n s[ 8] = (self.0[ 1] >> 12) as u8;\n s[ 9] = (self.0[ 1] >> 20) as u8;\n s[10] = (self.0[ 1] >> 28) as u8;\n s[11] = (self.0[ 1] >> 36) as u8;\n s[12] = (self.0[ 1] >> 44) as u8;\n s[13] = (self.0[ 2] >> 0) as u8;\n s[14] = (self.0[ 2] >> 8) as u8;\n s[15] = (self.0[ 2] >> 16) as u8;\n s[16] = (self.0[ 2] >> 24) as u8;\n s[17] = (self.0[ 2] >> 32) as u8;\n s[18] = (self.0[ 2] >> 40) as u8;\n s[19] = ((self.0[ 2] >> 48) | (self.0[ 3] << 4)) as u8;\n s[20] = (self.0[ 3] >> 4) as u8;\n s[21] = (self.0[ 3] >> 12) as u8;\n s[22] = (self.0[ 3] >> 20) as u8;\n s[23] = (self.0[ 3] >> 28) as u8;\n s[24] = (self.0[ 3] >> 36) as u8;\n s[25] = (self.0[ 3] >> 44) as u8;\n s[26] = (self.0[ 4] >> 0) as u8;\n s[27] = (self.0[ 4] >> 8) as u8;\n s[28] = (self.0[ 4] >> 16) as u8;\n s[29] = (self.0[ 4] >> 24) as u8;\n s[30] = (self.0[ 4] >> 32) as u8;\n s[31] = (self.0[ 4] >> 40) as u8;\n\n s\n }\n\n /// Compute `a + b` (mod l)\n pub fn add(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut sum = Scalar52::ZERO;\n let mask = (1u64 << 52) - 1;\n\n // a + b\n let mut carry: u64 = 0;\n for i in 0..5 {\n carry = a[i] + b[i] + (carry >> 52);\n sum[i] = carry & mask;\n }\n\n // subtract l if the sum is >= l\n Scalar52::sub(&sum, &constants::L)\n }\n\n /// Compute `a - b` (mod l)\n pub fn sub(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut difference = Scalar52::ZERO;\n let mask = (1u64 << 52) - 1;\n\n // a - b\n let mut borrow: u64 = 0;\n for i in 0..5 {\n borrow = a[i].wrapping_sub(b[i] + (borrow >> 63));\n difference[i] = borrow & mask;\n }\n\n // conditionally add l if the difference is negative\n difference.conditional_add_l(Choice::from((borrow >> 63) as u8));\n difference\n }\n\n pub(crate) fn conditional_add_l(&mut self, condition: Choice) -> u64 {\n let mut carry: u64 = 0;\n let mask = (1u64 << 52) - 1;\n\n for i in 0..5 {\n let addend = u64::conditional_select(&0, &constants::L[i], condition);\n carry = (carry >> 52) + self[i] + addend;\n self[i] = carry & mask;\n }\n\n carry\n }\n\n /// Compute a raw in-place carrying right shift over the limbs.\n #[inline(always)]\n pub(crate) fn shr1_assign(&mut self) -> u64 {\n let mut carry: u64 = 0;\n for i in (0..5).rev() {\n let limb = self[i];\n let next_carry = limb & 1;\n self[i] = (limb >> 1) | (carry << 51);\n carry = next_carry;\n }\n carry\n }\n\n /// Compute `a * b`\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of z[*] calculations\n pub (crate) fn mul_internal(a: &Scalar52, b: &Scalar52) -> [u128; 9] {\n let mut z = [0u128; 9];\n\n z[0] = m(a[0], b[0]);\n z[1] = m(a[0], b[1]) + m(a[1], b[0]);\n z[2] = m(a[0], b[2]) + m(a[1], b[1]) + m(a[2], b[0]);\n z[3] = m(a[0], b[3]) + m(a[1], b[2]) + m(a[2], b[1]) + m(a[3], b[0]);\n z[4] = m(a[0], b[4]) + m(a[1], b[3]) + m(a[2], b[2]) + m(a[3], b[1]) + m(a[4], b[0]);\n z[5] = m(a[1], b[4]) + m(a[2], b[3]) + m(a[3], b[2]) + m(a[4], b[1]);\n z[6] = m(a[2], b[4]) + m(a[3], b[3]) + m(a[4], b[2]);\n z[7] = m(a[3], b[4]) + m(a[4], b[3]);\n z[8] = m(a[4], b[4]);\n\n z\n }\n\n /// Compute `a^2`\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of return calculations\n fn square_internal(a: &Scalar52) -> [u128; 9] {\n let aa = [\n a[0] * 2,\n a[1] * 2,\n a[2] * 2,\n a[3] * 2,\n ];\n\n [\n m( a[0], a[0]),\n m(aa[0], a[1]),\n m(aa[0], a[2]) + m( a[1], a[1]),\n m(aa[0], a[3]) + m(aa[1], a[2]),\n m(aa[0], a[4]) + m(aa[1], a[3]) + m( a[2], a[2]),\n m(aa[1], a[4]) + m(aa[2], a[3]),\n m(aa[2], a[4]) + m( a[3], a[3]),\n m(aa[3], a[4]),\n m(a[4], a[4])\n ]\n }\n\n /// Compute `limbs/R` (mod l), where R is the Montgomery modulus 2^260\n #[inline(always)]\n #[rustfmt::skip] // keep alignment of n* and r* calculations\n pub (crate) fn montgomery_reduce(limbs: &[u128; 9]) -> Scalar52 {\n\n #[inline(always)]\n fn part1(sum: u128) -> (u128, u64) {\n let p = (sum as u64).wrapping_mul(constants::LFACTOR) & ((1u64 << 52) - 1);\n ((sum + m(p, constants::L[0])) >> 52, p)\n }\n\n #[inline(always)]\n fn part2(sum: u128) -> (u128, u64) {\n let w = (sum as u64) & ((1u64 << 52) - 1);\n (sum >> 52, w)\n }\n\n // note: l[3] is zero, so its multiples can be skipped\n let l = &constants::L;\n\n // the first half computes the Montgomery adjustment factor n, and begins adding n*l to make limbs divisible by R\n let (carry, n0) = part1( limbs[0]);\n let (carry, n1) = part1(carry + limbs[1] + m(n0, l[1]));\n let (carry, n2) = part1(carry + limbs[2] + m(n0, l[2]) + m(n1, l[1]));\n let (carry, n3) = part1(carry + limbs[3] + m(n1, l[2]) + m(n2, l[1]));\n let (carry, n4) = part1(carry + limbs[4] + m(n0, l[4]) + m(n2, l[2]) + m(n3, l[1]));\n\n // limbs is divisible by R now, so we can divide by R by simply storing the upper half as the result\n let (carry, r0) = part2(carry + limbs[5] + m(n1, l[4]) + m(n3, l[2]) + m(n4, l[1]));\n let (carry, r1) = part2(carry + limbs[6] + m(n2,l[4]) + m(n4, l[2]));\n let (carry, r2) = part2(carry + limbs[7] + m(n3, l[4]) );\n let (carry, r3) = part2(carry + limbs[8] + m(n4, l[4]));\n let r4 = carry as u64;\n\n // result may be >= l, so attempt to subtract l\n Scalar52::sub(&Scalar52([r0, r1, r2, r3, r4]), l)\n }\n\n /// Compute `a * b` (mod l)\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut ab_limbs = Scalar52::mul_internal(a, b);\n let mut ab = Scalar52::montgomery_reduce(&ab_limbs);\n let mut rr_limbs = Scalar52::mul_internal(&ab, &constants::RR);\n let product = Scalar52::montgomery_reduce(&rr_limbs);\n\n #[cfg(feature = \"zeroize\")]\n {\n ab_limbs.zeroize();\n ab.zeroize();\n rr_limbs.zeroize();\n }\n\n product\n }\n\n /// Compute `a^2` (mod l)\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n #[allow(dead_code)] // XXX we don't expose square() via the Scalar API\n pub fn square(&self) -> Scalar52 {\n let mut aa_limbs = Scalar52::square_internal(self);\n let mut aa = Scalar52::montgomery_reduce(&aa_limbs);\n let mut rr_limbs = Scalar52::mul_internal(&aa, &constants::RR);\n let square = Scalar52::montgomery_reduce(&rr_limbs);\n\n #[cfg(feature = \"zeroize\")]\n {\n aa_limbs.zeroize();\n aa.zeroize();\n rr_limbs.zeroize();\n }\n\n square\n }\n\n /// Compute `(a * b) / R` (mod l), where R is the Montgomery modulus 2^260\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn montgomery_mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {\n let mut limbs = Scalar52::mul_internal(a, b);\n let product = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n product\n }\n\n /// Compute `(a^2) / R` (mod l) in Montgomery form, where R is the Montgomery modulus 2^260\n #[inline(never)]\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n pub fn montgomery_square(&self) -> Scalar52 {\n let mut limbs = Scalar52::square_internal(self);\n let square = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n square\n }\n\n /// Puts a Scalar52 in to Montgomery form, i.e. computes `a*R (mod l)`\n #[inline(never)]\n pub fn as_montgomery(&self) -> Scalar52 {\n Scalar52::montgomery_mul(self, &constants::RR)\n }\n\n /// Takes a Scalar52 out of Montgomery form, i.e. computes `a/R (mod l)`\n #[allow(clippy::wrong_self_convention)]\n #[inline(never)]\n pub fn from_montgomery(&self) -> Scalar52 {\n let mut limbs = [0u128; 9];\n for i in 0..5 {\n limbs[i] = self[i] as u128;\n }\n let scalar = Scalar52::montgomery_reduce(&limbs);\n\n #[cfg(feature = \"zeroize\")]\n limbs.zeroize();\n\n scalar\n }\n}\n\n#[cfg(test)]\nmod test {\n use super::*;\n\n /// Note: x is 2^253-1 which is slightly larger than the largest scalar produced by\n /// this implementation (l-1), and should show there are no overflows for valid scalars\n ///\n /// x = 14474011154664524427946373126085988481658748083205070504932198000989141204991\n /// x = 7237005577332262213973186563042994240801631723825162898930247062703686954002 mod l\n /// x = 3057150787695215392275360544382990118917283750546154083604586903220563173085*R mod l in Montgomery form\n pub static X: Scalar52 = Scalar52([\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x000fffffffffffff,\n 0x00001fffffffffff,\n ]);\n\n /// x^2 = 3078544782642840487852506753550082162405942681916160040940637093560259278169 mod l\n pub static XX: Scalar52 = Scalar52([\n 0x0001668020217559,\n 0x000531640ffd0ec0,\n 0x00085fd6f9f38a31,\n 0x000c268f73bb1cf4,\n 0x000006ce65046df0,\n ]);\n\n /// x^2 = 4413052134910308800482070043710297189082115023966588301924965890668401540959*R mod l in Montgomery form\n pub static XX_MONT: Scalar52 = Scalar52([\n 0x000c754eea569a5c,\n 0x00063b6ed36cb215,\n 0x0008ffa36bf25886,\n 0x000e9183614e7543,\n 0x0000061db6c6f26f,\n ]);\n\n /// y = 6145104759870991071742105800796537629880401874866217824609283457819451087098\n pub static Y: Scalar52 = Scalar52([\n 0x000b75071e1458fa,\n 0x000bf9d75e1ecdac,\n 0x000433d2baf0672b,\n 0x0005fffcc11fad13,\n 0x00000d96018bb825,\n ]);\n\n /// x*y = 36752150652102274958925982391442301741 mod l\n pub static XY: Scalar52 = Scalar52([\n 0x000ee6d76ba7632d,\n 0x000ed50d71d84e02,\n 0x00000000001ba634,\n 0x0000000000000000,\n 0x0000000000000000,\n ]);\n\n /// x*y = 658448296334113745583381664921721413881518248721417041768778176391714104386*R mod l in Montgomery form\n pub static XY_MONT: Scalar52 = Scalar52([\n 0x0006d52bf200cfd5,\n 0x00033fb1d7021570,\n 0x000f201bc07139d8,\n 0x0001267e3e49169e,\n 0x000007b839c00268,\n ]);\n\n /// a = 2351415481556538453565687241199399922945659411799870114962672658845158063753\n pub static A: Scalar52 = Scalar52([\n 0x0005236c07b3be89,\n 0x0001bc3d2a67c0c4,\n 0x000a4aa782aae3ee,\n 0x0006b3f6e4fec4c4,\n 0x00000532da9fab8c,\n ]);\n\n /// b = 4885590095775723760407499321843594317911456947580037491039278279440296187236\n pub static B: Scalar52 = Scalar52([\n 0x000d3fae55421564,\n 0x000c2df24f65a4bc,\n 0x0005b5587d69fb0b,\n 0x00094c091b013b3b,\n 0x00000acd25605473,\n ]);\n\n /// a+b = 0\n /// a-b = 4702830963113076907131374482398799845891318823599740229925345317690316127506\n pub static AB: Scalar52 = Scalar52([\n 0x000a46d80f677d12,\n 0x0003787a54cf8188,\n 0x0004954f0555c7dc,\n 0x000d67edc9fd8989,\n 0x00000a65b53f5718,\n ]);\n\n // c = (2^512 - 1) % l = 1627715501170711445284395025044413883736156588369414752970002579683115011840\n pub static C: Scalar52 = Scalar52([\n 0x000611e3449c0f00,\n 0x000a768859347a40,\n 0x0007f5be65d00e1b,\n 0x0009a3dceec73d21,\n 0x00000399411b7c30,\n ]);\n\n #[test]\n fn mul_max() {\n let res = Scalar52::mul(&X, &X);\n for i in 0..5 {\n assert!(res[i] == XX[i]);\n }\n }\n\n #[test]\n fn square_max() {\n let res = X.square();\n for i in 0..5 {\n assert!(res[i] == XX[i]);\n }\n }\n\n #[test]\n fn montgomery_mul_max() {\n let res = Scalar52::montgomery_mul(&X, &X);\n for i in 0..5 {\n assert!(res[i] == XX_MONT[i]);\n }\n }\n\n #[test]\n fn montgomery_square_max() {\n let res = X.montgomery_square();\n for i in 0..5 {\n assert!(res[i] == XX_MONT[i]);\n }\n }\n\n #[test]\n fn mul() {\n let res = Scalar52::mul(&X, &Y);\n for i in 0..5 {\n assert!(res[i] == XY[i]);\n }\n }\n\n #[test]\n fn montgomery_mul() {\n let res = Scalar52::montgomery_mul(&X, &Y);\n for i in 0..5 {\n assert!(res[i] == XY_MONT[i]);\n }\n }\n\n #[test]\n fn add() {\n let res = Scalar52::add(&A, &B);\n let zero = Scalar52::ZERO;\n for i in 0..5 {\n assert!(res[i] == zero[i]);\n }\n }\n\n #[test]\n fn sub() {\n let res = Scalar52::sub(&A, &B);\n for i in 0..5 {\n assert!(res[i] == AB[i]);\n }\n }\n\n #[test]\n fn from_bytes_wide() {\n let bignum = [255u8; 64]; // 2^512 - 1\n let reduced = Scalar52::from_bytes_wide(&bignum);\n for i in 0..5 {\n assert!(reduced[i] == C[i]);\n }\n }\n}\n"},{"id":1,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2018 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! The `u64` backend uses `u64`s and a `(u64, u64) -> u128` multiplier.\n//!\n//! On x86_64, the idiom `(x as u128) * (y as u128)` lowers to `MUL`\n//! instructions taking 64-bit inputs and producing 128-bit outputs. On\n//! other platforms, this implementation is not recommended.\n//!\n//! On Haswell and newer, the BMI2 extension provides `MULX`, and on\n//! Broadwell and newer, the ADX extension provides `ADCX` and `ADOX`\n//! (allowing the CPU to compute two carry chains in parallel). These\n//! will be used if available.\n\npub mod field;\n\npub mod scalar;\n\npub mod constants;\n"},{"id":2,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Serial implementations of field, scalar, point arithmetic.\n//!\n//! When the vector backend is disabled, the crate uses the mixed-model strategy\n//! for implementing point operations and scalar multiplication; see the\n//! [`curve_models`] and [`scalar_mul`] documentation for more information.\n//!\n//! When the vector backend is enabled, the field and scalar\n//! implementations are still used for non-vectorized operations.\n\n#[doc(hidden)]\npub mod u64;\n\npub mod curve_models;\n\npub mod scalar_mul;\n"},{"id":3,"name":{"Local":"curve25519/solana-ed25519/src/backend.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! **INTERNALS:** Pluggable implementations for different architectures.\n//!\n//! The backend code is split into two parts: a serial backend,\n//! and a vector backend.\n//!\n//! The [`serial`] backend contains 32- and 64-bit implementations of\n//! field arithmetic and scalar arithmetic, as well as implementations\n//! of point operations using the mixed-model strategy (passing\n//! between different curve models depending on the operation).\n//!\n//! The [`vector`] backend contains implementations of vectorized\n//! field arithmetic, used to implement point operations using a novel\n//! implementation strategy derived from parallel formulas of Hisil,\n//! Wong, Carter, and Dawson.\n//!\n//! Because the two strategies give rise to different curve models,\n//! it's not possible to reuse exactly the same scalar multiplication\n//! code (or to write it generically), so both serial and vector\n//! backends contain matching implementations of scalar multiplication\n//! algorithms. These are intended to be selected by a `#[cfg]`-based\n//! type alias.\n//!\n//! The [`vector`] backend is selected by the `simd_backend` cargo\n//! feature; it uses the [`serial`] backend for non-vectorized operations.\n\nuse crate::EdwardsPoint;\nuse crate::Scalar;\n\npub mod serial;\n\n#[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\npub mod vector;\n\n#[derive(Copy, Clone)]\nenum BackendKind {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n Avx2,\n Serial,\n}\n\n#[inline]\nfn get_selected_backend() -> BackendKind {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n {\n cpufeatures::new!(cpuid_avx2, \"avx2\");\n let token_avx2: cpuid_avx2::InitToken = cpuid_avx2::init();\n if token_avx2.get() {\n return BackendKind::Avx2;\n }\n }\n\n BackendKind::Serial\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn pippenger_optional_multiscalar_mul(scalars: I, points: J) -> Option\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator>,\n{\n use crate::traits::VartimeMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::pippenger::spec_avx2::Pippenger::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::pippenger::Pippenger::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n }\n}\n\n#[cfg(feature = \"alloc\")]\npub(crate) enum VartimePrecomputedStraus {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n Avx2(vector::scalar_mul::precomputed_straus::spec_avx2::VartimePrecomputedStraus),\n Scalar(serial::scalar_mul::precomputed_straus::VartimePrecomputedStraus),\n}\n\n#[cfg(feature = \"alloc\")]\nimpl VartimePrecomputedStraus {\n pub fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => VartimePrecomputedStraus::Avx2(\n vector::scalar_mul::precomputed_straus::spec_avx2::VartimePrecomputedStraus::new(\n static_points,\n ),\n ),\n BackendKind::Serial => VartimePrecomputedStraus::Scalar(\n serial::scalar_mul::precomputed_straus::VartimePrecomputedStraus::new(\n static_points,\n ),\n ),\n }\n }\n\n /// Return the number of static points in the precomputation.\n pub fn len(&self) -> usize {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.len(),\n VartimePrecomputedStraus::Scalar(inner) => inner.len(),\n }\n }\n\n /// Determine if the precomputation is empty.\n pub fn is_empty(&self) -> bool {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.is_empty(),\n VartimePrecomputedStraus::Scalar(inner) => inner.is_empty(),\n }\n }\n\n pub fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator,\n J::Item: core::borrow::Borrow,\n K: IntoIterator>,\n {\n use crate::traits::VartimePrecomputedMultiscalarMul;\n\n match self {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n VartimePrecomputedStraus::Avx2(inner) => inner.optional_mixed_multiscalar_mul(\n static_scalars,\n dynamic_scalars,\n dynamic_points,\n ),\n VartimePrecomputedStraus::Scalar(inner) => inner.optional_mixed_multiscalar_mul(\n static_scalars,\n dynamic_scalars,\n dynamic_points,\n ),\n }\n }\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn straus_multiscalar_mul(scalars: I, points: J) -> EdwardsPoint\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator,\n J::Item: core::borrow::Borrow,\n{\n use crate::traits::MultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::straus::spec_avx2::Straus::multiscalar_mul::(scalars, points)\n }\n BackendKind::Serial => {\n serial::scalar_mul::straus::Straus::multiscalar_mul::(scalars, points)\n }\n }\n}\n\n#[allow(missing_docs)]\n#[cfg(feature = \"alloc\")]\npub fn straus_optional_multiscalar_mul(scalars: I, points: J) -> Option\nwhere\n I: IntoIterator,\n I::Item: core::borrow::Borrow,\n J: IntoIterator>,\n{\n use crate::traits::VartimeMultiscalarMul;\n\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::straus::spec_avx2::Straus::optional_multiscalar_mul::(\n scalars, points,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::straus::Straus::optional_multiscalar_mul::(scalars, points)\n }\n }\n}\n\n/// Perform constant-time, variable-base scalar multiplication.\npub fn variable_base_mul(point: &EdwardsPoint, scalar: &Scalar) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => vector::scalar_mul::variable_base::spec_avx2::mul(point, scalar),\n BackendKind::Serial => serial::scalar_mul::variable_base::mul(point, scalar),\n }\n}\n\n/// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n#[allow(non_snake_case)]\npub fn vartime_double_base_mul(a: &Scalar, A: &EdwardsPoint, b: &Scalar) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => vector::scalar_mul::vartime_double_base::spec_avx2::mul(a, A, b),\n BackendKind::Serial => serial::scalar_mul::vartime_double_base::mul(a, A, b),\n }\n}\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n///\n/// This function uses an optimized path when \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\),\n/// and falls back to general scalar multiplication otherwise.\n#[allow(non_snake_case)]\npub fn vartime_triple_base_mul_128_128_256(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n if !scalar_fits_in_128_bits(a1) || !scalar_fits_in_128_bits(a2) {\n return (a1 * A1) + (a2 * A2) + EdwardsPoint::mul_base(b);\n }\n\n vartime_triple_base_mul_128_128_256_prechecked(a1, A1, a2, A2, b)\n}\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) using the optimized 128/128/256-bit path.\n///\n/// Callers must ensure \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\).\n#[allow(non_snake_case)]\npub(crate) fn vartime_triple_base_mul_128_128_256_prechecked(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n match get_selected_backend() {\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n BackendKind::Avx2 => {\n vector::scalar_mul::vartime_triple_base::spec_avx2::mul_128_128_256_prechecked(\n a1, A1, a2, A2, b,\n )\n }\n BackendKind::Serial => {\n serial::scalar_mul::vartime_triple_base::mul_128_128_256_prechecked(a1, A1, a2, A2, b)\n }\n }\n}\n\n#[inline]\nfn scalar_fits_in_128_bits(scalar: &Scalar) -> bool {\n scalar.as_bytes()[16..32].iter().all(|&byte| byte == 0)\n}\n"},{"id":4,"name":{"Local":"curve25519/solana-ed25519/src/lib.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n#![no_std]\n#![cfg_attr(docsrs, feature(doc_cfg))]\n#![cfg_attr(docsrs, doc(auto_cfg(hide(docsrs))))]\n//------------------------------------------------------------------------\n// Documentation:\n//------------------------------------------------------------------------\n#![doc(\n html_logo_url = \"https://cdn.jsdelivr.net/gh/dalek-cryptography/curve25519/docs/assets/dalek-logo-clear.png\"\n)]\n#![doc = include_str!(\"../README.md\")]\n//------------------------------------------------------------------------\n// Linting:\n//------------------------------------------------------------------------\n#![cfg_attr(allow_unused_unsafe, allow(unused_unsafe))]\n#![warn(\n clippy::mod_module_files,\n clippy::unwrap_used,\n missing_docs,\n rust_2018_idioms,\n unused_lifetimes,\n unused_qualifications\n)]\n\n//------------------------------------------------------------------------\n// External dependencies:\n//------------------------------------------------------------------------\n\n#[cfg(feature = \"alloc\")]\n#[allow(unused_imports)]\n#[macro_use]\nextern crate alloc;\n\n// TODO: move std-dependent tests to `tests/`\n#[cfg(any(test, feature = \"std\"))]\n#[macro_use]\nextern crate std;\n\n#[cfg(feature = \"digest\")]\npub use digest;\n\n// Internal macros. Must come first!\n#[macro_use]\npub(crate) mod macros;\n\n//------------------------------------------------------------------------\n// curve25519 public modules\n//------------------------------------------------------------------------\n\n// Scalar arithmetic mod l = 2^252 + ..., the order of the Ristretto group\npub mod scalar;\n\n// Point operations on the Montgomery form of Curve25519\npub mod montgomery;\n\n// Point operations on the Edwards form of Curve25519\npub mod edwards;\n\n// Point operations on the short Weierstrass form of Curve25519\npub mod short_weierstrass;\n\n// Group operations on the Ristretto group\npub mod ristretto;\n\n// Useful constants, like the Ed25519 basepoint\npub mod constants;\n\n// External (and internal) traits.\npub mod traits;\n\n//------------------------------------------------------------------------\n// ed25519 public modules\n//------------------------------------------------------------------------\n\npub mod ed_sigs;\n\n//------------------------------------------------------------------------\n// curve25519 internal modules\n//------------------------------------------------------------------------\n\n// Finite field arithmetic mod p = 2^255 - 19\npub(crate) mod field;\n\n// Arithmetic backends (using u32, u64, etc) live here\n#[cfg(docsrs)]\npub mod backend;\n#[cfg(not(docsrs))]\npub(crate) mod backend;\n\n// Generic code for window lookups\npub(crate) mod window;\n\n#[cfg(feature = \"lizard\")]\nmod lizard;\n\npub use crate::{\n edwards::EdwardsPoint, montgomery::MontgomeryPoint, ristretto::RistrettoPoint, scalar::Scalar,\n};\n\n// Build time diagnostics for validation\n#[cfg(curve25519_diagnostics = \"build\")]\nmod diagnostics;\n"},{"id":5,"name":{"Local":"curve25519/solana-ed25519/src/scalar.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// Portions Copyright 2017 Brian Smith\n// See LICENSE for licensing information.\n//\n// Authors:\n// - Isis Agora Lovecruft \n// - Henry de Valence \n// - Brian Smith \n\n//! Arithmetic on scalars (integers mod the group order).\n//!\n//! Both the Ristretto group and the Ed25519 basepoint have prime order\n//! \\\\( \\ell = 2\\^{252} + 27742317777372353535851937790883648493 \\\\).\n//!\n//! This code is intended to be useful with both the Ristretto group\n//! (where everything is done modulo \\\\( \\ell \\\\)), and the X/Ed25519\n//! setting, which mandates specific bit-twiddles that are not\n//! well-defined modulo \\\\( \\ell \\\\).\n//!\n//! All arithmetic on `Scalars` is done modulo \\\\( \\ell \\\\).\n//!\n//! # Constructing a scalar\n//!\n//! To create a [`Scalar`](struct.Scalar.html) from a supposedly canonical encoding, use\n//! [`Scalar::from_canonical_bytes`](struct.Scalar.html#method.from_canonical_bytes).\n//!\n//! This function does input validation, ensuring that the input bytes\n//! are the canonical encoding of a `Scalar`.\n//! If they are, we'll get\n//! `Some(Scalar)` in return:\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let one_as_bytes: [u8; 32] = Scalar::ONE.to_bytes();\n//! let a: Option = Scalar::from_canonical_bytes(one_as_bytes).into();\n//!\n//! assert!(a.is_some());\n//! ```\n//!\n//! However, if we give it bytes representing a scalar larger than \\\\( \\ell \\\\)\n//! (in this case, \\\\( \\ell + 2 \\\\)), we'll get `None` back:\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let l_plus_two_bytes: [u8; 32] = [\n//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,\n//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,\n//! ];\n//! let a: Option = Scalar::from_canonical_bytes(l_plus_two_bytes).into();\n//!\n//! assert!(a.is_none());\n//! ```\n//!\n//! Another way to create a `Scalar` is by reducing a \\\\(256\\\\)-bit integer mod\n//! \\\\( \\ell \\\\), for which one may use the\n//! [`Scalar::from_bytes_mod_order`](struct.Scalar.html#method.from_bytes_mod_order)\n//! method. In the case of the second example above, this would reduce the\n//! resultant scalar \\\\( \\mod \\ell \\\\), producing \\\\( 2 \\\\):\n//!\n//! ```\n//! use curve25519::scalar::Scalar;\n//!\n//! let l_plus_two_bytes: [u8; 32] = [\n//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,\n//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,\n//! ];\n//! let a: Scalar = Scalar::from_bytes_mod_order(l_plus_two_bytes);\n//!\n//! let two: Scalar = Scalar::ONE + Scalar::ONE;\n//!\n//! assert!(a == two);\n//! ```\n//!\n//! There is also a constructor that reduces a \\\\(512\\\\)-bit integer,\n//! [`Scalar::from_bytes_mod_order_wide`].\n//!\n//! To construct a `Scalar` as the hash of some input data, use\n//! [`Scalar::hash_from_bytes`], which takes a buffer, or\n//! [`Scalar::from_hash`], which allows an IUF API.\n//!\n#![cfg_attr(feature = \"digest\", doc = \"```\")]\n#![cfg_attr(not(feature = \"digest\"), doc = \"```ignore\")]\n//! # fn main() {\n//! use sha2::{Digest, Sha512};\n//! use curve25519::scalar::Scalar;\n//!\n//! // Hashing a single byte slice\n//! let a = Scalar::hash_from_bytes::(b\"Abolish ICE\");\n//!\n//! // Streaming data into a hash object\n//! let mut hasher = Sha512::default();\n//! hasher.update(b\"Abolish \");\n//! hasher.update(b\"ICE\");\n//! let a2 = Scalar::from_hash(hasher);\n//!\n//! assert_eq!(a, a2);\n//! # }\n//! ```\n//!\n//! See also `Scalar::hash_from_bytes` and `Scalar::from_hash` that\n//! reduces a \\\\(512\\\\)-bit integer, if the optional `digest` feature\n//! has been enabled.\n\nuse core::borrow::Borrow;\nuse core::fmt::Debug;\nuse core::iter::{Product, Sum};\nuse core::ops::Index;\nuse core::ops::Neg;\nuse core::ops::{Add, AddAssign};\nuse core::ops::{Mul, MulAssign};\nuse core::ops::{Sub, SubAssign};\n\n#[cfg(feature = \"group\")]\nuse group::ff::{Field, FromUniformBytes, PrimeField};\n#[cfg(feature = \"group-bits\")]\nuse group::ff::{FieldBits, PrimeFieldBits};\n\n#[cfg(any(feature = \"group\", feature = \"rand_core\"))]\nuse rand_core::RngCore;\n\n#[cfg(feature = \"rand_core\")]\nuse rand_core::CryptoRng;\n\n#[cfg(feature = \"digest\")]\nuse digest::Digest;\n#[cfg(feature = \"digest\")]\nuse digest::array::typenum::U64;\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\nuse subtle::CtOption;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::backend;\nuse crate::constants;\nuse crate::scalar::heea::{I256, curve25519_heea_vartime};\nuse crate::traits::HEEADecomposition;\n\nmod heea;\npub(crate) use heea::HEEA_MAX_INDEX;\n\npub(crate) const NAF_128_SIZE: usize = HEEA_MAX_INDEX + 1;\n\n/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.\n///\n/// This is pinned to the 64-bit serial scalar backend.\ntype UnpackedScalar = backend::serial::u64::scalar::Scalar52;\n\n/// The `Scalar` struct holds an element of \\\\(\\mathbb Z / \\ell\\mathbb Z \\\\).\n#[allow(clippy::derived_hash_with_manual_eq)]\n#[derive(Copy, Clone, Hash)]\npub struct Scalar {\n /// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the\n /// group order.\n ///\n /// # Invariant #1\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255}\\\\). That is, the most\n /// significant bit of `bytes[31]` is 0.\n ///\n /// This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most\n /// integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`]).\n /// The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB\n /// is 0 (see `MontgomeryPoint::mul`).\n ///\n /// # Invariant #2 (weak)\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents\n /// a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is\n /// stronger than invariant #1. It also sometimes has to be broken.\n ///\n /// This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,\n /// mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and\n /// `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,\n /// scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant\n /// #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined.\n ///\n /// Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar.\n /// Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars.\n /// Multiplication is correct, but this is only due to a quirk of our implementation, and not\n /// guaranteed to hold in general in the future.\n ///\n /// Note: It is not possible to construct an unreduced `Scalar` from the public API unless the\n /// `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all\n /// public non-legacy uses, invariant #2\n /// always holds.\n ///\n pub(crate) bytes: [u8; 32],\n}\n\nimpl Scalar {\n /// Construct a `Scalar` by reducing a 256-bit little-endian integer\n /// modulo the group order \\\\( \\ell \\\\).\n pub fn from_bytes_mod_order(bytes: [u8; 32]) -> Scalar {\n // Temporarily allow s_unreduced.bytes > 2^255 ...\n let s_unreduced = Scalar { bytes };\n\n // Then reduce mod the group order and return the reduced representative.\n let s = s_unreduced.reduce();\n debug_assert_eq!(0u8, s[31] >> 7);\n\n s\n }\n\n /// Construct a `Scalar` by reducing a 512-bit little-endian integer\n /// modulo the group order \\\\( \\ell \\\\).\n pub fn from_bytes_mod_order_wide(input: &[u8; 64]) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut unpacked = UnpackedScalar::from_bytes_wide(input);\n let scalar = unpacked.pack();\n\n #[cfg(feature = \"zeroize\")]\n unpacked.zeroize();\n\n scalar\n }\n\n /// Attempt to construct a `Scalar` from a canonical byte representation.\n ///\n /// # Return\n ///\n /// - `Some(s)`, where `s` is the `Scalar` corresponding to `bytes`,\n /// if `bytes` is a canonical byte representation modulo the group order \\\\( \\ell \\\\);\n /// - `None` if `bytes` is not a canonical byte representation.\n pub fn from_canonical_bytes(bytes: [u8; 32]) -> CtOption {\n let high_bit_unset = (bytes[31] >> 7).ct_eq(&0);\n let candidate = Scalar { bytes };\n CtOption::new(candidate, high_bit_unset & candidate.is_canonical())\n }\n\n /// Construct a `Scalar` from bytes that are known to be canonical.\n #[inline]\n #[cfg(all(target_arch = \"x86_64\", not(curve25519_serial_only)))]\n pub(crate) const fn from_canonical_bytes_unchecked(bytes: [u8; 32]) -> Scalar {\n Scalar { bytes }\n }\n\n /// Construct a `Scalar` from the low 255 bits of a 256-bit integer. This breaks the invariant\n /// that scalars are always reduced. Scalar-scalar arithmetic, i.e., addition, subtraction,\n /// multiplication, **does not work** on scalars produced from this function. You may only use\n /// the output of this function for `EdwardsPoint::mul`, `MontgomeryPoint::mul`, and\n /// `EdwardsPoint::vartime_double_scalar_mul_basepoint`. **Do not use this function** unless\n /// you absolutely have to.\n #[cfg(feature = \"legacy_compatibility\")]\n pub const fn from_bits(bytes: [u8; 32]) -> Scalar {\n let mut s = Scalar { bytes };\n // Ensure invariant #1 holds. That is, make s < 2^255 by masking the high bit.\n s.bytes[31] &= 0b0111_1111;\n\n s\n }\n}\n\nimpl Debug for Scalar {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n f.write_str(\"Scalar{..}\")\n }\n}\n\nimpl Eq for Scalar {}\nimpl PartialEq for Scalar {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl ConstantTimeEq for Scalar {\n fn ct_eq(&self, other: &Self) -> Choice {\n self.bytes.ct_eq(&other.bytes)\n }\n}\n\nimpl Index for Scalar {\n type Output = u8;\n\n /// Index the bytes of the representative for this `Scalar`. Mutation is not permitted.\n fn index(&self, _index: usize) -> &u8 {\n &(self.bytes[_index])\n }\n}\n\nimpl<'a> MulAssign<&'a Scalar> for Scalar {\n fn mul_assign(&mut self, _rhs: &'a Scalar) {\n *self = UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack();\n }\n}\n\ndefine_mul_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Mul<&'a Scalar> for &Scalar {\n type Output = Scalar;\n fn mul(self, _rhs: &'a Scalar) -> Scalar {\n UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack()\n }\n}\n\ndefine_mul_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl<'a> AddAssign<&'a Scalar> for Scalar {\n fn add_assign(&mut self, _rhs: &'a Scalar) {\n *self = *self + _rhs;\n }\n}\n\ndefine_add_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Add<&'a Scalar> for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn add(self, _rhs: &'a Scalar) -> Scalar {\n // The UnpackedScalar::add function produces reduced outputs if the inputs are reduced. By\n // Scalar invariant #1, this is always the case.\n UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack()\n }\n}\n\ndefine_add_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl<'a> SubAssign<&'a Scalar> for Scalar {\n fn sub_assign(&mut self, _rhs: &'a Scalar) {\n *self = *self - _rhs;\n }\n}\n\ndefine_sub_assign_variants!(LHS = Scalar, RHS = Scalar);\n\nimpl<'a> Sub<&'a Scalar> for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn sub(self, rhs: &'a Scalar) -> Scalar {\n // The UnpackedScalar::sub function produces reduced outputs if the inputs are reduced. By\n // Scalar invariant #1, this is always the case.\n UnpackedScalar::sub(&self.unpack(), &rhs.unpack()).pack()\n }\n}\n\ndefine_sub_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);\n\nimpl Neg for &Scalar {\n type Output = Scalar;\n #[allow(non_snake_case)]\n fn neg(self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_unpacked = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_R = UnpackedScalar::mul_internal(&self_unpacked, &constants::R);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut self_mod_l = UnpackedScalar::montgomery_reduce(&self_R);\n let negated = UnpackedScalar::sub(&UnpackedScalar::ZERO, &self_mod_l).pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n self_unpacked.zeroize();\n self_R.zeroize();\n self_mod_l.zeroize();\n }\n\n negated\n }\n}\n\nimpl Neg for Scalar {\n type Output = Scalar;\n fn neg(self) -> Scalar {\n -&self\n }\n}\n\nimpl ConditionallySelectable for Scalar {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = u8::conditional_select(&a.bytes[i], &b.bytes[i], choice);\n }\n Scalar { bytes }\n }\n}\n\n#[cfg(feature = \"serde\")]\nuse serde::de::Visitor;\n#[cfg(feature = \"serde\")]\nuse serde::{Deserialize, Deserializer, Serialize, Serializer};\n\n#[cfg(feature = \"serde\")]\n#[cfg_attr(docsrs, doc(cfg(feature = \"serde\")))]\nimpl Serialize for Scalar {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\n#[cfg_attr(docsrs, doc(cfg(feature = \"serde\")))]\nimpl<'de> Deserialize<'de> for Scalar {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct ScalarVisitor;\n\n impl<'de> Visitor<'de> for ScalarVisitor {\n type Value = Scalar;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\n \"a sequence of 32 bytes whose little-endian interpretation is less than the \\\n basepoint order ℓ\",\n )\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n let scalar = Option::from(Scalar::from_canonical_bytes(bytes))\n .ok_or_else(|| serde::de::Error::custom(\"scalar was not canonically encoded\"));\n\n #[cfg(feature = \"zeroize\")]\n bytes.zeroize();\n\n scalar\n }\n }\n\n deserializer.deserialize_tuple(32, ScalarVisitor)\n }\n}\n\nimpl Product for Scalar\nwhere\n T: Borrow,\n{\n fn product(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(Scalar::ONE, |acc, item| acc * item.borrow())\n }\n}\n\nimpl Sum for Scalar\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(Scalar::ZERO, |acc, item| acc + item.borrow())\n }\n}\n\nimpl Default for Scalar {\n fn default() -> Scalar {\n Scalar::ZERO\n }\n}\n\nimpl From for Scalar {\n fn from(x: u8) -> Scalar {\n let mut s_bytes = [0u8; 32];\n s_bytes[0] = x;\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u16) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u32) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n /// Construct a scalar from the given `u64`.\n ///\n /// # Inputs\n ///\n /// An `u64` to convert to a `Scalar`.\n ///\n /// # Returns\n ///\n /// A `Scalar` corresponding to the input `u64`.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let fourtytwo = Scalar::from(42u64);\n /// let six = Scalar::from(6u64);\n /// let seven = Scalar::from(7u64);\n ///\n /// assert!(fourtytwo == six * seven);\n /// ```\n fn from(x: u64) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From for Scalar {\n fn from(x: u128) -> Scalar {\n let mut s_bytes = [0u8; 32];\n let x_bytes = x.to_le_bytes();\n s_bytes[0..x_bytes.len()].copy_from_slice(&x_bytes);\n Scalar { bytes: s_bytes }\n }\n}\n\nimpl From<&Scalar> for I256 {\n fn from(s: &Scalar) -> I256 {\n I256::from_le_bytes(s.to_bytes())\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for Scalar {\n fn zeroize(&mut self) {\n self.bytes.zeroize();\n }\n}\n\nimpl HEEADecomposition for Scalar {\n /// Generate approximately half-size non-negative scalars `(rho, tau)` for a\n /// given challenge scalar `h`.\n ///\n /// The signed HEEA output satisfies `rho_i = tau_i * h (mod ell)`. This\n /// method returns the absolute values of `rho_i` and `tau_i`, plus `flip_h`\n /// to record whether exactly one of the signed values was negative.\n ///\n /// If `flip_h` is false, then `rho = tau * h (mod ell)`. If `flip_h` is\n /// true, then `rho = -tau * h (mod ell)`.\n fn heea_decompose(&self) -> (Scalar, Scalar, bool) {\n // Convert h to I256\n let v = self.into();\n\n // Run the algorithm\n let (mut rho_i, mut tau_i) = curve25519_heea_vartime(v);\n\n // Convert back to Scalars and ensure shortness\n let mut flip_h = false;\n if rho_i < I256::ZERO {\n rho_i = -rho_i;\n flip_h = !flip_h;\n }\n if tau_i < I256::ZERO {\n tau_i = -tau_i;\n flip_h = !flip_h;\n }\n\n let rho = Scalar::from_positive_i256(&rho_i);\n let tau = Scalar::from_positive_i256(&tau_i);\n\n (rho, tau, flip_h)\n }\n}\n\nimpl Scalar {\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: Self = Self { bytes: [0u8; 32] };\n\n /// The scalar \\\\( 1 \\\\).\n pub const ONE: Self = Self {\n bytes: [\n 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n\n /// Return a `Scalar` chosen uniformly at random using a user-provided RNG.\n ///\n /// # Inputs\n ///\n /// * `rng`: any RNG which implements `CryptoRng` interface.\n ///\n /// # Returns\n ///\n /// A random scalar within \\\\(\\mathbb{Z} / \\ell\\mathbb{Z}\\\\).\n ///\n /// # Example\n ///\n /// ```\n /// # fn main() {\n /// use curve25519::scalar::Scalar;\n ///\n /// let mut csprng = rand::thread_rng();\n /// let a: Scalar = Scalar::random(&mut csprng);\n /// # }\n #[cfg(feature = \"rand_core\")]\n pub fn random(rng: &mut R) -> Self {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut scalar_bytes = [0u8; 64];\n rng.fill_bytes(&mut scalar_bytes);\n let scalar = Scalar::from_bytes_mod_order_wide(&scalar_bytes);\n\n #[cfg(feature = \"zeroize\")]\n scalar_bytes.zeroize();\n\n scalar\n }\n\n #[cfg(feature = \"digest\")]\n /// Hash a slice of bytes into a scalar.\n ///\n /// Takes a type parameter `D`, which is any `Digest` producing 64\n /// bytes (512 bits) of output.\n ///\n /// Convenience wrapper around `from_hash`.\n ///\n /// # Example\n ///\n #[cfg_attr(feature = \"digest\", doc = \"```\")]\n #[cfg_attr(not(feature = \"digest\"), doc = \"```ignore\")]\n /// # use curve25519::scalar::Scalar;\n /// use sha2::Sha512;\n ///\n /// # // Need fn main() here in comment so the doctest compiles\n /// # // See https://doc.rust-lang.org/book/documentation.html#documentation-as-tests\n /// # fn main() {\n /// let msg = \"To really appreciate architecture, you may even need to commit a murder\";\n /// let s = Scalar::hash_from_bytes::(msg.as_bytes());\n /// # }\n /// ```\n pub fn hash_from_bytes(input: &[u8]) -> Scalar\n where\n D: Digest + Default,\n {\n let mut hash = D::default();\n hash.update(input);\n Scalar::from_hash(hash)\n }\n\n #[cfg(feature = \"digest\")]\n /// Construct a scalar from an existing `Digest` instance.\n ///\n /// Use this instead of `hash_from_bytes` if it is more convenient\n /// to stream data into the `Digest` than to pass a single byte\n /// slice.\n ///\n /// # Example\n ///\n /// ```\n /// # use curve25519::scalar::Scalar;\n /// use curve25519::digest::Update;\n ///\n /// use sha2::Digest;\n /// use sha2::Sha512;\n ///\n /// # fn main() {\n /// let mut h = Sha512::new()\n /// .chain(\"To really appreciate architecture, you may even need to commit a murder.\")\n /// .chain(\"While the programs used for The Manhattan Transcripts are of the most extreme\")\n /// .chain(\"nature, they also parallel the most common formula plot: the archetype of\")\n /// .chain(\"murder. Other phantasms were occasionally used to underline the fact that\")\n /// .chain(\"perhaps all architecture, rather than being about functional standards, is\")\n /// .chain(\"about love and death.\");\n ///\n /// let s = Scalar::from_hash(h);\n ///\n /// println!(\"{:?}\", s.to_bytes());\n /// assert_eq!(\n /// s.to_bytes(),\n /// [ 21, 88, 208, 252, 63, 122, 210, 152,\n /// 154, 38, 15, 23, 16, 167, 80, 150,\n /// 192, 221, 77, 226, 62, 25, 224, 148,\n /// 239, 48, 176, 10, 185, 69, 168, 11, ],\n /// );\n /// # }\n /// ```\n pub fn from_hash(hash: D) -> Scalar\n where\n D: Digest,\n {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut digest = hash.finalize();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut output = [0u8; 64];\n output.copy_from_slice(digest.as_slice());\n let scalar = Scalar::from_bytes_mod_order_wide(&output);\n\n #[cfg(feature = \"zeroize\")]\n {\n digest.zeroize();\n output.zeroize();\n }\n\n scalar\n }\n\n /// Convert this `Scalar` to its underlying sequence of bytes.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let s: Scalar = Scalar::ZERO;\n ///\n /// assert!(s.to_bytes() == [0u8; 32]);\n /// ```\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.bytes\n }\n\n /// View the little-endian byte encoding of the integer representing this Scalar.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// let s: Scalar = Scalar::ZERO;\n ///\n /// assert!(s.as_bytes() == &[0u8; 32]);\n /// ```\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.bytes\n }\n\n /// Given a nonzero `Scalar`, compute its multiplicative inverse.\n ///\n /// # Warning\n ///\n /// `self` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n ///\n /// # Returns\n ///\n /// The multiplicative inverse of the this `Scalar`.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n ///\n /// // x = 2238329342913194256032495932344128051776374960164957527413114840482143558222\n /// let X: Scalar = Scalar::from_bytes_mod_order([\n /// 0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,\n /// 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,\n /// 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,\n /// 0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,\n /// ]);\n /// // 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244\n /// let XINV: Scalar = Scalar::from_bytes_mod_order([\n /// 0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb,\n /// 0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01, 0x63, 0x47,\n /// 0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96,\n /// 0xd5, 0x0b, 0xcd, 0x7a, 0x3f, 0x96, 0x2a, 0x0f,\n /// ]);\n ///\n /// let inv_X: Scalar = X.invert();\n /// assert!(XINV == inv_X);\n /// let should_be_one: Scalar = &inv_X * &X;\n /// assert!(should_be_one == Scalar::ONE);\n /// ```\n pub fn invert(&self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut unpacked = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = unpacked.invert();\n let scalar = inverse.pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n unpacked.zeroize();\n inverse.zeroize();\n }\n\n scalar\n }\n\n /// Given a slice of nonzero (possibly secret) `Scalar`s,\n /// compute their inverses in a batch.\n ///\n /// # Return\n ///\n /// Each element of `inputs` is replaced by its inverse.\n ///\n /// The product of all inverses is returned.\n ///\n /// # Warning\n ///\n /// All input `Scalars` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n ///\n /// # Example\n ///\n /// ```\n /// # use curve25519::scalar::Scalar;\n /// # fn main() {\n /// let mut scalars = [\n /// Scalar::from(3u64),\n /// Scalar::from(5u64),\n /// Scalar::from(7u64),\n /// Scalar::from(11u64),\n /// ];\n ///\n /// let allinv = Scalar::invert_batch(&mut scalars);\n ///\n /// assert_eq!(allinv, Scalar::from(3*5*7*11u64).invert());\n /// assert_eq!(scalars[0], Scalar::from(3u64).invert());\n /// assert_eq!(scalars[1], Scalar::from(5u64).invert());\n /// assert_eq!(scalars[2], Scalar::from(7u64).invert());\n /// assert_eq!(scalars[3], Scalar::from(11u64).invert());\n /// # }\n /// ```\n pub fn invert_batch(inputs: &mut [Scalar; N]) -> Scalar {\n let one: UnpackedScalar = Scalar::ONE.unpack().as_montgomery();\n\n let mut scratch = [one; N];\n\n Self::invert_batch_internal(inputs, &mut scratch)\n }\n\n /// Given a slice of nonzero (possibly secret) `Scalar`s, compute their inverses in a batch.\n /// This the allocating form of [`Self::invert_batch`]. See those docs for examples.\n ///\n /// # Return\n ///\n /// Each element of `inputs` is replaced by its inverse.\n ///\n /// The product of all inverses is returned.\n ///\n /// # Warning\n ///\n /// All input `Scalars` **MUST** be nonzero. If you cannot\n /// *prove* that this is the case, you **SHOULD NOT USE THIS\n /// FUNCTION**.\n #[cfg(feature = \"alloc\")]\n pub fn invert_batch_alloc(inputs: &mut [Scalar]) -> Scalar {\n let n = inputs.len();\n let one: UnpackedScalar = Scalar::ONE.unpack().as_montgomery();\n\n let mut scratch = vec![one; n];\n\n Self::invert_batch_internal(inputs, &mut scratch)\n }\n\n fn invert_batch_internal(inputs: &mut [Scalar], scratch: &mut [UnpackedScalar]) -> Scalar {\n // This code is essentially identical to the FieldElement\n // implementation, and is documented there. Unfortunately,\n // it's not easy to write it generically, since here we want\n // to use `UnpackedScalar`s internally, and `Scalar`s\n // externally, but there's no corresponding distinction for\n // field elements.\n\n // Keep an accumulator of all of the previous products\n let mut acc = Scalar::ONE.unpack().as_montgomery();\n\n // Pass through the input vector, recording the previous\n // products in the scratch space\n for (input, scratch) in inputs.iter_mut().zip(scratch.iter_mut()) {\n *scratch = acc;\n\n // Avoid unnecessary Montgomery multiplication in second pass by\n // keeping inputs in Montgomery form\n let tmp = input.unpack().as_montgomery();\n *input = tmp.pack();\n acc = UnpackedScalar::montgomery_mul(&acc, &tmp);\n }\n\n // acc is nonzero iff all inputs are nonzero\n debug_assert!(acc.pack() != Scalar::ZERO);\n\n // Compute the inverse of all products\n acc = acc.montgomery_invert().from_montgomery();\n\n // We need to return the product of all inverses later\n let ret = acc.pack();\n\n // Pass through the vector backwards to compute the inverses\n // in place\n for (input, scratch) in inputs.iter_mut().rev().zip(scratch.iter().rev()) {\n let tmp = UnpackedScalar::montgomery_mul(&acc, &input.unpack());\n *input = UnpackedScalar::montgomery_mul(&acc, scratch).pack();\n acc = tmp;\n }\n\n #[cfg(feature = \"zeroize\")]\n scratch.zeroize();\n\n ret\n }\n\n /// Compute `b` such that `b + b = a mod modulus`.\n pub fn div_by_2(&self) -> Self {\n // We are looking for such `b` that `b + b = a mod modulus`.\n // Two possibilities:\n // - if `a` is even, we can just divide by 2;\n // - if `a` is odd, we divide `(a + modulus)` by 2.\n let is_odd = Choice::from(self.as_bytes()[0] & 1);\n let mut scalar = self.unpack();\n scalar.conditional_add_l(is_odd);\n\n let carry = scalar.shr1_assign();\n debug_assert_eq!(carry, 0);\n\n scalar.pack()\n }\n\n /// Get the bits of the scalar, in little-endian order\n pub(crate) fn bits_le(&self) -> impl DoubleEndedIterator + '_ {\n (0..256).map(|i| {\n // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index\n // the byte. Since self.bytes is little-endian at the byte level, this iterator is\n // little-endian on the bit level\n ((self.bytes[i >> 3] >> (i & 7)) & 1u8) == 1\n })\n }\n\n /// Compute a width-\\\\(w\\\\) \"Non-Adjacent Form\" of this scalar.\n ///\n /// A width-\\\\(w\\\\) NAF of a positive integer \\\\(k\\\\) is an expression\n /// $$\n /// k = \\sum_{i=0}\\^m n\\_i 2\\^i,\n /// $$\n /// where each nonzero\n /// coefficient \\\\(n\\_i\\\\) is odd and bounded by \\\\(|n\\_i| < 2\\^{w-1}\\\\),\n /// \\\\(n\\_{m-1}\\\\) is nonzero, and at most one of any \\\\(w\\\\) consecutive\n /// coefficients is nonzero. (Hankerson, Menezes, Vanstone; def 3.32).\n ///\n /// The length of the NAF is at most one more than the length of\n /// the binary representation of \\\\(k\\\\). This is why the\n /// `Scalar` type maintains an invariant (invariant #1) that the top bit is\n /// \\\\(0\\\\), so that the NAF of a scalar has at most 256 digits.\n ///\n /// Intuitively, this is like a binary expansion, except that we\n /// allow some coefficients to grow in magnitude up to\n /// \\\\(2\\^{w-1}\\\\) so that the nonzero coefficients are as sparse\n /// as possible.\n ///\n /// When doing scalar multiplication, we can then use a lookup\n /// table of precomputed multiples of a point to add the nonzero\n /// terms \\\\( k_i P \\\\). Using signed digits cuts the table size\n /// in half, and using odd digits cuts the table size in half\n /// again.\n ///\n /// To compute a \\\\(w\\\\)-NAF, we use a modification of Algorithm 3.35 of HMV:\n ///\n /// 1. \\\\( i \\gets 0 \\\\)\n /// 2. While \\\\( k \\ge 1 \\\\):\n /// 1. If \\\\(k\\\\) is odd, \\\\( n_i \\gets k \\operatorname{mods} 2^w \\\\), \\\\( k \\gets k - n_i \\\\).\n /// 2. If \\\\(k\\\\) is even, \\\\( n_i \\gets 0 \\\\).\n /// 3. \\\\( k \\gets k / 2 \\\\), \\\\( i \\gets i + 1 \\\\).\n /// 3. Return \\\\( n_0, n_1, ... , \\\\)\n ///\n /// Here \\\\( \\bar x = x \\operatorname{mods} 2^w \\\\) means the\n /// \\\\( \\bar x \\\\) with \\\\( \\bar x \\equiv x \\pmod{2^w} \\\\) and\n /// \\\\( -2^{w-1} \\leq \\bar x < 2^{w-1} \\\\).\n ///\n /// We implement this by scanning across the bits of \\\\(k\\\\) from\n /// least-significant bit to most-significant-bit.\n /// Write the bits of \\\\(k\\\\) as\n /// $$\n /// k = \\sum\\_{i=0}\\^m k\\_i 2^i,\n /// $$\n /// and split the sum as\n /// $$\n /// k = \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// $$\n /// where the first part is \\\\( k \\mod 2^w \\\\).\n ///\n /// If \\\\( k \\mod 2^w\\\\) is odd, and \\\\( k \\mod 2^w < 2^{w-1} \\\\), then we emit\n /// \\\\( n_0 = k \\mod 2^w \\\\). Instead of computing\n /// \\\\( k - n_0 \\\\), we just advance \\\\(w\\\\) bits and reindex.\n ///\n /// If \\\\( k \\mod 2^w\\\\) is odd, and \\\\( k \\mod 2^w \\ge 2^{w-1} \\\\), then\n /// \\\\( n_0 = k \\operatorname{mods} 2^w = k \\mod 2^w - 2^w \\\\).\n /// The quantity \\\\( k - n_0 \\\\) is\n /// $$\n /// \\begin{aligned}\n /// k - n_0 &= \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// - \\sum\\_{i=0}^{w-1} k\\_i 2^i + 2^w \\\\\\\\\n /// &= 2^w + 2^w \\sum\\_{i=0} k\\_{i+w} 2^i\n /// \\end{aligned}\n /// $$\n /// so instead of computing the subtraction, we can set a carry\n /// bit, advance \\\\(w\\\\) bits, and reindex.\n ///\n /// If \\\\( k \\mod 2^w\\\\) is even, we emit \\\\(0\\\\), advance 1 bit\n /// and reindex. In fact, by setting all digits to \\\\(0\\\\)\n /// initially, we don't need to emit anything.\n pub(crate) fn non_adjacent_form(&self, w: usize) -> [i8; 256] {\n // required by the NAF definition\n debug_assert!(w >= 2);\n // required so that the NAF digits fit in i8\n debug_assert!(w <= 8);\n\n let mut naf = [0i8; 256];\n\n let mut x_u64 = [0u64; 5];\n // AENEAS-COMPAT: index-based little-endian load instead of\n // read_le_u64_into (its chunks/zip iterators are opaque to the\n // extraction). Pure refactor, semantics identical.\n let mut k: usize = 0;\n while k < 4 {\n let mut t: u64 = 0;\n let mut bi: usize = 0;\n while bi < 8 {\n t |= (self.bytes[8 * k + bi] as u64) << (8 * bi);\n bi += 1;\n }\n x_u64[k] = t;\n k += 1;\n }\n\n let width = 1 << w;\n let window_mask = width - 1;\n\n let mut pos = 0;\n let mut carry = 0;\n while pos < 256 {\n // Construct a buffer of bits of the scalar, starting at bit `pos`\n let u64_idx = pos / 64;\n let bit_idx = pos % 64;\n let bit_buf: u64 = if bit_idx < 64 - w {\n // This window's bits are contained in a single u64\n x_u64[u64_idx] >> bit_idx\n } else {\n // Combine the current u64's bits with the bits from the next u64\n (x_u64[u64_idx] >> bit_idx) | (x_u64[1 + u64_idx] << (64 - bit_idx))\n };\n\n // Add the carry into the current window\n let window = carry + (bit_buf & window_mask);\n\n if window & 1 == 0 {\n // If the window value is even, preserve the carry and continue.\n // Why is the carry preserved?\n // If carry == 0 and window & 1 == 0, then the next carry should be 0\n // If carry == 1 and window & 1 == 0, then bit_buf & 1 == 1 so the next carry should be 1\n pos += 1;\n continue;\n }\n\n if window < width / 2 {\n carry = 0;\n naf[pos] = window as i8;\n } else {\n carry = 1;\n naf[pos] = (window as i8).wrapping_sub(width as i8);\n }\n\n pos += w;\n }\n\n naf\n }\n\n /// Compute a width-\\\\(w\\\\) non-adjacent form for scalars known to fit in 128 bits.\n pub(crate) fn non_adjacent_form_128(&self, w: usize) -> [i8; NAF_128_SIZE] {\n // required by the NAF definition\n debug_assert!(w >= 2);\n // required so that the NAF digits fit in i8\n debug_assert!(w <= 8);\n debug_assert!(self.bytes[16..32].iter().all(|&b| b == 0));\n\n let mut naf = [0i8; NAF_128_SIZE];\n\n let mut x_u64 = [0u64; 3];\n read_le_u64_into(&self.bytes[..16], &mut x_u64[0..2]);\n\n let width = 1u64 << w;\n let window_mask = width - 1;\n\n let mut pos = 0;\n let mut carry = 0;\n while pos < HEEA_MAX_INDEX {\n let u64_idx = pos / 64;\n let bit_idx = pos % 64;\n let bit_buf: u64 = if bit_idx < 64 - w {\n x_u64[u64_idx] >> bit_idx\n } else {\n (x_u64[u64_idx] >> bit_idx) | (x_u64[1 + u64_idx] << (64 - bit_idx))\n };\n\n let window = carry + (bit_buf & window_mask);\n\n if window & 1 == 0 {\n pos += 1;\n continue;\n }\n\n if window < width / 2 {\n carry = 0;\n naf[pos] = window as i8;\n } else {\n carry = 1;\n naf[pos] = (window as i8).wrapping_sub(width as i8);\n }\n\n pos += w;\n }\n\n debug_assert_eq!(carry, 0);\n naf\n }\n\n /// Write this scalar in radix 16, with coefficients in \\\\([-8,8)\\\\),\n /// i.e., compute \\\\(a\\_i\\\\) such that\n /// $$\n /// a = a\\_0 + a\\_1 16\\^1 + \\cdots + a_{63} 16\\^{63},\n /// $$\n /// with \\\\(-8 \\leq a_i < 8\\\\) for \\\\(0 \\leq i < 63\\\\) and \\\\(-8 \\leq a_{63} \\leq 8\\\\).\n ///\n /// The largest value that can be decomposed like this is just over \\\\(2^{255}\\\\). Thus, in\n /// order to not error, the top bit MUST NOT be set, i.e., `Self` MUST be less than\n /// \\\\(2^{255}\\\\).\n pub(crate) fn as_radix_16(&self) -> [i8; 64] {\n debug_assert!(self[31] <= 127);\n let mut output = [0i8; 64];\n\n // Step 1: change radix.\n // Convert from radix 256 (bytes) to radix 16 (nibbles)\n #[allow(clippy::identity_op)]\n #[inline(always)]\n fn bot_half(x: u8) -> u8 {\n (x >> 0) & 15\n }\n #[inline(always)]\n fn top_half(x: u8) -> u8 {\n (x >> 4) & 15\n }\n\n for i in 0..32 {\n output[2 * i] = bot_half(self[i]) as i8;\n output[2 * i + 1] = top_half(self[i]) as i8;\n }\n // Precondition note: since self[31] <= 127, output[63] <= 7\n\n // Step 2: recenter coefficients from [0,16) to [-8,8)\n for i in 0..63 {\n let carry = (output[i] + 8) >> 4;\n output[i] -= carry << 4;\n output[i + 1] += carry;\n }\n // Precondition note: output[63] is not recentered. It\n // increases by carry <= 1. Thus output[63] <= 8.\n\n output\n }\n\n /// Returns a size hint indicating how many entries of the return\n /// value of `to_radix_2w` are nonzero.\n #[cfg(any(feature = \"alloc\", all(test, feature = \"precomputed-tables\")))]\n pub(crate) fn to_radix_2w_size_hint(w: usize) -> usize {\n debug_assert!(w >= 4);\n debug_assert!(w <= 8);\n\n let digits_count = match w {\n 4..=7 => 256_usize.div_ceil(w),\n // See comment in to_radix_2w on handling the terminal carry.\n 8 => 256_usize.div_ceil(w) + 1_usize,\n _ => panic!(\"invalid radix parameter\"),\n };\n\n debug_assert!(digits_count <= 64);\n digits_count\n }\n\n /// Creates a representation of a Scalar in radix \\\\( 2^w \\\\) with \\\\(w = 4, 5, 6, 7, 8\\\\) for\n /// use with the Pippenger algorithm. Higher radixes are not supported to save cache space.\n /// Radix 256 is near-optimal even for very large inputs.\n ///\n /// Radix below 16 or above 256 is prohibited.\n /// This method returns digits in a fixed-sized array, excess digits are zeroes.\n ///\n /// For radix 16, `Self` must be less than \\\\(2^{255}\\\\). This is because most integers larger\n /// than \\\\(2^{255}\\\\) are unrepresentable in the form described below for \\\\(w = 4\\\\). This\n /// would be true for \\\\(w = 8\\\\) as well, but it is compensated for by increasing the size\n /// hint by 1.\n ///\n /// ## Scalar representation\n ///\n /// Radix \\\\(2\\^w\\\\), with \\\\(n = ceil(256/w)\\\\) coefficients in \\\\([-(2\\^w)/2,(2\\^w)/2)\\\\),\n /// i.e., scalar is represented using digits \\\\(a\\_i\\\\) such that\n /// $$\n /// a = a\\_0 + a\\_1 2\\^1w + \\cdots + a_{n-1} 2\\^{w*(n-1)},\n /// $$\n /// with \\\\(-2\\^w/2 \\leq a_i < 2\\^w/2\\\\) for \\\\(0 \\leq i < (n-1)\\\\) and \\\\(-2\\^w/2 \\leq a_{n-1} \\leq 2\\^w/2\\\\).\n ///\n #[cfg(any(feature = \"alloc\", feature = \"precomputed-tables\"))]\n pub(crate) fn as_radix_2w(&self, w: usize) -> [i8; 64] {\n debug_assert!(w >= 4);\n debug_assert!(w <= 8);\n\n if w == 4 {\n return self.as_radix_16();\n }\n\n // Scalar formatted as four `u64`s with carry bit packed into the highest bit.\n let mut scalar64x4 = [0u64; 4];\n read_le_u64_into(&self.bytes, &mut scalar64x4[0..4]);\n\n let radix: u64 = 1 << w;\n let window_mask: u64 = radix - 1;\n\n let mut carry = 0u64;\n let mut digits = [0i8; 64];\n let digits_count = 256_usize.div_ceil(w);\n #[allow(clippy::needless_range_loop)]\n for i in 0..digits_count {\n // Construct a buffer of bits of the scalar, starting at `bit_offset`.\n let bit_offset = i * w;\n let u64_idx = bit_offset / 64;\n let bit_idx = bit_offset % 64;\n\n // Read the bits from the scalar\n let bit_buf: u64 = if bit_idx < 64 - w || u64_idx == 3 {\n // This window's bits are contained in a single u64,\n // or it's the last u64 anyway.\n scalar64x4[u64_idx] >> bit_idx\n } else {\n // Combine the current u64's bits with the bits from the next u64\n (scalar64x4[u64_idx] >> bit_idx) | (scalar64x4[1 + u64_idx] << (64 - bit_idx))\n };\n\n // Read the actual coefficient value from the window\n let coef = carry + (bit_buf & window_mask); // coef = [0, 2^r)\n\n // Recenter coefficients from [0,2^w) to [-2^w/2, 2^w/2)\n carry = (coef + (radix / 2)) >> w;\n digits[i] = ((coef as i64) - (carry << w) as i64) as i8;\n }\n\n // When 4 < w < 8, we can fold the final carry onto the last digit d,\n // because d < 2^w/2 so d + carry*2^w = d + 1*2^w < 2^(w+1) < 2^8.\n //\n // When w = 8, we can't fit carry*2^w into an i8. This should\n // not happen anyways, because the final carry will be 0 for\n // reduced scalars, but Scalar invariant #1 allows 255-bit scalars.\n // To handle this, we expand the size_hint by 1 when w=8,\n // and accumulate the final carry onto another digit.\n match w {\n 8 => digits[digits_count] += carry as i8,\n _ => digits[digits_count - 1] += (carry << w) as i8,\n }\n\n digits\n }\n\n /// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.\n pub(crate) fn unpack(&self) -> UnpackedScalar {\n UnpackedScalar::from_bytes(&self.bytes)\n }\n\n /// Create a `Scalar` from a positive `I256`.\n /// The caller must ensure val > 0.\n pub(crate) fn from_positive_i256(val: &I256) -> Self {\n debug_assert!(val >= &I256::ZERO);\n Scalar {\n bytes: val.to_le_bytes(),\n }\n }\n\n /// Reduce this `Scalar` modulo \\\\(\\ell\\\\).\n #[allow(non_snake_case)]\n fn reduce(&self) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut x = self.unpack();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut xR = UnpackedScalar::mul_internal(&x, &constants::R);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut x_mod_l = UnpackedScalar::montgomery_reduce(&xR);\n let scalar = x_mod_l.pack();\n\n #[cfg(feature = \"zeroize\")]\n {\n x.zeroize();\n xR.zeroize();\n x_mod_l.zeroize();\n }\n\n scalar\n }\n\n /// Check whether this `Scalar` is the canonical representative mod \\\\(\\ell\\\\). This is not\n /// public because any `Scalar` that is publicly observed is reduced, by scalar invariant #2.\n fn is_canonical(&self) -> Choice {\n self.ct_eq(&self.reduce())\n }\n}\n\nimpl UnpackedScalar {\n /// Pack the limbs of this `UnpackedScalar` into a `Scalar`.\n fn pack(&self) -> Scalar {\n Scalar {\n bytes: self.to_bytes(),\n }\n }\n\n /// Inverts an UnpackedScalar in Montgomery form.\n #[rustfmt::skip] // keep alignment of addition chain and squarings\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n #[allow(clippy::just_underscores_and_digits)]\n pub fn montgomery_invert(&self) -> UnpackedScalar {\n // Uses the addition chain from\n // https://briansmith.org/ecc-inversion-addition-chains-01#curve25519_scalar_inversion\n let mut _1 = *self;\n let mut _10 = _1.montgomery_square();\n let mut _100 = _10.montgomery_square();\n let mut _11 = UnpackedScalar::montgomery_mul(&_10, &_1);\n let mut _101 = UnpackedScalar::montgomery_mul(&_10, &_11);\n let mut _111 = UnpackedScalar::montgomery_mul(&_10, &_101);\n let mut _1001 = UnpackedScalar::montgomery_mul(&_10, &_111);\n let mut _1011 = UnpackedScalar::montgomery_mul(&_10, &_1001);\n let mut _1111 = UnpackedScalar::montgomery_mul(&_100, &_1011);\n\n // _10000\n let mut y = UnpackedScalar::montgomery_mul(&_1111, &_1);\n\n #[inline]\n fn square_multiply(y: &mut UnpackedScalar, squarings: usize, x: &UnpackedScalar) {\n for _ in 0..squarings {\n *y = y.montgomery_square();\n }\n *y = UnpackedScalar::montgomery_mul(y, x);\n }\n\n square_multiply(&mut y, 123 + 3, &_101);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 4, &_1001);\n square_multiply(&mut y, 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 1 + 3, &_101);\n square_multiply(&mut y, 3 + 3, &_101);\n square_multiply(&mut y, 3, &_111);\n square_multiply(&mut y, 1 + 4, &_1111);\n square_multiply(&mut y, 2 + 3, &_111);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1011);\n square_multiply(&mut y, 2 + 4, &_1011);\n square_multiply(&mut y, 6 + 4, &_1001);\n square_multiply(&mut y, 2 + 2, &_11);\n square_multiply(&mut y, 3 + 2, &_11);\n square_multiply(&mut y, 3 + 2, &_11);\n square_multiply(&mut y, 1 + 4, &_1001);\n square_multiply(&mut y, 1 + 3, &_111);\n square_multiply(&mut y, 2 + 4, &_1111);\n square_multiply(&mut y, 1 + 4, &_1011);\n square_multiply(&mut y, 3, &_101);\n square_multiply(&mut y, 2 + 4, &_1111);\n square_multiply(&mut y, 3, &_101);\n square_multiply(&mut y, 1 + 2, &_11);\n\n let inverse = y;\n\n #[cfg(feature = \"zeroize\")]\n {\n _1.zeroize();\n _10.zeroize();\n _100.zeroize();\n _11.zeroize();\n _101.zeroize();\n _111.zeroize();\n _1001.zeroize();\n _1011.zeroize();\n _1111.zeroize();\n y.zeroize();\n }\n\n inverse\n }\n\n /// Inverts an UnpackedScalar not in Montgomery form.\n pub fn invert(&self) -> UnpackedScalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut montgomery = self.as_montgomery();\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = montgomery.montgomery_invert();\n let result = inverse.from_montgomery();\n\n #[cfg(feature = \"zeroize\")]\n {\n montgomery.zeroize();\n inverse.zeroize();\n }\n\n result\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Field for Scalar {\n const ZERO: Self = Self::ZERO;\n const ONE: Self = Self::ONE;\n\n fn random(mut rng: impl RngCore) -> Self {\n // NOTE: this is duplicated due to different `rng` bounds\n let mut scalar_bytes = [0u8; 64];\n rng.fill_bytes(&mut scalar_bytes);\n let scalar = Self::from_bytes_mod_order_wide(&scalar_bytes);\n\n #[cfg(feature = \"zeroize\")]\n scalar_bytes.zeroize();\n\n scalar\n }\n\n fn square(&self) -> Self {\n self * self\n }\n\n fn double(&self) -> Self {\n self + self\n }\n\n fn invert(&self) -> CtOption {\n CtOption::new(self.invert(), !self.is_zero())\n }\n\n fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {\n #[allow(unused_qualifications)]\n group::ff::helpers::sqrt_ratio_generic(num, div)\n }\n\n fn sqrt(&self) -> CtOption {\n #[allow(unused_qualifications)]\n group::ff::helpers::sqrt_tonelli_shanks(\n self,\n [\n 0xcb02_4c63_4b9e_ba7d,\n 0x029b_df3b_d45e_f39a,\n 0x0000_0000_0000_0000,\n 0x0200_0000_0000_0000,\n ],\n )\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl PrimeField for Scalar {\n type Repr = [u8; 32];\n\n fn from_repr(repr: Self::Repr) -> CtOption {\n Self::from_canonical_bytes(repr)\n }\n\n fn from_repr_vartime(repr: Self::Repr) -> Option {\n // Check that the high bit is not set\n if (repr[31] >> 7) != 0u8 {\n return None;\n }\n\n let candidate = Scalar { bytes: repr };\n\n if candidate == candidate.reduce() {\n Some(candidate)\n } else {\n None\n }\n }\n\n fn to_repr(&self) -> Self::Repr {\n self.to_bytes()\n }\n\n fn is_odd(&self) -> Choice {\n Choice::from(self.as_bytes()[0] & 1)\n }\n\n const MODULUS: &'static str =\n \"0x1000000000000000000000000000000014def9dea2f79cd65812631a5cf5d3ed\";\n const NUM_BITS: u32 = 253;\n const CAPACITY: u32 = 252;\n\n const TWO_INV: Self = Self {\n bytes: [\n 0xf7, 0xe9, 0x7a, 0x2e, 0x8d, 0x31, 0x09, 0x2c, 0x6b, 0xce, 0x7b, 0x51, 0xef, 0x7c,\n 0x6f, 0x0a, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x08,\n ],\n };\n const MULTIPLICATIVE_GENERATOR: Self = Self {\n bytes: [\n 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n const S: u32 = 2;\n const ROOT_OF_UNITY: Self = Self {\n bytes: [\n 0xd4, 0x07, 0xbe, 0xeb, 0xdf, 0x75, 0x87, 0xbe, 0xfe, 0x83, 0xce, 0x42, 0x53, 0x56,\n 0xf0, 0x0e, 0x7a, 0xc2, 0xc1, 0xab, 0x60, 0x6d, 0x3d, 0x7d, 0xe7, 0x81, 0x79, 0xe0,\n 0x10, 0x73, 0x4a, 0x09,\n ],\n };\n const ROOT_OF_UNITY_INV: Self = Self {\n bytes: [\n 0x19, 0xcc, 0x37, 0x71, 0x3a, 0xed, 0x8a, 0x99, 0xd7, 0x18, 0x29, 0x60, 0x8b, 0xa3,\n 0xee, 0x05, 0x86, 0x3d, 0x3e, 0x54, 0x9f, 0x92, 0xc2, 0x82, 0x18, 0x7e, 0x86, 0x1f,\n 0xef, 0x8c, 0xb5, 0x06,\n ],\n };\n const DELTA: Self = Self {\n bytes: [\n 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ],\n };\n}\n\n#[cfg(feature = \"group-bits\")]\nimpl PrimeFieldBits for Scalar {\n type ReprBits = [u8; 32];\n\n fn to_le_bits(&self) -> FieldBits {\n self.to_repr().into()\n }\n\n fn char_le_bits() -> FieldBits {\n constants::BASEPOINT_ORDER.to_bytes().into()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl FromUniformBytes<64> for Scalar {\n fn from_uniform_bytes(bytes: &[u8; 64]) -> Self {\n Scalar::from_bytes_mod_order_wide(bytes)\n }\n}\n\n/// Read one or more u64s stored as little endian bytes.\n///\n/// ## Panics\n/// Panics if `src.len() != 8 * dst.len()`.\nfn read_le_u64_into(src: &[u8], dst: &mut [u64]) {\n assert!(\n src.len() == 8 * dst.len(),\n \"src.len() = {}, dst.len() = {}\",\n src.len(),\n dst.len()\n );\n for (bytes, val) in src.chunks(8).zip(dst.iter_mut()) {\n *val = u64::from_le_bytes(\n bytes\n .try_into()\n .expect(\"Incorrect src length, should be 8 * dst.len()\"),\n );\n }\n}\n\n/// _Clamps_ the given little-endian representation of a 32-byte integer. Clamping the value puts\n/// it in the range:\n///\n/// **n ∈ 2^254 + 8\\*{0, 1, 2, 3, . . ., 2^251 − 1}**\n///\n/// # Explanation of clamping\n///\n/// For Curve25519, h = 8, and multiplying by 8 is the same as a binary left-shift by 3 bits.\n/// If you take a secret scalar value between 2^251 and 2^252 – 1 and left-shift by 3 bits\n/// then you end up with a 255-bit number with the most significant bit set to 1 and\n/// the least-significant three bits set to 0.\n///\n/// The Curve25519 clamping operation takes **an arbitrary 256-bit random value** and\n/// clears the most-significant bit (making it a 255-bit number), sets the next bit, and then\n/// clears the 3 least-significant bits. In other words, it directly creates a scalar value that is\n/// in the right form and pre-multiplied by the cofactor.\n///\n/// See [here](https://neilmadden.blog/2020/05/28/whats-the-curve25519-clamping-all-about/) for\n/// more details.\n#[must_use]\npub const fn clamp_integer(mut bytes: [u8; 32]) -> [u8; 32] {\n bytes[0] &= 0b1111_1000;\n bytes[31] &= 0b0111_1111;\n bytes[31] |= 0b0100_0000;\n bytes\n}\n\n#[cfg(test)]\npub(crate) mod test {\n use super::*;\n\n #[cfg(feature = \"alloc\")]\n use alloc::vec::Vec;\n use rand::RngCore;\n\n /// x = 2238329342913194256032495932344128051776374960164957527413114840482143558222\n pub static X: Scalar = Scalar {\n bytes: [\n 0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84, 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2,\n 0x7d, 0x52, 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44, 0xd4, 0x49, 0xf4, 0xa8,\n 0x79, 0xd9, 0xf2, 0x04,\n ],\n };\n /// 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244\n pub static XINV: Scalar = Scalar {\n bytes: [\n 0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb, 0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01,\n 0x63, 0x47, 0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96, 0xd5, 0x0b, 0xcd, 0x7a,\n 0x3f, 0x96, 0x2a, 0x0f,\n ],\n };\n /// y = 2592331292931086675770238855846338635550719849568364935475441891787804997264\n pub static Y: Scalar = Scalar {\n bytes: [\n 0x90, 0x76, 0x33, 0xfe, 0x1c, 0x4b, 0x66, 0xa4, 0xa2, 0x8d, 0x2d, 0xd7, 0x67, 0x83,\n 0x86, 0xc3, 0x53, 0xd0, 0xde, 0x54, 0x55, 0xd4, 0xfc, 0x9d, 0xe8, 0xef, 0x7a, 0xc3,\n 0x1f, 0x35, 0xbb, 0x05,\n ],\n };\n\n /// The largest scalar that satisfies invariant #1, i.e., the largest scalar with the top bit\n /// set to 0. Since this scalar violates invariant #2, i.e., it's greater than the modulus `l`,\n /// addition and subtraction are broken. The only thing you can do with this is scalar-point\n /// multiplication (and actually also scalar-scalar multiplication, but that's just a quirk of\n /// our implementation).\n pub(crate) static LARGEST_UNREDUCED_SCALAR: Scalar = Scalar {\n bytes: [\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ],\n };\n\n /// x*y = 5690045403673944803228348699031245560686958845067437804563560795922180092780\n static X_TIMES_Y: Scalar = Scalar {\n bytes: [\n 0x6c, 0x33, 0x74, 0xa1, 0x89, 0x4f, 0x62, 0x21, 0x0a, 0xaa, 0x2f, 0xe1, 0x86, 0xa6,\n 0xf9, 0x2c, 0xe0, 0xaa, 0x75, 0xc2, 0x77, 0x95, 0x81, 0xc2, 0x95, 0xfc, 0x08, 0x17,\n 0x9a, 0x73, 0x94, 0x0c,\n ],\n };\n\n /// sage: l = 2^252 + 27742317777372353535851937790883648493\n /// sage: big = 2^256 - 1\n /// sage: repr((big % l).digits(256))\n static CANONICAL_2_256_MINUS_1: Scalar = Scalar {\n bytes: [\n 28, 149, 152, 141, 116, 49, 236, 214, 112, 207, 125, 115, 244, 91, 239, 198, 254, 255,\n 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 15,\n ],\n };\n\n static A_SCALAR: Scalar = Scalar {\n bytes: [\n 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x09,\n ],\n };\n\n static A_NAF: [i8; 256] = [\n 0, 13, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 3, 0, 0,\n 0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 11, 0, 0, 0, 0,\n 11, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,\n 0, -1, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, -15, 0, 0, 0, 0, -7, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 5,\n 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -11, 0, 0, 0, 0, -7, 0, 0, 0, 0, -13, 0, 0,\n 0, 0, 11, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 1, 0, 0, 0, 0,\n 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 15,\n 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -15, 0,\n 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,\n ];\n\n const BASEPOINT_ORDER_MINUS_ONE: Scalar = Scalar {\n bytes: [\n 0xec, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9,\n 0xde, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x10,\n ],\n };\n\n /// The largest clamped integer\n static LARGEST_CLAMPED_INTEGER: [u8; 32] = clamp_integer(LARGEST_UNREDUCED_SCALAR.bytes);\n\n #[test]\n fn fuzzer_testcase_reduction() {\n // LE bytes of 24519928653854221733733552434404946937899825954937634815\n let a_bytes = [\n 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255,\n 255, 255, 255, 255, 255, 255, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n ];\n // LE bytes of 4975441334397345751130612518500927154628011511324180036903450236863266160640\n let b_bytes = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 255, 210, 210,\n 210, 255, 255, 255, 255, 10,\n ];\n // LE bytes of 6432735165214683820902750800207468552549813371247423777071615116673864412038\n let c_bytes = [\n 134, 171, 119, 216, 180, 128, 178, 62, 171, 132, 32, 62, 34, 119, 104, 193, 47, 215,\n 181, 250, 14, 207, 172, 93, 75, 207, 211, 103, 144, 204, 56, 14,\n ];\n\n let a = Scalar::from_bytes_mod_order(a_bytes);\n let b = Scalar::from_bytes_mod_order(b_bytes);\n let c = Scalar::from_bytes_mod_order(c_bytes);\n\n let mut tmp = [0u8; 64];\n\n // also_a = (a mod l)\n tmp[0..32].copy_from_slice(&a_bytes[..]);\n let also_a = Scalar::from_bytes_mod_order_wide(&tmp);\n\n // also_b = (b mod l)\n tmp[0..32].copy_from_slice(&b_bytes[..]);\n let also_b = Scalar::from_bytes_mod_order_wide(&tmp);\n\n let expected_c = a * b;\n let also_expected_c = also_a * also_b;\n\n assert_eq!(c, expected_c);\n assert_eq!(c, also_expected_c);\n }\n\n #[test]\n fn non_adjacent_form_test_vector() {\n let naf = A_SCALAR.non_adjacent_form(5);\n for i in 0..256 {\n assert_eq!(naf[i], A_NAF[i]);\n }\n }\n\n #[test]\n fn non_adjacent_form_128_matches_generic() {\n let mut high_bit = [0u8; 32];\n high_bit[15] = 0x80;\n\n let cases = [\n Scalar::ZERO,\n Scalar::ONE,\n Scalar::from(u64::MAX),\n Scalar::from(0xfedc_ba98_7654_3210_0123_4567_89ab_cdefu128),\n Scalar::from(u128::MAX),\n Scalar { bytes: high_bit },\n ];\n\n for scalar in cases {\n for w in 2..=8 {\n let generic = scalar.non_adjacent_form(w);\n let naf_128 = scalar.non_adjacent_form_128(w);\n\n for i in 0..=HEEA_MAX_INDEX {\n assert_eq!(\n naf_128[i], generic[i],\n \"NAF mismatch at index {i} for width {w}\"\n );\n }\n\n assert!(\n generic[NAF_128_SIZE..].iter().all(|&digit| digit == 0),\n \"generic NAF has non-zero digits above 128-bit range for width {w}\"\n );\n }\n }\n }\n\n #[cfg(feature = \"rand_core\")]\n fn non_adjacent_form_iter(w: usize, x: &Scalar) {\n let naf = x.non_adjacent_form(w);\n\n // Reconstruct the scalar from the computed NAF\n let mut y = Scalar::ZERO;\n for i in (0..256).rev() {\n y += y;\n let digit = if naf[i] < 0 {\n -Scalar::from((-naf[i]) as u64)\n } else {\n Scalar::from(naf[i] as u64)\n };\n y += digit;\n }\n\n assert_eq!(*x, y);\n }\n\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn non_adjacent_form_random() {\n let mut rng = rand::thread_rng();\n for _ in 0..1_000 {\n let x = Scalar::random(&mut rng);\n for w in &[5, 6, 7, 8] {\n non_adjacent_form_iter(*w, &x);\n }\n }\n }\n\n #[test]\n fn from_u64() {\n let val: u64 = 0xdeadbeefdeadbeef;\n let s = Scalar::from(val);\n assert_eq!(s[7], 0xde);\n assert_eq!(s[6], 0xad);\n assert_eq!(s[5], 0xbe);\n assert_eq!(s[4], 0xef);\n assert_eq!(s[3], 0xde);\n assert_eq!(s[2], 0xad);\n assert_eq!(s[1], 0xbe);\n assert_eq!(s[0], 0xef);\n }\n\n #[test]\n fn scalar_mul_by_one() {\n let test_scalar = X * Scalar::ONE;\n for i in 0..32 {\n assert!(test_scalar[i] == X[i]);\n }\n }\n\n #[test]\n fn add_reduces() {\n // Check that addition wraps around the modulus\n assert_eq!(BASEPOINT_ORDER_MINUS_ONE + Scalar::ONE, Scalar::ZERO);\n }\n\n #[test]\n fn sub_reduces() {\n // Check that subtraction wraps around the modulus\n assert_eq!(Scalar::ZERO - Scalar::ONE, BASEPOINT_ORDER_MINUS_ONE);\n }\n\n #[test]\n fn impl_add() {\n let two = Scalar::from(2u64);\n let one = Scalar::ONE;\n let should_be_two = one + one;\n assert_eq!(should_be_two, two);\n }\n\n #[allow(non_snake_case)]\n #[test]\n fn impl_mul() {\n let should_be_X_times_Y = X * Y;\n assert_eq!(should_be_X_times_Y, X_TIMES_Y);\n }\n\n #[allow(non_snake_case)]\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_product() {\n // Test that product works for non-empty iterators\n let X_Y_vector = [X, Y];\n let should_be_X_times_Y: Scalar = X_Y_vector.iter().product();\n assert_eq!(should_be_X_times_Y, X_TIMES_Y);\n\n // Test that product works for the empty iterator\n let one = Scalar::ONE;\n let empty_vector = [];\n let should_be_one: Scalar = empty_vector.iter().product();\n assert_eq!(should_be_one, one);\n\n // Test that product works for iterators where Item = Scalar\n let xs = [Scalar::from(2u64); 10];\n let ys = [Scalar::from(3u64); 10];\n // now zs is an iterator with Item = Scalar\n let zs = xs.iter().zip(ys.iter()).map(|(x, y)| x * y);\n\n let x_prod: Scalar = xs.iter().product();\n let y_prod: Scalar = ys.iter().product();\n let z_prod: Scalar = zs.product();\n\n assert_eq!(x_prod, Scalar::from(1024u64));\n assert_eq!(y_prod, Scalar::from(59049u64));\n assert_eq!(z_prod, Scalar::from(60466176u64));\n assert_eq!(x_prod * y_prod, z_prod);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_sum() {\n // Test that sum works for non-empty iterators\n let two = Scalar::from(2u64);\n let one_vector = [Scalar::ONE, Scalar::ONE];\n let should_be_two: Scalar = one_vector.iter().sum();\n assert_eq!(should_be_two, two);\n\n // Test that sum works for the empty iterator\n let zero = Scalar::ZERO;\n let empty_vector = [];\n let should_be_zero: Scalar = empty_vector.iter().sum();\n assert_eq!(should_be_zero, zero);\n\n // Test that sum works for owned types\n let xs = [Scalar::from(1u64); 10];\n let ys = [Scalar::from(2u64); 10];\n // now zs is an iterator with Item = Scalar\n let zs = xs.iter().zip(ys.iter()).map(|(x, y)| x + y);\n\n let x_sum: Scalar = xs.iter().sum();\n let y_sum: Scalar = ys.iter().sum();\n let z_sum: Scalar = zs.sum();\n\n assert_eq!(x_sum, Scalar::from(10u64));\n assert_eq!(y_sum, Scalar::from(20u64));\n assert_eq!(z_sum, Scalar::from(30u64));\n assert_eq!(x_sum + y_sum, z_sum);\n }\n\n #[test]\n fn square() {\n let expected = X * X;\n let actual = X.unpack().square().pack();\n for i in 0..32 {\n assert!(expected[i] == actual[i]);\n }\n }\n\n #[test]\n fn div_by_2() {\n // test a range of small scalars\n for i in 0u64..32 {\n let scalar = Scalar::from(i);\n let dividend = scalar.div_by_2();\n assert_eq!(scalar, dividend + dividend);\n }\n\n // test a range of scalars near the modulus\n for i in 0u64..32 {\n let scalar = Scalar::ZERO - Scalar::from(i);\n let dividend = scalar.div_by_2();\n assert_eq!(scalar, dividend + dividend);\n }\n }\n\n #[test]\n fn reduce() {\n let biggest = Scalar::from_bytes_mod_order([0xff; 32]);\n assert_eq!(biggest, CANONICAL_2_256_MINUS_1);\n }\n\n #[test]\n fn from_bytes_mod_order_wide() {\n let mut bignum = [0u8; 64];\n // set bignum = x + 2^256x\n for i in 0..32 {\n bignum[i] = X[i];\n bignum[32 + i] = X[i];\n }\n // 3958878930004874126169954872055634648693766179881526445624823978500314864344\n // = x + 2^256x (mod l)\n let reduced = Scalar {\n bytes: [\n 216, 154, 179, 139, 210, 121, 2, 71, 69, 99, 158, 216, 23, 173, 63, 100, 204, 0,\n 91, 50, 219, 153, 57, 249, 28, 82, 31, 197, 100, 165, 192, 8,\n ],\n };\n let test_red = Scalar::from_bytes_mod_order_wide(&bignum);\n for i in 0..32 {\n assert!(test_red[i] == reduced[i]);\n }\n }\n\n #[allow(non_snake_case)]\n #[test]\n fn invert() {\n let inv_X = X.invert();\n assert_eq!(inv_X, XINV);\n let should_be_one = inv_X * X;\n assert_eq!(should_be_one, Scalar::ONE);\n }\n\n // Negating a scalar twice should result in the original scalar.\n #[allow(non_snake_case)]\n #[test]\n fn neg_twice_is_identity() {\n let negative_X = -&X;\n let should_be_X = -&negative_X;\n\n assert_eq!(should_be_X, X);\n }\n\n #[test]\n fn to_bytes_from_bytes_roundtrips() {\n let unpacked = X.unpack();\n let bytes = unpacked.to_bytes();\n let should_be_unpacked = UnpackedScalar::from_bytes(&bytes);\n\n assert_eq!(should_be_unpacked.0, unpacked.0);\n }\n\n #[test]\n fn montgomery_reduce_matches_from_bytes_mod_order_wide() {\n let mut bignum = [0u8; 64];\n\n // set bignum = x + 2^256x\n for i in 0..32 {\n bignum[i] = X[i];\n bignum[32 + i] = X[i];\n }\n // x + 2^256x (mod l)\n // = 3958878930004874126169954872055634648693766179881526445624823978500314864344\n let expected = Scalar {\n bytes: [\n 216, 154, 179, 139, 210, 121, 2, 71, 69, 99, 158, 216, 23, 173, 63, 100, 204, 0,\n 91, 50, 219, 153, 57, 249, 28, 82, 31, 197, 100, 165, 192, 8,\n ],\n };\n let reduced = Scalar::from_bytes_mod_order_wide(&bignum);\n\n // The reduced scalar should match the expected\n assert_eq!(reduced.bytes, expected.bytes);\n\n // (x + 2^256x) * R\n let interim =\n UnpackedScalar::mul_internal(&UnpackedScalar::from_bytes_wide(&bignum), &constants::R);\n // ((x + 2^256x) * R) / R (mod l)\n let montgomery_reduced = UnpackedScalar::montgomery_reduce(&interim);\n\n // The Montgomery reduced scalar should match the reduced one, as well as the expected\n assert_eq!(montgomery_reduced.0, reduced.unpack().0);\n assert_eq!(montgomery_reduced.0, expected.unpack().0)\n }\n\n #[test]\n fn canonical_decoding() {\n // canonical encoding of 1667457891\n let canonical_bytes = [\n 99, 99, 99, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0, 0,\n ];\n\n // encoding of\n // 7265385991361016183439748078976496179028704920197054998554201349516117938192\n // = 28380414028753969466561515933501938171588560817147392552250411230663687203 (mod l)\n // non_canonical because unreduced mod l\n let non_canonical_bytes_because_unreduced = [16; 32];\n\n // encoding with high bit set, to check that the parser isn't pre-masking the high bit\n let non_canonical_bytes_because_highbit = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 128,\n ];\n\n assert!(bool::from(\n Scalar::from_canonical_bytes(canonical_bytes).is_some()\n ));\n assert!(bool::from(\n Scalar::from_canonical_bytes(non_canonical_bytes_because_unreduced).is_none()\n ));\n assert!(bool::from(\n Scalar::from_canonical_bytes(non_canonical_bytes_because_highbit).is_none()\n ));\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_scalar_roundtrip() {\n use bincode;\n let encoded = bincode::serialize(&X).unwrap();\n let parsed: Scalar = bincode::deserialize(&encoded).unwrap();\n assert_eq!(parsed, X);\n\n // Check that the encoding is 32 bytes exactly\n assert_eq!(encoded.len(), 32);\n\n // Check that the encoding itself matches the usual one\n assert_eq!(X, bincode::deserialize(X.as_bytes()).unwrap(),);\n }\n\n #[cfg(debug_assertions)]\n #[test]\n #[should_panic]\n fn invert_batch_with_a_zero_input_panics() {\n let mut xs = [Scalar::ONE; 16];\n xs[3] = Scalar::ZERO;\n // This should panic in debug mode.\n Scalar::invert_batch(&mut xs);\n }\n\n #[test]\n fn invert_batch_empty() {\n assert_eq!(Scalar::ONE, Scalar::invert_batch(&mut []));\n }\n\n #[test]\n fn invert_batch_consistency() {\n let mut x = Scalar::from(1u64);\n let mut v1: [Scalar; 16] = core::array::from_fn(|_| {\n let tmp = x;\n x = x + x;\n tmp\n });\n let v2 = v1;\n\n let expected: Scalar = v1.iter().product();\n let expected = expected.invert();\n let ret = Scalar::invert_batch(&mut v1);\n assert_eq!(ret, expected);\n\n for (a, b) in v1.iter().zip(v2.iter()) {\n assert_eq!(a * b, Scalar::ONE);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn batch_vec_invert_consistency() {\n let mut x = Scalar::from(1u64);\n let mut v1: Vec<_> = (0..16)\n .map(|_| {\n let tmp = x;\n x = x + x;\n tmp\n })\n .collect();\n let v2 = v1.clone();\n\n let expected: Scalar = v1.iter().product();\n let expected = expected.invert();\n let ret = Scalar::invert_batch_alloc(&mut v1);\n assert_eq!(ret, expected);\n\n for (a, b) in v1.iter().zip(v2.iter()) {\n assert_eq!(a * b, Scalar::ONE);\n }\n }\n\n #[cfg(feature = \"precomputed-tables\")]\n fn test_pippenger_radix_iter(scalar: Scalar, w: usize) {\n let digits_count = Scalar::to_radix_2w_size_hint(w);\n let digits = scalar.as_radix_2w(w);\n\n let radix = Scalar::from((1 << w) as u64);\n let mut term = Scalar::ONE;\n let mut recovered_scalar = Scalar::ZERO;\n for digit in &digits[0..digits_count] {\n let digit = *digit;\n if digit != 0 {\n let sdigit = if digit < 0 {\n -Scalar::from((-(digit as i64)) as u64)\n } else {\n Scalar::from(digit as u64)\n };\n recovered_scalar += term * sdigit;\n }\n term *= radix;\n }\n // When the input is unreduced, we may only recover the scalar mod l.\n assert_eq!(recovered_scalar, scalar.reduce());\n }\n\n #[test]\n #[cfg(feature = \"precomputed-tables\")]\n fn test_pippenger_radix() {\n use core::iter;\n // For each valid radix it tests that 1000 random-ish scalars can be restored\n // from the produced representation precisely.\n let cases = (2..100)\n .map(|s| Scalar::from(s as u64).invert())\n // The largest unreduced scalar, s = 2^255-1. This is not reduced mod l. Scalar mult\n // still works though.\n .chain(iter::once(LARGEST_UNREDUCED_SCALAR));\n\n for scalar in cases {\n test_pippenger_radix_iter(scalar, 6);\n test_pippenger_radix_iter(scalar, 7);\n test_pippenger_radix_iter(scalar, 8);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn test_read_le_u64_into() {\n let cases: &[(&[u8], &[u64])] = &[\n (\n &[0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F, 0xF0],\n &[0xF00F_F11F_0110_EFFE],\n ),\n (\n &[\n 0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F, 0xF0, 0x12, 0x34, 0x56, 0x78, 0x9A,\n 0xBC, 0xDE, 0xF0,\n ],\n &[0xF00F_F11F_0110_EFFE, 0xF0DE_BC9A_7856_3412],\n ),\n ];\n\n for (src, expected) in cases {\n let mut dst = vec![0; expected.len()];\n read_le_u64_into(src, &mut dst);\n\n assert_eq!(&dst, expected, \"Expected {:x?} got {:x?}\", expected, dst);\n }\n }\n\n // Tests consistency of From<{integer}> impls for Scalar\n #[test]\n fn test_scalar_from_int() {\n let s1 = Scalar::ONE;\n\n // For `x` in `u8`, `u16`, `u32`, `u64`, and `u128`, check that\n // `Scalar::from(x + 1) == Scalar::from(x) + Scalar::from(1)`\n\n let x = 0x23u8;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323u16;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323u32;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323_2323_2323u64;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n\n let x = 0x2323_2323_2323_2323_2323_2323_2323_2323u128;\n let sx = Scalar::from(x);\n assert_eq!(sx + s1, Scalar::from(x + 1));\n }\n\n #[cfg(feature = \"group\")]\n #[test]\n fn ff_constants() {\n assert_eq!(Scalar::from(2u64) * Scalar::TWO_INV, Scalar::ONE);\n\n assert_eq!(\n Scalar::ROOT_OF_UNITY * Scalar::ROOT_OF_UNITY_INV,\n Scalar::ONE,\n );\n\n // ROOT_OF_UNITY^{2^s} mod m == 1\n assert_eq!(\n Scalar::ROOT_OF_UNITY.pow(&[1u64 << Scalar::S, 0, 0, 0]),\n Scalar::ONE,\n );\n\n // DELTA^{t} mod m == 1\n assert_eq!(\n Scalar::DELTA.pow(&[\n 0x9604_98c6_973d_74fb,\n 0x0537_be77_a8bd_e735,\n 0x0000_0000_0000_0000,\n 0x0400_0000_0000_0000,\n ]),\n Scalar::ONE,\n );\n }\n\n #[cfg(feature = \"group\")]\n #[test]\n fn ff_impls() {\n assert!(bool::from(Scalar::ZERO.is_even()));\n assert!(bool::from(Scalar::ONE.is_odd()));\n assert!(bool::from(Scalar::from(2u64).is_even()));\n assert!(bool::from(Scalar::DELTA.is_even()));\n\n assert!(bool::from(Field::invert(&Scalar::ZERO).is_none()));\n assert_eq!(Field::invert(&X).unwrap(), XINV);\n\n let x_sq = X.square();\n // We should get back either the positive or negative root.\n assert!([X, -X].contains(&x_sq.sqrt().unwrap()));\n\n assert_eq!(Scalar::from_repr_vartime(X.to_repr()), Some(X));\n assert_eq!(Scalar::from_repr_vartime([0xff; 32]), None);\n\n assert_eq!(Scalar::from_repr(X.to_repr()).unwrap(), X);\n assert!(bool::from(Scalar::from_repr([0xff; 32]).is_none()));\n }\n\n #[test]\n #[should_panic]\n fn test_read_le_u64_into_should_panic_on_bad_input() {\n let mut dst = [0_u64; 1];\n // One byte short\n read_le_u64_into(&[0xFE, 0xEF, 0x10, 0x01, 0x1F, 0xF1, 0x0F], &mut dst);\n }\n\n #[test]\n fn test_scalar_clamp() {\n let input = A_SCALAR.bytes;\n let expected = [\n 0x18, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x49,\n ];\n let actual = clamp_integer(input);\n assert_eq!(actual, expected);\n\n let expected = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0x40,\n ];\n let actual = clamp_integer([0; 32]);\n assert_eq!(expected, actual);\n let expected = [\n 0xf8, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ];\n let actual = clamp_integer([0xff; 32]);\n assert_eq!(actual, expected);\n\n assert_eq!(\n LARGEST_CLAMPED_INTEGER,\n clamp_integer(LARGEST_CLAMPED_INTEGER)\n );\n }\n\n // Check that a * b == a.reduce() * a.reduce() for ANY scalars a,b, even ones that violate\n // invariant #1, i.e., a,b > 2^255. Old versions of ed25519-dalek did multiplication where a\n // was reduced and b was clamped and unreduced. This checks that was always well-defined.\n #[test]\n fn test_mul_reduction_invariance() {\n let mut rng = rand::thread_rng();\n\n for _ in 0..10 {\n // Also define c that's clamped. We'll make sure that clamping doesn't affect\n // computation\n let (a, b, c) = {\n let mut a_bytes = [0u8; 32];\n let mut b_bytes = [0u8; 32];\n let mut c_bytes = [0u8; 32];\n rng.fill_bytes(&mut a_bytes);\n rng.fill_bytes(&mut b_bytes);\n rng.fill_bytes(&mut c_bytes);\n (\n Scalar { bytes: a_bytes },\n Scalar { bytes: b_bytes },\n Scalar {\n bytes: clamp_integer(c_bytes),\n },\n )\n };\n\n // Make sure this is the same product no matter how you cut it\n let reduced_mul_ab = a.reduce() * b.reduce();\n let reduced_mul_ac = a.reduce() * c.reduce();\n assert_eq!(a * b, reduced_mul_ab);\n assert_eq!(a.reduce() * b, reduced_mul_ab);\n assert_eq!(a * b.reduce(), reduced_mul_ab);\n assert_eq!(a * c, reduced_mul_ac);\n assert_eq!(a.reduce() * c, reduced_mul_ac);\n assert_eq!(a * c.reduce(), reduced_mul_ac);\n }\n }\n}\n"},{"id":6,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs/verification_key.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of solana-ed25519's ed_sigs module, forked from ed25519-zebra.\n// Original ed25519-zebra code: Copyright (c) Zcash Foundation contributors\n// Modifications for HEEA: Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE-APACHE and LICENSE-MIT for licensing information.\n//\n// Modifications from ed25519-zebra:\n// - Added `verify_zebra`, an accelerated verification path using the HEEA\n// scalar decomposition from curve25519-sol's `HEEADecomposition` trait.\n// See \"Accelerating EdDSA Signature Verification with Faster Scalar Size\n// Halving\" (TCHES 2025) for the algorithm.\n// - `verify` dispatches to `verify_zebra`, preserving ZIP-215 semantics.\n\nuse crate::{\n edwards::{CompressedEdwardsY, EdwardsPoint},\n scalar::Scalar,\n traits::{HEEADecomposition, IsIdentity},\n};\nuse core::convert::{TryFrom, TryInto};\nuse sha2::{Sha512, digest::Update};\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\nuse ed25519::{Signature, signature::Verifier};\n\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::der::asn1::BitStringRef;\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::spki::{\n AlgorithmIdentifierRef, DecodePublicKey, EncodePublicKey, Error as SpkiError,\n SubjectPublicKeyInfoRef,\n};\n#[cfg(feature = \"pkcs8\")]\nuse pkcs8::{Document, ObjectIdentifier};\n\nuse super::{Error, scalar_from_sha512};\n\n/// The length of an ed25519 `VerificationKey`, in bytes.\npub const VERIFICATION_KEY_LENGTH: usize = 32;\n\n#[cfg(feature = \"pkcs8\")]\nconst OID: ObjectIdentifier = ObjectIdentifier::new_unwrap(\"1.3.101.112\"); // RFC 8410\n#[cfg(feature = \"pkcs8\")]\nconst ALGORITHM_ID: AlgorithmIdentifierRef<'_> = AlgorithmIdentifierRef {\n oid: OID,\n parameters: None,\n};\n\nconst LEGACY_EXCLUDED_R_ENCODINGS: [[u8; 32]; 11] = [\n [\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00,\n ],\n [\n 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00,\n ],\n [\n 0x26, 0xe8, 0x95, 0x8f, 0xc2, 0xb2, 0x27, 0xb0, 0x45, 0xc3, 0xf4, 0x89, 0xf2, 0xef, 0x98,\n 0xf0, 0xd5, 0xdf, 0xac, 0x05, 0xd3, 0xc6, 0x33, 0x39, 0xb1, 0x38, 0x02, 0x88, 0x6d, 0x53,\n 0xfc, 0x05,\n ],\n [\n 0xc7, 0x17, 0x6a, 0x70, 0x3d, 0x4d, 0xd8, 0x4f, 0xba, 0x3c, 0x0b, 0x76, 0x0d, 0x10, 0x67,\n 0x0f, 0x2a, 0x20, 0x53, 0xfa, 0x2c, 0x39, 0xcc, 0xc6, 0x4e, 0xc7, 0xfd, 0x77, 0x92, 0xac,\n 0x03, 0x7a,\n ],\n [\n 0x13, 0xe8, 0x95, 0x8f, 0xc2, 0xb2, 0x27, 0xb0, 0x45, 0xc3, 0xf4, 0x89, 0xf2, 0xef, 0x98,\n 0xf0, 0xd5, 0xdf, 0xac, 0x05, 0xd3, 0xc6, 0x33, 0x39, 0xb1, 0x38, 0x02, 0x88, 0x6d, 0x53,\n 0xfc, 0x85,\n ],\n [\n 0xb4, 0x17, 0x6a, 0x70, 0x3d, 0x4d, 0xd8, 0x4f, 0xba, 0x3c, 0x0b, 0x76, 0x0d, 0x10, 0x67,\n 0x0f, 0x2a, 0x20, 0x53, 0xfa, 0x2c, 0x39, 0xcc, 0xc6, 0x4e, 0xc7, 0xfd, 0x77, 0x92, 0xac,\n 0x03, 0xfa,\n ],\n [\n 0xec, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xed, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0x7f,\n ],\n [\n 0xd9, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff,\n ],\n [\n 0xda, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff,\n ],\n];\n\n/// A container for the 32-byte encoded form of an Ed25519 verification key.\n///\n/// This type only checks or carries the byte length. It does not prove that the\n/// bytes decompress to a valid Ed25519 verification key. Convert it to\n/// [`VerificationKey`] to validate the encoded point and cache decoded state\n/// used in signature verification.\n///\n/// A `VerificationKeyBytes` can be used to verify a single signature using the\n/// following idiom:\n/// ```\n/// use core::convert::TryFrom;\n/// # use curve25519::ed_sigs::*;\n/// # let msg = b\"Zcash\";\n/// # let sk = SigningKey::from_bytes(&[1u8; 32]);\n/// # let sig = sk.sign(msg);\n/// # let vk_bytes = VerificationKeyBytes::from(&sk);\n/// VerificationKey::try_from(vk_bytes)\n/// .and_then(|vk| vk.verify(&sig, msg));\n/// ```\n#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\npub struct VerificationKeyBytes(pub(crate) [u8; VERIFICATION_KEY_LENGTH]);\n\nimpl core::fmt::Debug for VerificationKeyBytes {\n fn fmt(&self, fmt: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n fmt.debug_tuple(\"VerificationKeyBytes\")\n .field(&self.0)\n .finish()\n }\n}\n\nimpl AsRef<[u8]> for VerificationKeyBytes {\n fn as_ref(&self) -> &[u8] {\n &self.0[..]\n }\n}\n\nimpl TryFrom<&[u8]> for VerificationKeyBytes {\n type Error = Error;\n fn try_from(slice: &[u8]) -> Result {\n if slice.len() == 32 {\n let mut bytes = [0u8; 32];\n bytes[..].copy_from_slice(slice);\n Ok(bytes.into())\n } else {\n Err(Error::InvalidSliceLength)\n }\n }\n}\n\nimpl From<[u8; 32]> for VerificationKeyBytes {\n fn from(bytes: [u8; 32]) -> VerificationKeyBytes {\n VerificationKeyBytes(bytes)\n }\n}\n\nimpl From for [u8; 32] {\n fn from(refined: VerificationKeyBytes) -> [u8; 32] {\n refined.0\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl<'a> TryFrom> for VerificationKeyBytes {\n type Error = Error;\n\n fn try_from(spki: SubjectPublicKeyInfoRef<'a>) -> Result {\n verification_key_bytes_from_spki(spki).map_err(|_| Error::MalformedPublicKey)\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nfn verification_key_bytes_from_spki(\n spki: SubjectPublicKeyInfoRef<'_>,\n) -> Result {\n if spki.algorithm.oid != OID {\n return Err(SpkiError::OidUnknown {\n oid: spki.algorithm.oid,\n });\n }\n\n if spki.algorithm != ALGORITHM_ID {\n return Err(SpkiError::KeyMalformed);\n }\n\n let bytes = spki\n .subject_public_key\n .as_bytes()\n .ok_or(SpkiError::KeyMalformed)?;\n\n VerificationKeyBytes::try_from(bytes).map_err(|_| SpkiError::KeyMalformed)\n}\n\n/// A valid Ed25519 verification key.\n///\n/// This is also called a public key by other implementations.\n///\n/// This type holds decompressed state used in signature verification; if the\n/// verification key may not be used immediately, it is probably better to use\n/// [`VerificationKeyBytes`], which stores only the length-checked encoded bytes.\n///\n/// ## Zcash-specific consensus properties\n///\n/// Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in\n/// [ZIP 215]. The verification criteria for an (encoded) verification key `A_bytes` are:\n///\n/// * `A_bytes` MUST be an encoding of a point `A` on the twisted Edwards form of\n/// Curve25519, and non-canonical encodings MUST be accepted;\n///\n/// [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519\n#[derive(PartialEq, Eq, Copy, Clone, Debug)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\n#[cfg_attr(feature = \"serde\", serde(try_from = \"VerificationKeyBytes\"))]\n#[cfg_attr(feature = \"serde\", serde(into = \"VerificationKeyBytes\"))]\n#[allow(non_snake_case)]\npub struct VerificationKey {\n pub(crate) A_bytes: VerificationKeyBytes,\n pub(crate) minus_A: EdwardsPoint,\n}\n\nimpl From for VerificationKeyBytes {\n fn from(vk: VerificationKey) -> VerificationKeyBytes {\n vk.A_bytes\n }\n}\n\nimpl AsRef<[u8]> for VerificationKey {\n fn as_ref(&self) -> &[u8] {\n &self.A_bytes.0[..]\n }\n}\n\nimpl Default for VerificationKey {\n fn default() -> VerificationKey {\n let identity: EdwardsPoint = Default::default();\n let identity_bytes = identity.compress().to_bytes();\n\n VerificationKey {\n A_bytes: VerificationKeyBytes::from(identity_bytes),\n minus_A: -identity,\n }\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for VerificationKey {}\n\nimpl From for [u8; 32] {\n fn from(vk: VerificationKey) -> [u8; 32] {\n vk.A_bytes.0\n }\n}\n\nimpl TryFrom for VerificationKey {\n type Error = Error;\n #[allow(non_snake_case)]\n fn try_from(bytes: VerificationKeyBytes) -> Result {\n // * `A_bytes` and `R_bytes` MUST be encodings of points `A` and `R` respectively on the\n // twisted Edwards form of Curve25519, and non-canonical encodings MUST be accepted;\n let A = CompressedEdwardsY(bytes.0)\n .decompress()\n .ok_or(Error::MalformedPublicKey)?;\n\n Ok(VerificationKey {\n A_bytes: bytes,\n minus_A: -A,\n })\n }\n}\n\nimpl TryFrom<&[u8]> for VerificationKey {\n type Error = Error;\n fn try_from(slice: &[u8]) -> Result {\n VerificationKeyBytes::try_from(slice).and_then(|vkb| vkb.try_into())\n }\n}\n\nimpl TryFrom<[u8; 32]> for VerificationKey {\n type Error = Error;\n fn try_from(bytes: [u8; 32]) -> Result {\n VerificationKeyBytes::from(bytes).try_into()\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl EncodePublicKey for VerificationKey {\n /// Serialize [`VerificationKey`] to an ASN.1 DER-encoded document.\n fn to_public_key_der(&self) -> pkcs8::spki::Result {\n SubjectPublicKeyInfoRef {\n algorithm: ALGORITHM_ID,\n subject_public_key: BitStringRef::from_bytes(&self.A_bytes.0[..])?,\n }\n .try_into()\n }\n}\n\n#[cfg(feature = \"pkcs8\")]\nimpl DecodePublicKey for VerificationKey {\n /// Deserialize [`VerificationKey`] from ASN.1 DER bytes (32 bytes).\n fn from_public_key_der(bytes: &[u8]) -> Result {\n let spki = SubjectPublicKeyInfoRef::try_from(bytes)?;\n let pk_bytes = verification_key_bytes_from_spki(spki)?;\n Self::try_from(pk_bytes).map_err(|_| SpkiError::KeyMalformed)\n }\n}\n\nimpl Verifier for VerificationKey {\n /// Verify a [`Signature`] object against a given [`VerificationKey`].\n fn verify(\n &self,\n message: &[u8],\n signature: &Signature,\n ) -> Result<(), ed25519::signature::Error> {\n self.verify(signature, message)\n .map_err(|_| ed25519::signature::Error::new())\n }\n}\n\nimpl VerificationKey {\n fn challenge_scalar(&self, signature: &Signature, msg: &[u8]) -> Scalar {\n scalar_from_sha512(\n Sha512::default()\n .chain(&signature.r_bytes()[..])\n .chain(&self.A_bytes.0[..])\n .chain(msg),\n )\n }\n\n /// Verify a purported `signature` on the given `msg`.\n ///\n /// This is the default verification mode and uses the HEEA-accelerated\n /// verification path with Zebra / ZIP-215 semantics.\n ///\n /// ## Zcash-specific consensus properties\n ///\n /// Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in\n /// [ZIP215]. The verification criteria for an (encoded) signature `(R_bytes, s_bytes)` with\n /// (encoded) verification key `A_bytes` are:\n ///\n /// * `A_bytes` and `R_bytes` MUST be encodings of points `A` and `R` respectively on the\n /// twisted Edwards form of Curve25519, and non-canonical encodings MUST be accepted;\n ///\n /// * `s_bytes` MUST represent an integer `s` less than `l`, the order of the prime-order\n /// subgroup of Curve25519;\n ///\n /// * the verification equation `[8][s]B = [8]R + [8][k]A` MUST be satisfied;\n ///\n /// * the alternate verification equation `[s]B = R + [k]A`, allowed by RFC 8032, MUST NOT be\n /// used.\n ///\n /// [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519\n /// [ZIP215]: https://zips.z.cash/zip-0215\n pub fn verify(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_zebra(signature, msg)\n }\n\n /// Verify a signature using HEEA with Zebra / ZIP-215 semantics.\n ///\n /// This implements the algorithm from \"Accelerating EdDSA Signature Verification\n /// with Faster Scalar Size Halving\" (TCHES 2025).\n ///\n /// The decomposition returns ρ and τ such that either ρ ≡ τh (mod ℓ) or\n /// ρ ≡ -τh (mod ℓ). The standard verification equation sB = R + hA is\n /// multiplied by τ and the sign of A is selected according to `flip_h`.\n ///\n /// Both ρ and τ are approximately half the size of h.\n ///\n /// We then decompose τs into two 128-bit scalars:\n /// τs = τs_hi * 2^128 + τs_lo\n ///\n /// The resulting equation can be checked with a 4-variable MSM with\n /// half-size scalars.\n #[allow(non_snake_case)]\n pub fn verify_zebra(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_zebra_prehashed(signature, self.challenge_scalar(signature, msg))\n }\n\n #[allow(non_snake_case)]\n pub(crate) fn verify_zebra_prehashed(\n &self,\n signature: &Signature,\n h: Scalar,\n ) -> Result<(), Error> {\n // Generate half-size scalars ρ and τ. If flip_h is false, then\n // ρ ≡ τh (mod ℓ). If flip_h is true, then ρ ≡ -τh (mod ℓ), so the\n // sign of A is flipped below.\n let (rho, tau, flip_h) = h.heea_decompose();\n\n // Extract s from the signature\n let s = Option::::from(Scalar::from_canonical_bytes(*signature.s_bytes()))\n .ok_or(Error::InvalidSignature)?;\n\n // Decode R from the signature\n let neg_R = -CompressedEdwardsY(*signature.r_bytes())\n .decompress()\n .ok_or(Error::InvalidSignature)?;\n\n // Standard verification checks: sB = R + hA.\n //\n // We verify:\n // [8] τs B + [8] τ (-R) + [8] ρ A_term == 0\n // where A_term is -A when ρ ≡ τh and A when ρ ≡ -τh.\n\n // Compute τs\n let ts = tau * s;\n let A = if flip_h { -self.minus_A } else { self.minus_A };\n // HEEA decomposition guarantees tau and rho fit the optimized\n // 128/128/256-bit multiplication path.\n let result = crate::backend::vartime_triple_base_mul_128_128_256_prechecked(\n &tau, &neg_R, &rho, &A, &ts,\n );\n\n if result.mul_by_cofactor().is_identity() {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n\n /// Verify a signature with dalek-style canonical-`R` byte comparison.\n ///\n /// This recomputes the expected canonical `R` encoding and compares it to the\n /// signature's `R` bytes.\n ///\n /// This helper also preserves this crate's legacy compatibility filters: it\n /// rejects an all-zero encoded public key and the known legacy-excluded `R`\n /// encodings before running the canonical-`R` comparison. Because of those\n /// extra checks, this is not a byte-for-byte clone of every `ed25519-dalek`\n /// release.\n ///\n /// Note that dalek-style canonical-`R` comparison is incompatible with the HEEA\n /// transformed equation because the transformed check does not preserve the\n /// original `R` encoding needed for the byte comparison.\n #[allow(non_snake_case)]\n pub fn verify_dalek(&self, signature: &Signature, msg: &[u8]) -> Result<(), Error> {\n self.verify_dalek_prehashed(signature, self.challenge_scalar(signature, msg))\n }\n\n #[allow(non_snake_case)]\n fn verify_dalek_prehashed(&self, signature: &Signature, h: Scalar) -> Result<(), Error> {\n if self.A_bytes.0 == [0; 32] || LEGACY_EXCLUDED_R_ENCODINGS.contains(signature.r_bytes()) {\n return Err(Error::InvalidSignature);\n }\n\n let s = Option::::from(Scalar::from_canonical_bytes(*signature.s_bytes()))\n .ok_or(Error::InvalidSignature)?;\n\n let expected_R =\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&h, &self.minus_A, &s).compress();\n\n if expected_R.as_bytes() == signature.r_bytes() {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n}\n\n// ---------------------------------------------------------------------------\n// AENEAS-COMPAT verified-verification entry points.\n//\n// `verify_sha512` is semantically `verify_dalek` with each step spelled in\n// extractor-friendly form: the derived array `PartialEq`/`contains` become\n// explicit index loops, and `Scalar::from_canonical_bytes` (whose `subtle`\n// internals defeat the extractor) becomes an explicit `s < l` byte compare\n// followed by `Scalar::from_bytes_mod_order` (the identity on canonical\n// bytes). The verification path is variable-time throughout, as upstream's.\n// ---------------------------------------------------------------------------\n\n/// `bytes` interpreted little-endian is a canonical scalar (< l)? If so the\n/// scalar itself; value-level semantics identical to\n/// `Scalar::from_canonical_bytes(bytes).into()`.\nfn check_scalar_canonical(bytes: [u8; 32]) -> Result {\n /// l = 2^252 + 27742317777372353535851937790883648493, little-endian.\n const L_BYTES: [u8; 32] = [\n 237, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222,\n 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16,\n ];\n // bytes < l, most-significant byte first; the first differing byte decides.\n let mut lt = false;\n let mut decided = false;\n let mut i = 32;\n while i > 0 {\n let j = i - 1;\n if !decided {\n if bytes[j] < L_BYTES[j] {\n lt = true;\n decided = true;\n } else if bytes[j] > L_BYTES[j] {\n decided = true;\n }\n }\n i -= 1;\n }\n if lt {\n Ok(Scalar::from_bytes_mod_order(bytes))\n } else {\n Err(Error::InvalidSignature)\n }\n}\n\n/// Explicit-loop `LEGACY_EXCLUDED_R_ENCODINGS.contains(r)`.\nfn is_legacy_excluded_r(r: &[u8; 32]) -> bool {\n let mut found = false;\n let mut i = 0;\n while i < 11 {\n let mut eq = true;\n let mut j = 0;\n while j < 32 {\n if LEGACY_EXCLUDED_R_ENCODINGS[i][j] != r[j] {\n eq = false;\n }\n j += 1;\n }\n if eq {\n found = true;\n }\n i += 1;\n }\n found\n}\n\nimpl VerificationKey {\n /// Explicit-loop `self.A_bytes.0 != [0; 32]`.\n fn a_bytes_nonzero(&self) -> bool {\n let mut nonzero = false;\n let mut i = 0;\n while i < 32 {\n if self.A_bytes.0[i] != 0 {\n nonzero = true;\n }\n i += 1;\n }\n nonzero\n }\n\n /// Recompute the expected canonical `R` encoding:\n /// `compress([k]*(-A) + [s]*B)` with `k = SHA-512(R || A || msg) mod l`.\n fn recompute_r_sha512(\n &self,\n r_bytes: &[u8; 32],\n s: &Scalar,\n msg: &[u8],\n ) -> CompressedEdwardsY {\n let k = Scalar::from_bytes_mod_order_wide(&super::sha512_hash3(\n &r_bytes[..],\n &self.A_bytes.0[..],\n msg,\n ));\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&k, &self.minus_A, s).compress()\n }\n\n /// Semantically identical to [`Self::verify_dalek`]; see the module\n /// comment above for the extractor-friendly spellings. The signature's\n /// `R`/`s` accessors are each called exactly once.\n pub fn verify_sha512(&self, sig: &Signature, msg: &[u8]) -> Result<(), Error> {\n // (parameter named `sig`: the extractor's generated code would\n // otherwise shadow the `signature::` crate namespace)\n let r_bytes: [u8; 32] = *sig.r_bytes();\n let s_bytes: [u8; 32] = *sig.s_bytes();\n\n if !self.a_bytes_nonzero() {\n return Err(Error::InvalidSignature);\n }\n if is_legacy_excluded_r(&r_bytes) {\n return Err(Error::InvalidSignature);\n }\n let s = check_scalar_canonical(s_bytes)?;\n\n let expected_r = self.recompute_r_sha512(&r_bytes, &s, msg);\n let e = expected_r.as_bytes();\n let mut equal = true;\n let mut k = 0;\n while k < 32 {\n if e[k] != r_bytes[k] {\n equal = false;\n }\n k += 1;\n }\n if equal {\n Ok(())\n } else {\n Err(Error::InvalidSignature)\n }\n }\n}\n"},{"id":7,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs.rs"},"crate_name":"curve25519","contents":"//! Ed25519 signing and verification (ZIP-215 / HEEA-accelerated).\n\nuse crate::scalar::Scalar;\nuse sha2::{Digest, Sha512};\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n#[cfg(test)]\nmod tests;\n\n#[cfg(feature = \"alloc\")]\npub mod batch;\nmod error;\nmod signing_key;\nmod verification_key;\n\n// Allows importing traits used by `Signature`.\npub use ::ed25519;\npub use ::ed25519::Signature;\npub use error::Error;\npub use signing_key::SigningKey;\npub use verification_key::{VerificationKey, VerificationKeyBytes};\n\npub(crate) fn scalar_from_sha512(hash: Sha512) -> Scalar {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut output = hash.finalize();\n let mut bytes = [0u8; 64];\n bytes.copy_from_slice(output.as_slice());\n let scalar = Scalar::from_bytes_mod_order_wide(&bytes);\n\n #[cfg(feature = \"zeroize\")]\n {\n output.zeroize();\n bytes.zeroize();\n }\n\n scalar\n}\n\n/// AENEAS-COMPAT: single-call SHA-512 oracle for the verified verification\n/// path. Semantically `Sha512(r || a || m)` — identical to hashing via the\n/// incremental `Digest` API. A single monomorphic function whose signature\n/// carries no foreign types lets the extractor treat the whole hash\n/// computation as one opaque oracle.\npub(crate) fn sha512_hash3(r: &[u8], a: &[u8], m: &[u8]) -> [u8; 64] {\n let mut h = Sha512::default();\n Digest::update(&mut h, r);\n Digest::update(&mut h, a);\n Digest::update(&mut h, m);\n let output = h.finalize();\n let mut bytes = [0u8; 64];\n bytes.copy_from_slice(output.as_slice());\n bytes\n}\n"},{"id":8,"name":{"Local":"curve25519/solana-ed25519/src/field.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis agora lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - Isis Agora Lovecruft \n// - Henry de Valence \n\n//! Field arithmetic modulo \\\\(p = 2\\^{255} - 19\\\\).\n//!\n//! The `curve25519::field` module provides a type alias\n//! `curve25519::field::FieldElement` to a field element type\n//! defined in the `backend` module; either `FieldElement51` or\n//! `FieldElement2625`.\n//!\n//! Field operations defined in terms of machine\n//! operations, such as field multiplication or squaring, are defined in\n//! the backend implementation.\n//!\n//! Field operations defined in terms of other field operations, such as\n//! field inversion or square roots, are defined here.\n\n#![allow(unused_qualifications)]\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\nuse crate::backend;\nuse crate::constants;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n#[cfg(feature = \"digest\")]\nuse digest::{\n Digest, FixedOutput, HashMarker,\n array::{Array, typenum::U64},\n block_api::BlockSizeUser,\n typenum::{IsGreater, True},\n};\n\n/// A `FieldElement` represents an element of the field\n/// \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\).\n///\n/// The `FieldElement` type is pinned to the 64-bit serial implementation.\npub(crate) type FieldElement = backend::serial::u64::field::FieldElement51;\n\nimpl Eq for FieldElement {}\n\nimpl PartialEq for FieldElement {\n fn eq(&self, other: &FieldElement) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl ConstantTimeEq for FieldElement {\n /// Test equality between two `FieldElement`s. Since the\n /// internal representation is not canonical, the field elements\n /// are normalized to wire format before comparison.\n fn ct_eq(&self, other: &FieldElement) -> Choice {\n self.to_bytes().ct_eq(&other.to_bytes())\n }\n}\n\nimpl Default for FieldElement {\n fn default() -> Self {\n FieldElement::ZERO\n }\n}\n\nimpl FieldElement {\n /// Load a `FieldElement` from 64 bytes, by reducing modulo q.\n #[cfg(feature = \"digest\")]\n pub(crate) fn from_bytes_wide(bytes: &[u8; 64]) -> Self {\n let mut fl = [0u8; 32];\n let mut gl = [0u8; 32];\n fl.copy_from_slice(&bytes[..32]);\n gl.copy_from_slice(&bytes[32..]);\n // Mask off the top bits of both halves, since from_bytes masks them off anyway. We'll add\n // them back in later.\n let fl_top_bit = (fl[31] >> 7) as u16;\n let gl_top_bit = (gl[31] >> 7) as u16;\n fl[31] &= 0x7f;\n gl[31] &= 0x7f;\n\n // Interpret both sides as field elements\n let mut fe_f = Self::from_bytes(&fl);\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut fe_g = Self::from_bytes(&gl);\n\n // The full field elem is now fe_f + 2²⁵⁵ fl_top_bit + 2²⁵⁶ fe_g + 2⁵¹¹ gl_top_bit\n\n // Add the masked off bits back to fe_f. fl_top_bit, if set, is 2^255 ≡ 19 (mod q).\n // gl_top_bit, if set, is 2^511 ≡ 722 (mod q)\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut top_bits_sum = {\n // This only need to be a u16 because the max value is 741\n let addend: u16 = fl_top_bit * 19 + gl_top_bit * 722;\n let mut addend_bytes = [0u8; 32];\n addend_bytes[..2].copy_from_slice(&addend.to_le_bytes());\n let top_bits_sum = Self::from_bytes(&addend_bytes);\n\n #[cfg(feature = \"zeroize\")]\n addend_bytes.zeroize();\n\n top_bits_sum\n };\n fe_f += &top_bits_sum;\n\n // Now add the high half into fe_f. The RHS is multiplied by 2^256 ≡ 38 (mod q)\n const THIRTY_EIGHT: FieldElement = FieldElement::from_bytes(&[\n 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ]);\n fe_f += &(&THIRTY_EIGHT * &fe_g);\n\n #[cfg(feature = \"zeroize\")]\n {\n fl.zeroize();\n gl.zeroize();\n fe_g.zeroize();\n top_bits_sum.zeroize();\n }\n\n fe_f\n }\n\n /// Determine if this `FieldElement` is negative, in the sense\n /// used in the ed25519 paper: `x` is negative if the low bit is\n /// set.\n ///\n /// # Return\n ///\n /// If negative, return `Choice(1)`. Otherwise, return `Choice(0)`.\n pub(crate) fn is_negative(&self) -> Choice {\n let bytes = self.to_bytes();\n (bytes[0] & 1).into()\n }\n\n /// Determine if this `FieldElement` is zero.\n ///\n /// # Return\n ///\n /// If zero, return `Choice(1)`. Otherwise, return `Choice(0)`.\n pub(crate) fn is_zero(&self) -> Choice {\n let zero = [0u8; 32];\n let bytes = self.to_bytes();\n\n bytes.ct_eq(&zero)\n }\n\n /// Compute (self^(2^250-1), self^11), used as a helper function\n /// within invert() and pow22523().\n #[rustfmt::skip] // keep alignment of explanatory comments\n fn pow22501(&self) -> (FieldElement, FieldElement) {\n // Instead of managing which temporary variables are used\n // for what, we define as many as we need and leave stack\n // allocation to the compiler\n //\n // Each temporary variable t_i is of the form (self)^e_i.\n // Squaring t_i corresponds to multiplying e_i by 2,\n // so the pow2k function shifts e_i left by k places.\n // Multiplying t_i and t_j corresponds to adding e_i + e_j.\n //\n // Temporary t_i Nonzero bits of e_i\n //\n let t0 = self.square(); // 1 e_0 = 2^1\n let t1 = t0.square().square(); // 3 e_1 = 2^3\n let t2 = self * &t1; // 3,0 e_2 = 2^3 + 2^0\n let t3 = &t0 * &t2; // 3,1,0\n let t4 = t3.square(); // 4,2,1\n let t5 = &t2 * &t4; // 4,3,2,1,0\n let t6 = t5.pow2k(5); // 9,8,7,6,5\n let t7 = &t6 * &t5; // 9,8,7,6,5,4,3,2,1,0\n let t8 = t7.pow2k(10); // 19..10\n let t9 = &t8 * &t7; // 19..0\n let t10 = t9.pow2k(20); // 39..20\n let t11 = &t10 * &t9; // 39..0\n let t12 = t11.pow2k(10); // 49..10\n let t13 = &t12 * &t7; // 49..0\n let t14 = t13.pow2k(50); // 99..50\n let t15 = &t14 * &t13; // 99..0\n let t16 = t15.pow2k(100); // 199..100\n let t17 = &t16 * &t15; // 199..0\n let t18 = t17.pow2k(50); // 249..50\n let t19 = &t18 * &t13; // 249..0\n\n (t19, t3)\n }\n\n /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse.\n ///\n /// When an input `FieldElement` is zero, its value is unchanged.\n #[cfg(feature = \"alloc\")]\n pub(crate) fn invert_batch_alloc(inputs: &mut [FieldElement]) {\n let n = inputs.len();\n let mut scratch = vec![FieldElement::ONE; n];\n\n Self::internal_invert_batch(inputs, &mut scratch);\n\n #[cfg(feature = \"zeroize\")]\n scratch.zeroize();\n }\n\n /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse. `scratch` can\n /// contain anything, so long as its length is the same as `inputs`.\n ///\n /// When an input `FieldElement` is zero, its value is unchanged.\n ///\n /// # Panics\n /// Panics when `scratch.len() != inputs.len()`\n fn internal_invert_batch(inputs: &mut [FieldElement], scratch: &mut [FieldElement]) {\n // Montgomery’s Trick and Fast Implementation of Masked AES\n // Genelle, Prouff and Quisquater\n // Section 3.2\n\n debug_assert_eq!(inputs.len(), scratch.len());\n\n // Keep an accumulator of all of the previous products\n let mut acc = FieldElement::ONE;\n\n // Pass through the input vector, recording the previous\n // products in the scratch space\n for (input, scratch) in inputs.iter().zip(scratch.iter_mut()) {\n *scratch = acc;\n // acc <- acc * input, but skipping zeros (constant-time)\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut product = &acc * input;\n acc.conditional_assign(&product, !input.is_zero());\n\n #[cfg(feature = \"zeroize\")]\n product.zeroize();\n }\n\n // acc is nonzero because we skipped zeros in inputs\n assert!(bool::from(!acc.is_zero()));\n\n // Compute the inverse of all products\n acc = acc.invert();\n\n // Pass through the vector backwards to compute the inverses\n // in place\n for (input, scratch) in inputs.iter_mut().rev().zip(scratch.iter_mut().rev()) {\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut tmp = &acc * input;\n #[cfg_attr(not(feature = \"zeroize\"), allow(unused_mut))]\n let mut inverse = &acc * scratch;\n // input <- acc * scratch, then acc <- tmp\n // Again, we skip zeros in a constant-time way\n let nz = !input.is_zero();\n input.conditional_assign(&inverse, nz);\n acc.conditional_assign(&tmp, nz);\n\n #[cfg(feature = \"zeroize\")]\n {\n tmp.zeroize();\n inverse.zeroize();\n }\n }\n\n #[cfg(feature = \"zeroize\")]\n {\n acc.zeroize();\n scratch.zeroize();\n }\n }\n\n /// Given a nonzero field element, compute its inverse.\n ///\n /// The inverse is computed as self^(p-2), since\n /// x^(p-2)x = x^(p-1) = 1 (mod p).\n ///\n /// This function returns zero on input zero.\n #[rustfmt::skip] // keep alignment of explanatory comments\n #[allow(clippy::let_and_return)]\n pub(crate) fn invert(&self) -> FieldElement {\n // The bits of p-2 = 2^255 -19 -2 are 11010111111...11.\n //\n // nonzero bits of exponent\n let (t19, t3) = self.pow22501(); // t19: 249..0 ; t3: 3,1,0\n let t20 = t19.pow2k(5); // 254..5\n let t21 = &t20 * &t3; // 254..5,3,1,0\n\n t21\n }\n\n /// Raise this field element to the power (p-5)/8 = 2^252 -3.\n #[rustfmt::skip] // keep alignment of explanatory comments\n #[allow(clippy::let_and_return)]\n pub(crate) fn pow_p58(&self) -> FieldElement {\n // The bits of (p-5)/8 are 101111.....11.\n //\n // nonzero bits of exponent\n let (t19, _) = self.pow22501(); // 249..0\n let t20 = t19.pow2k(2); // 251..2\n let t21 = self * &t20; // 251..2,0\n\n t21\n }\n\n /// Given `FieldElements` `u` and `v`, compute either `sqrt(u/v)`\n /// or `sqrt(i*u/v)` in constant time.\n ///\n /// This function always returns the nonnegative square root.\n ///\n /// # Return\n ///\n /// - `(Choice(1), +sqrt(u/v)) ` if `v` is nonzero and `u/v` is square;\n /// - `(Choice(1), zero) ` if `u` is zero;\n /// - `(Choice(0), zero) ` if `v` is zero and `u` is nonzero;\n /// - `(Choice(0), +sqrt(i*u/v))` if `u/v` is nonsquare (so `i*u/v` is square).\n ///\n pub(crate) fn sqrt_ratio_i(u: &FieldElement, v: &FieldElement) -> (Choice, FieldElement) {\n // Using the same trick as in ed25519 decoding, we merge the\n // inversion, the square root, and the square test as follows.\n //\n // To compute sqrt(α), we can compute β = α^((p+3)/8).\n // Then β^2 = ±α, so multiplying β by sqrt(-1) if necessary\n // gives sqrt(α).\n //\n // To compute 1/sqrt(α), we observe that\n // 1/β = α^(p-1 - (p+3)/8) = α^((7p-11)/8)\n // = α^3 * (α^7)^((p-5)/8).\n //\n // We can therefore compute sqrt(u/v) = sqrt(u)/sqrt(v)\n // by first computing\n // r = u^((p+3)/8) v^(p-1-(p+3)/8)\n // = u u^((p-5)/8) v^3 (v^7)^((p-5)/8)\n // = (uv^3) (uv^7)^((p-5)/8).\n //\n // If v is nonzero and u/v is square, then r^2 = ±u/v,\n // so vr^2 = ±u.\n // If vr^2 = u, then sqrt(u/v) = r.\n // If vr^2 = -u, then sqrt(u/v) = r*sqrt(-1).\n //\n // If v is zero, r is also zero.\n\n let v3 = &v.square() * v;\n let v7 = &v3.square() * v;\n let mut r = &(u * &v3) * &(u * &v7).pow_p58();\n let check = v * &r.square();\n\n let i = &constants::SQRT_M1;\n\n let correct_sign_sqrt = check.ct_eq(u);\n let flipped_sign_sqrt = check.ct_eq(&(-u));\n let flipped_sign_sqrt_i = check.ct_eq(&(&(-u) * i));\n\n let r_prime = &constants::SQRT_M1 * &r;\n r.conditional_assign(&r_prime, flipped_sign_sqrt | flipped_sign_sqrt_i);\n\n // Choose the nonnegative square root.\n //\n // NOTE: written as negate-then-conditional-assign rather than\n // `r.conditional_negate(r_is_negative)`: semantically identical and\n // still constant-time, but it avoids subtle's `ConditionallyNegatable`\n // blanket impl (whose `for<'a> &'a T: Neg` bound breaks the Aeneas\n // verification toolchain; see verification/field/README.md).\n let r_is_negative = r.is_negative();\n let r_neg = -&r;\n r.conditional_assign(&r_neg, r_is_negative);\n\n let was_nonzero_square = correct_sign_sqrt | flipped_sign_sqrt;\n\n (was_nonzero_square, r)\n }\n\n /// Attempt to compute `sqrt(1/self)` in constant time.\n ///\n /// Convenience wrapper around `sqrt_ratio_i`.\n ///\n /// This function always returns the nonnegative square root.\n ///\n /// # Return\n ///\n /// - `(Choice(1), +sqrt(1/self)) ` if `self` is a nonzero square;\n /// - `(Choice(0), zero) ` if `self` is zero;\n /// - `(Choice(0), +sqrt(i/self)) ` if `self` is a nonzero nonsquare;\n ///\n pub(crate) fn invsqrt(&self) -> (Choice, FieldElement) {\n FieldElement::sqrt_ratio_i(&FieldElement::ONE, self)\n }\n\n #[cfg(feature = \"digest\")]\n /// Hashes the given message and domain separator to produce `COUNT` [`FieldElement`]s, per RFC\n /// 9380. The hash's input is the concatenation of the elements of `msg`. Likewise for the\n /// domain separator with `domain_sep`. At least one element of `domain_sep`, MUST be nonempty,\n /// its concatenation MUST NOT exceed 255 bytes, and `COUNT` MUST be 1 or 2.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`. Also panics if `COUNT > 2` or\n /// `COUNT == 0`.\n pub fn hash_to_field(\n msg: &[&[u8]],\n domain_sep: &[&[u8]],\n ) -> [Self; COUNT]\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // We only use `hash_to_field` for Elligator2, which uses 0 < COUNT <= 2\n assert!(COUNT == 1 || COUNT == 2);\n\n // §5.2, we generate count * m * L = COUNT * 1 * (256 + 128)/8 bytes\n let len_in_bytes = COUNT * 48;\n // Buffer should hold however many digests we need to produce len_in_bytes bytes.\n // len_in_bytes is at most 2*48=96, and the digest is 64B. So that's 2 digests, or 128B\n let mut buf = [0u8; 128];\n // We can call this without worrying about panics: len_in_bytes is at most 96, and buf is\n // bigger than it. Further, if `domain_sep` is too large, that's also a panic condition of\n // this function so it's fine\n let uniform_bytes = expand_msg_xmd::(msg, domain_sep, &mut buf, len_in_bytes);\n\n let mut result = [FieldElement::ONE; COUNT];\n for i in 0..COUNT {\n let mut bytes_wide = [0u8; 64];\n bytes_wide[..48].copy_from_slice(&uniform_bytes[48 * i..48 * (i + 1)]);\n bytes_wide[..48].reverse();\n\n result[i] = FieldElement::from_bytes_wide(&bytes_wide);\n }\n\n result\n }\n}\n\n/// Hashes the concatenation of the elements of `msg` with domain separator equal to the\n/// concatenation of `domain_sep`. The output is an `outlen`-length slice into `buf`. Follows\n/// https://www.rfc-editor.org/rfc/rfc9380.html#section-5.3.1\n///\n/// # Panics\n/// Panics if the domain separator is empty, if the total length of the domain separator is more\n/// than 255 bytes, if `outlen` is more than `255 * D::output_size()`, if `outlen is more than\n/// 65535, or if `dst.len() < outlen`.\n#[cfg(feature = \"digest\")]\nfn expand_msg_xmd<'a, D>(\n msg: &[&[u8]],\n domain_sep: &[&[u8]],\n buf: &'a mut [u8],\n outlen: usize,\n) -> &'a [u8]\nwhere\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n{\n use core::iter::once;\n\n // The notation we use in this function is the same as in the spec\n let len_in_bytes = u16::try_from(outlen).expect(\"outlen must not exceed 65535\");\n\n assert!(buf.len() >= outlen);\n let ell = u8::try_from((len_in_bytes as usize).div_ceil(D::output_size()))\n .expect(\"output length cannot exceed 255 times digest size\");\n let domain_sep_len = u8::try_from(domain_sep.iter().map(|c| c.len()).sum::())\n .expect(\"unexpected overflow from domain separator's size.\");\n assert_ne!(\n domain_sep_len, 0,\n \"domain separator MUST have nonzero length.\"\n );\n\n let domain_sep_len_slice = &[domain_sep_len][..];\n let dst_prime = domain_sep.iter().copied().chain(once(domain_sep_len_slice));\n let z_pad = Array::::default();\n let l_i_b_str = len_in_bytes.to_be_bytes();\n\n // Collect the components of msg_prime\n let msg_prime = once(z_pad.as_slice())\n .chain(msg.iter().copied())\n .chain(once(l_i_b_str.as_slice()))\n .chain(once(&[0u8][..]))\n .chain(dst_prime.clone());\n // Hash all of msg_prime\n let b_0 = msg_prime\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n\n // Collect the input components for the b_1 hash\n let b_1_input = once(b_0.as_slice())\n .chain(once(&[1u8][..]))\n .chain(dst_prime.clone());\n let b_1 = b_1_input\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n // Write b_1 to the output buffer\n buf[..D::output_size()].copy_from_slice(&b_1);\n\n for i in 2..=ell {\n // Get the last digest we produced\n let i_ = i as usize;\n let last_b = &buf[(i_ - 2) * D::output_size()..(i_ - 1) * D::output_size()];\n // XOR it with b_0\n let mut xor_bs = b_0.clone();\n xor_bs\n .as_mut_slice()\n .iter_mut()\n .zip(last_b)\n .for_each(|(l, r)| *l ^= *r);\n\n // Hash (b_0 ^ b_{i-1}) || i || dst_prime\n let i_slice = &[i][..];\n let b_i_input = once(xor_bs.as_slice())\n .chain(once(i_slice))\n .chain(dst_prime.clone());\n let b_i = b_i_input\n .fold(D::new(), |h, slice| h.chain_update(slice))\n .finalize();\n\n // Write b_i to the output buffer\n buf[(i_ - 1) * D::output_size()..i_ * D::output_size()].copy_from_slice(&b_i);\n }\n\n &buf[..outlen]\n}\n\n#[cfg(test)]\nmod test {\n use crate::field::*;\n use subtle::ConditionallyNegatable;\n\n /// Random element a of GF(2^255-19), from Sage\n /// a = 1070314506888354081329385823235218444233221\\\n /// 2228051251926706380353716438957572\n static A_BYTES: [u8; 32] = [\n 0x04, 0xfe, 0xdf, 0x98, 0xa7, 0xfa, 0x0a, 0x68, 0x84, 0x92, 0xbd, 0x59, 0x08, 0x07, 0xa7,\n 0x03, 0x9e, 0xd1, 0xf6, 0xf2, 0xe1, 0xd9, 0xe2, 0xa4, 0xa4, 0x51, 0x47, 0x36, 0xf3, 0xc3,\n 0xa9, 0x17,\n ];\n\n /// Byte representation of a**2\n static ASQ_BYTES: [u8; 32] = [\n 0x75, 0x97, 0x24, 0x9e, 0xe6, 0x06, 0xfe, 0xab, 0x24, 0x04, 0x56, 0x68, 0x07, 0x91, 0x2d,\n 0x5d, 0x0b, 0x0f, 0x3f, 0x1c, 0xb2, 0x6e, 0xf2, 0xe2, 0x63, 0x9c, 0x12, 0xba, 0x73, 0x0b,\n 0xe3, 0x62,\n ];\n\n /// Byte representation of 1/a\n static AINV_BYTES: [u8; 32] = [\n 0x96, 0x1b, 0xcd, 0x8d, 0x4d, 0x5e, 0xa2, 0x3a, 0xe9, 0x36, 0x37, 0x93, 0xdb, 0x7b, 0x4d,\n 0x70, 0xb8, 0x0d, 0xc0, 0x55, 0xd0, 0x4c, 0x1d, 0x7b, 0x90, 0x71, 0xd8, 0xe9, 0xb6, 0x18,\n 0xe6, 0x30,\n ];\n\n /// Byte representation of a^((p-5)/8)\n static AP58_BYTES: [u8; 32] = [\n 0x6a, 0x4f, 0x24, 0x89, 0x1f, 0x57, 0x60, 0x36, 0xd0, 0xbe, 0x12, 0x3c, 0x8f, 0xf5, 0xb1,\n 0x59, 0xe0, 0xf0, 0xb8, 0x1b, 0x20, 0xd2, 0xb5, 0x1f, 0x15, 0x21, 0xf9, 0xe3, 0xe1, 0x61,\n 0x21, 0x55,\n ];\n\n #[test]\n fn a_mul_a_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(asq, &a * &a);\n }\n\n #[test]\n fn a_square_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(asq, a.square());\n }\n\n #[test]\n fn a_square2_vs_a_squared_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n assert_eq!(a.square2(), &asq + &asq);\n }\n\n #[test]\n fn a_invert_vs_inverse_of_a_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n let should_be_inverse = a.invert();\n assert_eq!(ainv, should_be_inverse);\n assert_eq!(FieldElement::ONE, &a * &should_be_inverse);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn invert_batch_a_matches_nonbatched() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ap58 = FieldElement::from_bytes(&AP58_BYTES);\n let asq = FieldElement::from_bytes(&ASQ_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n let a0 = &a - &a;\n let a2 = &a + &a;\n let a_list = vec![a, ap58, asq, ainv, a0, a2];\n let mut ainv_list = a_list.clone();\n FieldElement::invert_batch_alloc(&mut ainv_list[..]);\n for i in 0..6 {\n assert_eq!(a_list[i].invert(), ainv_list[i]);\n }\n }\n\n #[test]\n fn sqrt_ratio_behavior() {\n let zero = FieldElement::ZERO;\n let one = FieldElement::ONE;\n let i = constants::SQRT_M1;\n let two = &one + &one; // 2 is nonsquare mod p.\n let four = &two + &two; // 4 is square mod p.\n\n // 0/0 should return (1, 0) since u is 0\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&zero, &zero);\n assert!(bool::from(choice));\n assert_eq!(sqrt, zero);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 1/0 should return (0, 0) since v is 0, u is nonzero\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &zero);\n assert!(bool::from(!choice));\n assert_eq!(sqrt, zero);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 2/1 is nonsquare, so we expect (0, sqrt(i*2))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&two, &one);\n assert!(bool::from(!choice));\n assert_eq!(sqrt.square(), &two * &i);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 4/1 is square, so we expect (1, sqrt(4))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&four, &one);\n assert!(bool::from(choice));\n assert_eq!(sqrt.square(), four);\n assert!(bool::from(!sqrt.is_negative()));\n\n // 1/4 is square, so we expect (1, 1/sqrt(4))\n let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &four);\n assert!(bool::from(choice));\n assert_eq!(&sqrt.square() * &four, one);\n assert!(bool::from(!sqrt.is_negative()));\n }\n\n #[test]\n fn a_p58_vs_ap58_constant() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ap58 = FieldElement::from_bytes(&AP58_BYTES);\n assert_eq!(ap58, a.pow_p58());\n }\n\n #[test]\n fn equality() {\n let a = FieldElement::from_bytes(&A_BYTES);\n let ainv = FieldElement::from_bytes(&AINV_BYTES);\n assert!(a == a);\n assert!(a != ainv);\n }\n\n /// Notice that the last element has the high bit set, which\n /// should be ignored\n static B_BYTES: [u8; 32] = [\n 113, 191, 169, 143, 91, 234, 121, 15, 241, 131, 217, 36, 230, 101, 92, 234, 8, 208, 170,\n 251, 97, 127, 70, 210, 58, 23, 166, 87, 240, 169, 184, 178,\n ];\n\n #[test]\n fn from_bytes_highbit_is_ignored() {\n let mut cleared_bytes = B_BYTES;\n cleared_bytes[31] &= 127u8;\n let with_highbit_set = FieldElement::from_bytes(&B_BYTES);\n let without_highbit_set = FieldElement::from_bytes(&cleared_bytes);\n assert_eq!(without_highbit_set, with_highbit_set);\n }\n\n #[test]\n fn conditional_negate() {\n let one = FieldElement::ONE;\n let minus_one = FieldElement::MINUS_ONE;\n let mut x = one;\n x.conditional_negate(Choice::from(1));\n assert_eq!(x, minus_one);\n x.conditional_negate(Choice::from(0));\n assert_eq!(x, minus_one);\n x.conditional_negate(Choice::from(1));\n assert_eq!(x, one);\n }\n\n #[test]\n fn encoding_is_canonical() {\n // Encode 1 wrongly as 1 + (2^255 - 19) = 2^255 - 18\n let one_encoded_wrongly_bytes: [u8; 32] = [\n 0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,\n 0xff, 0xff, 0xff, 0x7f,\n ];\n // Decode to a field element\n let one = FieldElement::from_bytes(&one_encoded_wrongly_bytes);\n // .. then check that the encoding is correct\n let one_bytes = one.to_bytes();\n assert_eq!(one_bytes[0], 1);\n for byte in &one_bytes[1..] {\n assert_eq!(*byte, 0);\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn invert_batch_empty() {\n FieldElement::invert_batch_alloc(&mut []);\n }\n\n // The following two consts were generated with the following sage script:\n //\n // import random\n //\n // F = GF(2**255 - 19)\n // # Use a seed to make sure we produce the same test vectors every time\n // random.seed(\"Ozamataz Buckshank\")\n //\n // # Generates test vectors, each of the form (input_bytes, reduced_field_elem_bytes),\n // # where input_bytes is length input_bytes_len\n // def gen_example(input_bytes_len):\n // # Generate random bytes\n // input_bytes = [random.randint(0, 255) for _ in range(input_bytes_len)]\n //\n // # Now convert to a field element and get the reduced byte representation\n // elem = F(int.from_bytes(input_bytes, byteorder='little'))\n // reduced_bytes = list(int(elem).to_bytes(32, byteorder='little'))\n //\n // # Format input and output as hex strings\n // input_bytes_hex = ''.join(f'{byte:02x}' for byte in input_bytes)\n // reduced_bytes_hex = ''.join(f'{byte:02x}' for byte in reduced_bytes)\n // return f\"(\\\"{input_bytes_hex}\\\", \\\"{reduced_bytes_hex}\\\")\"\n //\n // print(\"SET 1: Input bytes are length 64\")\n // for _ in range(5):\n // print(gen_example(64))\n //\n // print(\"SET 2: Input bytes are length 48\")\n // for _ in range(5):\n // print(gen_example(48))\n\n /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-64 bytestring,\n /// reduced field element)\n #[cfg(feature = \"digest\")]\n const FROM_BYTES_WIDE_KAT_BIG: &[(&str, &str)] = &[\n (\n \"77b663085cac0e916f40dbeea5116f201816406e68ccf01b32a97162ae1d5bf95d0d01c2c72fbeeb27a63\\\n 5b85b715d5ce6f74118a60a7aec53c798ad648a482f\",\n \"62b38bd402c4498f5cead14643e54dd649e20a0810610e36a73f1f27a0a81f7e\",\n ),\n (\n \"d437c75ec79886650243a79c62933bb307eb12ff16d05db4a6a8a877f4a91abb6eeb64d2e20519c021799\\\n 3a1dc5639283a06639985a2c892208171503335afb5\",\n \"3d2ec29972783de9043e8b982278beaba9d7c5c3ebef257e7cd38168928f1c33\",\n ),\n (\n \"6daa9e1abe6c604fb6e841c04bf90a6ef88aef6b1eab17dd44f7207ef472cd2d54bac849f703e64f36e56\\\n 77e7e86b82be7d26aa220daf1f208bb36dcc1a12338\",\n \"28546a0e7303852bc6eead8312f06eeb48d9ca87f60bfeec98ba402ebb751703\",\n ),\n (\n \"c3920e326dbf806a50105be78263c1dc9390fb4741587b250cd758c2bfa3ed70faedbbc5f9b1d024e00fe\\\n 7d7daf796866853f42e72d638e6533c5eb5b7caf3c6\",\n \"40eaf38b802a7be1956ba7f3fe2d2ad717f23f40342deb5180cb55ae04bb1d79\",\n ),\n (\n \"23f143c72ead6c0f336b4e746a06921f0eb180002e8ce916d196de16216788617c6aeb90a074a85196f03\\\n 81375011248927c1215e9ec65b382a6ec556fb3f504\",\n \"b1bf354a04fd6d2e8321c24ecb3d3ed2c42e3f21c7b60ab8374effd7a709011e\",\n ),\n ];\n\n /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-48 bytestring,\n /// reduced field element)\n #[cfg(feature = \"digest\")]\n const FROM_BYTES_WIDE_KAT_MEDIUM: &[(&str, &str)] = &[\n (\n \"82e9cbe4928e3d0bbf1f91824a91acfb30d929f7a2fa5cbcc967c63ea0f3357c29c19f1bc9dcad69d85c1\\\n c6265970685\",\n \"989582fe6c540cbbdee7c612570aa7ba44d929f7a2fa5cbcc967c63ea0f3357c\",\n ),\n (\n \"5480494df4fb3a3b19da17e1c8b9192ccb09ec76720321977079300c42c17b9e95b01eb37ffe7048fcd1c\\\n 9e6094da6c4\",\n \"85b6d7e3e8c200fc8b050d234129c95ce809ec76720321977079300c42c17b1e\",\n ),\n (\n \"93ec8a480dde098f74bcd341ef4f248f6440cc6e631d7000784f66975a4fd628438bb1350ba4c1421fec3\\\n 670decced06\",\n \"8598e540b737c87718c9fae9f3b870966540cc6e631d7000784f66975a4fd628\",\n ),\n (\n \"fd0154ff9a5c4c9ee4e8183c23db97018e0e6201a812f6d4faedda50652d51f65c110b9a1a100a3fc3ff1\\\n c4ea3cf22e4\",\n \"b895f8dc8dc0caf9dfdf66d460adc2deaf0e6201a812f6d4faedda50652d5176\",\n ),\n (\n \"0e829dc955e0a1e0dbda9849cb2022b295275782348bd6308b3d0c5836f3ca0130911a17fd54054c3a0f8\\\n b2486f8ce85\",\n \"2e0f8f37e77d6c29831d3db6b404db8ea9275782348bd6308b3d0c5836f3ca01\",\n ),\n ];\n\n #[cfg(feature = \"digest\")]\n #[test]\n fn from_bytes_wide() {\n // Do the 64-byte input ones first\n for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_BIG {\n let decoded_input =\n hex::decode(input_bytes).expect(\"64-byte reduction test vector hex must decode\");\n let decoded_input: [u8; 64] = decoded_input\n .as_slice()\n .try_into()\n .expect(\"64-byte reduction test vector must decode to 64 bytes\");\n let reduce_fe = FieldElement::from_bytes_wide(&decoded_input);\n assert_eq!(\n &reduce_fe.to_bytes(),\n hex::decode(expected_reduced)\n .expect(\"reduction result test vector hex must decode\")\n .as_slice()\n );\n }\n\n // Now do the 48-byte inputs\n for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_MEDIUM {\n let mut padded_input_bytes = [0u8; 64];\n padded_input_bytes[..48].copy_from_slice(\n &hex::decode(input_bytes).expect(\"48-byte reduction test vector hex must decode\"),\n );\n let reduce_fe = FieldElement::from_bytes_wide(&padded_input_bytes);\n assert_eq!(\n &reduce_fe.to_bytes(),\n hex::decode(expected_reduced)\n .expect(\"reduction result test vector hex must decode\")\n .as_slice()\n );\n }\n }\n\n #[cfg(feature = \"digest\")]\n fn fe_from_test_vector(expected_hex: &str) -> FieldElement {\n let mut expected_hash =\n hex::decode(expected_hex).expect(\"hash-to-field test vector hex must decode\");\n expected_hash.reverse();\n let expected_hash: [u8; 32] = expected_hash\n .try_into()\n .expect(\"hash-to-field test vector must decode to 32 bytes\");\n FieldElement::from_bytes(&expected_hash)\n }\n\n /// Hash to field test vectors from\n /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.2\n /// These are of the form (input_msg, output_field_elem)\n #[cfg(feature = \"digest\")]\n const RFC_HASH_TO_FIELD_KAT: &[(&[u8], &str)] = &[\n (\n b\"\",\n \"7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0374a30641d\"\n ),\n (\n b\"abc\",\n \"09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd201a556e253b\"\n ),\n (\n b\"abcdef0123456789\",\n \"475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75d33edba941\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf6929814ac95d03d\"\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f0750d24fa0\"\n )\n ];\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn hash_to_field() {\n use sha2::Sha512;\n let dst = \"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_\";\n\n for (msg, expected_hash_hex) in RFC_HASH_TO_FIELD_KAT {\n let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()])[0];\n let expected_fe = fe_from_test_vector(expected_hash_hex);\n\n assert_eq!(fe, expected_fe);\n }\n }\n\n /// Hash to field test vectors from\n /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.1\n /// These are of the form (input_msg, output_field_elem, output_field_elem)\n #[cfg(feature = \"digest\")]\n const RFC_HASH_TO_FIELD_KAT_2: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee712d203f3a\",\n \"780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae793e6a60c75\"\n ),\n (\n b\"abc\",\n \"5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78f9bd57c227\",\n \"005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119aa0b925b76\"\n ),\n (\n b\"abcdef0123456789\",\n \"285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642636a78d5b3\",\n \"2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b369760b81b31\"\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c533bf321af\",\n \"02f20716a5801b843987097a8276b6d869295b2e11253751ca72c109d37485a9\"\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06df4a4ea33b\",\n \"1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e84068f5fd96\"\n )\n ];\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn hash_to_field_2() {\n use sha2::Sha512;\n let dst = \"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_\";\n\n for (msg, expected_hash_hex_1, expected_hash_hex_2) in\n crate::field::test::RFC_HASH_TO_FIELD_KAT_2\n {\n let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()]);\n\n let expected_fe_1 = fe_from_test_vector(expected_hash_hex_1);\n assert_eq!(fe[0], expected_fe_1);\n\n let expected_fe_2 = fe_from_test_vector(expected_hash_hex_2);\n assert_eq!(fe[1], expected_fe_2);\n }\n }\n}\n"},{"id":9,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64/field.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Field arithmetic modulo \\\\(p = 2\\^{255} - 19\\\\), using \\\\(64\\\\)-bit\n//! limbs with \\\\(128\\\\)-bit products.\n\nuse core::fmt::Debug;\nuse core::ops::Neg;\nuse core::ops::{Add, AddAssign};\nuse core::ops::{Mul, MulAssign};\nuse core::ops::{Sub, SubAssign};\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\n/// A `FieldElement51` represents an element of the field\n/// \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\).\n///\n/// In the 64-bit implementation, a `FieldElement` is represented in\n/// radix \\\\(2\\^{51}\\\\) as five `u64`s; the coefficients are allowed to\n/// grow up to \\\\(2\\^{54}\\\\) between reductions modulo \\\\(p\\\\).\n///\n/// # Note\n///\n/// The `curve25519::field` module provides a type alias\n/// `curve25519::field::FieldElement` to either `FieldElement51`\n/// or `FieldElement2625`.\n///\n/// The backend-specific type `FieldElement51` should not be used\n/// outside of the `curve25519::field` module.\n#[derive(Copy, Clone)]\npub struct FieldElement51(pub(crate) [u64; 5]);\n\nimpl Debug for FieldElement51 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"FieldElement51({:?})\", &self.0[..])\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for FieldElement51 {}\n\nimpl<'a> AddAssign<&'a FieldElement51> for FieldElement51 {\n fn add_assign(&mut self, _rhs: &'a FieldElement51) {\n for i in 0..5 {\n self.0[i] += _rhs.0[i];\n }\n }\n}\n\nimpl<'a> Add<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn add(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n let mut output = *self;\n output += _rhs;\n output\n }\n}\n\nimpl<'a> SubAssign<&'a FieldElement51> for FieldElement51 {\n fn sub_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) - _rhs;\n self.0 = result.0;\n }\n}\n\nimpl<'a> Sub<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn sub(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n // To avoid underflow, first add a multiple of p.\n // Choose 16*p = p << 4 to be larger than 54-bit _rhs.\n //\n // If we could statically track the bitlengths of the limbs\n // of every FieldElement51, we could choose a multiple of p\n // just bigger than _rhs and avoid having to do a reduction.\n //\n // Since we don't yet have type-level integers to do this, we\n // have to add an explicit reduction call here.\n FieldElement51::reduce([\n (self.0[0] + 36028797018963664u64) - _rhs.0[0],\n (self.0[1] + 36028797018963952u64) - _rhs.0[1],\n (self.0[2] + 36028797018963952u64) - _rhs.0[2],\n (self.0[3] + 36028797018963952u64) - _rhs.0[3],\n (self.0[4] + 36028797018963952u64) - _rhs.0[4],\n ])\n }\n}\n\nimpl<'a> MulAssign<&'a FieldElement51> for FieldElement51 {\n fn mul_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) * _rhs;\n self.0 = result.0;\n }\n}\n\nimpl<'a> Mul<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n\n #[rustfmt::skip] // keep alignment of c* calculations\n fn mul(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n /// Helper function to multiply two 64-bit integers with 128\n /// bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }\n\n // Alias self, _rhs for more readable formulas\n let a: &[u64; 5] = &self.0;\n let b: &[u64; 5] = &_rhs.0;\n\n // Precondition: assume input limbs a[i], b[i] are bounded as\n //\n // a[i], b[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // 64-bit precomputations to avoid 128-bit multiplications.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let b1_19 = b[1] * 19;\n let b2_19 = b[2] * 19;\n let b3_19 = b[3] * 19;\n let b4_19 = b[4] * 19;\n\n // Multiply to get 128-bit coefficients of output\n let c0: u128 = m(a[0], b[0]) + m(a[4], b1_19) + m(a[3], b2_19) + m(a[2], b3_19) + m(a[1], b4_19);\n let mut c1: u128 = m(a[1], b[0]) + m(a[0], b[1]) + m(a[4], b2_19) + m(a[3], b3_19) + m(a[2], b4_19);\n let mut c2: u128 = m(a[2], b[0]) + m(a[1], b[1]) + m(a[0], b[2]) + m(a[4], b3_19) + m(a[3], b4_19);\n let mut c3: u128 = m(a[3], b[0]) + m(a[2], b[1]) + m(a[1], b[2]) + m(a[0], b[3]) + m(a[4], b4_19);\n let mut c4: u128 = m(a[4], b[0]) + m(a[3], b[1]) + m(a[2], b[2]) + m(a[1], b[3]) + m(a[0] , b[4]);\n\n // How big are the c[i]? We have\n //\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));\n\n // Casting to u64 and back tells the compiler that the carry is\n // bounded by 2^64, so that the addition is a u128 + u64 rather\n // than u128 + u128.\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n let mut out = [0u64; 5];\n\n c1 += ((c0 >> 51) as u64) as u128;\n out[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n out[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n out[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n out[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n out[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need out[0] + carry * 19 < 2^64.\n //\n // c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)\n // < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n out[0] += carry * 19;\n\n // Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n out[1] += out[0] >> 51;\n out[0] &= LOW_51_BIT_MASK;\n\n // Now out[i] < 2^(51 + epsilon) for all i.\n FieldElement51(out)\n }\n}\n\nimpl Neg for &FieldElement51 {\n type Output = FieldElement51;\n fn neg(self) -> FieldElement51 {\n let mut output = *self;\n output.negate();\n output\n }\n}\n\nimpl ConditionallySelectable for FieldElement51 {\n fn conditional_select(\n a: &FieldElement51,\n b: &FieldElement51,\n choice: Choice,\n ) -> FieldElement51 {\n FieldElement51([\n u64::conditional_select(&a.0[0], &b.0[0], choice),\n u64::conditional_select(&a.0[1], &b.0[1], choice),\n u64::conditional_select(&a.0[2], &b.0[2], choice),\n u64::conditional_select(&a.0[3], &b.0[3], choice),\n u64::conditional_select(&a.0[4], &b.0[4], choice),\n ])\n }\n\n fn conditional_swap(a: &mut FieldElement51, b: &mut FieldElement51, choice: Choice) {\n u64::conditional_swap(&mut a.0[0], &mut b.0[0], choice);\n u64::conditional_swap(&mut a.0[1], &mut b.0[1], choice);\n u64::conditional_swap(&mut a.0[2], &mut b.0[2], choice);\n u64::conditional_swap(&mut a.0[3], &mut b.0[3], choice);\n u64::conditional_swap(&mut a.0[4], &mut b.0[4], choice);\n }\n\n fn conditional_assign(&mut self, other: &FieldElement51, choice: Choice) {\n self.0[0].conditional_assign(&other.0[0], choice);\n self.0[1].conditional_assign(&other.0[1], choice);\n self.0[2].conditional_assign(&other.0[2], choice);\n self.0[3].conditional_assign(&other.0[3], choice);\n self.0[4].conditional_assign(&other.0[4], choice);\n }\n}\n\nimpl FieldElement51 {\n pub(crate) const fn from_limbs(limbs: [u64; 5]) -> FieldElement51 {\n FieldElement51(limbs)\n }\n\n /// The scalar \\\\( 0 \\\\).\n pub const ZERO: FieldElement51 = FieldElement51::from_limbs([0, 0, 0, 0, 0]);\n /// The scalar \\\\( 1 \\\\).\n pub const ONE: FieldElement51 = FieldElement51::from_limbs([1, 0, 0, 0, 0]);\n /// The scalar \\\\( -1 \\\\).\n pub const MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n ]);\n\n /// Invert the sign of this field element\n pub fn negate(&mut self) {\n // See commentary in the Sub impl\n let neg = FieldElement51::reduce([\n 36028797018963664u64 - self.0[0],\n 36028797018963952u64 - self.0[1],\n 36028797018963952u64 - self.0[2],\n 36028797018963952u64 - self.0[3],\n 36028797018963952u64 - self.0[4],\n ]);\n self.0 = neg.0;\n }\n\n /// Given 64-bit input limbs, reduce to enforce the bound 2^(51 + epsilon).\n #[inline(always)]\n fn reduce(mut limbs: [u64; 5]) -> FieldElement51 {\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n\n // Since the input limbs are bounded by 2^64, the biggest\n // carry-out is bounded by 2^13.\n //\n // The biggest carry-in is c4 * 19, resulting in\n //\n // 2^51 + 19*2^13 < 2^51.0000000001\n //\n // Because we don't need to canonicalize, only to reduce the\n // limb sizes, it's OK to do a \"weak reduction\", where we\n // compute the carry-outs in parallel.\n\n let c0 = limbs[0] >> 51;\n let c1 = limbs[1] >> 51;\n let c2 = limbs[2] >> 51;\n let c3 = limbs[3] >> 51;\n let c4 = limbs[4] >> 51;\n\n limbs[0] &= LOW_51_BIT_MASK;\n limbs[1] &= LOW_51_BIT_MASK;\n limbs[2] &= LOW_51_BIT_MASK;\n limbs[3] &= LOW_51_BIT_MASK;\n limbs[4] &= LOW_51_BIT_MASK;\n\n limbs[0] += c4 * 19;\n limbs[1] += c0;\n limbs[2] += c1;\n limbs[3] += c2;\n limbs[4] += c3;\n\n FieldElement51(limbs)\n }\n\n /// Load a `FieldElement51` from the low 255 bits of a 256-bit\n /// input.\n ///\n /// # Warning\n ///\n /// This function does not check that the input used the canonical\n /// representative. It masks the high bit, but it will happily\n /// decode 2^255 - 18 to 1. Applications that require a canonical\n /// encoding of every field element should decode, re-encode to\n /// the canonical encoding, and check that the input was\n /// canonical.\n ///\n #[rustfmt::skip] // keep alignment of bit shifts\n pub const fn from_bytes(bytes: &[u8; 32]) -> FieldElement51 {\n const fn load8_at(input: &[u8], i: usize) -> u64 {\n (input[i] as u64)\n | ((input[i + 1] as u64) << 8)\n | ((input[i + 2] as u64) << 16)\n | ((input[i + 3] as u64) << 24)\n | ((input[i + 4] as u64) << 32)\n | ((input[i + 5] as u64) << 40)\n | ((input[i + 6] as u64) << 48)\n | ((input[i + 7] as u64) << 56)\n }\n\n let low_51_bit_mask = (1u64 << 51) - 1;\n FieldElement51(\n // load bits [ 0, 64), no shift\n [ load8_at(bytes, 0) & low_51_bit_mask\n // load bits [ 48,112), shift to [ 51,112)\n , (load8_at(bytes, 6) >> 3) & low_51_bit_mask\n // load bits [ 96,160), shift to [102,160)\n , (load8_at(bytes, 12) >> 6) & low_51_bit_mask\n // load bits [152,216), shift to [153,216)\n , (load8_at(bytes, 19) >> 1) & low_51_bit_mask\n // load bits [192,256), shift to [204,112)\n , (load8_at(bytes, 24) >> 12) & low_51_bit_mask\n ])\n }\n\n /// Serialize this `FieldElement51` to a 32-byte array. The\n /// encoding is canonical.\n #[rustfmt::skip] // keep alignment of s[*] calculations\n pub fn to_bytes(self) -> [u8; 32] {\n // Let h = limbs[0] + limbs[1]*2^51 + ... + limbs[4]*2^204.\n //\n // Write h = pq + r with 0 <= r < p.\n //\n // We want to compute r = h mod p.\n //\n // If h < 2*p = 2^256 - 38,\n // then q = 0 or 1,\n //\n // with q = 0 when h < p\n // and q = 1 when h >= p.\n //\n // Notice that h >= p <==> h + 19 >= p + 19 <==> h + 19 >= 2^255.\n // Therefore q can be computed as the carry bit of h + 19.\n\n // First, reduce the limbs to ensure h < 2*p.\n let mut limbs = FieldElement51::reduce(self.0).0;\n\n let mut q = (limbs[0] + 19) >> 51;\n q = (limbs[1] + q) >> 51;\n q = (limbs[2] + q) >> 51;\n q = (limbs[3] + q) >> 51;\n q = (limbs[4] + q) >> 51;\n\n // Now we can compute r as r = h - pq = r - (2^255-19)q = r + 19q - 2^255q\n\n limbs[0] += 19 * q;\n\n // Now carry the result to compute r + 19q ...\n let low_51_bit_mask = (1u64 << 51) - 1;\n limbs[1] += limbs[0] >> 51;\n limbs[0] &= low_51_bit_mask;\n limbs[2] += limbs[1] >> 51;\n limbs[1] &= low_51_bit_mask;\n limbs[3] += limbs[2] >> 51;\n limbs[2] &= low_51_bit_mask;\n limbs[4] += limbs[3] >> 51;\n limbs[3] &= low_51_bit_mask;\n // ... but instead of carrying (limbs[4] >> 51) = 2^255q\n // into another limb, discard it, subtracting the value\n limbs[4] &= low_51_bit_mask;\n\n // Now arrange the bits of the limbs.\n let mut s = [0u8;32];\n s[ 0] = limbs[0] as u8;\n s[ 1] = (limbs[0] >> 8) as u8;\n s[ 2] = (limbs[0] >> 16) as u8;\n s[ 3] = (limbs[0] >> 24) as u8;\n s[ 4] = (limbs[0] >> 32) as u8;\n s[ 5] = (limbs[0] >> 40) as u8;\n s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;\n s[ 7] = (limbs[1] >> 5) as u8;\n s[ 8] = (limbs[1] >> 13) as u8;\n s[ 9] = (limbs[1] >> 21) as u8;\n s[10] = (limbs[1] >> 29) as u8;\n s[11] = (limbs[1] >> 37) as u8;\n s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;\n s[13] = (limbs[2] >> 2) as u8;\n s[14] = (limbs[2] >> 10) as u8;\n s[15] = (limbs[2] >> 18) as u8;\n s[16] = (limbs[2] >> 26) as u8;\n s[17] = (limbs[2] >> 34) as u8;\n s[18] = (limbs[2] >> 42) as u8;\n s[19] = ((limbs[2] >> 50) | (limbs[3] << 1)) as u8;\n s[20] = (limbs[3] >> 7) as u8;\n s[21] = (limbs[3] >> 15) as u8;\n s[22] = (limbs[3] >> 23) as u8;\n s[23] = (limbs[3] >> 31) as u8;\n s[24] = (limbs[3] >> 39) as u8;\n s[25] = ((limbs[3] >> 47) | (limbs[4] << 4)) as u8;\n s[26] = (limbs[4] >> 4) as u8;\n s[27] = (limbs[4] >> 12) as u8;\n s[28] = (limbs[4] >> 20) as u8;\n s[29] = (limbs[4] >> 28) as u8;\n s[30] = (limbs[4] >> 36) as u8;\n s[31] = (limbs[4] >> 44) as u8;\n\n // High bit should be zero.\n debug_assert!((s[31] & 0b1000_0000u8) == 0u8);\n\n s\n }\n\n /// Given `k > 0`, return `self^(2^k)`.\n #[rustfmt::skip] // keep alignment of c* calculations\n pub fn pow2k(&self, mut k: u32) -> FieldElement51 {\n\n debug_assert!( k > 0 );\n\n /// Multiply two 64-bit integers with 128 bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 {\n (x as u128) * (y as u128)\n }\n\n let mut a: [u64; 5] = self.0;\n\n loop {\n // Precondition: assume input limbs a[i] are bounded as\n //\n // a[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // Precomputation: 64-bit multiply by 19.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let a3_19 = 19 * a[3];\n let a4_19 = 19 * a[4];\n\n // Multiply to get 128-bit coefficients of output.\n //\n // The 128-bit multiplications by 2 turn into 1 slr + 1 slrd each,\n // which doesn't seem any better or worse than doing them as precomputations\n // on the 64-bit inputs.\n let c0: u128 = m(a[0], a[0]) + 2*( m(a[1], a4_19) + m(a[2], a3_19) );\n let mut c1: u128 = m(a[3], a3_19) + 2*( m(a[0], a[1]) + m(a[2], a4_19) );\n let mut c2: u128 = m(a[1], a[1]) + 2*( m(a[0], a[2]) + m(a[4], a3_19) );\n let mut c3: u128 = m(a[4], a4_19) + 2*( m(a[0], a[3]) + m(a[1], a[2]) );\n let mut c4: u128 = m(a[2], a[2]) + 2*( m(a[0], a[4]) + m(a[1], a[3]) );\n\n // Same bound as in multiply:\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54));\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n\n // Casting to u64 and back tells the compiler that the carry is bounded by 2^64, so\n // that the addition is a u128 + u64 rather than u128 + u128.\n c1 += ((c0 >> 51) as u64) as u128;\n a[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n a[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n a[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n a[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n a[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need a[0] + carry * 19 < 2^64.\n //\n // c4 < a2^2 + 2*a0*a4 + 2*a1*a3 + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // a[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n a[0] += carry * 19;\n\n // Now a[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n a[1] += a[0] >> 51;\n a[0] &= LOW_51_BIT_MASK;\n\n // Now all a[i] < 2^(51 + epsilon) and a = self^(2^k).\n\n k -= 1;\n if k == 0 {\n break;\n }\n }\n\n FieldElement51(a)\n }\n\n /// Returns the square of this field element.\n pub fn square(&self) -> FieldElement51 {\n self.pow2k(1)\n }\n\n /// Returns 2 times the square of this field element.\n pub fn square2(&self) -> FieldElement51 {\n let mut square = self.pow2k(1);\n for i in 0..5 {\n square.0[i] *= 2;\n }\n\n square\n }\n}\n"},{"id":10,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/curve_models.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Internal curve representations which are not part of the public API.\n//!\n//! # Curve representations\n//!\n//! Internally, we use several different models for the curve. Here\n//! is a sketch of the relationship between the models, following [a\n//! post][smith-moderncrypto]\n//! by Ben Smith on the `moderncrypto` mailing list. This is also briefly\n//! discussed in section 2.5 of [_Montgomery curves and their\n//! arithmetic_][costello-smith-2017] by Costello and Smith.\n//!\n//! Begin with the affine equation for the curve,\n//! $$\n//! -x\\^2 + y\\^2 = 1 + dx\\^2y\\^2.\n//! $$\n//! Next, pass to the projective closure \\\\(\\mathbb P\\^1 \\times \\mathbb\n//! P\\^1 \\\\) by setting \\\\(x=X/Z\\\\), \\\\(y=Y/T.\\\\) Clearing denominators\n//! gives the model\n//! $$\n//! -X\\^2T\\^2 + Y\\^2Z\\^2 = Z\\^2T\\^2 + dX\\^2Y\\^2.\n//! $$\n//! In `curve25519-dalek`, this is represented as the `CompletedPoint`\n//! struct.\n//! To map from \\\\(\\mathbb P\\^1 \\times \\mathbb P\\^1 \\\\), a product of\n//! two lines, to \\\\(\\mathbb P\\^3\\\\), we use the [Segre\n//! embedding](https://en.wikipedia.org/wiki/Segre_embedding)\n//! $$\n//! \\sigma : ((X:Z),(Y:T)) \\mapsto (XY:XT:ZY:ZT).\n//! $$\n//! Using coordinates \\\\( (W_0:W_1:W_2:W_3) \\\\) for \\\\(\\mathbb P\\^3\\\\),\n//! the image \\\\(\\sigma (\\mathbb P\\^1 \\times \\mathbb P\\^1) \\\\) is the\n//! surface defined by \\\\( W_0 W_3 = W_1 W_2 \\\\), and under \\\\(\n//! \\sigma\\\\), the equation above becomes\n//! $$\n//! -W\\_1\\^2 + W\\_2\\^2 = W\\_3\\^2 + dW\\_0\\^2,\n//! $$\n//! so that the curve is given by the pair of equations\n//! $$\n//! \\begin{aligned}\n//! -W\\_1\\^2 + W\\_2\\^2 &= W\\_3\\^2 + dW\\_0\\^2, \\\\\\\\ W_0 W_3 &= W_1 W_2.\n//! \\end{aligned}\n//! $$\n//! Up to variable naming, this is exactly the \"extended\" curve model\n//! introduced in [_Twisted Edwards Curves\n//! Revisited_][hisil-wong-carter-dawson-2008] by Hisil, Wong, Carter,\n//! and Dawson. In `curve25519-dalek`, it is represented as the\n//! `EdwardsPoint` struct. We can map from \\\\(\\mathbb P\\^3 \\\\) to\n//! \\\\(\\mathbb P\\^2 \\\\) by sending \\\\( (W\\_0:W\\_1:W\\_2:W\\_3) \\\\) to \\\\(\n//! (W\\_1:W\\_2:W\\_3) \\\\). Notice that\n//! $$\n//! \\frac {W\\_1} {W\\_3} = \\frac {XT} {ZT} = \\frac X Z = x,\n//! $$\n//! and\n//! $$\n//! \\frac {W\\_2} {W\\_3} = \\frac {YZ} {ZT} = \\frac Y T = y,\n//! $$\n//! so this is the same as if we had started with the affine model\n//! and passed to \\\\( \\mathbb P\\^2 \\\\) by setting \\\\( x = W\\_1 / W\\_3\n//! \\\\), \\\\(y = W\\_2 / W\\_3 \\\\).\n//! Up to variable naming, this is the projective representation\n//! introduced in [_Twisted Edwards\n//! Curves_][bernstein-birkner-joye-lange-peters-2008] by Bernstein,\n//! Birkner, Joye, Lange, and Peters. In `curve25519-dalek`, it is\n//! represented by the `ProjectivePoint` struct.\n//!\n//! # Passing between curve models\n//!\n//! Although the \\\\( \\mathbb P\\^3 \\\\) model provides faster addition\n//! formulas, the \\\\( \\mathbb P\\^2 \\\\) model provides faster doubling\n//! formulas. Hisil, Wong, Carter, and Dawson therefore suggest mixing\n//! coordinate systems for scalar multiplication, attributing the idea\n//! to [a 1998 paper][cohen-miyaji-ono-1998] of Cohen, Miyagi, and Ono.\n//!\n//! Their suggestion is to vary the formulas used by context, using a\n//! \\\\( \\mathbb P\\^2 \\rightarrow \\mathbb P\\^2 \\\\) doubling formula when\n//! a doubling is followed\n//! by another doubling, a \\\\( \\mathbb P\\^2 \\rightarrow \\mathbb P\\^3 \\\\)\n//! doubling formula when a doubling is followed by an addition, and\n//! computing point additions using a \\\\( \\mathbb P\\^3 \\times \\mathbb P\\^3\n//! \\rightarrow \\mathbb P\\^2 \\\\) formula.\n//!\n//! The `ref10` reference implementation of [Ed25519][ed25519], by\n//! Bernstein, Duif, Lange, Schwabe, and Yang, tweaks\n//! this strategy, factoring the addition formulas through the\n//! completion \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1 \\\\), so that the\n//! output of an addition or doubling always lies in \\\\( \\mathbb P\\^1 \\times\n//! \\mathbb P\\^1\\\\), and the choice of which formula to use is replaced\n//! by a choice of whether to convert the result to \\\\( \\mathbb P\\^2 \\\\)\n//! or \\\\(\\mathbb P\\^3 \\\\). However, this tweak is not described in\n//! their paper, only in their software.\n//!\n//! Our naming for the `CompletedPoint` (\\\\(\\mathbb P\\^1 \\times \\mathbb\n//! P\\^1 \\\\)), `ProjectivePoint` (\\\\(\\mathbb P\\^2 \\\\)), and\n//! `EdwardsPoint` (\\\\(\\mathbb P\\^3 \\\\)) structs follows the naming in\n//! Adam Langley's [Golang ed25519][agl-ed25519] implementation, which\n//! `curve25519-dalek` was originally derived from.\n//!\n//! Finally, to accelerate readditions, we use two cached point formats\n//! in \"Niels coordinates\", named for Niels Duif,\n//! one for the affine model and one for the \\\\( \\mathbb P\\^3 \\\\) model:\n//!\n//! * `AffineNielsPoint`: \\\\( (y+x, y-x, 2dxy) \\\\)\n//! * `ProjectiveNielsPoint`: \\\\( (Y+X, Y-X, Z, 2dXY) \\\\)\n//!\n//! [smith-moderncrypto]: https://moderncrypto.org/mail-archive/curves/2016/000807.html\n//! [costello-smith-2017]: https://eprint.iacr.org/2017/212\n//! [hisil-wong-carter-dawson-2008]: https://www.iacr.org/archive/asiacrypt2008/53500329/53500329.pdf\n//! [bernstein-birkner-joye-lange-peters-2008]: https://eprint.iacr.org/2008/013\n//! [cohen-miyaji-ono-1998]: https://link.springer.com/content/pdf/10.1007%2F3-540-49649-1_6.pdf\n//! [ed25519]: https://eprint.iacr.org/2011/368\n//! [agl-ed25519]: https://github.com/agl/ed25519\n\n#![allow(non_snake_case)]\n\nuse core::fmt::Debug;\nuse core::ops::{Add, Neg, Sub};\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::constants;\n\nuse crate::edwards::EdwardsPoint;\nuse crate::field::FieldElement;\nuse crate::traits::ValidityCheck;\n\n// ------------------------------------------------------------------------\n// Internal point representations\n// ------------------------------------------------------------------------\n\n/// A `ProjectivePoint` is a point \\\\((X:Y:Z)\\\\) on the \\\\(\\mathbb\n/// P\\^2\\\\) model of the curve.\n/// A point \\\\((x,y)\\\\) in the affine model corresponds to\n/// \\\\((x:y:1)\\\\).\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[allow(missing_docs)]\n#[derive(Copy, Clone)]\npub struct ProjectivePoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n}\n\n/// A `CompletedPoint` is a point \\\\(((X:Z), (Y:T))\\\\) on the \\\\(\\mathbb\n/// P\\^1 \\times \\mathbb P\\^1 \\\\) model of the curve.\n/// A point (x,y) in the affine model corresponds to \\\\( ((x:1),(y:1))\n/// \\\\).\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct CompletedPoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n pub T: FieldElement,\n}\n\n/// A pre-computed point in the affine model for the curve, represented as\n/// \\\\((y+x, y-x, 2dxy)\\\\) in \"Niels coordinates\".\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n// Safe to derive Eq because affine coordinates.\n#[derive(Copy, Clone, Eq, PartialEq)]\n#[allow(missing_docs)]\npub struct AffineNielsPoint {\n pub y_plus_x: FieldElement,\n pub y_minus_x: FieldElement,\n pub xy2d: FieldElement,\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for AffineNielsPoint {\n fn zeroize(&mut self) {\n self.y_plus_x.zeroize();\n self.y_minus_x.zeroize();\n self.xy2d.zeroize();\n }\n}\n\n/// A pre-computed point on the \\\\( \\mathbb P\\^3 \\\\) model for the\n/// curve, represented as \\\\((Y+X, Y-X, Z, 2dXY)\\\\) in \"Niels coordinates\".\n///\n/// More details on the relationships between the different curve models\n/// can be found in the module-level documentation.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct ProjectiveNielsPoint {\n pub Y_plus_X: FieldElement,\n pub Y_minus_X: FieldElement,\n pub Z: FieldElement,\n pub T2d: FieldElement,\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for ProjectiveNielsPoint {\n fn zeroize(&mut self) {\n self.Y_plus_X.zeroize();\n self.Y_minus_X.zeroize();\n self.Z.zeroize();\n self.T2d.zeroize();\n }\n}\n\n// ------------------------------------------------------------------------\n// Constructors\n// ------------------------------------------------------------------------\n\nuse crate::traits::Identity;\n\nimpl Identity for ProjectivePoint {\n fn identity() -> ProjectivePoint {\n ProjectivePoint {\n X: FieldElement::ZERO,\n Y: FieldElement::ONE,\n Z: FieldElement::ONE,\n }\n }\n}\n\nimpl Identity for ProjectiveNielsPoint {\n fn identity() -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::ONE,\n Y_minus_X: FieldElement::ONE,\n Z: FieldElement::ONE,\n T2d: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for ProjectiveNielsPoint {\n fn default() -> ProjectiveNielsPoint {\n ProjectiveNielsPoint::identity()\n }\n}\n\nimpl Identity for AffineNielsPoint {\n fn identity() -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: FieldElement::ONE,\n y_minus_x: FieldElement::ONE,\n xy2d: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for AffineNielsPoint {\n fn default() -> AffineNielsPoint {\n AffineNielsPoint::identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Validity checks (for debugging, not CT)\n// ------------------------------------------------------------------------\n\nimpl ValidityCheck for ProjectivePoint {\n fn is_valid(&self) -> bool {\n // Curve equation is -x^2 + y^2 = 1 + d*x^2*y^2,\n // homogenized as (-X^2 + Y^2)*Z^2 = Z^4 + d*X^2*Y^2\n let XX = self.X.square();\n let YY = self.Y.square();\n let ZZ = self.Z.square();\n let ZZZZ = ZZ.square();\n let lhs = &(&YY - &XX) * &ZZ;\n let rhs = &ZZZZ + &(&constants::EDWARDS_D * &(&XX * &YY));\n\n lhs == rhs\n }\n}\n\n// ------------------------------------------------------------------------\n// Constant-time assignment\n// ------------------------------------------------------------------------\n\nimpl ConditionallySelectable for ProjectiveNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::conditional_select(&a.Y_plus_X, &b.Y_plus_X, choice),\n Y_minus_X: FieldElement::conditional_select(&a.Y_minus_X, &b.Y_minus_X, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T2d: FieldElement::conditional_select(&a.T2d, &b.T2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.Y_plus_X.conditional_assign(&other.Y_plus_X, choice);\n self.Y_minus_X.conditional_assign(&other.Y_minus_X, choice);\n self.Z.conditional_assign(&other.Z, choice);\n self.T2d.conditional_assign(&other.T2d, choice);\n }\n}\n\nimpl ConditionallySelectable for AffineNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n AffineNielsPoint {\n y_plus_x: FieldElement::conditional_select(&a.y_plus_x, &b.y_plus_x, choice),\n y_minus_x: FieldElement::conditional_select(&a.y_minus_x, &b.y_minus_x, choice),\n xy2d: FieldElement::conditional_select(&a.xy2d, &b.xy2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.y_plus_x.conditional_assign(&other.y_plus_x, choice);\n self.y_minus_x.conditional_assign(&other.y_minus_x, choice);\n self.xy2d.conditional_assign(&other.xy2d, choice);\n }\n}\n\n// ------------------------------------------------------------------------\n// Point conversions\n// ------------------------------------------------------------------------\n\nimpl ProjectivePoint {\n /// Convert this point from the \\\\( \\mathbb P\\^2 \\\\) model to the\n /// \\\\( \\mathbb P\\^3 \\\\) model.\n ///\n /// This costs \\\\(3 \\mathrm M + 1 \\mathrm S\\\\).\n pub fn as_extended(&self) -> EdwardsPoint {\n EdwardsPoint {\n X: &self.X * &self.Z,\n Y: &self.Y * &self.Z,\n Z: self.Z.square(),\n T: &self.X * &self.Y,\n }\n }\n}\n\nimpl CompletedPoint {\n /// Convert this point from the \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1\n /// \\\\) model to the \\\\( \\mathbb P\\^2 \\\\) model.\n ///\n /// This costs \\\\(3 \\mathrm M \\\\).\n pub fn as_projective(&self) -> ProjectivePoint {\n ProjectivePoint {\n X: &self.X * &self.T,\n Y: &self.Y * &self.Z,\n Z: &self.Z * &self.T,\n }\n }\n\n /// Convert this point from the \\\\( \\mathbb P\\^1 \\times \\mathbb P\\^1\n /// \\\\) model to the \\\\( \\mathbb P\\^3 \\\\) model.\n ///\n /// This costs \\\\(4 \\mathrm M \\\\).\n pub fn as_extended(&self) -> EdwardsPoint {\n EdwardsPoint {\n X: &self.X * &self.T,\n Y: &self.Y * &self.Z,\n Z: &self.Z * &self.T,\n T: &self.X * &self.Y,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Doubling\n// ------------------------------------------------------------------------\n\nimpl ProjectivePoint {\n /// Double this point: return self + self\n pub fn double(&self) -> CompletedPoint {\n // Double()\n let XX = self.X.square();\n let YY = self.Y.square();\n let ZZ2 = self.Z.square2();\n let X_plus_Y = &self.X + &self.Y;\n let X_plus_Y_sq = X_plus_Y.square();\n let YY_plus_XX = &YY + &XX;\n let YY_minus_XX = &YY - &XX;\n\n CompletedPoint {\n X: &X_plus_Y_sq - &YY_plus_XX,\n Y: YY_plus_XX,\n Z: YY_minus_XX,\n T: &ZZ2 - &YY_minus_XX,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Addition and Subtraction\n// ------------------------------------------------------------------------\n\n// XXX(hdevalence) These were doc(hidden) so they don't appear in the\n// public API docs.\n// However, that prevents them being used with --document-private-items,\n// so comment out the doc(hidden) for now until this is resolved\n//\n// upstream rust issue: https://github.com/rust-lang/rust/issues/46380\n//#[doc(hidden)]\nimpl<'a> Add<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.Y_plus_X;\n let MM = &Y_minus_X * &other.Y_minus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &ZZ2 + &TT2d,\n T: &ZZ2 - &TT2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Sub<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.Y_minus_X;\n let MP = &Y_minus_X * &other.Y_plus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &ZZ2 - &TT2d,\n T: &ZZ2 + &TT2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Add<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.y_plus_x;\n let MM = &Y_minus_X * &other.y_minus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &Z2 + &Txy2d,\n T: &Z2 - &Txy2d,\n }\n }\n}\n\n//#[doc(hidden)]\nimpl<'a> Sub<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.y_minus_x;\n let MP = &Y_minus_X * &other.y_plus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &Z2 - &Txy2d,\n T: &Z2 + &Txy2d,\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Negation\n// ------------------------------------------------------------------------\n\nimpl Neg for &ProjectiveNielsPoint {\n type Output = ProjectiveNielsPoint;\n\n fn neg(self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: self.Y_minus_X,\n Y_minus_X: self.Y_plus_X,\n Z: self.Z,\n T2d: -(&self.T2d),\n }\n }\n}\n\nimpl Neg for &AffineNielsPoint {\n type Output = AffineNielsPoint;\n\n fn neg(self) -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: self.y_minus_x,\n y_minus_x: self.y_plus_x,\n xy2d: -(&self.xy2d),\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Debug traits\n// ------------------------------------------------------------------------\n\nimpl Debug for ProjectivePoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"ProjectivePoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z\n )\n }\n}\n\nimpl Debug for CompletedPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"CompletedPoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?},\\n\\tT: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z, &self.T\n )\n }\n}\n\nimpl Debug for AffineNielsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"AffineNielsPoint{{\\n\\ty_plus_x: {:?},\\n\\ty_minus_x: {:?},\\n\\txy2d: {:?}\\n}}\",\n &self.y_plus_x, &self.y_minus_x, &self.xy2d\n )\n }\n}\n\nimpl Debug for ProjectiveNielsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"ProjectiveNielsPoint{{\\n\\tY_plus_X: {:?},\\n\\tY_minus_X: {:?},\\n\\tZ: {:?},\\n\\tT2d: {:?}\\n}}\",\n &self.Y_plus_X, &self.Y_minus_X, &self.Z, &self.T2d\n )\n }\n}\n"},{"id":11,"name":{"Local":"curve25519/solana-ed25519/src/edwards.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2020 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Group operations for Curve25519, in Edwards form.\n//!\n//! ## Encoding and Decoding\n//!\n//! Encoding is done by converting to and from a `CompressedEdwardsY`\n//! struct, which is a typed wrapper around `[u8; 32]`.\n//!\n//! ## Equality Testing\n//!\n//! The `EdwardsPoint` struct implements the [`subtle::ConstantTimeEq`]\n//! trait for constant-time equality checking, and also uses this to\n//! ensure `Eq` equality checking runs in constant time.\n//!\n//! ## Cofactor-related functions\n//!\n//! The order of the group of points on the curve \\\\(\\mathcal E\\\\)\n//! is \\\\(|\\mathcal E| = 8\\ell \\\\), so its structure is \\\\( \\mathcal\n//! E = \\mathcal E\\[8\\] \\times \\mathcal E[\\ell]\\\\). The torsion\n//! subgroup \\\\( \\mathcal E\\[8\\] \\\\) consists of eight points of small\n//! order. Technically, all of \\\\(\\mathcal E\\\\) is torsion, but we\n//! use the word only to refer to the small \\\\(\\mathcal E\\[8\\]\\\\) part, not\n//! the large prime-order \\\\(\\mathcal E[\\ell]\\\\) part.\n//!\n//! To test if a point is in \\\\( \\mathcal E\\[8\\] \\\\), use\n//! [`EdwardsPoint::is_small_order`].\n//!\n//! To test if a point is in \\\\( \\mathcal E[\\ell] \\\\), use\n//! [`EdwardsPoint::is_torsion_free`].\n//!\n//! To multiply by the cofactor, use [`EdwardsPoint::mul_by_cofactor`].\n//!\n//! To avoid dealing with cofactors entirely, consider using Ristretto.\n//!\n//! ## Scalars\n//!\n//! Scalars are represented by the [`Scalar`] struct. To construct a scalar, see\n//! [`Scalar::from_canonical_bytes`] or [`Scalar::from_bytes_mod_order_wide`].\n//!\n//! ## Scalar Multiplication\n//!\n//! Scalar multiplication on Edwards points is provided by:\n//!\n//! * the `*` operator between a `Scalar` and a `EdwardsPoint`, which\n//! performs constant-time variable-base scalar multiplication;\n//!\n//! * the `*` operator between a `Scalar` and a\n//! `EdwardsBasepointTable`, which performs constant-time fixed-base\n//! scalar multiplication;\n//!\n//! * an implementation of the\n//! [`MultiscalarMul`](../traits/trait.MultiscalarMul.html) trait for\n//! constant-time variable-base multiscalar multiplication;\n//!\n//! * an implementation of the\n//! [`VartimeMultiscalarMul`](../traits/trait.VartimeMultiscalarMul.html)\n//! trait for variable-time variable-base multiscalar multiplication;\n//!\n//! ## Implementation\n//!\n//! The Edwards arithmetic is implemented using the “extended twisted\n//! coordinates” of Hisil, Wong, Carter, and Dawson, and the\n//! corresponding complete formulas. For more details,\n//! see the [`curve_models` submodule][curve_models]\n//! of the internal documentation.\n//!\n//! ## Validity Checking\n//!\n//! There is no function for checking whether a point is valid.\n//! Instead, the `EdwardsPoint` struct is guaranteed to hold a valid\n//! point on the curve.\n//!\n//! We use the Rust type system to make invalid points\n//! unrepresentable: `EdwardsPoint` objects can only be created via\n//! successful decompression of a compressed point, or else by\n//! operations on other (valid) `EdwardsPoint`s.\n//!\n//! [curve_models]: https://docs.rs/curve25519-dalek/latest/curve25519-dalek/backend/serial/curve_models/index.html\n\n// We allow non snake_case names because coordinates in projective space are\n// traditionally denoted by the capitalisation of their respective\n// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my\n// affine and projective cakes and eat both of them too.\n#![allow(non_snake_case)]\n\nmod affine;\n\nuse cfg_if::cfg_if;\nuse core::array::TryFromSliceError;\nuse core::borrow::Borrow;\nuse core::fmt::Debug;\nuse core::iter::Sum;\nuse core::ops::{Add, Neg, Sub};\nuse core::ops::{AddAssign, SubAssign};\nuse core::ops::{Mul, MulAssign};\n\n#[cfg(feature = \"digest\")]\nuse digest::{\n FixedOutput, HashMarker, array::typenum::U64, block_api::BlockSizeUser, consts::True,\n typenum::IsGreater,\n};\n\n#[cfg(feature = \"group\")]\nuse {\n group::{GroupEncoding, cofactor::CofactorGroup, prime::PrimeGroup},\n rand_core::RngCore as GroupRngCore,\n subtle::CtOption,\n};\n\n#[cfg(feature = \"rand_core\")]\nuse rand_core::{CryptoRng, RngCore};\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nuse crate::constants;\n\nuse crate::field::FieldElement;\nuse crate::scalar::{Scalar, clamp_integer};\n\nuse crate::montgomery::MontgomeryPoint;\n\nuse crate::backend::serial::curve_models::AffineNielsPoint;\nuse crate::backend::serial::curve_models::CompletedPoint;\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::backend::serial::curve_models::ProjectivePoint;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::window::{\n LookupTableRadix16, LookupTableRadix32, LookupTableRadix64, LookupTableRadix128,\n LookupTableRadix256,\n};\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::traits::BasepointTable;\n\nuse crate::traits::ValidityCheck;\nuse crate::traits::{Identity, IsIdentity};\n\nuse affine::AffinePoint;\n\n#[cfg(feature = \"alloc\")]\nuse crate::traits::MultiscalarMul;\n#[cfg(feature = \"alloc\")]\nuse crate::traits::{VartimeMultiscalarMul, VartimePrecomputedMultiscalarMul};\n#[cfg(feature = \"alloc\")]\nuse alloc::vec::Vec;\n\n// ------------------------------------------------------------------------\n// Compressed points\n// ------------------------------------------------------------------------\n\n/// In \"Edwards y\" / \"Ed25519\" format, the curve point \\\\((x,y)\\\\) is\n/// determined by the \\\\(y\\\\)-coordinate and the sign of \\\\(x\\\\).\n///\n/// The first 255 bits of a `CompressedEdwardsY` represent the\n/// \\\\(y\\\\)-coordinate. The high bit of the 32nd byte gives the sign of \\\\(x\\\\).\n#[allow(clippy::derived_hash_with_manual_eq)]\n#[derive(Copy, Clone, Hash)]\npub struct CompressedEdwardsY(pub [u8; 32]);\n\nimpl ConstantTimeEq for CompressedEdwardsY {\n fn ct_eq(&self, other: &CompressedEdwardsY) -> Choice {\n self.as_bytes().ct_eq(other.as_bytes())\n }\n}\n\nimpl Eq for CompressedEdwardsY {}\nimpl PartialEq for CompressedEdwardsY {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Debug for CompressedEdwardsY {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"CompressedEdwardsY: {:?}\", self.as_bytes())\n }\n}\n\nimpl CompressedEdwardsY {\n /// View this `CompressedEdwardsY` as an array of bytes.\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.0\n }\n\n /// Copy this `CompressedEdwardsY` to an array of bytes.\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.0\n }\n\n /// Attempt to decompress to an `EdwardsPoint`.\n ///\n /// Returns `None` if the input is not the \\\\(y\\\\)-coordinate of a\n /// curve point.\n pub fn decompress(&self) -> Option {\n let (is_valid_y_coord, X, Y, Z) = decompress::step_1(self);\n\n if is_valid_y_coord.into() {\n Some(decompress::step_2(self, X, Y, Z))\n } else {\n None\n }\n }\n}\n\nmod decompress {\n use super::*;\n\n #[rustfmt::skip] // keep alignment of explanatory comments\n pub(super) fn step_1(\n repr: &CompressedEdwardsY,\n ) -> (Choice, FieldElement, FieldElement, FieldElement) {\n let Y = FieldElement::from_bytes(repr.as_bytes());\n let Z = FieldElement::ONE;\n let YY = Y.square();\n let u = &YY - &Z; // u = y²-1\n let v = &(&YY * &constants::EDWARDS_D) + &Z; // v = dy²+1\n let (is_valid_y_coord, X) = FieldElement::sqrt_ratio_i(&u, &v);\n\n (is_valid_y_coord, X, Y, Z)\n }\n\n #[rustfmt::skip]\n pub(super) fn step_2(\n repr: &CompressedEdwardsY,\n mut X: FieldElement,\n Y: FieldElement,\n Z: FieldElement,\n ) -> EdwardsPoint {\n // FieldElement::sqrt_ratio_i always returns the nonnegative square root,\n // so we negate according to the supplied sign bit.\n let compressed_sign_bit = Choice::from(repr.as_bytes()[31] >> 7);\n // AENEAS-COMPAT: negate-then-conditional-assign instead of\n // `X.conditional_negate(...)` — semantically identical and still\n // constant-time, but avoids subtle's `ConditionallyNegatable`\n // blanket impl which breaks the verification toolchain (the same\n // documented rewrite as in `FieldElement::sqrt_ratio_i`).\n let X_neg = -&X;\n X.conditional_assign(&X_neg, compressed_sign_bit);\n\n EdwardsPoint {\n X,\n Y,\n Z,\n T: &X * &Y,\n }\n }\n}\n\nimpl TryFrom<&[u8]> for CompressedEdwardsY {\n type Error = TryFromSliceError;\n\n fn try_from(slice: &[u8]) -> Result {\n Self::from_slice(slice)\n }\n}\n\n// ------------------------------------------------------------------------\n// Serde support\n// ------------------------------------------------------------------------\n// Serializes to and from `EdwardsPoint` directly, doing compression\n// and decompression internally. This means that users can create\n// structs containing `EdwardsPoint`s and use Serde's derived\n// serializers to serialize those structures.\n\n#[cfg(feature = \"digest\")]\nuse constants::ED25519_SQRTAM2;\n#[cfg(feature = \"serde\")]\nuse serde::de::Visitor;\n#[cfg(feature = \"serde\")]\nuse serde::{Deserialize, Deserializer, Serialize, Serializer};\n\n#[cfg(feature = \"serde\")]\nimpl Serialize for EdwardsPoint {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.compress().as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl Serialize for CompressedEdwardsY {\n fn serialize(&self, serializer: S) -> Result\n where\n S: Serializer,\n {\n use serde::ser::SerializeTuple;\n let mut tup = serializer.serialize_tuple(32)?;\n for byte in self.as_bytes().iter() {\n tup.serialize_element(byte)?;\n }\n tup.end()\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl<'de> Deserialize<'de> for EdwardsPoint {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct EdwardsPointVisitor;\n\n impl<'de> Visitor<'de> for EdwardsPointVisitor {\n type Value = EdwardsPoint;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\"a valid point in Edwards y + sign format\")\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n CompressedEdwardsY(bytes)\n .decompress()\n .ok_or_else(|| serde::de::Error::custom(\"decompression failed\"))\n }\n }\n\n deserializer.deserialize_tuple(32, EdwardsPointVisitor)\n }\n}\n\n#[cfg(feature = \"serde\")]\nimpl<'de> Deserialize<'de> for CompressedEdwardsY {\n fn deserialize(deserializer: D) -> Result\n where\n D: Deserializer<'de>,\n {\n struct CompressedEdwardsYVisitor;\n\n impl<'de> Visitor<'de> for CompressedEdwardsYVisitor {\n type Value = CompressedEdwardsY;\n\n fn expecting(&self, formatter: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n formatter.write_str(\"32 bytes of data\")\n }\n\n fn visit_seq(self, mut seq: A) -> Result\n where\n A: serde::de::SeqAccess<'de>,\n {\n let mut bytes = [0u8; 32];\n #[allow(clippy::needless_range_loop)]\n for i in 0..32 {\n bytes[i] = seq\n .next_element()?\n .ok_or_else(|| serde::de::Error::invalid_length(i, &\"expected 32 bytes\"))?;\n }\n Ok(CompressedEdwardsY(bytes))\n }\n }\n\n deserializer.deserialize_tuple(32, CompressedEdwardsYVisitor)\n }\n}\n\n// ------------------------------------------------------------------------\n// Internal point representations\n// ------------------------------------------------------------------------\n\n/// An `EdwardsPoint` represents a point on the Edwards form of Curve25519.\n#[derive(Copy, Clone)]\n#[allow(missing_docs)]\npub struct EdwardsPoint {\n pub(crate) X: FieldElement,\n pub(crate) Y: FieldElement,\n pub(crate) Z: FieldElement,\n pub(crate) T: FieldElement,\n}\n\n// ------------------------------------------------------------------------\n// Constructors\n// ------------------------------------------------------------------------\n\nimpl Identity for CompressedEdwardsY {\n fn identity() -> CompressedEdwardsY {\n CompressedEdwardsY([\n 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0,\n ])\n }\n}\n\nimpl Default for CompressedEdwardsY {\n fn default() -> CompressedEdwardsY {\n CompressedEdwardsY::identity()\n }\n}\n\nimpl CompressedEdwardsY {\n /// Construct a `CompressedEdwardsY` from a slice of bytes.\n ///\n /// # Errors\n ///\n /// Returns [`TryFromSliceError`] if the input `bytes` slice does not have\n /// a length of 32.\n pub fn from_slice(bytes: &[u8]) -> Result {\n bytes.try_into().map(CompressedEdwardsY)\n }\n}\n\nimpl Identity for EdwardsPoint {\n fn identity() -> EdwardsPoint {\n EdwardsPoint {\n X: FieldElement::ZERO,\n Y: FieldElement::ONE,\n Z: FieldElement::ONE,\n T: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for EdwardsPoint {\n fn default() -> EdwardsPoint {\n EdwardsPoint::identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Zeroize implementations for wiping points from memory\n// ------------------------------------------------------------------------\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for CompressedEdwardsY {\n /// Reset this `CompressedEdwardsY` to the compressed form of the identity element.\n fn zeroize(&mut self) {\n self.0.zeroize();\n self.0[0] = 1;\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for EdwardsPoint {\n /// Reset this `EdwardsPoint` to the identity element.\n fn zeroize(&mut self) {\n self.X.zeroize();\n self.Y = FieldElement::ONE;\n self.Z = FieldElement::ONE;\n self.T.zeroize();\n }\n}\n\n// ------------------------------------------------------------------------\n// Validity checks (for debugging, not CT)\n// ------------------------------------------------------------------------\n\nimpl ValidityCheck for EdwardsPoint {\n fn is_valid(&self) -> bool {\n let point_on_curve = self.as_projective().is_valid();\n let on_segre_image = (&self.X * &self.Y) == (&self.Z * &self.T);\n\n point_on_curve && on_segre_image\n }\n}\n\n// ------------------------------------------------------------------------\n// Constant-time assignment\n// ------------------------------------------------------------------------\n\nimpl ConditionallySelectable for EdwardsPoint {\n fn conditional_select(a: &EdwardsPoint, b: &EdwardsPoint, choice: Choice) -> EdwardsPoint {\n EdwardsPoint {\n X: FieldElement::conditional_select(&a.X, &b.X, choice),\n Y: FieldElement::conditional_select(&a.Y, &b.Y, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T: FieldElement::conditional_select(&a.T, &b.T, choice),\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Equality\n// ------------------------------------------------------------------------\n\nimpl ConstantTimeEq for EdwardsPoint {\n fn ct_eq(&self, other: &EdwardsPoint) -> Choice {\n // We would like to check that the point (X/Z, Y/Z) is equal to\n // the point (X'/Z', Y'/Z') without converting into affine\n // coordinates (x, y) and (x', y'), which requires two inversions.\n // We have that X = xZ and X' = x'Z'. Thus, x = x' is equivalent to\n // (xZ)Z' = (x'Z')Z, and similarly for the y-coordinate.\n\n (&self.X * &other.Z).ct_eq(&(&other.X * &self.Z))\n & (&self.Y * &other.Z).ct_eq(&(&other.Y * &self.Z))\n }\n}\n\nimpl PartialEq for EdwardsPoint {\n fn eq(&self, other: &EdwardsPoint) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for EdwardsPoint {}\n\n// ------------------------------------------------------------------------\n// Point conversions\n// ------------------------------------------------------------------------\n\nimpl EdwardsPoint {\n /// Convert to a ProjectiveNielsPoint\n pub(crate) fn as_projective_niels(&self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: &self.Y + &self.X,\n Y_minus_X: &self.Y - &self.X,\n Z: self.Z,\n T2d: &self.T * &constants::EDWARDS_D2,\n }\n }\n\n /// Convert the representation of this point from extended\n /// coordinates to projective coordinates.\n ///\n /// Free.\n pub(crate) const fn as_projective(&self) -> ProjectivePoint {\n ProjectivePoint {\n X: self.X,\n Y: self.Y,\n Z: self.Z,\n }\n }\n\n /// Dehomogenize to a `AffineNielsPoint`.\n /// Mainly for testing.\n pub(crate) fn as_affine_niels(&self) -> AffineNielsPoint {\n let recip = self.Z.invert();\n let x = &self.X * &recip;\n let y = &self.Y * &recip;\n let xy2d = &(&x * &y) * &constants::EDWARDS_D2;\n AffineNielsPoint {\n y_plus_x: &y + &x,\n y_minus_x: &y - &x,\n xy2d,\n }\n }\n\n /// Dehomogenize to `AffinePoint`.\n pub(crate) fn to_affine(self) -> AffinePoint {\n let recip = self.Z.invert();\n let x = &self.X * &recip;\n let y = &self.Y * &recip;\n AffinePoint { x, y }\n }\n\n /// Convert this `EdwardsPoint` on the Edwards model to the\n /// corresponding `MontgomeryPoint` on the Montgomery model.\n ///\n /// This function has one exceptional case; the identity point of\n /// the Edwards curve is sent to the 2-torsion point \\\\((0,0)\\\\)\n /// on the Montgomery curve.\n ///\n /// Note that this is a one-way conversion, since the Montgomery\n /// model does not retain sign information.\n pub fn to_montgomery(&self) -> MontgomeryPoint {\n // We have u = (1+y)/(1-y) = (Z+Y)/(Z-Y).\n //\n // The denominator is zero only when y=1, the identity point of\n // the Edwards curve. Since 0.invert() = 0, in this case we\n // compute the 2-torsion point (0,0).\n let U = &self.Z + &self.Y;\n let W = &self.Z - &self.Y;\n let u = &U * &W.invert();\n MontgomeryPoint(u.to_bytes())\n }\n\n /// Converts a large batch of points to Edwards at once. This has the same\n /// behavior on identity elements as [`Self::to_montgomery`].\n #[cfg(feature = \"alloc\")]\n pub fn to_montgomery_batch(eds: &[Self]) -> Vec {\n // Do the same thing as the above function. u = (1+y)/(1-y) = (Z+Y)/(Z-Y).\n // We will do this in a batch, ie compute (Z-Y) for all the input\n // points, then invert them all at once\n\n // Compute the denominators in a batch\n let mut denominators = eds.iter().map(|p| &p.Z - &p.Y).collect::>();\n FieldElement::invert_batch_alloc(&mut denominators);\n\n // Now compute the Montgomery u coordinate for every point\n let mut ret = Vec::with_capacity(eds.len());\n for (ed, d) in eds.iter().zip(denominators.iter()) {\n let u = &(&ed.Z + &ed.Y) * d;\n ret.push(MontgomeryPoint(u.to_bytes()));\n }\n\n ret\n }\n\n /// Compress this point to `CompressedEdwardsY` format.\n pub fn compress(&self) -> CompressedEdwardsY {\n self.to_affine().compress()\n }\n\n /// Compress several `EdwardsPoint`s into `CompressedEdwardsY` format, using a batch inversion\n /// for a significant speedup.\n #[cfg(feature = \"alloc\")]\n pub fn compress_batch(inputs: &[EdwardsPoint]) -> Vec {\n let mut zs = inputs.iter().map(|input| input.Z).collect::>();\n FieldElement::invert_batch_alloc(&mut zs);\n\n inputs\n .iter()\n .zip(&zs)\n .map(|(input, recip)| {\n let x = &input.X * recip;\n let y = &input.Y * recip;\n AffinePoint { x, y }.compress()\n })\n .collect()\n }\n\n #[cfg(feature = \"digest\")]\n // The function `map_to_curve` calculates an [EdwardsPoint] from a [FieldElement].\n fn map_to_curve(fe: FieldElement) -> EdwardsPoint {\n let c1 = ED25519_SQRTAM2;\n\n // 1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u)\n let (xMn, xMd, yMn, yMd) = crate::montgomery::elligator_encode(&fe);\n // 2. xn = xMn * yMd\n let xn = &xMn * &yMd;\n // 3. xn = xn * c1\n let xn = &xn * &c1;\n // 4. xd = xMd * yMn\n let xd = &xMd * &yMn;\n // 5. yn = xMn - xMd\n let yn = &xMn - &xMd;\n // 6. yd = xMn + xMd\n let yd = &xMn + &xMd;\n // 7. tv1 = xd * yd\n let tv1 = &xd * &yd;\n // 8. e = tv1 == 0\n let e = tv1.ct_eq(&FieldElement::ZERO);\n // 9. xn = CMOV(xn, 0, e)\n let xn = FieldElement::conditional_select(&xn, &FieldElement::ZERO, e);\n // 10. xd = CMOV(xd, 1, e)\n let xd = FieldElement::conditional_select(&xd, &FieldElement::ONE, e);\n // 11. yn = CMOV(yn, 1, e)\n let yn = FieldElement::conditional_select(&yn, &FieldElement::ONE, e);\n // 12. yd = CMOV(yd, 1, e)\n let yd = FieldElement::conditional_select(&yd, &FieldElement::ONE, e);\n // 13. return (xn, xd, yn, yd)\n\n EdwardsPoint {\n X: &xn * &yd,\n Y: &xd * &yn,\n Z: &xd * &yd,\n T: &xn * &yn,\n }\n }\n\n #[cfg(feature = \"digest\")]\n /// Perform encode to curve per RFC 9380, with explicit hash function and domain separator\n /// `domain_sep`, using the Twisted Edwards Elligator 2 method. The input is the concatenation\n /// of the elements of `bytes`. Likewise for the domain separator with `domain_sep`. At least\n /// one element of `domain_sep`, MUST be nonempty, and the concatenation MUST NOT exceed 255\n /// bytes.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Warning\n /// `encode_to_curve` is a nonuniform encoding from byte strings to points in `G`. That is,\n /// the distribution of its output is not uniformly random in `G`: the set of possible outputs\n /// of encode_to_curve is only a fraction of the points in `G`, and some points in this set\n /// are more likely to be output than others.\n ///\n /// If your application needs the distribution of the output to be statistically close to\n /// uniform in `G`, use [Self::hash_to_curve] instead.\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`\n pub fn encode_to_curve(bytes: &[&[u8]], domain_sep: &[&[u8]]) -> EdwardsPoint\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // For reference see\n // https://www.rfc-editor.org/rfc/rfc9380.html#name-elligator-2-method-2\n\n let fe = FieldElement::hash_to_field::(bytes, domain_sep);\n let Q = Self::map_to_curve(fe[0]);\n Q.mul_by_cofactor()\n }\n\n #[cfg(feature = \"digest\")]\n /// Perform a hash to curve per RFC 9380, with explicit hash function and domain separator\n /// `domain_sep`, using the Twisted Edwards Elligator 2 method. The input is the concatenation\n /// of the elements of `bytes`. Likewise for the domain separator with `domain_sep`. At least\n /// one element of `domain_sep`, MUST be nonempty, and the concatenation MUST NOT exceed\n /// 255 bytes.\n ///\n /// The specification names SHA-512 as an example of a secure hash to use with this function,\n /// but you may use any 512-bit hash within reason (see the\n /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details).\n ///\n /// # Panics\n /// Panics if `domain_sep.collect().len() == 0` or `> 255`\n pub fn hash_to_curve(bytes: &[&[u8]], domain_sep: &[&[u8]]) -> EdwardsPoint\n where\n D: BlockSizeUser + Default + FixedOutput + HashMarker,\n D::BlockSize: IsGreater,\n {\n // For reference see\n // https://www.rfc-editor.org/rfc/rfc9380.html#name-elligator-2-method-2\n\n let fe = FieldElement::hash_to_field::(bytes, domain_sep);\n let Q0 = Self::map_to_curve(fe[0]);\n let Q1 = Self::map_to_curve(fe[1]);\n\n let R = Q0 + Q1;\n R.mul_by_cofactor()\n }\n\n /// Return an `EdwardsPoint` chosen uniformly at random using a user-provided RNG.\n ///\n /// # Inputs\n ///\n /// * `rng`: any RNG which implements `CryptoRng` and `RngCore`\n ///\n /// # Returns\n ///\n /// A random `EdwardsPoint`.\n ///\n /// # Implementation\n ///\n /// Uses rejection sampling, generating a random `CompressedEdwardsY` and then attempting point\n /// decompression, rejecting invalid points.\n #[cfg(feature = \"rand_core\")]\n pub fn random(rng: &mut R) -> Self {\n let mut repr = CompressedEdwardsY([0u8; 32]);\n loop {\n rng.fill_bytes(&mut repr.0);\n if let Some(p) = repr.decompress() {\n if !IsIdentity::is_identity(&p) {\n break p;\n }\n }\n }\n }\n}\n\n// ------------------------------------------------------------------------\n// Doubling\n// ------------------------------------------------------------------------\n\nimpl EdwardsPoint {\n /// Add this point to itself.\n pub(crate) fn double(&self) -> EdwardsPoint {\n self.as_projective().double().as_extended()\n }\n}\n\n// ------------------------------------------------------------------------\n// Addition and Subtraction\n// ------------------------------------------------------------------------\n\nimpl<'a> Add<&'a EdwardsPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn add(self, other: &'a EdwardsPoint) -> EdwardsPoint {\n (self + &other.as_projective_niels()).as_extended()\n }\n}\n\ndefine_add_variants!(\n LHS = EdwardsPoint,\n RHS = EdwardsPoint,\n Output = EdwardsPoint\n);\n\nimpl<'a> AddAssign<&'a EdwardsPoint> for EdwardsPoint {\n fn add_assign(&mut self, _rhs: &'a EdwardsPoint) {\n *self = (self as &EdwardsPoint) + _rhs;\n }\n}\n\ndefine_add_assign_variants!(LHS = EdwardsPoint, RHS = EdwardsPoint);\n\nimpl<'a> Sub<&'a EdwardsPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn sub(self, other: &'a EdwardsPoint) -> EdwardsPoint {\n (self - &other.as_projective_niels()).as_extended()\n }\n}\n\ndefine_sub_variants!(\n LHS = EdwardsPoint,\n RHS = EdwardsPoint,\n Output = EdwardsPoint\n);\n\nimpl<'a> SubAssign<&'a EdwardsPoint> for EdwardsPoint {\n fn sub_assign(&mut self, _rhs: &'a EdwardsPoint) {\n *self = (self as &EdwardsPoint) - _rhs;\n }\n}\n\ndefine_sub_assign_variants!(LHS = EdwardsPoint, RHS = EdwardsPoint);\n\nimpl Sum for EdwardsPoint\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n iter.fold(EdwardsPoint::identity(), |acc, item| acc + item.borrow())\n }\n}\n\n// ------------------------------------------------------------------------\n// Negation\n// ------------------------------------------------------------------------\n\nimpl Neg for &EdwardsPoint {\n type Output = EdwardsPoint;\n\n fn neg(self) -> EdwardsPoint {\n EdwardsPoint {\n X: -(&self.X),\n Y: self.Y,\n Z: self.Z,\n T: -(&self.T),\n }\n }\n}\n\nimpl Neg for EdwardsPoint {\n type Output = EdwardsPoint;\n\n fn neg(self) -> EdwardsPoint {\n -&self\n }\n}\n\n// ------------------------------------------------------------------------\n// Scalar multiplication\n// ------------------------------------------------------------------------\n\nimpl<'a> MulAssign<&'a Scalar> for EdwardsPoint {\n fn mul_assign(&mut self, scalar: &'a Scalar) {\n let result = (self as &EdwardsPoint) * scalar;\n *self = result;\n }\n}\n\ndefine_mul_assign_variants!(LHS = EdwardsPoint, RHS = Scalar);\n\ndefine_mul_variants!(LHS = EdwardsPoint, RHS = Scalar, Output = EdwardsPoint);\ndefine_mul_variants!(LHS = Scalar, RHS = EdwardsPoint, Output = EdwardsPoint);\n\nimpl<'a> Mul<&'a Scalar> for &EdwardsPoint {\n type Output = EdwardsPoint;\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, scalar: &'a Scalar) -> EdwardsPoint {\n crate::backend::variable_base_mul(self, scalar)\n }\n}\n\nimpl<'a> Mul<&'a EdwardsPoint> for &Scalar {\n type Output = EdwardsPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, point: &'a EdwardsPoint) -> EdwardsPoint {\n point * self\n }\n}\n\nimpl EdwardsPoint {\n /// Fixed-base scalar multiplication by the Ed25519 base point.\n ///\n /// Uses precomputed basepoint tables when the `precomputed-tables` feature\n /// is enabled, trading off increased code size for ~4x better performance.\n pub fn mul_base(scalar: &Scalar) -> Self {\n #[cfg(not(feature = \"precomputed-tables\"))]\n {\n scalar * constants::ED25519_BASEPOINT_POINT\n }\n\n #[cfg(feature = \"precomputed-tables\")]\n {\n scalar * constants::ED25519_BASEPOINT_TABLE\n }\n }\n\n /// Multiply this point by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_clamped(self, bytes: [u8; 32]) -> Self {\n // We have to construct a Scalar that is not reduced mod l, which breaks scalar invariant\n // #2. But #2 is not necessary for correctness of variable-base multiplication. All that\n // needs to hold is invariant #1, i.e., the scalar is less than 2^255. This is guaranteed\n // by clamping.\n // Further, we don't do any reduction or arithmetic with this clamped value, so there's no\n // issues arising from the fact that the curve point is not necessarily in the prime-order\n // subgroup.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n s * self\n }\n\n /// Multiply the basepoint by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_base_clamped(bytes: [u8; 32]) -> Self {\n // See reasoning in Self::mul_clamped why it is OK to make an unreduced Scalar here. We\n // note that fixed-base multiplication is also defined for all values of `bytes` less than\n // 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n Self::mul_base(&s)\n }\n}\n\n// ------------------------------------------------------------------------\n// Multiscalar Multiplication impls\n// ------------------------------------------------------------------------\n\n// These use the iterator's size hint and the target settings to\n// forward to a specific backend implementation.\n\n#[cfg(feature = \"alloc\")]\nimpl MultiscalarMul for EdwardsPoint {\n type Point = EdwardsPoint;\n\n fn multiscalar_mul(scalars: I, points: J) -> EdwardsPoint\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n {\n // Sanity-check lengths of input iterators\n let mut scalars = scalars.into_iter();\n let mut points = points.into_iter();\n\n // Lower and upper bounds on iterators\n let (s_lo, s_hi) = scalars.by_ref().size_hint();\n let (p_lo, p_hi) = points.by_ref().size_hint();\n\n // They should all be equal\n assert_eq!(s_lo, p_lo);\n assert_eq!(s_hi, Some(s_lo));\n assert_eq!(p_hi, Some(p_lo));\n\n // Now we know there's a single size. When we do\n // size-dependent algorithm dispatch, use this as the hint.\n let _size = s_lo;\n\n crate::backend::straus_multiscalar_mul(scalars, points)\n }\n}\n\n#[cfg(feature = \"alloc\")]\nimpl VartimeMultiscalarMul for EdwardsPoint {\n type Point = EdwardsPoint;\n\n fn optional_multiscalar_mul(scalars: I, points: J) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator>,\n {\n // Sanity-check lengths of input iterators\n let mut scalars = scalars.into_iter();\n let mut points = points.into_iter();\n\n // Lower and upper bounds on iterators\n let (s_lo, s_hi) = scalars.by_ref().size_hint();\n let (p_lo, p_hi) = points.by_ref().size_hint();\n\n // They should all be equal\n assert_eq!(s_lo, p_lo);\n assert_eq!(s_hi, Some(s_lo));\n assert_eq!(p_hi, Some(p_lo));\n\n // Now we know there's a single size.\n // Use this as the hint to decide which algorithm to use.\n let size = s_lo;\n\n if size < 190 {\n crate::backend::straus_optional_multiscalar_mul(scalars, points)\n } else {\n crate::backend::pippenger_optional_multiscalar_mul(scalars, points)\n }\n }\n}\n\n/// Precomputation for variable-time multiscalar multiplication with `EdwardsPoint`s.\n// This wraps the inner implementation in a facade type so that we can\n// decouple stability of the inner type from the stability of the\n// outer type.\n#[cfg(feature = \"alloc\")]\npub struct VartimeEdwardsPrecomputation(crate::backend::VartimePrecomputedStraus);\n\n#[cfg(feature = \"alloc\")]\nimpl VartimePrecomputedMultiscalarMul for VartimeEdwardsPrecomputation {\n type Point = EdwardsPoint;\n\n fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: Borrow,\n {\n Self(crate::backend::VartimePrecomputedStraus::new(static_points))\n }\n\n fn len(&self) -> usize {\n self.0.len()\n }\n\n fn is_empty(&self) -> bool {\n self.0.is_empty()\n }\n\n fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator>,\n {\n self.0\n .optional_mixed_multiscalar_mul(static_scalars, dynamic_scalars, dynamic_points)\n }\n}\n\nimpl EdwardsPoint {\n /// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n pub fn vartime_double_scalar_mul_basepoint(\n a: &Scalar,\n A: &EdwardsPoint,\n b: &Scalar,\n ) -> EdwardsPoint {\n crate::backend::vartime_double_base_mul(a, A, b)\n }\n\n /// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n ///\n /// This function is optimized for the case where \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\),\n /// and falls back to general scalar multiplication for full-width scalars.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::scalar::Scalar;\n /// use curve25519::constants::ED25519_BASEPOINT_POINT;\n /// use curve25519::edwards::EdwardsPoint;\n ///\n /// let a1 = Scalar::from(123u64);\n /// let a2 = Scalar::from(456u64);\n /// let b = Scalar::from(789u64);\n ///\n /// let A1 = &ED25519_BASEPOINT_POINT * &Scalar::from(2u64);\n /// let A2 = &ED25519_BASEPOINT_POINT * &Scalar::from(3u64);\n ///\n /// // Compute a1*A1 + a2*A2 + b*B efficiently\n /// let result = EdwardsPoint::vartime_triple_scalar_mul_basepoint(&a1, &A1, &a2, &A2, &b);\n /// ```\n #[allow(non_snake_case)]\n pub fn vartime_triple_scalar_mul_basepoint(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n ) -> EdwardsPoint {\n crate::backend::vartime_triple_base_mul_128_128_256(a1, A1, a2, A2, b)\n }\n}\n\n#[cfg(feature = \"precomputed-tables\")]\nmacro_rules! impl_basepoint_table {\n (Name = $name:ident, LookupTable = $table:ident, Point = $point:ty, Radix = $radix:expr, Additions = $adds:expr) => {\n /// A precomputed table of multiples of a basepoint, for accelerating\n /// fixed-base scalar multiplication. One table, for the Ed25519\n /// basepoint, is provided in the [`constants`] module.\n ///\n /// The basepoint tables are reasonably large, so they should probably be boxed.\n ///\n /// The sizes for the tables and the number of additions required for one scalar\n /// multiplication are as follows:\n ///\n /// * [`EdwardsBasepointTableRadix16`]: 30KB, 64A\n /// (this is the default size, and is used for\n /// [`constants::ED25519_BASEPOINT_TABLE`])\n /// * [`EdwardsBasepointTableRadix64`]: 120KB, 43A\n /// * [`EdwardsBasepointTableRadix128`]: 240KB, 37A\n /// * [`EdwardsBasepointTableRadix256`]: 480KB, 33A\n ///\n /// # Why 33 additions for radix-256?\n ///\n /// Normally, the radix-256 tables would allow for only 32 additions per scalar\n /// multiplication. However, due to the fact that standardised definitions of\n /// legacy protocols—such as x25519—require allowing unreduced 255-bit scalars\n /// invariants, when converting such an unreduced scalar's representation to\n /// radix-\\\\(2^{8}\\\\), we cannot guarantee the carry bit will fit in the last\n /// coefficient (the coefficients are `i8`s). When, \\\\(w\\\\), the power-of-2 of\n /// the radix, is \\\\(w < 8\\\\), we can fold the final carry onto the last\n /// coefficient, \\\\(d\\\\), because \\\\(d < 2^{w/2}\\\\), so\n /// $$\n /// d + carry \\cdot 2^{w} = d + 1 \\cdot 2^{w} < 2^{w+1} < 2^{8}\n /// $$\n /// When \\\\(w = 8\\\\), we can't fit \\\\(carry \\cdot 2^{w}\\\\) into an `i8`, so we\n /// add the carry bit onto an additional coefficient.\n #[derive(Clone)]\n #[repr(transparent)]\n pub struct $name(pub(crate) [$table; 32]);\n\n impl BasepointTable for $name {\n type Point = $point;\n\n /// Create a table of precomputed multiples of `basepoint`.\n fn create(basepoint: &$point) -> $name {\n // XXX use init_with\n let mut table = $name([$table::default(); 32]);\n let mut P = *basepoint;\n for i in 0..32 {\n // P = (2w)^i * B\n table.0[i] = $table::from(&P);\n P = P.mul_by_pow_2($radix + $radix);\n }\n table\n }\n\n /// Get the basepoint for this table as an `EdwardsPoint`.\n fn basepoint(&self) -> $point {\n // self.0[0].select(1) = 1*(16^2)^0*B\n // but as an `AffineNielsPoint`, so add identity to convert to extended.\n (&<$point>::identity() + &self.0[0].select(1)).as_extended()\n }\n\n /// The computation uses Pippeneger's algorithm, as described for the\n /// specific case of radix-16 on page 13 of the Ed25519 paper.\n ///\n /// # Piggenger's Algorithm Generalised\n ///\n /// Write the scalar \\\\(a\\\\) in radix-\\\\(w\\\\), where \\\\(w\\\\) is a power of\n /// 2, with coefficients in \\\\([\\frac{-w}{2},\\frac{w}{2})\\\\), i.e.,\n /// $$\n /// a = a\\_0 + a\\_1 w\\^1 + \\cdots + a\\_{x} w\\^{x},\n /// $$\n /// with\n /// $$\n /// \\begin{aligned}\n /// \\frac{-w}{2} \\leq a_i < \\frac{w}{2}\n /// &&\\cdots&&\n /// \\frac{-w}{2} \\leq a\\_{x} \\leq \\frac{w}{2}\n /// \\end{aligned}\n /// $$\n /// and the number of additions, \\\\(x\\\\), is given by\n /// \\\\(x = \\lceil \\frac{256}{w} \\rceil\\\\). Then\n /// $$\n /// a B = a\\_0 B + a\\_1 w\\^1 B + \\cdots + a\\_{x-1} w\\^{x-1} B.\n /// $$\n /// Grouping even and odd coefficients gives\n /// $$\n /// \\begin{aligned}\n /// a B = \\quad a\\_0 w\\^0 B +& a\\_2 w\\^2 B + \\cdots + a\\_{x-2} w\\^{x-2} B \\\\\\\\\n /// + a\\_1 w\\^1 B +& a\\_3 w\\^3 B + \\cdots + a\\_{x-1} w\\^{x-1} B \\\\\\\\\n /// = \\quad(a\\_0 w\\^0 B +& a\\_2 w\\^2 B + \\cdots + a\\_{x-2} w\\^{x-2} B) \\\\\\\\\n /// + w(a\\_1 w\\^0 B +& a\\_3 w\\^2 B + \\cdots + a\\_{x-1} w\\^{x-2} B). \\\\\\\\\n /// \\end{aligned}\n /// $$\n /// For each \\\\(i = 0 \\ldots 31\\\\), we create a lookup table of\n /// $$\n /// [w\\^{2i} B, \\ldots, \\frac{w}{2}\\cdot w\\^{2i} B],\n /// $$\n /// and use it to select \\\\( y \\cdot w\\^{2i} \\cdot B \\\\) in constant time.\n ///\n /// The radix-\\\\(w\\\\) representation requires that the scalar is bounded\n /// by \\\\(2\\^{255}\\\\), which is always the case.\n ///\n /// The above algorithm is trivially generalised to other powers-of-2 radices.\n fn mul_base(&self, scalar: &Scalar) -> $point {\n let a = scalar.as_radix_2w($radix);\n\n let tables = &self.0;\n let mut P = <$point>::identity();\n\n for i in (0..$adds).filter(|x| x % 2 == 1) {\n P = (&P + &tables[i / 2].select(a[i])).as_extended();\n }\n\n P = P.mul_by_pow_2($radix);\n\n for i in (0..$adds).filter(|x| x % 2 == 0) {\n P = (&P + &tables[i / 2].select(a[i])).as_extended();\n }\n\n P\n }\n }\n\n impl<'a, 'b> Mul<&'b Scalar> for &'a $name {\n type Output = $point;\n\n /// Construct an `EdwardsPoint` from a `Scalar` \\\\(a\\\\) by\n /// computing the multiple \\\\(aB\\\\) of this basepoint \\\\(B\\\\).\n fn mul(self, scalar: &'b Scalar) -> $point {\n // delegate to a private function so that its documentation appears in internal docs\n self.mul_base(scalar)\n }\n }\n\n impl<'a, 'b> Mul<&'a $name> for &'b Scalar {\n type Output = $point;\n\n /// Construct an `EdwardsPoint` from a `Scalar` \\\\(a\\\\) by\n /// computing the multiple \\\\(aB\\\\) of this basepoint \\\\(B\\\\).\n fn mul(self, basepoint_table: &'a $name) -> $point {\n basepoint_table * self\n }\n }\n\n impl Debug for $name {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"{:?}([\\n\", stringify!($name))?;\n for i in 0..32 {\n write!(f, \"\\t{:?},\\n\", &self.0[i])?;\n }\n write!(f, \"])\")\n }\n }\n };\n} // End macro_rules! impl_basepoint_table\n\n// The number of additions required is ceil(256/w) where w is the radix representation.\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n impl_basepoint_table! {\n Name = EdwardsBasepointTable,\n LookupTable = LookupTableRadix16,\n Point = EdwardsPoint,\n Radix = 4,\n Additions = 64\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix32,\n LookupTable = LookupTableRadix32,\n Point = EdwardsPoint,\n Radix = 5,\n Additions = 52\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix64,\n LookupTable = LookupTableRadix64,\n Point = EdwardsPoint,\n Radix = 6,\n Additions = 43\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix128,\n LookupTable = LookupTableRadix128,\n Point = EdwardsPoint,\n Radix = 7,\n Additions = 37\n }\n impl_basepoint_table! {\n Name = EdwardsBasepointTableRadix256,\n LookupTable = LookupTableRadix256,\n Point = EdwardsPoint,\n Radix = 8,\n Additions = 33\n }\n\n /// A type-alias for [`EdwardsBasepointTable`] because the latter is\n /// used as a constructor in the [`constants`] module.\n //\n // Same as for `LookupTableRadix16`, we have to define `EdwardsBasepointTable`\n // first, because it's used as a constructor, and then provide a type alias for\n // it.\n pub type EdwardsBasepointTableRadix16 = EdwardsBasepointTable;\n }\n}\n\n#[cfg(feature = \"precomputed-tables\")]\nmacro_rules! impl_basepoint_table_conversions {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl<'a> From<&'a $lhs> for $rhs {\n fn from(table: &'a $lhs) -> $rhs {\n <$rhs>::create(&table.basepoint())\n }\n }\n\n impl<'a> From<&'a $rhs> for $lhs {\n fn from(table: &'a $rhs) -> $lhs {\n <$lhs>::create(&table.basepoint())\n }\n }\n };\n}\n\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n // Conversions from radix 16\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix32\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix64\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix16,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 32\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix64\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix32,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 64\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix64,\n RHS = EdwardsBasepointTableRadix128\n }\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix64,\n RHS = EdwardsBasepointTableRadix256\n }\n\n // Conversions from radix 128\n impl_basepoint_table_conversions! {\n LHS = EdwardsBasepointTableRadix128,\n RHS = EdwardsBasepointTableRadix256\n }\n }\n}\n\nimpl EdwardsPoint {\n /// Multiply by the cofactor: return \\\\(\\[8\\]P\\\\).\n pub fn mul_by_cofactor(&self) -> EdwardsPoint {\n self.mul_by_pow_2(3)\n }\n\n /// Compute \\\\([2\\^k] P \\\\) by successive doublings. Requires \\\\( k > 0 \\\\).\n pub(crate) fn mul_by_pow_2(&self, k: u32) -> EdwardsPoint {\n debug_assert!(k > 0);\n let mut r: CompletedPoint;\n let mut s = self.as_projective();\n for _ in 0..(k - 1) {\n r = s.double();\n s = r.as_projective();\n }\n // Unroll last iteration so we can go directly as_extended()\n s.double().as_extended()\n }\n\n /// Determine if this point is of small order.\n ///\n /// # Return\n ///\n /// * `true` if `self` is in the torsion subgroup \\\\( \\mathcal E\\[8\\] \\\\);\n /// * `false` if `self` is not in the torsion subgroup \\\\( \\mathcal E\\[8\\] \\\\).\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::constants;\n ///\n /// // Generator of the prime-order subgroup\n /// let P = constants::ED25519_BASEPOINT_POINT;\n /// // Generator of the torsion subgroup\n /// let Q = constants::EIGHT_TORSION[1];\n ///\n /// // P has large order\n /// assert_eq!(P.is_small_order(), false);\n ///\n /// // Q has small order\n /// assert_eq!(Q.is_small_order(), true);\n /// ```\n pub fn is_small_order(&self) -> bool {\n self.mul_by_cofactor().is_identity()\n }\n\n /// Determine if this point is “torsion-free”, i.e., is contained in\n /// the prime-order subgroup.\n ///\n /// # Return\n ///\n /// * `true` if `self` has zero torsion component and is in the\n /// prime-order subgroup;\n /// * `false` if `self` has a nonzero torsion component and is not\n /// in the prime-order subgroup.\n ///\n /// # Example\n ///\n /// ```\n /// use curve25519::constants;\n ///\n /// // Generator of the prime-order subgroup\n /// let P = constants::ED25519_BASEPOINT_POINT;\n /// // Generator of the torsion subgroup\n /// let Q = constants::EIGHT_TORSION[1];\n ///\n /// // P is torsion-free\n /// assert_eq!(P.is_torsion_free(), true);\n ///\n /// // P + Q is not torsion-free\n /// assert_eq!((P+Q).is_torsion_free(), false);\n /// ```\n pub fn is_torsion_free(&self) -> bool {\n (self * constants::BASEPOINT_ORDER).is_identity()\n }\n}\n\n// ------------------------------------------------------------------------\n// Debug traits\n// ------------------------------------------------------------------------\n\nimpl Debug for EdwardsPoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"EdwardsPoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?},\\n\\tT: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z, &self.T\n )\n }\n}\n\n// ------------------------------------------------------------------------\n// group traits\n// ------------------------------------------------------------------------\n\n// Use the full trait path to avoid Group::identity overlapping Identity::identity in the\n// rest of the module (e.g. tests).\n#[cfg(feature = \"group\")]\nimpl group::Group for EdwardsPoint {\n type Scalar = Scalar;\n\n fn random(mut rng: impl GroupRngCore) -> Self {\n let mut repr = CompressedEdwardsY([0u8; 32]);\n loop {\n rng.fill_bytes(&mut repr.0);\n if let Some(p) = repr.decompress() {\n if !IsIdentity::is_identity(&p) {\n break p;\n }\n }\n }\n }\n\n fn identity() -> Self {\n Identity::identity()\n }\n\n fn generator() -> Self {\n constants::ED25519_BASEPOINT_POINT\n }\n\n fn is_identity(&self) -> Choice {\n self.ct_eq(&Identity::identity())\n }\n\n fn double(&self) -> Self {\n self.double()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl GroupEncoding for EdwardsPoint {\n type Repr = [u8; 32];\n\n fn from_bytes(bytes: &Self::Repr) -> CtOption {\n let repr = CompressedEdwardsY(*bytes);\n let (is_valid_y_coord, X, Y, Z) = decompress::step_1(&repr);\n CtOption::new(decompress::step_2(&repr, X, Y, Z), is_valid_y_coord)\n }\n\n fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption {\n // Just use the checked API; there are no checks we can skip.\n Self::from_bytes(bytes)\n }\n\n fn to_bytes(&self) -> Self::Repr {\n self.compress().to_bytes()\n }\n}\n\n/// A `SubgroupPoint` represents a point on the Edwards form of Curve25519, that is\n/// guaranteed to be in the prime-order subgroup.\n#[cfg(feature = \"group\")]\n#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)]\npub struct SubgroupPoint(EdwardsPoint);\n\n#[cfg(feature = \"group\")]\nimpl From for EdwardsPoint {\n fn from(p: SubgroupPoint) -> Self {\n p.0\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Neg for SubgroupPoint {\n type Output = Self;\n\n fn neg(self) -> Self::Output {\n SubgroupPoint(-self.0)\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Add<&SubgroupPoint> for &SubgroupPoint {\n type Output = SubgroupPoint;\n fn add(self, other: &SubgroupPoint) -> SubgroupPoint {\n SubgroupPoint(self.0 + other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_variants!(\n LHS = SubgroupPoint,\n RHS = SubgroupPoint,\n Output = SubgroupPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl Add<&SubgroupPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn add(self, other: &SubgroupPoint) -> EdwardsPoint {\n self + other.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_variants!(\n LHS = EdwardsPoint,\n RHS = SubgroupPoint,\n Output = EdwardsPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl AddAssign<&SubgroupPoint> for SubgroupPoint {\n fn add_assign(&mut self, rhs: &SubgroupPoint) {\n self.0 += rhs.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl AddAssign<&SubgroupPoint> for EdwardsPoint {\n fn add_assign(&mut self, rhs: &SubgroupPoint) {\n *self += rhs.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_add_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Sub<&SubgroupPoint> for &SubgroupPoint {\n type Output = SubgroupPoint;\n fn sub(self, other: &SubgroupPoint) -> SubgroupPoint {\n SubgroupPoint(self.0 - other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_variants!(\n LHS = SubgroupPoint,\n RHS = SubgroupPoint,\n Output = SubgroupPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl Sub<&SubgroupPoint> for &EdwardsPoint {\n type Output = EdwardsPoint;\n fn sub(self, other: &SubgroupPoint) -> EdwardsPoint {\n self - other.0\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_variants!(\n LHS = EdwardsPoint,\n RHS = SubgroupPoint,\n Output = EdwardsPoint\n);\n\n#[cfg(feature = \"group\")]\nimpl SubAssign<&SubgroupPoint> for SubgroupPoint {\n fn sub_assign(&mut self, rhs: &SubgroupPoint) {\n self.0 -= rhs.0;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl SubAssign<&SubgroupPoint> for EdwardsPoint {\n fn sub_assign(&mut self, rhs: &SubgroupPoint) {\n *self -= rhs.0;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_sub_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Sum for SubgroupPoint\nwhere\n T: Borrow,\n{\n fn sum(iter: I) -> Self\n where\n I: Iterator,\n {\n use group::Group;\n iter.fold(SubgroupPoint::identity(), |acc, item| acc + item.borrow())\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl Mul<&Scalar> for &SubgroupPoint {\n type Output = SubgroupPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, scalar: &Scalar) -> SubgroupPoint {\n SubgroupPoint(self.0 * scalar)\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_variants!(LHS = Scalar, RHS = SubgroupPoint, Output = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl Mul<&SubgroupPoint> for &Scalar {\n type Output = SubgroupPoint;\n\n /// Scalar multiplication: compute `scalar * self`.\n ///\n /// For scalar multiplication of a basepoint,\n /// `EdwardsBasepointTable` is approximately 4x faster.\n fn mul(self, point: &SubgroupPoint) -> SubgroupPoint {\n point * self\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_variants!(LHS = SubgroupPoint, RHS = Scalar, Output = SubgroupPoint);\n\n#[cfg(feature = \"group\")]\nimpl MulAssign<&Scalar> for SubgroupPoint {\n fn mul_assign(&mut self, scalar: &Scalar) {\n self.0 *= scalar;\n }\n}\n\n#[cfg(feature = \"group\")]\ndefine_mul_assign_variants!(LHS = SubgroupPoint, RHS = Scalar);\n\n#[cfg(feature = \"group\")]\nimpl ConstantTimeEq for SubgroupPoint {\n fn ct_eq(&self, other: &SubgroupPoint) -> Choice {\n self.0.ct_eq(&other.0)\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl ConditionallySelectable for SubgroupPoint {\n fn conditional_select(a: &SubgroupPoint, b: &SubgroupPoint, choice: Choice) -> SubgroupPoint {\n SubgroupPoint(EdwardsPoint::conditional_select(&a.0, &b.0, choice))\n }\n}\n\n#[cfg(all(feature = \"group\", feature = \"zeroize\"))]\nimpl Zeroize for SubgroupPoint {\n fn zeroize(&mut self) {\n self.0.zeroize();\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl group::Group for SubgroupPoint {\n type Scalar = Scalar;\n\n fn random(mut rng: impl GroupRngCore) -> Self {\n use group::ff::Field;\n\n // This will almost never loop, but `Group::random` is documented as returning a\n // non-identity element.\n let s = loop {\n let s: Scalar = Field::random(&mut rng);\n if !s.is_zero_vartime() {\n break s;\n }\n };\n\n // This gives an element of the prime-order subgroup.\n Self::generator() * s\n }\n\n fn identity() -> Self {\n SubgroupPoint(Identity::identity())\n }\n\n fn generator() -> Self {\n SubgroupPoint(EdwardsPoint::generator())\n }\n\n fn is_identity(&self) -> Choice {\n self.0.ct_eq(&Identity::identity())\n }\n\n fn double(&self) -> Self {\n SubgroupPoint(self.0.double())\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl GroupEncoding for SubgroupPoint {\n type Repr = ::Repr;\n\n fn from_bytes(bytes: &Self::Repr) -> CtOption {\n EdwardsPoint::from_bytes(bytes).and_then(|p| p.into_subgroup())\n }\n\n fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption {\n EdwardsPoint::from_bytes_unchecked(bytes).and_then(|p| p.into_subgroup())\n }\n\n fn to_bytes(&self) -> Self::Repr {\n self.0.compress().to_bytes()\n }\n}\n\n#[cfg(feature = \"group\")]\nimpl PrimeGroup for SubgroupPoint {}\n\n#[cfg(feature = \"group\")]\nimpl CofactorGroup for EdwardsPoint {\n type Subgroup = SubgroupPoint;\n\n fn clear_cofactor(&self) -> Self::Subgroup {\n SubgroupPoint(self.mul_by_cofactor())\n }\n\n fn into_subgroup(self) -> CtOption {\n CtOption::new(SubgroupPoint(self), CofactorGroup::is_torsion_free(&self))\n }\n\n fn is_torsion_free(&self) -> Choice {\n (self * constants::BASEPOINT_ORDER).ct_eq(&Self::identity())\n }\n}\n\n// ------------------------------------------------------------------------\n// Tests\n// ------------------------------------------------------------------------\n\n#[cfg(test)]\nmod test {\n use super::*;\n use rand::Rng;\n\n #[cfg(feature = \"precomputed-tables\")]\n use crate::constants::ED25519_BASEPOINT_TABLE;\n\n /// X coordinate of the basepoint.\n /// = 15112221349535400772501151409588531511454012693041857206046113283949847762202\n static BASE_X_COORD_BYTES: [u8; 32] = [\n 0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c,\n 0x69, 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36,\n 0x69, 0x21,\n ];\n\n /// Compressed Edwards Y form of 2*basepoint.\n static BASE2_CMPRSSD: CompressedEdwardsY = CompressedEdwardsY([\n 0xc9, 0xa3, 0xf8, 0x6a, 0xae, 0x46, 0x5f, 0xe, 0x56, 0x51, 0x38, 0x64, 0x51, 0x0f, 0x39,\n 0x97, 0x56, 0x1f, 0xa2, 0xc9, 0xe8, 0x5e, 0xa2, 0x1d, 0xc2, 0x29, 0x23, 0x09, 0xf3, 0xcd,\n 0x60, 0x22,\n ]);\n\n /// Compressed Edwards Y form of 16*basepoint.\n static BASE16_CMPRSSD: CompressedEdwardsY = CompressedEdwardsY([\n 0xeb, 0x27, 0x67, 0xc1, 0x37, 0xab, 0x7a, 0xd8, 0x27, 0x9c, 0x07, 0x8e, 0xff, 0x11, 0x6a,\n 0xb0, 0x78, 0x6e, 0xad, 0x3a, 0x2e, 0x0f, 0x98, 0x9f, 0x72, 0xc3, 0x7f, 0x82, 0xf2, 0x96,\n 0x96, 0x70,\n ]);\n\n /// 4493907448824000747700850167940867464579944529806937181821189941592931634714\n pub static A_SCALAR: Scalar = Scalar {\n bytes: [\n 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8,\n 0x26, 0x4d, 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, 0x58, 0x9e, 0x7b, 0x7f,\n 0x23, 0x76, 0xef, 0x09,\n ],\n };\n\n /// 2506056684125797857694181776241676200180934651973138769173342316833279714961\n pub static B_SCALAR: Scalar = Scalar {\n bytes: [\n 0x91, 0x26, 0x7a, 0xcf, 0x25, 0xc2, 0x09, 0x1b, 0xa2, 0x17, 0x74, 0x7b, 0x66, 0xf0,\n 0xb3, 0x2e, 0x9d, 0xf2, 0xa5, 0x67, 0x41, 0xcf, 0xda, 0xc4, 0x56, 0xa7, 0xd4, 0xaa,\n 0xb8, 0x60, 0x8a, 0x05,\n ],\n };\n\n /// A_SCALAR * basepoint, computed with ed25519.py\n pub static A_TIMES_BASEPOINT: CompressedEdwardsY = CompressedEdwardsY([\n 0xea, 0x27, 0xe2, 0x60, 0x53, 0xdf, 0x1b, 0x59, 0x56, 0xf1, 0x4d, 0x5d, 0xec, 0x3c, 0x34,\n 0xc3, 0x84, 0xa2, 0x69, 0xb7, 0x4c, 0xc3, 0x80, 0x3e, 0xa8, 0xe2, 0xe7, 0xc9, 0x42, 0x5e,\n 0x40, 0xa5,\n ]);\n\n /// A_SCALAR * (A_TIMES_BASEPOINT) + B_SCALAR * BASEPOINT\n /// computed with ed25519.py\n static DOUBLE_SCALAR_MULT_RESULT: CompressedEdwardsY = CompressedEdwardsY([\n 0x7d, 0xfd, 0x6c, 0x45, 0xaf, 0x6d, 0x6e, 0x0e, 0xba, 0x20, 0x37, 0x1a, 0x23, 0x64, 0x59,\n 0xc4, 0xc0, 0x46, 0x83, 0x43, 0xde, 0x70, 0x4b, 0x85, 0x09, 0x6f, 0xfe, 0x35, 0x4f, 0x13,\n 0x2b, 0x42,\n ]);\n\n /// Test round-trip decompression for the basepoint.\n #[test]\n fn basepoint_decompression_compression() {\n let base_X = FieldElement::from_bytes(&BASE_X_COORD_BYTES);\n let bp = constants::ED25519_BASEPOINT_COMPRESSED\n .decompress()\n .expect(\"basepoint should decompress\");\n assert!(bp.is_valid());\n // Check that decompression actually gives the correct X coordinate\n assert_eq!(base_X, bp.X);\n assert_eq!(bp.compress(), constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test sign handling in decompression\n #[test]\n fn decompression_sign_handling() {\n // Manually set the high bit of the last byte to flip the sign\n let mut minus_basepoint_bytes = *constants::ED25519_BASEPOINT_COMPRESSED.as_bytes();\n minus_basepoint_bytes[31] |= 1 << 7;\n let minus_basepoint = CompressedEdwardsY(minus_basepoint_bytes)\n .decompress()\n .expect(\"minus basepoint should decompress\");\n // Test projective coordinates exactly since we know they should\n // only differ by a flipped sign.\n assert_eq!(minus_basepoint.X, -(&constants::ED25519_BASEPOINT_POINT.X));\n assert_eq!(minus_basepoint.Y, constants::ED25519_BASEPOINT_POINT.Y);\n assert_eq!(minus_basepoint.Z, constants::ED25519_BASEPOINT_POINT.Z);\n assert_eq!(minus_basepoint.T, -(&constants::ED25519_BASEPOINT_POINT.T));\n }\n\n /// Test that computing 1*basepoint gives the correct basepoint.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_mult_one_vs_basepoint() {\n let bp = ED25519_BASEPOINT_TABLE * &Scalar::ONE;\n let compressed = bp.compress();\n assert_eq!(compressed, constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test that `EdwardsBasepointTable::basepoint()` gives the correct basepoint.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_table_basepoint_function_correct() {\n let bp = ED25519_BASEPOINT_TABLE.basepoint();\n assert_eq!(bp.compress(), constants::ED25519_BASEPOINT_COMPRESSED);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using basepoint + basepoint versus the 2*basepoint constant.\n #[test]\n fn basepoint_plus_basepoint_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_added = bp + bp;\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using the basepoint, basepoint2 constants\n #[test]\n fn basepoint_plus_basepoint_projective_niels_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_added = (&bp + &bp.as_projective_niels()).as_extended();\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Test `impl Add for EdwardsPoint`\n /// using the basepoint, basepoint2 constants\n #[test]\n fn basepoint_plus_basepoint_affine_niels_vs_basepoint2() {\n let bp = constants::ED25519_BASEPOINT_POINT;\n let bp_affine_niels = bp.as_affine_niels();\n let bp_added = (&bp + &bp_affine_niels).as_extended();\n assert_eq!(bp_added.compress(), BASE2_CMPRSSD);\n }\n\n /// Check that equality of `EdwardsPoints` handles projective\n /// coordinates correctly.\n #[test]\n fn extended_point_equality_handles_scaling() {\n let mut two_bytes = [0u8; 32];\n two_bytes[0] = 2;\n let id1 = EdwardsPoint::identity();\n let id2 = EdwardsPoint {\n X: FieldElement::ZERO,\n Y: FieldElement::from_bytes(&two_bytes),\n Z: FieldElement::from_bytes(&two_bytes),\n T: FieldElement::ZERO,\n };\n assert!(bool::from(id1.ct_eq(&id2)));\n }\n\n /// Sanity check for conversion to precomputed points\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn to_affine_niels_clears_denominators() {\n // construct a point as aB so it has denominators (ie. Z != 1)\n let aB = ED25519_BASEPOINT_TABLE * &A_SCALAR;\n let aB_affine_niels = aB.as_affine_niels();\n let also_aB = (&EdwardsPoint::identity() + &aB_affine_niels).as_extended();\n assert_eq!(aB.compress(), also_aB.compress());\n }\n\n /// Test mul_base versus a known scalar multiple from ed25519.py\n #[test]\n fn basepoint_mult_vs_ed25519py() {\n let aB = EdwardsPoint::mul_base(&A_SCALAR);\n assert_eq!(aB.compress(), A_TIMES_BASEPOINT);\n }\n\n /// Test that multiplication by the basepoint order kills the basepoint\n #[test]\n fn basepoint_mult_by_basepoint_order() {\n let should_be_id = EdwardsPoint::mul_base(&constants::BASEPOINT_ORDER);\n assert!(should_be_id.is_identity());\n }\n\n /// Test precomputed basepoint mult\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn test_precomputed_basepoint_mult() {\n let aB_1 = ED25519_BASEPOINT_TABLE * &A_SCALAR;\n let aB_2 = constants::ED25519_BASEPOINT_POINT * A_SCALAR;\n assert_eq!(aB_1.compress(), aB_2.compress());\n }\n\n /// Test scalar_mul versus a known scalar multiple from ed25519.py\n #[test]\n fn scalar_mul_vs_ed25519py() {\n let aB = constants::ED25519_BASEPOINT_POINT * A_SCALAR;\n assert_eq!(aB.compress(), A_TIMES_BASEPOINT);\n }\n\n /// Test basepoint.double() versus the 2*basepoint constant.\n #[test]\n fn basepoint_double_vs_basepoint2() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT.double().compress(),\n BASE2_CMPRSSD\n );\n }\n\n /// Test that computing 2*basepoint is the same as basepoint.double()\n #[test]\n fn basepoint_mult_two_vs_basepoint2() {\n let two = Scalar::from(2u64);\n let bp2 = EdwardsPoint::mul_base(&two);\n assert_eq!(bp2.compress(), BASE2_CMPRSSD);\n }\n\n /// Test that all the basepoint table types compute the same results.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_tables() {\n let P = &constants::ED25519_BASEPOINT_POINT;\n let a = A_SCALAR;\n\n let table_radix16 = EdwardsBasepointTableRadix16::create(P);\n let table_radix32 = EdwardsBasepointTableRadix32::create(P);\n let table_radix64 = EdwardsBasepointTableRadix64::create(P);\n let table_radix128 = EdwardsBasepointTableRadix128::create(P);\n let table_radix256 = EdwardsBasepointTableRadix256::create(P);\n\n let aP = (ED25519_BASEPOINT_TABLE * &a).compress();\n let aP16 = (&table_radix16 * &a).compress();\n let aP32 = (&table_radix32 * &a).compress();\n let aP64 = (&table_radix64 * &a).compress();\n let aP128 = (&table_radix128 * &a).compress();\n let aP256 = (&table_radix256 * &a).compress();\n\n assert_eq!(aP, aP16);\n assert_eq!(aP16, aP32);\n assert_eq!(aP32, aP64);\n assert_eq!(aP64, aP128);\n assert_eq!(aP128, aP256);\n }\n\n /// Check unreduced scalar multiplication by the basepoint tables is the same no matter what\n /// radix the table is.\n #[cfg(feature = \"precomputed-tables\")]\n #[test]\n fn basepoint_tables_unreduced_scalar() {\n let P = &constants::ED25519_BASEPOINT_POINT;\n let a = crate::scalar::test::LARGEST_UNREDUCED_SCALAR;\n\n let table_radix16 = EdwardsBasepointTableRadix16::create(P);\n let table_radix32 = EdwardsBasepointTableRadix32::create(P);\n let table_radix64 = EdwardsBasepointTableRadix64::create(P);\n let table_radix128 = EdwardsBasepointTableRadix128::create(P);\n let table_radix256 = EdwardsBasepointTableRadix256::create(P);\n\n let aP = (ED25519_BASEPOINT_TABLE * &a).compress();\n let aP16 = (&table_radix16 * &a).compress();\n let aP32 = (&table_radix32 * &a).compress();\n let aP64 = (&table_radix64 * &a).compress();\n let aP128 = (&table_radix128 * &a).compress();\n let aP256 = (&table_radix256 * &a).compress();\n\n assert_eq!(aP, aP16);\n assert_eq!(aP16, aP32);\n assert_eq!(aP32, aP64);\n assert_eq!(aP64, aP128);\n assert_eq!(aP128, aP256);\n }\n\n /// Check that converting to projective and then back to extended round-trips.\n #[test]\n fn basepoint_projective_extended_round_trip() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT\n .as_projective()\n .as_extended()\n .compress(),\n constants::ED25519_BASEPOINT_COMPRESSED\n );\n }\n\n /// Test computing 16*basepoint vs mul_by_pow_2(4)\n #[test]\n fn basepoint16_vs_mul_by_pow_2_4() {\n let bp16 = constants::ED25519_BASEPOINT_POINT.mul_by_pow_2(4);\n assert_eq!(bp16.compress(), BASE16_CMPRSSD);\n }\n\n /// Check that mul_base_clamped and mul_clamped agree\n #[test]\n fn mul_base_clamped() {\n let mut csprng = rand::thread_rng();\n\n // Make a random curve point in the curve. Give it torsion to make things interesting.\n #[cfg(feature = \"precomputed-tables\")]\n let random_point = {\n let mut b = [0u8; 32];\n csprng.fill(&mut b);\n EdwardsPoint::mul_base_clamped(b) + constants::EIGHT_TORSION[1]\n };\n // Make a basepoint table from the random point. We'll use this with mul_base_clamped\n #[cfg(feature = \"precomputed-tables\")]\n let random_table = EdwardsBasepointTableRadix256::create(&random_point);\n\n // Now test scalar mult. agreement on the default basepoint as well as random_point\n\n // Test that mul_base_clamped and mul_clamped agree on a large integer. Even after\n // clamping, this integer is not reduced mod l.\n let a_bytes = [0xff; 32];\n assert_eq!(\n EdwardsPoint::mul_base_clamped(a_bytes),\n constants::ED25519_BASEPOINT_POINT.mul_clamped(a_bytes)\n );\n #[cfg(feature = \"precomputed-tables\")]\n assert_eq!(\n random_table.mul_base_clamped(a_bytes),\n random_point.mul_clamped(a_bytes)\n );\n\n // Test agreement on random integers\n for _ in 0..100 {\n // This will be reduced mod l with probability l / 2^256 ≈ 6.25%\n let mut a_bytes = [0u8; 32];\n csprng.fill(&mut a_bytes);\n\n assert_eq!(\n EdwardsPoint::mul_base_clamped(a_bytes),\n constants::ED25519_BASEPOINT_POINT.mul_clamped(a_bytes)\n );\n #[cfg(feature = \"precomputed-tables\")]\n assert_eq!(\n random_table.mul_base_clamped(a_bytes),\n random_point.mul_clamped(a_bytes)\n );\n }\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn impl_sum() {\n // Test that sum works for non-empty iterators\n let BASE = constants::ED25519_BASEPOINT_POINT;\n\n let s1 = Scalar::from(999u64);\n let P1 = BASE * s1;\n\n let s2 = Scalar::from(333u64);\n let P2 = BASE * s2;\n\n let arr = [P1, P2];\n let sum: EdwardsPoint = arr.iter().sum();\n\n assert_eq!(sum, P1 + P2);\n\n // Test that sum works for the empty iterator\n let empty_array: [EdwardsPoint; 0] = [];\n let sum: EdwardsPoint = empty_array.iter().sum();\n\n assert_eq!(sum, EdwardsPoint::identity());\n\n // Test that sum works on owning iterators\n let s = Scalar::from(2u64);\n let mapped = arr.iter().map(|x| x * s);\n let sum: EdwardsPoint = mapped.sum();\n\n assert_eq!(sum, P1 * s + P2 * s);\n }\n\n /// Test that the conditional assignment trait works for AffineNielsPoints.\n #[test]\n fn conditional_assign_for_affine_niels_point() {\n let id = AffineNielsPoint::identity();\n let mut p1 = AffineNielsPoint::identity();\n let bp = constants::ED25519_BASEPOINT_POINT.as_affine_niels();\n\n p1.conditional_assign(&bp, Choice::from(0));\n assert_eq!(p1, id);\n p1.conditional_assign(&bp, Choice::from(1));\n assert_eq!(p1, bp);\n }\n\n #[test]\n fn is_small_order() {\n // The basepoint has large prime order\n assert!(!constants::ED25519_BASEPOINT_POINT.is_small_order());\n // constants::EIGHT_TORSION has all points of small order.\n for torsion_point in &constants::EIGHT_TORSION {\n assert!(torsion_point.is_small_order());\n }\n }\n\n #[test]\n fn compressed_identity() {\n assert_eq!(\n EdwardsPoint::identity().compress(),\n CompressedEdwardsY::identity()\n );\n\n #[cfg(feature = \"alloc\")]\n {\n let compressed = EdwardsPoint::compress_batch(&[EdwardsPoint::identity()]);\n assert_eq!(&compressed, &[CompressedEdwardsY::identity()]);\n }\n }\n\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n #[test]\n fn compress_batch() {\n let mut rng = rand::thread_rng();\n\n // TODO(tarcieri): proptests?\n // Make some points deterministically then randomly\n let mut points = (1u64..16)\n .map(|n| constants::ED25519_BASEPOINT_POINT * Scalar::from(n))\n .collect::>();\n points.extend(core::iter::repeat_with(|| EdwardsPoint::random(&mut rng)).take(100));\n let compressed = EdwardsPoint::compress_batch(&points);\n\n // Check that the batch-compressed points match the individually compressed ones\n for (point, compressed) in points.iter().zip(&compressed) {\n assert_eq!(&point.compress(), compressed);\n }\n }\n\n #[test]\n fn is_identity() {\n assert!(EdwardsPoint::identity().is_identity());\n assert!(!constants::ED25519_BASEPOINT_POINT.is_identity());\n }\n\n /// Rust's debug builds have overflow and underflow trapping,\n /// and enable `debug_assert!()`. This performs many scalar\n /// multiplications to attempt to trigger possible overflows etc.\n ///\n /// For instance, the `u64` `Mul` implementation for\n /// `FieldElements` requires the input `Limb`s to be bounded by\n /// 2^54, but we cannot enforce this dynamically at runtime, or\n /// statically at compile time (until Rust gets type-level\n /// integers, at which point we can encode \"bits of headroom\" into\n /// the type system and prove correctness).\n #[test]\n fn monte_carlo_overflow_underflow_debug_assert_test() {\n let mut P = constants::ED25519_BASEPOINT_POINT;\n // N.B. each scalar_mul does 1407 field mults, 1024 field squarings,\n // so this does ~ 1M of each operation.\n for _ in 0..1_000 {\n P *= &A_SCALAR;\n }\n }\n\n #[test]\n fn scalarmult_extended_point_works_both_ways() {\n let G: EdwardsPoint = constants::ED25519_BASEPOINT_POINT;\n let s: Scalar = A_SCALAR;\n\n let P1 = G * s;\n let P2 = s * G;\n\n assert!(P1.compress().to_bytes() == P2.compress().to_bytes());\n }\n\n // A single iteration of a consistency check for MSM.\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_iter(n: usize) {\n let mut rng = rand::thread_rng();\n\n // Construct random coefficients x0, ..., x_{n-1},\n // followed by some extra hardcoded ones.\n let xs = (0..n).map(|_| Scalar::random(&mut rng)).collect::>();\n let check = xs.iter().map(|xi| xi * xi).sum::();\n\n // Construct points G_i = x_i * B\n let Gs = xs.iter().map(EdwardsPoint::mul_base).collect::>();\n\n // Compute H1 = (consttime)\n let H1 = EdwardsPoint::multiscalar_mul(&xs, &Gs);\n // Compute H2 = (vartime)\n let H2 = EdwardsPoint::vartime_multiscalar_mul(&xs, &Gs);\n // Compute H3 = = sum(xi^2) * B\n let H3 = EdwardsPoint::mul_base(&check);\n\n assert_eq!(H1, H3);\n assert_eq!(H2, H3);\n }\n\n // Use different multiscalar sizes to hit different internal\n // parameters.\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_100() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(100);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_250() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(250);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_500() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(500);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn multiscalar_consistency_n_1000() {\n let iters = 50;\n for _ in 0..iters {\n multiscalar_consistency_iter(1000);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn batch_to_montgomery() {\n let mut rng = rand::thread_rng();\n\n let scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let points = scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n\n let single_monts = points\n .iter()\n .map(EdwardsPoint::to_montgomery)\n .collect::>();\n\n for i in [0, 1, 2, 3, 10, 50, 128] {\n let invs = EdwardsPoint::to_montgomery_batch(&points[..i]);\n assert_eq!(&invs, &single_monts[..i]);\n }\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"rand_core\"))]\n fn vartime_precomputed_vs_nonprecomputed_multiscalar() {\n let mut rng = rand::thread_rng();\n\n let static_scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let dynamic_scalars = (0..128)\n .map(|_| Scalar::random(&mut rng))\n .collect::>();\n\n let check_scalar: Scalar = static_scalars\n .iter()\n .chain(dynamic_scalars.iter())\n .map(|s| s * s)\n .sum();\n\n let static_points = static_scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n let dynamic_points = dynamic_scalars\n .iter()\n .map(EdwardsPoint::mul_base)\n .collect::>();\n\n let precomputation = VartimeEdwardsPrecomputation::new(static_points.iter());\n\n assert_eq!(precomputation.len(), 128);\n assert!(!precomputation.is_empty());\n\n let P = precomputation.vartime_mixed_multiscalar_mul(\n &static_scalars,\n &dynamic_scalars,\n &dynamic_points,\n );\n\n use crate::traits::VartimeMultiscalarMul;\n let Q = EdwardsPoint::vartime_multiscalar_mul(\n static_scalars.iter().chain(dynamic_scalars.iter()),\n static_points.iter().chain(dynamic_points.iter()),\n );\n\n let R = EdwardsPoint::mul_base(&check_scalar);\n\n assert_eq!(P.compress(), R.compress());\n assert_eq!(Q.compress(), R.compress());\n }\n\n mod vartime {\n use super::super::*;\n use super::{A_SCALAR, A_TIMES_BASEPOINT, B_SCALAR, DOUBLE_SCALAR_MULT_RESULT};\n\n /// Test double_scalar_mul_vartime vs ed25519.py\n #[test]\n fn double_scalar_mul_basepoint_vs_ed25519py() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result =\n EdwardsPoint::vartime_double_scalar_mul_basepoint(&A_SCALAR, &A, &B_SCALAR);\n assert_eq!(result.compress(), DOUBLE_SCALAR_MULT_RESULT);\n }\n\n #[test]\n fn triple_scalar_mul_basepoint_accepts_full_width_scalars() {\n let mut a1_bytes = [0u8; 32];\n a1_bytes[0] = 7;\n a1_bytes[16] = 1;\n let a1 = Scalar::from_canonical_bytes(a1_bytes).unwrap();\n\n let mut a2_bytes = [0u8; 32];\n a2_bytes[0] = 11;\n a2_bytes[24] = 1;\n let a2 = Scalar::from_canonical_bytes(a2_bytes).unwrap();\n\n let b = B_SCALAR;\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(17u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(19u64);\n\n let result = EdwardsPoint::vartime_triple_scalar_mul_basepoint(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1) + (a2 * A2) + EdwardsPoint::mul_base(&b);\n\n assert_eq!(result, expected);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn multiscalar_mul_vs_ed25519py() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result = EdwardsPoint::vartime_multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n assert_eq!(result.compress(), DOUBLE_SCALAR_MULT_RESULT);\n }\n\n #[test]\n #[cfg(feature = \"alloc\")]\n fn multiscalar_mul_vartime_vs_consttime() {\n let A = A_TIMES_BASEPOINT\n .decompress()\n .expect(\"test point should decompress\");\n let result_vartime = EdwardsPoint::vartime_multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n let result_consttime = EdwardsPoint::multiscalar_mul(\n &[A_SCALAR, B_SCALAR],\n &[A, constants::ED25519_BASEPOINT_POINT],\n );\n\n assert_eq!(result_vartime.compress(), result_consttime.compress());\n }\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_basepoint_roundtrip() {\n use bincode;\n\n let encoded = bincode::serialize(&constants::ED25519_BASEPOINT_POINT).unwrap();\n let enc_compressed = bincode::serialize(&constants::ED25519_BASEPOINT_COMPRESSED).unwrap();\n assert_eq!(encoded, enc_compressed);\n\n // Check that the encoding is 32 bytes exactly\n assert_eq!(encoded.len(), 32);\n\n let dec_uncompressed: EdwardsPoint = bincode::deserialize(&encoded).unwrap();\n let dec_compressed: CompressedEdwardsY = bincode::deserialize(&encoded).unwrap();\n\n assert_eq!(dec_uncompressed, constants::ED25519_BASEPOINT_POINT);\n assert_eq!(dec_compressed, constants::ED25519_BASEPOINT_COMPRESSED);\n\n // Check that the encoding itself matches the usual one\n let raw_bytes = constants::ED25519_BASEPOINT_COMPRESSED.as_bytes();\n let bp: EdwardsPoint = bincode::deserialize(raw_bytes).unwrap();\n assert_eq!(bp, constants::ED25519_BASEPOINT_POINT);\n }\n\n // Hash-to-curve test vectors from\n // https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.2\n // These are of the form (input_msg, output_x, output_y)\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n const RFC_ENCODE_TO_CURVE_KAT: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"1ff2b70ecf862799e11b7ae744e3489aa058ce805dd323a936375a84695e76da\",\n \"222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab2d687f0f9b\",\n ),\n (\n b\"abc\",\n \"5f13cc69c891d86927eb37bd4afc6672360007c63f68a33ab423a3aa040fd2a8\",\n \"67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1f5c827fa42\",\n ),\n (\n b\"abcdef0123456789\",\n \"1dd2fefce934ecfd7aae6ec998de088d7dd03316aa1847198aecf699ba6613f1\",\n \"2f8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a0a8e1a86fb\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"35fbdc5143e8a97afd3096f2b843e07df72e15bfca2eaf6879bf97c5d3362f73\",\n \"2af6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91f57a603450\",\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"6e5e1f37e99345887fc12111575fc1c3e36df4b289b8759d23af14d774b66bff\",\n \"2c90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a4245891a37\"\n )\n ];\n\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn hex_str_to_fe(hex_str: &str) -> FieldElement {\n let mut bytes = hex::decode(hex_str).expect(\"RFC test vector hex must decode\");\n bytes.reverse();\n let bytes: [u8; 32] = bytes\n .try_into()\n .expect(\"RFC test vector field element must be 32 bytes\");\n FieldElement::from_bytes(&bytes)\n }\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn elligator_encode_to_curve_test_vectors() {\n let dst = b\"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_\";\n for (index, vector) in RFC_ENCODE_TO_CURVE_KAT.iter().enumerate() {\n let input = vector.0;\n\n let expected_output = {\n let x = hex_str_to_fe(vector.1);\n let y = hex_str_to_fe(vector.2);\n AffinePoint { x, y }.to_edwards()\n };\n\n let computed = EdwardsPoint::encode_to_curve::(&[input], &[dst]);\n assert_eq!(computed, expected_output, \"Failed in test {}\", index);\n }\n }\n\n // Hash-to-curve test vectors from\n // https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.1\n // These are of the form (input_msg, output_x, output_y)\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n const RFC_HASH_TO_CURVE_KAT: &[(&[u8], &str, &str)] = &[\n (\n b\"\",\n \"3c3da6925a3c3c268448dcabb47ccde5439559d9599646a8260e47b1e4822fc6\",\n \"09a6c8561a0b22bef63124c588ce4c62ea83a3c899763af26d795302e115dc21\",\n ),\n\n (\n b\"abc\",\n \"608040b42285cc0d72cbb3985c6b04c935370c7361f4b7fbdb1ae7f8c1a8ecad\",\n \"1a8395b88338f22e435bbd301183e7f20a5f9de643f11882fb237f88268a5531\",\n ),\n\n (\n b\"abcdef0123456789\",\n \"6d7fabf47a2dc03fe7d47f7dddd21082c5fb8f86743cd020f3fb147d57161472\",\n \"53060a3d140e7fbcda641ed3cf42c88a75411e648a1add71217f70ea8ec561a6\",\n ),\n (\n b\"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\\\n qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\",\n \"5fb0b92acedd16f3bcb0ef83f5c7b7a9466b5f1e0d8d217421878ea3686f8524\",\n \"2eca15e355fcfa39d2982f67ddb0eea138e2994f5956ed37b7f72eea5e89d2f7\",\n ),\n (\n b\"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\\n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\",\n \"0efcfde5898a839b00997fbe40d2ebe950bc81181afbd5cd6b9618aa336c1e8c\",\n \"6dc2fc04f266c5c27f236a80b14f92ccd051ef1ff027f26a07f8c0f327d8f995\"\n )\n ];\n\n #[test]\n #[cfg(all(feature = \"alloc\", feature = \"digest\"))]\n fn elligator_hash_to_curve_test_vectors() {\n let dst = b\"QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_\";\n for (index, vector) in RFC_HASH_TO_CURVE_KAT.iter().enumerate() {\n let input = vector.0;\n\n let expected_output = {\n let x = hex_str_to_fe(vector.1);\n let y = hex_str_to_fe(vector.2);\n\n AffinePoint { x, y }.to_edwards()\n };\n\n let computed = EdwardsPoint::hash_to_curve::(&[input], &[dst]);\n\n assert_eq!(expected_output, computed, \"Failed in test {}\", index);\n }\n }\n}\n"},{"id":12,"name":{"Local":"curve25519/solana-ed25519/src/macros.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis agora lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Internal macros.\n\n/// Define borrow and non-borrow variants of `Add`.\nmacro_rules! define_add_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Add<&'b $rhs> for $lhs {\n type Output = $out;\n fn add(self, rhs: &'b $rhs) -> $out {\n &self + rhs\n }\n }\n\n impl<'a> Add<$rhs> for &'a $lhs {\n type Output = $out;\n fn add(self, rhs: $rhs) -> $out {\n self + &rhs\n }\n }\n\n impl Add<$rhs> for $lhs {\n type Output = $out;\n fn add(self, rhs: $rhs) -> $out {\n &self + &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `AddAssign`.\nmacro_rules! define_add_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl AddAssign<$rhs> for $lhs {\n fn add_assign(&mut self, rhs: $rhs) {\n *self += &rhs;\n }\n }\n };\n}\n\n/// Define borrow and non-borrow variants of `Sub`.\nmacro_rules! define_sub_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Sub<&'b $rhs> for $lhs {\n type Output = $out;\n fn sub(self, rhs: &'b $rhs) -> $out {\n &self - rhs\n }\n }\n\n impl<'a> Sub<$rhs> for &'a $lhs {\n type Output = $out;\n fn sub(self, rhs: $rhs) -> $out {\n self - &rhs\n }\n }\n\n impl Sub<$rhs> for $lhs {\n type Output = $out;\n fn sub(self, rhs: $rhs) -> $out {\n &self - &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `SubAssign`.\nmacro_rules! define_sub_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl SubAssign<$rhs> for $lhs {\n fn sub_assign(&mut self, rhs: $rhs) {\n *self -= &rhs;\n }\n }\n };\n}\n\n/// Define borrow and non-borrow variants of `Mul`.\nmacro_rules! define_mul_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {\n impl<'b> Mul<&'b $rhs> for $lhs {\n type Output = $out;\n fn mul(self, rhs: &'b $rhs) -> $out {\n &self * rhs\n }\n }\n\n impl<'a> Mul<$rhs> for &'a $lhs {\n type Output = $out;\n fn mul(self, rhs: $rhs) -> $out {\n self * &rhs\n }\n }\n\n impl Mul<$rhs> for $lhs {\n type Output = $out;\n fn mul(self, rhs: $rhs) -> $out {\n &self * &rhs\n }\n }\n };\n}\n\n/// Define non-borrow variants of `MulAssign`.\nmacro_rules! define_mul_assign_variants {\n (LHS = $lhs:ty, RHS = $rhs:ty) => {\n impl MulAssign<$rhs> for $lhs {\n fn mul_assign(&mut self, rhs: $rhs) {\n *self *= &rhs;\n }\n }\n };\n}\n"},{"id":13,"name":{"Local":"/rustc/library/core/src/ops/range.rs"},"crate_name":"core","contents":null},{"id":14,"name":{"Local":"/rustc/library/core/src/ops/mod.rs"},"crate_name":"core","contents":null},{"id":15,"name":{"Local":"/rustc/library/core/src/lib.rs"},"crate_name":"core","contents":null},{"id":16,"name":{"Local":"/rustc/library/core/src/option.rs"},"crate_name":"core","contents":null},{"id":17,"name":{"Local":"/rustc/library/core/src/iter/traits/collect.rs"},"crate_name":"core","contents":null},{"id":18,"name":{"Local":"/rustc/library/core/src/iter/traits/mod.rs"},"crate_name":"core","contents":null},{"id":19,"name":{"Local":"/rustc/library/core/src/iter/mod.rs"},"crate_name":"core","contents":null},{"id":20,"name":{"Local":"/rustc/library/core/src/iter/traits/iterator.rs"},"crate_name":"core","contents":null},{"id":21,"name":{"Local":"/rustc/library/core/src/iter/range.rs"},"crate_name":"core","contents":null},{"id":22,"name":{"Local":"/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs"},"crate_name":"subtle","contents":null},{"id":23,"name":{"Local":"/rustc/library/core/src/num/uint_macros.rs"},"crate_name":"core","contents":null},{"id":24,"name":{"Local":"/rustc/library/core/src/num/mod.rs"},"crate_name":"core","contents":null},{"id":25,"name":{"Local":"/rustc/library/core/src/macros/mod.rs"},"crate_name":"core","contents":null},{"id":26,"name":{"Local":"/rustc/library/core/src/panicking.rs"},"crate_name":"core","contents":null},{"id":27,"name":{"Local":"/rustc/library/core/src/fmt/mod.rs"},"crate_name":"core","contents":null},{"id":28,"name":{"Local":"/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/ed25519-2.2.3/src/lib.rs"},"crate_name":"ed25519","contents":null},{"id":29,"name":{"Local":"/rustc/library/core/src/result.rs"},"crate_name":"core","contents":null},{"id":30,"name":{"Local":"curve25519/solana-ed25519/src/ed_sigs/error.rs"},"crate_name":"curve25519","contents":"use core::fmt;\n\n/// An error related to Ed25519 signatures.\n#[derive(Debug, Copy, Clone, Eq, PartialEq)]\npub enum Error {\n /// The encoding of a secret key was malformed.\n MalformedSecretKey,\n /// The encoding of a public key was malformed.\n MalformedPublicKey,\n /// Signature verification failed.\n InvalidSignature,\n /// A byte slice of the wrong length was supplied during parsing.\n InvalidSliceLength,\n}\n\nimpl fmt::Display for Error {\n fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {\n let msg = match self {\n Self::MalformedSecretKey => \"Malformed secret key encoding.\",\n Self::MalformedPublicKey => \"Malformed public key encoding.\",\n Self::InvalidSignature => \"Invalid signature.\",\n Self::InvalidSliceLength => \"Invalid length when parsing byte slice.\",\n };\n\n msg.fmt(f)\n }\n}\n\n#[cfg(feature = \"std\")]\nimpl std::error::Error for Error {}\n"},{"id":31,"name":{"Local":"/rustc/library/core/src/ops/control_flow.rs"},"crate_name":"core","contents":null},{"id":32,"name":{"Local":"/rustc/library/core/src/convert/mod.rs"},"crate_name":"core","contents":null},{"id":33,"name":{"Local":"/rustc/library/core/src/cmp.rs"},"crate_name":"core","contents":null},{"id":34,"name":{"Local":"/rustc/library/core/src/default.rs"},"crate_name":"core","contents":null},{"id":35,"name":{"Local":"/rustc/library/core/src/marker.rs"},"crate_name":"core","contents":null},{"id":36,"name":{"Local":"/rustc/library/core/src/clone.rs"},"crate_name":"core","contents":null},{"id":37,"name":{"Local":"/rustc/library/core/src/ops/arith.rs"},"crate_name":"core","contents":null},{"id":38,"name":{"Local":"curve25519/solana-ed25519/src/traits.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek / curve25519-sol.\n// Original curve25519-dalek code:\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// Additions for curve25519-sol:\n// Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE for licensing information.\n//\n// Original authors:\n// - isis agora lovecruft \n// - Henry de Valence \n//\n// Additions in curve25519-sol:\n// - `HEEADecomposition` trait for half-extended Euclidean scalar decomposition.\n\n//! Module for common traits.\n\n#![allow(non_snake_case)]\n\nuse core::borrow::Borrow;\n\nuse crate::scalar::{Scalar, clamp_integer};\nuse subtle::ConstantTimeEq;\n\n// ------------------------------------------------------------------------\n// Public Traits\n// ------------------------------------------------------------------------\n\n/// Trait for getting the identity element of a point type.\npub trait Identity {\n /// Returns the identity element of the curve.\n /// Can be used as a constructor.\n fn identity() -> Self;\n}\n\n/// Trait for testing if a curve point is equivalent to the identity point.\npub trait IsIdentity {\n /// Return true if this element is the identity element of the curve.\n fn is_identity(&self) -> bool;\n}\n\n/// Implement generic identity equality testing for a point representations\n/// which have constant-time equality testing and a defined identity\n/// constructor.\nimpl IsIdentity for T\nwhere\n T: ConstantTimeEq + Identity,\n{\n fn is_identity(&self) -> bool {\n self.ct_eq(&T::identity()).into()\n }\n}\n\n/// A precomputed table of basepoints, for optimising scalar multiplications.\npub trait BasepointTable {\n /// The type of point contained within this table.\n type Point;\n\n /// Generate a new precomputed basepoint table from the given basepoint.\n fn create(basepoint: &Self::Point) -> Self;\n\n /// Retrieve the original basepoint from this table.\n fn basepoint(&self) -> Self::Point;\n\n /// Multiply a `scalar` by this precomputed basepoint table, in constant time.\n fn mul_base(&self, scalar: &Scalar) -> Self::Point;\n\n /// Multiply `clamp_integer(bytes)` by this precomputed basepoint table, in constant time. For\n /// a description of clamping, see [`clamp_integer`].\n fn mul_base_clamped(&self, bytes: [u8; 32]) -> Self::Point {\n // Basepoint multiplication is defined for all values of `bytes` up to and including\n // 2^255 - 1. The limit comes from the fact that scalar.as_radix_16() doesn't work for\n // most scalars larger than 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n self.mul_base(&s)\n }\n}\n\n/// A trait for constant-time multiscalar multiplication without precomputation.\npub trait MultiscalarMul {\n /// The type of point being multiplied, e.g., `RistrettoPoint`.\n type Point;\n\n /// Given an iterator of (possibly secret) scalars and an iterator of\n /// public points, compute\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n.\n /// $$\n ///\n /// It is an error to call this function with two iterators of different lengths.\n ///\n /// # Examples\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n ///\n /// ```\n /// # #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::MultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n ///\n /// // A1 = a*P + b*Q + c*R\n /// let abc = [a,b,c];\n /// let A1 = RistrettoPoint::multiscalar_mul(&abc, &[P,Q,R]);\n /// // Note: (&abc).into_iter(): Iterator\n ///\n /// // A2 = (-a)*P + (-b)*Q + (-c)*R\n /// let minus_abc = abc.iter().map(|x| -x);\n /// let A2 = RistrettoPoint::multiscalar_mul(minus_abc, &[P,Q,R]);\n /// // Note: minus_abc.into_iter(): Iterator\n ///\n /// assert_eq!(A1.compress(), (-A2).compress());\n /// # }\n /// ```\n fn multiscalar_mul(scalars: I, points: J) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow;\n}\n\n/// A trait for variable-time multiscalar multiplication without precomputation.\npub trait VartimeMultiscalarMul {\n /// The type of point being multiplied, e.g., `RistrettoPoint`.\n type Point;\n\n /// Given an iterator of public scalars and an iterator of\n /// `Option`s of points, compute either `Some(Q)`, where\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n,\n /// $$\n /// if all points were `Some(P_i)`, or else return `None`.\n ///\n /// This function is particularly useful when verifying statements\n /// involving compressed points. Accepting `Option` allows\n /// inlining point decompression into the multiscalar call,\n /// avoiding the need for temporary buffers.\n ///\n /// ```\n /// #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::VartimeMultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n /// let abc = [a,b,c];\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n /// let PQR = [P, Q, R];\n ///\n /// let compressed = [P.compress(), Q.compress(), R.compress()];\n ///\n /// // Now we can compute A1 = a*P + b*Q + c*R using P, Q, R:\n /// let A1 = RistrettoPoint::vartime_multiscalar_mul(&abc, &PQR);\n ///\n /// // Or using the compressed points:\n /// let A2 = RistrettoPoint::optional_multiscalar_mul(\n /// &abc,\n /// compressed.iter().map(|pt| pt.decompress()),\n /// );\n ///\n /// assert_eq!(A2, Some(A1));\n ///\n /// // It's also possible to mix compressed and uncompressed points:\n /// let A3 = RistrettoPoint::optional_multiscalar_mul(\n /// abc.iter()\n /// .chain(abc.iter()),\n /// compressed.iter().map(|pt| pt.decompress())\n /// .chain(PQR.iter().map(|&pt| Some(pt))),\n /// );\n ///\n /// assert_eq!(A3, Some(A1+A1));\n /// # }\n /// ```\n fn optional_multiscalar_mul(scalars: I, points: J) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator>;\n\n /// Given an iterator of public scalars and an iterator of\n /// public points, compute\n /// $$\n /// Q = c\\_1 P\\_1 + \\cdots + c\\_n P\\_n,\n /// $$\n /// using variable-time operations.\n ///\n /// It is an error to call this function with two iterators of different lengths.\n ///\n /// # Examples\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n ///\n /// ```\n /// #[cfg(feature = \"alloc\")]\n /// # {\n /// use curve25519::constants;\n /// use curve25519::traits::VartimeMultiscalarMul;\n /// use curve25519::ristretto::RistrettoPoint;\n /// use curve25519::scalar::Scalar;\n ///\n /// // Some scalars\n /// let a = Scalar::from(87329482u64);\n /// let b = Scalar::from(37264829u64);\n /// let c = Scalar::from(98098098u64);\n ///\n /// // Some points\n /// let P = constants::RISTRETTO_BASEPOINT_POINT;\n /// let Q = P + P;\n /// let R = P + Q;\n ///\n /// // A1 = a*P + b*Q + c*R\n /// let abc = [a,b,c];\n /// let A1 = RistrettoPoint::vartime_multiscalar_mul(&abc, &[P,Q,R]);\n /// // Note: (&abc).into_iter(): Iterator\n ///\n /// // A2 = (-a)*P + (-b)*Q + (-c)*R\n /// let minus_abc = abc.iter().map(|x| -x);\n /// let A2 = RistrettoPoint::vartime_multiscalar_mul(minus_abc, &[P,Q,R]);\n /// // Note: minus_abc.into_iter(): Iterator\n ///\n /// assert_eq!(A1.compress(), (-A2).compress());\n /// # }\n /// ```\n fn vartime_multiscalar_mul(scalars: I, points: J) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n Self::Point: Clone,\n {\n Self::optional_multiscalar_mul(\n scalars,\n points.into_iter().map(|P| Some(P.borrow().clone())),\n )\n .expect(\"should return some point\")\n }\n}\n\n/// A trait for variable-time multiscalar multiplication with precomputation.\n///\n/// A general multiscalar multiplication with precomputation can be written as\n/// $$\n/// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n/// $$\n/// where the \\\\(B_i\\\\) are *static* points, for which precomputation\n/// is possible, and the \\\\(A_j\\\\) are *dynamic* points, for which\n/// precomputation is not possible.\n///\n/// This trait has three methods for performing this computation:\n///\n/// * [`Self::vartime_multiscalar_mul`], which handles the special case where\n/// \\\\(n = 0\\\\) and there are no dynamic points;\n///\n/// * [`Self::vartime_mixed_multiscalar_mul`], which takes the dynamic points as\n/// already-validated `Point`s and is infallible;\n///\n/// * [`Self::optional_mixed_multiscalar_mul`], which takes the dynamic points\n/// as `Option`s and returns an `Option`, allowing decompression\n/// to be composed into the input iterators.\n///\n/// All methods require that the lengths of the input iterators be\n/// known, as if they were `ExactSizeIterator`s. (It\n/// does not require `ExactSizeIterator` only because that trait is\n/// broken).\npub trait VartimePrecomputedMultiscalarMul: Sized {\n /// The type of point to be multiplied, e.g., `RistrettoPoint`.\n type Point: Clone;\n\n /// Given the static points \\\\( B_i \\\\), perform precomputation\n /// and return the precomputation data.\n fn new(static_points: I) -> Self\n where\n I: IntoIterator,\n I::Item: Borrow;\n\n /// Return the number of static points in the precomputation.\n fn len(&self) -> usize;\n\n /// Determine if the precomputation is empty.\n fn is_empty(&self) -> bool;\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), compute\n /// $$\n /// Q = b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\).\n ///\n /// The trait bound aims for maximum flexibility: the input must\n /// be convertible to iterators (`I: IntoIter`), and the\n /// iterator's items must be `Borrow`, to allow iterators\n /// returning either `Scalar`s or `&Scalar`s.\n fn vartime_multiscalar_mul(&self, static_scalars: I) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n {\n use core::iter;\n\n Self::vartime_mixed_multiscalar_mul(\n self,\n static_scalars,\n iter::empty::(),\n iter::empty::(),\n )\n }\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), `dynamic_scalars`, an iterator of public scalars\n /// \\\\(a_i\\\\), and `dynamic_points`, an iterator of points\n /// \\\\(A_i\\\\), compute\n /// $$\n /// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\), or if \\\\(a_i\\\\) and \\\\(A_i\\\\) do not have\n /// the same length.\n ///\n /// The trait bound aims for maximum flexibility: the inputs must be\n /// convertible to iterators (`I: IntoIter`), and the iterator's items\n /// must be `Borrow` (or `Borrow`), to allow\n /// iterators returning either `Scalar`s or `&Scalar`s.\n fn vartime_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Self::Point\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator,\n K::Item: Borrow,\n {\n Self::optional_mixed_multiscalar_mul(\n self,\n static_scalars,\n dynamic_scalars,\n dynamic_points.into_iter().map(|P| Some(P.borrow().clone())),\n )\n .expect(\"should return some point\")\n }\n\n /// Given `static_scalars`, an iterator of public scalars\n /// \\\\(b_i\\\\), `dynamic_scalars`, an iterator of public scalars\n /// \\\\(a_i\\\\), and `dynamic_points`, an iterator of points\n /// \\\\(A_i\\\\), compute\n /// $$\n /// Q = a_1 A_1 + \\cdots + a_n A_n + b_1 B_1 + \\cdots + b_m B_m,\n /// $$\n /// where the \\\\(B_j\\\\) are the points that were supplied to `new`.\n ///\n /// If any of the dynamic points were `None`, return `None`.\n ///\n /// It is valid for \\\\(b_i\\\\) to have a shorter length than \\\\(B_j\\\\).\n /// In this case, any \"unused\" points are ignored in the computation.\n /// It is an error to call this function if \\\\(b_i\\\\) has a longer\n /// length than \\\\(B_j\\\\), or if \\\\(a_i\\\\) and \\\\(A_i\\\\) do not have\n /// the same length.\n ///\n /// This function is particularly useful when verifying statements\n /// involving compressed points. Accepting `Option` allows\n /// inlining point decompression into the multiscalar call,\n /// avoiding the need for temporary buffers.\n fn optional_mixed_multiscalar_mul(\n &self,\n static_scalars: I,\n dynamic_scalars: J,\n dynamic_points: K,\n ) -> Option\n where\n I: IntoIterator,\n I::Item: Borrow,\n J: IntoIterator,\n J::Item: Borrow,\n K: IntoIterator>;\n}\n\n/// A trait for decomposing scalars for the HEEA method.\n///\n/// The HEEA method is used by EdDSA verification to replace multiplication by\n/// a full-width challenge scalar with multiplications by two shorter scalars.\n/// For a challenge scalar `h`, the decomposition returns non-negative scalars\n/// `rho` and `tau` that satisfy either `rho == tau * h (mod ell)` or\n/// `rho == -tau * h (mod ell)`.\n///\n/// This trait provides a method for performing this decomposition.\npub trait HEEADecomposition {\n /// Decomposes a scalar `h` into two approximately half-size scalars.\n ///\n /// Returns `(rho, tau, flip_h)`. If `flip_h` is false, then\n /// `rho == tau * h (mod ell)`. If `flip_h` is true, then\n /// `rho == -tau * h (mod ell)`.\n fn heea_decompose(&self) -> (Scalar, Scalar, bool);\n}\n\n// ------------------------------------------------------------------------\n// Private Traits\n// ------------------------------------------------------------------------\n\n/// Trait for checking whether a point is on the curve.\n///\n/// This trait is only for debugging/testing, since it should be\n/// impossible for a `curve25519-dalek` user to construct an invalid\n/// point.\n#[allow(dead_code)]\npub(crate) trait ValidityCheck {\n /// Checks whether the point is on the curve. Not CT.\n fn is_valid(&self) -> bool;\n}\n"},{"id":39,"name":{"Local":"curve25519/solana-ed25519/src/edwards/affine.rs"},"crate_name":"curve25519","contents":"use super::{CompressedEdwardsY, EdwardsPoint};\nuse crate::traits::Identity;\nuse crate::{Scalar, field::FieldElement};\nuse core::ops::Mul;\nuse subtle::{Choice, ConditionallySelectable, ConstantTimeEq};\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::DefaultIsZeroes;\n\n/// Affine Edwards point on untwisted curve.\n#[derive(Copy, Clone, Debug)]\npub struct AffinePoint {\n pub(super) x: FieldElement,\n pub(super) y: FieldElement,\n}\n\nimpl ConstantTimeEq for AffinePoint {\n fn ct_eq(&self, other: &Self) -> Choice {\n self.x.ct_eq(&other.x) & self.y.ct_eq(&other.y)\n }\n}\n\nimpl ConditionallySelectable for AffinePoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n Self {\n x: FieldElement::conditional_select(&a.x, &b.x, choice),\n y: FieldElement::conditional_select(&a.y, &b.y, choice),\n }\n }\n}\n\nimpl Default for AffinePoint {\n fn default() -> AffinePoint {\n AffinePoint::identity()\n }\n}\n\nimpl Identity for AffinePoint {\n fn identity() -> AffinePoint {\n AffinePoint {\n x: FieldElement::ZERO,\n y: FieldElement::ONE,\n }\n }\n}\n\nimpl PartialEq for AffinePoint {\n fn eq(&self, other: &Self) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for AffinePoint {}\n\n#[cfg(feature = \"zeroize\")]\nimpl DefaultIsZeroes for AffinePoint {}\n\nimpl AffinePoint {\n /// Convert to extended coordinates.\n pub fn to_edwards(self) -> EdwardsPoint {\n EdwardsPoint {\n X: self.x,\n Y: self.y,\n Z: FieldElement::ONE,\n T: &self.x * &self.y,\n }\n }\n\n /// Compress affine Edwards coordinates into `CompressedEdwardsY` format.\n #[inline]\n pub fn compress(self) -> CompressedEdwardsY {\n let mut s = self.y.to_bytes();\n s[31] ^= self.x.is_negative().unwrap_u8() << 7;\n CompressedEdwardsY(s)\n }\n}\n\nimpl Mul for Scalar {\n type Output = EdwardsPoint;\n\n #[inline]\n fn mul(self, rhs: AffinePoint) -> EdwardsPoint {\n self * &rhs\n }\n}\n\nimpl Mul<&AffinePoint> for Scalar {\n type Output = EdwardsPoint;\n\n #[inline]\n fn mul(self, rhs: &AffinePoint) -> EdwardsPoint {\n rhs.to_edwards() * self\n }\n}\n\n#[cfg(test)]\nmod tests {\n use super::{AffinePoint, EdwardsPoint, Identity};\n use crate::constants;\n\n #[test]\n fn identity_conversion() {\n assert_eq!(\n AffinePoint::identity().to_edwards(),\n EdwardsPoint::identity()\n );\n }\n\n #[test]\n fn generator_round_trip() {\n let basepoint = constants::ED25519_BASEPOINT_POINT;\n assert_eq!(basepoint.to_affine().to_edwards(), basepoint);\n }\n}\n"},{"id":40,"name":{"Local":"/rustc/library/core/src/hash/mod.rs"},"crate_name":"core","contents":null},{"id":41,"name":{"Local":"/rustc/library/core/src/array/mod.rs"},"crate_name":"core","contents":null},{"id":42,"name":{"Local":"/rustc/library/core/src/borrow.rs"},"crate_name":"core","contents":null},{"id":43,"name":{"Local":"/rustc/library/core/src/iter/traits/accum.rs"},"crate_name":"core","contents":null},{"id":44,"name":{"Local":"/rustc/library/core/src/array/iter.rs"},"crate_name":"core","contents":null},{"id":45,"name":{"Local":"/rustc/library/core/src/num/nonzero.rs"},"crate_name":"core","contents":null},{"id":46,"name":{"Local":"/rustc/library/core/src/iter/adapters/step_by.rs"},"crate_name":"core","contents":null},{"id":47,"name":{"Local":"/rustc/library/core/src/iter/adapters/mod.rs"},"crate_name":"core","contents":null},{"id":48,"name":{"Local":"/rustc/library/core/src/iter/adapters/chain.rs"},"crate_name":"core","contents":null},{"id":49,"name":{"Local":"/rustc/library/core/src/iter/adapters/zip.rs"},"crate_name":"core","contents":null},{"id":50,"name":{"Local":"/rustc/library/core/src/iter/adapters/intersperse.rs"},"crate_name":"core","contents":null},{"id":51,"name":{"Local":"/rustc/library/core/src/ops/function.rs"},"crate_name":"core","contents":null},{"id":52,"name":{"Local":"/rustc/library/core/src/iter/adapters/map.rs"},"crate_name":"core","contents":null},{"id":53,"name":{"Local":"/rustc/library/core/src/iter/adapters/filter.rs"},"crate_name":"core","contents":null},{"id":54,"name":{"Local":"/rustc/library/core/src/iter/adapters/filter_map.rs"},"crate_name":"core","contents":null},{"id":55,"name":{"Local":"/rustc/library/core/src/iter/adapters/enumerate.rs"},"crate_name":"core","contents":null},{"id":56,"name":{"Local":"/rustc/library/core/src/iter/adapters/peekable.rs"},"crate_name":"core","contents":null},{"id":57,"name":{"Local":"/rustc/library/core/src/iter/adapters/skip_while.rs"},"crate_name":"core","contents":null},{"id":58,"name":{"Local":"/rustc/library/core/src/iter/adapters/take_while.rs"},"crate_name":"core","contents":null},{"id":59,"name":{"Local":"/rustc/library/core/src/iter/adapters/map_while.rs"},"crate_name":"core","contents":null},{"id":60,"name":{"Local":"/rustc/library/core/src/iter/adapters/skip.rs"},"crate_name":"core","contents":null},{"id":61,"name":{"Local":"/rustc/library/core/src/iter/adapters/take.rs"},"crate_name":"core","contents":null},{"id":62,"name":{"Local":"/rustc/library/core/src/iter/adapters/scan.rs"},"crate_name":"core","contents":null},{"id":63,"name":{"Local":"/rustc/library/core/src/iter/adapters/flatten.rs"},"crate_name":"core","contents":null},{"id":64,"name":{"Local":"/rustc/library/core/src/iter/adapters/map_windows.rs"},"crate_name":"core","contents":null},{"id":65,"name":{"Local":"/rustc/library/core/src/iter/adapters/fuse.rs"},"crate_name":"core","contents":null},{"id":66,"name":{"Local":"/rustc/library/core/src/iter/adapters/inspect.rs"},"crate_name":"core","contents":null},{"id":67,"name":{"Local":"/rustc/library/core/src/ops/try_trait.rs"},"crate_name":"core","contents":null},{"id":68,"name":{"Local":"/rustc/library/core/src/iter/traits/double_ended.rs"},"crate_name":"core","contents":null},{"id":69,"name":{"Local":"/rustc/library/core/src/iter/traits/exact_size.rs"},"crate_name":"core","contents":null},{"id":70,"name":{"Local":"/rustc/library/core/src/iter/adapters/rev.rs"},"crate_name":"core","contents":null},{"id":71,"name":{"Local":"/rustc/library/core/src/iter/adapters/copied.rs"},"crate_name":"core","contents":null},{"id":72,"name":{"Local":"/rustc/library/core/src/iter/adapters/cloned.rs"},"crate_name":"core","contents":null},{"id":73,"name":{"Local":"/rustc/library/core/src/iter/adapters/cycle.rs"},"crate_name":"core","contents":null},{"id":74,"name":{"Local":"/rustc/library/core/src/iter/adapters/array_chunks.rs"},"crate_name":"core","contents":null},{"id":75,"name":{"Local":"/rustc/library/core/src/ops/index.rs"},"crate_name":"core","contents":null},{"id":76,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/u64/constants.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! This module contains backend-specific constant values, such as the 64-bit limbs of curve constants.\n\nuse super::field::FieldElement51;\nuse super::scalar::Scalar52;\nuse crate::edwards::EdwardsPoint;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::{\n backend::serial::curve_models::{AffineNielsPoint, ProjectiveNielsPoint},\n edwards::EdwardsBasepointTable,\n window::{LookupTable, NafLookupTable5, NafLookupTable8},\n};\n\n/// The value of minus one, equal to `-&FieldElement::ONE`\npub(crate) const MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n]);\n\n/// sqrt(-486664)\n#[cfg(feature = \"digest\")]\npub(crate) const ED25519_SQRTAM2: FieldElement51 = FieldElement51::from_limbs([\n 1693982333959686,\n 608509411481997,\n 2235573344831311,\n 947681270984193,\n 266558006233600,\n]);\n\n/// Edwards `d` value, equal to `-121665/121666 mod p`.\npub(crate) const EDWARDS_D: FieldElement51 = FieldElement51::from_limbs([\n 929955233495203,\n 466365720129213,\n 1662059464998953,\n 2033849074728123,\n 1442794654840575,\n]);\n\n/// Edwards `2*d` value, equal to `2*(-121665/121666) mod p`.\npub(crate) const EDWARDS_D2: FieldElement51 = FieldElement51::from_limbs([\n 1859910466990425,\n 932731440258426,\n 1072319116312658,\n 1815898335770999,\n 633789495995903,\n]);\n\n/// One minus edwards `d` value squared, equal to `(1 - (-121665/121666) mod p) pow 2`\npub(crate) const ONE_MINUS_EDWARDS_D_SQUARED: FieldElement51 = FieldElement51::from_limbs([\n 1136626929484150,\n 1998550399581263,\n 496427632559748,\n 118527312129759,\n 45110755273534,\n]);\n\n/// Edwards `d` value minus one squared, equal to `(((-121665/121666) mod p) - 1) pow 2`\npub(crate) const EDWARDS_D_MINUS_ONE_SQUARED: FieldElement51 = FieldElement51::from_limbs([\n 1507062230895904,\n 1572317787530805,\n 683053064812840,\n 317374165784489,\n 1572899562415810,\n]);\n\n/// `= sqrt(a*d - 1)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.\npub(crate) const SQRT_AD_MINUS_ONE: FieldElement51 = FieldElement51::from_limbs([\n 2241493124984347,\n 425987919032274,\n 2207028919301688,\n 1220490630685848,\n 974799131293748,\n]);\n\n/// `= 1/sqrt(a-d)`, where `a = -1 (mod p)`, `d` are the Edwards curve parameters.\npub(crate) const INVSQRT_A_MINUS_D: FieldElement51 = FieldElement51::from_limbs([\n 278908739862762,\n 821645201101625,\n 8113234426968,\n 1777959178193151,\n 2118520810568447,\n]);\n\n/// Precomputed value of one of the square roots of -1 (mod p)\npub(crate) const SQRT_M1: FieldElement51 = FieldElement51::from_limbs([\n 1718705420411056,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n]);\n\n/// `APLUS2_OVER_FOUR` is (A+2)/4. (This is used internally within the Montgomery ladder.)\npub(crate) const APLUS2_OVER_FOUR: FieldElement51 =\n FieldElement51::from_limbs([121666, 0, 0, 0, 0]);\n\n#[cfg(feature = \"digest\")]\n/// `MONTGOMERY_A` is equal to 486662, which is a constant of the curve equation\n/// for Curve25519 in its Montgomery form. (This is used internally within the\n/// Elligator map.)\npub(crate) const MONTGOMERY_A: FieldElement51 = FieldElement51::from_limbs([486662, 0, 0, 0, 0]);\n\n#[cfg(feature = \"digest\")]\n/// `MONTGOMERY_A_NEG` is equal to -486662. (This is used internally within the\n/// Elligator map.)\npub(crate) const MONTGOMERY_A_NEG: FieldElement51 = FieldElement51::from_limbs([\n 2251799813198567,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n]);\n\n/// `L` is the order of base point, i.e. 2^252 + 27742317777372353535851937790883648493\npub(crate) const L: Scalar52 = Scalar52([\n 0x0002631a5cf5d3ed,\n 0x000dea2f79cd6581,\n 0x000000000014def9,\n 0x0000000000000000,\n 0x0000100000000000,\n]);\n\n/// `L` * `LFACTOR` = -1 (mod 2^52)\npub(crate) const LFACTOR: u64 = 0x51da312547e1b;\n\n/// `R` = R % L where R = 2^260\npub(crate) const R: Scalar52 = Scalar52([\n 0x000f48bd6721e6ed,\n 0x0003bab5ac67e45a,\n 0x000fffffeb35e51b,\n 0x000fffffffffffff,\n 0x00000fffffffffff,\n]);\n\n/// `RR` = (R^2) % L where R = 2^260\npub(crate) const RR: Scalar52 = Scalar52([\n 0x0009d265e952d13b,\n 0x000d63c715bea69f,\n 0x0005be65cb687604,\n 0x0003dceec73d217f,\n 0x000009411b7c309a,\n]);\n\n/// The Ed25519 basepoint, as an `EdwardsPoint`.\n///\n/// This is called `_POINT` to distinguish it from\n/// `ED25519_BASEPOINT_TABLE`, which should be used for scalar\n/// multiplication (it's much faster).\npub const ED25519_BASEPOINT_POINT: EdwardsPoint = EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1738742601995546,\n 1146398526822698,\n 2070867633025821,\n 562264141797630,\n 587772402128613,\n ]),\n Y: FieldElement51::from_limbs([\n 1801439850948184,\n 1351079888211148,\n 450359962737049,\n 900719925474099,\n 1801439850948198,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1841354044333475,\n 16398895984059,\n 755974180946558,\n 900171276175154,\n 1821297809914039,\n ]),\n};\n\n/// The Ed25519 basepoint, mul by 2^128, as an `EdwardsPoint`.\npub const ED25519_BASEPOINT_128_POINT: EdwardsPoint = EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1573694877509226,\n 2118644427590872,\n 940662180141619,\n 707110682864191,\n 1255997186674731,\n ]),\n Y: FieldElement51::from_limbs([\n 1162474291335259,\n 820106152083656,\n 523232807607258,\n 2134608004007539,\n 729879497843472,\n ]),\n Z: FieldElement51::from_limbs([\n 445134475408207,\n 1820906444767788,\n 40212681131971,\n 1448661247201044,\n 965705781338211,\n ]),\n T: FieldElement51::from_limbs([\n 1855274855831337,\n 1818119365171423,\n 269991889323070,\n 1370944576683708,\n 140790155072480,\n ]),\n};\n\n/// The 8-torsion subgroup \\\\(\\mathcal E \\[8\\]\\\\).\n///\n/// In the case of Curve25519, it is cyclic; the \\\\(i\\\\)-th element of\n/// the array is \\\\(\\[i\\]P\\\\), where \\\\(P\\\\) is a point of order \\\\(8\\\\)\n/// generating \\\\(\\mathcal E\\[8\\]\\\\).\n///\n/// Thus \\\\(\\mathcal E\\[4\\]\\\\) is the points indexed by `0,2,4,6`, and\n/// \\\\(\\mathcal E\\[2\\]\\\\) is the points indexed by `0,4`.\npub const EIGHT_TORSION: [EdwardsPoint; 8] = EIGHT_TORSION_INNER_DOC_HIDDEN;\n\n/// Inner item used to hide limb constants from cargo doc output.\n#[doc(hidden)]\npub const EIGHT_TORSION_INNER_DOC_HIDDEN: [EdwardsPoint; 8] = [\n EdwardsPoint {\n X: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Y: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 358744748052810,\n 1691584618240980,\n 977650209285361,\n 1429865912637724,\n 560044844278676,\n ]),\n Y: FieldElement51::from_limbs([\n 84926274344903,\n 473620666599931,\n 365590438845504,\n 1028470286882429,\n 2146499180330972,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1448326834587521,\n 1857896831960481,\n 1093722731865333,\n 1677408490711241,\n 1915505153018406,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 533094393274173,\n 2016890930128738,\n 18285341111199,\n 134597186663265,\n 1486323764102114,\n ]),\n Y: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 358744748052810,\n 1691584618240980,\n 977650209285361,\n 1429865912637724,\n 560044844278676,\n ]),\n Y: FieldElement51::from_limbs([\n 2166873539340326,\n 1778179147085316,\n 1886209374839743,\n 1223329526802818,\n 105300633354275,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 803472979097708,\n 393902981724766,\n 1158077081819914,\n 574391322974006,\n 336294660666841,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Y: FieldElement51::from_limbs([\n 2251799813685228,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n 2251799813685247,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1893055065632419,\n 560215195444267,\n 1274149604399886,\n 821933901047523,\n 1691754969406571,\n ]),\n Y: FieldElement51::from_limbs([\n 2166873539340326,\n 1778179147085316,\n 1886209374839743,\n 1223329526802818,\n 105300633354275,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 1448326834587521,\n 1857896831960481,\n 1093722731865333,\n 1677408490711241,\n 1915505153018406,\n ]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1718705420411056,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n ]),\n Y: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([0, 0, 0, 0, 0]),\n },\n EdwardsPoint {\n X: FieldElement51::from_limbs([\n 1893055065632419,\n 560215195444267,\n 1274149604399886,\n 821933901047523,\n 1691754969406571,\n ]),\n Y: FieldElement51::from_limbs([\n 84926274344903,\n 473620666599931,\n 365590438845504,\n 1028470286882429,\n 2146499180330972,\n ]),\n Z: FieldElement51::from_limbs([1, 0, 0, 0, 0]),\n T: FieldElement51::from_limbs([\n 803472979097708,\n 393902981724766,\n 1158077081819914,\n 574391322974006,\n 336294660666841,\n ]),\n },\n];\n\n/// Table containing precomputed multiples of the Ed25519 basepoint \\\\(B = (x, 4/5)\\\\).\n#[cfg(feature = \"precomputed-tables\")]\npub static ED25519_BASEPOINT_TABLE: &EdwardsBasepointTable =\n &ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN;\n\n/// Inner constant, used to avoid filling the docs with precomputed points.\n#[doc(hidden)]\n#[cfg(feature = \"precomputed-tables\")]\nstatic ED25519_BASEPOINT_TABLE_INNER_DOC_HIDDEN: EdwardsBasepointTable = EdwardsBasepointTable([\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3540182452943730,\n 2497478415033846,\n 2521227595762870,\n 1462984067271729,\n 2389212253076811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 62697248952638,\n 204681361388450,\n 631292143396476,\n 338455783676468,\n 1213667448819585,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301289933810280,\n 1259582250014073,\n 1422107436869536,\n 796239922652654,\n 1953934009299142,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3632771708514775,\n 790832306631235,\n 2067202295274102,\n 1995808275510000,\n 1566530869037010,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 463307831301544,\n 432984605774163,\n 1610641361907204,\n 750899048855000,\n 1894842303421586,\n ]),\n xy2d: FieldElement51::from_limbs([\n 748439484463711,\n 1033211726465151,\n 1396005112841647,\n 1611506220286469,\n 1972177495910992,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1601611775252272,\n 1720807796594148,\n 1132070835939856,\n 3512254832574799,\n 2147779492816910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 316559037616741,\n 2177824224946892,\n 1459442586438991,\n 1461528397712656,\n 751590696113597,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850748884277385,\n 1200145853858453,\n 1068094770532492,\n 672251375690438,\n 1586055907191707,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 934282339813791,\n 1846903124198670,\n 1172395437954843,\n 1007037127761661,\n 1830588347719256,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1694390458783935,\n 1735906047636159,\n 705069562067493,\n 648033061693059,\n 696214010414170,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1121406372216585,\n 192876649532226,\n 190294192191717,\n 1994165897297032,\n 2245000007398739,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 769950342298400,\n 2384754244604994,\n 3095885746880802,\n 3225892188161580,\n 2977876099231263,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 425251763115706,\n 608463272472562,\n 442562545713235,\n 837766094556764,\n 374555092627893,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1086255230780037,\n 274979815921559,\n 1960002765731872,\n 929474102396301,\n 1190409889297339,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1388594989461809,\n 316767091099457,\n 2646098655878230,\n 1230079486801004,\n 1440737038838979,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 7380825640100,\n 146210432690483,\n 304903576448906,\n 1198869323871120,\n 997689833219095,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1181317918772081,\n 114573476638901,\n 262805072233344,\n 265712217171332,\n 294181933805782,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2916800678241215,\n 2065379846933858,\n 2622030924071124,\n 2602788184473875,\n 1233371373142984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2019367628972465,\n 676711900706637,\n 110710997811333,\n 1108646842542025,\n 517791959672113,\n ]),\n xy2d: FieldElement51::from_limbs([\n 965130719900578,\n 247011430587952,\n 526356006571389,\n 91986625355052,\n 2157223321444601,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4320419353804412,\n 4218074731744053,\n 957728544705548,\n 729906502578991,\n 2411634706750414,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2073601412052185,\n 31021124762708,\n 264500969797082,\n 248034690651703,\n 1030252227928288,\n ]),\n xy2d: FieldElement51::from_limbs([\n 551790716293402,\n 1989538725166328,\n 801169423371717,\n 2052451893578887,\n 678432056995012,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1368953770187805,\n 3042147450398169,\n 2689308289352409,\n 2142576377050579,\n 1932081720066286,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 953638594433374,\n 1092333936795051,\n 1419774766716690,\n 805677984380077,\n 859228993502513,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1200766035879111,\n 20142053207432,\n 1465634435977050,\n 1645256912097844,\n 295121984874596,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1735718747031538,\n 1248237894295956,\n 1204753118328107,\n 976066523550493,\n 2317743583219840,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1060098822528990,\n 1586825862073490,\n 212301317240126,\n 1975302711403555,\n 666724059764335,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1091990273418756,\n 1572899409348578,\n 80968014455247,\n 306009358661350,\n 1520450739132526,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3732317023121341,\n 1511153322193951,\n 3496143672676420,\n 2556587964178488,\n 2620936670181690,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2151330273626164,\n 762045184746182,\n 1688074332551515,\n 823046109005759,\n 907602769079491,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2047386910586836,\n 168470092900250,\n 1552838872594810,\n 340951180073789,\n 360819374702533,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1982622644432037,\n 2014393600336956,\n 2380709022489462,\n 3869592437614438,\n 2357094095599062,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 980234343912898,\n 1712256739246056,\n 588935272190264,\n 204298813091998,\n 841798321043288,\n ]),\n xy2d: FieldElement51::from_limbs([\n 197561292938973,\n 454817274782871,\n 1963754960082318,\n 2113372252160468,\n 971377527342673,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2416499262514576,\n 2254927265442919,\n 3451304785234000,\n 1766155447043651,\n 1899238924683527,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 732262946680281,\n 1674412764227063,\n 2182456405662809,\n 1350894754474250,\n 558458873295247,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2103305098582922,\n 1960809151316468,\n 715134605001343,\n 1454892949167181,\n 40827143824949,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1239289043050193,\n 1744654158124578,\n 758702410031698,\n 4048562808759936,\n 2253402870349013,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2232056027107988,\n 987343914584615,\n 2115594492994461,\n 1819598072792159,\n 1119305654014850,\n ]),\n xy2d: FieldElement51::from_limbs([\n 320153677847348,\n 939613871605645,\n 641883205761567,\n 1930009789398224,\n 329165806634126,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3232730304159378,\n 1242488692177892,\n 1251446316964684,\n 1086618677993530,\n 1961430968465772,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 276821765317453,\n 1536835591188030,\n 1305212741412361,\n 61473904210175,\n 2051377036983058,\n ]),\n xy2d: FieldElement51::from_limbs([\n 833449923882501,\n 1750270368490475,\n 1123347002068295,\n 185477424765687,\n 278090826653186,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 794524995833413,\n 1849907304548286,\n 2305148486158393,\n 1272368559505216,\n 1147304168324779,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1504846112759364,\n 1203096289004681,\n 562139421471418,\n 274333017451844,\n 1284344053775441,\n ]),\n xy2d: FieldElement51::from_limbs([\n 483048732424432,\n 2116063063343382,\n 30120189902313,\n 292451576741007,\n 1156379271702225,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3180171966714267,\n 2147692869914563,\n 1455665844462196,\n 1986737809425946,\n 2437006863943337,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 137732961814206,\n 706670923917341,\n 1387038086865771,\n 1965643813686352,\n 1384777115696347,\n ]),\n xy2d: FieldElement51::from_limbs([\n 481144981981577,\n 2053319313589856,\n 2065402289827512,\n 617954271490316,\n 1106602634668125,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2948097833334040,\n 3145099472726142,\n 1148636718636008,\n 2278533891034865,\n 2203955659340680,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 657390353372855,\n 998499966885562,\n 991893336905797,\n 810470207106761,\n 343139804608786,\n ]),\n xy2d: FieldElement51::from_limbs([\n 791736669492960,\n 934767652997115,\n 824656780392914,\n 1759463253018643,\n 361530362383518,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2022541353055578,\n 4346500076272714,\n 3802807888710933,\n 2494585331103411,\n 2947785218648809,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1287487199965223,\n 2215311941380308,\n 1552928390931986,\n 1664859529680196,\n 1125004975265243,\n ]),\n xy2d: FieldElement51::from_limbs([\n 677434665154918,\n 989582503122485,\n 1817429540898386,\n 1052904935475344,\n 1143826298169798,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2619066141993637,\n 2570231002607651,\n 2947429167440602,\n 2885885471266079,\n 2276381426249673,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 773360688841258,\n 1815381330538070,\n 363773437667376,\n 539629987070205,\n 783280434248437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 180820816194166,\n 168937968377394,\n 748416242794470,\n 1227281252254508,\n 1567587861004268,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2730575372268893,\n 2062896624554806,\n 2951191072970647,\n 2609899222113120,\n 1277310261461760,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1984740906540026,\n 1079164179400229,\n 1056021349262661,\n 1659958556483663,\n 1088529069025527,\n ]),\n xy2d: FieldElement51::from_limbs([\n 580736401511151,\n 1842931091388998,\n 1177201471228238,\n 2075460256527244,\n 1301133425678027,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1515728832059163,\n 1575261009617579,\n 1510246567196186,\n 2442877836294952,\n 2368461529974388,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295295738269652,\n 1714742313707026,\n 545583042462581,\n 2034411676262552,\n 1513248090013606,\n ]),\n xy2d: FieldElement51::from_limbs([\n 230710545179830,\n 30821514358353,\n 760704303452229,\n 390668103790604,\n 573437871383156,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3421179921230875,\n 2514967047430861,\n 4274701112739695,\n 3071700566936367,\n 4275698278559832,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2102254323485823,\n 1570832666216754,\n 34696906544624,\n 1993213739807337,\n 70638552271463,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894132856735058,\n 548675863558441,\n 845349339503395,\n 1942269668326667,\n 1615682209874691,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3539470031223082,\n 1222355136884919,\n 1846481788678694,\n 1150426571265110,\n 1613523400722047,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 793388516527298,\n 1315457083650035,\n 1972286999342417,\n 1901825953052455,\n 338269477222410,\n ]),\n xy2d: FieldElement51::from_limbs([\n 550201530671806,\n 778605267108140,\n 2063911101902983,\n 115500557286349,\n 2041641272971022,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 717255318455100,\n 519313764361315,\n 2080406977303708,\n 541981206705521,\n 774328150311600,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 261715221532238,\n 1795354330069993,\n 1496878026850283,\n 499739720521052,\n 389031152673770,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1997217696294013,\n 1717306351628065,\n 1684313917746180,\n 1644426076011410,\n 1857378133465451,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3727234538477877,\n 2328731709971226,\n 3368528843456914,\n 2002544139318041,\n 2977347647489186,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2022306639183567,\n 726296063571875,\n 315345054448644,\n 1058733329149221,\n 1448201136060677,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1710065158525665,\n 1895094923036397,\n 123988286168546,\n 1145519900776355,\n 1607510767693874,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2813405189107769,\n 1071733543815036,\n 2383296312486238,\n 1946868434569998,\n 3079937947649451,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1548495173745801,\n 442310529226540,\n 998072547000384,\n 553054358385281,\n 644824326376171,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1445526537029440,\n 2225519789662536,\n 914628859347385,\n 1064754194555068,\n 1660295614401091,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3451490036797185,\n 2275827949507588,\n 2318438102929588,\n 2309425969971222,\n 2816893781664854,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 876926774220824,\n 554618976488214,\n 1012056309841565,\n 839961821554611,\n 1414499340307677,\n ]),\n xy2d: FieldElement51::from_limbs([\n 703047626104145,\n 1266841406201770,\n 165556500219173,\n 486991595001879,\n 1011325891650656,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1622861044480487,\n 1156394801573634,\n 4120932379100752,\n 2578903799462977,\n 2095342781472283,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 334886927423922,\n 489511099221528,\n 129160865966726,\n 1720809113143481,\n 619700195649254,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1646545795166119,\n 1758370782583567,\n 714746174550637,\n 1472693650165135,\n 898994790308209,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2585203586724508,\n 2547572356138185,\n 1693106465353609,\n 912330357530760,\n 2723035471635610,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1811196219982022,\n 1068969825533602,\n 289602974833439,\n 1988956043611592,\n 863562343398367,\n ]),\n xy2d: FieldElement51::from_limbs([\n 906282429780072,\n 2108672665779781,\n 432396390473936,\n 150625823801893,\n 1708930497638539,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 925664675702309,\n 2273216662253932,\n 4083236455546587,\n 601157008940112,\n 2623617868729744,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1479786007267725,\n 1738881859066675,\n 68646196476567,\n 2146507056100328,\n 1247662817535471,\n ]),\n xy2d: FieldElement51::from_limbs([\n 52035296774456,\n 939969390708103,\n 312023458773250,\n 59873523517659,\n 1231345905848899,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2895154920100990,\n 2541986621181021,\n 2013561737429022,\n 2571447883196794,\n 2645536492181409,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 129358342392716,\n 1932811617704777,\n 1176749390799681,\n 398040349861790,\n 1170779668090425,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2051980782668029,\n 121859921510665,\n 2048329875753063,\n 1235229850149665,\n 519062146124755,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3859970785658325,\n 2667608874045675,\n 1350468408164765,\n 2038620059057678,\n 3278704299674360,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1837656083115103,\n 1510134048812070,\n 906263674192061,\n 1821064197805734,\n 565375124676301,\n ]),\n xy2d: FieldElement51::from_limbs([\n 578027192365650,\n 2034800251375322,\n 2128954087207123,\n 478816193810521,\n 2196171989962750,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1633188840273120,\n 3104586986058956,\n 1548762607215795,\n 1266275218902681,\n 3359018017010381,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 462189358480054,\n 1784816734159228,\n 1611334301651368,\n 1303938263943540,\n 707589560319424,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1038829280972848,\n 38176604650029,\n 753193246598573,\n 1136076426528122,\n 595709990562434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3660251634545082,\n 2194984964010832,\n 2198361797561729,\n 1061962440055713,\n 1645147963442934,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 4701053362120,\n 1647641066302348,\n 1047553002242085,\n 1923635013395977,\n 206970314902065,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1750479161778571,\n 1362553355169293,\n 1891721260220598,\n 966109370862782,\n 1024913988299801,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2464498862816952,\n 1117950018299774,\n 1873945661751056,\n 3655602735669306,\n 2382695896337945,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 636808533673210,\n 1262201711667560,\n 390951380330599,\n 1663420692697294,\n 561951321757406,\n ]),\n xy2d: FieldElement51::from_limbs([\n 520731594438141,\n 1446301499955692,\n 273753264629267,\n 1565101517999256,\n 1019411827004672,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3178327305714638,\n 3443653291096626,\n 734233225181170,\n 2435838701226518,\n 4042225960010590,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1464651961852572,\n 1483737295721717,\n 1519450561335517,\n 1161429831763785,\n 405914998179977,\n ]),\n xy2d: FieldElement51::from_limbs([\n 996126634382301,\n 796204125879525,\n 127517800546509,\n 344155944689303,\n 615279846169038,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2990523894660505,\n 2188666632415295,\n 1961313708559162,\n 1506545807547587,\n 3403101452654988,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622917337413835,\n 1218989177089035,\n 1284857712846592,\n 970502061709359,\n 351025208117090,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2067814584765580,\n 1677855129927492,\n 2086109782475197,\n 235286517313238,\n 1416314046739645,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2838644076315587,\n 2559244195637442,\n 458399356043425,\n 2853867838192310,\n 3280348017100490,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 678489922928203,\n 2016657584724032,\n 90977383049628,\n 1026831907234582,\n 615271492942522,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301225714012278,\n 1094837270268560,\n 1202288391010439,\n 644352775178361,\n 1647055902137983,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1210746697896459,\n 1416608304244708,\n 2938287290903104,\n 3496931005119382,\n 3303038150540984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1135604073198207,\n 1683322080485474,\n 769147804376683,\n 2086688130589414,\n 900445683120379,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1971518477615628,\n 401909519527336,\n 448627091057375,\n 1409486868273821,\n 1214789035034363,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1364039144731711,\n 1897497433586190,\n 2203097701135459,\n 2397261210496499,\n 1349844460790698,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1045230323257973,\n 818206601145807,\n 630513189076103,\n 1672046528998132,\n 807204017562437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 439961968385997,\n 386362664488986,\n 1382706320807688,\n 309894000125359,\n 2207801346498567,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3480804500082836,\n 3172443782216110,\n 2375775707596425,\n 2933223806901024,\n 1400559197080972,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2003766096898049,\n 170074059235165,\n 1141124258967971,\n 1485419893480973,\n 1573762821028725,\n ]),\n xy2d: FieldElement51::from_limbs([\n 729905708611432,\n 1270323270673202,\n 123353058984288,\n 426460209632942,\n 2195574535456672,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1271140255321216,\n 2044363183174497,\n 2303925201319937,\n 3696920060379952,\n 3194341800024331,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1761608437466135,\n 583360847526804,\n 1586706389685493,\n 2157056599579261,\n 1170692369685772,\n ]),\n xy2d: FieldElement51::from_limbs([\n 871476219910823,\n 1878769545097794,\n 2241832391238412,\n 548957640601001,\n 690047440233174,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2548994545820755,\n 1366347803776819,\n 3552985325930849,\n 561849853336293,\n 1533554921345731,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 999628998628371,\n 1132836708493400,\n 2084741674517453,\n 469343353015612,\n 678782988708035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2189427607417022,\n 699801937082607,\n 412764402319267,\n 1478091893643349,\n 2244675696854460,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3964091869651792,\n 2456213404310121,\n 3657538451018088,\n 2660781114515010,\n 3112882032961968,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 508561155940631,\n 966928475686665,\n 2236717801150132,\n 424543858577297,\n 2089272956986143,\n ]),\n xy2d: FieldElement51::from_limbs([\n 221245220129925,\n 1156020201681217,\n 491145634799213,\n 542422431960839,\n 828100817819207,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2405556784925632,\n 1299874139923976,\n 2644898978945750,\n 1058234455773021,\n 996989038681183,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 559086812798481,\n 573177704212711,\n 1629737083816402,\n 1399819713462595,\n 1646954378266038,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1887963056288059,\n 228507035730124,\n 1468368348640282,\n 930557653420194,\n 613513962454686,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1224529808187534,\n 1577022856702685,\n 2206946542980843,\n 625883007765001,\n 2531730607197406,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1076287717051609,\n 1114455570543035,\n 187297059715481,\n 250446884292121,\n 1885187512550540,\n ]),\n xy2d: FieldElement51::from_limbs([\n 902497362940219,\n 76749815795675,\n 1657927525633846,\n 1420238379745202,\n 1340321636548352,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1129576631190765,\n 3533793823712575,\n 996844254743017,\n 2509676177174497,\n 3402650555740265,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 628740660038789,\n 1943038498527841,\n 467786347793886,\n 1093341428303375,\n 235413859513003,\n ]),\n xy2d: FieldElement51::from_limbs([\n 237425418909360,\n 469614029179605,\n 1512389769174935,\n 1241726368345357,\n 441602891065214,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3988217766743784,\n 726531315520507,\n 1833335034432527,\n 1629442561574747,\n 2876218732971333,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1960754663920689,\n 497040957888962,\n 1909832851283095,\n 1271432136996826,\n 2219780368020940,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1537037379417136,\n 1358865369268262,\n 2130838645654099,\n 828733687040705,\n 1999987652890901,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 629042105241795,\n 1098854999137608,\n 887281544569320,\n 3674901833560025,\n 2259711072636808,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1811562332665373,\n 1501882019007673,\n 2213763501088999,\n 359573079719636,\n 36370565049116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 218907117361280,\n 1209298913016966,\n 1944312619096112,\n 1130690631451061,\n 1342327389191701,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1369976867854685,\n 1396479602419169,\n 4017456468084104,\n 2203659200586298,\n 3250127649802489,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2230701885562825,\n 1348173180338974,\n 2172856128624598,\n 1426538746123771,\n 444193481326151,\n ]),\n xy2d: FieldElement51::from_limbs([\n 784210426627951,\n 918204562375674,\n 1284546780452985,\n 1324534636134684,\n 1872449409642708,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2571438643225542,\n 2848082470493653,\n 2037902696412607,\n 1557219121643918,\n 341938082688094,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1901860206695915,\n 2004489122065736,\n 1625847061568236,\n 973529743399879,\n 2075287685312905,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1371853944110545,\n 1042332820512553,\n 1949855697918254,\n 1791195775521505,\n 37487364849293,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 687200189577836,\n 1082536651125675,\n 2896024754556794,\n 2592723009743198,\n 2595381160432643,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2082717129583892,\n 27829425539422,\n 145655066671970,\n 1690527209845512,\n 1865260509673478,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1059729620568824,\n 2163709103470266,\n 1440302280256872,\n 1769143160546397,\n 869830310425069,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3861316033464273,\n 777277757338816,\n 2101121130363987,\n 550762194946473,\n 1905542338659364,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2024821921041576,\n 426948675450149,\n 595133284085473,\n 471860860885970,\n 600321679413000,\n ]),\n xy2d: FieldElement51::from_limbs([\n 598474602406721,\n 1468128276358244,\n 1191923149557635,\n 1501376424093216,\n 1281662691293476,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1721138489890688,\n 1264336102277790,\n 2684864359106535,\n 1359988423149465,\n 3813671107094695,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 719520245587143,\n 393380711632345,\n 132350400863381,\n 1543271270810729,\n 1819543295798660,\n ]),\n xy2d: FieldElement51::from_limbs([\n 396397949784152,\n 1811354474471839,\n 1362679985304303,\n 2117033964846756,\n 498041172552279,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1812471844975748,\n 1856491995543149,\n 126579494584102,\n 3288044672967868,\n 1975108050082549,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 650623932407995,\n 1137551288410575,\n 2125223403615539,\n 1725658013221271,\n 2134892965117796,\n ]),\n xy2d: FieldElement51::from_limbs([\n 522584000310195,\n 1241762481390450,\n 1743702789495384,\n 2227404127826575,\n 1686746002148897,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 427904865186293,\n 1703211129693455,\n 1585368107547509,\n 3688784302429584,\n 3012988348299225,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 318101947455002,\n 248138407995851,\n 1481904195303927,\n 309278454311197,\n 1258516760217879,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1275068538599310,\n 513726919533379,\n 349926553492294,\n 688428871968420,\n 1702400196000666,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3313663849950481,\n 3213411074010628,\n 2573659446386085,\n 3297400443644764,\n 1985130202504037,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1558816436882417,\n 1962896332636523,\n 1337709822062152,\n 1501413830776938,\n 294436165831932,\n ]),\n xy2d: FieldElement51::from_limbs([\n 818359826554971,\n 1862173000996177,\n 626821592884859,\n 573655738872376,\n 1749691246745455,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1988022651432119,\n 3333911312271288,\n 1834020786104820,\n 3706626690108935,\n 692929915223121,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2146513703733331,\n 584788900394667,\n 464965657279958,\n 2183973639356127,\n 238371159456790,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1129007025494441,\n 2197883144413266,\n 265142755578169,\n 971864464758890,\n 1983715884903702,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1291366624493056,\n 2633256531874362,\n 1711482489312443,\n 1815233647702022,\n 3144079596677715,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 444548969917454,\n 1452286453853356,\n 2113731441506810,\n 645188273895859,\n 810317625309512,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2242724082797924,\n 1373354730327868,\n 1006520110883049,\n 2147330369940688,\n 1151816104883620,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3997520014069025,\n 4163522956860564,\n 2056329390702073,\n 2607026987995097,\n 3131032608056347,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 163723479936298,\n 115424889803150,\n 1156016391581227,\n 1894942220753364,\n 1970549419986329,\n ]),\n xy2d: FieldElement51::from_limbs([\n 681981452362484,\n 267208874112496,\n 1374683991933094,\n 638600984916117,\n 646178654558546,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2265178468539480,\n 2358037120714814,\n 1944412051589650,\n 4093776581610705,\n 2482502633520820,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 260683893467075,\n 854060306077237,\n 913639551980112,\n 4704576840123,\n 280254810808712,\n ]),\n xy2d: FieldElement51::from_limbs([\n 715374893080287,\n 1173334812210491,\n 1806524662079626,\n 1894596008000979,\n 398905715033393,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2751826223412909,\n 3848231101880618,\n 1420380351989369,\n 3237011375206737,\n 392444930785632,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2096421546958141,\n 1922523000950363,\n 789831022876840,\n 427295144688779,\n 320923973161730,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1927770723575450,\n 1485792977512719,\n 1850996108474547,\n 551696031508956,\n 2126047405475647,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2112099158080129,\n 2994370617594963,\n 2258284371762679,\n 1951119898618915,\n 2344890196388664,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 383905201636970,\n 859946997631870,\n 855623867637644,\n 1017125780577795,\n 794250831877809,\n ]),\n xy2d: FieldElement51::from_limbs([\n 77571826285752,\n 999304298101753,\n 487841111777762,\n 1038031143212339,\n 339066367948762,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2926794589205781,\n 2517835660016036,\n 826951213393477,\n 1405007746162285,\n 1781791018620876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1001412661522686,\n 348196197067298,\n 1666614366723946,\n 888424995032760,\n 580747687801357,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1939560076207777,\n 1409892634407635,\n 552574736069277,\n 383854338280405,\n 190706709864139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2177087163428741,\n 1439255351721944,\n 3459870654068041,\n 2230616362004768,\n 1396886392021913,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 676962063230039,\n 1880275537148808,\n 2046721011602706,\n 888463247083003,\n 1318301552024067,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1466980508178206,\n 617045217998949,\n 652303580573628,\n 757303753529064,\n 207583137376902,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3762856566592150,\n 2357202940576524,\n 2745234706458093,\n 1091943425335975,\n 1802717338077427,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1853982405405128,\n 1878664056251147,\n 1528011020803992,\n 1019626468153565,\n 1128438412189035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1963939888391106,\n 293456433791664,\n 697897559513649,\n 985882796904380,\n 796244541237972,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2668570812315008,\n 2641455366112301,\n 1314476859406755,\n 1749382513022778,\n 3413705412424739,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1428358296490651,\n 1027115282420478,\n 304840698058337,\n 441410174026628,\n 1819358356278573,\n ]),\n xy2d: FieldElement51::from_limbs([\n 204943430200135,\n 1554861433819175,\n 216426658514651,\n 264149070665950,\n 2047097371738319,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1934415182909015,\n 1393285083565062,\n 2768209145458208,\n 3409490548679139,\n 2372839480279515,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 662035583584445,\n 286736105093098,\n 1131773000510616,\n 818494214211439,\n 472943792054479,\n ]),\n xy2d: FieldElement51::from_limbs([\n 665784778135882,\n 1893179629898606,\n 808313193813106,\n 276797254706413,\n 1563426179676396,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 945205108984213,\n 2778077376644543,\n 1324180513733565,\n 1666970227868664,\n 2405347422974421,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2031433403516252,\n 203996615228162,\n 170487168837083,\n 981513604791390,\n 843573964916831,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1476570093962618,\n 838514669399805,\n 1857930577281364,\n 2017007352225784,\n 317085545220047,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1461557121912823,\n 1600674043318359,\n 2157134900399597,\n 1670641601940616,\n 2379565397488531,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1293543509393474,\n 2143624609202546,\n 1058361566797508,\n 214097127393994,\n 946888515472729,\n ]),\n xy2d: FieldElement51::from_limbs([\n 357067959932916,\n 1290876214345711,\n 521245575443703,\n 1494975468601005,\n 800942377643885,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2817916472785262,\n 820247422481739,\n 994464017954148,\n 2578957425371613,\n 2344391131796991,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 617256647603209,\n 1652107761099439,\n 1857213046645471,\n 1085597175214970,\n 817432759830522,\n ]),\n xy2d: FieldElement51::from_limbs([\n 771808161440705,\n 1323510426395069,\n 680497615846440,\n 851580615547985,\n 1320806384849017,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1219260086131896,\n 2898968820282063,\n 2331400938444953,\n 2161724213426747,\n 2656661710745446,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1327968293887866,\n 1335500852943256,\n 1401587164534264,\n 558137311952440,\n 1551360549268902,\n ]),\n xy2d: FieldElement51::from_limbs([\n 417621685193956,\n 1429953819744454,\n 396157358457099,\n 1940470778873255,\n 214000046234152,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1268047918491954,\n 2172375426948536,\n 1533916099229249,\n 1761293575457130,\n 3842422480712013,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1627072914981959,\n 2211603081280073,\n 1912369601616504,\n 1191770436221309,\n 2187309757525860,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1149147819689533,\n 378692712667677,\n 828475842424202,\n 2218619146419342,\n 70688125792186,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3551539230764990,\n 3690416477138006,\n 3788528892189659,\n 2053896748919837,\n 3260220846276494,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2040723824657366,\n 399555637875075,\n 632543375452995,\n 872649937008051,\n 1235394727030233,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2211311599327900,\n 2139787259888175,\n 938706616835350,\n 12609661139114,\n 2081897930719789,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1324994503390431,\n 2588782144267879,\n 1183998925654176,\n 3343454479598522,\n 2300527487656566,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1845522914617879,\n 1222198248335542,\n 150841072760134,\n 1927029069940982,\n 1189913404498011,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1079559557592645,\n 2215338383666441,\n 1903569501302605,\n 49033973033940,\n 305703433934152,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2346453219102138,\n 3637921163538246,\n 3313930291577009,\n 2288353761164521,\n 3085469462634093,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1432015813136298,\n 440364795295369,\n 1395647062821501,\n 1976874522764578,\n 934452372723352,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1296625309219774,\n 2068273464883862,\n 1858621048097805,\n 1492281814208508,\n 2235868981918946,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1490330266465551,\n 1858795661361448,\n 3688040948655011,\n 2546373032584894,\n 3459939824714180,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1282462923712748,\n 741885683986255,\n 2027754642827561,\n 518989529541027,\n 1826610009555945,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1525827120027511,\n 723686461809551,\n 1597702369236987,\n 244802101764964,\n 1502833890372311,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2365421849929742,\n 3485539881431101,\n 2925909765963743,\n 2114345180342964,\n 2418564326541511,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2041668749310338,\n 2184405322203901,\n 1633400637611036,\n 2110682505536899,\n 2048144390084644,\n ]),\n xy2d: FieldElement51::from_limbs([\n 503058759232932,\n 760293024620937,\n 2027152777219493,\n 666858468148475,\n 1539184379870952,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1916168475367211,\n 3167426246226591,\n 883217071712574,\n 363427871374304,\n 1976029821251593,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 678039535434506,\n 570587290189340,\n 1605302676614120,\n 2147762562875701,\n 1706063797091704,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1439489648586438,\n 2194580753290951,\n 832380563557396,\n 561521973970522,\n 584497280718389,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2439789269177838,\n 681223515948274,\n 1933493571072456,\n 1872921007304880,\n 2739962177820919,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1413466089534451,\n 410844090765630,\n 1397263346404072,\n 408227143123410,\n 1594561803147811,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2102170800973153,\n 719462588665004,\n 1479649438510153,\n 1097529543970028,\n 1302363283777685,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3193865531532443,\n 3321113493038208,\n 2007341951411050,\n 2322773230131539,\n 1419433790163705,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1146565545556377,\n 1661971299445212,\n 406681704748893,\n 564452436406089,\n 1109109865829139,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2214421081775077,\n 1165671861210569,\n 1890453018796184,\n 3556249878661,\n 442116172656317,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3005630360306059,\n 1666955059895018,\n 1530775289309243,\n 3371786842789394,\n 2164156153857579,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 615171919212796,\n 1523849404854568,\n 854560460547503,\n 2067097370290715,\n 1765325848586042,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1094538949313667,\n 1796592198908825,\n 870221004284388,\n 2025558921863561,\n 1699010892802384,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1951351290725195,\n 1916457206844795,\n 2449824998123274,\n 1909076887557594,\n 1938542290318919,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1014323197538413,\n 869150639940606,\n 1756009942696599,\n 1334952557375672,\n 1544945379082874,\n ]),\n xy2d: FieldElement51::from_limbs([\n 764055910920305,\n 1603590757375439,\n 146805246592357,\n 1843313433854297,\n 954279890114939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 80113526615731,\n 764536758732259,\n 3306939158785481,\n 2721052465444637,\n 2869697326116762,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 74497112547268,\n 740094153192149,\n 1745254631717581,\n 727713886503130,\n 1283034364416928,\n ]),\n xy2d: FieldElement51::from_limbs([\n 525892105991110,\n 1723776830270342,\n 1476444848991936,\n 573789489857760,\n 133864092632978,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2794411533877810,\n 1986812262899320,\n 1162535242465837,\n 2733298779828712,\n 2796400347268869,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 64123227344372,\n 1239927720647794,\n 1360722983445904,\n 222610813654661,\n 62429487187991,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1793193323953132,\n 91096687857833,\n 70945970938921,\n 2158587638946380,\n 1537042406482111,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1895854577604590,\n 3646695522634664,\n 1728548428495943,\n 3392664713925397,\n 2815445147288308,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 141358280486863,\n 91435889572504,\n 1087208572552643,\n 1829599652522921,\n 1193307020643647,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1611230858525381,\n 950720175540785,\n 499589887488610,\n 2001656988495019,\n 88977313255908,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3440880315164906,\n 2184348804772596,\n 3292618539427567,\n 2018318290311833,\n 1712060030915354,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 873966876953756,\n 1090638350350440,\n 1708559325189137,\n 672344594801910,\n 1320437969700239,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1508590048271766,\n 1131769479776094,\n 101550868699323,\n 428297785557897,\n 561791648661744,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3008217384184691,\n 2489682092917849,\n 2136263418594015,\n 1701968045454886,\n 2955512998822720,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1781187809325462,\n 1697624151492346,\n 1381393690939988,\n 175194132284669,\n 1483054666415238,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2175517777364616,\n 708781536456029,\n 955668231122942,\n 1967557500069555,\n 2021208005604118,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3366935780292116,\n 2476017186636029,\n 915967306279221,\n 593866251291540,\n 2813546907893254,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1443163092879439,\n 391875531646162,\n 2180847134654632,\n 464538543018753,\n 1594098196837178,\n ]),\n xy2d: FieldElement51::from_limbs([\n 850858855888869,\n 319436476624586,\n 327807784938441,\n 740785849558761,\n 17128415486016,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2132756334090048,\n 2788047633840893,\n 2300706964962114,\n 2860273011285942,\n 3513489358708031,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1525176236978354,\n 974205476721062,\n 293436255662638,\n 148269621098039,\n 137961998433963,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1121075518299410,\n 2071745529082111,\n 1265567917414828,\n 1648196578317805,\n 496232102750820,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2374121042985030,\n 3274721891178932,\n 2001275453369483,\n 2017441881607947,\n 3245005694463250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 654925550560074,\n 1168810995576858,\n 575655959430926,\n 905758704861388,\n 496774564663534,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1954109525779738,\n 2117022646152485,\n 338102630417180,\n 1194140505732026,\n 107881734943492,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1714785840001267,\n 4288299832366837,\n 1876380234251965,\n 2056717182974196,\n 1645855254384642,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 106431476499341,\n 62482972120563,\n 1513446655109411,\n 807258751769522,\n 538491469114,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2002850762893643,\n 1243624520538135,\n 1486040410574605,\n 2184752338181213,\n 378495998083531,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 922510868424903,\n 1089502620807680,\n 402544072617374,\n 1131446598479839,\n 1290278588136533,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1867998812076769,\n 715425053580701,\n 39968586461416,\n 2173068014586163,\n 653822651801304,\n ]),\n xy2d: FieldElement51::from_limbs([\n 162892278589453,\n 182585796682149,\n 75093073137630,\n 497037941226502,\n 133871727117371,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4166396390264918,\n 1608999621851577,\n 1987629837704609,\n 1519655314857977,\n 1819193753409464,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1949315551096831,\n 1069003344994464,\n 1939165033499916,\n 1548227205730856,\n 1933767655861407,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1730519386931635,\n 1393284965610134,\n 1597143735726030,\n 416032382447158,\n 1429665248828629,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 360275475604546,\n 2799635544748326,\n 2467160717872776,\n 2848446553564254,\n 2584509464110332,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 47602113726801,\n 1522314509708010,\n 437706261372925,\n 814035330438027,\n 335930650933545,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1291597595523886,\n 1058020588994081,\n 402837842324045,\n 1363323695882781,\n 2105763393033193,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2361321796251793,\n 3967057562270386,\n 1112231216891515,\n 2046641005101484,\n 2386048970842261,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2156991030936798,\n 2227544497153325,\n 1869050094431622,\n 754875860479115,\n 1754242344267058,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1846089562873800,\n 98894784984326,\n 1412430299204844,\n 171351226625762,\n 1100604760929008,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2335972195815721,\n 2751510784385293,\n 425749630620777,\n 1762872794206857,\n 2864642415813208,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 868309334532756,\n 1703010512741873,\n 1952690008738057,\n 4325269926064,\n 2071083554962116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 523094549451158,\n 401938899487815,\n 1407690589076010,\n 2022387426254453,\n 158660516411257,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 612867287630009,\n 2700012425789062,\n 2823428891104443,\n 1466796750919375,\n 1728478129663858,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1723848973783452,\n 2208822520534681,\n 1718748322776940,\n 1974268454121942,\n 1194212502258141,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1254114807944608,\n 977770684047110,\n 2010756238954993,\n 1783628927194099,\n 1525962994408256,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2484263871921055,\n 1948628555342433,\n 1835348780427694,\n 1031609499437291,\n 2316271920603621,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 767338676040683,\n 754089548318405,\n 1523192045639075,\n 435746025122062,\n 512692508440385,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1255955808701983,\n 1700487367990941,\n 1166401238800299,\n 1175121994891534,\n 1190934801395380,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2600943821853521,\n 1337012557669161,\n 1475912332999108,\n 3573418268585706,\n 2299411105589567,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 877519947135419,\n 2172838026132651,\n 272304391224129,\n 1655143327559984,\n 886229406429814,\n ]),\n xy2d: FieldElement51::from_limbs([\n 375806028254706,\n 214463229793940,\n 572906353144089,\n 572168269875638,\n 697556386112979,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1168827102357825,\n 823864273033637,\n 4323338565789945,\n 788062026895923,\n 2851378154428610,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1948116082078088,\n 2054898304487796,\n 2204939184983900,\n 210526805152138,\n 786593586607626,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1915320147894736,\n 156481169009469,\n 655050471180417,\n 592917090415421,\n 2165897438660879,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1726336468579724,\n 1119932070398949,\n 1929199510967666,\n 2285718602008207,\n 1836837863503149,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 829996854845988,\n 217061778005138,\n 1686565909803640,\n 1346948817219846,\n 1723823550730181,\n ]),\n xy2d: FieldElement51::from_limbs([\n 384301494966394,\n 687038900403062,\n 2211195391021739,\n 254684538421383,\n 1245698430589680,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1247567493562669,\n 4229981908141095,\n 2435671288478202,\n 806570235643434,\n 2540261331753164,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1449077384734201,\n 38285445457996,\n 2136537659177832,\n 2146493000841573,\n 725161151123125,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1201928866368855,\n 800415690605445,\n 1703146756828343,\n 997278587541744,\n 1858284414104014,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2608268623334125,\n 3034173730618399,\n 1718002439402869,\n 3644022065904502,\n 663171266061950,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 759628738230460,\n 1012693474275852,\n 353780233086498,\n 246080061387552,\n 2030378857679162,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2040672435071076,\n 888593182036908,\n 1298443657189359,\n 1804780278521327,\n 354070726137060,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1894938527423184,\n 3715012855162525,\n 2726210319182898,\n 2499094776718546,\n 877975941029127,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 207937160991127,\n 12966911039119,\n 820997788283092,\n 1010440472205286,\n 1701372890140810,\n ]),\n xy2d: FieldElement51::from_limbs([\n 218882774543183,\n 533427444716285,\n 1233243976733245,\n 435054256891319,\n 1509568989549904,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4140638349397055,\n 3303977572025869,\n 3465353617009382,\n 2420981822812579,\n 2715174081801119,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 299137589460312,\n 1594371588983567,\n 868058494039073,\n 257771590636681,\n 1805012993142921,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1806842755664364,\n 2098896946025095,\n 1356630998422878,\n 1458279806348064,\n 347755825962072,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1402334161391744,\n 3811883484731547,\n 1008585416617746,\n 1147797150908892,\n 1420416683642459,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 665506704253369,\n 273770475169863,\n 799236974202630,\n 848328990077558,\n 1811448782807931,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1468412523962641,\n 771866649897997,\n 1931766110147832,\n 799561180078482,\n 524837559150077,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2223212657821831,\n 2882216061048914,\n 2144451165500327,\n 3068710944633039,\n 3276150872095279,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1266603897524861,\n 156378408858100,\n 1275649024228779,\n 447738405888420,\n 253186462063095,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2022215964509735,\n 136144366993649,\n 1800716593296582,\n 1193970603800203,\n 871675847064218,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1862751661970309,\n 851596246739884,\n 1519315554814041,\n 3794598280232697,\n 3669775149586767,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1228168094547481,\n 334133883362894,\n 587567568420081,\n 433612590281181,\n 603390400373205,\n ]),\n xy2d: FieldElement51::from_limbs([\n 121893973206505,\n 1843345804916664,\n 1703118377384911,\n 497810164760654,\n 101150811654673,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2710146069631716,\n 2542709749304591,\n 1452768413850678,\n 2802722688939463,\n 1537286854336537,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 584322311184395,\n 380661238802118,\n 114839394528060,\n 655082270500073,\n 2111856026034852,\n ]),\n xy2d: FieldElement51::from_limbs([\n 996965581008991,\n 2148998626477022,\n 1012273164934654,\n 1073876063914522,\n 1688031788934939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3175286832534829,\n 2085106799623354,\n 2779882615305384,\n 1606206360876187,\n 2987706905397772,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1697697887804317,\n 1335343703828273,\n 831288615207040,\n 949416685250051,\n 288760277392022,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1419122478109648,\n 1325574567803701,\n 602393874111094,\n 2107893372601700,\n 1314159682671307,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2201150872731785,\n 2180241023425241,\n 2349463270108411,\n 1633405770247823,\n 3100744856129234,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1173339555550611,\n 818605084277583,\n 47521504364289,\n 924108720564965,\n 735423405754506,\n ]),\n xy2d: FieldElement51::from_limbs([\n 830104860549448,\n 1886653193241086,\n 1600929509383773,\n 1475051275443631,\n 286679780900937,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3828911108518224,\n 3282698983453994,\n 2396700729978777,\n 4216472406664814,\n 2820189914640497,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 278388655910247,\n 487143369099838,\n 927762205508727,\n 181017540174210,\n 1616886700741287,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1191033906638969,\n 940823957346562,\n 1606870843663445,\n 861684761499847,\n 658674867251089,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1875032594195527,\n 1427106132796197,\n 2976536204647406,\n 3153660325729987,\n 2887068310954007,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622869792298357,\n 1903919278950367,\n 1922588621661629,\n 1520574711600434,\n 1087100760174640,\n ]),\n xy2d: FieldElement51::from_limbs([\n 25465949416618,\n 1693639527318811,\n 1526153382657203,\n 125943137857169,\n 145276964043999,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2466539671654587,\n 920212862967914,\n 4191701364657517,\n 3463662605460468,\n 2336897329405367,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2006245852772938,\n 734762734836159,\n 254642929763427,\n 1406213292755966,\n 239303749517686,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1619678837192149,\n 1919424032779215,\n 1357391272956794,\n 1525634040073113,\n 1310226789796241,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3292563523447371,\n 1704449869235351,\n 2857062884141577,\n 1998838089036354,\n 1312142911487502,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1996723311435669,\n 1844342766567060,\n 985455700466044,\n 1165924681400960,\n 311508689870129,\n ]),\n xy2d: FieldElement51::from_limbs([\n 43173156290518,\n 2202883069785309,\n 1137787467085917,\n 1733636061944606,\n 1394992037553852,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 670078326344559,\n 2807454838744604,\n 2723759199967685,\n 2141455487356408,\n 849015953823125,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2197214573372804,\n 794254097241315,\n 1030190060513737,\n 267632515541902,\n 2040478049202624,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1812516004670529,\n 1609256702920783,\n 1706897079364493,\n 258549904773295,\n 996051247540686,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1540374301420565,\n 1764656898914615,\n 1810104162020396,\n 3175608592848336,\n 2916189887881826,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1323460699404750,\n 1262690757880991,\n 871777133477900,\n 1060078894988977,\n 1712236889662886,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1696163952057966,\n 1391710137550823,\n 608793846867416,\n 1034391509472039,\n 1780770894075012,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1367603834210822,\n 4383788460268472,\n 890353773628143,\n 1908908219165595,\n 2522636708938139,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 597536315471731,\n 40375058742586,\n 1942256403956049,\n 1185484645495932,\n 312666282024145,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1919411405316294,\n 1234508526402192,\n 1066863051997083,\n 1008444703737597,\n 1348810787701552,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2102881477513865,\n 3822074379630609,\n 1573617900503707,\n 2270462449417831,\n 2232324307922097,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1853931367696942,\n 8107973870707,\n 350214504129299,\n 775206934582587,\n 1752317649166792,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1417148368003523,\n 721357181628282,\n 505725498207811,\n 373232277872983,\n 261634707184480,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2186733281493248,\n 2250694917008620,\n 1014829812957440,\n 2731797975137637,\n 2335366007561721,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1268116367301224,\n 560157088142809,\n 802626839600444,\n 2210189936605713,\n 1129993785579988,\n ]),\n xy2d: FieldElement51::from_limbs([\n 615183387352312,\n 917611676109240,\n 878893615973325,\n 978940963313282,\n 938686890583575,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 522024729211672,\n 3296859129001056,\n 1892245413707789,\n 1907891107684253,\n 2059998109500714,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1799679152208884,\n 912132775900387,\n 25967768040979,\n 432130448590461,\n 274568990261996,\n ]),\n xy2d: FieldElement51::from_limbs([\n 98698809797682,\n 2144627600856209,\n 1907959298569602,\n 811491302610148,\n 1262481774981493,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1791451399743152,\n 1713538728337276,\n 2370149810942738,\n 1882306388849953,\n 158235232210248,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1217809823321928,\n 2173947284933160,\n 1986927836272325,\n 1388114931125539,\n 12686131160169,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1650875518872272,\n 1136263858253897,\n 1732115601395988,\n 734312880662190,\n 1252904681142109,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2624786269799113,\n 2777230729143418,\n 2116279931702134,\n 2753222527273063,\n 1907002872974924,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 803147181835288,\n 868941437997146,\n 316299302989663,\n 943495589630550,\n 571224287904572,\n ]),\n xy2d: FieldElement51::from_limbs([\n 227742695588364,\n 1776969298667369,\n 628602552821802,\n 457210915378118,\n 2041906378111140,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 815000523470260,\n 3164885502413555,\n 3303859931956420,\n 1345536665214222,\n 541623413135555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1580216071604333,\n 1877997504342444,\n 857147161260913,\n 703522726778478,\n 2182763974211603,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1870080310923419,\n 71988220958492,\n 1783225432016732,\n 615915287105016,\n 1035570475990230,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2982787564515398,\n 857613889540279,\n 1083813157271766,\n 1002817255970169,\n 1719228484436074,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 377616581647602,\n 1581980403078513,\n 804044118130621,\n 2034382823044191,\n 643844048472185,\n ]),\n xy2d: FieldElement51::from_limbs([\n 176957326463017,\n 1573744060478586,\n 528642225008045,\n 1816109618372371,\n 1515140189765006,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1888911448245718,\n 3638910709296328,\n 4176303607751676,\n 1731539523700948,\n 2230378382645454,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 443392177002051,\n 233793396845137,\n 2199506622312416,\n 1011858706515937,\n 974676837063129,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1846351103143623,\n 1949984838808427,\n 671247021915253,\n 1946756846184401,\n 1929296930380217,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 849646212451983,\n 1410198775302919,\n 2325567699868943,\n 1641663456615811,\n 3014056086137659,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 692017667358279,\n 723305578826727,\n 1638042139863265,\n 748219305990306,\n 334589200523901,\n ]),\n xy2d: FieldElement51::from_limbs([\n 22893968530686,\n 2235758574399251,\n 1661465835630252,\n 925707319443452,\n 1203475116966621,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3053098849470395,\n 3985092410411378,\n 1664508947088595,\n 2719548934677170,\n 3899298398220870,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 903105258014366,\n 427141894933047,\n 561187017169777,\n 1884330244401954,\n 1914145708422219,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1344191060517578,\n 1960935031767890,\n 1518838929955259,\n 1781502350597190,\n 1564784025565682,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2925523165433334,\n 1979969272514922,\n 3427087126180756,\n 1187589090978665,\n 1881897672213940,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1917185587363432,\n 1098342571752737,\n 5935801044414,\n 2000527662351839,\n 1538640296181569,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2495540013192,\n 678856913479236,\n 224998292422872,\n 219635787698590,\n 1972465269000940,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 271413961212179,\n 3604851875156899,\n 2596511104968730,\n 2014925838520661,\n 2006221033113941,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 194583029968109,\n 514316781467765,\n 829677956235672,\n 1676415686873082,\n 810104584395840,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1980510813313589,\n 1948645276483975,\n 152063780665900,\n 129968026417582,\n 256984195613935,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1860190562533083,\n 1936576191345085,\n 2712900106391212,\n 1811043097042829,\n 3209286562992083,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 796664815624365,\n 1543160838872951,\n 1500897791837765,\n 1667315977988401,\n 599303877030711,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1151480509533204,\n 2136010406720455,\n 738796060240027,\n 319298003765044,\n 1150614464349587,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1731069268103131,\n 2987442261301335,\n 1364750481334267,\n 2669032653668119,\n 3178908082812908,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1017222050227968,\n 1987716148359,\n 2234319589635701,\n 621282683093392,\n 2132553131763026,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1567828528453324,\n 1017807205202360,\n 565295260895298,\n 829541698429100,\n 307243822276582,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 249079270936229,\n 1501514259790706,\n 3199709537890096,\n 944551802437486,\n 2804458577667728,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2089966982947227,\n 1854140343916181,\n 2151980759220007,\n 2139781292261749,\n 158070445864917,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1338766321464554,\n 1906702607371284,\n 1519569445519894,\n 115384726262267,\n 1393058953390992,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3616421371950629,\n 3764188048593604,\n 1926731583198685,\n 2041482526432505,\n 3172200936019022,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1884844597333588,\n 601480070269079,\n 620203503079537,\n 1079527400117915,\n 1202076693132015,\n ]),\n xy2d: FieldElement51::from_limbs([\n 840922919763324,\n 727955812569642,\n 1303406629750194,\n 522898432152867,\n 294161410441865,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2605560604520539,\n 1598361541848742,\n 3374705511887547,\n 4174333403844152,\n 2670907514351827,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 359856369838236,\n 180914355488683,\n 861726472646627,\n 218807937262986,\n 575626773232501,\n ]),\n xy2d: FieldElement51::from_limbs([\n 755467689082474,\n 909202735047934,\n 730078068932500,\n 936309075711518,\n 2007798262842972,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1609384177904054,\n 2614544999293875,\n 1335318541768200,\n 3052765584121496,\n 2799677792952659,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 984339177776787,\n 815727786505884,\n 1645154585713747,\n 1659074964378553,\n 1686601651984156,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1697863093781930,\n 599794399429786,\n 1104556219769607,\n 830560774794755,\n 12812858601017,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1168737550514982,\n 897832437380552,\n 463140296333799,\n 2554364413707795,\n 2008360505135500,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1856930662813910,\n 678090852002597,\n 1920179140755167,\n 1259527833759868,\n 55540971895511,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1158643631044921,\n 476554103621892,\n 178447851439725,\n 1305025542653569,\n 103433927680625,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2176793111709008,\n 3828525530035639,\n 2009350167273522,\n 2012390194631546,\n 2125297410909580,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 825403285195098,\n 2144208587560784,\n 1925552004644643,\n 1915177840006985,\n 1015952128947864,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1807108316634472,\n 1534392066433717,\n 347342975407218,\n 1153820745616376,\n 7375003497471,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3234860815484973,\n 2683011703586488,\n 2201903782961092,\n 3069193724749589,\n 2214616493042166,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 228567918409756,\n 865093958780220,\n 358083886450556,\n 159617889659320,\n 1360637926292598,\n ]),\n xy2d: FieldElement51::from_limbs([\n 234147501399755,\n 2229469128637390,\n 2175289352258889,\n 1397401514549353,\n 1885288963089922,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3363562226636810,\n 2504649386192636,\n 3300514047508588,\n 2397910909286693,\n 1237505378776769,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1113790697840279,\n 1051167139966244,\n 1045930658550944,\n 2011366241542643,\n 1686166824620755,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1054097349305049,\n 1872495070333352,\n 182121071220717,\n 1064378906787311,\n 100273572924182,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3558210666856834,\n 1627717417672446,\n 2302783034773665,\n 1109249951172249,\n 3122001602766640,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 104233794644221,\n 1548919791188248,\n 2224541913267306,\n 2054909377116478,\n 1043803389015153,\n ]),\n xy2d: FieldElement51::from_limbs([\n 216762189468802,\n 707284285441622,\n 190678557969733,\n 973969342604308,\n 1403009538434867,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3530824104723725,\n 2596576648903557,\n 2525521909702446,\n 4086000250496689,\n 634517197663803,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 343805853118335,\n 1302216857414201,\n 566872543223541,\n 2051138939539004,\n 321428858384280,\n ]),\n xy2d: FieldElement51::from_limbs([\n 470067171324852,\n 1618629234173951,\n 2000092177515639,\n 7307679772789,\n 1117521120249968,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2529951391976704,\n 1810282338562946,\n 1771599529530998,\n 3635459223356879,\n 2937173228157088,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 577009397403102,\n 1791440261786291,\n 2177643735971638,\n 174546149911960,\n 1412505077782326,\n ]),\n xy2d: FieldElement51::from_limbs([\n 893719721537457,\n 1201282458018197,\n 1522349501711173,\n 58011597740583,\n 1130406465887139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 412607348255434,\n 1280455764199780,\n 2233277987330768,\n 2265979894086913,\n 2583384512102412,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 262483770854550,\n 990511055108216,\n 526885552771698,\n 571664396646158,\n 354086190278723,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1820352417585487,\n 24495617171480,\n 1547899057533253,\n 10041836186225,\n 480457105094042,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2023310314989233,\n 2889705151211129,\n 2106474638900686,\n 2809620524769320,\n 1687858215057825,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1144168702609745,\n 604444390410187,\n 1544541121756138,\n 1925315550126027,\n 626401428894002,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1922168257351784,\n 2018674099908659,\n 1776454117494445,\n 956539191509034,\n 36031129147635,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2796444352433270,\n 1039872944430373,\n 3128550222815858,\n 2962457525011798,\n 3468752501170219,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 58242421545916,\n 2035812695641843,\n 2118491866122923,\n 1191684463816273,\n 46921517454099,\n ]),\n xy2d: FieldElement51::from_limbs([\n 272268252444639,\n 1374166457774292,\n 2230115177009552,\n 1053149803909880,\n 1354288411641016,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1857910905368338,\n 1754729879288912,\n 3137745277795125,\n 1516096106802165,\n 1602902393369811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1193437069800958,\n 901107149704790,\n 999672920611411,\n 477584824802207,\n 364239578697845,\n ]),\n xy2d: FieldElement51::from_limbs([\n 886299989548838,\n 1538292895758047,\n 1590564179491896,\n 1944527126709657,\n 837344427345298,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3006358179063534,\n 1712186480903617,\n 3955456640022779,\n 3002110732175033,\n 2770795853936147,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1309847803895382,\n 1462151862813074,\n 211370866671570,\n 1544595152703681,\n 1027691798954090,\n ]),\n xy2d: FieldElement51::from_limbs([\n 803217563745370,\n 1884799722343599,\n 1357706345069218,\n 2244955901722095,\n 730869460037413,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2941099284981214,\n 1831210565161070,\n 3626987155270686,\n 3358084791231418,\n 1893781834054268,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 696351368613042,\n 1494385251239250,\n 738037133616932,\n 636385507851544,\n 927483222611406,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1949114198209333,\n 1104419699537997,\n 783495707664463,\n 1747473107602770,\n 2002634765788641,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1607325776830197,\n 2782683755100581,\n 1451089452727894,\n 3833490970768671,\n 496100432831153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1068900648804224,\n 2006891997072550,\n 1134049269345549,\n 1638760646180091,\n 2055396084625778,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2222475519314561,\n 1870703901472013,\n 1884051508440561,\n 1344072275216753,\n 1318025677799069,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 155711679280637,\n 681100400509288,\n 389811735211209,\n 2135723811340709,\n 2660533024889373,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 7813206966729,\n 194444201427550,\n 2071405409526507,\n 1065605076176312,\n 1645486789731291,\n ]),\n xy2d: FieldElement51::from_limbs([\n 16625790644959,\n 1647648827778410,\n 1579910185572704,\n 436452271048548,\n 121070048451050,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3289062842237779,\n 2820185594063076,\n 2549752917829677,\n 3810384325616458,\n 2238221839292470,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 190565267697443,\n 672855706028058,\n 338796554369226,\n 337687268493904,\n 853246848691734,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1763863028400139,\n 766498079432444,\n 1321118624818005,\n 69494294452268,\n 858786744165651,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3543856582248253,\n 1456632109855637,\n 3352431060735432,\n 1386133165675320,\n 3484698163879000,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 366253102478259,\n 525676242508811,\n 1449610995265438,\n 1183300845322183,\n 185960306491545,\n ]),\n xy2d: FieldElement51::from_limbs([\n 28315355815982,\n 460422265558930,\n 1799675876678724,\n 1969256312504498,\n 1051823843138725,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2408714813047231,\n 3857948219405196,\n 1665208410108429,\n 2569443092377519,\n 1383783705665319,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 54684536365732,\n 2210010038536222,\n 1194984798155308,\n 535239027773705,\n 1516355079301361,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1484387703771650,\n 198537510937949,\n 2186282186359116,\n 617687444857508,\n 647477376402122,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2147715541830533,\n 2751832352131065,\n 2898179830570073,\n 2604027669016369,\n 1488268620408051,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 159386186465542,\n 1877626593362941,\n 618737197060512,\n 1026674284330807,\n 1158121760792685,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1744544377739822,\n 1964054180355661,\n 1685781755873170,\n 2169740670377448,\n 1286112621104591,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2333777063470241,\n 3919742931398333,\n 3920783633320113,\n 1605016835177614,\n 1353960708075544,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1602253788689063,\n 439542044889886,\n 2220348297664483,\n 657877410752869,\n 157451572512238,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1029287186166717,\n 65860128430192,\n 525298368814832,\n 1491902500801986,\n 1461064796385400,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2660016802414475,\n 2121095722306988,\n 913562102267595,\n 1879708920318308,\n 2492861262121979,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1185483484383269,\n 1356339572588553,\n 584932367316448,\n 102132779946470,\n 1792922621116791,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1966196870701923,\n 2230044620318636,\n 1425982460745905,\n 261167817826569,\n 46517743394330,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2358877405280588,\n 3136759755857592,\n 2279106683482647,\n 2224911448949389,\n 3216151871930471,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1730194207717538,\n 431790042319772,\n 1831515233279467,\n 1372080552768581,\n 1074513929381760,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1450880638731607,\n 1019861580989005,\n 1229729455116861,\n 1174945729836143,\n 826083146840706,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1899935429242705,\n 1602068751520477,\n 940583196550370,\n 2334230882739107,\n 1540863155745695,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2136688454840028,\n 2099509000964294,\n 1690800495246475,\n 1217643678575476,\n 828720645084218,\n ]),\n xy2d: FieldElement51::from_limbs([\n 765548025667841,\n 462473984016099,\n 998061409979798,\n 546353034089527,\n 2212508972466858,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2298375097456408,\n 3144370785258318,\n 1281983193144089,\n 1491520128287375,\n 75847005908304,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1801436127943107,\n 1734436817907890,\n 1268728090345068,\n 167003097070711,\n 2233597765834956,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1997562060465113,\n 1048700225534011,\n 7615603985628,\n 1855310849546841,\n 2242557647635213,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1161017320376250,\n 2744424393854291,\n 2169815802355236,\n 3228296595417790,\n 1770879511019628,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1357044908364776,\n 729130645262438,\n 1762469072918979,\n 1365633616878458,\n 181282906404941,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1080413443139865,\n 1155205815510486,\n 1848782073549786,\n 622566975152580,\n 124965574467971,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1184526762066993,\n 247622751762817,\n 2943928830891604,\n 3071818503097743,\n 2188697339828084,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2020536369003019,\n 202261491735136,\n 1053169669150884,\n 2056531979272544,\n 778165514694311,\n ]),\n xy2d: FieldElement51::from_limbs([\n 237404399610207,\n 1308324858405118,\n 1229680749538400,\n 720131409105291,\n 1958958863624906,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2767383321724075,\n 2269456792542436,\n 1717918437373988,\n 1568052070792483,\n 2298775616809171,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 281527309158085,\n 36970532401524,\n 866906920877543,\n 2222282602952734,\n 1289598729589882,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1278207464902042,\n 494742455008756,\n 1262082121427081,\n 1577236621659884,\n 1888786707293291,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 353042527954210,\n 1830056151907359,\n 1111731275799225,\n 2426760769524072,\n 404312815582674,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2064251142068628,\n 1666421603389706,\n 1419271365315441,\n 468767774902855,\n 191535130366583,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1716987058588002,\n 1859366439773457,\n 1767194234188234,\n 64476199777924,\n 1117233614485261,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3236091949205521,\n 2386938060636506,\n 2220652137473166,\n 1722843421165029,\n 2442282371698157,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 298845952651262,\n 1166086588952562,\n 1179896526238434,\n 1347812759398693,\n 1412945390096208,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1143239552672925,\n 906436640714209,\n 2177000572812152,\n 2075299936108548,\n 325186347798433,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2972824668060020,\n 2936287674948563,\n 3625238557779406,\n 2193186935276994,\n 1387043709851261,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 418098668140962,\n 715065997721283,\n 1471916138376055,\n 2168570337288357,\n 937812682637044,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1043584187226485,\n 2143395746619356,\n 2209558562919611,\n 482427979307092,\n 847556718384018,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1248731221520740,\n 1465200936117687,\n 2792603306395388,\n 2304778448366139,\n 2513234303861356,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1057329623869501,\n 620334067429122,\n 461700859268034,\n 2012481616501857,\n 297268569108938,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1055352180870759,\n 1553151421852298,\n 1510903185371259,\n 1470458349428097,\n 1226259419062731,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3744788603986897,\n 3042126439258578,\n 3441906842094992,\n 3641194565844440,\n 3872208010289441,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 47000654413729,\n 1004754424173864,\n 1868044813557703,\n 173236934059409,\n 588771199737015,\n ]),\n xy2d: FieldElement51::from_limbs([\n 30498470091663,\n 1082245510489825,\n 576771653181956,\n 806509986132686,\n 1317634017056939,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2672107869436803,\n 3745154677001249,\n 2417006535213335,\n 4136645508605033,\n 2065456951573058,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1115636332012334,\n 1854340990964155,\n 83792697369514,\n 1972177451994021,\n 457455116057587,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1698968457310898,\n 1435137169051090,\n 1083661677032510,\n 938363267483709,\n 340103887207182,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1995325341336555,\n 911500251774648,\n 2415810569088940,\n 855378419194761,\n 3825401211214090,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 241719380661528,\n 310028521317150,\n 1215881323380194,\n 1408214976493624,\n 2141142156467363,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1315157046163473,\n 727368447885818,\n 1363466668108618,\n 1668921439990361,\n 1398483384337907,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2326829491984875,\n 3267188020145720,\n 1849729037055211,\n 4191614430138232,\n 2696204044080201,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2053597130993710,\n 2024431685856332,\n 2233550957004860,\n 2012407275509545,\n 872546993104440,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1217269667678610,\n 599909351968693,\n 1390077048548598,\n 1471879360694802,\n 739586172317596,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3970118453066023,\n 1560510726633957,\n 3156262694845170,\n 1418028351780051,\n 2346204163137185,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2132502667405250,\n 214379346175414,\n 1502748313768060,\n 1960071701057800,\n 1353971822643138,\n ]),\n xy2d: FieldElement51::from_limbs([\n 319394212043702,\n 2127459436033571,\n 717646691535162,\n 663366796076914,\n 318459064945314,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2657789238608841,\n 1960452633787082,\n 2919148848086913,\n 3744474074452359,\n 1451061489880786,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 947085906234007,\n 323284730494107,\n 1485778563977200,\n 728576821512394,\n 901584347702286,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1575783124125742,\n 2126210792434375,\n 1569430791264065,\n 1402582372904727,\n 1891780248341114,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3090232019245924,\n 4249503325136911,\n 3270591693593114,\n 1662001808174330,\n 2330127946643001,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 739152638255629,\n 2074935399403557,\n 505483666745895,\n 1611883356514088,\n 628654635394878,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1822054032121349,\n 643057948186973,\n 7306757352712,\n 577249257962099,\n 284735863382083,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3618358370049178,\n 1448606567552085,\n 3730680834630016,\n 2417602993041145,\n 1115718458123497,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 204146226972102,\n 1630511199034723,\n 2215235214174763,\n 174665910283542,\n 956127674017216,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1562934578796716,\n 1070893489712745,\n 11324610642270,\n 958989751581897,\n 2172552325473805,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1770564423056008,\n 2987323445349813,\n 1326060113795288,\n 1509650369341127,\n 2317692235267932,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 623682558650637,\n 1337866509471512,\n 990313350206649,\n 1314236615762469,\n 1164772974270275,\n ]),\n xy2d: FieldElement51::from_limbs([\n 223256821462517,\n 723690150104139,\n 1000261663630601,\n 933280913953265,\n 254872671543046,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1969087237026022,\n 2876595539132372,\n 1335555107635968,\n 2069986355593023,\n 3963899963027150,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1236103475266979,\n 1837885883267218,\n 1026072585230455,\n 1025865513954973,\n 1801964901432134,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1115241013365517,\n 1712251818829143,\n 2148864332502771,\n 2096001471438138,\n 2235017246626125,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3551068012286861,\n 2047148477845620,\n 2165648650132450,\n 1612539282026145,\n 2765997725314138,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 118352772338543,\n 1067608711804704,\n 1434796676193498,\n 1683240170548391,\n 230866769907437,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850689576796636,\n 1601590730430274,\n 1139674615958142,\n 1954384401440257,\n 76039205311,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1723387471374172,\n 3249101280723658,\n 2785727448808904,\n 2272728458379212,\n 1756575222802512,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2146711623855116,\n 503278928021499,\n 625853062251406,\n 1109121378393107,\n 1033853809911861,\n ]),\n xy2d: FieldElement51::from_limbs([\n 571005965509422,\n 2005213373292546,\n 1016697270349626,\n 56607856974274,\n 914438579435146,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1346698876211176,\n 2076651707527589,\n 3336561384795453,\n 2517134292513653,\n 1068954492309670,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1769967932677654,\n 1695893319756416,\n 1151863389675920,\n 1781042784397689,\n 400287774418285,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1851867764003121,\n 403841933237558,\n 820549523771987,\n 761292590207581,\n 1743735048551143,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 410915148140008,\n 2107072311871739,\n 3256167275561751,\n 2351484709082008,\n 1180818713503223,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 285945406881439,\n 648174397347453,\n 1098403762631981,\n 1366547441102991,\n 1505876883139217,\n ]),\n xy2d: FieldElement51::from_limbs([\n 672095903120153,\n 1675918957959872,\n 636236529315028,\n 1569297300327696,\n 2164144194785875,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1902708175321798,\n 3287143344600686,\n 1178560808893262,\n 2552895497743394,\n 1280977479761117,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1615357281742403,\n 404257611616381,\n 2160201349780978,\n 1160947379188955,\n 1578038619549541,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2013087639791217,\n 822734930507457,\n 1785668418619014,\n 1668650702946164,\n 389450875221715,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2705718263383616,\n 2358206633614248,\n 2072540975937134,\n 308588860670238,\n 1304394580755385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295082798350326,\n 2091844511495996,\n 1851348972587817,\n 3375039684596,\n 789440738712837,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2083069137186154,\n 848523102004566,\n 993982213589257,\n 1405313299916317,\n 1532824818698468,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3747761112537659,\n 1397203457344778,\n 4026750030752190,\n 2391102557240943,\n 2318403398028034,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1782411379088302,\n 1096724939964781,\n 27593390721418,\n 542241850291353,\n 1540337798439873,\n ]),\n xy2d: FieldElement51::from_limbs([\n 693543956581437,\n 171507720360750,\n 1557908942697227,\n 1074697073443438,\n 1104093109037196,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 345288228393400,\n 3351443383432420,\n 2386681722088990,\n 1740551994106739,\n 2500011992985018,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 231429562203065,\n 1526290236421172,\n 2021375064026423,\n 1520954495658041,\n 806337791525116,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1079623667189886,\n 872403650198613,\n 766894200588288,\n 2163700860774109,\n 2023464507911816,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 854645372543796,\n 1936406001954827,\n 2403260476226501,\n 3077125552956802,\n 1554306377287555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1497138821904622,\n 1044820250515590,\n 1742593886423484,\n 1237204112746837,\n 849047450816987,\n ]),\n xy2d: FieldElement51::from_limbs([\n 667962773375330,\n 1897271816877105,\n 1399712621683474,\n 1143302161683099,\n 2081798441209593,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2378947665252234,\n 1936114012888109,\n 1704424366552046,\n 3108474694401560,\n 2968403435020606,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1072409664800960,\n 2146937497077528,\n 1508780108920651,\n 935767602384853,\n 1112800433544068,\n ]),\n xy2d: FieldElement51::from_limbs([\n 333549023751292,\n 280219272863308,\n 2104176666454852,\n 1036466864875785,\n 536135186520207,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2625466093568366,\n 2398257055215356,\n 2555916080813104,\n 2667888562832962,\n 3510376944868638,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1186115062588401,\n 2251609796968486,\n 1098944457878953,\n 1153112761201374,\n 1791625503417267,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1870078460219737,\n 2129630962183380,\n 852283639691142,\n 292865602592851,\n 401904317342226,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1361070124828016,\n 815664541425524,\n 3278598711049919,\n 1951790935390646,\n 2807674705520038,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1546301003424277,\n 459094500062839,\n 1097668518375311,\n 1780297770129643,\n 720763293687608,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1212405311403990,\n 1536693382542438,\n 61028431067459,\n 1863929423417129,\n 1223219538638038,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1294303766540260,\n 3435357279640341,\n 3134071170918340,\n 2315654383110622,\n 2213283684565086,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 339050984211414,\n 601386726509773,\n 413735232134068,\n 966191255137228,\n 1839475899458159,\n ]),\n xy2d: FieldElement51::from_limbs([\n 235605972169408,\n 2174055643032978,\n 1538335001838863,\n 1281866796917192,\n 1815940222628465,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1632352921721536,\n 1833328609514701,\n 2092779091951987,\n 4175756015558474,\n 2210068022482918,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 35271216625062,\n 1712350667021807,\n 983664255668860,\n 98571260373038,\n 1232645608559836,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1998172393429622,\n 1798947921427073,\n 784387737563581,\n 1589352214827263,\n 1589861734168180,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1733739258725305,\n 2283515530744786,\n 2453769758904107,\n 3243892858242237,\n 1194308773174555,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 846415389605137,\n 746163495539180,\n 829658752826080,\n 592067705956946,\n 957242537821393,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1758148849754419,\n 619249044817679,\n 168089007997045,\n 1371497636330523,\n 1867101418880350,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2578433797894864,\n 2513559319756263,\n 1700682323676192,\n 1577907266349064,\n 3469447477068264,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1714182387328607,\n 1477856482074168,\n 574895689942184,\n 2159118410227270,\n 1555532449716575,\n ]),\n xy2d: FieldElement51::from_limbs([\n 853828206885131,\n 998498946036955,\n 1835887550391235,\n 207627336608048,\n 258363815956050,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2392941288336925,\n 3488528558590503,\n 2894901233585134,\n 1646615130509172,\n 1208239602291765,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1501663228068911,\n 1354879465566912,\n 1444432675498247,\n 897812463852601,\n 855062598754348,\n ]),\n xy2d: FieldElement51::from_limbs([\n 714380763546606,\n 1032824444965790,\n 1774073483745338,\n 1063840874947367,\n 1738680636537158,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1640635546696233,\n 2884968766877360,\n 2212651044092395,\n 2282390772269100,\n 2620315074574625,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1171650314802029,\n 1567085444565577,\n 1453660792008405,\n 757914533009261,\n 1619511342778196,\n ]),\n xy2d: FieldElement51::from_limbs([\n 420958967093237,\n 971103481109486,\n 2169549185607107,\n 1301191633558497,\n 1661514101014240,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3158923465503550,\n 1332556122804145,\n 4075855067109735,\n 3619414031128206,\n 1982558335973171,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1121533090144639,\n 1021251337022187,\n 110469995947421,\n 1511059774758394,\n 2110035908131662,\n ]),\n xy2d: FieldElement51::from_limbs([\n 303213233384524,\n 2061932261128138,\n 352862124777736,\n 40828818670255,\n 249879468482660,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 856559257852200,\n 2760317478634258,\n 3629993581580163,\n 3975258940632376,\n 1962275756614520,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1445691340537320,\n 40614383122127,\n 402104303144865,\n 485134269878232,\n 1659439323587426,\n ]),\n xy2d: FieldElement51::from_limbs([\n 20057458979482,\n 1183363722525800,\n 2140003847237215,\n 2053873950687614,\n 2112017736174909,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2228654250927986,\n 3735391177100515,\n 1368661293910955,\n 3328311098862539,\n 526650682059607,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 709481497028540,\n 531682216165724,\n 316963769431931,\n 1814315888453765,\n 258560242424104,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1053447823660455,\n 1955135194248683,\n 1010900954918985,\n 1182614026976701,\n 1240051576966610,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1957943897155478,\n 1788667368028035,\n 2389492723714354,\n 2252839333292309,\n 3078204576998275,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1848942433095597,\n 1582009882530495,\n 1849292741020143,\n 1068498323302788,\n 2001402229799484,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1528282417624269,\n 2142492439828191,\n 2179662545816034,\n 362568973150328,\n 1591374675250271,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2411826493119617,\n 2484141002903963,\n 2149181472355544,\n 598041771119831,\n 2435658815595421,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2013278155187349,\n 662660471354454,\n 793981225706267,\n 411706605985744,\n 804490933124791,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2051892037280204,\n 488391251096321,\n 2230187337030708,\n 930221970662692,\n 679002758255210,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1530723630438670,\n 875873929577927,\n 2593359947955236,\n 2701702933216000,\n 1055551308214178,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1461835919309432,\n 1955256480136428,\n 180866187813063,\n 1551979252664528,\n 557743861963950,\n ]),\n xy2d: FieldElement51::from_limbs([\n 359179641731115,\n 1324915145732949,\n 902828372691474,\n 294254275669987,\n 1887036027752957,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4295071423139571,\n 2038225437857463,\n 1317528426475850,\n 1398989128982787,\n 2027639881006861,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2072902725256516,\n 312132452743412,\n 309930885642209,\n 996244312618453,\n 1590501300352303,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1397254305160710,\n 695734355138021,\n 2233992044438756,\n 1776180593969996,\n 1085588199351115,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2692366865016258,\n 2506694600041928,\n 2745669038615469,\n 1556322069683365,\n 3819256354004466,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1950722461391320,\n 1907845598854797,\n 1822757481635527,\n 2121567704750244,\n 73811931471221,\n ]),\n xy2d: FieldElement51::from_limbs([\n 387139307395758,\n 2058036430315676,\n 1220915649965325,\n 1794832055328951,\n 1230009312169328,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1765973779329498,\n 2911143873132225,\n 2271621715291913,\n 3553728154996461,\n 3368065817761132,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1127572801181483,\n 1224743760571696,\n 1276219889847274,\n 1529738721702581,\n 1589819666871853,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2181229378964934,\n 2190885205260020,\n 1511536077659137,\n 1246504208580490,\n 668883326494241,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2689666469258543,\n 2920826224880015,\n 2333696811665585,\n 523874406393177,\n 2496851874620484,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1975438052228868,\n 1071801519999806,\n 594652299224319,\n 1877697652668809,\n 1489635366987285,\n ]),\n xy2d: FieldElement51::from_limbs([\n 958592545673770,\n 233048016518599,\n 851568750216589,\n 567703851596087,\n 1740300006094761,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2014540178270324,\n 192672779514432,\n 2465676996326778,\n 2194819933853410,\n 1716422829364835,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1540769606609725,\n 2148289943846077,\n 1597804156127445,\n 1230603716683868,\n 815423458809453,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1738560251245018,\n 1779576754536888,\n 1783765347671392,\n 1880170990446751,\n 1088225159617541,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2911103727614740,\n 1956447718227572,\n 1830568515922666,\n 3092868863429656,\n 1669607124206367,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1143465490433355,\n 1532194726196059,\n 1093276745494697,\n 481041706116088,\n 2121405433561163,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1686424298744462,\n 1451806974487153,\n 266296068846582,\n 1834686947542675,\n 1720762336132256,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3141016840074207,\n 3295090436969907,\n 3107924901237156,\n 1669272323124635,\n 1603340330827879,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1206396181488998,\n 333158148435054,\n 1402633492821422,\n 1120091191722026,\n 1945474114550509,\n ]),\n xy2d: FieldElement51::from_limbs([\n 766720088232571,\n 1512222781191002,\n 1189719893490790,\n 2091302129467914,\n 2141418006894941,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2671463460991841,\n 1998875112167986,\n 3678399683938955,\n 3406728169064757,\n 2738338345823434,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 938160078005954,\n 1421776319053174,\n 1941643234741774,\n 180002183320818,\n 1414380336750546,\n ]),\n xy2d: FieldElement51::from_limbs([\n 398001940109652,\n 1577721237663248,\n 1012748649830402,\n 1540516006905144,\n 1011684812884559,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1653276489969611,\n 2257881638852872,\n 1921777941170835,\n 1604139841794531,\n 3113010867325889,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 996661541407379,\n 1455877387952927,\n 744312806857277,\n 139213896196746,\n 1000282908547789,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1450817495603008,\n 1476865707053229,\n 1030490562252053,\n 620966950353376,\n 1744760161539058,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2811528223687828,\n 2288856475326432,\n 2038622963352005,\n 1637244893271723,\n 3278365165924196,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 962165956135846,\n 1116599660248791,\n 182090178006815,\n 1455605467021751,\n 196053588803284,\n ]),\n xy2d: FieldElement51::from_limbs([\n 796863823080135,\n 1897365583584155,\n 420466939481601,\n 2165972651724672,\n 932177357788289,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 877047233620613,\n 1375632631944375,\n 2895573425567369,\n 2911822552533124,\n 2271153746017078,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2216943882299338,\n 394841323190322,\n 2222656898319671,\n 558186553950529,\n 1077236877025190,\n ]),\n xy2d: FieldElement51::from_limbs([\n 801118384953213,\n 1914330175515892,\n 574541023311511,\n 1471123787903705,\n 1526158900256288,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3201417702772463,\n 2207116611267330,\n 3164719852826535,\n 2752958352884036,\n 2314162374456719,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1474518386765335,\n 1760793622169197,\n 1157399790472736,\n 1622864308058898,\n 165428294422792,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1961673048027128,\n 102619413083113,\n 1051982726768458,\n 1603657989805485,\n 1941613251499678,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1401939116319247,\n 2587106153588320,\n 2323846009771033,\n 862423201496005,\n 3102318568216632,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1234706593321979,\n 1083343891215917,\n 898273974314935,\n 1640859118399498,\n 157578398571149,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1143483057726416,\n 1992614991758919,\n 674268662140796,\n 1773370048077526,\n 674318359920189,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1835401379538542,\n 173900035308392,\n 818247630716732,\n 4013900225838034,\n 1021506399448290,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1506632088156630,\n 2127481795522179,\n 513812919490255,\n 140643715928370,\n 442476620300318,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2056683376856736,\n 219094741662735,\n 2193541883188309,\n 1841182310235800,\n 556477468664293,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3566819241596075,\n 1049075855992602,\n 4318372866671791,\n 2518704280870781,\n 2040482348591519,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 94096246544434,\n 922482381166992,\n 24517828745563,\n 2139430508542503,\n 2097139044231004,\n ]),\n xy2d: FieldElement51::from_limbs([\n 537697207950515,\n 1399352016347350,\n 1563663552106345,\n 2148749520888918,\n 549922092988516,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1747985413252415,\n 680511052635695,\n 1809559829982725,\n 2846074064615302,\n 2453472984431229,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 323583936109569,\n 1973572998577657,\n 1192219029966558,\n 79354804385273,\n 1374043025560347,\n ]),\n xy2d: FieldElement51::from_limbs([\n 213277331329947,\n 416202017849623,\n 1950535221091783,\n 1313441578103244,\n 2171386783823658,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2440888617915079,\n 993969372859109,\n 3147669935222235,\n 3799101348983503,\n 1477373024911349,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1620578418245010,\n 541035331188469,\n 2235785724453865,\n 2154865809088198,\n 1974627268751826,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1346805451740245,\n 1350981335690626,\n 942744349501813,\n 2155094562545502,\n 1012483751693409,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2107080134091762,\n 1132567062788208,\n 1824935377687210,\n 769194804343737,\n 1857941799971888,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1074666112436467,\n 249279386739593,\n 1174337926625354,\n 1559013532006480,\n 1472287775519121,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1872620123779532,\n 1892932666768992,\n 1921559078394978,\n 1270573311796160,\n 1438913646755037,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3089190001333428,\n 3264053113908846,\n 989780015893986,\n 1351393287739814,\n 2580427560230798,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1028328827183114,\n 1711043289969857,\n 1350832470374933,\n 1923164689604327,\n 1495656368846911,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1900828492104143,\n 430212361082163,\n 687437570852799,\n 832514536673512,\n 1685641495940794,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3094432661621646,\n 605670026766215,\n 290836444839585,\n 2415010588577604,\n 2213815011799644,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1176336383453996,\n 1725477294339771,\n 12700622672454,\n 678015708818208,\n 162724078519879,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1448049969043497,\n 1789411762943521,\n 385587766217753,\n 90201620913498,\n 832999441066823,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2767886146978542,\n 2240508292484615,\n 3603469341851756,\n 3475055379001735,\n 3002035638112385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1263624896582495,\n 1102602401673328,\n 526302183714372,\n 2152015839128799,\n 1483839308490010,\n ]),\n xy2d: FieldElement51::from_limbs([\n 442991718646863,\n 1599275157036458,\n 1925389027579192,\n 899514691371390,\n 350263251085160,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1689713572022124,\n 2845654372939621,\n 3229894858477217,\n 1985127338729498,\n 3927868934032873,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1557207018622683,\n 340631692799603,\n 1477725909476187,\n 614735951619419,\n 2033237123746766,\n ]),\n xy2d: FieldElement51::from_limbs([\n 968764929340557,\n 1225534776710944,\n 662967304013036,\n 1155521416178595,\n 791142883466590,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1487081286167458,\n 3244839255500182,\n 1792378982844639,\n 2950452258685122,\n 2153908693179753,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1123181311102823,\n 685575944875442,\n 507605465509927,\n 1412590462117473,\n 568017325228626,\n ]),\n xy2d: FieldElement51::from_limbs([\n 560258797465417,\n 2193971151466401,\n 1824086900849026,\n 579056363542056,\n 1690063960036441,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1918407319222397,\n 2605567366745211,\n 1930426334528098,\n 1564816146005724,\n 4113142195393344,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2131325168777276,\n 1176636658428908,\n 1756922641512981,\n 1390243617176012,\n 1966325177038383,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2063958120364491,\n 2140267332393533,\n 699896251574968,\n 273268351312140,\n 375580724713232,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2024297515263178,\n 2668759143407935,\n 3330814048702549,\n 2423412039258430,\n 1031677520051052,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2033900009388450,\n 1744902869870788,\n 2190580087917640,\n 1949474984254121,\n 231049754293748,\n ]),\n xy2d: FieldElement51::from_limbs([\n 343868674606581,\n 550155864008088,\n 1450580864229630,\n 481603765195050,\n 896972360018042,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2151139328380127,\n 2566545695770176,\n 2311556639460451,\n 1676664391494650,\n 2048348075599360,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1528930066340597,\n 1605003907059576,\n 1055061081337675,\n 1458319101947665,\n 1234195845213142,\n ]),\n xy2d: FieldElement51::from_limbs([\n 830430507734812,\n 1780282976102377,\n 1425386760709037,\n 362399353095425,\n 2168861579799910,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3407562046415562,\n 980662895504005,\n 2053766700883521,\n 2742766027762854,\n 2762205690726604,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1683750316716132,\n 652278688286128,\n 1221798761193539,\n 1897360681476669,\n 319658166027343,\n ]),\n xy2d: FieldElement51::from_limbs([\n 618808732869972,\n 72755186759744,\n 2060379135624181,\n 1730731526741822,\n 48862757828238,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3714971784278753,\n 3394840525452699,\n 614590986558882,\n 1409210575145591,\n 1882816996436803,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2230133264691131,\n 563950955091024,\n 2042915975426398,\n 827314356293472,\n 672028980152815,\n ]),\n xy2d: FieldElement51::from_limbs([\n 264204366029760,\n 1654686424479449,\n 2185050199932931,\n 2207056159091748,\n 506015669043634,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1784446333136550,\n 1973746527984364,\n 334856327359575,\n 3408569589569858,\n 3275749938360725,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2065270940578383,\n 31477096270353,\n 306421879113491,\n 181958643936686,\n 1907105536686083,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1496516440779464,\n 1748485652986458,\n 872778352227340,\n 818358834654919,\n 97932669284220,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2723435829455580,\n 2924255216478824,\n 1804995246884102,\n 1842309243470804,\n 3753662318666930,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1013216974933691,\n 538921919682598,\n 1915776722521558,\n 1742822441583877,\n 1886550687916656,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2094270000643336,\n 303971879192276,\n 40801275554748,\n 649448917027930,\n 1818544418535447,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2241737709499146,\n 549397817447461,\n 838180519319392,\n 1725686958520781,\n 3957438894582995,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1216074541925116,\n 50120933933509,\n 1565829004133810,\n 721728156134580,\n 349206064666188,\n ]),\n xy2d: FieldElement51::from_limbs([\n 948617110470858,\n 346222547451945,\n 1126511960599975,\n 1759386906004538,\n 493053284802266,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1454933046815146,\n 3126495827951610,\n 1467170975468587,\n 1432316382418897,\n 2111710746366763,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2105387117364450,\n 1996463405126433,\n 1303008614294500,\n 851908115948209,\n 1353742049788635,\n ]),\n xy2d: FieldElement51::from_limbs([\n 750300956351719,\n 1487736556065813,\n 15158817002104,\n 1511998221598392,\n 971739901354129,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1874648163531674,\n 2124487685930551,\n 1810030029384882,\n 918400043048335,\n 2838148440985898,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1235084464747900,\n 1166111146432082,\n 1745394857881591,\n 1405516473883040,\n 4463504151617,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1663810156463827,\n 327797390285791,\n 1341846161759410,\n 1964121122800605,\n 1747470312055380,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 660005247548214,\n 2071860029952887,\n 3610548013635355,\n 911703252219106,\n 3266179736709079,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2206641276178231,\n 1690587809721504,\n 1600173622825126,\n 2156096097634421,\n 1106822408548216,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1344788193552206,\n 1949552134239140,\n 1735915881729557,\n 675891104100469,\n 1834220014427292,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1920949492387945,\n 2410685102072778,\n 2322108077349280,\n 2877838278583064,\n 3719881539786256,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 622221042073383,\n 1210146474039168,\n 1742246422343683,\n 1403839361379025,\n 417189490895736,\n ]),\n xy2d: FieldElement51::from_limbs([\n 22727256592983,\n 168471543384997,\n 1324340989803650,\n 1839310709638189,\n 504999476432775,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3565040332441556,\n 1721896294296941,\n 2304063388272514,\n 2065069734239231,\n 3056710287109878,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1337466662091884,\n 1287645354669772,\n 2018019646776184,\n 652181229374245,\n 898011753211715,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1969792547910734,\n 779969968247557,\n 2011350094423418,\n 1823964252907487,\n 1058949448296945,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2459143550747250,\n 1118176942430252,\n 3010694408233412,\n 806764629546265,\n 1157700123092949,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1273565321399022,\n 1638509681964574,\n 759235866488935,\n 666015124346707,\n 897983460943405,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1717263794012298,\n 1059601762860786,\n 1837819172257618,\n 1054130665797229,\n 680893204263559,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2237039662793603,\n 2249022333361206,\n 2058613546633703,\n 2401253908530527,\n 2215176649164581,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 79472182719605,\n 1851130257050174,\n 1825744808933107,\n 821667333481068,\n 781795293511946,\n ]),\n xy2d: FieldElement51::from_limbs([\n 755822026485370,\n 152464789723500,\n 1178207602290608,\n 410307889503239,\n 156581253571278,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3669985309815545,\n 2736319981413860,\n 3898537095128197,\n 3653287498355512,\n 1349185550126960,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1495380034400429,\n 325049476417173,\n 46346894893933,\n 1553408840354856,\n 828980101835683,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1280337889310282,\n 2070832742866672,\n 1640940617225222,\n 2098284908289951,\n 450929509534434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2659503167684029,\n 2378371955168899,\n 2537839641198868,\n 1999255076709337,\n 2030511179441770,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1254958221100483,\n 1153235960999843,\n 942907704968834,\n 637105404087392,\n 1149293270147267,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894249020470196,\n 400291701616810,\n 406878712230981,\n 1599128793487393,\n 1145868722604026,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3749755063888563,\n 2361916158338507,\n 1128535642171975,\n 1900106496009660,\n 2381592531146157,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 452487513298665,\n 1352120549024569,\n 1173495883910956,\n 1999111705922009,\n 367328130454226,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1717539401269642,\n 1475188995688487,\n 891921989653942,\n 836824441505699,\n 1885988485608364,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3493583935107776,\n 2439136865632830,\n 3370281625921440,\n 2680547565621609,\n 2282158712612572,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2022432361201842,\n 1088816090685051,\n 1977843398539868,\n 1854834215890724,\n 564238862029357,\n ]),\n xy2d: FieldElement51::from_limbs([\n 938868489100585,\n 1100285072929025,\n 1017806255688848,\n 1957262154788833,\n 152787950560442,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3119119231364171,\n 2872271776627789,\n 2477832016990963,\n 2593801257642876,\n 1761675818237335,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1295072362439987,\n 931227904689414,\n 1355731432641687,\n 922235735834035,\n 892227229410209,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1680989767906154,\n 535362787031440,\n 2136691276706570,\n 1942228485381244,\n 1267350086882274,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2617818047455756,\n 2684460443440843,\n 2378209521329782,\n 1973842949591661,\n 2897427157127624,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 535509430575217,\n 546885533737322,\n 1524675609547799,\n 2138095752851703,\n 1260738089896827,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1159906385590467,\n 2198530004321610,\n 714559485023225,\n 81880727882151,\n 1484020820037082,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1377485731340769,\n 2046328105512000,\n 1802058637158797,\n 2313945950453421,\n 1356993908853900,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2013612215646735,\n 1830770575920375,\n 536135310219832,\n 609272325580394,\n 270684344495013,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1237542585982777,\n 2228682050256790,\n 1385281931622824,\n 593183794882890,\n 493654978552689,\n ]),\n },\n ]),\n LookupTable([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2299141301692989,\n 1891414891220256,\n 983894663308928,\n 2427961581972066,\n 3378060928864955,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1694030170963455,\n 502038567066200,\n 1691160065225467,\n 949628319562187,\n 275110186693066,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1124515748676336,\n 1661673816593408,\n 1499640319059718,\n 1584929449166988,\n 558148594103306,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1784525599998356,\n 1619698033617383,\n 2097300287550715,\n 2510065271789004,\n 1905684794832757,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1288941072872766,\n 931787902039402,\n 190731008859042,\n 2006859954667190,\n 1005931482221702,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1465551264822703,\n 152905080555927,\n 680334307368453,\n 173227184634745,\n 666407097159852,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2111017076203943,\n 3630560299479595,\n 1248583954016455,\n 3604089008549670,\n 1895180776543895,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 171348223915638,\n 662766099800389,\n 462338943760497,\n 466917763340314,\n 656911292869115,\n ]),\n xy2d: FieldElement51::from_limbs([\n 488623681976577,\n 866497561541722,\n 1708105560937768,\n 1673781214218839,\n 1506146329818807,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2412225278142205,\n 950394373239688,\n 2682296937026182,\n 711676555398831,\n 320964687779005,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 988979367990485,\n 1359729327576302,\n 1301834257246029,\n 294141160829308,\n 29348272277475,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1434382743317910,\n 100082049942065,\n 221102347892623,\n 186982837860588,\n 1305765053501834,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2205916462268190,\n 2751663643476068,\n 961960554686615,\n 2409862576442233,\n 1841471168298304,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1191737341426592,\n 1847042034978363,\n 1382213545049056,\n 1039952395710448,\n 788812858896859,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1346965964571152,\n 1291881610839830,\n 2142916164336056,\n 786821641205979,\n 1571709146321039,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 787164375951248,\n 2454669019058437,\n 3608390234717387,\n 1431233331032509,\n 786341368775957,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 492448143532951,\n 304105152670757,\n 1761767168301056,\n 233782684697790,\n 1981295323106089,\n ]),\n xy2d: FieldElement51::from_limbs([\n 665807507761866,\n 1343384868355425,\n 895831046139653,\n 439338948736892,\n 1986828765695105,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3007896024559801,\n 1721699973539148,\n 2510565115413133,\n 1390588532210644,\n 1212530909934781,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 852891097972275,\n 1816988871354562,\n 1543772755726524,\n 1174710635522444,\n 202129090724628,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1205281565824323,\n 22430498399418,\n 992947814485516,\n 1392458699738672,\n 688441466734558,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3302427242100220,\n 1955849529137134,\n 2171162376368357,\n 2343545681983462,\n 447733118757825,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1287181461435438,\n 622722465530711,\n 880952150571872,\n 741035693459198,\n 311565274989772,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1003649078149734,\n 545233927396469,\n 1849786171789880,\n 1318943684880434,\n 280345687170552,\n ]),\n },\n ]),\n]);\n\n/// Odd multiples of the basepoint `[B, 3B, 5B, 7B, 9B, 11B, 13B, 15B, ..., 127B]`.\n#[cfg(feature = \"precomputed-tables\")]\n#[allow(dead_code)]\npub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT: NafLookupTable8 =\n NafLookupTable8([\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3540182452943730,\n 2497478415033846,\n 2521227595762870,\n 1462984067271729,\n 2389212253076811,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 62697248952638,\n 204681361388450,\n 631292143396476,\n 338455783676468,\n 1213667448819585,\n ]),\n xy2d: FieldElement51::from_limbs([\n 301289933810280,\n 1259582250014073,\n 1422107436869536,\n 796239922652654,\n 1953934009299142,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1601611775252272,\n 1720807796594148,\n 1132070835939856,\n 3512254832574799,\n 2147779492816910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 316559037616741,\n 2177824224946892,\n 1459442586438991,\n 1461528397712656,\n 751590696113597,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1850748884277385,\n 1200145853858453,\n 1068094770532492,\n 672251375690438,\n 1586055907191707,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 769950342298400,\n 2384754244604994,\n 3095885746880802,\n 3225892188161580,\n 2977876099231263,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 425251763115706,\n 608463272472562,\n 442562545713235,\n 837766094556764,\n 374555092627893,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1086255230780037,\n 274979815921559,\n 1960002765731872,\n 929474102396301,\n 1190409889297339,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2916800678241215,\n 2065379846933858,\n 2622030924071124,\n 2602788184473875,\n 1233371373142984,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2019367628972465,\n 676711900706637,\n 110710997811333,\n 1108646842542025,\n 517791959672113,\n ]),\n xy2d: FieldElement51::from_limbs([\n 965130719900578,\n 247011430587952,\n 526356006571389,\n 91986625355052,\n 2157223321444601,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1802695059464988,\n 1664899123557221,\n 2845359304426105,\n 2160434469266658,\n 3179370264440279,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1725674970513508,\n 1933645953859181,\n 1542344539275782,\n 1767788773573747,\n 1297447965928905,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1381809363726107,\n 1430341051343062,\n 2061843536018959,\n 1551778050872521,\n 2036394857967624,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4222693909998302,\n 2779866139518454,\n 1619374932191226,\n 2207306624415883,\n 1169170329061080,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2070390218572616,\n 1458919061857835,\n 624171843017421,\n 1055332792707765,\n 433987520732508,\n ]),\n xy2d: FieldElement51::from_limbs([\n 893653801273833,\n 1168026499324677,\n 1242553501121234,\n 1306366254304474,\n 1086752658510815,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2465253816303469,\n 3191571337672685,\n 1159882208056013,\n 2569188183312765,\n 621213314200686,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1971678598905747,\n 338026507889165,\n 762398079972271,\n 655096486107477,\n 42299032696322,\n ]),\n xy2d: FieldElement51::from_limbs([\n 177130678690680,\n 1754759263300204,\n 1864311296286618,\n 1180675631479880,\n 1292726903152791,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1913163449625248,\n 2712579013977241,\n 2193883288642313,\n 1008900146920800,\n 1721983679009502,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1070401523076875,\n 1272492007800961,\n 1910153608563310,\n 2075579521696771,\n 1191169788841221,\n ]),\n xy2d: FieldElement51::from_limbs([\n 692896803108118,\n 500174642072499,\n 2068223309439677,\n 1162190621851337,\n 1426986007309901,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1819621230288238,\n 2735700366193240,\n 1755134670739586,\n 3080648199451191,\n 4172807995775876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 992069868904071,\n 799011518185730,\n 1777586403832768,\n 1134820506145684,\n 1999461475558530,\n ]),\n xy2d: FieldElement51::from_limbs([\n 425204543703124,\n 2040469794090382,\n 1651690622153809,\n 1500530168597569,\n 1253908377065966,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2105824306960939,\n 1387520302709358,\n 3633176580451016,\n 2211816663841753,\n 1629085891776489,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1485201376284999,\n 1022406647424656,\n 504181009209019,\n 962621520820995,\n 590876713147230,\n ]),\n xy2d: FieldElement51::from_limbs([\n 265873406365287,\n 1192742653492898,\n 88553098803050,\n 525037770869640,\n 1266933811251234,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3552316659826612,\n 1254279525791875,\n 1609927932077699,\n 3578654071679972,\n 3750681296069893,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 37186803519861,\n 1404297334376301,\n 578519728836650,\n 1740727951192592,\n 2095534282477028,\n ]),\n xy2d: FieldElement51::from_limbs([\n 833234263154399,\n 2023862470013762,\n 1854137933982069,\n 853924318090959,\n 1589812702805850,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3679150557957763,\n 1319179453661745,\n 497496853611112,\n 2665464286942351,\n 1208137952365560,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1654513078530905,\n 907489875842908,\n 126098711296368,\n 1726320004173677,\n 28269495058173,\n ]),\n xy2d: FieldElement51::from_limbs([\n 114436686957443,\n 532739313025996,\n 115428841215897,\n 2191499400074366,\n 370280402676434,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1111146849833253,\n 2016430049079759,\n 1860522747477948,\n 3537164738290194,\n 4137142824844184,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 429069864577128,\n 975327637149449,\n 237881983565075,\n 1654761232378630,\n 2122527599091807,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2093793463548278,\n 754827233241879,\n 1420389751719629,\n 1829952782588138,\n 2011865756773717,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 676293365438898,\n 2850296017886344,\n 1205350322490195,\n 2763699392265669,\n 2133931188538142,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 48340340349120,\n 1299261101494832,\n 1137329686775218,\n 1534848106674340,\n 1351662218216799,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1904520614137939,\n 1590301001714014,\n 215781420985270,\n 2043534301034629,\n 1970888949300424,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2365217962409710,\n 2061307169694064,\n 1887478590157603,\n 2169639621284316,\n 2373810867477200,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1020052624656948,\n 1260412094216707,\n 366721640607121,\n 585331442306596,\n 345876457758061,\n ]),\n xy2d: FieldElement51::from_limbs([\n 975390299880933,\n 1066555195234642,\n 12651997758352,\n 1184252205433068,\n 1058378155074223,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1431537716602643,\n 2024827957433813,\n 3746434518400495,\n 1087794891033550,\n 2156817571680455,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 929288033346881,\n 255179964546973,\n 711057989588035,\n 208899572612840,\n 185348357387383,\n ]),\n xy2d: FieldElement51::from_limbs([\n 823689746424808,\n 47266130989546,\n 209403309368097,\n 1100966895202707,\n 710792075292719,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2311213117823762,\n 3296668540922318,\n 2004276520649823,\n 1861500579441125,\n 3148029033359833,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1563693677475261,\n 1843782073741194,\n 1950700654453170,\n 911540858113949,\n 2085151496302359,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1427880892005482,\n 106216431121745,\n 42608394782284,\n 1217295886989793,\n 1514235272796882,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3544335535746750,\n 2367994491347456,\n 2567261456502612,\n 1854058085060971,\n 2263545563461076,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 787426011300053,\n 2105981035769060,\n 1130476291127206,\n 1748659348100075,\n 53470983013756,\n ]),\n xy2d: FieldElement51::from_limbs([\n 553548273865386,\n 5927805718390,\n 65184587381926,\n 633576679686953,\n 576048559439973,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 993787326657446,\n 3868807161609258,\n 1615796046728943,\n 2514644292681953,\n 2059021068660907,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 251010270518880,\n 1681684095763484,\n 1521949356387564,\n 431593457045116,\n 1855308922422910,\n ]),\n xy2d: FieldElement51::from_limbs([\n 618490909691959,\n 1257497595618257,\n 202952467594088,\n 35577762721238,\n 1494883566841973,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1673474571932262,\n 2409784519770613,\n 2636095316260487,\n 2761112584601925,\n 3333713288149876,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1600640202645197,\n 1019569075331823,\n 1041916487915822,\n 1680448171313267,\n 2126903137527901,\n ]),\n xy2d: FieldElement51::from_limbs([\n 894964745143659,\n 106116880092678,\n 1009869382959477,\n 317866368542032,\n 1986983122763912,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1765281781276487,\n 2863247187455184,\n 2589075472439062,\n 1386435905543054,\n 2182338478845320,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1144730936996693,\n 2213315231278180,\n 1489676672185125,\n 665039429138074,\n 1131283313040268,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2004734176670602,\n 1738311085075235,\n 418866995976618,\n 1050782508034394,\n 577747313404652,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2185209688340293,\n 1309276076461009,\n 2514740038571278,\n 3994889904012999,\n 3018098826231021,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1405936970888515,\n 1754621155316654,\n 1211862168554999,\n 1813045702919083,\n 997853418197172,\n ]),\n xy2d: FieldElement51::from_limbs([\n 82037622045021,\n 1646398333621944,\n 613095452763466,\n 1312329542583705,\n 81014679202721,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2389287991277873,\n 403851022333257,\n 1597473361477193,\n 2953351602509212,\n 2135174663049062,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1826548187201150,\n 302299893734126,\n 1475477168615781,\n 842617616347376,\n 1438600873676130,\n ]),\n xy2d: FieldElement51::from_limbs([\n 663049852468609,\n 1649295727846569,\n 1048009692742781,\n 628866177992421,\n 1914360327429204,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1795645928096646,\n 306878154408959,\n 2924901319092394,\n 2801261341654799,\n 1653782432983523,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2077597317438627,\n 212642017882064,\n 674844477518888,\n 875487498687554,\n 2060550250171182,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1420448018683809,\n 1032663994771382,\n 1341927003385267,\n 1340360916546159,\n 1988547473895228,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1082660122598844,\n 2545055705583789,\n 3888919679589007,\n 1670283344995811,\n 3403239134794618,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 90430593339788,\n 1838338032241275,\n 571293238480915,\n 1639938867416883,\n 257378872001111,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1528535658865034,\n 1516636853043960,\n 787000569996728,\n 1464531394704506,\n 1684822625133795,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 811329918113934,\n 2783463529007378,\n 1769095754634835,\n 2970819621866866,\n 881037178164325,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1784566501964517,\n 433890943689325,\n 1186055625589419,\n 1496077405487512,\n 1731807117886548,\n ]),\n xy2d: FieldElement51::from_limbs([\n 424909811816304,\n 1355993963741797,\n 409606483251841,\n 455665350637068,\n 1617009023642808,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2478728492077816,\n 2780289048655501,\n 2328687177473769,\n 4107341333582032,\n 1316147724308250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1617420574301156,\n 1741273341070467,\n 667135503486508,\n 2100436564640123,\n 1032223920000865,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1753947659404033,\n 247279202390193,\n 1819288880178945,\n 737334285670249,\n 1037873664856104,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1762568490530034,\n 673742465299012,\n 2054571050635888,\n 2040165159255111,\n 3040123733327257,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1627187989987422,\n 1686331580821752,\n 1309895873498183,\n 719718719104086,\n 300063199808722,\n ]),\n xy2d: FieldElement51::from_limbs([\n 238176707016164,\n 1440454788877048,\n 203336037573144,\n 1437789888677072,\n 101522256664211,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1895216760098480,\n 1934324337975022,\n 3677350688973167,\n 2536415965456176,\n 714678003308640,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 508185358728815,\n 1691320535341855,\n 2168887448239256,\n 1035124393070661,\n 1936603999698584,\n ]),\n xy2d: FieldElement51::from_limbs([\n 390562831571647,\n 1390223890708972,\n 1383183990676371,\n 435998174196410,\n 1882086414390730,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3747620842612921,\n 2081794785291195,\n 3284594056262745,\n 2090090346797895,\n 2581692978935809,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 244144781251265,\n 1290834426417077,\n 1888701171101942,\n 1233922456644870,\n 241117402207491,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1266169390045455,\n 1148042013187970,\n 878921907853942,\n 1815738019658093,\n 908920199341621,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2521768507305118,\n 953557056811112,\n 2015863732865770,\n 1358382511861315,\n 2835421647899992,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2239837206240498,\n 330928973149665,\n 422268062913642,\n 1481280019493032,\n 619879520439841,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1360166735366017,\n 1770556573948510,\n 1395061284191031,\n 1814003148068126,\n 522781147076884,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2611794802645686,\n 707234844948070,\n 1314059396506491,\n 2919250341703934,\n 2161831667832785,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 934831784182383,\n 433734253968318,\n 1660867106725771,\n 1968393082772831,\n 873946300968490,\n ]),\n xy2d: FieldElement51::from_limbs([\n 26306827827554,\n 430884999378685,\n 1504310424376419,\n 1761358720837522,\n 542195685418530,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1762131062631725,\n 3123952634417535,\n 3619918390837537,\n 2909990877347294,\n 1411594230004385,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 538272372224622,\n 1425714779586199,\n 588313661410172,\n 1497062084392578,\n 1602174047128512,\n ]),\n xy2d: FieldElement51::from_limbs([\n 907490361939255,\n 1963620338391363,\n 626927432296975,\n 1250748516081414,\n 959901171882527,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1335066153744413,\n 2887804660779657,\n 2653073855954038,\n 2765226981667422,\n 938831784476763,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 296699434737224,\n 2047543711075683,\n 2076451038937139,\n 227783599906901,\n 1602062110967627,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1574834773194203,\n 1384279952062839,\n 393652417255803,\n 2166968242848859,\n 1552890441390820,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1619646774410947,\n 1576090644023562,\n 3035228391320965,\n 1735328519940543,\n 2355324535937066,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1024074573633446,\n 957088456885874,\n 1690425531356997,\n 2102187380180052,\n 1082544623222033,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1871906170635853,\n 1719383891167200,\n 1584032250247862,\n 823764804192117,\n 2244048510084261,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 642147846489775,\n 3334304977145699,\n 305205716788147,\n 2589176626729533,\n 2224680511484174,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1734162377166545,\n 260713621840346,\n 157174591942595,\n 952544272517991,\n 222818702471733,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1213115494182947,\n 286778704335711,\n 2130189536016490,\n 308349182281342,\n 1217623948685491,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3360052266973635,\n 1843486583624091,\n 1561693837124349,\n 1084041964025479,\n 1866270922024009,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 460705465481210,\n 1968151453817859,\n 497005926994844,\n 625618055866751,\n 2176893440866887,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1655800250476757,\n 2036588542300609,\n 666447448675243,\n 1615721995750683,\n 1508669225186765,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2245948203759141,\n 1058306669699396,\n 1452898014240582,\n 3961024141962768,\n 1633235287338608,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 986647273684279,\n 1507266907811370,\n 1260572633649005,\n 2071672342077446,\n 695976026010857,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1312356620823495,\n 1635278548098567,\n 901946076841033,\n 585120475533168,\n 1240667113237384,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2313723935779695,\n 1506054666773895,\n 996040223525031,\n 636592914999692,\n 1497801917020297,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 292042016419794,\n 1158932298133044,\n 2062611870323738,\n 1946058478962569,\n 1749165808126286,\n ]),\n xy2d: FieldElement51::from_limbs([\n 654683942212830,\n 1526897351349087,\n 2006818439922838,\n 2194919327350361,\n 1451960776874416,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3015041017808905,\n 2951823141773809,\n 2584865668253675,\n 2508192032998563,\n 2582137700042019,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1628123495344283,\n 2072923641214546,\n 1647225812023982,\n 855655925244679,\n 1758126430071140,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1615895096489599,\n 275295258643784,\n 937665541219916,\n 1313496726746346,\n 1186468946422626,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1603070202850694,\n 2072127623773242,\n 1692648737212158,\n 2493373404187852,\n 1248948672117105,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 11167836031898,\n 596565174397990,\n 2196351068723859,\n 314744641791907,\n 1102014997250781,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1409047922401191,\n 69960384467966,\n 688103515547600,\n 1309746102488044,\n 150292892873778,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1986083055103168,\n 691715819340300,\n 1361811659746933,\n 3459052030333434,\n 1063594696046061,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1201987338414749,\n 2198784582460616,\n 1203335513981498,\n 489243077045066,\n 2205278143582433,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2034744376624534,\n 2077387101466387,\n 148448542974969,\n 1502697574577258,\n 473186584705655,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 472016956315960,\n 720786972252993,\n 2840633661190043,\n 3150798753357827,\n 2816563335499153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 253464247569755,\n 168314237403057,\n 511780806170295,\n 1058862316549135,\n 1646858476817137,\n ]),\n xy2d: FieldElement51::from_limbs([\n 595092995922219,\n 1491311840717691,\n 291581784452778,\n 1569186646367854,\n 1031385061400544,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3483137021572755,\n 1526955102024322,\n 2778006642704458,\n 457549634924205,\n 1097420237736736,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1246991699537710,\n 81367319519439,\n 530844036072196,\n 163656863755855,\n 1950742455979290,\n ]),\n xy2d: FieldElement51::from_limbs([\n 191532664076407,\n 539378506082089,\n 1021612562876554,\n 1026603384732632,\n 1773368780410653,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 4144620731387879,\n 590179521333342,\n 4034023318016108,\n 2255745030335426,\n 2699746851701250,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2206599697359952,\n 553895797384417,\n 181689161933786,\n 1153123447919104,\n 778568064152659,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1706307000059211,\n 1885601289314487,\n 889758608505788,\n 550131729999853,\n 1006862664714268,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3210197754285058,\n 2048500453422630,\n 3403309827888207,\n 927154428508963,\n 4199813798872019,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 992058915374933,\n 476120535358775,\n 1973648780784340,\n 2025282643598818,\n 2182318983793230,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1343440812005821,\n 1316045839091795,\n 1884951299078063,\n 1765919609219175,\n 2197567554627988,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3129247779382818,\n 4415026969054274,\n 1900265885969643,\n 1528796215447059,\n 2172730393748688,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1773355092297603,\n 64654329538271,\n 1332124041660957,\n 748492100858001,\n 895500006200535,\n ]),\n xy2d: FieldElement51::from_limbs([\n 2000840647851980,\n 546565968824914,\n 420633283457524,\n 195470736374507,\n 1958689297569520,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 743138980705446,\n 3411117504637167,\n 2591389959690621,\n 2380042066577202,\n 3022267940115114,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 165947002229363,\n 115186103724967,\n 1068573292121517,\n 1842565776920938,\n 1969395681111987,\n ]),\n xy2d: FieldElement51::from_limbs([\n 553322266190633,\n 234265665613185,\n 484544650202821,\n 1238773526575826,\n 2017991917953668,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2581954631514051,\n 1245093644265357,\n 3537016673825374,\n 1834216551713857,\n 923978372152807,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1855378315339552,\n 890045579230758,\n 1764718173975590,\n 197904186055854,\n 1718129022310327,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1278162928734862,\n 1894118254109862,\n 987503995465517,\n 177406744098996,\n 781538103127693,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1996603431230215,\n 1191888797552937,\n 1207440075928499,\n 2765853449051137,\n 2525314961343288,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 808903879370889,\n 990820108751280,\n 1084429472258867,\n 1078562781312589,\n 254514692695625,\n ]),\n xy2d: FieldElement51::from_limbs([\n 615855140068469,\n 586046731175395,\n 693470779212674,\n 1964537100203868,\n 1350330550265229,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3344544372023708,\n 720386671449874,\n 2480841360702110,\n 2036034126860286,\n 2015744690201389,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1337446193390478,\n 1984110761311871,\n 746489405020285,\n 407347127604128,\n 1740475330360596,\n ]),\n xy2d: FieldElement51::from_limbs([\n 140840424783613,\n 1063284623568331,\n 1136446106453878,\n 372042229029799,\n 442607248430694,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2330781679120937,\n 376801425148230,\n 2032603686676107,\n 1488926293635130,\n 1317278311532959,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1290116731380016,\n 2166899563471713,\n 831997001838078,\n 870954980505220,\n 2108537278055823,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1912719171026343,\n 846194720551034,\n 2043988124740726,\n 993234269653961,\n 421229796383281,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2651184584992902,\n 2775702557638963,\n 2539786009779572,\n 2575974880015305,\n 2122619079836732,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1154054290132562,\n 931753998725577,\n 1647742001778052,\n 865765466488226,\n 1083816107290025,\n ]),\n xy2d: FieldElement51::from_limbs([\n 986341121095108,\n 1522330369638573,\n 1990880546211047,\n 501525962272123,\n 198539304862139,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1496414019192687,\n 3991034436173951,\n 3380311659062196,\n 2854747485359158,\n 3346958036643152,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 805612068303425,\n 1891790027761335,\n 1587008567571549,\n 722120737390201,\n 378156757163816,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1588994517921951,\n 977362751042302,\n 1329302387067714,\n 2069348224564088,\n 1586007159625211,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2490539421551682,\n 1985699850375015,\n 2331762317128172,\n 4145097393776678,\n 2521049460190674,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 615817553313996,\n 2245962768078178,\n 482564324326173,\n 2101336843140780,\n 1240914880829407,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1438242482238189,\n 874267817785463,\n 1620810389770625,\n 866155221338671,\n 1040426546798301,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 2403083624110300,\n 2548561409802975,\n 2492699136535911,\n 2358289519456539,\n 3203964320363148,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1913986535403097,\n 1977163223054199,\n 1972905914623196,\n 1650122133472502,\n 1905849310819035,\n ]),\n xy2d: FieldElement51::from_limbs([\n 858174816360838,\n 614595356564037,\n 1099584959044836,\n 636998087084906,\n 1070393269058348,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3666695924830668,\n 3585640662737501,\n 2372994528684236,\n 2628565977288995,\n 3482812783469694,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1994161359147952,\n 2198039369802658,\n 62790022842537,\n 1522306785848169,\n 951223194802833,\n ]),\n xy2d: FieldElement51::from_limbs([\n 852296621440717,\n 431889737774209,\n 370755457746189,\n 437604073958073,\n 627857326892757,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1794955764684156,\n 2586904290013612,\n 1322647643615887,\n 856117964085888,\n 2652432778663153,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 933592377399646,\n 78031722952813,\n 926049890685253,\n 1471649501316246,\n 33789909190376,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1479319468832059,\n 203906207621608,\n 659828362330083,\n 44358398435755,\n 1273573524210803,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1592342143350813,\n 3227219208247713,\n 2345240352078765,\n 2577750109932929,\n 2933512841197243,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2184946892642995,\n 1517382324576002,\n 1557940277419806,\n 2170635134813213,\n 747314658627002,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1823193620577742,\n 1135817878516419,\n 1731253819308581,\n 1031652967267804,\n 2123506616999453,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1346190246005805,\n 2052692552023851,\n 1718128041785940,\n 2491557332978474,\n 3474370880388305,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 424776012994573,\n 281050757243423,\n 626466040846420,\n 990194703866532,\n 38571969885982,\n ]),\n xy2d: FieldElement51::from_limbs([\n 192408346595466,\n 1054889725292349,\n 584097975693004,\n 1447909807397749,\n 2134645004369136,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3169895788615063,\n 3503097743181446,\n 601598510029975,\n 1422812237223371,\n 2121009661378329,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1603348391996783,\n 2066143816131699,\n 1789627290363958,\n 2145705961178118,\n 1985578641438222,\n ]),\n xy2d: FieldElement51::from_limbs([\n 352633958653380,\n 856927627345554,\n 793925083122702,\n 93551575767286,\n 1222010153634215,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1756866499986349,\n 911731956999969,\n 2707505543214075,\n 4006920335263786,\n 822501008147910,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1094036422864347,\n 1897208881572508,\n 1503607738246960,\n 1901060196071406,\n 294068411105729,\n ]),\n xy2d: FieldElement51::from_limbs([\n 587776484399576,\n 1116861711228807,\n 343398777436088,\n 936544065763093,\n 1643746750211060,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 3477749685790410,\n 267997399528836,\n 2953780922004404,\n 3252368924080907,\n 3787792887348381,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 2042368155872443,\n 41662387210459,\n 1676313264498480,\n 1333968523426810,\n 1765708383352310,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1453394896690938,\n 1585795827439909,\n 1469309456804303,\n 1294645324464404,\n 2042954198665899,\n ]),\n },\n AffineNielsPoint {\n y_plus_x: FieldElement51::from_limbs([\n 1810069207599881,\n 1358344669503239,\n 1989371257548167,\n 2316270051121225,\n 3019675451276507,\n ]),\n y_minus_x: FieldElement51::from_limbs([\n 1866114438287676,\n 1663420339568364,\n 1437691317033088,\n 538298302628038,\n 1212711449614363,\n ]),\n xy2d: FieldElement51::from_limbs([\n 1769235035677897,\n 1562012115317882,\n 31277513664750,\n 536198657928416,\n 1976134212537183,\n ]),\n },\n ]);\n\n/// Precomputed NafLookupTable5 for ED25519_BASEPOINT_128_POINT\n/// Contains odd multiples [1B', 3B', 5B', 7B', 9B', 11B', 13B', 15B']\n/// where B' = B * 2^128\n#[cfg(feature = \"precomputed-tables\")]\npub(crate) const AFFINE_ODD_MULTIPLES_OF_BASEPOINT_128: NafLookupTable5 =\n NafLookupTable5([\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2736169168844485,\n 2938750579674528,\n 1463894987748877,\n 2841718686871730,\n 1985876684518203,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1840579227511262,\n 953261538178031,\n 1834370441150886,\n 1427497321143347,\n 1725682124853989,\n ]),\n Z: FieldElement51::from_limbs([\n 445134475408207,\n 1820906444767788,\n 40212681131971,\n 1448661247201044,\n 965705781338211,\n ]),\n T2d: FieldElement51::from_limbs([\n 773498901840598,\n 1322244500169520,\n 2247887491217152,\n 438555850765425,\n 2003166138168448,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 4052979773311382,\n 2779314608743045,\n 2025864608050267,\n 4239124274918649,\n 2047715283211872,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 46708290287836,\n 854678853922811,\n 473843612020374,\n 93416542836491,\n 36887865609850,\n ]),\n Z: FieldElement51::from_limbs([\n 1415283798768778,\n 558152993015714,\n 1009343863091429,\n 202683238470016,\n 1437237189863098,\n ]),\n T2d: FieldElement51::from_limbs([\n 798901183048341,\n 2088882963129737,\n 1959766260461794,\n 785983100035341,\n 1898332425873393,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 1485884064040933,\n 3544995893699855,\n 3336498043486695,\n 1676518325863799,\n 3924305903599551,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 773665168496622,\n 1762929435191211,\n 1242066992989018,\n 976919579682600,\n 1755245776095155,\n ]),\n Z: FieldElement51::from_limbs([\n 364043412623079,\n 1742274550500525,\n 1826931625194320,\n 1613487341804061,\n 1706694369057824,\n ]),\n T2d: FieldElement51::from_limbs([\n 612881112626481,\n 1851646735104528,\n 442596889481956,\n 1350586815509012,\n 139462310881258,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 1368397133953923,\n 257295989327092,\n 2151475747930859,\n 4157544636401722,\n 2431868518957253,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1086943799443606,\n 2156492892436322,\n 1304912429279094,\n 2236825713899676,\n 956336603451346,\n ]),\n Z: FieldElement51::from_limbs([\n 1435533833442335,\n 2143693239375794,\n 284803180569965,\n 1526121151845459,\n 1146944306085907,\n ]),\n T2d: FieldElement51::from_limbs([\n 746985711541365,\n 1325962113197906,\n 2066433367845509,\n 352065757310156,\n 1030077083459185,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 3242158246905157,\n 1655285217271758,\n 979176760705656,\n 2239696862950768,\n 2581654422853147,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1546547740539545,\n 1712475305386299,\n 1360625165812971,\n 1346151673524691,\n 1382871190753080,\n ]),\n Z: FieldElement51::from_limbs([\n 1301309314400143,\n 754500097438053,\n 1939065018255086,\n 827616229574117,\n 877330897268831,\n ]),\n T2d: FieldElement51::from_limbs([\n 1197188348633815,\n 659807996092455,\n 1711852247689765,\n 558583377269877,\n 99988890374241,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2217993410898728,\n 2786874281837735,\n 277442167258821,\n 2384554073376980,\n 2273131766789461,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 109972040445858,\n 950690845410797,\n 17504561734371,\n 523897781536666,\n 750596827295120,\n ]),\n Z: FieldElement51::from_limbs([\n 54891282232941,\n 1760155098862594,\n 331491218058708,\n 1646635945464764,\n 1039579529390631,\n ]),\n T2d: FieldElement51::from_limbs([\n 1742119867631979,\n 1920835943375882,\n 1823982243967588,\n 8235830273089,\n 868879199166457,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 3550694783929828,\n 2336047885420235,\n 1565306465484489,\n 3136433856071917,\n 1830368149412737,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1774723137461971,\n 2098439526999162,\n 1474163710169650,\n 779998216757295,\n 806943695140007,\n ]),\n Z: FieldElement51::from_limbs([\n 2031052346670915,\n 25708785675631,\n 1400901524447834,\n 748277746457979,\n 1216527237136724,\n ]),\n T2d: FieldElement51::from_limbs([\n 1889193855982501,\n 1710313274081557,\n 2073215834484008,\n 1044022860024178,\n 773075805930498,\n ]),\n },\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement51::from_limbs([\n 2127006969359196,\n 2117739369058606,\n 3577576537383903,\n 2671841131559038,\n 4063127780311805,\n ]),\n Y_minus_X: FieldElement51::from_limbs([\n 1402473726670403,\n 2118415406774282,\n 135582654390618,\n 892801005112916,\n 2121295222735022,\n ]),\n Z: FieldElement51::from_limbs([\n 1578281559293828,\n 2156602763973009,\n 2209233981554667,\n 288157485447820,\n 729847524631017,\n ]),\n T2d: FieldElement51::from_limbs([\n 1551520416538582,\n 1392706341252901,\n 2044032904069908,\n 1190995531234982,\n 442600800298007,\n ]),\n },\n ]);\n"},{"id":77,"name":{"Local":"/rustc/library/core/src/slice/index.rs"},"crate_name":"core","contents":null},{"id":78,"name":{"Local":"/rustc/library/core/src/slice/mod.rs"},"crate_name":"core","contents":null},{"id":79,"name":{"Local":"/rustc/library/core/src/fmt/rt.rs"},"crate_name":"core","contents":null},{"id":80,"name":{"Local":"/rustc/library/core/src/fmt/num.rs"},"crate_name":"core","contents":null},{"id":81,"name":{"Local":"curve25519/solana-ed25519/src/montgomery.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Scalar multiplication on the Montgomery form of Curve25519.\n//!\n//! To avoid notational confusion with the Edwards code, we use\n//! variables \\\\( u, v \\\\) for the Montgomery curve, so that “Montgomery\n//! \\\\(u\\\\)” here corresponds to “Montgomery \\\\(x\\\\)” elsewhere.\n//!\n//! Montgomery arithmetic works not on the curve itself, but on the\n//! \\\\(u\\\\)-line, which discards sign information and unifies the curve\n//! and its quadratic twist. See [_Montgomery curves and their\n//! arithmetic_][costello-smith] by Costello and Smith for more details.\n//!\n//! The `MontgomeryPoint` struct contains the affine \\\\(u\\\\)-coordinate\n//! \\\\(u\\_0(P)\\\\) of a point \\\\(P\\\\) on either the curve or the twist.\n//! Here the map \\\\(u\\_0 : \\mathcal M \\rightarrow \\mathbb F\\_p \\\\) is\n//! defined by \\\\(u\\_0((u,v)) = u\\\\); \\\\(u\\_0(\\mathcal O) = 0\\\\). See\n//! section 5.4 of Costello-Smith for more details.\n//!\n//! # Scalar Multiplication\n//!\n//! Scalar multiplication on `MontgomeryPoint`s is provided by the `*`\n//! operator, which implements the Montgomery ladder.\n//!\n//! # Edwards Conversion\n//!\n//! The \\\\(2\\\\)-to-\\\\(1\\\\) map from the Edwards model to the Montgomery\n//! \\\\(u\\\\)-line is provided by `EdwardsPoint::to_montgomery()`.\n//!\n//! To lift a `MontgomeryPoint` to an `EdwardsPoint`, use\n//! `MontgomeryPoint::to_edwards()`, which takes a sign parameter.\n//! This function rejects `MontgomeryPoints` which correspond to points\n//! on the twist.\n//!\n//! [costello-smith]: https://eprint.iacr.org/2017/212.pdf\n\n// We allow non snake_case names because coordinates in projective space are\n// traditionally denoted by the capitalisation of their respective\n// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my\n// affine and projective cakes and eat both of them too.\n#![allow(non_snake_case)]\n\nuse core::{\n hash::{Hash, Hasher},\n ops::{Mul, MulAssign},\n};\n\nuse crate::constants::APLUS2_OVER_FOUR;\n#[cfg(feature = \"digest\")]\nuse crate::constants::{MONTGOMERY_A, MONTGOMERY_A_NEG, SQRT_M1};\nuse crate::edwards::{CompressedEdwardsY, EdwardsPoint};\nuse crate::field::FieldElement;\nuse crate::scalar::{Scalar, clamp_integer};\n\nuse crate::traits::Identity;\n\nuse subtle::Choice;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\n// We need the const 2^((p+3)/8) for elligator_encode. This def is checked in tests::consts()\n#[cfg(feature = \"digest\")]\nconst FE_C2: FieldElement = FieldElement::from_limbs([\n 1718705420411057,\n 234908883556509,\n 2233514472574048,\n 2117202627021982,\n 765476049583133,\n]);\n\n/// Holds the \\\\(u\\\\)-coordinate of a point on the Montgomery form of\n/// Curve25519 or its twist.\n#[derive(Copy, Clone, Debug, Default)]\n#[cfg_attr(feature = \"serde\", derive(serde::Serialize, serde::Deserialize))]\npub struct MontgomeryPoint(pub [u8; 32]);\n\n/// Equality of `MontgomeryPoint`s is defined mod p.\nimpl ConstantTimeEq for MontgomeryPoint {\n fn ct_eq(&self, other: &MontgomeryPoint) -> Choice {\n let self_fe = FieldElement::from_bytes(&self.0);\n let other_fe = FieldElement::from_bytes(&other.0);\n\n self_fe.ct_eq(&other_fe)\n }\n}\n\nimpl ConditionallySelectable for MontgomeryPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n Self(<[u8; 32]>::conditional_select(&a.0, &b.0, choice))\n }\n}\n\nimpl PartialEq for MontgomeryPoint {\n fn eq(&self, other: &MontgomeryPoint) -> bool {\n self.ct_eq(other).into()\n }\n}\n\nimpl Eq for MontgomeryPoint {}\n\n// Equal MontgomeryPoints must hash to the same value. So we have to get them into a canonical\n// encoding first\nimpl Hash for MontgomeryPoint {\n fn hash(&self, state: &mut H) {\n // Do a round trip through a `FieldElement`. `as_bytes` is guaranteed to give a canonical\n // 32-byte encoding\n let canonical_bytes = FieldElement::from_bytes(&self.0).to_bytes();\n canonical_bytes.hash(state);\n }\n}\n\nimpl Identity for MontgomeryPoint {\n /// Return the group identity element, which has order 4.\n fn identity() -> MontgomeryPoint {\n MontgomeryPoint([0u8; 32])\n }\n}\n\n#[cfg(feature = \"zeroize\")]\nimpl Zeroize for MontgomeryPoint {\n fn zeroize(&mut self) {\n self.0.zeroize();\n }\n}\n\nimpl MontgomeryPoint {\n /// Fixed-base scalar multiplication (i.e. multiplication by the base point).\n pub fn mul_base(scalar: &Scalar) -> Self {\n EdwardsPoint::mul_base(scalar).to_montgomery()\n }\n\n /// Multiply this point by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_clamped(self, bytes: [u8; 32]) -> Self {\n // We have to construct a Scalar that is not reduced mod l, which breaks scalar invariant\n // #2. But #2 is not necessary for correctness of variable-base multiplication. All that\n // needs to hold is invariant #1, i.e., the scalar is less than 2^255. This is guaranteed\n // by clamping.\n // Further, we don't do any reduction or arithmetic with this clamped value, so there's no\n // issues arising from the fact that the curve point is not necessarily in the prime-order\n // subgroup.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n s * self\n }\n\n /// Multiply the basepoint by `clamp_integer(bytes)`. For a description of clamping, see\n /// [`clamp_integer`].\n pub fn mul_base_clamped(bytes: [u8; 32]) -> Self {\n // See reasoning in Self::mul_clamped why it is OK to make an unreduced Scalar here. We\n // note that fixed-base multiplication is also defined for all values of `bytes` less than\n // 2^255.\n let s = Scalar {\n bytes: clamp_integer(bytes),\n };\n Self::mul_base(&s)\n }\n\n /// Given `self` \\\\( = u\\_0(P) \\\\), and a big-endian bit representation of an integer\n /// \\\\(n\\\\), return \\\\( u\\_0(\\[n\\]P) \\\\). This is constant time in the length of `bits`.\n ///\n /// **NOTE:** You probably do not want to use this function. Almost every protocol built on\n /// Curve25519 uses _clamped multiplication_, explained\n /// [here](https://neilmadden.blog/2020/05/28/whats-the-curve25519-clamping-all-about/).\n /// When in doubt, use [`Self::mul_clamped`].\n pub fn mul_bits_be(&self, bits: impl Iterator) -> MontgomeryPoint {\n // Algorithm 8 of Costello-Smith 2017\n let affine_u = FieldElement::from_bytes(&self.0);\n let mut x0 = ProjectivePoint::identity();\n let mut x1 = ProjectivePoint {\n U: affine_u,\n W: FieldElement::ONE,\n };\n\n // Go through the bits from most to least significant, using a sliding window of 2\n let mut prev_bit = false;\n for cur_bit in bits {\n let choice: u8 = (prev_bit ^ cur_bit) as u8;\n\n debug_assert!(choice == 0 || choice == 1);\n\n ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());\n differential_add_and_double(&mut x0, &mut x1, &affine_u);\n\n prev_bit = cur_bit;\n }\n // The final value of prev_bit above is scalar.bits()[0], i.e., the LSB of scalar\n ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(prev_bit as u8));\n // Don't leave the bit in the stack\n #[cfg(feature = \"zeroize\")]\n prev_bit.zeroize();\n\n x0.as_affine()\n }\n\n /// View this `MontgomeryPoint` as an array of bytes.\n pub const fn as_bytes(&self) -> &[u8; 32] {\n &self.0\n }\n\n /// Convert this `MontgomeryPoint` to an array of bytes.\n pub const fn to_bytes(&self) -> [u8; 32] {\n self.0\n }\n\n /// Attempt to convert to an `EdwardsPoint`, using the supplied\n /// choice of sign for the `EdwardsPoint`.\n ///\n /// # Inputs\n ///\n /// * `sign`: a `u8` donating the desired sign of the resulting\n /// `EdwardsPoint`. `0` denotes positive and `1` negative.\n ///\n /// # Return\n ///\n /// * `Some(EdwardsPoint)` if `self` is the \\\\(u\\\\)-coordinate of a\n /// point on (the Montgomery form of) Curve25519;\n ///\n /// * `None` if `self` is the \\\\(u\\\\)-coordinate of a point on the\n /// twist of (the Montgomery form of) Curve25519;\n ///\n pub fn to_edwards(&self, sign: u8) -> Option {\n // To decompress the Montgomery u coordinate to an\n // `EdwardsPoint`, we apply the birational map to obtain the\n // Edwards y coordinate, then do Edwards decompression.\n //\n // The birational map is y = (u-1)/(u+1).\n //\n // The exceptional points are the zeros of the denominator,\n // i.e., u = -1.\n //\n // But when u = -1, v^2 = u*(u^2+486662*u+1) = 486660.\n //\n // Since this is nonsquare mod p, u = -1 corresponds to a point\n // on the twist, not the curve, so we can reject it early.\n\n let u = FieldElement::from_bytes(&self.0);\n\n if u == FieldElement::MINUS_ONE {\n return None;\n }\n\n let one = FieldElement::ONE;\n\n let y = &(&u - &one) * &(&u + &one).invert();\n\n let mut y_bytes = y.to_bytes();\n y_bytes[31] ^= sign << 7;\n\n CompressedEdwardsY(y_bytes).decompress()\n }\n}\n\n#[cfg(feature = \"digest\")]\n/// Perform the Elligator2 mapping to a tuple `(xn, xd, yn, yd)` such that\n/// `(xn / xd, yn / yd)` is a point on curve25519.\n///\n/// See \npub(crate) fn elligator_encode(\n u: &FieldElement,\n) -> (FieldElement, FieldElement, FieldElement, FieldElement) {\n // We follow https://www.rfc-editor.org/rfc/rfc9380.html#appendix-G.2.1\n\n use core::ops::Neg;\n let one = FieldElement::ONE;\n let c2 = FE_C2;\n\n // 1. tv1 = u^2\n // 2. tv1 = 2 * tv1\n let tv1 = u.square2();\n // 3. xd = tv1 + 1\n let xd = &one + &tv1;\n // 4. x1n = -J\n let x1n = MONTGOMERY_A_NEG;\n // 5. tv2 = xd^2\n let tv2 = xd.square();\n // 6. gxd = tv2 * xd\n let gxd = &tv2 * &xd;\n // 7. gx1 = J * tv1\n let gx1 = &MONTGOMERY_A * &tv1;\n // 8. gx1 = gx1 * x1n\n let gx1 = &gx1 * &x1n;\n // 9. gx1 = gx1 + tv2\n let gx1 = &gx1 + &tv2;\n // 10. gx1 = gx1 * x1n\n let gx1 = &gx1 * &x1n;\n // 11. tv3 = gxd^2\n let tv3 = gxd.square();\n // 12. tv2 = tv3^2\n let tv2 = tv3.square();\n // 13. tv3 = tv3 * gxd\n let tv3 = &tv3 * &gxd;\n // 14. tv3 = tv3 * gx1\n let tv3 = &tv3 * &gx1;\n // 15. tv2 = tv2 * tv3\n let tv2 = &tv2 * &tv3;\n // 16. y11 = tv2^c4\n let y11 = tv2.pow_p58();\n // 17. y11 = y11 * tv3\n let y11 = &y11 * &tv3;\n // 18. y12 = y11 * c3\n let y12 = &y11 * &SQRT_M1;\n // 19. tv2 = y11^2\n let tv2 = y11.square();\n // 20. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 21. e1 = tv2 == gx1\n let e1 = tv2.ct_eq(&gx1);\n // 22. y1 = CMOV(y12, y11, e1)\n let y1 = FieldElement::conditional_select(&y12, &y11, e1);\n // 23. x2n = x1n * tv1\n let x2n = &x1n * &tv1;\n // 24. y21 = y11 * u\n let y21 = &y11 * u;\n // 25. y21 = y21 * c2\n let y21 = &y21 * &c2;\n // 26. y22 = y21 * c3\n let y22 = &y21 * &SQRT_M1;\n // 27. gx2 = gx1 * tv1\n let gx2 = &gx1 * &tv1;\n // 28. tv2 = y21^2\n let tv2 = y21.square();\n // 29. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 30. e2 = tv2 == gx2\n let e2 = tv2.ct_eq(&gx2);\n // 31. y2 = CMOV(y22, y21, e2)\n let y2 = FieldElement::conditional_select(&y22, &y21, e2);\n // 32. tv2 = y1^2\n let tv2 = y1.square();\n // 33. tv2 = tv2 * gxd\n let tv2 = &tv2 * &gxd;\n // 34. e3 = tv2 == gx1\n let e3 = tv2.ct_eq(&gx1);\n // 35. xn = CMOV(x2n, x1n, e3)\n let xn = FieldElement::conditional_select(&x2n, &x1n, e3);\n // 36. y = CMOV(y2, y1, e3)\n let y = FieldElement::conditional_select(&y2, &y1, e3);\n // 37. e4 = sgn0(y) == 1\n let e4 = y.is_negative();\n // 38. y = CMOV(y, -y, e3 XOR e4)\n let y = FieldElement::conditional_select(&y, &y.neg(), e3 ^ e4);\n // 39. return (xn, xd, y, 1)\n\n (xn, xd, y, one)\n}\n\n/// A `ProjectivePoint` holds a point on the projective line\n/// \\\\( \\mathbb P(\\mathbb F\\_p) \\\\), which we identify with the Kummer\n/// line of the Montgomery curve.\n#[derive(Copy, Clone, Debug)]\nstruct ProjectivePoint {\n pub U: FieldElement,\n pub W: FieldElement,\n}\n\nimpl Identity for ProjectivePoint {\n fn identity() -> ProjectivePoint {\n ProjectivePoint {\n U: FieldElement::ONE,\n W: FieldElement::ZERO,\n }\n }\n}\n\nimpl Default for ProjectivePoint {\n fn default() -> ProjectivePoint {\n ProjectivePoint::identity()\n }\n}\n\nimpl ConditionallySelectable for ProjectivePoint {\n fn conditional_select(\n a: &ProjectivePoint,\n b: &ProjectivePoint,\n choice: Choice,\n ) -> ProjectivePoint {\n ProjectivePoint {\n U: FieldElement::conditional_select(&a.U, &b.U, choice),\n W: FieldElement::conditional_select(&a.W, &b.W, choice),\n }\n }\n}\n\nimpl ProjectivePoint {\n /// Dehomogenize this point to affine coordinates.\n ///\n /// # Return\n ///\n /// * \\\\( u = U / W \\\\) if \\\\( W \\neq 0 \\\\);\n /// * \\\\( 0 \\\\) if \\\\( W \\eq 0 \\\\);\n pub fn as_affine(&self) -> MontgomeryPoint {\n let u = &self.U * &self.W.invert();\n MontgomeryPoint(u.to_bytes())\n }\n}\n\n/// Perform the double-and-add step of the Montgomery ladder.\n///\n/// Given projective points\n/// \\\\( (U\\_P : W\\_P) = u(P) \\\\),\n/// \\\\( (U\\_Q : W\\_Q) = u(Q) \\\\),\n/// and the affine difference\n/// \\\\( u\\_{P-Q} = u(P-Q) \\\\), set\n/// $$\n/// (U\\_P : W\\_P) \\gets u(\\[2\\]P)\n/// $$\n/// and\n/// $$\n/// (U\\_Q : W\\_Q) \\gets u(P + Q).\n/// $$\n#[rustfmt::skip] // keep alignment of explanatory comments\nfn differential_add_and_double(\n P: &mut ProjectivePoint,\n Q: &mut ProjectivePoint,\n affine_PmQ: &FieldElement,\n) {\n let t0 = &P.U + &P.W;\n let t1 = &P.U - &P.W;\n let t2 = &Q.U + &Q.W;\n let t3 = &Q.U - &Q.W;\n\n let t4 = t0.square(); // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2\n let t5 = t1.square(); // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2\n\n let t6 = &t4 - &t5; // 4 U_P W_P\n\n let t7 = &t0 * &t3; // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q\n let t8 = &t1 * &t2; // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q\n\n let t9 = &t7 + &t8; // 2 (U_P U_Q - W_P W_Q)\n let t10 = &t7 - &t8; // 2 (W_P U_Q - U_P W_Q)\n\n let t11 = t9.square(); // 4 (U_P U_Q - W_P W_Q)^2\n let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2\n\n let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q\n\n let t14 = &t4 * &t5; // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2\n let t15 = &t13 + &t5; // (U_P - W_P)^2 + (A + 2) U_P W_P\n\n let t16 = &t6 * &t15; // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)\n\n let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2\n let t18 = t11; // W_D * 4 (U_P U_Q - W_P W_Q)^2\n\n P.U = t14; // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2\n P.W = t16; // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)\n Q.U = t18; // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2\n Q.W = t17; // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2\n}\n\ndefine_mul_assign_variants!(LHS = MontgomeryPoint, RHS = Scalar);\n\ndefine_mul_variants!(\n LHS = MontgomeryPoint,\n RHS = Scalar,\n Output = MontgomeryPoint\n);\ndefine_mul_variants!(\n LHS = Scalar,\n RHS = MontgomeryPoint,\n Output = MontgomeryPoint\n);\n\n/// Multiply this `MontgomeryPoint` by a `Scalar`.\nimpl Mul<&Scalar> for &MontgomeryPoint {\n type Output = MontgomeryPoint;\n\n /// Given `self` \\\\( = u\\_0(P) \\\\), and a `Scalar` \\\\(n\\\\), return \\\\( u\\_0(\\[n\\]P) \\\\)\n fn mul(self, scalar: &Scalar) -> MontgomeryPoint {\n // We multiply by the integer representation of the given Scalar. By scalar invariant #1,\n // the MSB is 0, so we can skip it.\n self.mul_bits_be(scalar.bits_le().rev().skip(1))\n }\n}\n\nimpl MulAssign<&Scalar> for MontgomeryPoint {\n fn mul_assign(&mut self, scalar: &Scalar) {\n *self = (self as &MontgomeryPoint) * scalar;\n }\n}\n\nimpl Mul<&MontgomeryPoint> for &Scalar {\n type Output = MontgomeryPoint;\n\n fn mul(self, point: &MontgomeryPoint) -> MontgomeryPoint {\n point * self\n }\n}\n\n// ------------------------------------------------------------------------\n// Tests\n// ------------------------------------------------------------------------\n\n#[cfg(test)]\nmod test {\n use super::*;\n use crate::constants;\n\n #[cfg(feature = \"rand_core\")]\n use rand::CryptoRng;\n use rand::RngCore;\n\n #[test]\n fn identity_in_different_coordinates() {\n let id_projective = ProjectivePoint::identity();\n let id_montgomery = id_projective.as_affine();\n\n assert!(id_montgomery == MontgomeryPoint::identity());\n }\n\n #[test]\n fn identity_in_different_models() {\n assert!(EdwardsPoint::identity().to_montgomery() == MontgomeryPoint::identity());\n }\n\n #[test]\n #[cfg(feature = \"serde\")]\n fn serde_bincode_basepoint_roundtrip() {\n use bincode;\n\n let encoded = bincode::serialize(&constants::X25519_BASEPOINT).unwrap();\n let decoded: MontgomeryPoint = bincode::deserialize(&encoded).unwrap();\n\n assert_eq!(encoded.len(), 32);\n assert_eq!(decoded, constants::X25519_BASEPOINT);\n\n let raw_bytes = constants::X25519_BASEPOINT.as_bytes();\n let bp: MontgomeryPoint = bincode::deserialize(raw_bytes).unwrap();\n assert_eq!(bp, constants::X25519_BASEPOINT);\n }\n\n /// Test Montgomery -> Edwards on the X/Ed25519 basepoint\n #[test]\n fn basepoint_montgomery_to_edwards() {\n // sign bit = 0 => basepoint\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT,\n constants::X25519_BASEPOINT\n .to_edwards(0)\n .expect(\"sign bit 0 should decompress\")\n );\n // sign bit = 1 => minus basepoint\n assert_eq!(\n -constants::ED25519_BASEPOINT_POINT,\n constants::X25519_BASEPOINT\n .to_edwards(1)\n .expect(\"sign bit 1 should decompress\")\n );\n }\n\n /// Test Edwards -> Montgomery on the X/Ed25519 basepoint\n #[test]\n fn basepoint_edwards_to_montgomery() {\n assert_eq!(\n constants::ED25519_BASEPOINT_POINT.to_montgomery(),\n constants::X25519_BASEPOINT\n );\n }\n\n /// Check that Montgomery -> Edwards fails for points on the twist.\n #[test]\n fn montgomery_to_edwards_rejects_twist() {\n let one = FieldElement::ONE;\n\n // u = 2 corresponds to a point on the twist.\n let two = MontgomeryPoint((&one + &one).to_bytes());\n\n assert!(two.to_edwards(0).is_none());\n\n // u = -1 corresponds to a point on the twist, but should be\n // checked explicitly because it's an exceptional point for the\n // birational map. For instance, libsignal will accept it.\n let minus_one = MontgomeryPoint((-&one).to_bytes());\n\n assert!(minus_one.to_edwards(0).is_none());\n }\n\n #[test]\n fn eq_defined_mod_p() {\n let mut u18_bytes = [0u8; 32];\n u18_bytes[0] = 18;\n let u18 = MontgomeryPoint(u18_bytes);\n let u18_unred = MontgomeryPoint([255; 32]);\n\n assert_eq!(u18, u18_unred);\n }\n\n /// Returns a random point on the prime-order subgroup\n #[cfg(feature = \"rand_core\")]\n fn rand_prime_order_point(rng: &mut R) -> EdwardsPoint {\n let s: Scalar = Scalar::random(rng);\n EdwardsPoint::mul_base(&s)\n }\n\n /// Given a bytestring that's little-endian at the byte level, return an iterator over all the\n /// bits, in little-endian order.\n fn bytestring_bits_le(x: &[u8]) -> impl DoubleEndedIterator + Clone + '_ {\n let bitlen = x.len() * 8;\n (0..bitlen).map(|i| {\n // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index\n // the byte. Since self.bytes is little-endian at the byte level, this iterator is\n // little-endian on the bit level\n ((x[i >> 3] >> (i & 7)) & 1u8) == 1\n })\n }\n\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn montgomery_ladder_matches_edwards_scalarmult() {\n let mut csprng = rand::thread_rng();\n\n for _ in 0..100 {\n let p_edwards = rand_prime_order_point(&mut csprng);\n let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();\n\n let s: Scalar = Scalar::random(&mut csprng);\n let expected = s * p_edwards;\n let result = s * p_montgomery;\n\n assert_eq!(result, expected.to_montgomery())\n }\n }\n\n // Tests that, on the prime-order subgroup, MontgomeryPoint::mul_bits_be is the same as\n // multiplying by the Scalar representation of the same bits\n #[cfg(feature = \"rand_core\")]\n #[test]\n fn montgomery_mul_bits_be() {\n let mut csprng = rand::thread_rng();\n\n for _ in 0..100 {\n // Make a random prime-order point P\n let p_edwards = rand_prime_order_point(&mut csprng);\n let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();\n\n // Make a random integer b\n let mut bigint = [0u8; 64];\n csprng.fill_bytes(&mut bigint[..]);\n let bigint_bits_be = bytestring_bits_le(&bigint).rev();\n\n // Check that bP is the same whether calculated as scalar-times-edwards or\n // integer-times-montgomery.\n let expected = Scalar::from_bytes_mod_order_wide(&bigint) * p_edwards;\n let result = p_montgomery.mul_bits_be(bigint_bits_be);\n assert_eq!(result, expected.to_montgomery())\n }\n }\n\n // Tests that MontgomeryPoint::mul_bits_be is consistent on any point, even ones that might be\n // on the curve's twist. Specifically, this tests that b₁(b₂P) == b₂(b₁P) for random\n // integers b₁, b₂ and random (curve or twist) point P.\n #[test]\n fn montgomery_mul_bits_be_twist() {\n let mut csprng = rand::rngs::OsRng;\n\n for _ in 0..100 {\n // Make a random point P on the curve or its twist\n let p_montgomery = {\n let mut buf = [0u8; 32];\n csprng.fill_bytes(&mut buf);\n MontgomeryPoint(buf)\n };\n\n // Compute two big integers b₁ and b₂\n let mut bigint1 = [0u8; 64];\n let mut bigint2 = [0u8; 64];\n csprng.fill_bytes(&mut bigint1[..]);\n csprng.fill_bytes(&mut bigint2[..]);\n\n // Compute b₁P and b₂P\n let bigint1_bits_be = bytestring_bits_le(&bigint1).rev();\n let bigint2_bits_be = bytestring_bits_le(&bigint2).rev();\n let prod1 = p_montgomery.mul_bits_be(bigint1_bits_be.clone());\n let prod2 = p_montgomery.mul_bits_be(bigint2_bits_be.clone());\n\n // Check that b₁(b₂P) == b₂(b₁P)\n assert_eq!(\n prod1.mul_bits_be(bigint2_bits_be),\n prod2.mul_bits_be(bigint1_bits_be)\n );\n }\n }\n\n /// Check that mul_base_clamped and mul_clamped agree\n #[test]\n fn mul_base_clamped() {\n let mut csprng = rand::rngs::OsRng;\n\n // Test agreement on a large integer. Even after clamping, this is not reduced mod l.\n let a_bytes = [0xff; 32];\n assert_eq!(\n MontgomeryPoint::mul_base_clamped(a_bytes),\n constants::X25519_BASEPOINT.mul_clamped(a_bytes)\n );\n\n // Test agreement on random integers\n for _ in 0..100 {\n // This will be reduced mod l with probability l / 2^256 ≈ 6.25%\n let mut a_bytes = [0u8; 32];\n csprng.fill_bytes(&mut a_bytes);\n\n assert_eq!(\n MontgomeryPoint::mul_base_clamped(a_bytes),\n constants::X25519_BASEPOINT.mul_clamped(a_bytes)\n );\n }\n }\n\n #[cfg(feature = \"alloc\")]\n #[cfg(feature = \"digest\")]\n const ELLIGATOR_CORRECT_OUTPUT: [u8; 32] = [\n 0x5f, 0x35, 0x20, 0x00, 0x1c, 0x6c, 0x99, 0x36, 0xa3, 0x12, 0x06, 0xaf, 0xe7, 0xc7, 0xac,\n 0x22, 0x4e, 0x88, 0x61, 0x61, 0x9b, 0xf9, 0x88, 0x72, 0x44, 0x49, 0x15, 0x89, 0x9d, 0x95,\n 0xf4, 0x6e,\n ];\n\n #[test]\n #[cfg(feature = \"alloc\")]\n #[cfg(feature = \"digest\")]\n fn montgomery_elligator_correct() {\n use alloc::vec::Vec;\n let bytes: Vec = (0u8..32u8).collect();\n let bits_in: [u8; 32] = (&bytes[..]).try_into().expect(\"Range invariant broken\");\n\n let fe = FieldElement::from_bytes(&bits_in);\n let (un, ud, ..) = elligator_encode(&fe);\n let u = &un * &ud.invert();\n assert_eq!(u.to_bytes(), ELLIGATOR_CORRECT_OUTPUT);\n }\n\n #[test]\n #[cfg(feature = \"digest\")]\n fn montgomery_elligator_zero_zero() {\n let zero = [0u8; 32];\n let fe = FieldElement::from_bytes(&zero);\n let (un, ud, ..) = elligator_encode(&fe);\n let u = &un * &ud.invert();\n assert_eq!(u.to_bytes(), zero);\n }\n\n // Check that FE_C2 is correctly defined\n #[test]\n #[cfg(feature = \"digest\")]\n fn c2() {\n let one = FieldElement::ONE;\n let two = &one + &one;\n // c2 = 2^((p+3)/8) = 2^((p-5)/8 + 8/8) = 2*2^((p-5)/8)\n let c2 = &two * &(two.pow_p58());\n\n assert_eq!(c2, FE_C2);\n }\n}\n"},{"id":82,"name":{"Local":"curve25519/solana-ed25519/src/constants.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n//! Various constants, such as the Ristretto and Ed25519 basepoints.\n\n#![allow(non_snake_case)]\n\nuse crate::edwards::CompressedEdwardsY;\nuse crate::montgomery::MontgomeryPoint;\nuse crate::ristretto::{CompressedRistretto, RistrettoPoint};\nuse crate::scalar::Scalar;\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::edwards::EdwardsBasepointTable;\n\npub use crate::backend::serial::u64::constants::*;\n\n/// The Ed25519 basepoint, in `CompressedEdwardsY` format.\n///\n/// This is the little-endian byte encoding of \\\\( 4/5 \\pmod p \\\\),\n/// which is the \\\\(y\\\\)-coordinate of the Ed25519 basepoint.\n///\n/// The sign bit is 0 since the basepoint has \\\\(x\\\\) chosen to be positive.\npub const ED25519_BASEPOINT_COMPRESSED: CompressedEdwardsY = CompressedEdwardsY([\n 0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,\n 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,\n]);\n\n/// The X25519 basepoint, in `MontgomeryPoint` format.\npub const X25519_BASEPOINT: MontgomeryPoint = MontgomeryPoint([\n 0x09, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n]);\n\n/// The Ristretto basepoint, in `CompressedRistretto` format.\npub const RISTRETTO_BASEPOINT_COMPRESSED: CompressedRistretto = CompressedRistretto([\n 0xe2, 0xf2, 0xae, 0x0a, 0x6a, 0xbc, 0x4e, 0x71, 0xa8, 0x84, 0xa9, 0x61, 0xc5, 0x00, 0x51, 0x5f,\n 0x58, 0xe3, 0x0b, 0x6a, 0xa5, 0x82, 0xdd, 0x8d, 0xb6, 0xa6, 0x59, 0x45, 0xe0, 0x8d, 0x2d, 0x76,\n]);\n\n/// The Ristretto basepoint, as a `RistrettoPoint`.\n///\n/// This is called `_POINT` to distinguish it from `_TABLE`, which\n/// provides fast scalar multiplication.\npub const RISTRETTO_BASEPOINT_POINT: RistrettoPoint = RistrettoPoint(ED25519_BASEPOINT_POINT);\n\n/// `BASEPOINT_ORDER` is the order of the Ristretto group and of the Ed25519 basepoint, i.e.,\n/// $$\n/// \\ell = 2^\\{252\\} + 27742317777372353535851937790883648493.\n/// $$\npub(crate) const BASEPOINT_ORDER: Scalar = Scalar {\n bytes: [\n 0xed, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde,\n 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x10,\n ],\n};\n\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::ristretto::RistrettoBasepointTable;\n\n/// The Ristretto basepoint, as a `RistrettoBasepointTable` for scalar multiplication.\n#[cfg(feature = \"precomputed-tables\")]\npub static RISTRETTO_BASEPOINT_TABLE: &RistrettoBasepointTable = unsafe {\n // SAFETY: `RistrettoBasepointTable` is a `#[repr(transparent)]` newtype of\n // `EdwardsBasepointTable`\n &*(ED25519_BASEPOINT_TABLE as *const EdwardsBasepointTable as *const RistrettoBasepointTable)\n};\n\n#[cfg(test)]\nmod test {\n use crate::constants;\n use crate::field::FieldElement;\n use crate::traits::{IsIdentity, ValidityCheck};\n\n #[test]\n fn test_eight_torsion() {\n for i in 0..8 {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(3);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n #[test]\n fn test_four_torsion() {\n for i in (0..8).filter(|i| i % 2 == 0) {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(2);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n #[test]\n fn test_two_torsion() {\n for i in (0..8).filter(|i| i % 4 == 0) {\n let Q = constants::EIGHT_TORSION[i].mul_by_pow_2(1);\n assert!(Q.is_valid());\n assert!(Q.is_identity());\n }\n }\n\n /// Test that SQRT_M1 is the positive square root of -1\n #[test]\n fn test_sqrt_minus_one() {\n let minus_one = FieldElement::MINUS_ONE;\n let sqrt_m1_sq = &constants::SQRT_M1 * &constants::SQRT_M1;\n assert_eq!(minus_one, sqrt_m1_sq);\n assert!(bool::from(!constants::SQRT_M1.is_negative()));\n }\n\n #[test]\n fn test_sqrt_constants_sign() {\n let minus_one = FieldElement::MINUS_ONE;\n let (was_nonzero_square, invsqrt_m1) = minus_one.invsqrt();\n assert!(bool::from(was_nonzero_square));\n let sign_test_sqrt = &invsqrt_m1 * &constants::SQRT_M1;\n assert_eq!(sign_test_sqrt, minus_one);\n }\n\n /// Test that d = -121665/121666\n #[test]\n fn test_d_vs_ratio() {\n use crate::backend::serial::u64::field::FieldElement51;\n let a = -&FieldElement51([121665, 0, 0, 0, 0]);\n let b = FieldElement51([121666, 0, 0, 0, 0]);\n let d = &a * &b.invert();\n let d2 = &d + &d;\n assert_eq!(d, constants::EDWARDS_D);\n assert_eq!(d2, constants::EDWARDS_D2);\n }\n\n #[test]\n fn test_sqrt_ad_minus_one() {\n let a = FieldElement::MINUS_ONE;\n let ad_minus_one = &(&a * &constants::EDWARDS_D) + &a;\n let should_be_ad_minus_one = constants::SQRT_AD_MINUS_ONE.square();\n assert_eq!(should_be_ad_minus_one, ad_minus_one);\n }\n\n /// Test that ED25519_SQRTAM2 squared is MONTGOMERY_A_NEG - 2\n #[test]\n #[cfg(feature = \"digest\")]\n fn test_sqrt_a_minus_2() {\n let one = FieldElement::ONE;\n let a_minus_two = &(&constants::MONTGOMERY_A_NEG - &one) - &one;\n\n assert_eq!(constants::ED25519_SQRTAM2.square(), a_minus_two)\n }\n}\n"},{"id":83,"name":{"Local":"/rustc/library/core/src/num/niche_types.rs"},"crate_name":"core","contents":null},{"id":84,"name":{"Local":"/rustc/library/core/src/ops/bit.rs"},"crate_name":"core","contents":null},{"id":85,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/variable_base.rs"},"crate_name":"curve25519","contents":"#![allow(non_snake_case)]\n\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::Scalar;\nuse crate::traits::Identity;\nuse crate::window::LookupTable;\n\n/// Perform constant-time, variable-base scalar multiplication.\n#[rustfmt::skip] // keep alignment of explanatory comments\npub(crate) fn mul(point: &EdwardsPoint, scalar: &Scalar) -> EdwardsPoint {\n // Construct a lookup table of [P,2P,3P,4P,5P,6P,7P,8P]\n let lookup_table = LookupTable::::from(point);\n // Setting s = scalar, compute\n //\n // s = s_0 + s_1*16^1 + ... + s_63*16^63,\n //\n // with `-8 ≤ s_i < 8` for `0 ≤ i < 63` and `-8 ≤ s_63 ≤ 8`.\n // This decomposition requires s < 2^255, which is guaranteed by Scalar invariant #1.\n let scalar_digits = scalar.as_radix_16();\n // Compute s*P as\n //\n // s*P = P*(s_0 + s_1*16^1 + s_2*16^2 + ... + s_63*16^63)\n // s*P = P*s_0 + P*s_1*16^1 + P*s_2*16^2 + ... + P*s_63*16^63\n // s*P = P*s_0 + 16*(P*s_1 + 16*(P*s_2 + 16*( ... + P*s_63)...))\n //\n // We sum right-to-left.\n\n // Unwrap first loop iteration to save computing 16*identity\n let mut tmp2;\n let mut tmp3 = EdwardsPoint::identity();\n let mut tmp1 = &tmp3 + &lookup_table.select(scalar_digits[63]);\n // Now tmp1 = s_63*P in P1xP1 coords\n for i in (0..63).rev() {\n tmp2 = tmp1.as_projective(); // tmp2 = (prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 2*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 2*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 4*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 4*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 8*(prev) in P1xP1 coords\n tmp2 = tmp1.as_projective(); // tmp2 = 8*(prev) in P2 coords\n tmp1 = tmp2.double(); // tmp1 = 16*(prev) in P1xP1 coords\n tmp3 = tmp1.as_extended(); // tmp3 = 16*(prev) in P3 coords\n tmp1 = &tmp3 + &lookup_table.select(scalar_digits[i]);\n // Now tmp1 = s_i*P + 16*(prev) in P1xP1 coords\n }\n tmp1.as_extended()\n}\n"},{"id":86,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Implementations of various scalar multiplication algorithms.\n//!\n//! Note that all of these implementations use serial code for field\n//! arithmetic with the multi-model strategy described in the\n//! `curve_models` module. The vectorized AVX2 backend has its own\n//! scalar multiplication implementations, since it only uses one\n//! curve model.\n\n#[allow(missing_docs)]\npub mod variable_base;\n\n#[allow(missing_docs)]\npub mod vartime_double_base;\n\n#[allow(missing_docs)]\npub mod vartime_triple_base;\n\n#[cfg(feature = \"alloc\")]\npub mod straus;\n\n#[cfg(feature = \"alloc\")]\npub mod precomputed_straus;\n\n#[cfg(feature = \"alloc\")]\npub mod pippenger;\n"},{"id":87,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/vartime_double_base.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-dalek.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n#![allow(non_snake_case)]\n\nuse core::cmp::Ordering;\n\nuse crate::backend::serial::curve_models::{CompletedPoint, ProjectiveNielsPoint, ProjectivePoint};\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::backend::serial::curve_models::AffineNielsPoint;\n#[cfg(feature = \"precomputed-tables\")]\nuse crate::window::NafLookupTable8;\nuse crate::constants;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::Scalar;\nuse crate::traits::Identity;\nuse crate::window::NafLookupTable5;\n\n/// Compute \\\\(aA + bB\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n\n/// AENEAS-COMPAT helper: highest index with a nonzero NAF digit (or 0).\nfn dsm_top_index(a_naf: &[i8; 256], b_naf: &[i8; 256]) -> usize {\n // AENEAS-COMPAT: always process all 256 digits. Leading zero digits\n // double the identity (a no-op), so the result of `mul` is unchanged;\n // only the variable-time skip is dropped. (The conditional-fold scan\n // fails Aeneas' loop fixed point.)\n let _ = a_naf;\n let _ = b_naf;\n 255\n}\n\n/// AENEAS-COMPAT helper: the double-and-add main loop over indices i..0,\n/// with all borrows rooted in parameters (pure refactor).\n#[cfg(feature = \"precomputed-tables\")]\nfn dsm_loop(\n i: usize,\n a_naf: &[i8; 256],\n b_naf: &[i8; 256],\n table_a: &NafLookupTable5,\n table_b: &NafLookupTable8,\n) -> ProjectivePoint {\n let mut r = ProjectivePoint::identity();\n let mut j: usize = i + 1;\n while j > 0 {\n let idx = j - 1;\n let t0 = r.double();\n let t1 = dsm_step_p(t0, table_a, a_naf[idx]);\n let t2 = dsm_step_b(t1, table_b, b_naf[idx]);\n r = t2.as_projective();\n j -= 1;\n }\n r\n}\n\n#[cfg(not(feature = \"precomputed-tables\"))]\nfn dsm_loop(\n i: usize,\n a_naf: &[i8; 256],\n b_naf: &[i8; 256],\n table_a: &NafLookupTable5,\n table_b: &NafLookupTable5,\n) -> ProjectivePoint {\n let mut r = ProjectivePoint::identity();\n let mut j: usize = i + 1;\n while j > 0 {\n let idx = j - 1;\n let t0 = r.double();\n let t1 = dsm_step_p(t0, table_a, a_naf[idx]);\n let t2 = dsm_step_b(t1, table_b, b_naf[idx]);\n r = t2.as_projective();\n j -= 1;\n }\n r\n}\n\n/// AENEAS-COMPAT helper: one NAF-digit conditional add against a\n/// ProjectiveNiels table (hoisted out of the main loop so its body is\n/// single-assignment; pure refactor, semantics identical).\nfn dsm_step_p(\n t: CompletedPoint,\n table: &NafLookupTable5,\n d: i8,\n) -> CompletedPoint {\n if d > 0 {\n &t.as_extended() + &table.select(d as usize)\n } else if d < 0 {\n &t.as_extended() - &table.select((-d) as usize)\n } else {\n t\n }\n}\n\n/// AENEAS-COMPAT helper: as `dsm_step_p`, for the basepoint table.\n#[cfg(feature = \"precomputed-tables\")]\nfn dsm_step_b(\n t: CompletedPoint,\n table: &NafLookupTable8,\n d: i8,\n) -> CompletedPoint {\n if d > 0 {\n &t.as_extended() + &table.select(d as usize)\n } else if d < 0 {\n &t.as_extended() - &table.select((-d) as usize)\n } else {\n t\n }\n}\n\n/// AENEAS-COMPAT helper: as `dsm_step_p` (no precomputed tables).\n#[cfg(not(feature = \"precomputed-tables\"))]\nfn dsm_step_b(\n t: CompletedPoint,\n table: &NafLookupTable5,\n d: i8,\n) -> CompletedPoint {\n dsm_step_p(t, table, d)\n}\n\npub fn mul(a: &Scalar, A: &EdwardsPoint, b: &Scalar) -> EdwardsPoint {\n let a_naf = a.non_adjacent_form(5);\n\n #[cfg(feature = \"precomputed-tables\")]\n let b_naf = b.non_adjacent_form(8);\n #[cfg(not(feature = \"precomputed-tables\"))]\n let b_naf = b.non_adjacent_form(5);\n\n // Find starting index (AENEAS-COMPAT: helper, upward fold)\n let i = dsm_top_index(&a_naf, &b_naf);\n\n let table_A = NafLookupTable5::::from(A);\n #[cfg(feature = \"precomputed-tables\")]\n let table_B = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_POINT);\n\n let r = dsm_loop(i, &a_naf, &b_naf, &table_A, table_B);\n\n r.as_extended()\n}\n"},{"id":88,"name":{"Local":"curve25519/solana-ed25519/src/window.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519.\n// Copyright (c) 2016-2021 isis lovecruft\n// Copyright (c) 2016-2019 Henry de Valence\n// See LICENSE for licensing information.\n//\n// Authors:\n// - isis agora lovecruft \n// - Henry de Valence \n\n//! Code for fixed- and sliding-window functionality\n\n#![allow(non_snake_case)]\n\nuse core::fmt::Debug;\n\n#[cfg(feature = \"alloc\")]\nuse alloc::vec::Vec;\nuse cfg_if::cfg_if;\n#[cfg(feature = \"alloc\")]\nuse core::borrow::Borrow;\n\nuse subtle::Choice;\nuse subtle::ConditionallyNegatable;\nuse subtle::ConditionallySelectable;\nuse subtle::ConstantTimeEq;\n\nuse crate::traits::Identity;\n\nuse crate::backend::serial::curve_models::AffineNielsPoint;\nuse crate::backend::serial::curve_models::ProjectiveNielsPoint;\nuse crate::edwards::EdwardsPoint;\n\n#[cfg(feature = \"alloc\")]\npub(crate) fn build_lookup_tables(points: I) -> Vec\nwhere\n T: for<'a> From<&'a EdwardsPoint>,\n I: IntoIterator,\n I::Item: Borrow,\n{\n points\n .into_iter()\n .map(|point| T::from(point.borrow()))\n .collect()\n}\n\n#[cfg(feature = \"alloc\")]\npub(crate) fn build_lookup_tables_for_optional_points(points: I) -> Option>\nwhere\n T: for<'a> From<&'a EdwardsPoint>,\n I: IntoIterator>,\n P: Borrow,\n{\n points\n .into_iter()\n .map(|point| point.map(|point| T::from(point.borrow())))\n .collect()\n}\n\n#[cfg(feature = \"zeroize\")]\nuse zeroize::Zeroize;\n\nmacro_rules! impl_lookup_table {\n (Name = $name:ident, Size = $size:expr, SizeNeg = $neg:expr, SizeRange = $range:expr, ConversionRange = $conv_range:expr) => {\n /// A lookup table of precomputed multiples of a point \\\\(P\\\\), used to\n /// compute \\\\( xP \\\\) for \\\\( -8 \\leq x \\leq 8 \\\\).\n ///\n /// The computation of \\\\( xP \\\\) is done in constant time by the `select` function.\n ///\n /// Since `LookupTable` does not implement `Index`, it's more difficult\n /// to accidentally use the table directly. Unfortunately the table is\n /// only `pub(crate)` so that we can write hardcoded constants, so it's\n /// still technically possible. It would be nice to prevent direct\n /// access to the table.\n #[derive(Copy, Clone)]\n pub struct $name(pub(crate) [T; $size]);\n\n impl $name\n where\n T: Identity + ConditionallySelectable + ConditionallyNegatable,\n {\n /// Given \\\\(-8 \\leq x \\leq 8\\\\), return \\\\(xP\\\\) in constant time.\n pub fn select(&self, x: i8) -> T {\n debug_assert!(x >= $neg);\n debug_assert!(x as i16 <= $size as i16); // XXX We have to convert to i16s here for the radix-256 case.. this is wrong.\n\n // Compute xabs = |x|\n let xmask = x as i16 >> 7;\n let xabs = (x as i16 + xmask) ^ xmask;\n\n // Set t = 0 * P = identity\n let mut t = T::identity();\n for j in $range {\n // Copy `points[j-1] == j*P` onto `t` in constant time if `|x| == j`.\n let c = (xabs as u16).ct_eq(&(j as u16));\n t.conditional_assign(&self.0[j - 1], c);\n }\n // Now t == |x| * P.\n\n let neg_mask = Choice::from((xmask & 1) as u8);\n t.conditional_negate(neg_mask);\n // Now t == x * P.\n\n t\n }\n }\n\n impl Default for $name {\n fn default() -> $name {\n $name([T::default(); $size])\n }\n }\n\n impl Debug for $name {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"{:?}(\", stringify!($name))?;\n\n for x in self.0.iter() {\n write!(f, \"{:?}\", x)?;\n }\n\n write!(f, \")\")\n }\n }\n\n impl<'a> From<&'a EdwardsPoint> for $name {\n fn from(P: &'a EdwardsPoint) -> Self {\n let mut points = [P.as_projective_niels(); $size];\n for j in $conv_range {\n points[j + 1] = (P + &points[j]).as_extended().as_projective_niels();\n }\n $name(points)\n }\n }\n\n impl<'a> From<&'a EdwardsPoint> for $name {\n fn from(P: &'a EdwardsPoint) -> Self {\n let mut points = [P.as_affine_niels(); $size];\n // XXX batch inversion would be good if perf mattered here\n for j in $conv_range {\n points[j + 1] = (P + &points[j]).as_extended().as_affine_niels()\n }\n $name(points)\n }\n }\n\n #[cfg(feature = \"zeroize\")]\n impl Zeroize for $name\n where\n T: Copy + Default + Zeroize,\n {\n fn zeroize(&mut self) {\n self.0.iter_mut().zeroize();\n }\n }\n };\n} // End macro_rules! impl_lookup_table\n\n// The first one has to be named \"LookupTable\" because it's used as a constructor for consts.\n// This is radix-16\nimpl_lookup_table! {\n Name = LookupTable,\n Size = 8,\n SizeNeg = -8,\n SizeRange = 1..9,\n ConversionRange = 0..7\n}\n\n// The rest only get used to make basepoint tables\ncfg_if! {\n if #[cfg(feature = \"precomputed-tables\")] {\n // radix-32\n impl_lookup_table! {\n Name = LookupTableRadix32,\n Size = 16,\n SizeNeg = -16,\n SizeRange = 1..17,\n ConversionRange = 0..15\n }\n // radix-64\n impl_lookup_table! {\n Name = LookupTableRadix64,\n Size = 32,\n SizeNeg = -32,\n SizeRange = 1..33,\n ConversionRange = 0..31\n }\n // radix-128\n impl_lookup_table! {\n Name = LookupTableRadix128,\n Size = 64,\n SizeNeg = -64,\n SizeRange = 1..65,\n ConversionRange = 0..63\n }\n // radix-256\n impl_lookup_table! {\n Name = LookupTableRadix256,\n Size = 128,\n SizeNeg = -128,\n SizeRange = 1..129,\n ConversionRange = 0..127\n }\n\n // For homogeneity we then alias it to \"LookupTableRadix16\".\n pub(crate) type LookupTableRadix16 = LookupTable;\n }\n}\n\n/// Holds odd multiples 1A, 3A, ..., 15A of a point A.\n#[derive(Copy, Clone)]\npub(crate) struct NafLookupTable5(pub(crate) [T; 8]);\n\nimpl NafLookupTable5 {\n /// Given public, odd \\\\( x \\\\) with \\\\( 0 < x < 2^4 \\\\), return \\\\(xA\\\\).\n pub fn select(&self, x: usize) -> T {\n debug_assert_eq!(x & 1, 1);\n debug_assert!(x < 16);\n\n self.0[x / 2]\n }\n}\n\nimpl Debug for NafLookupTable5 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"NafLookupTable5({:?})\", self.0)\n }\n}\n\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable5 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_projective_niels(); 8];\n let A2 = A.double();\n for i in 0..7 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_projective_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A]\n NafLookupTable5(Ai)\n }\n}\n\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable5 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_affine_niels(); 8];\n let A2 = A.double();\n for i in 0..7 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_affine_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A]\n NafLookupTable5(Ai)\n }\n}\n\n/// Holds stuff up to 8. The only time we use tables this big is for precomputed basepoint tables\n/// and multiscalar multiplication (which requires alloc).\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\n#[derive(Copy, Clone)]\npub(crate) struct NafLookupTable8(pub(crate) [T; 64]);\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl NafLookupTable8 {\n pub fn select(&self, x: usize) -> T {\n debug_assert_eq!(x & 1, 1);\n debug_assert!(x < 128);\n\n self.0[x / 2]\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl Debug for NafLookupTable8 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n writeln!(f, \"NafLookupTable8([\")?;\n for i in 0..64 {\n writeln!(f, \"\\t{:?},\", &self.0[i])?;\n }\n write!(f, \"])\")\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable8 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_projective_niels(); 64];\n let A2 = A.double();\n for i in 0..63 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_projective_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A, ..., 127A]\n NafLookupTable8(Ai)\n }\n}\n\n#[cfg(any(feature = \"precomputed-tables\", feature = \"alloc\"))]\nimpl<'a> From<&'a EdwardsPoint> for NafLookupTable8 {\n fn from(A: &'a EdwardsPoint) -> Self {\n let mut Ai = [A.as_affine_niels(); 64];\n let A2 = A.double();\n for i in 0..63 {\n Ai[i + 1] = (&A2 + &Ai[i]).as_extended().as_affine_niels();\n }\n // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A, ..., 127A]\n NafLookupTable8(Ai)\n }\n}\n"},{"id":89,"name":{"Local":"curve25519/solana-ed25519/src/backend/serial/scalar_mul/vartime_triple_base.rs"},"crate_name":"curve25519","contents":"// -*- mode: rust; -*-\n//\n// This file is part of curve25519-sol.\n// Copyright (c) 2025 curve25519-sol contributors\n// See LICENSE for licensing information.\n//\n// This file is new in curve25519-sol and has no upstream equivalent in\n// curve25519-dalek. It implements a 4-variable multi-scalar multiplication\n// optimised for two 128-bit scalars and one 256-bit scalar, designed to\n// support the HEEA-accelerated Ed25519 verification path.\n#![allow(non_snake_case)]\n\nuse core::cmp::Ordering;\n\nuse crate::backend::serial::curve_models::{ProjectiveNielsPoint, ProjectivePoint};\nuse crate::constants;\nuse crate::edwards::EdwardsPoint;\nuse crate::scalar::{HEEA_MAX_INDEX, Scalar};\nuse crate::traits::Identity;\nuse crate::window::NafLookupTable5;\n\nconst DYNAMIC_NAF_WINDOW: usize = 5;\n\n// This intentionally differs from the vector backend when precomputed tables\n// are enabled. The serial backend can select width-5 b_lo digits from the\n// larger width-8 basepoint table, but keeps the smaller NAF window here as a\n// backend-specific performance tradeoff. Revisit this only with comparative\n// serial/vector benchmarks.\nconst B_LO_NAF_WINDOW: usize = DYNAMIC_NAF_WINDOW;\n\n/// Compute \\\\(a_1 A_1 + a_2 A_2 + b B\\\\) in variable time, where \\\\(B\\\\) is the Ed25519 basepoint.\n///\n/// This function is optimized for the case where \\\\(a_1\\\\) and \\\\(a_2\\\\) are known to be less than\n/// \\\\(2^{128}\\\\), while \\\\(b\\\\) is a full 256-bit scalar.\n///\n/// # Precondition\n///\n/// Callers must ensure \\\\(a_1\\\\) and \\\\(a_2\\\\) are less than \\\\(2^{128}\\\\). Use\n/// `vartime_triple_base_mul_128_128_256` for a checked wrapper that falls back\n/// to general scalar multiplication for full-width scalars.\n///\n/// # Optimization Strategy\n///\n/// The function decomposes the 256-bit scalar \\\\(b\\\\) as \\\\(b = b_{lo} + b_{hi} \\cdot 2^{128}\\\\),\n/// where both \\\\(b_{lo}\\\\) and \\\\(b_{hi}\\\\) are 128-bit values. This allows computing:\n///\n/// \\\\[\n/// a_1 A_1 + a_2 A_2 + b_{lo} B + b_{hi} B'\n/// \\\\]\n///\n/// where \\\\(B' = B \\cdot 2^{128}\\\\) is a precomputed constant. Now all four scalars\n/// (\\\\(a_1, a_2, b_{lo}, b_{hi}\\\\)) are 128 bits, and two of the bases (\\\\(B\\\\) and \\\\(B'\\\\))\n/// use precomputed tables.\n///\n/// # Implementation\n///\n/// - For \\\\(A_1\\\\) and \\\\(A_2\\\\): NAF with window width 5 (8 precomputed points each)\n/// - For \\\\(B\\\\): NAF with window width 5. When precomputed tables are enabled,\n/// these width-5 digits are selected from the larger width-8 basepoint table.\n/// - For \\\\(B'\\\\): NAF with window width 5.\n///\n/// The vector backend uses width 8 for \\\\(b_{lo}\\\\) when precomputed tables are\n/// enabled. This serial backend keeps width 5 for \\\\(b_{lo}\\\\) as a\n/// backend-specific performance tradeoff.\n///\n/// The algorithm shares doublings across all four scalar multiplications, processing\n/// only 128 bits instead of 256, providing approximately 2x speedup over the naive approach.\npub(crate) fn mul_128_128_256_prechecked(\n a1: &Scalar,\n A1: &EdwardsPoint,\n a2: &Scalar,\n A2: &EdwardsPoint,\n b: &Scalar,\n) -> EdwardsPoint {\n // Decompose b into b_lo (lower 128 bits) and b_hi (upper 128 bits)\n // b = b_lo + b_hi * 2^128\n let b_bytes = b.as_bytes();\n\n let mut b_lo_bytes = [0u8; 32];\n let mut b_hi_bytes = [0u8; 32];\n\n // Copy lower 16 bytes to b_lo, upper 16 bytes to b_hi\n b_lo_bytes[..16].copy_from_slice(&b_bytes[..16]);\n b_hi_bytes[..16].copy_from_slice(&b_bytes[16..]);\n\n let b_lo = Scalar::from_canonical_bytes(b_lo_bytes).unwrap();\n let b_hi = Scalar::from_canonical_bytes(b_hi_bytes).unwrap();\n\n // Compute NAF representations (all scalars are now ~128 bits).\n //\n // The serial backend keeps b_lo at width 5 even when table_B is the larger\n // precomputed width-8 basepoint table. The vector backend uses width 8 in\n // that configuration.\n let a1_naf = a1.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n let a2_naf = a2.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n let b_lo_naf = b_lo.non_adjacent_form_128(B_LO_NAF_WINDOW);\n let b_hi_naf = b_hi.non_adjacent_form_128(DYNAMIC_NAF_WINDOW);\n\n // Find starting index - check all NAFs up to bit 127\n // (with potential carry to bit 128 or 129)\n let mut i: usize = HEEA_MAX_INDEX;\n for j in (0..=HEEA_MAX_INDEX).rev() {\n i = j;\n if a1_naf[i] != 0 || a2_naf[i] != 0 || b_lo_naf[i] != 0 || b_hi_naf[i] != 0 {\n break;\n }\n }\n\n // Create lookup tables\n let table_A1 = NafLookupTable5::::from(A1);\n let table_A2 = NafLookupTable5::::from(A2);\n\n #[cfg(feature = \"precomputed-tables\")]\n let table_B = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_POINT);\n\n // B' = B * 2^128 (precomputed constant point)\n #[cfg(feature = \"precomputed-tables\")]\n let table_B_128 = &constants::AFFINE_ODD_MULTIPLES_OF_BASEPOINT_128;\n #[cfg(not(feature = \"precomputed-tables\"))]\n let table_B_128 =\n &NafLookupTable5::::from(&constants::ED25519_BASEPOINT_128_POINT);\n\n let mut r = ProjectivePoint::identity();\n\n loop {\n let mut t = r.double();\n\n // Add contributions from a1*A1\n match a1_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_A1.select(a1_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_A1.select(-a1_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from a2*A2\n match a2_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_A2.select(a2_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_A2.select(-a2_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from b_lo*B\n match b_lo_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_B.select(b_lo_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_B.select(-b_lo_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n // Add contributions from b_hi*B' where B' = B * 2^128\n match b_hi_naf[i].cmp(&0) {\n Ordering::Greater => t = &t.as_extended() + &table_B_128.select(b_hi_naf[i] as usize),\n Ordering::Less => t = &t.as_extended() - &table_B_128.select(-b_hi_naf[i] as usize),\n Ordering::Equal => {}\n }\n\n r = t.as_projective();\n\n if i == 0 {\n break;\n }\n i -= 1;\n }\n\n r.as_extended()\n}\n\n#[cfg(test)]\nmod test {\n\n use rand::{RngCore, thread_rng};\n\n use super::*;\n use crate::scalar::Scalar;\n\n fn random_scalar() -> Scalar {\n let mut wide = [0u8; 64];\n let mut rng = thread_rng();\n rng.fill_bytes(&mut wide);\n Scalar::from_bytes_mod_order_wide(&wide)\n }\n\n #[test]\n fn test_triple_base_multiplication() {\n // Test vectors with random scalars\n let a1 = Scalar::from(12345u64);\n let a2 = Scalar::from(67890u64);\n let b = random_scalar();\n\n // Random points (using scalar multiplication of basepoint)\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(2u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n // Compute using the optimized triple-base function\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute using naive addition\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_multiplication_128() {\n // Test with 128-bit scalars for a1 and a2\n // Create a 128-bit scalar (use lower half)\n let a1_bytes = [\n 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,\n 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00,\n ];\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n\n let a2_bytes = [\n 0x12, 0x34, 0x56, 0x78, 0x9A, 0xBC, 0xDE, 0xF0, 0x11, 0x22, 0x33, 0x44, 0x55, 0x66,\n 0x77, 0x88, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,\n 0x00, 0x00, 0x00, 0x00,\n ];\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n // Full 256-bit scalar for b\n let b = random_scalar();\n\n // Test points\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(5u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(7u64);\n\n // Test the optimized 128-bit version\n let result_128 = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute expected result\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result_128, expected, \"Optimized 128-bit version failed\");\n }\n\n #[test]\n fn test_triple_base_with_zero_scalars() {\n let a1 = Scalar::ZERO;\n let a2 = Scalar::from(123u64);\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(2u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = a2 * A2 + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_with_identity_points() {\n let a1 = Scalar::from(111u64);\n let a2 = Scalar::from(222u64);\n let b = random_scalar();\n\n let A1 = EdwardsPoint::identity();\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(3u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = a2 * A2 + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_triple_base_consistency() {\n // Test that both functions give the same result for 128-bit inputs\n let a1 = Scalar::from(0x123456789ABCDEFu64);\n let a2 = Scalar::from(0xFEDCBA987654321u64);\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(11u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(13u64);\n\n let result_optimized = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let result_general = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result_optimized, result_general);\n }\n\n #[test]\n fn test_triple_base_large_scalars() {\n // Test with large scalars\n let mut a1_bytes = [0xFFu8; 32];\n a1_bytes[16..].fill(0x00);\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n\n let mut a2_bytes = [0xAAu8; 32];\n a2_bytes[16..].fill(0x00);\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n let b = random_scalar();\n\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(17u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(19u64);\n\n let result = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n #[test]\n fn test_checked_wrapper_accepts_full_width_scalars() {\n let mut a1_bytes = [0u8; 32];\n a1_bytes[0] = 7;\n a1_bytes[16] = 1;\n let a1 = Scalar::from_canonical_bytes(a1_bytes).unwrap();\n\n let mut a2_bytes = [0u8; 32];\n a2_bytes[0] = 11;\n a2_bytes[24] = 1;\n let a2 = Scalar::from_canonical_bytes(a2_bytes).unwrap();\n\n let b = random_scalar();\n let A1 = constants::ED25519_BASEPOINT_POINT * Scalar::from(23u64);\n let A2 = constants::ED25519_BASEPOINT_POINT * Scalar::from(29u64);\n\n let result = crate::backend::vartime_triple_base_mul_128_128_256(&a1, &A1, &a2, &A2, &b);\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n assert_eq!(result, expected);\n }\n\n // Proptest for vartime_triple_scalar_mul_basepoint equivalence\n proptest::proptest! {\n #[test]\n fn proptest_triple_scalar_mul_equivalence(\n a1_bytes_16 in proptest::array::uniform16(proptest::num::u8::ANY),\n a2_bytes_16 in proptest::array::uniform16(proptest::num::u8::ANY),\n b_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n A1_scalar_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n A2_scalar_bytes in proptest::array::uniform32(proptest::num::u8::ANY),\n ) {\n // Construct 128-bit scalars a1 and a2 (upper 16 bytes are zero)\n let mut a1_bytes = [0u8; 32];\n let mut a2_bytes = [0u8; 32];\n a1_bytes[..16].copy_from_slice(&a1_bytes_16);\n a2_bytes[..16].copy_from_slice(&a2_bytes_16);\n\n let a1 = Scalar::from_bytes_mod_order(a1_bytes);\n let a2 = Scalar::from_bytes_mod_order(a2_bytes);\n\n // Construct full 256-bit scalar b\n let b = Scalar::from_bytes_mod_order(b_bytes);\n\n // Generate random points A1 and A2 using scalar multiplication of basepoint\n let A1_scalar = Scalar::from_bytes_mod_order(A1_scalar_bytes);\n let A2_scalar = Scalar::from_bytes_mod_order(A2_scalar_bytes);\n let A1 = constants::ED25519_BASEPOINT_POINT * A1_scalar;\n let A2 = constants::ED25519_BASEPOINT_POINT * A2_scalar;\n\n // Compute using the optimized triple-base function\n let result_optimized = mul_128_128_256_prechecked(&a1, &A1, &a2, &A2, &b);\n\n // Compute using raw operations: a1*A1 + a2*A2 + b*B\n let expected = (a1 * A1 + a2 * A2) + b * constants::ED25519_BASEPOINT_POINT;\n\n proptest::prop_assert_eq!(result_optimized, expected, \"Optimized triple scalar mul should equal raw operations\");\n }\n 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limbs."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":0,"beg":{"line":26,"col":20},"end":{"line":26,"col":32}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":null,"ty":{"HashConsedValue":[1785,{"Array":[{"Deduplicated":1783},{"kind":{"Literal":{"Scalar":{"Unsigned":["Usize","5"]}}},"ty":{"HashConsedValue":[1784,{"Literal":{"UInt":"Usize"}}]}}]}]}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":40,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":1,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["scalar",0]},{"Ident":["Scalar",0]}],"span":{"data":{"file_id":5,"beg":{"line":165,"col":0},"end":{"line":202,"col":1}},"generated_from_span":null},"source_text":"pub struct Scalar {\n /// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the\n /// group order.\n ///\n /// # Invariant #1\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255}\\\\). That is, the most\n /// significant bit of `bytes[31]` is 0.\n ///\n /// This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most\n /// integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`]).\n /// The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB\n /// is 0 (see `MontgomeryPoint::mul`).\n ///\n /// # Invariant #2 (weak)\n ///\n /// The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents\n /// a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is\n /// stronger than invariant #1. It also sometimes has to be broken.\n ///\n /// This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,\n /// mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and\n /// `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,\n /// scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant\n /// #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined.\n ///\n /// Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar.\n /// Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars.\n /// Multiplication is correct, but this is only due to a quirk of our implementation, and not\n /// guaranteed to hold in general in the future.\n ///\n /// Note: It is not possible to construct an unreduced `Scalar` from the public API unless the\n /// `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all\n /// public non-legacy uses, invariant #2\n /// always holds.\n ///\n pub(crate) bytes: [u8; 32],\n}","attr_info":{"attributes":[{"DocComment":" The `Scalar` struct holds an element of \\\\(\\mathbb Z / \\ell\\mathbb Z \\\\)."},{"Unknown":{"path":"allow","args":"clippy::derived_hash_with_manual_eq"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":5,"beg":{"line":201,"col":4},"end":{"line":201,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the"},{"DocComment":" group order."},{"DocComment":""},{"DocComment":" # Invariant #1"},{"DocComment":""},{"DocComment":" The integer representing this scalar is less than \\\\(2\\^{255}\\\\). That is, the most"},{"DocComment":" significant bit of `bytes[31]` is 0."},{"DocComment":""},{"DocComment":" This is required for `EdwardsPoint` variable- and fixed-base multiplication, because most"},{"DocComment":" integers above 2^255 are unrepresentable in our radix-16 NAF (see [`Self::as_radix_16`])."},{"DocComment":" The invariant is also required because our `MontgomeryPoint` multiplication assumes the MSB"},{"DocComment":" is 0 (see `MontgomeryPoint::mul`)."},{"DocComment":""},{"DocComment":" # Invariant #2 (weak)"},{"DocComment":""},{"DocComment":" The integer representing this scalar is less than \\\\(2\\^{255} - 19 \\\\), i.e., it represents"},{"DocComment":" a canonical representative of an element of \\\\( \\mathbb Z / \\ell\\mathbb Z \\\\). This is"},{"DocComment":" stronger than invariant #1. It also sometimes has to be broken."},{"DocComment":""},{"DocComment":" This invariant is deliberately broken in the implementation of `EdwardsPoint::{mul_clamped,"},{"DocComment":" mul_base_clamped}`, `MontgomeryPoint::{mul_clamped, mul_base_clamped}`, and"},{"DocComment":" `BasepointTable::mul_base_clamped`. This is not an issue though. As mentioned above,"},{"DocComment":" scalar-point multiplication is defined for any choice of `bytes` that satisfies invariant"},{"DocComment":" #1. Since clamping guarantees invariant #1 is satisfied, these operations are well defined."},{"DocComment":""},{"DocComment":" Note: Scalar-point mult is the _only_ thing you can do safely with an unreduced scalar."},{"DocComment":" Scalar-scalar addition and subtraction are NOT correct when using unreduced scalars."},{"DocComment":" Multiplication is correct, but this is only due to a quirk of our implementation, and not"},{"DocComment":" guaranteed to hold in general in the future."},{"DocComment":""},{"DocComment":" Note: It is not possible to construct an unreduced `Scalar` from the public API unless the"},{"DocComment":" `legacy_compatibility` is enabled (thus making `Scalar::from_bits` public). Thus, for all"},{"DocComment":" public non-legacy uses, invariant #2"},{"DocComment":" always holds."},{"DocComment":""}],"inline":null,"rename":null,"public":false},"name":"bytes","ty":{"HashConsedValue":[2743,{"Array":[{"HashConsedValue":[2322,{"Literal":{"UInt":"U8"}}]},{"kind":{"Literal":{"Scalar":{"Unsigned":["Usize","32"]}}},"ty":{"Deduplicated":1784}}]}]}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":32,"align":1,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":2,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["ed_sigs",0]},{"Ident":["verification_key",0]},{"Ident":["VerificationKey",0]}],"span":{"data":{"file_id":6,"beg":{"line":223,"col":0},"end":{"line":226,"col":1}},"generated_from_span":null},"source_text":"pub struct VerificationKey {\n pub(crate) A_bytes: VerificationKeyBytes,\n pub(crate) minus_A: EdwardsPoint,\n}","attr_info":{"attributes":[{"DocComment":" A valid Ed25519 verification key."},{"DocComment":""},{"DocComment":" This is also called a public key by other implementations."},{"DocComment":""},{"DocComment":" This type holds decompressed state used in signature verification; if the"},{"DocComment":" verification key may not be used immediately, it is probably better to use"},{"DocComment":" [`VerificationKeyBytes`], which stores only the length-checked encoded bytes."},{"DocComment":""},{"DocComment":" ## Zcash-specific consensus properties"},{"DocComment":""},{"DocComment":" Ed25519 checks are described in [§5.4.5][ps] of the Zcash protocol specification and in"},{"DocComment":" [ZIP 215]. The verification criteria for an (encoded) verification key `A_bytes` are:"},{"DocComment":""},{"DocComment":" * `A_bytes` MUST be an encoding of a point `A` on the twisted Edwards form of"},{"DocComment":" Curve25519, and non-canonical encodings MUST be accepted;"},{"DocComment":""},{"DocComment":" [ps]: https://zips.z.cash/protocol/protocol.pdf#concreteed25519"},{"Unknown":{"path":"allow","args":"non_snake_case"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":6,"beg":{"line":224,"col":4},"end":{"line":224,"col":44}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"A_bytes","ty":{"HashConsedValue":[2917,{"Adt":{"id":{"Adt":16},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}]}},{"span":{"data":{"file_id":6,"beg":{"line":225,"col":4},"end":{"line":225,"col":36}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"minus_A","ty":{"Deduplicated":2919}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":192,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,32],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":3,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Ident":["FieldElement",0]}],"span":{"data":{"file_id":8,"beg":{"line":50,"col":0},"end":{"line":50,"col":75}},"generated_from_span":null},"source_text":"pub(crate) type FieldElement = backend::serial::u64::field::FieldElement51;","attr_info":{"attributes":[{"DocComment":" A `FieldElement` represents an element of the field"},{"DocComment":" \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\)."},{"DocComment":""},{"DocComment":" The `FieldElement` type is pinned to the 64-bit serial implementation."}],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Alias":{"Deduplicated":3140}},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":40,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":4,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Ident":["FieldElement51",0]}],"span":{"data":{"file_id":9,"beg":{"line":43,"col":0},"end":{"line":43,"col":47}},"generated_from_span":null},"source_text":"pub struct FieldElement51(pub(crate) [u64; 5]);","attr_info":{"attributes":[{"DocComment":" A `FieldElement51` represents an element of the field"},{"DocComment":" \\\\( \\mathbb Z / (2\\^{255} - 19)\\\\)."},{"DocComment":""},{"DocComment":" In the 64-bit implementation, a `FieldElement` is represented in"},{"DocComment":" radix \\\\(2\\^{51}\\\\) as five `u64`s; the coefficients are allowed to"},{"DocComment":" grow up to \\\\(2\\^{54}\\\\) between reductions modulo \\\\(p\\\\)."},{"DocComment":""},{"DocComment":" # Note"},{"DocComment":""},{"DocComment":" The `curve25519::field` module provides a type alias"},{"DocComment":" `curve25519::field::FieldElement` to either `FieldElement51`"},{"DocComment":" or `FieldElement2625`."},{"DocComment":""},{"DocComment":" The backend-specific type `FieldElement51` should not be used"},{"DocComment":" outside of the `curve25519::field` module."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":9,"beg":{"line":43,"col":26},"end":{"line":43,"col":45}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":null,"ty":{"Deduplicated":1785}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":40,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":5,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["ProjectivePoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":154,"col":0},"end":{"line":158,"col":1}},"generated_from_span":null},"source_text":"pub struct ProjectivePoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A `ProjectivePoint` is a point \\\\((X:Y:Z)\\\\) on the \\\\(\\mathbb"},{"DocComment":" P\\^2\\\\) model of the curve."},{"DocComment":" A point \\\\((x,y)\\\\) in the affine model corresponds to"},{"DocComment":" \\\\((x:y:1)\\\\)."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":155,"col":4},"end":{"line":155,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"X","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":156,"col":4},"end":{"line":156,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Y","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":157,"col":4},"end":{"line":157,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Z","ty":{"Deduplicated":3140}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":120,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":6,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["CompletedPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":169,"col":0},"end":{"line":174,"col":1}},"generated_from_span":null},"source_text":"pub struct CompletedPoint {\n pub X: FieldElement,\n pub Y: FieldElement,\n pub Z: FieldElement,\n pub T: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A `CompletedPoint` is a point \\\\(((X:Z), (Y:T))\\\\) on the \\\\(\\mathbb"},{"DocComment":" P\\^1 \\times \\mathbb P\\^1 \\\\) model of the curve."},{"DocComment":" A point (x,y) in the affine model corresponds to \\\\( ((x:1),(y:1))"},{"DocComment":" \\\\)."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":170,"col":4},"end":{"line":170,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"X","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":171,"col":4},"end":{"line":171,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Y","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":172,"col":4},"end":{"line":172,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"Z","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":173,"col":4},"end":{"line":173,"col":23}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"T","ty":{"Deduplicated":3140}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":160,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80,120],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":7,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["AffineNielsPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":184,"col":0},"end":{"line":188,"col":1}},"generated_from_span":null},"source_text":"pub struct AffineNielsPoint {\n pub y_plus_x: FieldElement,\n pub y_minus_x: FieldElement,\n pub xy2d: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A pre-computed point in the affine model for the curve, represented as"},{"DocComment":" \\\\((y+x, y-x, 2dxy)\\\\) in \"Niels coordinates\"."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level documentation."},{"Unknown":{"path":"allow","args":"missing_docs"}}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":10,"beg":{"line":185,"col":4},"end":{"line":185,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"y_plus_x","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":186,"col":4},"end":{"line":186,"col":31}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"y_minus_x","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":10,"beg":{"line":187,"col":4},"end":{"line":187,"col":26}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":"xy2d","ty":{"Deduplicated":3140}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":120,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40,80],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":8,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["curve_models",0]},{"Ident":["ProjectiveNielsPoint",0]}],"span":{"data":{"file_id":10,"beg":{"line":206,"col":0},"end":{"line":211,"col":1}},"generated_from_span":null},"source_text":"pub struct ProjectiveNielsPoint {\n pub Y_plus_X: FieldElement,\n pub Y_minus_X: FieldElement,\n pub Z: FieldElement,\n pub T2d: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" A pre-computed point on the \\\\( \\mathbb P\\^3 \\\\) model for the"},{"DocComment":" curve, represented as \\\\((Y+X, Y-X, Z, 2dXY)\\\\) in \"Niels coordinates\"."},{"DocComment":""},{"DocComment":" More details on the relationships between the different curve models"},{"DocComment":" can be found in the module-level 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Used to tell an operation whether it should exit early or go on as usual."},{"DocComment":""},{"DocComment":" This is used when exposing things (like graph traversals or visitors) where"},{"DocComment":" you want the user to be able to choose whether to exit early."},{"DocComment":" Having the enum makes it clearer -- no more wondering \"wait, what did `false`"},{"DocComment":" mean again?\" -- and allows including a value."},{"DocComment":""},{"DocComment":" Similar to [`Option`] and [`Result`], this enum can be used with the `?` operator"},{"DocComment":" to return immediately if the [`Break`] variant is present or otherwise continue normally"},{"DocComment":" with the value inside the [`Continue`] variant."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Early-exiting from [`Iterator::try_for_each`]:"},{"DocComment":" ```"},{"DocComment":" use std::ops::ControlFlow;"},{"DocComment":""},{"DocComment":" let r = (2..100).try_for_each(|x| {"},{"DocComment":" if 403 % x == 0 {"},{"DocComment":" return ControlFlow::Break(x)"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" });"},{"DocComment":" assert_eq!(r, ControlFlow::Break(13));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" A basic tree traversal:"},{"DocComment":" ```"},{"DocComment":" use std::ops::ControlFlow;"},{"DocComment":""},{"DocComment":" pub struct TreeNode {"},{"DocComment":" value: T,"},{"DocComment":" left: Option>>,"},{"DocComment":" right: Option>>,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl TreeNode {"},{"DocComment":" pub fn traverse_inorder(&self, f: &mut impl FnMut(&T) -> ControlFlow) -> ControlFlow {"},{"DocComment":" if let Some(left) = &self.left {"},{"DocComment":" left.traverse_inorder(f)?;"},{"DocComment":" }"},{"DocComment":" f(&self.value)?;"},{"DocComment":" if let Some(right) = &self.right {"},{"DocComment":" right.traverse_inorder(f)?;"},{"DocComment":" }"},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" }"},{"DocComment":" fn leaf(value: T) -> Option>> {"},{"DocComment":" Some(Box::new(Self { value, left: None, right: None }))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let node = TreeNode {"},{"DocComment":" value: 0,"},{"DocComment":" left: TreeNode::leaf(1),"},{"DocComment":" right: Some(Box::new(TreeNode {"},{"DocComment":" value: -1,"},{"DocComment":" left: TreeNode::leaf(5),"},{"DocComment":" right: TreeNode::leaf(2),"},{"DocComment":" }))"},{"DocComment":" };"},{"DocComment":" let mut sum = 0;"},{"DocComment":""},{"DocComment":" let res = node.traverse_inorder(&mut |val| {"},{"DocComment":" if *val < 0 {"},{"DocComment":" ControlFlow::Break(*val)"},{"DocComment":" } else {"},{"DocComment":" sum += *val;"},{"DocComment":" ControlFlow::Continue(())"},{"DocComment":" }"},{"DocComment":" });"},{"DocComment":" assert_eq!(res, ControlFlow::Break(-1));"},{"DocComment":" assert_eq!(sum, 6);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`Break`]: ControlFlow::Break"},{"DocComment":" [`Continue`]: ControlFlow::Continue"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ControlFlow"},"generics":{"regions":[],"types":[{"index":0,"name":"B"},{"index":1,"name":"C"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Enum":[{"id":0,"span":{"data":{"file_id":31,"beg":{"line":93,"col":4},"end":{"line":93,"col":12}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" Move on to the next phase of the operation as normal."}],"inline":null,"rename":null,"public":true},"name":"Continue","fields":[{"span":{"data":{"file_id":31,"beg":{"line":93,"col":13},"end":{"line":93,"col":14}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":null,"ty":{"Deduplicated":11756}}],"discriminant":{"Scalar":{"Signed":["Isize","0"]}}},{"id":1,"span":{"data":{"file_id":31,"beg":{"line":97,"col":4},"end":{"line":97,"col":9}},"generated_from_span":null},"attr_info":{"attributes":[{"DocComment":" Exit the operation without running subsequent phases."}],"inline":null,"rename":null,"public":true},"name":"Break","fields":[{"span":{"data":{"file_id":31,"beg":{"line":97,"col":10},"end":{"line":97,"col":11}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"name":null,"ty":{"Deduplicated":11755}}],"discriminant":{"Scalar":{"Signed":["Isize","1"]}}}]},"layout":[],"ptr_metadata":"None"},{"def_id":21,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["Infallible",0]}],"span":{"data":{"file_id":32,"beg":{"line":930,"col":0},"end":{"line":930,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The error type for errors that can never happen."},{"DocComment":""},{"DocComment":" Since this enum has no variant, a value of this type can never actually exist."},{"DocComment":" This can be useful for generic APIs that use [`Result`] and parameterize the error type,"},{"DocComment":" to indicate that the result is always [`Ok`]."},{"DocComment":""},{"DocComment":" For example, the [`TryFrom`] trait (conversion that returns a [`Result`])"},{"DocComment":" has a blanket implementation for all types where a reverse [`Into`] implementation exists."},{"DocComment":""},{"DocComment":" ```ignore (illustrates std code, duplicating the impl in a doctest would be an error)"},{"DocComment":" impl TryFrom for T where U: Into {"},{"DocComment":" type Error = Infallible;"},{"DocComment":""},{"DocComment":" fn try_from(value: U) -> Result {"},{"DocComment":" Ok(U::into(value)) // Never returns `Err`"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Future compatibility"},{"DocComment":""},{"DocComment":" This enum has the same role as [the `!` “never” type][never],"},{"DocComment":" which is unstable in this version of Rust."},{"DocComment":" When `!` is stabilized, we plan to make `Infallible` a type alias to it:"},{"DocComment":""},{"DocComment":" ```ignore (illustrates future std change)"},{"DocComment":" pub type Infallible = !;"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" … and eventually deprecate `Infallible`."},{"DocComment":""},{"DocComment":" However there is one case where `!` syntax can be used"},{"DocComment":" before `!` is stabilized as a full-fledged type: in the position of a function’s return type."},{"DocComment":" Specifically, it is possible to have implementations for two different function pointer types:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" trait MyTrait {}"},{"DocComment":" impl MyTrait for fn() -> ! {}"},{"DocComment":" impl MyTrait for fn() -> std::convert::Infallible {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" With `Infallible` being an enum, this code is valid."},{"DocComment":" However when `Infallible` becomes an alias for the never type,"},{"DocComment":" the two `impl`s will start to overlap"},{"DocComment":" and therefore will be disallowed by the language’s trait coherence rules."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Enum":[]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":0,"align":1,"discriminator":null,"uninhabited":true,"variant_layouts":[],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},null,{"def_id":23,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["edwards",0]},{"Ident":["affine",0]},{"Ident":["AffinePoint",0]}],"span":{"data":{"file_id":39,"beg":{"line":12,"col":0},"end":{"line":15,"col":1}},"generated_from_span":null},"source_text":"pub struct AffinePoint {\n pub(super) x: FieldElement,\n pub(super) y: FieldElement,\n}","attr_info":{"attributes":[{"DocComment":" Affine Edwards point on untwisted curve."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":{"Struct":[{"span":{"data":{"file_id":39,"beg":{"line":13,"col":4},"end":{"line":13,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"x","ty":{"Deduplicated":3140}},{"span":{"data":{"file_id":39,"beg":{"line":14,"col":4},"end":{"line":14,"col":30}},"generated_from_span":null},"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"name":"y","ty":{"Deduplicated":3140}}]},"layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":80,"align":8,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0,40],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},{"def_id":24,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["array",0]},{"Ident":["TryFromSliceError",0]}],"span":{"data":{"file_id":41,"beg":{"line":183,"col":0},"end":{"line":183,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The error type returned when a conversion from a slice to an array fails."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[{"key":"x86_64-unknown-linux-gnu","value":{"size":0,"align":1,"discriminator":{"Known":0},"uninhabited":false,"variant_layouts":[{"field_offsets":[0],"uninhabited":false,"tagger":[]}],"repr":{"repr_algo":"Rust","align_modif":null,"transparent":false,"explicit_discr_type":false}}}],"ptr_metadata":"None"},null,{"def_id":26,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["array",0]},{"Ident":["iter",0]},{"Ident":["IntoIter",0]}],"span":{"data":{"file_id":44,"beg":{"line":20,"col":0},"end":{"line":20,"col":38}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A by-value [array] iterator."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ArrayIntoIter"},"generics":{"regions":[],"types":[{"index":0,"name":"T"}],"const_generics":[{"index":0,"name":"N","ty":{"Deduplicated":1784}}],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":27,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["num",0]},{"Ident":["nonzero",0]},{"Ident":["NonZero",0]}],"span":{"data":{"file_id":45,"beg":{"line":128,"col":0},"end":{"line":128,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A value that is known not to equal zero."},{"DocComment":""},{"DocComment":" This enables some memory layout optimization."},{"DocComment":" For example, `Option>` is the same size as `u32`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use core::{num::NonZero};"},{"DocComment":""},{"DocComment":" assert_eq!(size_of::>>(), size_of::());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Layout"},{"DocComment":""},{"DocComment":" `NonZero` is guaranteed to have the same layout and bit validity as `T`"},{"DocComment":" with the exception that the all-zero bit pattern is invalid."},{"DocComment":" `Option>` is guaranteed to be compatible with `T`, including in"},{"DocComment":" FFI."},{"DocComment":""},{"DocComment":" Thanks to the [null pointer optimization], `NonZero` and"},{"DocComment":" `Option>` are guaranteed to have the same size and alignment:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::num::NonZero;"},{"DocComment":""},{"DocComment":" assert_eq!(size_of::>(), size_of::>>());"},{"DocComment":" assert_eq!(align_of::>(), align_of::>>());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [null pointer optimization]: crate::option#representation"},{"DocComment":""},{"DocComment":" # Note on generic usage"},{"DocComment":""},{"DocComment":" `NonZero` can only be used with some standard library primitive types"},{"DocComment":" (such as `u8`, `i32`, and etc.). The type parameter `T` must implement the"},{"DocComment":" internal trait [`ZeroablePrimitive`], which is currently permanently unstable"},{"DocComment":" and cannot be implemented by users. Therefore, you cannot use `NonZero`"},{"DocComment":" with your own types, nor can you implement traits for all `NonZero`,"},{"DocComment":" only for concrete types."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"NonZero"},"generics":{"regions":[],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_NonZeroInner"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":28,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["step_by",0]},{"Ident":["StepBy",0]}],"span":{"data":{"file_id":46,"beg":{"line":16,"col":0},"end":{"line":16,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator for stepping iterators by a custom amount."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`step_by`] method on [`Iterator`]. See"},{"DocComment":" its documentation for more."},{"DocComment":""},{"DocComment":" [`step_by`]: Iterator::step_by"},{"DocComment":" [`Iterator`]: trait.Iterator.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":29,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["chain",0]},{"Ident":["Chain",0]}],"span":{"data":{"file_id":48,"beg":{"line":23,"col":0},"end":{"line":23,"col":22}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that links two iterators together, in a chain."},{"DocComment":""},{"DocComment":" This `struct` is created by [`chain`] or [`Iterator::chain`]. See their"},{"DocComment":" documentation for more."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::iter::Chain;"},{"DocComment":" use std::slice::Iter;"},{"DocComment":""},{"DocComment":" let a1 = [1, 2, 3];"},{"DocComment":" let a2 = [4, 5, 6];"},{"DocComment":" let iter: Chain, Iter<'_, _>> = a1.iter().chain(a2.iter());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"A"},{"index":1,"name":"B"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":30,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["zip",0]},{"Ident":["Zip",0]}],"span":{"data":{"file_id":49,"beg":{"line":13,"col":0},"end":{"line":13,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that iterates two other iterators simultaneously."},{"DocComment":""},{"DocComment":" This `struct` is created by [`zip`] or [`Iterator::zip`]."},{"DocComment":" See their documentation for more."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"A"},{"index":1,"name":"B"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":31,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["intersperse",0]},{"Ident":["Intersperse",0]}],"span":{"data":{"file_id":50,"beg":{"line":9,"col":0},"end":{"line":9,"col":35}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator adapter that places a separator between all elements."},{"DocComment":""},{"DocComment":" This `struct` is created by [`Iterator::intersperse`]. See its documentation"},{"DocComment":" for more information."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":32,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["intersperse",0]},{"Ident":["IntersperseWith",0]}],"span":{"data":{"file_id":50,"beg":{"line":83,"col":0},"end":{"line":83,"col":32}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator adapter that places a separator between all elements."},{"DocComment":""},{"DocComment":" This `struct` is created by [`Iterator::intersperse_with`]. See its"},{"DocComment":" documentation for more information."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"G"},{"index":2,"name":"Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":33,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["map",0]},{"Ident":["Map",0]}],"span":{"data":{"file_id":52,"beg":{"line":61,"col":0},"end":{"line":61,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that maps the values of `iter` with `f`."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`map`] method on [`Iterator`]. See its"},{"DocComment":" documentation for more."},{"DocComment":""},{"DocComment":" [`map`]: Iterator::map"},{"DocComment":" [`Iterator`]: trait.Iterator.html"},{"DocComment":""},{"DocComment":" # Notes about side effects"},{"DocComment":""},{"DocComment":" The [`map`] iterator implements [`DoubleEndedIterator`], meaning that"},{"DocComment":" you can also [`map`] backwards:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let v: Vec = [1, 2, 3].into_iter().map(|x| x + 1).rev().collect();"},{"DocComment":""},{"DocComment":" assert_eq!(v, [4, 3, 2]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`DoubleEndedIterator`]: trait.DoubleEndedIterator.html"},{"DocComment":""},{"DocComment":" But if your closure has state, iterating backwards may act in a way you do"},{"DocComment":" not expect. Let's go through an example. First, in the forward direction:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let mut c = 0;"},{"DocComment":""},{"DocComment":" for pair in ['a', 'b', 'c'].into_iter()"},{"DocComment":" .map(|letter| { c += 1; (letter, c) }) {"},{"DocComment":" println!(\"{pair:?}\");"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" This will print `('a', 1), ('b', 2), ('c', 3)`."},{"DocComment":""},{"DocComment":" Now consider this twist where we add a call to `rev`. This version will"},{"DocComment":" print `('c', 1), ('b', 2), ('a', 3)`. Note that the letters are reversed,"},{"DocComment":" but the values of the counter still go in order. This is because `map()` is"},{"DocComment":" still being called lazily on each item, but we are popping items off the"},{"DocComment":" back of the vector now, instead of shifting them from the front."},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" let mut c = 0;"},{"DocComment":""},{"DocComment":" for pair in ['a', 'b', 'c'].into_iter()"},{"DocComment":" .map(|letter| { c += 1; (letter, c) })"},{"DocComment":" .rev() {"},{"DocComment":" println!(\"{pair:?}\");"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"I"},{"index":1,"name":"F"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"src":"TopLevel","kind":"Opaque","layout":[],"ptr_metadata":"None"},{"def_id":34,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["filter",0]},{"Ident":["Filter",0]}],"span":{"data":{"file_id":53,"beg":{"line":21,"col":0},"end":{"line":21,"col":23}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that filters the elements of `iter` with `predicate`."},{"DocComment":""},{"DocComment":" This `struct` is created by the [`filter`] method on [`Iterator`]. 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value\n limbs[4] &= low_51_bit_mask;\n\n // Now arrange the bits of the limbs.\n let mut s = [0u8;32];\n s[ 0] = limbs[0] as u8;\n s[ 1] = (limbs[0] >> 8) as u8;\n s[ 2] = (limbs[0] >> 16) as u8;\n s[ 3] = (limbs[0] >> 24) as u8;\n s[ 4] = (limbs[0] >> 32) as u8;\n s[ 5] = (limbs[0] >> 40) as u8;\n s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;\n s[ 7] = (limbs[1] >> 5) as u8;\n s[ 8] = (limbs[1] >> 13) as u8;\n s[ 9] = (limbs[1] >> 21) as u8;\n s[10] = (limbs[1] >> 29) as u8;\n s[11] = (limbs[1] >> 37) as u8;\n s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;\n s[13] = (limbs[2] >> 2) as u8;\n s[14] = (limbs[2] >> 10) as u8;\n s[15] = (limbs[2] >> 18) as u8;\n s[16] = (limbs[2] >> 26) as u8;\n s[17] = (limbs[2] >> 34) as u8;\n s[18] = (limbs[2] >> 42) as u8;\n s[19] = ((limbs[2] >> 50) | (limbs[3] << 1)) as u8;\n s[20] = (limbs[3] >> 7) as u8;\n s[21] = (limbs[3] >> 15) as u8;\n s[22] = (limbs[3] >> 23) as u8;\n s[23] = (limbs[3] >> 31) as u8;\n s[24] = (limbs[3] >> 39) as 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operation."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Clause0_Clause0_Output"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":30,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11936},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"src":{"TraitDecl":{"trait_ref":{"id":30,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},{"def_id":361,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["function",0]},{"Ident":["FnOnce",0]},{"Ident":["call_once",0]}],"span":{"data":{"file_id":51,"beg":{"line":250,"col":4},"end":{"line":250,"col":70}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Performs the call operation."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Clause0_Output"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"src":{"TraitDecl":{"trait_ref":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},{"def_id":362,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["FromIterator",0]},{"Ident":["from_iter",0]}],"span":{"data":{"file_id":17,"beg":{"line":152,"col":4},"end":{"line":152,"col":61}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Creates a value from an iterator."},{"DocComment":""},{"DocComment":" See the [module-level documentation] for more."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v = Vec::from_iter(five_fives);"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"from_iter_fn"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"},{"index":2,"name":"T"},{"index":3,"name":"Clause1_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":32,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":17,"beg":{"line":152,"col":20},"end":{"line":152,"col":42}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":11793},{"Deduplicated":11756},{"HashConsedValue":[11994,{"TypeVar":{"Free":3}}]}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11793}],"output":{"Deduplicated":11755}},"src":{"TraitDecl":{"trait_ref":{"id":32,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},{"def_id":363,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Try",0]},{"Ident":["from_output",0]}],"span":{"data":{"file_id":67,"beg":{"line":192,"col":4},"end":{"line":192,"col":49}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Constructs the type from its `Output` type."},{"DocComment":""},{"DocComment":" This should be implemented consistently with the `branch` method"},{"DocComment":" such that applying the `?` operator will get back the original value:"},{"DocComment":" `Try::from_output(x).branch() --> ControlFlow::Continue(x)`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::Try;"},{"DocComment":""},{"DocComment":" assert_eq!( as Try>::from_output(3), Ok(3));"},{"DocComment":" assert_eq!( as Try>::from_output(4), Some(4));"},{"DocComment":" assert_eq!("},{"DocComment":" as Try>::from_output(5),"},{"DocComment":" std::ops::ControlFlow::Continue(5),"},{"DocComment":" );"},{"DocComment":""},{"DocComment":" # fn make_question_mark_work() -> Option<()> {"},{"DocComment":" assert_eq!(Option::from_output(4)?, 4);"},{"DocComment":" # None }"},{"DocComment":" # make_question_mark_work();"},{"DocComment":""},{"DocComment":" // This is used, for example, on the accumulator in `try_fold`:"},{"DocComment":" let r = std::iter::empty().try_fold(4, |_, ()| -> Option<_> { unreachable!() });"},{"DocComment":" assert_eq!(r, Some(4));"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"from_output"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"HashConsedValue":[11996,{"TraitType":[{"HashConsedValue":[11995,{"kind":{"Clause":{"Free":0}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]},0,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]}],"output":{"Deduplicated":11755}},"src":{"TraitDecl":{"trait_ref":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},{"def_id":364,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Try",0]},{"Ident":["branch",0]}],"span":{"data":{"file_id":67,"beg":{"line":219,"col":4},"end":{"line":219,"col":65}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used in `?` to decide whether the operator should produce a value"},{"DocComment":" (because this returned [`ControlFlow::Continue`])"},{"DocComment":" or propagate a value back to the caller"},{"DocComment":" (because this returned [`ControlFlow::Break`])."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::{ControlFlow, Try};"},{"DocComment":""},{"DocComment":" assert_eq!(Ok::<_, String>(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(Err::(3).branch(), ControlFlow::Break(Err(3)));"},{"DocComment":""},{"DocComment":" assert_eq!(Some(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(None::.branch(), ControlFlow::Break(None));"},{"DocComment":""},{"DocComment":" assert_eq!(ControlFlow::::Continue(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!("},{"DocComment":" ControlFlow::<_, String>::Break(3).branch(),"},{"DocComment":" ControlFlow::Break(ControlFlow::Break(3)),"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"branch"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755}],"output":{"HashConsedValue":[11998,{"Adt":{"id":{"Adt":20},"generics":{"regions":[],"types":[{"HashConsedValue":[11997,{"TraitType":[{"Deduplicated":11995},1,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]},{"Deduplicated":11996}],"const_generics":[],"trait_refs":[]}}}]}},"src":{"TraitDecl":{"trait_ref":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":1},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},{"def_id":365,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["Extend",0]},{"Ident":["extend",0]}],"span":{"data":{"file_id":17,"beg":{"line":416,"col":4},"end":{"line":416,"col":61}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Extends a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" As this is the only required method for this trait, the [trait-level] docs"},{"DocComment":" contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: Extend"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"abc\");"},{"DocComment":""},{"DocComment":" message.extend(['d', 'e', 'f'].iter());"},{"DocComment":""},{"DocComment":" assert_eq!(\"abcdef\", &message);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"},{"index":2,"name":"T"},{"index":3,"name":"Clause1_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":35,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":17,"beg":{"line":416,"col":17},"end":{"line":416,"col":39}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":11793},{"Deduplicated":11756},{"Deduplicated":11994}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11936},{"Deduplicated":11793}],"output":{"Deduplicated":1801}},"src":{"TraitDecl":{"trait_ref":{"id":35,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},null,null,null,{"def_id":369,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["double_ended",0]},{"Ident":["DoubleEndedIterator",0]},{"Ident":["next_back",0]}],"span":{"data":{"file_id":68,"beg":{"line":96,"col":4},"end":{"line":96,"col":50}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Removes and returns an element from the end of the iterator."},{"DocComment":""},{"DocComment":" Returns `None` when there are no more elements."},{"DocComment":""},{"DocComment":" The [trait-level] docs contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: DoubleEndedIterator"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Remarks"},{"DocComment":""},{"DocComment":" The elements yielded by `DoubleEndedIterator`'s methods may differ from"},{"DocComment":" the ones yielded by [`Iterator`]'s methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let vec = vec![(1, 'a'), (1, 'b'), (1, 'c'), (2, 'a'), (2, 'b')];"},{"DocComment":" let uniq_by_fst_comp = || {"},{"DocComment":" let mut seen = std::collections::HashSet::new();"},{"DocComment":" vec.iter().copied().filter(move |x| seen.insert(x.0))"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" assert_eq!(uniq_by_fst_comp().last(), Some((2, 'a')));"},{"DocComment":" assert_eq!(uniq_by_fst_comp().next_back(), Some((2, 'b')));"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().fold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(1, 'a'), (2, 'a')]"},{"DocComment":" );"},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().rfold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(2, 'b'), (1, 'c')]"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Clause0_Clause0_Item"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11936}],"output":{"Deduplicated":11823}},"src":{"TraitDecl":{"trait_ref":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},null,null,null,null,null,null,null,{"def_id":377,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Ord",0]},{"Ident":["cmp",0]}],"span":{"data":{"file_id":33,"beg":{"line":1000,"col":4},"end":{"line":1000,"col":44}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" This method returns an [`Ordering`] between `self` and `other`."},{"DocComment":""},{"DocComment":" By convention, `self.cmp(&other)` returns the ordering matching the expression"},{"DocComment":" `self other` if true."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" assert_eq!(5.cmp(&10), Ordering::Less);"},{"DocComment":" assert_eq!(10.cmp(&5), Ordering::Greater);"},{"DocComment":" assert_eq!(5.cmp(&5), Ordering::Equal);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"ord_cmp_method"},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":4,"beg":{"line":1,"col":0},"end":{"line":1,"col":0}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11813},{"Deduplicated":11913}],"output":{"HashConsedValue":[5649,{"Adt":{"id":{"Adt":49},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}]}},"src":{"TraitDecl":{"trait_ref":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}},"item_id":{"Method":0},"has_default":false}},"is_global_initializer":null,"body":"Opaque"},null,null,null,{"def_id":381,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Product",0]},{"Ident":["product",0]}],"span":{"data":{"file_id":43,"beg":{"line":44,"col":4},"end":{"line":44,"col":55}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by multiplying"},{"DocComment":" the 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For more about the concept of iterators"},{"DocComment":" generally, please see the [module-level documentation]. In particular, you"},{"DocComment":" may want to know how to [implement `Iterator`][impl]."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":" [impl]: crate::iter#implementing-iterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"iterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"next","attr_info":{"attributes":[{"DocComment":" Advances the iterator and returns the next value."},{"DocComment":""},{"DocComment":" Returns [`None`] when iteration is finished. Individual iterator"},{"DocComment":" implementations may choose to resume iteration, and so calling `next()`"},{"DocComment":" again may or may not eventually start returning [`Some(Item)`] again at some"},{"DocComment":" point."},{"DocComment":""},{"DocComment":" [`Some(Item)`]: Some"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let a = [1, 2, 3];"},{"DocComment":""},{"DocComment":" let mut iter = a.into_iter();"},{"DocComment":""},{"DocComment":" // A call to next() returns the next value..."},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":""},{"DocComment":" // ... and then None once it's over."},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":""},{"DocComment":" // More calls may or may not return `None`. Here, they always will."},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"HashConsedValue":[11822,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11755},"Mut"]}]}],"output":{"Deduplicated":11823}},"item":{"id":177,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11824,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[0,0]}},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":25},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":1,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["range",0]},{"Ident":["Step",0]}],"span":{"data":{"file_id":21,"beg":{"line":34,"col":0},"end":{"line":34,"col":64}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Objects that have a notion of *successor* and *predecessor* operations."},{"DocComment":""},{"DocComment":" The *successor* operation moves towards values that compare greater."},{"DocComment":" The *predecessor* operation moves towards values that compare lesser."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"range_step"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":21,"beg":{"line":34,"col":22},"end":{"line":34,"col":35}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":21,"beg":{"line":34,"col":38},"end":{"line":34,"col":56}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"steps_between","attr_info":{"attributes":[{"DocComment":" Returns the bounds on the number of *successor* steps required to get from `start` to `end`"},{"DocComment":" like [`Iterator::size_hint()`][Iterator::size_hint()]."},{"DocComment":""},{"DocComment":" Returns `(usize::MAX, None)` if the number of steps would overflow `usize`, or is infinite."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `b`, and `n`:"},{"DocComment":""},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` if and only if `Step::forward_checked(&a, n) == Some(b)`"},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` if and only if `Step::backward_checked(&b, n) == Some(a)`"},{"DocComment":" * `steps_between(&a, &b) == (n, Some(n))` only if `a <= b`"},{"DocComment":" * Corollary: `steps_between(&a, &b) == (0, Some(0))` if and only if `a == b`"},{"DocComment":" * `steps_between(&a, &b) == (0, None)` if `a > b`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"HashConsedValue":[11825,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11755},"Shared"]}]},{"HashConsedValue":[11826,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11755},"Shared"]}]}],"output":{"Deduplicated":4223}},"item":{"id":263,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11827,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":1,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[1,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"forward_checked","attr_info":{"attributes":[{"DocComment":" Returns the value that would be obtained by taking the *successor*"},{"DocComment":" of `self` `count` times."},{"DocComment":""},{"DocComment":" If this would overflow the range of values supported by `Self`, returns `None`."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `n`, and `m`:"},{"DocComment":""},{"DocComment":" * `Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == Step::forward_checked(a, m).and_then(|x| Step::forward_checked(x, n))`"},{"DocComment":" * `Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == try { Step::forward_checked(a, n.checked_add(m)) }`"},{"DocComment":""},{"DocComment":" For any `a` and `n`:"},{"DocComment":""},{"DocComment":" * `Step::forward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::forward_checked(&x, 1))`"},{"DocComment":" * Corollary: `Step::forward_checked(a, 0) == Some(a)`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":1784}],"output":{"Deduplicated":11828}},"item":{"id":264,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11827}]}}},"kind":{"TraitMethod":[1,1]}},null,null,{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"backward_checked","attr_info":{"attributes":[{"DocComment":" Returns the value that would be obtained by taking the *predecessor*"},{"DocComment":" of `self` `count` times."},{"DocComment":""},{"DocComment":" If this would overflow the range of values supported by `Self`, returns `None`."},{"DocComment":""},{"DocComment":" # Invariants"},{"DocComment":""},{"DocComment":" For any `a`, `n`, and `m`:"},{"DocComment":""},{"DocComment":" * `Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == n.checked_add(m).and_then(|x| Step::backward_checked(a, x))`"},{"DocComment":" * `Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == try { Step::backward_checked(a, n.checked_add(m)?) }`"},{"DocComment":""},{"DocComment":" For any `a` and `n`:"},{"DocComment":""},{"DocComment":" * `Step::backward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::backward_checked(x, 1))`"},{"DocComment":" * Corollary: `Step::backward_checked(a, 0) == Some(a)`"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":1784}],"output":{"Deduplicated":11828}},"item":{"id":267,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11827}]}}},"kind":{"TraitMethod":[1,4]}},null,null],"vtable":null},{"def_id":2,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["From",0]}],"span":{"data":{"file_id":32,"beg":{"line":587,"col":0},"end":{"line":587,"col":30}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used to do value-to-value conversions while consuming the input value. It is the reciprocal of"},{"DocComment":" [`Into`]."},{"DocComment":""},{"DocComment":" One should always prefer implementing `From` over [`Into`]"},{"DocComment":" because implementing `From` automatically provides one with an implementation of [`Into`]"},{"DocComment":" thanks to the blanket implementation in the standard library."},{"DocComment":""},{"DocComment":" Only implement [`Into`] when targeting a version prior to Rust 1.41 and converting to a type"},{"DocComment":" outside the current crate."},{"DocComment":" `From` was not able to do these types of conversions in earlier versions because of Rust's"},{"DocComment":" orphaning rules."},{"DocComment":" See [`Into`] for more details."},{"DocComment":""},{"DocComment":" Prefer using [`Into`] over [`From`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`Into`] can be used as well."},{"DocComment":""},{"DocComment":" The `From` trait is also very useful when performing error handling. When constructing a function"},{"DocComment":" that is capable of failing, the return type will generally be of the form `Result`."},{"DocComment":" `From` simplifies error handling by allowing a function to return a single error type"},{"DocComment":" that encapsulates multiple error types. See the \"Examples\" section and [the book][book] for more"},{"DocComment":" details."},{"DocComment":""},{"DocComment":" **Note: This trait must not fail**. The `From` trait is intended for perfect conversions."},{"DocComment":" If the conversion can fail or is not perfect, use [`TryFrom`]."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - `From for U` implies [`Into`]` for T`"},{"DocComment":" - `From` is reflexive, which means that `From for T` is implemented"},{"DocComment":""},{"DocComment":" # When to implement `From`"},{"DocComment":""},{"DocComment":" While there's no technical restrictions on which conversions can be done using"},{"DocComment":" a `From` implementation, the general expectation is that the conversions"},{"DocComment":" should typically be restricted as follows:"},{"DocComment":""},{"DocComment":" * The conversion is *infallible*: if the conversion can fail, use [`TryFrom`]"},{"DocComment":" instead; don't provide a `From` impl that panics."},{"DocComment":""},{"DocComment":" * The conversion is *lossless*: semantically, it should not lose or discard"},{"DocComment":" information. For example, `i32: From` exists, where the original"},{"DocComment":" value can be recovered using `u16: TryFrom`. And `String: From<&str>`"},{"DocComment":" exists, where you can get something equivalent to the original value via"},{"DocComment":" `Deref`. But `From` cannot be used to convert from `u32` to `u16`, since"},{"DocComment":" that cannot succeed in a lossless way. (There's some wiggle room here for"},{"DocComment":" information not considered semantically relevant. For example,"},{"DocComment":" `Box<[T]>: From>` exists even though it might not preserve capacity,"},{"DocComment":" like how two vectors can be equal despite differing capacities.)"},{"DocComment":""},{"DocComment":" * The conversion is *value-preserving*: the conceptual kind and meaning of"},{"DocComment":" the resulting value is the same, even though the Rust type and technical"},{"DocComment":" representation might be different. For example `-1_i8 as u8` is *lossless*,"},{"DocComment":" since `as` casting back can recover the original value, but that conversion"},{"DocComment":" is *not* available via `From` because `-1` and `255` are different conceptual"},{"DocComment":" values (despite being identical bit patterns technically). But"},{"DocComment":" `f32: From` *is* available because `1_i16` and `1.0_f32` are conceptually"},{"DocComment":" the same real number (despite having very different bit patterns technically)."},{"DocComment":" `String: From` is available because they're both *text*, but"},{"DocComment":" `String: From` is *not* available, since `1` (a number) and `\"1\"`"},{"DocComment":" (text) are too different. (Converting values to text is instead covered"},{"DocComment":" by the [`Display`](crate::fmt::Display) trait.)"},{"DocComment":""},{"DocComment":" * The conversion is *obvious*: it's the only reasonable conversion between"},{"DocComment":" the two types. Otherwise it's better to have it be a named method or"},{"DocComment":" constructor, like how [`str::as_bytes`] is a method and how integers have"},{"DocComment":" methods like [`u32::from_ne_bytes`], [`u32::from_le_bytes`], and"},{"DocComment":" [`u32::from_be_bytes`], none of which are `From` implementations. Whereas"},{"DocComment":" there's only one reasonable way to wrap an [`Ipv6Addr`](crate::net::Ipv6Addr)"},{"DocComment":" into an [`IpAddr`](crate::net::IpAddr), thus `IpAddr: From` exists."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" [`String`] implements `From<&str>`:"},{"DocComment":""},{"DocComment":" An explicit conversion from a `&str` to a String is done as follows:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let string = \"hello\".to_string();"},{"DocComment":" let other_string = String::from(\"hello\");"},{"DocComment":""},{"DocComment":" assert_eq!(string, other_string);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" While performing error handling it is often useful to implement `From` for your own error type."},{"DocComment":" By converting underlying error types to our own custom error type that encapsulates the"},{"DocComment":" underlying error type, we can return a single error type without losing information on the"},{"DocComment":" underlying cause. The '?' operator automatically converts the underlying error type to our"},{"DocComment":" custom error type with `From::from`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fs;"},{"DocComment":" use std::io;"},{"DocComment":" use std::num;"},{"DocComment":""},{"DocComment":" enum CliError {"},{"DocComment":" IoError(io::Error),"},{"DocComment":" ParseError(num::ParseIntError),"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl From for CliError {"},{"DocComment":" fn from(error: io::Error) -> Self {"},{"DocComment":" CliError::IoError(error)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl From for CliError {"},{"DocComment":" fn from(error: num::ParseIntError) -> Self {"},{"DocComment":" CliError::ParseError(error)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn open_and_parse_file(file_name: &str) -> Result {"},{"DocComment":" let mut contents = fs::read_to_string(&file_name)?;"},{"DocComment":" let num: i32 = contents.trim().parse()?;"},{"DocComment":" Ok(num)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`from`]: From::from"},{"DocComment":" [book]: ../../book/ch09-00-error-handling.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"From"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from","attr_info":{"attributes":[{"DocComment":" Converts to this type from the input type."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11756}],"output":{"Deduplicated":11755}},"item":{"id":294,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11829,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":2,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[2,0]}}],"vtable":null},{"def_id":3,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Eq",0]}],"span":{"data":{"file_id":33,"beg":{"line":338,"col":0},"end":{"line":338,"col":58}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for comparisons corresponding to [equivalence relations]("},{"DocComment":" https://en.wikipedia.org/wiki/Equivalence_relation)."},{"DocComment":""},{"DocComment":" The primary difference to [`PartialEq`] is the additional requirement for reflexivity. A type"},{"DocComment":" that implements [`PartialEq`] guarantees that for all `a`, `b` and `c`:"},{"DocComment":""},{"DocComment":" - symmetric: `a == b` implies `b == a`"},{"DocComment":" - transitive: `a == b` and `b == c` implies `a == c`"},{"DocComment":" - consistent: `a != b` if and only if `!(a == b)`"},{"DocComment":""},{"DocComment":" `Eq`, which builds on top of [`PartialEq`] also implies:"},{"DocComment":""},{"DocComment":" - reflexive: `a == a`"},{"DocComment":""},{"DocComment":" This property cannot be checked by the compiler, and therefore `Eq` is a trait without methods."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" Floating point types such as [`f32`] and [`f64`] implement only [`PartialEq`] but *not* `Eq`"},{"DocComment":" because `NaN` != `NaN`."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`. When `derive`d, because `Eq` has no extra methods, it"},{"DocComment":" is only informing the compiler that this is an equivalence relation rather than a partial"},{"DocComment":" equivalence relation. Note that the `derive` strategy requires all fields are `Eq`, which isn't"},{"DocComment":" always desired."},{"DocComment":""},{"DocComment":" ## How can I implement `Eq`?"},{"DocComment":""},{"DocComment":" If you cannot use the `derive` strategy, specify that your type implements `Eq`, which has no"},{"DocComment":" extra methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.isbn == other.isbn"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Book {}"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Eq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":338,"col":20},"end":{"line":338,"col":43}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[null,{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"assert_fields_are_eq","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"Deduplicated":1801}},"item":{"id":41,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11830,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[3,1]}}],"vtable":null},{"def_id":4,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["PartialEq",0]}],"span":{"data":{"file_id":33,"beg":{"line":250,"col":0},"end":{"line":250,"col":65}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for comparisons using the equality operator."},{"DocComment":""},{"DocComment":" Implementing this trait for types provides the `==` and `!=` operators for"},{"DocComment":" those types."},{"DocComment":""},{"DocComment":" `x.eq(y)` can also be written `x == y`, and `x.ne(y)` can be written `x != y`."},{"DocComment":" We use the easier-to-read infix notation in the remainder of this documentation."},{"DocComment":""},{"DocComment":" This trait allows for comparisons using the equality operator, for types"},{"DocComment":" that do not have a full equivalence relation. For example, in floating point"},{"DocComment":" numbers `NaN != NaN`, so floating point types implement `PartialEq` but not"},{"DocComment":" [`trait@Eq`]. Formally speaking, when `Rhs == Self`, this trait corresponds"},{"DocComment":" to a [partial equivalence relation]."},{"DocComment":""},{"DocComment":" [partial equivalence relation]: https://en.wikipedia.org/wiki/Partial_equivalence_relation"},{"DocComment":""},{"DocComment":" Implementations must ensure that `eq` and `ne` are consistent with each other:"},{"DocComment":""},{"DocComment":" - `a != b` if and only if `!(a == b)`."},{"DocComment":""},{"DocComment":" The default implementation of `ne` provides this consistency and is almost"},{"DocComment":" always sufficient. It should not be overridden without very good reason."},{"DocComment":""},{"DocComment":" If [`PartialOrd`] or [`Ord`] are also implemented for `Self` and `Rhs`, their methods must also"},{"DocComment":" be consistent with `PartialEq` (see the documentation of those traits for the exact"},{"DocComment":" requirements). It's easy to accidentally make them disagree by deriving some of the traits and"},{"DocComment":" manually implementing others."},{"DocComment":""},{"DocComment":" The equality relation `==` must satisfy the following conditions"},{"DocComment":" (for all `a`, `b`, `c` of type `A`, `B`, `C`):"},{"DocComment":""},{"DocComment":" - **Symmetry**: if `A: PartialEq` and `B: PartialEq`, then **`a == b`"},{"DocComment":" implies `b == a`**; and"},{"DocComment":""},{"DocComment":" - **Transitivity**: if `A: PartialEq` and `B: PartialEq` and `A:"},{"DocComment":" PartialEq`, then **`a == b` and `b == c` implies `a == c`**."},{"DocComment":" This must also work for longer chains, such as when `A: PartialEq`, `B: PartialEq`,"},{"DocComment":" `C: PartialEq`, and `A: PartialEq` all exist."},{"DocComment":""},{"DocComment":" Note that the `B: PartialEq` (symmetric) and `A: PartialEq`"},{"DocComment":" (transitive) impls are not forced to exist, but these requirements apply"},{"DocComment":" whenever they do exist."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Cross-crate considerations"},{"DocComment":""},{"DocComment":" Upholding the requirements stated above can become tricky when one crate implements `PartialEq`"},{"DocComment":" for a type of another crate (i.e., to allow comparing one of its own types with a type from the"},{"DocComment":" standard library). The recommendation is to never implement this trait for a foreign type. In"},{"DocComment":" other words, such a crate should do `impl PartialEq for LocalType`, but it should"},{"DocComment":" *not* do `impl PartialEq for ForeignType`."},{"DocComment":""},{"DocComment":" This avoids the problem of transitive chains that criss-cross crate boundaries: for all local"},{"DocComment":" types `T`, you may assume that no other crate will add `impl`s that allow comparing `T == U`. In"},{"DocComment":" other words, if other crates add `impl`s that allow building longer transitive chains `U1 == ..."},{"DocComment":" == T == V1 == ...`, then all the types that appear to the right of `T` must be types that the"},{"DocComment":" crate defining `T` already knows about. This rules out transitive chains where downstream crates"},{"DocComment":" can add new `impl`s that \"stitch together\" comparisons of foreign types in ways that violate"},{"DocComment":" transitivity."},{"DocComment":""},{"DocComment":" Not having such foreign `impl`s also avoids forward compatibility issues where one crate adding"},{"DocComment":" more `PartialEq` implementations can cause build failures in downstream crates."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`. When `derive`d on structs, two"},{"DocComment":" instances are equal if all fields are equal, and not equal if any fields"},{"DocComment":" are not equal. When `derive`d on enums, two instances are equal if they"},{"DocComment":" are the same variant and all fields are equal."},{"DocComment":""},{"DocComment":" ## How can I implement `PartialEq`?"},{"DocComment":""},{"DocComment":" An example implementation for a domain in which two books are considered"},{"DocComment":" the same book if their ISBN matches, even if the formats differ:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.isbn == other.isbn"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let b1 = Book { isbn: 3, format: BookFormat::Paperback };"},{"DocComment":" let b2 = Book { isbn: 3, format: BookFormat::Ebook };"},{"DocComment":" let b3 = Book { isbn: 10, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == b2);"},{"DocComment":" assert!(b1 != b3);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I compare two different types?"},{"DocComment":""},{"DocComment":" The type you can compare with is controlled by `PartialEq`'s type parameter."},{"DocComment":" For example, let's tweak our previous code a bit:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // The derive implements == comparisons"},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Implement == comparisons"},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &BookFormat) -> bool {"},{"DocComment":" self.format == *other"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Implement == comparisons"},{"DocComment":" impl PartialEq for BookFormat {"},{"DocComment":" fn eq(&self, other: &Book) -> bool {"},{"DocComment":" *self == other.format"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let b1 = Book { isbn: 3, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == BookFormat::Paperback);"},{"DocComment":" assert!(BookFormat::Ebook != b1);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" By changing `impl PartialEq for Book` to `impl PartialEq for Book`,"},{"DocComment":" we allow `BookFormat`s to be compared with `Book`s."},{"DocComment":""},{"DocComment":" A comparison like the one above, which ignores some fields of the struct,"},{"DocComment":" can be dangerous. It can easily lead to an unintended violation of the"},{"DocComment":" requirements for a partial equivalence relation. For example, if we kept"},{"DocComment":" the above implementation of `PartialEq` for `BookFormat` and added an"},{"DocComment":" implementation of `PartialEq` for `Book` (either via a `#[derive]` or"},{"DocComment":" via the manual implementation from the first example) then the result would"},{"DocComment":" violate transitivity:"},{"DocComment":""},{"DocComment":" ```should_panic"},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" enum BookFormat {"},{"DocComment":" Paperback,"},{"DocComment":" Hardback,"},{"DocComment":" Ebook,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" #[derive(PartialEq)]"},{"DocComment":" struct Book {"},{"DocComment":" isbn: i32,"},{"DocComment":" format: BookFormat,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Book {"},{"DocComment":" fn eq(&self, other: &BookFormat) -> bool {"},{"DocComment":" self.format == *other"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for BookFormat {"},{"DocComment":" fn eq(&self, other: &Book) -> bool {"},{"DocComment":" *self == other.format"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn main() {"},{"DocComment":" let b1 = Book { isbn: 1, format: BookFormat::Paperback };"},{"DocComment":" let b2 = Book { isbn: 2, format: BookFormat::Paperback };"},{"DocComment":""},{"DocComment":" assert!(b1 == BookFormat::Paperback);"},{"DocComment":" assert!(BookFormat::Paperback == b2);"},{"DocComment":""},{"DocComment":" // The following should hold by transitivity but doesn't."},{"DocComment":" assert!(b1 == b2); // <-- PANICS"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: u32 = 0;"},{"DocComment":" let y: u32 = 1;"},{"DocComment":""},{"DocComment":" assert_eq!(x == y, false);"},{"DocComment":" assert_eq!(x.eq(&y), false);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`eq`]: PartialEq::eq"},{"DocComment":" [`ne`]: PartialEq::ne"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"eq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"eq","attr_info":{"attributes":[{"DocComment":" Tests for `self` and `other` values to be equal, and is used by `==`."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"HashConsedValue":[11831,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11756},"Shared"]}]}],"output":{"Deduplicated":1796}},"item":{"id":43,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11832,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[4,0]}},null],"vtable":{"id":{"Adt":22},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":5,"item_meta":{"name":[{"Ident":["subtle",0]},{"Ident":["ConstantTimeEq",0]}],"span":{"data":{"file_id":22,"beg":{"line":262,"col":0},"end":{"line":262,"col":24}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An `Eq`-like trait that produces a `Choice` instead of a `bool`."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConstantTimeEq;"},{"DocComment":" let x: u8 = 5;"},{"DocComment":" let y: u8 = 13;"},{"DocComment":""},{"DocComment":" assert_eq!(x.ct_eq(&y).unwrap_u8(), 0);"},{"DocComment":" assert_eq!(x.ct_eq(&x).unwrap_u8(), 1);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"ct_eq","attr_info":{"attributes":[{"DocComment":" Determine if two items are equal."},{"DocComment":""},{"DocComment":" The `ct_eq` function should execute in constant time."},{"DocComment":""},{"DocComment":" # Returns"},{"DocComment":""},{"DocComment":" * `Choice(1u8)` if `self == other`;"},{"DocComment":" * `Choice(0u8)` if `self != other`."}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"Deduplicated":11826}],"output":{"Deduplicated":2321}},"item":{"id":46,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11833,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":5,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[5,0]}},null],"vtable":null},{"def_id":6,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["default",0]},{"Ident":["Default",0]}],"span":{"data":{"file_id":34,"beg":{"line":107,"col":0},"end":{"line":107,"col":30}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for giving a type a useful default value."},{"DocComment":""},{"DocComment":" Sometimes, you want to fall back to some kind of default value, and"},{"DocComment":" don't particularly care what it is. This comes up often with `struct`s"},{"DocComment":" that define a set of options:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" How can we define some default values? You can use `Default`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Default)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn main() {"},{"DocComment":" let options: SomeOptions = Default::default();"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Now, you get all of the default values. Rust implements `Default` for various primitive types."},{"DocComment":""},{"DocComment":" If you want to override a particular option, but still retain the other defaults:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" # #[derive(Default)]"},{"DocComment":" # struct SomeOptions {"},{"DocComment":" # foo: i32,"},{"DocComment":" # bar: f32,"},{"DocComment":" # }"},{"DocComment":" fn main() {"},{"DocComment":" let options = SomeOptions { foo: 42, ..Default::default() };"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all of the type's fields implement"},{"DocComment":" `Default`. When `derive`d, it will use the default value for each field's type."},{"DocComment":""},{"DocComment":" ### `enum`s"},{"DocComment":""},{"DocComment":" When using `#[derive(Default)]` on an `enum`, you need to choose which unit variant will be"},{"DocComment":" default. You do this by placing the `#[default]` attribute on the variant."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Default)]"},{"DocComment":" enum Kind {"},{"DocComment":" #[default]"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You cannot use the `#[default]` attribute on non-unit or non-exhaustive variants."},{"DocComment":""},{"DocComment":" The `#[default]` attribute was stabilized in Rust 1.62.0."},{"DocComment":""},{"DocComment":" ## How can I implement `Default`?"},{"DocComment":""},{"DocComment":" Provide an implementation for the `default()` method that returns the value of"},{"DocComment":" your type that should be the default:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" enum Kind {"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Default for Kind {"},{"DocComment":" fn default() -> Self { Kind::A }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Default)]"},{"DocComment":" struct SomeOptions {"},{"DocComment":" foo: i32,"},{"DocComment":" bar: f32,"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Default"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"default","attr_info":{"attributes":[{"DocComment":" Returns the \"default value\" for a type."},{"DocComment":""},{"DocComment":" Default values are often some kind of initial value, identity value, or anything else that"},{"DocComment":" may make sense as a default."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Using built-in default values:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let i: i8 = Default::default();"},{"DocComment":" let (x, y): (Option, f64) = Default::default();"},{"DocComment":" let (a, b, (c, d)): (i32, u32, (bool, bool)) = Default::default();"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Making your own:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" enum Kind {"},{"DocComment":" A,"},{"DocComment":" B,"},{"DocComment":" C,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Default for Kind {"},{"DocComment":" fn default() -> Self { Kind::A }"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[],"output":{"Deduplicated":11755}},"item":{"id":298,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11834,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":6,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[6,0]}}],"vtable":null},{"def_id":7,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["Copy",0]}],"span":{"data":{"file_id":35,"beg":{"line":454,"col":0},"end":{"line":454,"col":21}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Types whose values can be duplicated simply by copying bits."},{"DocComment":""},{"DocComment":" By default, variable bindings have 'move semantics.' In other"},{"DocComment":" words:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Foo;"},{"DocComment":""},{"DocComment":" let x = Foo;"},{"DocComment":""},{"DocComment":" let y = x;"},{"DocComment":""},{"DocComment":" // `x` has moved into `y`, and so cannot be used"},{"DocComment":""},{"DocComment":" // println!(\"{x:?}\"); // error: use of moved value"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, if a type implements `Copy`, it instead has 'copy semantics':"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // We can derive a `Copy` implementation. `Clone` is also required, as it's"},{"DocComment":" // a supertrait of `Copy`."},{"DocComment":" #[derive(Debug, Copy, Clone)]"},{"DocComment":" struct Foo;"},{"DocComment":""},{"DocComment":" let x = Foo;"},{"DocComment":""},{"DocComment":" let y = x;"},{"DocComment":""},{"DocComment":" // `y` is a copy of `x`"},{"DocComment":""},{"DocComment":" println!(\"{x:?}\"); // A-OK!"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It's important to note that in these two examples, the only difference is whether you"},{"DocComment":" are allowed to access `x` after the assignment. Under the hood, both a copy and a move"},{"DocComment":" can result in bits being copied in memory, although this is sometimes optimized away."},{"DocComment":""},{"DocComment":" ## How can I implement `Copy`?"},{"DocComment":""},{"DocComment":" There are two ways to implement `Copy` on your type. The simplest is to use `derive`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct MyStruct;"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You can also implement `Copy` and `Clone` manually:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct MyStruct;"},{"DocComment":""},{"DocComment":" impl Copy for MyStruct { }"},{"DocComment":""},{"DocComment":" impl Clone for MyStruct {"},{"DocComment":" fn clone(&self) -> MyStruct {"},{"DocComment":" *self"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" There is a small difference between the two. The `derive` strategy will also place a `Copy`"},{"DocComment":" bound on type parameters:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Clone)]"},{"DocComment":" struct MyStruct(T);"},{"DocComment":""},{"DocComment":" impl Copy for MyStruct { }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" This isn't always desired. For example, shared references (`&T`) can be copied regardless of"},{"DocComment":" whether `T` is `Copy`. Likewise, a generic struct containing markers such as [`PhantomData`]"},{"DocComment":" could potentially be duplicated with a bit-wise copy."},{"DocComment":""},{"DocComment":" ## What's the difference between `Copy` and `Clone`?"},{"DocComment":""},{"DocComment":" Copies happen implicitly, for example as part of an assignment `y = x`. The behavior of"},{"DocComment":" `Copy` is not overloadable; it is always a simple bit-wise copy."},{"DocComment":""},{"DocComment":" Cloning is an explicit action, `x.clone()`. The implementation of [`Clone`] can"},{"DocComment":" provide any type-specific behavior necessary to duplicate values safely. For example,"},{"DocComment":" the implementation of [`Clone`] for [`String`] needs to copy the pointed-to string"},{"DocComment":" buffer in the heap. A simple bitwise copy of [`String`] values would merely copy the"},{"DocComment":" pointer, leading to a double free down the line. For this reason, [`String`] is [`Clone`]"},{"DocComment":" but not `Copy`."},{"DocComment":""},{"DocComment":" [`Clone`] is a supertrait of `Copy`, so everything which is `Copy` must also implement"},{"DocComment":" [`Clone`]. If a type is `Copy` then its [`Clone`] implementation only needs to return `*self`"},{"DocComment":" (see the example above)."},{"DocComment":""},{"DocComment":" ## When can my type be `Copy`?"},{"DocComment":""},{"DocComment":" A type can implement `Copy` if all of its components implement `Copy`. For example, this"},{"DocComment":" struct can be `Copy`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #[allow(dead_code)]"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" A struct can be `Copy`, and [`i32`] is `Copy`, therefore `Point` is eligible to be `Copy`."},{"DocComment":" By contrast, consider"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" # struct Point;"},{"DocComment":" struct PointList {"},{"DocComment":" points: Vec,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The struct `PointList` cannot implement `Copy`, because [`Vec`] is not `Copy`. If we"},{"DocComment":" attempt to derive a `Copy` implementation, we'll get an error:"},{"DocComment":""},{"DocComment":" ```text"},{"DocComment":" the trait `Copy` cannot be implemented for this type; field `points` does not implement `Copy`"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Shared references (`&T`) are also `Copy`, so a type can be `Copy`, even when it holds"},{"DocComment":" shared references of types `T` that are *not* `Copy`. Consider the following struct,"},{"DocComment":" which can implement `Copy`, because it only holds a *shared reference* to our non-`Copy`"},{"DocComment":" type `PointList` from above:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(dead_code)]"},{"DocComment":" # struct PointList;"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct PointListWrapper<'a> {"},{"DocComment":" point_list_ref: &'a PointList,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## When *can't* my type be `Copy`?"},{"DocComment":""},{"DocComment":" Some types can't be copied safely. For example, copying `&mut T` would create an aliased"},{"DocComment":" mutable reference. Copying [`String`] would duplicate responsibility for managing the"},{"DocComment":" [`String`]'s buffer, leading to a double free."},{"DocComment":""},{"DocComment":" Generalizing the latter case, any type implementing [`Drop`] can't be `Copy`, because it's"},{"DocComment":" managing some resource besides its own [`size_of::`] bytes."},{"DocComment":""},{"DocComment":" If you try to implement `Copy` on a struct or enum containing non-`Copy` data, you will get"},{"DocComment":" the error [E0204]."},{"DocComment":""},{"DocComment":" [E0204]: ../../error_codes/E0204.html"},{"DocComment":""},{"DocComment":" ## When *should* my type be `Copy`?"},{"DocComment":""},{"DocComment":" Generally speaking, if your type _can_ implement `Copy`, it should. Keep in mind, though,"},{"DocComment":" that implementing `Copy` is part of the public API of your type. If the type might become"},{"DocComment":" non-`Copy` in the future, it could be prudent to omit the `Copy` implementation now, to"},{"DocComment":" avoid a breaking API change."},{"DocComment":""},{"DocComment":" ## Additional implementors"},{"DocComment":""},{"DocComment":" In addition to the [implementors listed below][impls],"},{"DocComment":" the following types also implement `Copy`:"},{"DocComment":""},{"DocComment":" * Function item types (i.e., the distinct types defined for each function)"},{"DocComment":" * Function pointer types (e.g., `fn() -> i32`)"},{"DocComment":" * Closure types, if they capture no value from the environment"},{"DocComment":" or if all such captured values implement `Copy` themselves."},{"DocComment":" Note that variables captured by shared reference always implement `Copy`"},{"DocComment":" (even if the referent doesn't),"},{"DocComment":" while variables captured by mutable reference never implement `Copy`."},{"DocComment":""},{"DocComment":" [`Vec`]: ../../std/vec/struct.Vec.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`size_of::`]: size_of"},{"DocComment":" [impls]: #implementors"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"copy"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":35,"beg":{"line":454,"col":16},"end":{"line":454,"col":21}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[],"vtable":null},{"def_id":8,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["clone",0]},{"Ident":["Clone",0]}],"span":{"data":{"file_id":36,"beg":{"line":194,"col":0},"end":{"line":194,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A common trait that allows explicit creation of a duplicate value."},{"DocComment":""},{"DocComment":" Calling [`clone`] always produces a new value."},{"DocComment":" However, for types that are references to other data (such as smart pointers or references),"},{"DocComment":" the new value may still point to the same underlying data, rather than duplicating it."},{"DocComment":" See [`Clone::clone`] for more details."},{"DocComment":""},{"DocComment":" This distinction is especially important when using `#[derive(Clone)]` on structs containing"},{"DocComment":" smart pointers like `Arc>` - the cloned struct will share mutable state with the"},{"DocComment":" original."},{"DocComment":""},{"DocComment":" Differs from [`Copy`] in that [`Copy`] is implicit and an inexpensive bit-wise copy, while"},{"DocComment":" `Clone` is always explicit and may or may not be expensive. [`Copy`] has no methods, so you"},{"DocComment":" cannot change its behavior, but when implementing `Clone`, the `clone` method you provide"},{"DocComment":" may run arbitrary code."},{"DocComment":""},{"DocComment":" Since `Clone` is a supertrait of [`Copy`], any type that implements `Copy` must also implement"},{"DocComment":" `Clone`."},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all fields are `Clone`. The `derive`d"},{"DocComment":" implementation of [`Clone`] calls [`clone`] on each field."},{"DocComment":""},{"DocComment":" [`clone`]: Clone::clone"},{"DocComment":""},{"DocComment":" For a generic struct, `#[derive]` implements `Clone` conditionally by adding bound `Clone` on"},{"DocComment":" generic parameters."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // `derive` implements Clone for Reading when T is Clone."},{"DocComment":" #[derive(Clone)]"},{"DocComment":" struct Reading {"},{"DocComment":" frequency: T,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I implement `Clone`?"},{"DocComment":""},{"DocComment":" Types that are [`Copy`] should have a trivial implementation of `Clone`. More formally:"},{"DocComment":" if `T: Copy`, `x: T`, and `y: &T`, then `let x = y.clone();` is equivalent to `let x = *y;`."},{"DocComment":" Manual implementations should be careful to uphold this invariant; however, unsafe code"},{"DocComment":" must not rely on it to ensure memory safety."},{"DocComment":""},{"DocComment":" An example is a generic struct holding a function pointer. In this case, the"},{"DocComment":" implementation of `Clone` cannot be `derive`d, but can be implemented as:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" impl Copy for Generate {}"},{"DocComment":""},{"DocComment":" impl Clone for Generate {"},{"DocComment":" fn clone(&self) -> Self {"},{"DocComment":" *self"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If we `derive`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" the auto-derived implementations will have unnecessary `T: Copy` and `T: Clone` bounds:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" // Automatically derived"},{"DocComment":" impl Copy for Generate { }"},{"DocComment":""},{"DocComment":" // Automatically derived"},{"DocComment":" impl Clone for Generate {"},{"DocComment":" fn clone(&self) -> Generate {"},{"DocComment":" Generate(Clone::clone(&self.0))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The bounds are unnecessary because clearly the function itself should be"},{"DocComment":" copy- and cloneable even if its return type is not:"},{"DocComment":""},{"DocComment":" ```compile_fail,E0599"},{"DocComment":" #[derive(Copy, Clone)]"},{"DocComment":" struct Generate(fn() -> T);"},{"DocComment":""},{"DocComment":" struct NotCloneable;"},{"DocComment":""},{"DocComment":" fn generate_not_cloneable() -> NotCloneable {"},{"DocComment":" NotCloneable"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" Generate(generate_not_cloneable).clone(); // error: trait bounds were not satisfied"},{"DocComment":" // Note: With the manual implementations the above line will compile."},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## `Clone` and `PartialEq`/`Eq`"},{"DocComment":" `Clone` is intended for the duplication of objects. Consequently, when implementing"},{"DocComment":" both `Clone` and [`PartialEq`], the following property is expected to hold:"},{"DocComment":" ```text"},{"DocComment":" x == x -> x.clone() == x"},{"DocComment":" ```"},{"DocComment":" In other words, if an object compares equal to itself,"},{"DocComment":" its clone must also compare equal to the original."},{"DocComment":""},{"DocComment":" For types that also implement [`Eq`] – for which `x == x` always holds –"},{"DocComment":" this implies that `x.clone() == x` must always be true."},{"DocComment":" Standard library collections such as"},{"DocComment":" [`HashMap`], [`HashSet`], [`BTreeMap`], [`BTreeSet`] and [`BinaryHeap`]"},{"DocComment":" rely on their keys respecting this property for correct behavior."},{"DocComment":" Furthermore, these collections require that cloning a key preserves the outcome of the"},{"DocComment":" [`Hash`] and [`Ord`] methods. Thankfully, this follows automatically from `x.clone() == x`"},{"DocComment":" if `Hash` and `Ord` are correctly implemented according to their own requirements."},{"DocComment":""},{"DocComment":" When deriving both `Clone` and [`PartialEq`] using `#[derive(Clone, PartialEq)]`"},{"DocComment":" or when additionally deriving [`Eq`] using `#[derive(Clone, PartialEq, Eq)]`,"},{"DocComment":" then this property is automatically upheld – provided that it is satisfied by"},{"DocComment":" the underlying types."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on this property"},{"DocComment":" being satisfied."},{"DocComment":""},{"DocComment":" ## Additional implementors"},{"DocComment":""},{"DocComment":" In addition to the [implementors listed below][impls],"},{"DocComment":" the following types also implement `Clone`:"},{"DocComment":""},{"DocComment":" * Function item types (i.e., the distinct types defined for each function)"},{"DocComment":" * Function pointer types (e.g., `fn() -> i32`)"},{"DocComment":" * Closure types, if they capture no value from the environment"},{"DocComment":" or if all such captured values implement `Clone` themselves."},{"DocComment":" Note that variables captured by shared reference always implement `Clone`"},{"DocComment":" (even if the referent doesn't),"},{"DocComment":" while variables captured by mutable reference never implement `Clone`."},{"DocComment":""},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`HashSet`]: ../../std/collections/struct.HashSet.html"},{"DocComment":" [`BTreeMap`]: ../../std/collections/struct.BTreeMap.html"},{"DocComment":" [`BTreeSet`]: ../../std/collections/struct.BTreeSet.html"},{"DocComment":" [`BinaryHeap`]: ../../std/collections/struct.BinaryHeap.html"},{"DocComment":" [impls]: #implementors"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"clone"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"clone","attr_info":{"attributes":[{"DocComment":" Returns a duplicate of the value."},{"DocComment":""},{"DocComment":" Note that what \"duplicate\" means varies by type:"},{"DocComment":" - For most types, this creates a deep, independent copy"},{"DocComment":" - For reference types like `&T`, this creates another reference to the same value"},{"DocComment":" - For smart pointers like [`Arc`] or [`Rc`], this increments the reference count"},{"DocComment":" but still points to the same underlying data"},{"DocComment":""},{"DocComment":" [`Arc`]: ../../std/sync/struct.Arc.html"},{"DocComment":" [`Rc`]: ../../std/rc/struct.Rc.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(noop_method_call)]"},{"DocComment":" let hello = \"Hello\"; // &str implements Clone"},{"DocComment":""},{"DocComment":" assert_eq!(\"Hello\", hello.clone());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Example with a reference-counted type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::sync::{Arc, Mutex};"},{"DocComment":""},{"DocComment":" let data = Arc::new(Mutex::new(vec![1, 2, 3]));"},{"DocComment":" let data_clone = data.clone(); // Creates another Arc pointing to the same Mutex"},{"DocComment":""},{"DocComment":" {"},{"DocComment":" let mut lock = data.lock().unwrap();"},{"DocComment":" lock.push(4);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Changes are visible through the clone because they share the same underlying data"},{"DocComment":" assert_eq!(*data_clone.lock().unwrap(), vec![1, 2, 3, 4]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"Deduplicated":11755}},"item":{"id":58,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11835,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[8,0]}},null],"vtable":null},null,{"def_id":10,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["Destruct",0]}],"span":{"data":{"file_id":35,"beg":{"line":1062,"col":0},"end":{"line":1062,"col":38}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A marker for types that can be dropped."},{"DocComment":""},{"DocComment":" This should be used for `[const]` bounds,"},{"DocComment":" as non-const bounds will always hold for every type."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"destruct"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"drop_glue","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":true,"abi":"Rust","inputs":[{"Deduplicated":11822}],"output":{"Deduplicated":1801}},"item":{"id":300,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}},"kind":{"TraitMethod":[10,0]}}],"vtable":null},{"def_id":11,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["fmt",0]},{"Ident":["Debug",0]}],"span":{"data":{"file_id":27,"beg":{"line":1053,"col":0},"end":{"line":1053,"col":29}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" `?` formatting."},{"DocComment":""},{"DocComment":" `Debug` should format the output in a programmer-facing, debugging context."},{"DocComment":""},{"DocComment":" Generally speaking, you should just `derive` a `Debug` implementation."},{"DocComment":""},{"DocComment":" When used with the alternate format specifier `#?`, the output is pretty-printed."},{"DocComment":""},{"DocComment":" For more information on formatters, see [the module-level documentation][module]."},{"DocComment":""},{"DocComment":" [module]: ../../std/fmt/index.html"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]` if all fields implement `Debug`. When"},{"DocComment":" `derive`d for structs, it will use the name of the `struct`, then `{`, then a"},{"DocComment":" comma-separated list of each field's name and `Debug` value, then `}`. For"},{"DocComment":" `enum`s, it will use the name of the variant and, if applicable, `(`, then the"},{"DocComment":" `Debug` values of the fields, then `)`."},{"DocComment":""},{"DocComment":" # Stability"},{"DocComment":""},{"DocComment":" Derived `Debug` formats are not stable, and so may change with future Rust"},{"DocComment":" versions. Additionally, `Debug` implementations of types provided by the"},{"DocComment":" standard library (`std`, `core`, `alloc`, etc.) are not stable, and"},{"DocComment":" may also change with future Rust versions."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Deriving an implementation:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" format!(\"The origin is: {origin:?}\"),"},{"DocComment":" \"The origin is: Point { x: 0, y: 0 }\","},{"DocComment":" );"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Manually implementing:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fmt;"},{"DocComment":""},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl fmt::Debug for Point {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" f.debug_struct(\"Point\")"},{"DocComment":" .field(\"x\", &self.x)"},{"DocComment":" .field(\"y\", &self.y)"},{"DocComment":" .finish()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" format!(\"The origin is: {origin:?}\"),"},{"DocComment":" \"The origin is: Point { x: 0, y: 0 }\","},{"DocComment":" );"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" There are a number of helper methods on the [`Formatter`] struct to help you with manual"},{"DocComment":" implementations, such as [`debug_struct`]."},{"DocComment":""},{"DocComment":" [`debug_struct`]: Formatter::debug_struct"},{"DocComment":""},{"DocComment":" Types that do not wish to use the standard suite of debug representations"},{"DocComment":" provided by the `Formatter` trait (`debug_struct`, `debug_tuple`,"},{"DocComment":" `debug_list`, `debug_set`, `debug_map`) can do something totally custom by"},{"DocComment":" manually writing an arbitrary representation to the `Formatter`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # use std::fmt;"},{"DocComment":" # struct Point {"},{"DocComment":" # x: i32,"},{"DocComment":" # y: i32,"},{"DocComment":" # }"},{"DocComment":" #"},{"DocComment":" impl fmt::Debug for Point {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" write!(f, \"Point [{} {}]\", self.x, self.y)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" `Debug` implementations using either `derive` or the debug builder API"},{"DocComment":" on [`Formatter`] support pretty-printing using the alternate flag: `{:#?}`."},{"DocComment":""},{"DocComment":" Pretty-printing with `#?`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let origin = Point { x: 0, y: 0 };"},{"DocComment":""},{"DocComment":" let expected = \"The origin is: Point {"},{"DocComment":" x: 0,"},{"DocComment":" y: 0,"},{"DocComment":" }\";"},{"DocComment":" assert_eq!(format!(\"The origin is: {origin:#?}\"), expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Debug"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"},{"index":2,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"fmt","attr_info":{"attributes":[{"DocComment":"Formats the value using the given formatter.\n\n# Errors\n\nThis function should return [`Err`] if, and only if, the provided [`Formatter`] returns [`Err`].\nString formatting is considered an infallible operation; this function only\nreturns a [`Result`] because writing to the underlying stream might fail and it must\nprovide a way to propagate the fact that an error has occurred back up the stack.\n"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::fmt;"},{"DocComment":""},{"DocComment":" struct Position {"},{"DocComment":" longitude: f32,"},{"DocComment":" latitude: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl fmt::Debug for Position {"},{"DocComment":" fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {"},{"DocComment":" f.debug_tuple(\"\")"},{"DocComment":" .field(&self.longitude)"},{"DocComment":" .field(&self.latitude)"},{"DocComment":" .finish()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let position = Position { longitude: 1.987, latitude: 2.983 };"},{"DocComment":" assert_eq!(format!(\"{position:?}\"), \"(1.987, 2.983)\");"},{"DocComment":""},{"DocComment":" assert_eq!(format!(\"{position:#?}\"), \"("},{"DocComment":" 1.987,"},{"DocComment":" 2.983,"},{"DocComment":" )\");"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"HashConsedValue":[7101,{"Ref":[{"Var":{"Bound":[0,1]}},{"HashConsedValue":[7100,{"Adt":{"id":{"Adt":57},"generics":{"regions":[{"Var":{"Bound":[0,2]}}],"types":[],"const_generics":[],"trait_refs":[]}}}]},"Mut"]}]}],"output":{"Deduplicated":7103}},"item":{"id":301,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}},{"Var":{"Bound":[0,2]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11836,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":11,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[11,0]}}],"vtable":{"id":{"Adt":56},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":12,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["AddAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":768,"col":0},"end":{"line":768,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The addition assignment operator `+=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" This example creates a `Point` struct that implements the `AddAssign`"},{"DocComment":" trait, and then demonstrates add-assigning to a mutable `Point`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::AddAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl AddAssign for Point {"},{"DocComment":" fn add_assign(&mut self, other: Self) {"},{"DocComment":" *self = Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" };"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut point = Point { x: 1, y: 0 };"},{"DocComment":" point += Point { x: 2, y: 3 };"},{"DocComment":" assert_eq!(point, Point { x: 3, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"add_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"add_assign","attr_info":{"attributes":[{"DocComment":" Performs the `+=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x += 1;"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":11756}],"output":{"Deduplicated":1801}},"item":{"id":305,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11837,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":12,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[12,0]}}],"vtable":{"id":{"Adt":60},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":13,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Add",0]}],"span":{"data":{"file_id":37,"beg":{"line":76,"col":0},"end":{"line":76,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The addition operator `+`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory. For"},{"DocComment":" example, [`std::time::SystemTime`] implements `Add`, which permits"},{"DocComment":" operations of the form `SystemTime = SystemTime + Duration`."},{"DocComment":""},{"DocComment":" [`std::time::SystemTime`]: ../../std/time/struct.SystemTime.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Add`able points"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Add;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Add for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn add(self, other: Self) -> Self {"},{"DocComment":" Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 1, y: 0 } + Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 3, y: 3 });"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Implementing `Add` with generics"},{"DocComment":""},{"DocComment":" Here is an example of the same `Point` struct implementing the `Add` trait"},{"DocComment":" using generics."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Add;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: T,"},{"DocComment":" y: T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Notice that the implementation uses the associated type `Output`."},{"DocComment":" impl> Add for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn add(self, other: Self) -> Self::Output {"},{"DocComment":" Self {"},{"DocComment":" x: self.x + other.x,"},{"DocComment":" y: self.y + other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 1, y: 0 } + Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 3, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"add"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"add","attr_info":{"attributes":[{"DocComment":" Performs the `+` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 + 1, 13);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":306,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11838,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":13,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[13,0]}}],"vtable":{"id":{"Adt":61},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":14,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["SubAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":839,"col":0},"end":{"line":839,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The subtraction assignment operator `-=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" This example creates a `Point` struct that implements the `SubAssign`"},{"DocComment":" trait, and then demonstrates sub-assigning to a mutable `Point`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::SubAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl SubAssign for Point {"},{"DocComment":" fn sub_assign(&mut self, other: Self) {"},{"DocComment":" *self = Self {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" };"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut point = Point { x: 3, y: 3 };"},{"DocComment":" point -= Point { x: 2, y: 3 };"},{"DocComment":" assert_eq!(point, Point {x: 1, y: 0});"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"sub_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sub_assign","attr_info":{"attributes":[{"DocComment":" Performs the `-=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x -= 1;"},{"DocComment":" assert_eq!(x, 11);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":11756}],"output":{"Deduplicated":1801}},"item":{"id":307,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11839,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":14,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[14,0]}}],"vtable":{"id":{"Adt":62},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":15,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Sub",0]}],"span":{"data":{"file_id":37,"beg":{"line":188,"col":0},"end":{"line":188,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The subtraction operator `-`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory. For"},{"DocComment":" example, [`std::time::SystemTime`] implements `Sub`, which permits"},{"DocComment":" operations of the form `SystemTime = SystemTime - Duration`."},{"DocComment":""},{"DocComment":" [`std::time::SystemTime`]: ../../std/time/struct.SystemTime.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Sub`tractable points"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Sub;"},{"DocComment":""},{"DocComment":" #[derive(Debug, Copy, Clone, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: i32,"},{"DocComment":" y: i32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Sub for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn sub(self, other: Self) -> Self::Output {"},{"DocComment":" Self {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 3, y: 3 } - Point { x: 2, y: 3 },"},{"DocComment":" Point { x: 1, y: 0 });"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Implementing `Sub` with generics"},{"DocComment":""},{"DocComment":" Here is an example of the same `Point` struct implementing the `Sub` trait"},{"DocComment":" using generics."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Sub;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Point {"},{"DocComment":" x: T,"},{"DocComment":" y: T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Notice that the implementation uses the associated type `Output`."},{"DocComment":" impl> Sub for Point {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn sub(self, other: Self) -> Self::Output {"},{"DocComment":" Point {"},{"DocComment":" x: self.x - other.x,"},{"DocComment":" y: self.y - other.y,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Point { x: 2, y: 3 } - Point { x: 1, y: 0 },"},{"DocComment":" Point { x: 1, y: 3 });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"sub"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sub","attr_info":{"attributes":[{"DocComment":" Performs the `-` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 - 1, 11);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":308,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11840,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":15,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[15,0]}}],"vtable":{"id":{"Adt":63},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":16,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["MulAssign",0]}],"span":{"data":{"file_id":37,"beg":{"line":901,"col":0},"end":{"line":901,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The multiplication assignment operator `*=`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::MulAssign;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Frequency { hertz: f64 }"},{"DocComment":""},{"DocComment":" impl MulAssign for Frequency {"},{"DocComment":" fn mul_assign(&mut self, rhs: f64) {"},{"DocComment":" self.hertz *= rhs;"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut frequency = Frequency { hertz: 50.0 };"},{"DocComment":" frequency *= 4.0;"},{"DocComment":" assert_eq!(Frequency { hertz: 200.0 }, frequency);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"mul_assign"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"mul_assign","attr_info":{"attributes":[{"DocComment":" Performs the `*=` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x: u32 = 12;"},{"DocComment":" x *= 2;"},{"DocComment":" assert_eq!(x, 24);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":11756}],"output":{"Deduplicated":1801}},"item":{"id":309,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11841,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":16,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[16,0]}}],"vtable":{"id":{"Adt":64},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":17,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Mul",0]}],"span":{"data":{"file_id":37,"beg":{"line":322,"col":0},"end":{"line":322,"col":31}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The multiplication operator `*`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## `Mul`tipliable rational numbers"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Mul;"},{"DocComment":""},{"DocComment":" // By the fundamental theorem of arithmetic, rational numbers in lowest"},{"DocComment":" // terms are unique. So, by keeping `Rational`s in reduced form, we can"},{"DocComment":" // derive `Eq` and `PartialEq`."},{"DocComment":" #[derive(Debug, Eq, PartialEq)]"},{"DocComment":" struct Rational {"},{"DocComment":" numerator: usize,"},{"DocComment":" denominator: usize,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Rational {"},{"DocComment":" fn new(numerator: usize, denominator: usize) -> Self {"},{"DocComment":" if denominator == 0 {"},{"DocComment":" panic!(\"Zero is an invalid denominator!\");"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Reduce to lowest terms by dividing by the greatest common"},{"DocComment":" // divisor."},{"DocComment":" let gcd = gcd(numerator, denominator);"},{"DocComment":" Self {"},{"DocComment":" numerator: numerator / gcd,"},{"DocComment":" denominator: denominator / gcd,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Mul for Rational {"},{"DocComment":" // The multiplication of rational numbers is a closed operation."},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn mul(self, rhs: Self) -> Self {"},{"DocComment":" let numerator = self.numerator * rhs.numerator;"},{"DocComment":" let denominator = self.denominator * rhs.denominator;"},{"DocComment":" Self::new(numerator, denominator)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Euclid's two-thousand-year-old algorithm for finding the greatest common"},{"DocComment":" // divisor."},{"DocComment":" fn gcd(x: usize, y: usize) -> usize {"},{"DocComment":" let mut x = x;"},{"DocComment":" let mut y = y;"},{"DocComment":" while y != 0 {"},{"DocComment":" let t = y;"},{"DocComment":" y = x % y;"},{"DocComment":" x = t;"},{"DocComment":" }"},{"DocComment":" x"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Rational::new(1, 2), Rational::new(2, 4));"},{"DocComment":" assert_eq!(Rational::new(2, 3) * Rational::new(3, 4),"},{"DocComment":" Rational::new(1, 2));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Multiplying vectors by scalars as in linear algebra"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Mul;"},{"DocComment":""},{"DocComment":" struct Scalar { value: usize }"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Vector { value: Vec }"},{"DocComment":""},{"DocComment":" impl Mul for Vector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn mul(self, rhs: Scalar) -> Self::Output {"},{"DocComment":" Self { value: self.value.iter().map(|v| v * rhs.value).collect() }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let vector = Vector { value: vec![2, 4, 6] };"},{"DocComment":" let scalar = Scalar { value: 3 };"},{"DocComment":" assert_eq!(vector * scalar, Vector { value: vec![6, 12, 18] });"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"mul"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"mul","attr_info":{"attributes":[{"DocComment":" Performs the `*` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(12 * 2, 24);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":310,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11842,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":17,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[17,0]}}],"vtable":{"id":{"Adt":65},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":18,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["arith",0]},{"Ident":["Neg",0]}],"span":{"data":{"file_id":37,"beg":{"line":690,"col":0},"end":{"line":690,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The unary negation operator `-`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `Neg` for `Sign`, which allows the use of `-` to"},{"DocComment":" negate its value."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Neg;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" enum Sign {"},{"DocComment":" Negative,"},{"DocComment":" Zero,"},{"DocComment":" Positive,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Neg for Sign {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn neg(self) -> Self::Output {"},{"DocComment":" match self {"},{"DocComment":" Sign::Negative => Sign::Positive,"},{"DocComment":" Sign::Zero => Sign::Zero,"},{"DocComment":" Sign::Positive => Sign::Negative,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // A negative positive is a negative."},{"DocComment":" assert_eq!(-Sign::Positive, Sign::Negative);"},{"DocComment":" // A double negative is a positive."},{"DocComment":" assert_eq!(-Sign::Negative, Sign::Positive);"},{"DocComment":" // Zero is its own negation."},{"DocComment":" assert_eq!(-Sign::Zero, Sign::Zero);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"neg"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"neg","attr_info":{"attributes":[{"DocComment":" Performs the unary `-` operation."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: i32 = 12;"},{"DocComment":" assert_eq!(-x, -12);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755}],"output":{"Deduplicated":11756}},"item":{"id":312,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11843,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":18,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[18,0]}}],"vtable":{"id":{"Adt":66},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":19,"item_meta":{"name":[{"Ident":["subtle",0]},{"Ident":["ConditionallySelectable",0]}],"span":{"data":{"file_id":22,"beg":{"line":393,"col":0},"end":{"line":393,"col":39}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A type which can be conditionally selected in constant time."},{"DocComment":""},{"DocComment":" This trait also provides generic implementations of conditional"},{"DocComment":" assignment and conditional swaps."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":22,"beg":{"line":393,"col":35},"end":{"line":393,"col":39}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_select","attr_info":{"attributes":[{"DocComment":" Select `a` or `b` according to `choice`."},{"DocComment":""},{"DocComment":" # Returns"},{"DocComment":""},{"DocComment":" * `a` if `choice == Choice(0)`;"},{"DocComment":" * `b` if `choice == Choice(1)`."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let x: u8 = 13;"},{"DocComment":" let y: u8 = 42;"},{"DocComment":""},{"DocComment":" let z = u8::conditional_select(&x, &y, 0.into());"},{"DocComment":" assert_eq!(z, x);"},{"DocComment":" let z = u8::conditional_select(&x, &y, 1.into());"},{"DocComment":" assert_eq!(z, y);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"Deduplicated":11826},{"Deduplicated":2321}],"output":{"Deduplicated":11755}},"item":{"id":93,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11844,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":19,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[19,0]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_assign","attr_info":{"attributes":[{"DocComment":" Conditionally assign `other` to `self`, according to `choice`."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let mut x: u8 = 13;"},{"DocComment":" let mut y: u8 = 42;"},{"DocComment":""},{"DocComment":" x.conditional_assign(&y, 0.into());"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" x.conditional_assign(&y, 1.into());"},{"DocComment":" assert_eq!(x, 42);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":11826},{"Deduplicated":2321}],"output":{"Deduplicated":1801}},"item":{"id":94,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11844}]}}},"kind":{"TraitMethod":[19,1]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"conditional_swap","attr_info":{"attributes":[{"DocComment":" Conditionally swap `self` and `other` if `choice == 1`; otherwise,"},{"DocComment":" reassign both unto themselves."},{"DocComment":""},{"DocComment":" This function should execute in constant time."},{"DocComment":""},{"DocComment":" # Example"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use subtle::ConditionallySelectable;"},{"DocComment":" #"},{"DocComment":" # fn main() {"},{"DocComment":" let mut x: u8 = 13;"},{"DocComment":" let mut y: u8 = 42;"},{"DocComment":""},{"DocComment":" u8::conditional_swap(&mut x, &mut y, 0.into());"},{"DocComment":" assert_eq!(x, 13);"},{"DocComment":" assert_eq!(y, 42);"},{"DocComment":" u8::conditional_swap(&mut x, &mut y, 1.into());"},{"DocComment":" assert_eq!(x, 42);"},{"DocComment":" assert_eq!(y, 13);"},{"DocComment":" # }"},{"DocComment":" ```"}],"inline":"Hint","rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"HashConsedValue":[11845,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":11755},"Mut"]}]},{"Deduplicated":2321}],"output":{"Deduplicated":1801}},"item":{"id":95,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11844}]}}},"kind":{"TraitMethod":[19,2]}}],"vtable":null},{"def_id":20,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["marker",0]},{"Ident":["StructuralPartialEq",0]}],"span":{"data":{"file_id":35,"beg":{"line":257,"col":0},"end":{"line":257,"col":29}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Required trait for constants used in pattern matches."},{"DocComment":""},{"DocComment":" Constants are only allowed as patterns if (a) their type implements"},{"DocComment":" `PartialEq`, and (b) interpreting the value of the constant as a pattern"},{"DocComment":" is equivalent to calling `PartialEq`. This ensures that constants used as"},{"DocComment":" patterns cannot expose implementation details in an unexpected way or"},{"DocComment":" cause semver hazards."},{"DocComment":""},{"DocComment":" This trait ensures point (b)."},{"DocComment":" Any type that derives `PartialEq` automatically implements this trait."},{"DocComment":""},{"DocComment":" Implementing this trait (which is unstable) is a way for type authors to explicitly allow"},{"DocComment":" comparing const values of this type; that operation will recursively compare all fields"},{"DocComment":" (including private fields), even if that behavior differs from `PartialEq`. This can make it"},{"DocComment":" semver-breaking to add further private fields to a type."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"structural_peq"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[],"vtable":{"id":{"Adt":67},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":21,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["Identity",0]}],"span":{"data":{"file_id":38,"beg":{"line":32,"col":0},"end":{"line":36,"col":1}},"generated_from_span":null},"source_text":"pub trait Identity {\n /// Returns the identity element of the curve.\n /// Can be used as a constructor.\n fn identity() -> Self;\n}","attr_info":{"attributes":[{"DocComment":" Trait for getting the identity element of a point type."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"identity","attr_info":{"attributes":[{"DocComment":" Returns the identity element of the curve."},{"DocComment":" Can be used as a constructor."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[],"output":{"Deduplicated":11755}},"item":{"id":320,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11846,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":21,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[21,0]}}],"vtable":null},{"def_id":22,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["ValidityCheck",0]}],"span":{"data":{"file_id":38,"beg":{"line":451,"col":0},"end":{"line":454,"col":1}},"generated_from_span":null},"source_text":"pub(crate) trait ValidityCheck {\n /// Checks whether the point is on the curve. Not CT.\n fn is_valid(&self) -> bool;\n}","attr_info":{"attributes":[{"DocComment":" Trait for checking whether a point is on the curve."},{"DocComment":""},{"DocComment":" This trait is only for debugging/testing, since it should be"},{"DocComment":" impossible for a `curve25519-dalek` user to construct an invalid"},{"DocComment":" point."},{"Unknown":{"path":"allow","args":"dead_code"}}],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"is_valid","attr_info":{"attributes":[{"DocComment":" Checks whether the point is on the curve. Not CT."}],"inline":null,"rename":null,"public":false},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"Deduplicated":1796}},"item":{"id":321,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11847,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":22,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[22,0]}}],"vtable":{"id":{"Adt":68},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":23,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["hash",0]},{"Ident":["Hash",0]}],"span":{"data":{"file_id":40,"beg":{"line":185,"col":0},"end":{"line":185,"col":36}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A hashable type."},{"DocComment":""},{"DocComment":" Types implementing `Hash` are able to be [`hash`]ed with an instance of"},{"DocComment":" [`Hasher`]."},{"DocComment":""},{"DocComment":" ## Implementing `Hash`"},{"DocComment":""},{"DocComment":" You can derive `Hash` with `#[derive(Hash)]` if all fields implement `Hash`."},{"DocComment":" The resulting hash will be the combination of the values from calling"},{"DocComment":" [`hash`] on each field."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(Hash)]"},{"DocComment":" struct Rustacean {"},{"DocComment":" name: String,"},{"DocComment":" country: String,"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If you need more control over how a value is hashed, you can of course"},{"DocComment":" implement the `Hash` trait yourself:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{Hash, Hasher};"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" phone: u64,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Hash for Person {"},{"DocComment":" fn hash(&self, state: &mut H) {"},{"DocComment":" self.id.hash(state);"},{"DocComment":" self.phone.hash(state);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## `Hash` and `Eq`"},{"DocComment":""},{"DocComment":" When implementing both `Hash` and [`Eq`], it is important that the following"},{"DocComment":" property holds:"},{"DocComment":""},{"DocComment":" ```text"},{"DocComment":" k1 == k2 -> hash(k1) == hash(k2)"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" In other words, if two keys are equal, their hashes must also be equal."},{"DocComment":" [`HashMap`] and [`HashSet`] both rely on this behavior."},{"DocComment":""},{"DocComment":" Thankfully, you won't need to worry about upholding this property when"},{"DocComment":" deriving both [`Eq`] and `Hash` with `#[derive(PartialEq, Eq, Hash)]`."},{"DocComment":""},{"DocComment":" Violating this property is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Prefix collisions"},{"DocComment":""},{"DocComment":" Implementations of `hash` should ensure that the data they"},{"DocComment":" pass to the `Hasher` are prefix-free. That is,"},{"DocComment":" values which are not equal should cause two different sequences of values to be written,"},{"DocComment":" and neither of the two sequences should be a prefix of the other."},{"DocComment":""},{"DocComment":" For example, the standard implementation of [`Hash` for `&str`][impl] passes an extra"},{"DocComment":" `0xFF` byte to the `Hasher` so that the values `(\"ab\", \"c\")` and `(\"a\","},{"DocComment":" \"bc\")` hash differently."},{"DocComment":""},{"DocComment":" ## Portability"},{"DocComment":""},{"DocComment":" Due to differences in endianness and type sizes, data fed by `Hash` to a `Hasher`"},{"DocComment":" should not be considered portable across platforms. Additionally the data passed by most"},{"DocComment":" standard library types should not be considered stable between compiler versions."},{"DocComment":""},{"DocComment":" This means tests shouldn't probe hard-coded hash values or data fed to a `Hasher` and"},{"DocComment":" instead should check consistency with `Eq`."},{"DocComment":""},{"DocComment":" Serialization formats intended to be portable between platforms or compiler versions should"},{"DocComment":" either avoid encoding hashes or only rely on `Hash` and `Hasher` implementations that"},{"DocComment":" provide additional guarantees."},{"DocComment":""},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`HashSet`]: ../../std/collections/struct.HashSet.html"},{"DocComment":" [`hash`]: Hash::hash"},{"DocComment":" [impl]: ../../std/primitive.str.html#impl-Hash-for-str"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Hash"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"H"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":40,"beg":{"line":198,"col":15},"end":{"line":198,"col":21}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":3180}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"hash","attr_info":{"attributes":[{"DocComment":" Feeds this value into the given [`Hasher`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hash, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" 7920.hash(&mut hasher);"},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"HashConsedValue":[3795,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":3181},"Mut"]}]}],"output":{"Deduplicated":1801}},"item":{"id":114,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755},{"Deduplicated":3181}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11848,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":23,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[3774,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":3180}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[23,0]}},null],"vtable":null},{"def_id":24,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["hash",0]},{"Ident":["Hasher",0]}],"span":{"data":{"file_id":40,"beg":{"line":312,"col":0},"end":{"line":312,"col":16}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for hashing an arbitrary stream of bytes."},{"DocComment":""},{"DocComment":" Instances of `Hasher` usually represent state that is changed while hashing"},{"DocComment":" data."},{"DocComment":""},{"DocComment":" `Hasher` provides a fairly basic interface for retrieving the generated hash"},{"DocComment":" (with [`finish`]), and writing integers as well as slices of bytes into an"},{"DocComment":" instance (with [`write`] and [`write_u8`] etc.). Most of the time, `Hasher`"},{"DocComment":" instances are used in conjunction with the [`Hash`] trait."},{"DocComment":""},{"DocComment":" This trait provides no guarantees about how the various `write_*` methods are"},{"DocComment":" defined and implementations of [`Hash`] should not assume that they work one"},{"DocComment":" way or another. You cannot assume, for example, that a [`write_u32`] call is"},{"DocComment":" equivalent to four calls of [`write_u8`]. Nor can you assume that adjacent"},{"DocComment":" `write` calls are merged, so it's possible, for example, that"},{"DocComment":" ```"},{"DocComment":" # fn foo(hasher: &mut impl std::hash::Hasher) {"},{"DocComment":" hasher.write(&[1, 2]);"},{"DocComment":" hasher.write(&[3, 4, 5, 6]);"},{"DocComment":" # }"},{"DocComment":" ```"},{"DocComment":" and"},{"DocComment":" ```"},{"DocComment":" # fn foo(hasher: &mut impl std::hash::Hasher) {"},{"DocComment":" hasher.write(&[1, 2, 3, 4]);"},{"DocComment":" hasher.write(&[5, 6]);"},{"DocComment":" # }"},{"DocComment":" ```"},{"DocComment":" end up producing different hashes."},{"DocComment":""},{"DocComment":" Thus to produce the same hash value, [`Hash`] implementations must ensure"},{"DocComment":" for equivalent items that exactly the same sequence of calls is made -- the"},{"DocComment":" same methods with the same parameters in the same order."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":""},{"DocComment":" hasher.write_u32(1989);"},{"DocComment":" hasher.write_u8(11);"},{"DocComment":" hasher.write_u8(9);"},{"DocComment":" hasher.write(b\"Huh?\");"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`finish`]: Hasher::finish"},{"DocComment":" [`write`]: Hasher::write"},{"DocComment":" [`write_u8`]: Hasher::write_u8"},{"DocComment":" [`write_u32`]: Hasher::write_u32"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"finish","attr_info":{"attributes":[{"DocComment":" Returns the hash value for the values written so far."},{"DocComment":""},{"DocComment":" Despite its name, the method does not reset the hasher’s internal"},{"DocComment":" state. Additional [`write`]s will continue from the current value."},{"DocComment":" If you need to start a fresh hash value, you will have to create"},{"DocComment":" a new hasher."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" hasher.write(b\"Cool!\");"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`write`]: Hasher::write"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"Deduplicated":1783}},"item":{"id":333,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11849,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":24,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[24,0]}},{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"write","attr_info":{"attributes":[{"DocComment":" Writes some data into this `Hasher`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::hash::{DefaultHasher, Hasher};"},{"DocComment":""},{"DocComment":" let mut hasher = DefaultHasher::new();"},{"DocComment":" let data = [0x01, 0x23, 0x45, 0x67, 0x89, 0xab, 0xcd, 0xef];"},{"DocComment":""},{"DocComment":" hasher.write(&data);"},{"DocComment":""},{"DocComment":" println!(\"Hash is {:x}!\", hasher.finish());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Note to Implementers"},{"DocComment":""},{"DocComment":" You generally should not do length-prefixing as part of implementing"},{"DocComment":" this method. It's up to the [`Hash`] implementation to call"},{"DocComment":" [`Hasher::write_length_prefix`] before sequences that need it."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"HashConsedValue":[7753,{"Ref":[{"Var":{"Bound":[0,1]}},{"Deduplicated":2928},"Shared"]}]}],"output":{"Deduplicated":1801}},"item":{"id":334,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11849}]}}},"kind":{"TraitMethod":[24,1]}},null,null,null,null,null,null,null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":69},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":25,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["convert",0]},{"Ident":["TryFrom",0]}],"span":{"data":{"file_id":32,"beg":{"line":694,"col":0},"end":{"line":694,"col":33}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Simple and safe type conversions that may fail in a controlled"},{"DocComment":" way under some circumstances. It is the reciprocal of [`TryInto`]."},{"DocComment":""},{"DocComment":" This is useful when you are doing a type conversion that may"},{"DocComment":" trivially succeed but may also need special handling."},{"DocComment":" For example, there is no way to convert an [`i64`] into an [`i32`]"},{"DocComment":" using the [`From`] trait, because an [`i64`] may contain a value"},{"DocComment":" that an [`i32`] cannot represent and so the conversion would lose data."},{"DocComment":" This might be handled by truncating the [`i64`] to an [`i32`] or by"},{"DocComment":" simply returning [`i32::MAX`], or by some other method. The [`From`]"},{"DocComment":" trait is intended for perfect conversions, so the `TryFrom` trait"},{"DocComment":" informs the programmer when a type conversion could go bad and lets"},{"DocComment":" them decide how to handle it."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - `TryFrom for U` implies [`TryInto`]` for T`"},{"DocComment":" - [`try_from`] is reflexive, which means that `TryFrom for T`"},{"DocComment":" is implemented and cannot fail -- the associated `Error` type for"},{"DocComment":" calling `T::try_from()` on a value of type `T` is [`Infallible`]."},{"DocComment":" When the [`!`] type is stabilized [`Infallible`] and [`!`] will be"},{"DocComment":" equivalent."},{"DocComment":""},{"DocComment":" Prefer using [`TryInto`] over [`TryFrom`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`TryInto`] can be used as well."},{"DocComment":""},{"DocComment":" `TryFrom` can be implemented as follows:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct GreaterThanZero(i32);"},{"DocComment":""},{"DocComment":" impl TryFrom for GreaterThanZero {"},{"DocComment":" type Error = &'static str;"},{"DocComment":""},{"DocComment":" fn try_from(value: i32) -> Result {"},{"DocComment":" if value <= 0 {"},{"DocComment":" Err(\"GreaterThanZero only accepts values greater than zero!\")"},{"DocComment":" } else {"},{"DocComment":" Ok(GreaterThanZero(value))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" As described, [`i32`] implements `TryFrom<`[`i64`]`>`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let big_number = 1_000_000_000_000i64;"},{"DocComment":" // Silently truncates `big_number`, requires detecting"},{"DocComment":" // and handling the truncation after the fact."},{"DocComment":" let smaller_number = big_number as i32;"},{"DocComment":" assert_eq!(smaller_number, -727379968);"},{"DocComment":""},{"DocComment":" // Returns an error because `big_number` is too big to"},{"DocComment":" // fit in an `i32`."},{"DocComment":" let try_smaller_number = i32::try_from(big_number);"},{"DocComment":" assert!(try_smaller_number.is_err());"},{"DocComment":""},{"DocComment":" // Returns `Ok(3)`."},{"DocComment":" let try_successful_smaller_number = i32::try_from(3);"},{"DocComment":" assert!(try_successful_smaller_number.is_ok());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`try_from`]: TryFrom::try_from"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"TryFrom"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"},{"index":2,"name":"Self_Error"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"try_from","attr_info":{"attributes":[{"DocComment":" Performs the conversion."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11756}],"output":{"Deduplicated":11794}},"item":{"id":350,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11850,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":25,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[25,0]}}],"vtable":null},{"def_id":26,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["borrow",0]},{"Ident":["Borrow",0]}],"span":{"data":{"file_id":42,"beg":{"line":158,"col":0},"end":{"line":158,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A trait for borrowing data."},{"DocComment":""},{"DocComment":" In Rust, it is common to provide different representations of a type for"},{"DocComment":" different use cases. For instance, storage location and management for a"},{"DocComment":" value can be specifically chosen as appropriate for a particular use via"},{"DocComment":" pointer types such as [`Box`] or [`Rc`]. Beyond these generic"},{"DocComment":" wrappers that can be used with any type, some types provide optional"},{"DocComment":" facets providing potentially costly functionality. An example for such a"},{"DocComment":" type is [`String`] which adds the ability to extend a string to the basic"},{"DocComment":" [`str`]. This requires keeping additional information unnecessary for a"},{"DocComment":" simple, immutable string."},{"DocComment":""},{"DocComment":" These types provide access to the underlying data through references"},{"DocComment":" to the type of that data. They are said to be ‘borrowed as’ that type."},{"DocComment":" For instance, a [`Box`] can be borrowed as `T` while a [`String`]"},{"DocComment":" can be borrowed as `str`."},{"DocComment":""},{"DocComment":" Types express that they can be borrowed as some type `T` by implementing"},{"DocComment":" `Borrow`, providing a reference to a `T` in the trait’s"},{"DocComment":" [`borrow`] method. A type is free to borrow as several different types."},{"DocComment":" If it wishes to mutably borrow as the type, allowing the underlying data"},{"DocComment":" to be modified, it can additionally implement [`BorrowMut`]."},{"DocComment":""},{"DocComment":" Further, when providing implementations for additional traits, it needs"},{"DocComment":" to be considered whether they should behave identically to those of the"},{"DocComment":" underlying type as a consequence of acting as a representation of that"},{"DocComment":" underlying type. Generic code typically uses `Borrow` when it relies"},{"DocComment":" on the identical behavior of these additional trait implementations."},{"DocComment":" These traits will likely appear as additional trait bounds."},{"DocComment":""},{"DocComment":" In particular `Eq`, `Ord` and `Hash` must be equivalent for"},{"DocComment":" borrowed and owned values: `x.borrow() == y.borrow()` should give the"},{"DocComment":" same result as `x == y`."},{"DocComment":""},{"DocComment":" If generic code merely needs to work for all types that can"},{"DocComment":" provide a reference to related type `T`, it is often better to use"},{"DocComment":" [`AsRef`] as more types can safely implement it."},{"DocComment":""},{"DocComment":" [`Box`]: ../../std/boxed/struct.Box.html"},{"DocComment":" [`Mutex`]: ../../std/sync/struct.Mutex.html"},{"DocComment":" [`Rc`]: ../../std/rc/struct.Rc.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`borrow`]: Borrow::borrow"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" As a data collection, [`HashMap`] owns both keys and values. If"},{"DocComment":" the key’s actual data is wrapped in a managing type of some kind, it"},{"DocComment":" should, however, still be possible to search for a value using a"},{"DocComment":" reference to the key’s data. For instance, if the key is a string, then"},{"DocComment":" it is likely stored with the hash map as a [`String`], while it should"},{"DocComment":" be possible to search using a [`&str`][`str`]. Thus, `insert` needs to"},{"DocComment":" operate on a `String` while `get` needs to be able to use a `&str`."},{"DocComment":""},{"DocComment":" Slightly simplified, the relevant parts of `HashMap` look like"},{"DocComment":" this:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::borrow::Borrow;"},{"DocComment":" use std::hash::Hash;"},{"DocComment":""},{"DocComment":" pub struct HashMap {"},{"DocComment":" # marker: ::std::marker::PhantomData<(K, V)>,"},{"DocComment":" // fields omitted"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl HashMap {"},{"DocComment":" pub fn insert(&self, key: K, value: V) -> Option"},{"DocComment":" where K: Hash + Eq"},{"DocComment":" {"},{"DocComment":" # unimplemented!()"},{"DocComment":" // ..."},{"DocComment":" }"},{"DocComment":""},{"DocComment":" pub fn get(&self, k: &Q) -> Option<&V>"},{"DocComment":" where"},{"DocComment":" K: Borrow,"},{"DocComment":" Q: Hash + Eq + ?Sized"},{"DocComment":" {"},{"DocComment":" # unimplemented!()"},{"DocComment":" // ..."},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The entire hash map is generic over a key type `K`. Because these keys"},{"DocComment":" are stored with the hash map, this type has to own the key’s data."},{"DocComment":" When inserting a key-value pair, the map is given such a `K` and needs"},{"DocComment":" to find the correct hash bucket and check if the key is already present"},{"DocComment":" based on that `K`. It therefore requires `K: Hash + Eq`."},{"DocComment":""},{"DocComment":" When searching for a value in the map, however, having to provide a"},{"DocComment":" reference to a `K` as the key to search for would require to always"},{"DocComment":" create such an owned value. For string keys, this would mean a `String`"},{"DocComment":" value needs to be created just for the search for cases where only a"},{"DocComment":" `str` is available."},{"DocComment":""},{"DocComment":" Instead, the `get` method is generic over the type of the underlying key"},{"DocComment":" data, called `Q` in the method signature above. It states that `K`"},{"DocComment":" borrows as a `Q` by requiring that `K: Borrow`. By additionally"},{"DocComment":" requiring `Q: Hash + Eq`, it signals the requirement that `K` and `Q`"},{"DocComment":" have implementations of the `Hash` and `Eq` traits that produce identical"},{"DocComment":" results."},{"DocComment":""},{"DocComment":" The implementation of `get` relies in particular on identical"},{"DocComment":" implementations of `Hash` by determining the key’s hash bucket by calling"},{"DocComment":" `Hash::hash` on the `Q` value even though it inserted the key based on"},{"DocComment":" the hash value calculated from the `K` value."},{"DocComment":""},{"DocComment":" As a consequence, the hash map breaks if a `K` wrapping a `Q` value"},{"DocComment":" produces a different hash than `Q`. For instance, imagine you have a"},{"DocComment":" type that wraps a string but compares ASCII letters ignoring their case:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" pub struct CaseInsensitiveString(String);"},{"DocComment":""},{"DocComment":" impl PartialEq for CaseInsensitiveString {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.0.eq_ignore_ascii_case(&other.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for CaseInsensitiveString { }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Because two equal values need to produce the same hash value, the"},{"DocComment":" implementation of `Hash` needs to ignore ASCII case, too:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # use std::hash::{Hash, Hasher};"},{"DocComment":" # pub struct CaseInsensitiveString(String);"},{"DocComment":" impl Hash for CaseInsensitiveString {"},{"DocComment":" fn hash(&self, state: &mut H) {"},{"DocComment":" for c in self.0.as_bytes() {"},{"DocComment":" c.to_ascii_lowercase().hash(state)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Can `CaseInsensitiveString` implement `Borrow`? It certainly can"},{"DocComment":" provide a reference to a string slice via its contained owned string."},{"DocComment":" But because its `Hash` implementation differs, it behaves differently"},{"DocComment":" from `str` and therefore must not, in fact, implement `Borrow`."},{"DocComment":" If it wants to allow others access to the underlying `str`, it can do"},{"DocComment":" that via `AsRef` which doesn’t carry any extra requirements."},{"DocComment":""},{"DocComment":" [`Hash`]: crate::hash::Hash"},{"DocComment":" [`HashMap`]: ../../std/collections/struct.HashMap.html"},{"DocComment":" [`String`]: ../../std/string/struct.String.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Borrow"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Borrowed"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"borrow","attr_info":{"attributes":[{"DocComment":" Immutably borrows from an owned value."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::borrow::Borrow;"},{"DocComment":""},{"DocComment":" fn check>(s: T) {"},{"DocComment":" assert_eq!(\"Hello\", s.borrow());"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let s = \"Hello\".to_string();"},{"DocComment":""},{"DocComment":" check(s);"},{"DocComment":""},{"DocComment":" let s = \"Hello\";"},{"DocComment":""},{"DocComment":" check(s);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"HashConsedValue":[11851,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11756},"Shared"]}]}},"item":{"id":352,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11852,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":26,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[26,0]}}],"vtable":{"id":{"Adt":71},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":27,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Sum",0]}],"span":{"data":{"file_id":43,"beg":{"line":18,"col":0},"end":{"line":18,"col":36}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait to represent types that can be created by summing up an iterator."},{"DocComment":""},{"DocComment":" This trait is used to implement [`Iterator::sum()`]. Types which implement"},{"DocComment":" this trait can be generated by using the [`sum()`] method on an iterator."},{"DocComment":" Like [`FromIterator`], this trait should rarely be called directly."},{"DocComment":""},{"DocComment":" [`sum()`]: Iterator::sum"},{"DocComment":" [`FromIterator`]: iter::FromIterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":43,"beg":{"line":22,"col":14},"end":{"line":22,"col":32}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"sum","attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by \"summing up\""},{"DocComment":" the items."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3181}],"output":{"Deduplicated":11755}},"item":{"id":353,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":3181}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11853,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":27,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[11854,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[27,0]}}],"vtable":null},{"def_id":28,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["IntoIterator",0]}],"span":{"data":{"file_id":17,"beg":{"line":283,"col":0},"end":{"line":283,"col":28}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Conversion into an [`Iterator`]."},{"DocComment":""},{"DocComment":" By implementing `IntoIterator` for a type, you define how it will be"},{"DocComment":" converted to an iterator. This is common for types which describe a"},{"DocComment":" collection of some kind."},{"DocComment":""},{"DocComment":" One benefit of implementing `IntoIterator` is that your type will [work"},{"DocComment":" with Rust's `for` loop syntax](crate::iter#for-loops-and-intoiterator)."},{"DocComment":""},{"DocComment":" See also: [`FromIterator`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let v = [1, 2, 3];"},{"DocComment":" let mut iter = v.into_iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"},{"DocComment":" Implementing `IntoIterator` for your type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // and we'll implement IntoIterator"},{"DocComment":" impl IntoIterator for MyCollection {"},{"DocComment":" type Item = i32;"},{"DocComment":" type IntoIter = std::vec::IntoIter;"},{"DocComment":""},{"DocComment":" fn into_iter(self) -> Self::IntoIter {"},{"DocComment":" self.0.into_iter()"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Now we can make a new collection..."},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" // ... add some stuff to it ..."},{"DocComment":" c.add(0);"},{"DocComment":" c.add(1);"},{"DocComment":" c.add(2);"},{"DocComment":""},{"DocComment":" // ... and then turn it into an Iterator:"},{"DocComment":" for (i, n) in c.into_iter().enumerate() {"},{"DocComment":" assert_eq!(i as i32, n);"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It is common to use `IntoIterator` as a trait bound. This allows"},{"DocComment":" the input collection type to change, so long as it is still an"},{"DocComment":" iterator. Additional bounds can be specified by restricting on"},{"DocComment":" `Item`:"},{"DocComment":""},{"DocComment":" ```rust"},{"DocComment":" fn collect_as_strings(collection: T) -> Vec"},{"DocComment":" where"},{"DocComment":" T: IntoIterator,"},{"DocComment":" T::Item: std::fmt::Debug,"},{"DocComment":" {"},{"DocComment":" collection"},{"DocComment":" .into_iter()"},{"DocComment":" .map(|item| format!(\"{item:?}\"))"},{"DocComment":" .collect()"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"IntoIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Item"},{"index":2,"name":"Self_IntoIter"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":291,"col":19},"end":{"line":291,"col":46}},"generated_from_span":null},"origin":{"TraitItem":1},"trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11793},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null,null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"into_iter","attr_info":{"attributes":[{"DocComment":" Creates an iterator from a value."},{"DocComment":""},{"DocComment":" See the [module-level documentation] for more."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let v = [1, 2, 3];"},{"DocComment":" let mut iter = v.into_iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(1), iter.next());"},{"DocComment":" assert_eq!(Some(2), iter.next());"},{"DocComment":" assert_eq!(Some(3), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755}],"output":{"Deduplicated":11793}},"item":{"id":359,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11855,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[28,0]}}],"vtable":{"id":{"Adt":72},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":29,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["num",0]},{"Ident":["nonzero",0]},{"Ident":["ZeroablePrimitive",0]}],"span":{"data":{"file_id":45,"beg":{"line":34,"col":0},"end":{"line":34,"col":66}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A marker trait for primitive types which can be zero."},{"DocComment":""},{"DocComment":" This is an implementation detail for [NonZero]\\ which may disappear or be replaced at any time."},{"DocComment":""},{"DocComment":" # Safety"},{"DocComment":""},{"DocComment":" Types implementing this trait must be primitives that are valid when zeroed."},{"DocComment":""},{"DocComment":" The associated `Self::NonZeroInner` type must have the same size+align as `Self`,"},{"DocComment":" but with a niche and bit validity making it so the following `transmutes` are sound:"},{"DocComment":""},{"DocComment":" - `Self::NonZeroInner` to `Option`"},{"DocComment":" - `Option` to `Self`"},{"DocComment":""},{"DocComment":" (And, consequently, `Self::NonZeroInner` to `Self`.)"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_NonZeroInner"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":45,"beg":{"line":34,"col":44},"end":{"line":34,"col":48}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":45,"beg":{"line":34,"col":51},"end":{"line":34,"col":66}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":46,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":2,"span":{"data":{"file_id":45,"beg":{"line":36,"col":31},"end":{"line":36,"col":35}},"generated_from_span":null},"origin":{"TraitItem":0},"trait_":{"regions":[],"skip_binder":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null],"methods":[],"vtable":null},{"def_id":30,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["function",0]},{"Ident":["FnMut",0]}],"span":{"data":{"file_id":51,"beg":{"line":163,"col":0},"end":{"line":163,"col":48}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The version of the call operator that takes a mutable receiver."},{"DocComment":""},{"DocComment":" Instances of `FnMut` can be called repeatedly and may mutate state."},{"DocComment":""},{"DocComment":" `FnMut` is implemented automatically by closures which take mutable"},{"DocComment":" references to captured variables, as well as all types that implement"},{"DocComment":" [`Fn`], e.g., (safe) [function pointers] (since `FnMut` is a supertrait of"},{"DocComment":" [`Fn`]). Additionally, for any type `F` that implements `FnMut`, `&mut F`"},{"DocComment":" implements `FnMut`, too."},{"DocComment":""},{"DocComment":" Since [`FnOnce`] is a supertrait of `FnMut`, any instance of `FnMut` can be"},{"DocComment":" used where a [`FnOnce`] is expected, and since [`Fn`] is a subtrait of"},{"DocComment":" `FnMut`, any instance of [`Fn`] can be used where `FnMut` is expected."},{"DocComment":""},{"DocComment":" Use `FnMut` as a bound when you want to accept a parameter of function-like"},{"DocComment":" type and need to call it repeatedly, while allowing it to mutate state."},{"DocComment":" If you don't want the parameter to mutate state, use [`Fn`] as a"},{"DocComment":" bound; if you don't need to call it repeatedly, use [`FnOnce`]."},{"DocComment":""},{"DocComment":" See the [chapter on closures in *The Rust Programming Language*][book] for"},{"DocComment":" some more information on this topic."},{"DocComment":""},{"DocComment":" Also of note is the special syntax for `Fn` traits (e.g."},{"DocComment":" `Fn(usize, bool) -> usize`). Those interested in the technical details of"},{"DocComment":" this can refer to [the relevant section in the *Rustonomicon*][nomicon]."},{"DocComment":""},{"DocComment":" [book]: ../../book/ch13-01-closures.html"},{"DocComment":" [function pointers]: fn"},{"DocComment":" [nomicon]: ../../nomicon/hrtb.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## Calling a mutably capturing closure"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let mut x = 5;"},{"DocComment":" {"},{"DocComment":" let mut square_x = || x *= x;"},{"DocComment":" square_x();"},{"DocComment":" }"},{"DocComment":" assert_eq!(x, 25);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Using a `FnMut` parameter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn do_twice(mut func: F)"},{"DocComment":" where F: FnMut()"},{"DocComment":" {"},{"DocComment":" func();"},{"DocComment":" func();"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut x: usize = 1;"},{"DocComment":" {"},{"DocComment":" let add_two_to_x = || x += 2;"},{"DocComment":" do_twice(add_two_to_x);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(x, 5);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"fn_mut"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Self_Clause0_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":51,"beg":{"line":163,"col":36},"end":{"line":163,"col":48}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"call_mut","attr_info":{"attributes":[{"DocComment":" Performs the call operation."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11822},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":360,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11856,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":30,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[30,0]}}],"vtable":{"id":{"Adt":74},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":31,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["function",0]},{"Ident":["FnOnce",0]}],"span":{"data":{"file_id":51,"beg":{"line":242,"col":0},"end":{"line":242,"col":35}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The version of the call operator that takes a by-value receiver."},{"DocComment":""},{"DocComment":" Instances of `FnOnce` can be called, but might not be callable multiple"},{"DocComment":" times. Because of this, if the only thing known about a type is that it"},{"DocComment":" implements `FnOnce`, it can only be called once."},{"DocComment":""},{"DocComment":" `FnOnce` is implemented automatically by closures that might consume captured"},{"DocComment":" variables, as well as all types that implement [`FnMut`], e.g., (safe)"},{"DocComment":" [function pointers] (since `FnOnce` is a supertrait of [`FnMut`])."},{"DocComment":""},{"DocComment":" Since both [`Fn`] and [`FnMut`] are subtraits of `FnOnce`, any instance of"},{"DocComment":" [`Fn`] or [`FnMut`] can be used where a `FnOnce` is expected."},{"DocComment":""},{"DocComment":" Use `FnOnce` as a bound when you want to accept a parameter of function-like"},{"DocComment":" type and only need to call it once. If you need to call the parameter"},{"DocComment":" repeatedly, use [`FnMut`] as a bound; if you also need it to not mutate"},{"DocComment":" state, use [`Fn`]."},{"DocComment":""},{"DocComment":" See the [chapter on closures in *The Rust Programming Language*][book] for"},{"DocComment":" some more information on this topic."},{"DocComment":""},{"DocComment":" Also of note is the special syntax for `Fn` traits (e.g."},{"DocComment":" `Fn(usize, bool) -> usize`). Those interested in the technical details of"},{"DocComment":" this can refer to [the relevant section in the *Rustonomicon*][nomicon]."},{"DocComment":""},{"DocComment":" [book]: ../../book/ch13-01-closures.html"},{"DocComment":" [function pointers]: fn"},{"DocComment":" [nomicon]: ../../nomicon/hrtb.html"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ## Using a `FnOnce` parameter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn consume_with_relish(func: F)"},{"DocComment":" where F: FnOnce() -> String"},{"DocComment":" {"},{"DocComment":" // `func` consumes its captured variables, so it cannot be run more"},{"DocComment":" // than once."},{"DocComment":" println!(\"Consumed: {}\", func());"},{"DocComment":""},{"DocComment":" println!(\"Delicious!\");"},{"DocComment":""},{"DocComment":" // Attempting to invoke `func()` again will throw a `use of moved"},{"DocComment":" // value` error for `func`."},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let x = String::from(\"x\");"},{"DocComment":" let consume_and_return_x = move || x;"},{"DocComment":" consume_with_relish(consume_and_return_x);"},{"DocComment":""},{"DocComment":" // `consume_and_return_x` can no longer be invoked at this point"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"fn_once"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Args"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"call_once","attr_info":{"attributes":[{"DocComment":" Performs the call operation."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":{"Other":"rust-call"},"inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":361,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11857,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":31,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[31,0]}}],"vtable":{"id":{"Adt":75},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":32,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["FromIterator",0]}],"span":{"data":{"file_id":17,"beg":{"line":134,"col":0},"end":{"line":134,"col":32}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Conversion from an [`Iterator`]."},{"DocComment":""},{"DocComment":" By implementing `FromIterator` for a type, you define how it will be"},{"DocComment":" created from an iterator. This is common for types which describe a"},{"DocComment":" collection of some kind."},{"DocComment":""},{"DocComment":" If you want to create a collection from the contents of an iterator, the"},{"DocComment":" [`Iterator::collect()`] method is preferred. However, when you need to"},{"DocComment":" specify the container type, [`FromIterator::from_iter()`] can be more"},{"DocComment":" readable than using a turbofish (e.g. `::>()`). See the"},{"DocComment":" [`Iterator::collect()`] documentation for more examples of its use."},{"DocComment":""},{"DocComment":" See also: [`IntoIterator`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v = Vec::from_iter(five_fives);"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Using [`Iterator::collect()`] to implicitly use `FromIterator`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v: Vec = five_fives.collect();"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Using [`FromIterator::from_iter()`] as a more readable alternative to"},{"DocComment":" [`Iterator::collect()`]:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::collections::VecDeque;"},{"DocComment":" let first = (0..10).collect::>();"},{"DocComment":" let second = VecDeque::from_iter(0..10);"},{"DocComment":""},{"DocComment":" assert_eq!(first, second);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Implementing `FromIterator` for your type:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // and we'll implement FromIterator"},{"DocComment":" impl FromIterator for MyCollection {"},{"DocComment":" fn from_iter>(iter: I) -> Self {"},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" for i in iter {"},{"DocComment":" c.add(i);"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" c"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // Now we can make a new iterator..."},{"DocComment":" let iter = (0..5).into_iter();"},{"DocComment":""},{"DocComment":" // ... and make a MyCollection out of it"},{"DocComment":" let c = MyCollection::from_iter(iter);"},{"DocComment":""},{"DocComment":" assert_eq!(c.0, vec![0, 1, 2, 3, 4]);"},{"DocComment":""},{"DocComment":" // collect works too!"},{"DocComment":""},{"DocComment":" let iter = (0..5).into_iter();"},{"DocComment":" let c: MyCollection = iter.collect();"},{"DocComment":""},{"DocComment":" assert_eq!(c.0, vec![0, 1, 2, 3, 4]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"FromIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":152,"col":20},"end":{"line":152,"col":42}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756},{"HashConsedValue":[3202,{"TypeVar":{"Bound":[1,1]}}]}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_iter","attr_info":{"attributes":[{"DocComment":" Creates a value from an iterator."},{"DocComment":""},{"DocComment":" See the [module-level documentation] for more."},{"DocComment":""},{"DocComment":" [module-level documentation]: crate::iter"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let five_fives = std::iter::repeat(5).take(5);"},{"DocComment":""},{"DocComment":" let v = Vec::from_iter(five_fives);"},{"DocComment":""},{"DocComment":" assert_eq!(v, vec![5, 5, 5, 5, 5]);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3181}],"output":{"Deduplicated":11755}},"item":{"id":362,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":3181},{"HashConsedValue":[3201,{"TypeVar":{"Bound":[0,1]}}]}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11858,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":32,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]},{"HashConsedValue":[11859,{"kind":{"Clause":{"Bound":[0,0]}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756},{"Deduplicated":3202}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[32,0]}}],"vtable":null},{"def_id":33,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Try",0]}],"span":{"data":{"file_id":67,"beg":{"line":133,"col":0},"end":{"line":133,"col":41}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The `?` operator and `try {}` blocks."},{"DocComment":""},{"DocComment":" `try_*` methods typically involve a type implementing this trait. For"},{"DocComment":" example, the closures passed to [`Iterator::try_fold`] and"},{"DocComment":" [`Iterator::try_for_each`] must return such a type."},{"DocComment":""},{"DocComment":" `Try` types are typically those containing two or more categories of values,"},{"DocComment":" some subset of which are so commonly handled via early returns that it's"},{"DocComment":" worth providing a terse (but still visible) syntax to make that easy."},{"DocComment":""},{"DocComment":" This is most often seen for error handling with [`Result`] and [`Option`]."},{"DocComment":" The quintessential implementation of this trait is on [`ControlFlow`]."},{"DocComment":""},{"DocComment":" # Using `Try` in Generic Code"},{"DocComment":""},{"DocComment":" `Iterator::try_fold` was stabilized to call back in Rust 1.27, but"},{"DocComment":" this trait is much newer. To illustrate the various associated types and"},{"DocComment":" methods, let's implement our own version."},{"DocComment":""},{"DocComment":" As a reminder, an infallible version of a fold looks something like this:"},{"DocComment":" ```"},{"DocComment":" fn simple_fold("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> A,"},{"DocComment":" ) -> A {"},{"DocComment":" for x in iter {"},{"DocComment":" accum = f(accum, x);"},{"DocComment":" }"},{"DocComment":" accum"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" So instead of `f` returning just an `A`, we'll need it to return some other"},{"DocComment":" type that produces an `A` in the \"don't short circuit\" path. Conveniently,"},{"DocComment":" that's also the type we need to return from the function."},{"DocComment":""},{"DocComment":" Let's add a new generic parameter `R` for that type, and bound it to the"},{"DocComment":" output type that we want:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::Try;"},{"DocComment":" fn simple_try_fold_1>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" todo!()"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If we get through the entire iterator, we need to wrap up the accumulator"},{"DocComment":" into the return type using [`Try::from_output`]:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::{ControlFlow, Try};"},{"DocComment":" fn simple_try_fold_2>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" let cf = f(accum, x).branch();"},{"DocComment":" match cf {"},{"DocComment":" ControlFlow::Continue(a) => accum = a,"},{"DocComment":" ControlFlow::Break(_) => todo!(),"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" We'll also need [`FromResidual::from_residual`] to turn the residual back"},{"DocComment":" into the original type. But because it's a supertrait of `Try`, we don't"},{"DocComment":" need to mention it in the bounds. All types which implement `Try` can be"},{"DocComment":" recreated from their corresponding residual, so we'll just call it:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::{ControlFlow, Try};"},{"DocComment":" pub fn simple_try_fold_3>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" let cf = f(accum, x).branch();"},{"DocComment":" match cf {"},{"DocComment":" ControlFlow::Continue(a) => accum = a,"},{"DocComment":" ControlFlow::Break(r) => return R::from_residual(r),"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" But this \"call `branch`, then `match` on it, and `return` if it was a"},{"DocComment":" `Break`\" is exactly what happens inside the `?` operator. So rather than"},{"DocComment":" do all this manually, we can just use `?` instead:"},{"DocComment":" ```"},{"DocComment":" # #![feature(try_trait_v2)]"},{"DocComment":" # use std::ops::Try;"},{"DocComment":" fn simple_try_fold>("},{"DocComment":" iter: impl Iterator,"},{"DocComment":" mut accum: A,"},{"DocComment":" mut f: impl FnMut(A, T) -> R,"},{"DocComment":" ) -> R {"},{"DocComment":" for x in iter {"},{"DocComment":" accum = f(accum, x)?;"},{"DocComment":" }"},{"DocComment":" R::from_output(accum)"},{"DocComment":" }"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Try"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":67,"beg":{"line":133,"col":21},"end":{"line":133,"col":41}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":44,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"HashConsedValue":[11861,{"TraitType":[{"HashConsedValue":[11860,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]},1,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":67,"beg":{"line":160,"col":19},"end":{"line":160,"col":41}},"generated_from_span":null},"origin":{"TraitItem":1},"trait_":{"regions":[],"skip_binder":{"id":34,"generics":{"regions":[],"types":[{"Deduplicated":11861},{"HashConsedValue":[11862,{"TraitType":[{"Deduplicated":11860},0,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]},{"HashConsedValue":[11863,{"TraitType":[{"Deduplicated":11860},2,{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}]}]}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Output","attr_info":{"attributes":[{"DocComment":" The type of the value produced by `?` when *not* short-circuiting."}],"inline":null,"rename":null,"public":false},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Residual","attr_info":{"attributes":[{"DocComment":" The type of the value passed to [`FromResidual::from_residual`]"},{"DocComment":" as part of `?` when short-circuiting."},{"DocComment":""},{"DocComment":" This represents the possible values of the `Self` type which are *not*"},{"DocComment":" represented by the `Output` type."},{"DocComment":""},{"DocComment":" # Note to Implementors"},{"DocComment":""},{"DocComment":" The choice of this type is critical to interconversion."},{"DocComment":" Unlike the `Output` type, which will often be a raw generic type,"},{"DocComment":" this type is typically a newtype of some sort to \"color\" the type"},{"DocComment":" so that it's distinguishable from the residuals of other types."},{"DocComment":""},{"DocComment":" This is why `Result::Residual` is not `E`, but `Result`."},{"DocComment":" That way it's distinct from `ControlFlow::Residual`, for example,"},{"DocComment":" and thus `?` on `ControlFlow` cannot be used in a method returning `Result`."},{"DocComment":""},{"DocComment":" If you're making a generic type `Foo` that implements `Try`,"},{"DocComment":" then typically you can use `Foo` as its `Residual`"},{"DocComment":" type: that type will have a \"hole\" in the correct place, and will maintain the"},{"DocComment":" \"foo-ness\" of the residual so other types need to opt-in to interconversion."}],"inline":null,"rename":null,"public":false},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,1]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"Self_Clause1_TryType","attr_info":{"attributes":[],"inline":null,"rename":null,"public":true},"default":null,"implied_clauses":[]},"kind":{"TraitType":[33,2]}}],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_output","attr_info":{"attributes":[{"DocComment":" Constructs the type from its `Output` type."},{"DocComment":""},{"DocComment":" This should be implemented consistently with the `branch` method"},{"DocComment":" such that applying the `?` operator will get back the original value:"},{"DocComment":" `Try::from_output(x).branch() --> ControlFlow::Continue(x)`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::Try;"},{"DocComment":""},{"DocComment":" assert_eq!( as Try>::from_output(3), Ok(3));"},{"DocComment":" assert_eq!( as Try>::from_output(4), Some(4));"},{"DocComment":" assert_eq!("},{"DocComment":" as Try>::from_output(5),"},{"DocComment":" std::ops::ControlFlow::Continue(5),"},{"DocComment":" );"},{"DocComment":""},{"DocComment":" # fn make_question_mark_work() -> Option<()> {"},{"DocComment":" assert_eq!(Option::from_output(4)?, 4);"},{"DocComment":" # None }"},{"DocComment":" # make_question_mark_work();"},{"DocComment":""},{"DocComment":" // This is used, for example, on the accumulator in `try_fold`:"},{"DocComment":" let r = std::iter::empty().try_fold(4, |_, ()| -> Option<_> { unreachable!() });"},{"DocComment":" assert_eq!(r, Some(4));"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11862}],"output":{"Deduplicated":11755}},"item":{"id":363,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11860}]}}},"kind":{"TraitMethod":[33,0]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"branch","attr_info":{"attributes":[{"DocComment":" Used in `?` to decide whether the operator should produce a value"},{"DocComment":" (because this returned [`ControlFlow::Continue`])"},{"DocComment":" or propagate a value back to the caller"},{"DocComment":" (because this returned [`ControlFlow::Break`])."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::{ControlFlow, Try};"},{"DocComment":""},{"DocComment":" assert_eq!(Ok::<_, String>(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(Err::(3).branch(), ControlFlow::Break(Err(3)));"},{"DocComment":""},{"DocComment":" assert_eq!(Some(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!(None::.branch(), ControlFlow::Break(None));"},{"DocComment":""},{"DocComment":" assert_eq!(ControlFlow::::Continue(3).branch(), ControlFlow::Continue(3));"},{"DocComment":" assert_eq!("},{"DocComment":" ControlFlow::<_, String>::Break(3).branch(),"},{"DocComment":" ControlFlow::Break(ControlFlow::Break(3)),"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755}],"output":{"HashConsedValue":[11864,{"Adt":{"id":{"Adt":20},"generics":{"regions":[],"types":[{"Deduplicated":11861},{"Deduplicated":11862}],"const_generics":[],"trait_refs":[]}}}]}},"item":{"id":364,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"Deduplicated":11860}]}}},"kind":{"TraitMethod":[33,1]}}],"vtable":null},{"def_id":34,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["Residual",0]}],"span":{"data":{"file_id":67,"beg":{"line":364,"col":0},"end":{"line":364,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Allows retrieving the canonical type implementing [`Try`] that has this type"},{"DocComment":" as its residual and allows it to hold an `O` as its output."},{"DocComment":""},{"DocComment":" If you think of the `Try` trait as splitting a type into its [`Try::Output`]"},{"DocComment":" and [`Try::Residual`] components, this allows putting them back together."},{"DocComment":""},{"DocComment":" For example,"},{"DocComment":" `Result: Try>`,"},{"DocComment":" and in the other direction,"},{"DocComment":" ` as Residual>::TryType = Result`."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"O"},{"index":2,"name":"Self_TryType"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[{"regions":[],"skip_binder":{"trait_ref":{"HashConsedValue":[11866,{"kind":{"ParentClause":[{"HashConsedValue":[11865,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":34,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]},0]},"trait_decl_ref":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]},"type_id":0,"ty":{"Deduplicated":11756}}},{"regions":[],"skip_binder":{"trait_ref":{"Deduplicated":11866},"type_id":1,"ty":{"Deduplicated":11755}}}]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":67,"beg":{"line":368,"col":18},"end":{"line":368,"col":58}},"generated_from_span":null},"origin":{"TraitItem":0},"trait_":{"regions":[],"skip_binder":{"id":33,"generics":{"regions":[],"types":[{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[null],"methods":[],"vtable":null},{"def_id":35,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["collect",0]},{"Ident":["Extend",0]}],"span":{"data":{"file_id":17,"beg":{"line":397,"col":0},"end":{"line":397,"col":19}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Extend a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" Iterators produce a series of values, and collections can also be thought"},{"DocComment":" of as a series of values. The `Extend` trait bridges this gap, allowing you"},{"DocComment":" to extend a collection by including the contents of that iterator. When"},{"DocComment":" extending a collection with an already existing key, that entry is updated"},{"DocComment":" or, in the case of collections that permit multiple entries with equal"},{"DocComment":" keys, that entry is inserted."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"The first three letters are: \");"},{"DocComment":""},{"DocComment":" message.extend(&['a', 'b', 'c']);"},{"DocComment":""},{"DocComment":" assert_eq!(\"abc\", &message[29..32]);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" Implementing `Extend`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // A sample collection, that's just a wrapper over Vec"},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct MyCollection(Vec);"},{"DocComment":""},{"DocComment":" // Let's give it some methods so we can create one and add things"},{"DocComment":" // to it."},{"DocComment":" impl MyCollection {"},{"DocComment":" fn new() -> MyCollection {"},{"DocComment":" MyCollection(Vec::new())"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" fn add(&mut self, elem: i32) {"},{"DocComment":" self.0.push(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // since MyCollection has a list of i32s, we implement Extend for i32"},{"DocComment":" impl Extend for MyCollection {"},{"DocComment":""},{"DocComment":" // This is a bit simpler with the concrete type signature: we can call"},{"DocComment":" // extend on anything which can be turned into an Iterator which gives"},{"DocComment":" // us i32s. Because we need i32s to put into MyCollection."},{"DocComment":" fn extend>(&mut self, iter: T) {"},{"DocComment":""},{"DocComment":" // The implementation is very straightforward: loop through the"},{"DocComment":" // iterator, and add() each element to ourselves."},{"DocComment":" for elem in iter {"},{"DocComment":" self.add(elem);"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut c = MyCollection::new();"},{"DocComment":""},{"DocComment":" c.add(5);"},{"DocComment":" c.add(6);"},{"DocComment":" c.add(7);"},{"DocComment":""},{"DocComment":" // let's extend our collection with three more numbers"},{"DocComment":" c.extend(vec![1, 2, 3]);"},{"DocComment":""},{"DocComment":" // we've added these elements onto the end"},{"DocComment":" assert_eq!(\"MyCollection([5, 6, 7, 1, 2, 3])\", format!(\"{c:?}\"));"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[{"index":0,"name":"T"},{"index":1,"name":"Clause0_IntoIter"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":17,"beg":{"line":416,"col":17},"end":{"line":416,"col":39}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":28,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756},{"Deduplicated":3202}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"extend","attr_info":{"attributes":[{"DocComment":" Extends a collection with the contents of an iterator."},{"DocComment":""},{"DocComment":" As this is the only required method for this trait, the [trait-level] docs"},{"DocComment":" contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: Extend"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // You can extend a String with some chars:"},{"DocComment":" let mut message = String::from(\"abc\");"},{"DocComment":""},{"DocComment":" message.extend(['d', 'e', 'f'].iter());"},{"DocComment":""},{"DocComment":" assert_eq!(\"abcdef\", &message);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":3181}],"output":{"Deduplicated":1801}},"item":{"id":365,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":3181},{"Deduplicated":3201}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11867,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":35,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]},{"Deduplicated":11859}]}}},"kind":{"TraitMethod":[35,0]}},null,null,null],"vtable":null},{"def_id":36,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["double_ended",0]},{"Ident":["DoubleEndedIterator",0]}],"span":{"data":{"file_id":68,"beg":{"line":43,"col":0},"end":{"line":43,"col":53}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator able to yield elements from both ends."},{"DocComment":""},{"DocComment":" Something that implements `DoubleEndedIterator` has one extra capability"},{"DocComment":" over something that implements [`Iterator`]: the ability to also take"},{"DocComment":" `Item`s from the back, as well as the front."},{"DocComment":""},{"DocComment":" It is important to note that both back and forth work on the same range,"},{"DocComment":" and do not cross: iteration is over when they meet in the middle."},{"DocComment":""},{"DocComment":" In a similar fashion to the [`Iterator`] protocol, once a"},{"DocComment":" `DoubleEndedIterator` returns [`None`] from a [`next_back()`], calling it"},{"DocComment":" again may or may not ever return [`Some`] again. [`next()`] and"},{"DocComment":" [`next_back()`] are interchangeable for this purpose."},{"DocComment":""},{"DocComment":" [`next_back()`]: DoubleEndedIterator::next_back"},{"DocComment":" [`next()`]: Iterator::next"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"DoubleEndedIterator"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":68,"beg":{"line":43,"col":37},"end":{"line":43,"col":53}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"next_back","attr_info":{"attributes":[{"DocComment":" Removes and returns an element from the end of the iterator."},{"DocComment":""},{"DocComment":" Returns `None` when there are no more elements."},{"DocComment":""},{"DocComment":" The [trait-level] docs contain more details."},{"DocComment":""},{"DocComment":" [trait-level]: DoubleEndedIterator"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let numbers = vec![1, 2, 3, 4, 5, 6];"},{"DocComment":""},{"DocComment":" let mut iter = numbers.iter();"},{"DocComment":""},{"DocComment":" assert_eq!(Some(&1), iter.next());"},{"DocComment":" assert_eq!(Some(&6), iter.next_back());"},{"DocComment":" assert_eq!(Some(&5), iter.next_back());"},{"DocComment":" assert_eq!(Some(&2), iter.next());"},{"DocComment":" assert_eq!(Some(&3), iter.next());"},{"DocComment":" assert_eq!(Some(&4), iter.next());"},{"DocComment":" assert_eq!(None, iter.next());"},{"DocComment":" assert_eq!(None, iter.next_back());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Remarks"},{"DocComment":""},{"DocComment":" The elements yielded by `DoubleEndedIterator`'s methods may differ from"},{"DocComment":" the ones yielded by [`Iterator`]'s methods:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let vec = vec![(1, 'a'), (1, 'b'), (1, 'c'), (2, 'a'), (2, 'b')];"},{"DocComment":" let uniq_by_fst_comp = || {"},{"DocComment":" let mut seen = std::collections::HashSet::new();"},{"DocComment":" vec.iter().copied().filter(move |x| seen.insert(x.0))"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" assert_eq!(uniq_by_fst_comp().last(), Some((2, 'a')));"},{"DocComment":" assert_eq!(uniq_by_fst_comp().next_back(), Some((2, 'b')));"},{"DocComment":""},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().fold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(1, 'a'), (2, 'a')]"},{"DocComment":" );"},{"DocComment":" assert_eq!("},{"DocComment":" uniq_by_fst_comp().rfold(vec![], |mut v, x| {v.push(x); v}),"},{"DocComment":" vec![(2, 'b'), (1, 'c')]"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822}],"output":{"Deduplicated":11823}},"item":{"id":369,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11868,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":36,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[36,0]}},null,null,null,null,null],"vtable":{"id":{"Adt":76},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":37,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["exact_size",0]},{"Ident":["ExactSizeIterator",0]}],"span":{"data":{"file_id":69,"beg":{"line":86,"col":0},"end":{"line":86,"col":37}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" An iterator that knows its exact length."},{"DocComment":""},{"DocComment":" Many [`Iterator`]s don't know how many times they will iterate, but some do."},{"DocComment":" If an iterator knows how many times it can iterate, providing access to"},{"DocComment":" that information can be useful. For example, if you want to iterate"},{"DocComment":" backwards, a good start is to know where the end is."},{"DocComment":""},{"DocComment":" When implementing an `ExactSizeIterator`, you must also implement"},{"DocComment":" [`Iterator`]. When doing so, the implementation of [`Iterator::size_hint`]"},{"DocComment":" *must* return the exact size of the iterator."},{"DocComment":""},{"DocComment":" The [`len`] method has a default implementation, so you usually shouldn't"},{"DocComment":" implement it. However, you may be able to provide a more performant"},{"DocComment":" implementation than the default, so overriding it in this case makes sense."},{"DocComment":""},{"DocComment":" Note that this trait is a safe trait and as such does *not* and *cannot*"},{"DocComment":" guarantee that the returned length is correct. This means that `unsafe`"},{"DocComment":" code **must not** rely on the correctness of [`Iterator::size_hint`]. The"},{"DocComment":" unstable and unsafe [`TrustedLen`](super::marker::TrustedLen) trait gives"},{"DocComment":" this additional guarantee."},{"DocComment":""},{"DocComment":" [`len`]: ExactSizeIterator::len"},{"DocComment":""},{"DocComment":" # When *shouldn't* an adapter be `ExactSizeIterator`?"},{"DocComment":""},{"DocComment":" If an adapter makes an iterator *longer*, then it's usually incorrect for"},{"DocComment":" that adapter to implement `ExactSizeIterator`. The inner exact-sized"},{"DocComment":" iterator might already be `usize::MAX`-long, and thus the length of the"},{"DocComment":" longer adapted iterator would no longer be exactly representable in `usize`."},{"DocComment":""},{"DocComment":" This is why [`Chain`](crate::iter::Chain) isn't `ExactSizeIterator`,"},{"DocComment":" even when `A` and `B` are both `ExactSizeIterator`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" Basic usage:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" // a finite range knows exactly how many times it will iterate"},{"DocComment":" let five = 0..5;"},{"DocComment":""},{"DocComment":" assert_eq!(5, five.len());"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" In the [module-level docs], we implemented an [`Iterator`], `Counter`."},{"DocComment":" Let's implement `ExactSizeIterator` for it as well:"},{"DocComment":""},{"DocComment":" [module-level docs]: crate::iter"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # struct Counter {"},{"DocComment":" # count: usize,"},{"DocComment":" # }"},{"DocComment":" # impl Counter {"},{"DocComment":" # fn new() -> Counter {"},{"DocComment":" # Counter { count: 0 }"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" # impl Iterator for Counter {"},{"DocComment":" # type Item = usize;"},{"DocComment":" # fn next(&mut self) -> Option {"},{"DocComment":" # self.count += 1;"},{"DocComment":" # if self.count < 6 {"},{"DocComment":" # Some(self.count)"},{"DocComment":" # } else {"},{"DocComment":" # None"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" # }"},{"DocComment":" impl ExactSizeIterator for Counter {"},{"DocComment":" // We can easily calculate the remaining number of iterations."},{"DocComment":" fn len(&self) -> usize {"},{"DocComment":" 5 - self.count"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // And now we can use it!"},{"DocComment":""},{"DocComment":" let mut counter = Counter::new();"},{"DocComment":""},{"DocComment":" assert_eq!(5, counter.len());"},{"DocComment":" let _ = counter.next();"},{"DocComment":" assert_eq!(4, counter.len());"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Self_Clause0_Item"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":69,"beg":{"line":86,"col":29},"end":{"line":86,"col":37}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[null,null],"vtable":{"id":{"Adt":77},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":38,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["Ord",0]}],"span":{"data":{"file_id":33,"beg":{"line":982,"col":0},"end":{"line":982,"col":73}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for types that form a [total order](https://en.wikipedia.org/wiki/Total_order)."},{"DocComment":""},{"DocComment":" Implementations must be consistent with the [`PartialOrd`] implementation, and ensure `max`,"},{"DocComment":" `min`, and `clamp` are consistent with `cmp`:"},{"DocComment":""},{"DocComment":" - `partial_cmp(a, b) == Some(cmp(a, b))`."},{"DocComment":" - `max(a, b) == max_by(a, b, cmp)` (ensured by the default implementation)."},{"DocComment":" - `min(a, b) == min_by(a, b, cmp)` (ensured by the default implementation)."},{"DocComment":" - For `a.clamp(min, max)`, see the [method docs](#method.clamp) (ensured by the default"},{"DocComment":" implementation)."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Corollaries"},{"DocComment":""},{"DocComment":" From the above and the requirements of `PartialOrd`, it follows that for all `a`, `b` and `c`:"},{"DocComment":""},{"DocComment":" - exactly one of `a < b`, `a == b` or `a > b` is true; and"},{"DocComment":" - `<` is transitive: `a < b` and `b < c` implies `a < c`. The same must hold for both `==` and"},{"DocComment":" `>`."},{"DocComment":""},{"DocComment":" Mathematically speaking, the `<` operator defines a strict [weak order]. In cases where `==`"},{"DocComment":" conforms to mathematical equality, it also defines a strict [total order]."},{"DocComment":""},{"DocComment":" [weak order]: https://en.wikipedia.org/wiki/Weak_ordering"},{"DocComment":" [total order]: https://en.wikipedia.org/wiki/Total_order"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`."},{"DocComment":""},{"DocComment":" When `derive`d on structs, it will produce a"},{"DocComment":" [lexicographic](https://en.wikipedia.org/wiki/Lexicographic_order) ordering based on the"},{"DocComment":" top-to-bottom declaration order of the struct's members."},{"DocComment":""},{"DocComment":" When `derive`d on enums, variants are ordered primarily by their discriminants. Secondarily,"},{"DocComment":" they are ordered by their fields. By default, the discriminant is smallest for variants at the"},{"DocComment":" top, and largest for variants at the bottom. Here's an example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, Eq, PartialOrd, Ord)]"},{"DocComment":" enum E {"},{"DocComment":" Top,"},{"DocComment":" Bottom,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Top < E::Bottom);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, manually setting the discriminants can override this default behavior:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, Eq, PartialOrd, Ord)]"},{"DocComment":" enum E {"},{"DocComment":" Top = 2,"},{"DocComment":" Bottom = 1,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Bottom < E::Top);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Lexicographical comparison"},{"DocComment":""},{"DocComment":" Lexicographical comparison is an operation with the following properties:"},{"DocComment":" - Two sequences are compared element by element."},{"DocComment":" - The first mismatching element defines which sequence is lexicographically less or greater"},{"DocComment":" than the other."},{"DocComment":" - If one sequence is a prefix of another, the shorter sequence is lexicographically less than"},{"DocComment":" the other."},{"DocComment":" - If two sequences have equivalent elements and are of the same length, then the sequences are"},{"DocComment":" lexicographically equal."},{"DocComment":" - An empty sequence is lexicographically less than any non-empty sequence."},{"DocComment":" - Two empty sequences are lexicographically equal."},{"DocComment":""},{"DocComment":" ## How can I implement `Ord`?"},{"DocComment":""},{"DocComment":" `Ord` requires that the type also be [`PartialOrd`], [`PartialEq`], and [`Eq`]."},{"DocComment":""},{"DocComment":" Because `Ord` implies a stronger ordering relationship than [`PartialOrd`], and both `Ord` and"},{"DocComment":" [`PartialOrd`] must agree, you must choose how to implement `Ord` **first**. You can choose to"},{"DocComment":" derive it, or implement it manually. If you derive it, you should derive all four traits. If you"},{"DocComment":" implement it manually, you should manually implement all four traits, based on the"},{"DocComment":" implementation of `Ord`."},{"DocComment":""},{"DocComment":" Here's an example where you want to define the `Character` comparison by `health` and"},{"DocComment":" `experience` only, disregarding the field `mana`:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" mana: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> Ordering {"},{"DocComment":" self.experience"},{"DocComment":" .cmp(&other.experience)"},{"DocComment":" .then(self.health.cmp(&other.health))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.health == other.health && self.experience == other.experience"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" If all you need is to `slice::sort` a type by a field value, it can be simpler to use"},{"DocComment":" `slice::sort_by_key`."},{"DocComment":""},{"DocComment":" ## Examples of incorrect `Ord` implementations"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: f32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> std::cmp::Ordering {"},{"DocComment":" if self.health < other.health {"},{"DocComment":" Ordering::Less"},{"DocComment":" } else if self.health > other.health {"},{"DocComment":" Ordering::Greater"},{"DocComment":" } else {"},{"DocComment":" Ordering::Equal"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.health == other.health"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":""},{"DocComment":" let a = Character { health: 4.5 };"},{"DocComment":" let b = Character { health: f32::NAN };"},{"DocComment":""},{"DocComment":" // Mistake: floating-point values do not form a total order and using the built-in comparison"},{"DocComment":" // operands to implement `Ord` irregardless of that reality does not change it. Use"},{"DocComment":" // `f32::total_cmp` if you need a total order for floating-point values."},{"DocComment":""},{"DocComment":" // Reflexivity requirement of `Ord` is not given."},{"DocComment":" assert!(a == a);"},{"DocComment":" assert!(b != b);"},{"DocComment":""},{"DocComment":" // Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be"},{"DocComment":" // true, not both or neither."},{"DocComment":" assert_eq!((a < b) as u8 + (b < a) as u8, 0);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Character {"},{"DocComment":" fn cmp(&self, other: &Self) -> std::cmp::Ordering {"},{"DocComment":" if self.health < 50 {"},{"DocComment":" self.health.cmp(&other.health)"},{"DocComment":" } else {"},{"DocComment":" self.experience.cmp(&other.experience)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" // For performance reasons implementing `PartialEq` this way is not the idiomatic way, but it"},{"DocComment":" // ensures consistent behavior between `PartialEq`, `PartialOrd` and `Ord` in this example."},{"DocComment":" impl PartialEq for Character {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.cmp(other) == Ordering::Equal"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Character {}"},{"DocComment":""},{"DocComment":" let a = Character {"},{"DocComment":" health: 3,"},{"DocComment":" experience: 5,"},{"DocComment":" };"},{"DocComment":" let b = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 77,"},{"DocComment":" };"},{"DocComment":" let c = Character {"},{"DocComment":" health: 143,"},{"DocComment":" experience: 2,"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // Mistake: The implementation of `Ord` compares different fields depending on the value of"},{"DocComment":" // `self.health`, the resulting order is not total."},{"DocComment":""},{"DocComment":" // Transitivity requirement of `Ord` is not given. If a is smaller than b and b is smaller than"},{"DocComment":" // c, by transitive property a must also be smaller than c."},{"DocComment":" assert!(a < b && b < c && c < a);"},{"DocComment":""},{"DocComment":" // Antisymmetry requirement of `Ord` is not given. Only one of a < c and c < a is allowed to be"},{"DocComment":" // true, not both or neither."},{"DocComment":" assert_eq!((a < c) as u8 + (c < a) as u8, 2);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" The documentation of [`PartialOrd`] contains further examples, for example it's wrong for"},{"DocComment":" [`PartialOrd`] and [`PartialEq`] to disagree."},{"DocComment":""},{"DocComment":" [`cmp`]: Ord::cmp"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Ord"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":982,"col":21},"end":{"line":982,"col":31}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}},{"clause_id":1,"span":{"data":{"file_id":33,"beg":{"line":982,"col":34},"end":{"line":982,"col":58}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"cmp","attr_info":{"attributes":[{"DocComment":" This method returns an [`Ordering`] between `self` and `other`."},{"DocComment":""},{"DocComment":" By convention, `self.cmp(&other)` returns the ordering matching the expression"},{"DocComment":" `self other` if true."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" assert_eq!(5.cmp(&10), Ordering::Less);"},{"DocComment":" assert_eq!(10.cmp(&5), Ordering::Greater);"},{"DocComment":" assert_eq!(5.cmp(&5), Ordering::Equal);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"Deduplicated":11826}],"output":{"Deduplicated":5649}},"item":{"id":377,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11869,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":38,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[38,0]}},null,null,null],"vtable":null},{"def_id":39,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["traits",0]},{"Ident":["accum",0]},{"Ident":["Product",0]}],"span":{"data":{"file_id":43,"beg":{"line":40,"col":0},"end":{"line":40,"col":40}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait to represent types that can be created by multiplying elements of an"},{"DocComment":" iterator."},{"DocComment":""},{"DocComment":" This trait is used to implement [`Iterator::product()`]. Types which implement"},{"DocComment":" this trait can be generated by using the [`product()`] method on an iterator."},{"DocComment":" Like [`FromIterator`], this trait should rarely be called directly."},{"DocComment":""},{"DocComment":" [`product()`]: Iterator::product"},{"DocComment":" [`FromIterator`]: iter::FromIterator"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"A"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[{"index":0,"name":"I"}],"const_generics":[],"trait_clauses":[{"clause_id":0,"span":{"data":{"file_id":43,"beg":{"line":44,"col":18},"end":{"line":44,"col":36}},"generated_from_span":null},"origin":"WhereClauseOnFn","trait_":{"regions":[],"skip_binder":{"id":0,"generics":{"regions":[],"types":[{"Deduplicated":3180},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"product","attr_info":{"attributes":[{"DocComment":" Takes an iterator and generates `Self` from the elements by multiplying"},{"DocComment":" the items."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":3181}],"output":{"Deduplicated":11755}},"item":{"id":381,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":3181}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11870,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":39,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]},{"Deduplicated":11854}]}}},"kind":{"TraitMethod":[39,0]}}],"vtable":null},{"def_id":40,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["cmp",0]},{"Ident":["PartialOrd",0]}],"span":{"data":{"file_id":33,"beg":{"line":1366,"col":0},"end":{"line":1367,"col":41}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Trait for types that form a [partial order](https://en.wikipedia.org/wiki/Partial_order)."},{"DocComment":""},{"DocComment":" The `lt`, `le`, `gt`, and `ge` methods of this trait can be called using the `<`, `<=`, `>`, and"},{"DocComment":" `>=` operators, respectively."},{"DocComment":""},{"DocComment":" This trait should **only** contain the comparison logic for a type **if one plans on only"},{"DocComment":" implementing `PartialOrd` but not [`Ord`]**. Otherwise the comparison logic should be in [`Ord`]"},{"DocComment":" and this trait implemented with `Some(self.cmp(other))`."},{"DocComment":""},{"DocComment":" The methods of this trait must be consistent with each other and with those of [`PartialEq`]."},{"DocComment":" The following conditions must hold:"},{"DocComment":""},{"DocComment":" 1. `a == b` if and only if `partial_cmp(a, b) == Some(Equal)`."},{"DocComment":" 2. `a < b` if and only if `partial_cmp(a, b) == Some(Less)`"},{"DocComment":" 3. `a > b` if and only if `partial_cmp(a, b) == Some(Greater)`"},{"DocComment":" 4. `a <= b` if and only if `a < b || a == b`"},{"DocComment":" 5. `a >= b` if and only if `a > b || a == b`"},{"DocComment":" 6. `a != b` if and only if `!(a == b)`."},{"DocComment":""},{"DocComment":" Conditions 2–5 above are ensured by the default implementation. Condition 6 is already ensured"},{"DocComment":" by [`PartialEq`]."},{"DocComment":""},{"DocComment":" If [`Ord`] is also implemented for `Self` and `Rhs`, it must also be consistent with"},{"DocComment":" `partial_cmp` (see the documentation of that trait for the exact requirements). It's easy to"},{"DocComment":" accidentally make them disagree by deriving some of the traits and manually implementing others."},{"DocComment":""},{"DocComment":" The comparison relations must satisfy the following conditions (for all `a`, `b`, `c` of type"},{"DocComment":" `A`, `B`, `C`):"},{"DocComment":""},{"DocComment":" - **Transitivity**: if `A: PartialOrd` and `B: PartialOrd` and `A: PartialOrd`, then `a"},{"DocComment":" < b` and `b < c` implies `a < c`. The same must hold for both `==` and `>`. This must also"},{"DocComment":" work for longer chains, such as when `A: PartialOrd`, `B: PartialOrd`, `C:"},{"DocComment":" PartialOrd`, and `A: PartialOrd` all exist."},{"DocComment":" - **Duality**: if `A: PartialOrd` and `B: PartialOrd`, then `a < b` if and only if `b >"},{"DocComment":" a`."},{"DocComment":""},{"DocComment":" Note that the `B: PartialOrd` (dual) and `A: PartialOrd` (transitive) impls are not forced"},{"DocComment":" to exist, but these requirements apply whenever they do exist."},{"DocComment":""},{"DocComment":" Violating these requirements is a logic error. The behavior resulting from a logic error is not"},{"DocComment":" specified, but users of the trait must ensure that such logic errors do *not* result in"},{"DocComment":" undefined behavior. This means that `unsafe` code **must not** rely on the correctness of these"},{"DocComment":" methods."},{"DocComment":""},{"DocComment":" ## Cross-crate considerations"},{"DocComment":""},{"DocComment":" Upholding the requirements stated above can become tricky when one crate implements `PartialOrd`"},{"DocComment":" for a type of another crate (i.e., to allow comparing one of its own types with a type from the"},{"DocComment":" standard library). The recommendation is to never implement this trait for a foreign type. In"},{"DocComment":" other words, such a crate should do `impl PartialOrd for LocalType`, but it should"},{"DocComment":" *not* do `impl PartialOrd for ForeignType`."},{"DocComment":""},{"DocComment":" This avoids the problem of transitive chains that criss-cross crate boundaries: for all local"},{"DocComment":" types `T`, you may assume that no other crate will add `impl`s that allow comparing `T < U`. In"},{"DocComment":" other words, if other crates add `impl`s that allow building longer transitive chains `U1 < ..."},{"DocComment":" < T < V1 < ...`, then all the types that appear to the right of `T` must be types that the crate"},{"DocComment":" defining `T` already knows about. This rules out transitive chains where downstream crates can"},{"DocComment":" add new `impl`s that \"stitch together\" comparisons of foreign types in ways that violate"},{"DocComment":" transitivity."},{"DocComment":""},{"DocComment":" Not having such foreign `impl`s also avoids forward compatibility issues where one crate adding"},{"DocComment":" more `PartialOrd` implementations can cause build failures in downstream crates."},{"DocComment":""},{"DocComment":" ## Corollaries"},{"DocComment":""},{"DocComment":" The following corollaries follow from the above requirements:"},{"DocComment":""},{"DocComment":" - irreflexivity of `<` and `>`: `!(a < a)`, `!(a > a)`"},{"DocComment":" - transitivity of `>`: if `a > b` and `b > c` then `a > c`"},{"DocComment":" - duality of `partial_cmp`: `partial_cmp(a, b) == partial_cmp(b, a).map(Ordering::reverse)`"},{"DocComment":""},{"DocComment":" ## Strict and non-strict partial orders"},{"DocComment":""},{"DocComment":" The `<` and `>` operators behave according to a *strict* partial order. However, `<=` and `>=`"},{"DocComment":" do **not** behave according to a *non-strict* partial order. That is because mathematically, a"},{"DocComment":" non-strict partial order would require reflexivity, i.e. `a <= a` would need to be true for"},{"DocComment":" every `a`. This isn't always the case for types that implement `PartialOrd`, for example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let a = f64::NAN;"},{"DocComment":" assert_eq!(a <= a, false);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Derivable"},{"DocComment":""},{"DocComment":" This trait can be used with `#[derive]`."},{"DocComment":""},{"DocComment":" When `derive`d on structs, it will produce a"},{"DocComment":" [lexicographic](https://en.wikipedia.org/wiki/Lexicographic_order) ordering based on the"},{"DocComment":" top-to-bottom declaration order of the struct's members."},{"DocComment":""},{"DocComment":" When `derive`d on enums, variants are primarily ordered by their discriminants. Secondarily,"},{"DocComment":" they are ordered by their fields. By default, the discriminant is smallest for variants at the"},{"DocComment":" top, and largest for variants at the bottom. Here's an example:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, PartialOrd)]"},{"DocComment":" enum E {"},{"DocComment":" Top,"},{"DocComment":" Bottom,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Top < E::Bottom);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" However, manually setting the discriminants can override this default behavior:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #[derive(PartialEq, PartialOrd)]"},{"DocComment":" enum E {"},{"DocComment":" Top = 2,"},{"DocComment":" Bottom = 1,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert!(E::Bottom < E::Top);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## How can I implement `PartialOrd`?"},{"DocComment":""},{"DocComment":" `PartialOrd` only requires implementation of the [`partial_cmp`] method, with the others"},{"DocComment":" generated from default implementations."},{"DocComment":""},{"DocComment":" However it remains possible to implement the others separately for types which do not have a"},{"DocComment":" total order. For example, for floating point numbers, `NaN < 0 == false` and `NaN >= 0 == false`"},{"DocComment":" (cf. IEEE 754-2008 section 5.11)."},{"DocComment":""},{"DocComment":" `PartialOrd` requires your type to be [`PartialEq`]."},{"DocComment":""},{"DocComment":" If your type is [`Ord`], you can implement [`partial_cmp`] by using [`cmp`]:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" height: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Person {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.cmp(other))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Ord for Person {"},{"DocComment":" fn cmp(&self, other: &Self) -> Ordering {"},{"DocComment":" self.height.cmp(&other.height)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Person {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.height == other.height"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Eq for Person {}"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" You may also find it useful to use [`partial_cmp`] on your type's fields. Here is an example of"},{"DocComment":" `Person` types who have a floating-point `height` field that is the only field to be used for"},{"DocComment":" sorting:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" struct Person {"},{"DocComment":" id: u32,"},{"DocComment":" name: String,"},{"DocComment":" height: f64,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Person {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" self.height.partial_cmp(&other.height)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialEq for Person {"},{"DocComment":" fn eq(&self, other: &Self) -> bool {"},{"DocComment":" self.height == other.height"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" ## Examples of incorrect `PartialOrd` implementations"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" #[derive(PartialEq, Debug)]"},{"DocComment":" struct Character {"},{"DocComment":" health: u32,"},{"DocComment":" experience: u32,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl PartialOrd for Character {"},{"DocComment":" fn partial_cmp(&self, other: &Self) -> Option {"},{"DocComment":" Some(self.health.cmp(&other.health))"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let a = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 5,"},{"DocComment":" };"},{"DocComment":" let b = Character {"},{"DocComment":" health: 10,"},{"DocComment":" experience: 77,"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // Mistake: `PartialEq` and `PartialOrd` disagree with each other."},{"DocComment":""},{"DocComment":" assert_eq!(a.partial_cmp(&b).unwrap(), Ordering::Equal); // a == b according to `PartialOrd`."},{"DocComment":" assert_ne!(a, b); // a != b according to `PartialEq`."},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let x: u32 = 0;"},{"DocComment":" let y: u32 = 1;"},{"DocComment":""},{"DocComment":" assert_eq!(x < y, true);"},{"DocComment":" assert_eq!(x.lt(&y), true);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`partial_cmp`]: PartialOrd::partial_cmp"},{"DocComment":" [`cmp`]: Ord::cmp"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"partial_ord"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":33,"beg":{"line":1367,"col":4},"end":{"line":1367,"col":26}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"partial_cmp","attr_info":{"attributes":[{"DocComment":" This method returns an ordering between `self` and `other` values if one exists."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::cmp::Ordering;"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&2.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Less));"},{"DocComment":""},{"DocComment":" let result = 1.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Equal));"},{"DocComment":""},{"DocComment":" let result = 2.0.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, Some(Ordering::Greater));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" When comparison is impossible:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" let result = f64::NAN.partial_cmp(&1.0);"},{"DocComment":" assert_eq!(result, None);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"Deduplicated":11831}],"output":{"Deduplicated":5943}},"item":{"id":382,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11871,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":40,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[40,0]}},null,null,null,null,null,null,null,null],"vtable":{"id":{"Adt":78},"generics":{"regions":[],"types":[{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}},{"def_id":41,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["iter",0]},{"Ident":["adapters",0]},{"Ident":["zip",0]},{"Ident":["TrustedRandomAccessNoCoerce",0]}],"span":{"data":{"file_id":49,"beg":{"line":576,"col":0},"end":{"line":576,"col":51}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Like [`TrustedRandomAccess`] but without any of the requirements / guarantees around"},{"DocComment":" coercions to supertypes after `__iterator_get_unchecked` (they aren’t allowed here!), and"},{"DocComment":" without the requirement that subtypes / supertypes implement `TrustedRandomAccessNoCoerce`."},{"DocComment":""},{"DocComment":" This trait was created in PR #85874 to fix soundness issue #85873 without performance regressions."},{"DocComment":" It is subject to change as we might want to build a more generally useful (for performance"},{"DocComment":" optimizations) and more sophisticated trait or trait hierarchy that replaces or extends"},{"DocComment":" [`TrustedRandomAccess`] and `TrustedRandomAccessNoCoerce`."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[{"name":"MAY_HAVE_SIDE_EFFECT","attr_info":{"attributes":[{"DocComment":" `true` if getting an iterator element may have side effects."},{"DocComment":" Remember to take inner iterators into account."}],"inline":null,"rename":null,"public":true},"ty":{"Deduplicated":1796},"default":null}],"types":[],"methods":[null],"vtable":null},{"def_id":42,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["index",0]},{"Ident":["Index",0]}],"span":{"data":{"file_id":75,"beg":{"line":59,"col":0},"end":{"line":59,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used for indexing operations (`container[index]`) in immutable contexts."},{"DocComment":""},{"DocComment":" `container[index]` is actually syntactic sugar for `*container.index(index)`,"},{"DocComment":" but only when used as an immutable value. If a mutable value is requested,"},{"DocComment":" [`IndexMut`] is used instead. This allows nice things such as"},{"DocComment":" `let value = v[index]` if the type of `value` implements [`Copy`]."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" The following example implements `Index` on a read-only `NucleotideCount`"},{"DocComment":" container, enabling individual counts to be retrieved with index syntax."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::Index;"},{"DocComment":""},{"DocComment":" enum Nucleotide {"},{"DocComment":" A,"},{"DocComment":" C,"},{"DocComment":" G,"},{"DocComment":" T,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct NucleotideCount {"},{"DocComment":" a: usize,"},{"DocComment":" c: usize,"},{"DocComment":" g: usize,"},{"DocComment":" t: usize,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Index for NucleotideCount {"},{"DocComment":" type Output = usize;"},{"DocComment":""},{"DocComment":" fn index(&self, nucleotide: Nucleotide) -> &Self::Output {"},{"DocComment":" match nucleotide {"},{"DocComment":" Nucleotide::A => &self.a,"},{"DocComment":" Nucleotide::C => &self.c,"},{"DocComment":" Nucleotide::G => &self.g,"},{"DocComment":" Nucleotide::T => &self.t,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let nucleotide_count = NucleotideCount {a: 14, c: 9, g: 10, t: 12};"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::A], 14);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::C], 9);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::G], 10);"},{"DocComment":" assert_eq!(nucleotide_count[Nucleotide::T], 12);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"index"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Idx"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"index","attr_info":{"attributes":[{"DocComment":" Performs the indexing (`container[index]`) operation."},{"DocComment":""},{"DocComment":" # Panics"},{"DocComment":""},{"DocComment":" May panic if the index is out of bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825},{"Deduplicated":11756}],"output":{"HashConsedValue":[11872,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11793},"Shared"]}]}},"item":{"id":402,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11873,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":42,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[42,0]}}],"vtable":{"id":{"Adt":79},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":43,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["index",0]},{"Ident":["IndexMut",0]}],"span":{"data":{"file_id":75,"beg":{"line":170,"col":0},"end":{"line":170,"col":57}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used for indexing operations (`container[index]`) in mutable contexts."},{"DocComment":""},{"DocComment":" `container[index]` is actually syntactic sugar for"},{"DocComment":" `*container.index_mut(index)`, but only when used as a mutable value. If"},{"DocComment":" an immutable value is requested, the [`Index`] trait is used instead. This"},{"DocComment":" allows nice things such as `v[index] = value`."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" A very simple implementation of a `Balance` struct that has two sides, where"},{"DocComment":" each can be indexed mutably and immutably."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::{Index, IndexMut};"},{"DocComment":""},{"DocComment":" #[derive(Debug)]"},{"DocComment":" enum Side {"},{"DocComment":" Left,"},{"DocComment":" Right,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" enum Weight {"},{"DocComment":" Kilogram(f32),"},{"DocComment":" Pound(f32),"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" struct Balance {"},{"DocComment":" pub left: Weight,"},{"DocComment":" pub right: Weight,"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl Index for Balance {"},{"DocComment":" type Output = Weight;"},{"DocComment":""},{"DocComment":" fn index(&self, index: Side) -> &Self::Output {"},{"DocComment":" println!(\"Accessing {index:?}-side of balance immutably\");"},{"DocComment":" match index {"},{"DocComment":" Side::Left => &self.left,"},{"DocComment":" Side::Right => &self.right,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" impl IndexMut for Balance {"},{"DocComment":" fn index_mut(&mut self, index: Side) -> &mut Self::Output {"},{"DocComment":" println!(\"Accessing {index:?}-side of balance mutably\");"},{"DocComment":" match index {"},{"DocComment":" Side::Left => &mut self.left,"},{"DocComment":" Side::Right => &mut self.right,"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let mut balance = Balance {"},{"DocComment":" right: Weight::Kilogram(2.5),"},{"DocComment":" left: Weight::Pound(1.5),"},{"DocComment":" };"},{"DocComment":""},{"DocComment":" // In this case, `balance[Side::Right]` is sugar for"},{"DocComment":" // `*balance.index(Side::Right)`, since we are only *reading*"},{"DocComment":" // `balance[Side::Right]`, not writing it."},{"DocComment":" assert_eq!(balance[Side::Right], Weight::Kilogram(2.5));"},{"DocComment":""},{"DocComment":" // However, in this case `balance[Side::Left]` is sugar for"},{"DocComment":" // `*balance.index_mut(Side::Left)`, since we are writing"},{"DocComment":" // `balance[Side::Left]`."},{"DocComment":" balance[Side::Left] = Weight::Kilogram(3.0);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"index_mut"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Idx"},{"index":2,"name":"Self_Clause0_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[{"clause_id":0,"span":{"data":{"file_id":75,"beg":{"line":170,"col":39},"end":{"line":170,"col":57}},"generated_from_span":null},"origin":"WhereClauseOnTrait","trait_":{"regions":[],"skip_binder":{"id":42,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"index_mut","attr_info":{"attributes":[{"DocComment":" Performs the mutable indexing (`container[index]`) operation."},{"DocComment":""},{"DocComment":" # Panics"},{"DocComment":""},{"DocComment":" May panic if the index is out of bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11822},{"Deduplicated":11756}],"output":{"HashConsedValue":[11874,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11793},"Mut"]}]}},"item":{"id":403,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11875,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":43,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[43,0]}}],"vtable":{"id":{"Adt":80},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":44,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["try_trait",0]},{"Ident":["FromResidual",0]}],"span":{"data":{"file_id":67,"beg":{"line":310,"col":0},"end":{"line":310,"col":57}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" Used to specify which residuals can be converted into which [`crate::ops::Try`] types."},{"DocComment":""},{"DocComment":" Every `Try` type needs to be recreatable from its own associated"},{"DocComment":" `Residual` type, but can also have additional `FromResidual` implementations"},{"DocComment":" to support interconversion with other `Try` types."}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"FromResidual"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"R"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"from_residual","attr_info":{"attributes":[{"DocComment":" Constructs the type from a compatible `Residual` type."},{"DocComment":""},{"DocComment":" This should be implemented consistently with the `branch` method such"},{"DocComment":" that applying the `?` operator will get back an equivalent residual:"},{"DocComment":" `FromResidual::from_residual(r).branch() --> ControlFlow::Break(r)`."},{"DocComment":" (The residual is not mandated to be *identical* when interconversion is involved.)"},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" #![feature(try_trait_v2)]"},{"DocComment":" use std::ops::{ControlFlow, FromResidual};"},{"DocComment":""},{"DocComment":" assert_eq!(Result::::from_residual(Err(3_u8)), Err(3));"},{"DocComment":" assert_eq!(Option::::from_residual(None), None);"},{"DocComment":" assert_eq!("},{"DocComment":" ControlFlow::<_, String>::from_residual(ControlFlow::Break(5)),"},{"DocComment":" ControlFlow::Break(5),"},{"DocComment":" );"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11756}],"output":{"Deduplicated":11755}},"item":{"id":412,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11876,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":44,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[44,0]}}],"vtable":null},{"def_id":45,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["slice",0]},{"Ident":["index",0]},{"Ident":["SliceIndex",0]}],"span":{"data":{"file_id":77,"beg":{"line":163,"col":0},"end":{"line":163,"col":73}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" A helper trait used for indexing operations."},{"DocComment":""},{"DocComment":" Implementations of this trait have to promise that if the argument"},{"DocComment":" to `get_unchecked(_mut)` is a safe reference, then so is the 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Returns a mutable reference to the output at this location, if in"},{"DocComment":" bounds."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"HashConsedValue":[11879,{"Ref":[{"Var":{"Bound":[0,0]}},{"Deduplicated":11756},"Mut"]}]}],"output":{"HashConsedValue":[11880,{"Adt":{"id":{"Adt":12},"generics":{"regions":[],"types":[{"Deduplicated":11874}],"const_generics":[],"trait_refs":[]}}}]}},"item":{"id":415,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"Deduplicated":11878}]}}},"kind":{"TraitMethod":[45,1]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"get_unchecked","attr_info":{"attributes":[{"DocComment":" Returns a pointer to the output at this location, 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https://doc.rust-lang.org/reference/behavior-considered-undefined.html"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":true,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11881}],"output":{"Deduplicated":11882}},"item":{"id":416,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"Deduplicated":11878}]}}},"kind":{"TraitMethod":[45,2]}},{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"get_unchecked_mut","attr_info":{"attributes":[{"DocComment":" Returns a mutable pointer to the output at this location, without"},{"DocComment":" performing any bounds checking."},{"DocComment":""},{"DocComment":" Calling this method with an out-of-bounds index or a dangling `slice` pointer"},{"DocComment":" is *[undefined behavior]* even if the resulting pointer is 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A value-to-value conversion that consumes the input value. The"},{"DocComment":" opposite of [`From`]."},{"DocComment":""},{"DocComment":" One should avoid implementing [`Into`] and implement [`From`] instead."},{"DocComment":" Implementing [`From`] automatically provides one with an implementation of [`Into`]"},{"DocComment":" thanks to the blanket implementation in the standard library."},{"DocComment":""},{"DocComment":" Prefer using [`Into`] over [`From`] when specifying trait bounds on a generic function"},{"DocComment":" to ensure that types that only implement [`Into`] can be used as well."},{"DocComment":""},{"DocComment":" **Note: This trait must not fail**. If the conversion can fail, use [`TryInto`]."},{"DocComment":""},{"DocComment":" # Generic Implementations"},{"DocComment":""},{"DocComment":" - [`From`]` for U` implies `Into for T`"},{"DocComment":" - [`Into`] is reflexive, which means that `Into for T` is implemented"},{"DocComment":""},{"DocComment":" # Implementing [`Into`] for conversions to external types in old versions of Rust"},{"DocComment":""},{"DocComment":" Prior to Rust 1.41, if the destination type was not part of the current crate"},{"DocComment":" then you couldn't implement [`From`] directly."},{"DocComment":" For example, take this code:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" # #![allow(non_local_definitions)]"},{"DocComment":" struct Wrapper(Vec);"},{"DocComment":" impl From> for Vec {"},{"DocComment":" fn from(w: Wrapper) -> Vec {"},{"DocComment":" w.0"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":" This will fail to compile in older versions of the language because Rust's orphaning rules"},{"DocComment":" used to be a little bit more strict. To bypass this, you could implement [`Into`] directly:"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" struct Wrapper(Vec);"},{"DocComment":" impl Into> for Wrapper {"},{"DocComment":" fn into(self) -> Vec {"},{"DocComment":" self.0"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" It is important to understand that [`Into`] does not provide a [`From`] implementation"},{"DocComment":" (as [`From`] does with [`Into`]). Therefore, you should always try to implement [`From`]"},{"DocComment":" and then fall back to [`Into`] if [`From`] can't be implemented."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" [`String`] implements [`Into`]`<`[`Vec`]`<`[`u8`]`>>`:"},{"DocComment":""},{"DocComment":" In order to express that we want a generic function to take all arguments that can be"},{"DocComment":" converted to a specified type `T`, we can use a trait bound of [`Into`]``."},{"DocComment":" For example: The function `is_hello` takes all arguments that can be converted into a"},{"DocComment":" [`Vec`]`<`[`u8`]`>`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" fn is_hello>>(s: T) {"},{"DocComment":" let bytes = b\"hello\".to_vec();"},{"DocComment":" assert_eq!(bytes, s.into());"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let s = \"hello\".to_string();"},{"DocComment":" is_hello(s);"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" [`String`]: ../../std/string/struct.String.html"},{"DocComment":" [`Vec`]: ../../std/vec/struct.Vec.html"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"Into"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"T"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"into","attr_info":{"attributes":[{"DocComment":" Converts this type into the (usually inferred) input type."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755}],"output":{"Deduplicated":11756}},"item":{"id":466,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11885,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":48,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[48,0]}}],"vtable":null},{"def_id":49,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["bit",0]},{"Ident":["BitOr",0]}],"span":{"data":{"file_id":84,"beg":{"line":254,"col":0},"end":{"line":254,"col":33}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The bitwise OR operator `|`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `BitOr` for a wrapper around `bool`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitOr;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Scalar(bool);"},{"DocComment":""},{"DocComment":" impl BitOr for Scalar {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" // rhs is the \"right-hand side\" of the expression `a | b`"},{"DocComment":" fn bitor(self, rhs: Self) -> Self::Output {"},{"DocComment":" Self(self.0 | rhs.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Scalar(true) | Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(true) | Scalar(false), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(false) | Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(false) | Scalar(false), Scalar(false));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" An implementation of `BitOr` for a wrapper around `Vec`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitOr;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct BooleanVector(Vec);"},{"DocComment":""},{"DocComment":" impl BitOr for BooleanVector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn bitor(self, Self(rhs): Self) -> Self::Output {"},{"DocComment":" let Self(lhs) = self;"},{"DocComment":" assert_eq!(lhs.len(), rhs.len());"},{"DocComment":" Self("},{"DocComment":" lhs.iter()"},{"DocComment":" .zip(rhs.iter())"},{"DocComment":" .map(|(x, y)| *x | *y)"},{"DocComment":" .collect()"},{"DocComment":" )"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let bv1 = BooleanVector(vec![true, true, false, false]);"},{"DocComment":" let bv2 = BooleanVector(vec![true, false, true, false]);"},{"DocComment":" let expected = BooleanVector(vec![true, true, true, false]);"},{"DocComment":" assert_eq!(bv1 | bv2, expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"bitor"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"bitor","attr_info":{"attributes":[{"DocComment":" Performs the `|` operation."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(true | false, true);"},{"DocComment":" assert_eq!(false | false, false);"},{"DocComment":" assert_eq!(5u8 | 1u8, 5);"},{"DocComment":" assert_eq!(5u8 | 2u8, 7);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":467,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11886,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":49,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[49,0]}}],"vtable":{"id":{"Adt":85},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":50,"item_meta":{"name":[{"Ident":["core",0]},{"Ident":["ops",0]},{"Ident":["bit",0]},{"Ident":["BitAnd",0]}],"span":{"data":{"file_id":84,"beg":{"line":150,"col":0},"end":{"line":150,"col":34}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[{"DocComment":" The bitwise AND operator `&`."},{"DocComment":""},{"DocComment":" Note that `Rhs` is `Self` by default, but this is not mandatory."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" An implementation of `BitAnd` for a wrapper around `bool`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitAnd;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct Scalar(bool);"},{"DocComment":""},{"DocComment":" impl BitAnd for Scalar {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" // rhs is the \"right-hand side\" of the expression `a & b`"},{"DocComment":" fn bitand(self, rhs: Self) -> Self::Output {"},{"DocComment":" Self(self.0 & rhs.0)"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" assert_eq!(Scalar(true) & Scalar(true), Scalar(true));"},{"DocComment":" assert_eq!(Scalar(true) & Scalar(false), Scalar(false));"},{"DocComment":" assert_eq!(Scalar(false) & Scalar(true), Scalar(false));"},{"DocComment":" assert_eq!(Scalar(false) & Scalar(false), Scalar(false));"},{"DocComment":" ```"},{"DocComment":""},{"DocComment":" An implementation of `BitAnd` for a wrapper around `Vec`."},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" use std::ops::BitAnd;"},{"DocComment":""},{"DocComment":" #[derive(Debug, PartialEq)]"},{"DocComment":" struct BooleanVector(Vec);"},{"DocComment":""},{"DocComment":" impl BitAnd for BooleanVector {"},{"DocComment":" type Output = Self;"},{"DocComment":""},{"DocComment":" fn bitand(self, Self(rhs): Self) -> Self::Output {"},{"DocComment":" let Self(lhs) = self;"},{"DocComment":" assert_eq!(lhs.len(), rhs.len());"},{"DocComment":" Self("},{"DocComment":" lhs.iter()"},{"DocComment":" .zip(rhs.iter())"},{"DocComment":" .map(|(x, y)| *x & *y)"},{"DocComment":" .collect()"},{"DocComment":" )"},{"DocComment":" }"},{"DocComment":" }"},{"DocComment":""},{"DocComment":" let bv1 = BooleanVector(vec![true, true, false, false]);"},{"DocComment":" let bv2 = BooleanVector(vec![true, false, true, false]);"},{"DocComment":" let expected = BooleanVector(vec![true, false, false, false]);"},{"DocComment":" assert_eq!(bv1 & bv2, expected);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"is_local":false,"opacity":"Foreign","lang_item":"bitand"},"generics":{"regions":[],"types":[{"index":0,"name":"Self"},{"index":1,"name":"Rhs"},{"index":2,"name":"Self_Output"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"bitand","attr_info":{"attributes":[{"DocComment":" Performs the `&` operation."},{"DocComment":""},{"DocComment":" # Examples"},{"DocComment":""},{"DocComment":" ```"},{"DocComment":" assert_eq!(true & false, false);"},{"DocComment":" assert_eq!(true & true, true);"},{"DocComment":" assert_eq!(5u8 & 1u8, 1);"},{"DocComment":" assert_eq!(5u8 & 2u8, 0);"},{"DocComment":" ```"}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11755},{"Deduplicated":11756}],"output":{"Deduplicated":11793}},"item":{"id":470,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11887,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":50,"generics":{"regions":[],"types":[{"Deduplicated":11755},{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[50,0]}}],"vtable":{"id":{"Adt":86},"generics":{"regions":[],"types":[{"Deduplicated":11756},{"Deduplicated":11793}],"const_generics":[],"trait_refs":[]}}},{"def_id":51,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["traits",0]},{"Ident":["IsIdentity",0]}],"span":{"data":{"file_id":38,"beg":{"line":39,"col":0},"end":{"line":42,"col":1}},"generated_from_span":null},"source_text":"pub trait IsIdentity {\n /// Return true if this element is the identity element of the curve.\n fn is_identity(&self) -> bool;\n}","attr_info":{"attributes":[{"DocComment":" Trait for testing if a curve point is equivalent to the identity point."}],"inline":null,"rename":null,"public":true},"is_local":true,"opacity":"Transparent","lang_item":null},"generics":{"regions":[],"types":[{"index":0,"name":"Self"}],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_clauses":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"name":"is_identity","attr_info":{"attributes":[{"DocComment":" Return true if this element is the identity element of the curve."}],"inline":null,"rename":null,"public":true},"signature":{"is_unsafe":false,"abi":"Rust","inputs":[{"Deduplicated":11825}],"output":{"Deduplicated":1796}},"item":{"id":486,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[{"HashConsedValue":[11888,{"kind":"SelfId","trait_decl_ref":{"regions":[],"skip_binder":{"id":51,"generics":{"regions":[],"types":[{"Deduplicated":11755}],"const_generics":[],"trait_refs":[]}}}}]}]}}},"kind":{"TraitMethod":[51,0]}}],"vtable":{"id":{"Adt":88},"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}}],"trait_impls":[{"def_id":0,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":0}}],"span":{"data":{"file_id":8,"beg":{"line":52,"col":0},"end":{"line":52,"col":27}},"generated_from_span":null},"source_text":"impl Eq for FieldElement {}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":3,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[{"HashConsedValue":[11758,{"kind":{"TraitImpl":{"id":1,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":3140},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}}}}]}],"consts":[],"types":[],"methods":[null,{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":500,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[3,1]}}],"vtable":null},{"def_id":1,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":1}}],"span":{"data":{"file_id":8,"beg":{"line":54,"col":0},"end":{"line":58,"col":1}},"generated_from_span":null},"source_text":"impl PartialEq for FieldElement {\n fn eq(&self, other: &FieldElement) -> bool {\n self.ct_eq(other).into()\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":4,"generics":{"regions":[],"types":[{"Deduplicated":3140},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":42,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[4,0]}},null],"vtable":{"id":9,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":2,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":2}}],"span":{"data":{"file_id":8,"beg":{"line":60,"col":0},"end":{"line":67,"col":1}},"generated_from_span":null},"source_text":"impl ConstantTimeEq for FieldElement {\n /// Test equality between two `FieldElement`s. Since the\n /// internal representation is not canonical, the field elements\n /// are normalized to wire format before comparison.\n fn ct_eq(&self, other: &FieldElement) -> Choice {\n self.to_bytes().ct_eq(&other.to_bytes())\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":5,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":45,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[5,0]}},null],"vtable":null},{"def_id":3,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["field",0]},{"Impl":{"Trait":3}}],"span":{"data":{"file_id":8,"beg":{"line":69,"col":0},"end":{"line":73,"col":1}},"generated_from_span":null},"source_text":"impl Default for FieldElement {\n fn default() -> Self {\n FieldElement::ZERO\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":6,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":48,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[6,0]}}],"vtable":null},{"def_id":4,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":4}}],"span":{"data":{"file_id":9,"beg":{"line":42,"col":9},"end":{"line":42,"col":13}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":7,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[{"HashConsedValue":[11759,{"kind":{"TraitImpl":{"id":6,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},"trait_decl_ref":{"regions":[],"skip_binder":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}}}}]}],"consts":[],"types":[],"methods":[],"vtable":null},null,{"def_id":6,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":6}}],"span":{"data":{"file_id":9,"beg":{"line":42,"col":15},"end":{"line":42,"col":20}},"generated_from_span":null},"source_text":null,"attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":8,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":57,"generics":{"regions":[{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[8,0]}},null],"vtable":null},{"def_id":7,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":7}}],"span":{"data":{"file_id":9,"beg":{"line":45,"col":0},"end":{"line":49,"col":1}},"generated_from_span":null},"source_text":"impl Debug for FieldElement51 {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(f, \"FieldElement51({:?})\", &self.0[..])\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":11,"generics":{"regions":[],"types":[{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"},{"index":2,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":60,"generics":{"regions":[{"Var":{"Bound":[0,0]}},{"Var":{"Bound":[0,1]}},{"Var":{"Bound":[0,2]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[11,0]}}],"vtable":{"id":10,"generics":{"regions":[],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":8,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":8}}],"span":{"data":{"file_id":9,"beg":{"line":54,"col":0},"end":{"line":60,"col":1}},"generated_from_span":null},"source_text":"impl<'a> AddAssign<&'a FieldElement51> for FieldElement51 {\n fn add_assign(&mut self, _rhs: &'a FieldElement51) {\n for i in 0..5 {\n self.0[i] += _rhs.0[i];\n }\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":12,"generics":{"regions":[],"types":[{"Deduplicated":3140},{"Deduplicated":11760}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":61,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[12,0]}}],"vtable":{"id":11,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":9,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":9}}],"span":{"data":{"file_id":9,"beg":{"line":62,"col":0},"end":{"line":69,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Add<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn add(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n let mut output = *self;\n output += _rhs;\n output\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":13,"generics":{"regions":[],"types":[{"Deduplicated":11761},{"Deduplicated":11760},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":62,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[13,0]}}],"vtable":{"id":12,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":10,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":10}}],"span":{"data":{"file_id":9,"beg":{"line":71,"col":0},"end":{"line":76,"col":1}},"generated_from_span":null},"source_text":"impl<'a> SubAssign<&'a FieldElement51> for FieldElement51 {\n fn sub_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) - _rhs;\n self.0 = result.0;\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":14,"generics":{"regions":[],"types":[{"Deduplicated":3140},{"Deduplicated":11760}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":63,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[14,0]}}],"vtable":{"id":13,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":11,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":11}}],"span":{"data":{"file_id":9,"beg":{"line":78,"col":0},"end":{"line":98,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Sub<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n fn sub(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n // To avoid underflow, first add a multiple of p.\n // Choose 16*p = p << 4 to be larger than 54-bit _rhs.\n //\n // If we could statically track the bitlengths of the limbs\n // of every FieldElement51, we could choose a multiple of p\n // just bigger than _rhs and avoid having to do a reduction.\n //\n // Since we don't yet have type-level integers to do this, we\n // have to add an explicit reduction call here.\n FieldElement51::reduce([\n (self.0[0] + 36028797018963664u64) - _rhs.0[0],\n (self.0[1] + 36028797018963952u64) - _rhs.0[1],\n (self.0[2] + 36028797018963952u64) - _rhs.0[2],\n (self.0[3] + 36028797018963952u64) - _rhs.0[3],\n (self.0[4] + 36028797018963952u64) - _rhs.0[4],\n ])\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":15,"generics":{"regions":[],"types":[{"Deduplicated":11761},{"Deduplicated":11760},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":64,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[15,0]}}],"vtable":{"id":14,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":12,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":12}}],"span":{"data":{"file_id":9,"beg":{"line":100,"col":0},"end":{"line":105,"col":1}},"generated_from_span":null},"source_text":"impl<'a> MulAssign<&'a FieldElement51> for FieldElement51 {\n fn mul_assign(&mut self, _rhs: &'a FieldElement51) {\n let result = (self as &FieldElement51) * _rhs;\n self.0 = result.0;\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":16,"generics":{"regions":[],"types":[{"Deduplicated":3140},{"Deduplicated":11760}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[],"methods":[{"params":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":65,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Bound":[0,0]}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[16,0]}}],"vtable":{"id":15,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":13,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":13}}],"span":{"data":{"file_id":9,"beg":{"line":107,"col":0},"end":{"line":210,"col":1}},"generated_from_span":null},"source_text":"impl<'a> Mul<&'a FieldElement51> for &FieldElement51 {\n type Output = FieldElement51;\n\n #[rustfmt::skip] // keep alignment of c* calculations\n fn mul(self, _rhs: &'a FieldElement51) -> FieldElement51 {\n /// Helper function to multiply two 64-bit integers with 128\n /// bits of output.\n #[inline(always)]\n fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }\n\n // Alias self, _rhs for more readable formulas\n let a: &[u64; 5] = &self.0;\n let b: &[u64; 5] = &_rhs.0;\n\n // Precondition: assume input limbs a[i], b[i] are bounded as\n //\n // a[i], b[i] < 2^(51 + b)\n //\n // where b is a real parameter measuring the \"bit excess\" of the limbs.\n\n // 64-bit precomputations to avoid 128-bit multiplications.\n //\n // This fits into a u64 whenever 51 + b + lg(19) < 64.\n //\n // Since 51 + b + lg(19) < 51 + 4.25 + b\n // = 55.25 + b,\n // this fits if b < 8.75.\n let b1_19 = b[1] * 19;\n let b2_19 = b[2] * 19;\n let b3_19 = b[3] * 19;\n let b4_19 = b[4] * 19;\n\n // Multiply to get 128-bit coefficients of output\n let c0: u128 = m(a[0], b[0]) + m(a[4], b1_19) + m(a[3], b2_19) + m(a[2], b3_19) + m(a[1], b4_19);\n let mut c1: u128 = m(a[1], b[0]) + m(a[0], b[1]) + m(a[4], b2_19) + m(a[3], b3_19) + m(a[2], b4_19);\n let mut c2: u128 = m(a[2], b[0]) + m(a[1], b[1]) + m(a[0], b[2]) + m(a[4], b3_19) + m(a[3], b4_19);\n let mut c3: u128 = m(a[3], b[0]) + m(a[2], b[1]) + m(a[1], b[2]) + m(a[0], b[3]) + m(a[4], b4_19);\n let mut c4: u128 = m(a[4], b[0]) + m(a[3], b[1]) + m(a[2], b[2]) + m(a[1], b[3]) + m(a[0] , b[4]);\n\n // How big are the c[i]? We have\n //\n // c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)\n // < 2^(102 + lg(1 + 4*19) + 2*b)\n // < 2^(108.27 + 2*b)\n //\n // The carry (c[i] >> 51) fits into a u64 when\n // 108.27 + 2*b - 51 < 64\n // 2*b < 6.73\n // b < 3.365.\n //\n // So we require b < 3 to ensure this fits.\n debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));\n debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));\n debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));\n debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));\n debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));\n\n // Casting to u64 and back tells the compiler that the carry is\n // bounded by 2^64, so that the addition is a u128 + u64 rather\n // than u128 + u128.\n\n const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;\n let mut out = [0u64; 5];\n\n c1 += ((c0 >> 51) as u64) as u128;\n out[0] = (c0 as u64) & LOW_51_BIT_MASK;\n\n c2 += ((c1 >> 51) as u64) as u128;\n out[1] = (c1 as u64) & LOW_51_BIT_MASK;\n\n c3 += ((c2 >> 51) as u64) as u128;\n out[2] = (c2 as u64) & LOW_51_BIT_MASK;\n\n c4 += ((c3 >> 51) as u64) as u128;\n out[3] = (c3 as u64) & LOW_51_BIT_MASK;\n\n let carry: u64 = (c4 >> 51) as u64;\n out[4] = (c4 as u64) & LOW_51_BIT_MASK;\n\n // To see that this does not overflow, we need out[0] + carry * 19 < 2^64.\n //\n // c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)\n // < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)\n // < 2^(102 + 2*b + lg(5)) + 2^64.\n //\n // When b < 3 we get\n //\n // c4 < 2^110.33 so that carry < 2^59.33\n //\n // so that\n //\n // out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58\n //\n // and there is no overflow.\n out[0] += carry * 19;\n\n // Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).\n out[1] += out[0] >> 51;\n out[0] &= LOW_51_BIT_MASK;\n\n // Now out[i] < 2^(51 + epsilon) for all i.\n FieldElement51(out)\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":17,"generics":{"regions":[],"types":[{"Deduplicated":11761},{"Deduplicated":11760},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":"'a","mutability":"Unknown"},{"index":1,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":66,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[17,0]}}],"vtable":{"id":16,"generics":{"regions":[{"Var":{"Free":0}},{"Var":{"Free":1}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":14,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":14}}],"span":{"data":{"file_id":9,"beg":{"line":212,"col":0},"end":{"line":219,"col":1}},"generated_from_span":null},"source_text":"impl Neg for &FieldElement51 {\n type Output = FieldElement51;\n fn neg(self) -> FieldElement51 {\n let mut output = *self;\n output.negate();\n output\n }\n}","attr_info":{"attributes":[],"inline":null,"rename":null,"public":false},"is_local":true,"opacity":"Transparent","lang_item":null},"impl_trait":{"id":18,"generics":{"regions":[],"types":[{"Deduplicated":11760},{"Deduplicated":3140}],"const_generics":[],"trait_refs":[]}},"generics":{"regions":[{"index":0,"name":null,"mutability":"Unknown"}],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"implied_trait_refs":[],"consts":[],"types":[null],"methods":[{"params":{"regions":[],"types":[],"const_generics":[],"trait_clauses":[],"regions_outlive":[],"types_outlive":[],"trait_type_constraints":[]},"skip_binder":{"id":67,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}},"kind":{"TraitMethod":[18,0]}}],"vtable":{"id":17,"generics":{"regions":[{"Var":{"Free":0}}],"types":[],"const_generics":[],"trait_refs":[]}}},{"def_id":15,"item_meta":{"name":[{"Ident":["curve25519",0]},{"Ident":["backend",0]},{"Ident":["serial",0]},{"Ident":["u64",0]},{"Ident":["field",0]},{"Impl":{"Trait":15}}],"span":{"data":{"file_id":9,"beg":{"line":221,"col":0},"end":{"line":251,"col":1}},"generated_from_span":null},"source_text":"impl ConditionallySelectable for FieldElement51 {\n fn conditional_select(\n a: &FieldElement51,\n b: &FieldElement51,\n choice: Choice,\n ) -> FieldElement51 {\n FieldElement51([\n u64::conditional_select(&a.0[0], &b.0[0], choice),\n u64::conditional_select(&a.0[1], &b.0[1], choice),\n u64::conditional_select(&a.0[2], &b.0[2], choice),\n u64::conditional_select(&a.0[3], &b.0[3], choice),\n u64::conditional_select(&a.0[4], &b.0[4], choice),\n ])\n }\n\n fn conditional_swap(a: &mut FieldElement51, b: &mut FieldElement51, choice: Choice) {\n u64::conditional_swap(&mut a.0[0], &mut b.0[0], choice);\n u64::conditional_swap(&mut a.0[1], &mut b.0[1], choice);\n u64::conditional_swap(&mut a.0[2], &mut b.0[2], choice);\n u64::conditional_swap(&mut a.0[3], &mut b.0[3], choice);\n u64::conditional_swap(&mut a.0[4], &mut b.0[4], choice);\n }\n\n fn conditional_assign(&mut self, other: &FieldElement51, choice: Choice) {\n self.0[0].conditional_assign(&other.0[0], choice);\n self.0[1].conditional_assign(&other.0[1], choice);\n self.0[2].conditional_assign(&other.0[2], choice);\n self.0[3].conditional_assign(&other.0[3], choice);\n self.0[4].conditional_assign(&other.0[4], choice);\n 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ConditionallySelectable for ProjectiveNielsPoint {\n fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {\n ProjectiveNielsPoint {\n Y_plus_X: FieldElement::conditional_select(&a.Y_plus_X, &b.Y_plus_X, choice),\n Y_minus_X: FieldElement::conditional_select(&a.Y_minus_X, &b.Y_minus_X, choice),\n Z: FieldElement::conditional_select(&a.Z, &b.Z, choice),\n T2d: FieldElement::conditional_select(&a.T2d, &b.T2d, choice),\n }\n }\n\n fn conditional_assign(&mut self, other: &Self, choice: Choice) {\n self.Y_plus_X.conditional_assign(&other.Y_plus_X, choice);\n self.Y_minus_X.conditional_assign(&other.Y_minus_X, choice);\n self.Z.conditional_assign(&other.Z, choice);\n self.T2d.conditional_assign(&other.T2d, choice);\n 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Add<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.Y_plus_X;\n let MM = &Y_minus_X * &other.Y_minus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &ZZ2 + &TT2d,\n T: &ZZ2 - &TT2d,\n }\n 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Sub<&'a ProjectiveNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a ProjectiveNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.Y_minus_X;\n let MP = &Y_minus_X * &other.Y_plus_X;\n let TT2d = &self.T * &other.T2d;\n let ZZ = &self.Z * &other.Z;\n let ZZ2 = &ZZ + &ZZ;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &ZZ2 - &TT2d,\n T: &ZZ2 + &TT2d,\n }\n 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Add<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn add(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PP = &Y_plus_X * &other.y_plus_x;\n let MM = &Y_minus_X * &other.y_minus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PP - &MM,\n Y: &PP + &MM,\n Z: &Z2 + &Txy2d,\n T: &Z2 - &Txy2d,\n }\n 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Sub<&'a AffineNielsPoint> for &EdwardsPoint {\n type Output = CompletedPoint;\n\n fn sub(self, other: &'a AffineNielsPoint) -> CompletedPoint {\n let Y_plus_X = &self.Y + &self.X;\n let Y_minus_X = &self.Y - &self.X;\n let PM = &Y_plus_X * &other.y_minus_x;\n let MP = &Y_minus_X * &other.y_plus_x;\n let Txy2d = &self.T * &other.xy2d;\n let Z2 = &self.Z + &self.Z;\n\n CompletedPoint {\n X: &PM - &MP,\n Y: &PM + &MP,\n Z: &Z2 - &Txy2d,\n T: &Z2 + &Txy2d,\n }\n 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Neg for &ProjectiveNielsPoint {\n type Output = ProjectiveNielsPoint;\n\n fn neg(self) -> ProjectiveNielsPoint {\n ProjectiveNielsPoint {\n Y_plus_X: self.Y_minus_X,\n Y_minus_X: self.Y_plus_X,\n Z: self.Z,\n T2d: -(&self.T2d),\n }\n 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Neg for &AffineNielsPoint {\n type Output = AffineNielsPoint;\n\n fn neg(self) -> AffineNielsPoint {\n AffineNielsPoint {\n y_plus_x: self.y_minus_x,\n y_minus_x: self.y_plus_x,\n xy2d: -(&self.xy2d),\n }\n 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Debug for ProjectivePoint {\n fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {\n write!(\n f,\n \"ProjectivePoint{{\\n\\tX: {:?},\\n\\tY: {:?},\\n\\tZ: {:?}\\n}}\",\n &self.X, &self.Y, &self.Z\n )\n 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\ No newline at end of file diff --git a/verification/Proofs/DecompressMain.lean b/verification/Proofs/DecompressMain.lean new file mode 100644 index 0000000..bbeec41 --- /dev/null +++ b/verification/Proofs/DecompressMain.lean @@ -0,0 +1,323 @@ +/- ────────────────────────────────────────────────────────────────────────────── + Proofs/DecompressMain.lean — phase 2, decompress step 3: THE CONSTRUCTIVE + DECOMPRESSION THEOREM. + + `decompress_of_canonical`: for a valid on-curve point Q whose canonical + encoding is the bytes rb, the extracted `CompressedEdwardsY::decompress` + succeeds with `some P` — a valid on-curve point denoting exactly Q. + + The chain: from_bytes recovers the y-residue (the sign bit at 2²⁵⁵ is + discarded — from_bytes_spec is exact below it); u = y²−1 and + v = d·y²+1 are built by certified ops with ⟪EDWARDS_D⟫ = d + (edwards_d_spec + edD_char, cancelled by 121666 ≠ 0); Q's own + x-coordinate witnesses that u/v is a square (x_sq_of_onCurve), so + sqrt_ratio_i succeeds with the even-parity root; the sign bit — Q's + x-parity, extracted from byte 31 — selects between ±root, and the + parity-injectivity argument (enc_inj_coord, on-curve invariance of x²) + pins the selected root to edX Q. The result point {X, Y, 1, X·Y} is + ExtValid, on-curve, and denotes Q. + ────────────────────────────────────────────────────────────────────────────── -/ +import Proofs.FromBytesSpec +open Aeneas Aeneas.Std Result +open curve25519 + +set_option maxHeartbeats 8000000 +set_option linter.unusedSimpArgs false +set_option maxRecDepth 8000 + +namespace CurveFieldProofs + +open Aeneas.Std.WP + +/-- ⟪EDWARDS_D⟫ is THE curve constant d. -/ +theorem edwards_d_denote : + backend.serial.u64.constants.EDWARDS_D ⦃ D => Bnd D (2^52) ∧ ⟪D⟫ = edD ⦄ := by + apply spec_mono edwards_d_spec + intro D ⟨hb, hd⟩ + refine ⟨hb, ?_⟩ + have h121666 : (121666 : Fp) ≠ 0 := by + have h : ((121666 : ℕ) : Fp) ≠ 0 := natCast_ne_zero_of_mod (by decide) + exact_mod_cast h + have hchar := edD_char + have : (121666 : Fp) * (⟪D⟫ - edD) = 0 := by linear_combination hd - hchar + rcases mul_eq_zero.mp this with h | h + · exact absurd h h121666 + · linear_combination h + +open ed_sigs ed_sigs.verification_key in +/-- **THE CONSTRUCTIVE DECOMPRESSION THEOREM**: canonical encodings of + valid on-curve points decompress to them. -/ +theorem decompress_of_canonical (Q : EdPoint) (hQv : ExtValid Q) (hQc : OnCurveExt Q) + (rb : Std.Array Std.U8 32#usize) + (b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 : Std.U8) + (hbl : (↑rb : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31]) + (henc : bytesVal rb = (edY Q).val + ((edX Q).val % 2) * 2^255) : + edwards.CompressedEdwardsY.decompress rb ⦃ o => + ∃ Pt : EdPoint, o = some Pt ∧ ExtValid Pt ∧ OnCurveExt Pt ∧ edPt Pt = edPt Q ⦄ := by + haveI : NeZero P := ⟨by unfold P; norm_num⟩ + unfold edwards.CompressedEdwardsY.decompress curve25519.edwards.decompress.step_1 + -- as_bytes is the identity; parse y + simp only [edwards.CompressedEdwardsY.as_bytes, bind_tc_ok] + step with (from_bytes_spec rb b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 + b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 hbl) + as ⟨Y, hbY, hYv⟩ + -- the parsed field element denotes edY Q + have hyresid : (edY Q).val < 2^255 := + lt_of_lt_of_le (ZMod.val_lt _) (by unfold P; norm_num) + have hparle : (edX Q).val % 2 ≤ 1 := Nat.le_of_lt_succ (Nat.mod_lt _ (by norm_num)) + have hYval : feVal Y = (edY Q).val := by + rw [hYv, henc] + have : ((edY Q).val + (edX Q).val % 2 * 2^255) % 2^255 = (edY Q).val := by + rcases Nat.mod_two_eq_zero_or_one (edX Q).val with h | h <;> rw [h] <;> omega + exact this + have hYden : ⟪Y⟫ = edY Q := by + apply ZMod.val_injective + show (⟪Y⟫).val = (edY Q).val + unfold denote + rw [ZMod.val_natCast, hYval, Nat.mod_eq_of_lt (ZMod.val_lt _)] + -- Z = 1 + step with one_spec as ⟨Z, hbZ, hZv⟩ + -- YY = y², u = y² − 1 + step with (square_spec' Y (Bnd.mono hbY (by norm_num))) as ⟨YY, hbYY, hYY⟩ + obtain ⟨yy0, yy1, yy2, yy3, yy4, hyyl⟩ := Fe.exists_limbs YY + obtain ⟨z0, z1, z2, z3, z4, hzl⟩ := Fe.exists_limbs Z + step with (sub_spec YY Z yy0 yy1 yy2 yy3 yy4 z0 z1 z2 z3 z4 hyyl hzl + (Bnd.mono hbYY (by norm_num)) (Bnd.mono hbZ (by norm_num))) as ⟨u, hbu, huv⟩ + -- D, then v = d·y² + 1 + step with edwards_d_denote as ⟨D, hbD, hDv⟩ + step with (mul_spec' YY D (Bnd.mono hbYY (by norm_num)) (Bnd.mono hbD (by norm_num))) + as ⟨vd, hbvd, hvd⟩ + step with (add_spec'' vd Z (Bnd.mono hbvd (by norm_num)) (Bnd.mono hbZ (by norm_num))) + as ⟨v, hbv, hvv⟩ + -- interpreted u, v + have huval : ⟪u⟫ = (edY Q)^2 - 1 := by + rw [huv, hYY, hYden, hZv] + ring + have hvval : ⟪v⟫ = 1 + edD * (edY Q)^2 := by + rw [hvv, hvd, hYY, hYden, hDv, hZv] + ring + -- Q's x witnesses the square; v never vanishes + have hvne : ⟪v⟫ ≠ 0 := by rw [hvval]; exact one_add_d_y_sq_ne_zero _ + have hwit : (edX Q) ^ 2 * ⟪v⟫ = ⟪u⟫ := by + rw [hvval, huval] + exact x_sq_of_onCurve hQc + -- the square root succeeds with the even-parity root + step with (sqrt_ratio_i_sq_spec u v (Bnd.mono hbu (by norm_num)) + (Bnd.mono hbv (by norm_num)) hvne (edX Q) hwit) as ⟨sc, sr, hc1, hbr, hrsq, hrpar⟩ + -- the validity Choice converts to true + simp only [core.convert.IntoFrom.into, Bool.Insts.CoreConvertFromChoice.from, + bind_tc_ok] + rw [show (sc.val != 0) = true from by simp [hc1]] + rw [if_pos rfl] + -- ── step_2: the sign select ────────────────────────────────────────────── + unfold curve25519.edwards.decompress.step_2 + simp only [edwards.CompressedEdwardsY.as_bytes, bind_tc_ok] + step as ⟨t31, ht31⟩ + simp [hbl] at ht31 + step as ⟨sgn, hsgn⟩ + simp only [subtle.Choice.Insts.CoreConvertFromU8.from, bind_tc_ok] + -- the sign bit is Q's x-parity + have hb31v : b31.val = (edY Q).val / 2^248 + ((edX Q).val % 2) * 2^7 := by + have hexp : bytesVal rb = b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248 := by + simp only [bytesVal, hbl] + rw [henc] at hexp + have hB0 : b0.val < 256 := by scalar_tac + have hB1 : b1.val < 256 := by scalar_tac + have hB2 : b2.val < 256 := by scalar_tac + have hB3 : b3.val < 256 := by scalar_tac + have hB4 : b4.val < 256 := by scalar_tac + have hB5 : b5.val < 256 := by scalar_tac + have hB6 : b6.val < 256 := by scalar_tac + have hB7 : b7.val < 256 := by scalar_tac + have hB8 : b8.val < 256 := by scalar_tac + have hB9 : b9.val < 256 := by scalar_tac + have hB10 : b10.val < 256 := by scalar_tac + have hB11 : b11.val < 256 := by scalar_tac + have hB12 : b12.val < 256 := by scalar_tac + have hB13 : b13.val < 256 := by scalar_tac + have hB14 : b14.val < 256 := by scalar_tac + have hB15 : b15.val < 256 := by scalar_tac + have hB16 : b16.val < 256 := by scalar_tac + have hB17 : b17.val < 256 := by scalar_tac + have hB18 : b18.val < 256 := by scalar_tac + have hB19 : b19.val < 256 := by scalar_tac + have hB20 : b20.val < 256 := by scalar_tac + have hB21 : b21.val < 256 := by scalar_tac + have hB22 : b22.val < 256 := by scalar_tac + have hB23 : b23.val < 256 := by scalar_tac + have hB24 : b24.val < 256 := by scalar_tac + have hB25 : b25.val < 256 := by scalar_tac + have hB26 : b26.val < 256 := by scalar_tac + have hB27 : b27.val < 256 := by scalar_tac + have hB28 : b28.val < 256 := by scalar_tac + have hB29 : b29.val < 256 := by scalar_tac + have hB30 : b30.val < 256 := by scalar_tac + have hB31 : b31.val < 256 := by scalar_tac + omega + have hsgnv : sgn.val = (edX Q).val % 2 := by + rw [hsgn, ht31, nat_shr, hb31v] + have : (edY Q).val / 2^248 < 2^7 := by + have := hyresid + omega + omega + -- −root, then select + obtain ⟨r0, r1, r2, r3, r4, hrl⟩ := Fe.exists_limbs sr + unfold Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg + step with (neg_spec sr r0 r1 r2 r3 r4 hrl (Bnd.mono hbr (by norm_num))) + as ⟨Xn, hbXn, hXnv⟩ + obtain ⟨n0, n1, n2, n3, n4, hnl⟩ := Fe.exists_limbs Xn + step with (fe_cond_assign_spec sr Xn sgn r0 r1 r2 r3 r4 n0 n1 n2 n3 n4 hrl hnl) + as ⟨X1, hX1l⟩ + -- Bnd X1 first (needed by the T-multiply) + have hbX1 : Bnd X1 (2^52) := by + have hb1 : Bnd sr (2^52) := hbr + have hb2 : Bnd Xn (2^52) := hbXn + split at hX1l + · rw [Bnd_eq X1 r0 r1 r2 r3 r4 _ (by rw [hX1l])] + rw [Bnd_eq sr r0 r1 r2 r3 r4 _ hrl] at hb1 + exact hb1 + · rw [Bnd_eq X1 n0 n1 n2 n3 n4 _ (by rw [hX1l])] + rw [Bnd_eq Xn n0 n1 n2 n3 n4 _ hnl] at hb2 + exact hb2 + -- T = X1·Y + step with (mul_spec' X1 Y (Bnd.mono hbX1 (by norm_num)) (Bnd.mono hbY (by norm_num))) + as ⟨T, hbT, hTv⟩ + try simp only [spec_ok] + -- ── the selected root IS edX Q ─────────────────────────────────────────── + -- the root is on-curve (only x² appears in the equation) + have hrsq' : ⟪sr⟫ ^ 2 = (edX Q) ^ 2 := by + have h := hrsq + rw [hwit.symm] at h + have hcancel : (⟪sr⟫ ^ 2 - (edX Q) ^ 2) * ⟪v⟫ = 0 := by linear_combination h + rcases mul_eq_zero.mp hcancel with h' | h' + · linear_combination h' + · exact absurd h' hvne + have hronc : OnCurve ⟪sr⟫ (edY Q) := by + show -(⟪sr⟫^2) + (edY Q)^2 = 1 + edD * ⟪sr⟫^2 * (edY Q)^2 + rw [hrsq'] + exact hQc + have hX1den : ⟪X1⟫ = ⟪sr⟫ ∨ ⟪X1⟫ = -⟪sr⟫ := by + split at hX1l + · left + unfold denote + rw [feVal_eq X1 r0 r1 r2 r3 r4 (by rw [hX1l]), + feVal_eq sr r0 r1 r2 r3 r4 hrl] + · right + have : ⟪X1⟫ = ⟪Xn⟫ := by + unfold denote + rw [feVal_eq X1 n0 n1 n2 n3 n4 (by rw [hX1l]), + feVal_eq Xn n0 n1 n2 n3 n4 hnl] + rw [this, hXnv] + have hX1x : ⟪X1⟫ = edX Q := by + rcases Nat.mod_two_eq_zero_or_one (edX Q).val with hx | hx + · -- x has even parity: no flip (sgn = 0), root already matches by parity + have hs0 : sgn.val = 0 := by rw [hsgnv, hx] + have hkeep : ⟪X1⟫ = ⟪sr⟫ := by + split at hX1l + · unfold denote + rw [feVal_eq X1 r0 r1 r2 r3 r4 (by rw [hX1l]), + feVal_eq sr r0 r1 r2 r3 r4 hrl] + · exact absurd hs0 (by assumption) + rw [hkeep] + exact enc_inj_coord hronc hQc (by rw [hrpar, hx]) + · -- x odd: the flip fires; −root has odd parity (root even, nonzero) + have hs1 : sgn.val ≠ 0 := by rw [hsgnv, hx]; norm_num + have hflip : ⟪X1⟫ = -⟪sr⟫ := by + split at hX1l + · exact absurd (by assumption) hs1 + · have : ⟪X1⟫ = ⟪Xn⟫ := by + unfold denote + rw [feVal_eq X1 n0 n1 n2 n3 n4 (by rw [hX1l]), + feVal_eq Xn n0 n1 n2 n3 n4 hnl] + rw [this, hXnv] + have hrnz : ⟪sr⟫ ≠ 0 := by + intro hz + rw [hz] at hrsq' + have hxz : edX Q = 0 := by + have := hrsq'.symm + have h2 : (edX Q)^2 = 0 := by linear_combination -hrsq' + exact pow_eq_zero_iff (n := 2) (by norm_num) |>.mp h2 + rw [hxz] at hx + simp at hx + have hnegonc : OnCurve (-⟪sr⟫) (edY Q) := by + show -((-⟪sr⟫)^2) + (edY Q)^2 = 1 + edD * (-⟪sr⟫)^2 * (edY Q)^2 + have : (-⟪sr⟫)^2 = ⟪sr⟫^2 := by ring + rw [this, hrsq'] + exact hQc + have hnegpar : (-⟪sr⟫).val % 2 = 1 := by + rw [ZMod.neg_val, if_neg hrnz] + have hlt := ZMod.val_lt ⟪sr⟫ + have hpodd : P % 2 = 1 := by unfold P; norm_num + have hpos : 0 < (⟪sr⟫).val := by + rcases Nat.eq_zero_or_pos (⟪sr⟫).val with h | h + · exact absurd ((ZMod.val_eq_zero _).mp h) hrnz + · exact h + omega + rw [hflip] + exact enc_inj_coord hnegonc hQc (by rw [hnegpar, hx]) + -- ── assemble the point ─────────────────────────────────────────────────── + refine ⟨_, rfl, ?_, ?_, ?_⟩ + · -- ExtValid + refine ⟨?_, Bnd.mono hbY (by norm_num), Bnd.mono hbZ (by norm_num), ?_, ?_, ?_⟩ + · exact hbX1 + · exact Bnd.mono hbT (by norm_num) + · rw [hZv]; norm_num + · -- coherence X·Y = Z·T + show ⟪X1⟫ * ⟪Y⟫ = ⟪Z⟫ * ⟪T⟫ + rw [hTv, hZv] + ring + · -- on-curve + show OnCurve (⟪X1⟫ / ⟪Z⟫) (⟪Y⟫ / ⟪Z⟫) + rw [hZv, div_one, div_one, hX1x, hYden] + exact hQc + · -- denotes Q + show (⟪X1⟫ / ⟪Z⟫, ⟪Y⟫ / ⟪Z⟫) = edPt Q + rw [hZv, div_one, div_one, hX1x, hYden] + rfl + +open ed_sigs ed_sigs.verification_key in +/-- **THE FULL POINT-LEVEL LIFT** (anza). Under the point-equation premises, + the signature's R bytes DECOMPRESS to a valid on-curve point Pt, and the + verifier accepts **iff** Pt equals the recomputed [k]·minus_A + [s]·B: + + accept ⇔ decompress(R) = [k]·minus_A + [s]·B (as points). + + This is the constructive capstone of phase 2: byte comparison ↔ + canonical-encoding equality ↔ point equality ↔ decompressed-point + equality, every link machine-checked over the extracted code. -/ +theorem verify_accepts_iff_decompress + (self : verification_key.VerificationKey) + (sig : ed25519.Signature) (msg : Slice Std.U8) + (rb sb : Std.Array Std.U8 32#usize) (s : scalar.Scalar) + (er : edwards.CompressedEdwardsY) (e : Std.Array Std.U8 32#usize) + (hrb : ed25519.Signature.r_bytes sig = ok rb) + (hsb : ed25519.Signature.s_bytes sig = ok sb) + (hA : VerificationKey.a_bytes_nonzero self = ok true) + (hleg : is_legacy_excluded_r rb = ok false) + (hs : check_scalar_canonical sb = ok (core.result.Result.Ok s)) + (hrec : VerificationKey.recompute_r_sha512 self rb s msg = ok er) + (he : edwards.CompressedEdwardsY.as_bytes er = ok e) + (hkv : ExtValid self.minus_A) (hkc : OnCurveExt self.minus_A) + (t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21 t22 t23 t24 t25 t26 t27 t28 t29 t30 t31 : Std.U8) + (hsbytes : (↑s.bytes : List Std.U8) = [t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16, t17, t18, t19, t20, t21, t22, t23, t24, t25, t26, t27, t28, t29, t30, t31]) + (Vs : ℕ) (hVs : Vs = t0.val + t1.val * 2^8 + t2.val * 2^16 + t3.val * 2^24 + t4.val * 2^32 + t5.val * 2^40 + t6.val * 2^48 + t7.val * 2^56 + t8.val * 2^64 + t9.val * 2^72 + t10.val * 2^80 + t11.val * 2^88 + t12.val * 2^96 + t13.val * 2^104 + t14.val * 2^112 + t15.val * 2^120 + t16.val * 2^128 + t17.val * 2^136 + t18.val * 2^144 + t19.val * 2^152 + t20.val * 2^160 + t21.val * 2^168 + t22.val * 2^176 + t23.val * 2^184 + t24.val * 2^192 + t25.val * 2^200 + t26.val * 2^208 + t27.val * 2^216 + t28.val * 2^224 + t29.val * 2^232 + t30.val * 2^240 + t31.val * 2^248) + (hVslt : Vs < 2^253) + (Q : EdPoint) (hQv : ExtValid Q) (hQc : OnCurveExt Q) + (b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 : Std.U8) + (hbl : (↑rb : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31]) + (henc : bytesVal rb = (edY Q).val + ((edX Q).val % 2) * 2^255) : + ∃ (R' Pt : EdPoint), ExtValid R' ∧ OnCurveExt R' ∧ + curve25519.edwards.CompressedEdwardsY.decompress rb = ok (some Pt) ∧ + ExtValid Pt ∧ OnCurveExt Pt ∧ + (VerificationKey.verify_sha512 self sig msg = ok (core.result.Result.Ok ()) + ↔ edPt Pt = edPt R') := by + obtain ⟨R', hRv, hRc, hiff⟩ := verify_accepts_iff_point_eq self sig msg rb sb s er e + hrb hsb hA hleg hs hrec he hkv hkc + t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21 t22 t23 t24 t25 t26 t27 t28 t29 t30 t31 hsbytes Vs hVs hVslt Q hQv hQc henc + obtain ⟨o, ho, Pt, hosome, hPv, hPc, hPQ⟩ := spec_imp_exists + (decompress_of_canonical Q hQv hQc rb b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 hbl henc) + refine ⟨R', Pt, hRv, hRc, ?_, hPv, hPc, ?_⟩ + · rw [ho, hosome] + · rw [hiff, hPQ] + +end CurveFieldProofs diff --git a/verification/Proofs/DecompressSpec.lean b/verification/Proofs/DecompressSpec.lean new file mode 100644 index 0000000..919ab4c --- /dev/null +++ b/verification/Proofs/DecompressSpec.lean @@ -0,0 +1,509 @@ +/- ────────────────────────────────────────────────────────────────────────────── + Proofs/DecompressSpec.lean — phase 2, the constructive decompress chain, + part 1: the arithmetic ingredients of `sqrt_ratio_i`. + + · `pow_p58_spec` — a^((p−5)/8) via the pow22501 chain (Fermat-style, + the invert_spec pattern with exponent 2²⁵² − 3); + · √−1 — already certified (ConstSpecs.sqrt_m1_spec); + · `fe_ct_eq_spec` — the constant-time field comparison decides + denotational equality: to_bytes is CANONICAL + (to_bytes_spec), so byte equality is residue + equality in both directions. + + Part 2 (sequel): the sqrt_ratio_i success-case walk, from_bytes, and + `decompress_of_canonical` — the constructive upgrade of the point-level + verification equation. + ────────────────────────────────────────────────────────────────────────────── -/ +import Proofs.PointEqSpec +import Proofs.InvertSpec +open Aeneas Aeneas.Std Result +open curve25519 + +set_option maxHeartbeats 4000000 +set_option linter.unusedSimpArgs false +set_option exponentiation.threshold 600 + +namespace CurveFieldProofs + +open Aeneas.Std.WP + +/-- a^((p−5)/8) = a^(2²⁵² − 3): the pow22501 chain squared twice and folded + once more with a — the invert_spec pattern. -/ +theorem pow_p58_spec (a : Fe) (hba : Bnd a (2^54)) : + field.FieldElement51.pow_p58 a ⦃ r => Bnd r (2^52) ∧ ⟪r⟫ = ⟪a⟫ ^ (2^252 - 3) ⦄ := by + unfold field.FieldElement51.pow_p58 + let* ⟨ t19, t3, h1, h2, h3, h4 ⟩ ← pow22501_spec by bnd + let* ⟨ t20, t20_post1, t20_post2 ⟩ ← pow2k_spec' by bnd + let* ⟨ r, r_post1, r_post2 ⟩ ← mul_spec' by bnd + refine ⟨by bnd, ?_⟩ + rw [r_post2, t20_post2, h3] + rw [← pow_mul, ← pow_succ'] + congr 1 + +/- √−1: `sqrt_m1_spec` (ConstSpecs.lean) already pins the SQRT_M1 constant: + Bnd s (2⁵²) ∧ ⟪s⟫·⟪s⟫ = −1 — reused as-is by the sqrt walk below. -/ + +/-- Byte-array value equality forces list equality (the converse of congr): + little-endian digits are unique. -/ +theorem bytesVal_inj (sa sb : Std.Array Std.U8 32#usize) + (h : bytesVal sa = bytesVal sb) : (↑sa : List Std.U8) = (↑sb : List Std.U8) := by + obtain ⟨e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13, e14, e15, + e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, + hel⟩ := Bytes32.exists_bytes sa + obtain ⟨r0, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, + r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, + hrl⟩ := Bytes32.exists_bytes sb + have hrq := (rangeEq_iff_bytesVal sa sb).mpr h + have hpt : ∀ j, j < 32 → sa.val[j]! = sb.val[j]! := fun j hj => hrq j (Nat.zero_le _) hj + have h0 : e0 = r0 := by simpa [hel, hrl] using hpt 0 (by norm_num) + have h1 : e1 = r1 := by simpa [hel, hrl] using hpt 1 (by norm_num) + have h2 : e2 = r2 := by simpa [hel, hrl] using hpt 2 (by norm_num) + have h3 : e3 = r3 := by simpa [hel, hrl] using hpt 3 (by norm_num) + have h4 : e4 = r4 := by simpa [hel, hrl] using hpt 4 (by norm_num) + have h5 : e5 = r5 := by simpa [hel, hrl] using hpt 5 (by norm_num) + have h6 : e6 = r6 := by simpa [hel, hrl] using hpt 6 (by norm_num) + have h7 : e7 = r7 := by simpa [hel, hrl] using hpt 7 (by norm_num) + have h8 : e8 = r8 := by simpa [hel, hrl] using hpt 8 (by norm_num) + have h9 : e9 = r9 := by simpa [hel, hrl] using hpt 9 (by norm_num) + have h10 : e10 = r10 := by simpa [hel, hrl] using hpt 10 (by norm_num) + have h11 : e11 = r11 := by simpa [hel, hrl] using hpt 11 (by norm_num) + have h12 : e12 = r12 := by simpa [hel, hrl] using hpt 12 (by norm_num) + have h13 : e13 = r13 := by simpa [hel, hrl] using hpt 13 (by norm_num) + have h14 : e14 = r14 := by simpa [hel, hrl] using hpt 14 (by norm_num) + have h15 : e15 = r15 := by simpa [hel, hrl] using hpt 15 (by norm_num) + have h16 : e16 = r16 := by simpa [hel, hrl] using hpt 16 (by norm_num) + have h17 : e17 = r17 := by simpa [hel, hrl] using hpt 17 (by norm_num) + have h18 : e18 = r18 := by simpa [hel, hrl] using hpt 18 (by norm_num) + have h19 : e19 = r19 := by simpa [hel, hrl] using hpt 19 (by norm_num) + have h20 : e20 = r20 := by simpa [hel, hrl] using hpt 20 (by norm_num) + have h21 : e21 = r21 := by simpa [hel, hrl] using hpt 21 (by norm_num) + have h22 : e22 = r22 := by simpa [hel, hrl] using hpt 22 (by norm_num) + have h23 : e23 = r23 := by simpa [hel, hrl] using hpt 23 (by norm_num) + have h24 : e24 = r24 := by simpa [hel, hrl] using hpt 24 (by norm_num) + have h25 : e25 = r25 := by simpa [hel, hrl] using hpt 25 (by norm_num) + have h26 : e26 = r26 := by simpa [hel, hrl] using hpt 26 (by norm_num) + have h27 : e27 = r27 := by simpa [hel, hrl] using hpt 27 (by norm_num) + have h28 : e28 = r28 := by simpa [hel, hrl] using hpt 28 (by norm_num) + have h29 : e29 = r29 := by simpa [hel, hrl] using hpt 29 (by norm_num) + have h30 : e30 = r30 := by simpa [hel, hrl] using hpt 30 (by norm_num) + have h31 : e31 = r31 := by simpa [hel, hrl] using hpt 31 (by norm_num) + rw [hel, hrl, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9, h10, h11, h12, h13, + h14, h15, h16, h17, h18, h19, h20, h21, h22, h23, h24, h25, h26, h27, + h28, h29, h30, h31] + +/-- Lists determine `bytesVal`. -/ +theorem bytesVal_congr {sa sb : Std.Array Std.U8 32#usize} + (h : (↑sa : List Std.U8) = ↑sb) : bytesVal sa = bytesVal sb := by + unfold bytesVal + rw [h] + +/-- **The canonical-bytes bridge**: for canonical serializations, byte-list + equality IS denotational equality. -/ +theorem bytes_eq_iff_denote {a b : Fe} {sa sb : Std.Array Std.U8 32#usize} + (hsa : bytesVal sa = feVal a % P) (hsb : bytesVal sb = feVal b % P) : + (↑sa : List Std.U8) = ↑sb ↔ ⟪a⟫ = ⟪b⟫ := by + haveI : NeZero P := ⟨by unfold P; norm_num⟩ + have hmod : ⟪a⟫ = ⟪b⟫ ↔ feVal a % P = feVal b % P := by + unfold denote + rw [ZMod.natCast_eq_natCast_iff] + exact ⟨fun h => h, fun h => h⟩ + constructor + · intro h + rw [hmod, ← hsa, ← hsb] + exact bytesVal_congr h + · intro h + apply bytesVal_inj + rw [hsa, hsb] + exact hmod.mp h + +/-- **The constant-time field comparison decides denotational equality**: + to_bytes is canonical, so byte equality IS residue equality. -/ +theorem fe_ct_eq_spec (a b : Fe) : + backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq a b + ⦃ c => (c.val = 0 ∨ c.val = 1) ∧ (c.val = 1 ↔ ⟪a⟫ = ⟪b⟫) ⦄ := by + unfold backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq + step with (to_bytes_spec' a) as ⟨sa, hsa⟩ + step as ⟨la, hla⟩ + step with (to_bytes_spec' b) as ⟨sb, hsb⟩ + step as ⟨lb, hlb⟩ + simp only [Slice.Insts.SubtleConstantTimeEq.ct_eq] + try simp only [spec_ok] + have hlav : la.val = sa.val := by rw [hla]; rfl + have hlbv : lb.val = sb.val := by rw [hlb]; rfl + rw [hlav, hlbv] + have hbridge := bytes_eq_iff_denote hsa hsb + by_cases heq : (↑sa : List Std.U8) = ↑sb + · rw [if_pos heq] + exact ⟨Or.inr rfl, fun _ => hbridge.mp heq, fun _ => rfl⟩ + · rw [if_neg heq] + refine ⟨Or.inl rfl, fun h01 => absurd h01 (by norm_num), fun hab => ?_⟩ + exact absurd (hbridge.mpr hab) heq + +/-- u64 constant-time assign keeps `self` iff the choice is 0 (rfl on the + FunsExternal model; restated locally — Proofs.Basic is a parallel root + that clashes with the ConstSpecs chain). -/ +theorem u64_cond_assign (a b : Std.U64) (c : subtle.Choice) : + U64.Insts.SubtleConditionallySelectable.conditional_assign a b c + = ok (if c.val = 0 then a else b) := rfl + +/-- **Limb-wise constant-time selection on field elements**: keeps `self` + iff the choice is 0 — the in-place flavor `sqrt_ratio_i` uses twice + (root flip and sign normalization). -/ +theorem fe_cond_assign_spec (a b : Fe) (c : subtle.Choice) + (x0 x1 x2 x3 x4 y0 y1 y2 y3 y4 : U64) + (ha : (↑a : List U64) = [x0, x1, x2, x3, x4]) + (hb : (↑b : List U64) = [y0, y1, y2, y3, y4]) : + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_assign + a b c + ⦃ r => (↑r : List U64) + = if c.val = 0 then [x0, x1, x2, x3, x4] else [y0, y1, y2, y3, y4] ⦄ := by + unfold backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_assign + step as ⟨i0, back0, hi0, hback0⟩ + step as ⟨i1, hi1⟩ + try simp only [u64_cond_assign, bind_tc_ok] + step as ⟨i3, back1, hi3, hback1⟩ + try simp only [hback0] at * + step as ⟨i4, hi4⟩ + try simp only [u64_cond_assign, bind_tc_ok] + step as ⟨i6, back2, hi6, hback2⟩ + try simp only [hback1] at * + step as ⟨i7, hi7⟩ + try simp only [u64_cond_assign, bind_tc_ok] + step as ⟨i9, back3, hi9, hback3⟩ + try simp only [hback2] at * + step as ⟨i10, hi10⟩ + try simp only [u64_cond_assign, bind_tc_ok] + step as ⟨i12, back4, hi12, hback4⟩ + try simp only [hback3] at * + step as ⟨i13, hi13⟩ + try simp only [u64_cond_assign, bind_tc_ok] + try simp only [spec_ok] + by_cases hc : c.val = 0 + · simp only [hc, if_pos rfl] at * + simp_all [Array.set_val_eq, ha, hb] + · simp only [if_neg hc] at * + simp_all [Array.set_val_eq, ha, hb] + +/-- **THE SQUARE-ROOT CORE** (pure 𝔽_p): if u/v is a square (witness x) + with v ≠ 0, the candidate r = (u·v³)·(u·v⁷)^((p−5)/8) satisfies + v·r² = ±u — the algebraic heart of `sqrt_ratio_i`. The v-part of the + exponent collapses by Fermat; the residual x^((p−1)/2) is ±1. -/ +theorem sqrt_core (u v x : Fp) (hv : v ≠ 0) (hx : x ^ 2 * v = u) : + v * (u * v^3 * (u * v^7)^(2^252 - 3))^2 = u ∨ + v * (u * v^3 * (u * v^7)^(2^252 - 3))^2 = -u := by + haveI : Fact (Nat.Prime P) := ⟨P_prime⟩ + by_cases hx0 : x = 0 + · -- x = 0 forces u = 0 and the candidate is 0 = u + left + have hu : u = 0 := by rw [← hx, hx0]; ring + rw [hu] + ring + · set w : Fp := u * v^7 with hwdef + have hw : w = x^2 * v^8 := by rw [hwdef, ← hx]; ring + have hfer_v : v ^ (P - 1) = 1 := ZMod.pow_card_sub_one_eq_one hv + have hfer_x2 : (x ^ ((P-1)/2))^2 = 1 := by + rw [← pow_mul] + have he : (P-1)/2 * 2 = P - 1 := by unfold P; norm_num + rw [he] + exact ZMod.pow_card_sub_one_eq_one hx0 + have hpm : x ^ ((P-1)/2) = 1 ∨ x ^ ((P-1)/2) = -1 := by + have hfac : (x ^ ((P-1)/2) - 1) * (x ^ ((P-1)/2) + 1) = 0 := by + linear_combination hfer_x2 + rcases mul_eq_zero.mp hfac with h' | h' + · left; linear_combination h' + · right; linear_combination h' + have hkey : v * (u * v^3 * w^(2^252 - 3))^2 = u * x^((P-1)/2) := by + have h1 : v * (u * v^3 * w^(2^252-3))^2 = u * w * (w^(2^252-3))^2 := by + rw [hwdef]; ring + have h2 : (w^(2^252-3) : Fp)^2 = w^(2^253-6) := by + rw [← pow_mul] + norm_num + have h3 : (u * w * w^(2^253-6) : Fp) = u * w^(2^253-5) := by + have : (w * w^(2^253-6) : Fp) = w^(2^253-5) := by + rw [← pow_succ'] + norm_num + rw [mul_assoc, this] + rw [h1, h2, h3, hw] + have h4 : ((x^2 * v^8 : Fp))^(2^253-5) = x^(2^254-10) * v^(2^256-40) := by + rw [mul_pow, ← pow_mul, ← pow_mul] + norm_num + rw [h4] + have h5 : (v : Fp)^(2^256-40) = 1 := by + have he : (2^256 - 40 : ℕ) = (P - 1) * 2 := by unfold P; norm_num + rw [he, pow_mul, hfer_v, one_pow] + have h6 : (2^254 - 10 : ℕ) = (P-1)/2 := by unfold P; norm_num + rw [h5, h6] + ring + rcases hpm with h | h + · left; rw [hkey, h, mul_one] + · right; rw [hkey, h]; ring + +/-- In 𝔽_p (p odd), an element equal to its own negative is zero. -/ +theorem eq_neg_self_iff_zero (a : Fp) : a = -a ↔ a = 0 := by + constructor + · intro h + have h2 : (2 : Fp) * a = 0 := by linear_combination h + rcases mul_eq_zero.mp h2 with h' | h' + · exact absurd h' two_ne_zero_Fp + · exact h' + · intro h; rw [h]; ring + +/-- 1 + √−1 does not vanish (else −1 = 1, contradicting p odd). -/ +theorem one_add_i_ne_zero {i : Fp} (hi : i * i = -1) : (1 : Fp) + i ≠ 0 := by + intro h + have him : i = -1 := by linear_combination h + rw [him] at hi + have : (2 : Fp) = 0 := by linear_combination hi + exact two_ne_zero_Fp this + +/-- **THE SQUARE-ROOT WALK** (success case): if u/v is a square (witness x, + v ≠ 0), `sqrt_ratio_i` returns choice 1 and the even-parity root: + r² · v = u with r's canonical residue even. -/ +theorem sqrt_ratio_i_sq_spec (u v : Fe) (hbu : Bnd u (2^54)) (hbv : Bnd v (2^54)) + (hvne : ⟪v⟫ ≠ 0) (x : Fp) (hx : x ^ 2 * ⟪v⟫ = ⟪u⟫) : + field.FieldElement51.sqrt_ratio_i u v ⦃ cr => + cr.1.val = 1 ∧ Bnd cr.2 (2^52) ∧ + ⟪cr.2⟫ ^ 2 * ⟪v⟫ = ⟪u⟫ ∧ (⟪cr.2⟫).val % 2 = 0 ⦄ := by + haveI : NeZero P := ⟨by unfold P; norm_num⟩ + unfold field.FieldElement51.sqrt_ratio_i + -- the arithmetic chain: v³, v⁷, u·v³, u·v⁷, (u·v⁷)^((p−5)/8), r, r², check + let* ⟨ fe, hbfe, hfe ⟩ ← square_spec' by bnd + let* ⟨ v3, hbv3, hv3 ⟩ ← mul_spec' by bnd + let* ⟨ fe1, hbfe1, hfe1 ⟩ ← square_spec' by bnd + let* ⟨ v7, hbv7, hv7 ⟩ ← mul_spec' by bnd + let* ⟨ fe2, hbfe2, hfe2 ⟩ ← mul_spec' by bnd + let* ⟨ fe3, hbfe3, hfe3 ⟩ ← mul_spec' by bnd + let* ⟨ fe4, hbfe4, hfe4 ⟩ ← pow_p58_spec by bnd + let* ⟨ r, hbr, hr ⟩ ← mul_spec' by bnd + let* ⟨ fe5, hbfe5, hfe5 ⟩ ← square_spec' by bnd + let* ⟨ check, hbcheck, hcheck ⟩ ← mul_spec' by bnd + -- √−1 + step with sqrt_m1_spec as ⟨im, hbim, him⟩ + -- the three constant-time checks + step with (fe_ct_eq_spec check u) as ⟨correct, hc01, hciff⟩ + -- −u (needs u's limbs) + obtain ⟨u0, u1, u2, u3, u4, hul⟩ := Fe.exists_limbs u + unfold Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg + step with (neg_spec u u0 u1 u2 u3 u4 hul (by bnd)) as ⟨fe6, hbfe6, hfe6⟩ + step with (fe_ct_eq_spec check fe6) as ⟨flipped, hf01, hfiff⟩ + let* ⟨ fe7, hbfe7, hfe7 ⟩ ← mul_spec' by bnd + step with (fe_ct_eq_spec check fe7) as ⟨flipped_i, hfi01, hfiiff⟩ + -- r′ = √−1 · r + let* ⟨ r_prime, hbrp, hrp ⟩ ← mul_spec' by bnd + -- the flip choice and the root flip + simp only [subtle.Choice.Insts.CoreOpsBitBitOrChoiceChoice.bitor, bind_tc_ok] + obtain ⟨rr0, rr1, rr2, rr3, rr4, hrl⟩ := Fe.exists_limbs r + obtain ⟨rp0, rp1, rp2, rp3, rp4, hrpl⟩ := Fe.exists_limbs r_prime + step with (fe_cond_assign_spec r r_prime _ rr0 rr1 rr2 rr3 rr4 rp0 rp1 rp2 rp3 rp4 hrl hrpl) + as ⟨r1, hr1l⟩ + -- sign normalization + step with (is_negative_spec r1) as ⟨rneg, hrneg⟩ + obtain ⟨q0, q1, q2, q3, q4, hq⟩ := Fe.exists_limbs r1 + -- Bnd r1 (2^52): its list is one of the two bounded lists + have hbr1 : Bnd r1 (2^52) := by + have hb1 : Bnd r (2^52) := Bnd.mono hbr (by norm_num) + have hb2 : Bnd r_prime (2^52) := Bnd.mono hbrp (by norm_num) + split at hr1l + · rw [Bnd_eq r1 rr0 rr1 rr2 rr3 rr4 _ (by rw [hr1l])] + rw [Bnd_eq r rr0 rr1 rr2 rr3 rr4 _ hrl] at hb1 + exact hb1 + · rw [Bnd_eq r1 rp0 rp1 rp2 rp3 rp4 _ (by rw [hr1l])] + rw [Bnd_eq r_prime rp0 rp1 rp2 rp3 rp4 _ hrpl] at hb2 + exact hb2 + -- −r1 and the parity select + step with (neg_spec r1 q0 q1 q2 q3 q4 hq (Bnd.mono hbr1 (by norm_num))) + as ⟨r_neg, hbrn, hrn⟩ + obtain ⟨n0, n1, n2, n3, n4, hnl⟩ := Fe.exists_limbs r_neg + step with (fe_cond_assign_spec r1 r_neg _ q0 q1 q2 q3 q4 n0 n1 n2 n3 n4 hq hnl) + as ⟨r2, hr2l⟩ + try simp only [spec_ok] + -- ── interpreted values ─────────────────────────────────────────────────── + have hfe2v : ⟪fe2⟫ = ⟪u⟫ * ⟪v⟫^3 := by rw [hfe2, hv3, hfe]; ring + have hfe3v : ⟪fe3⟫ = ⟪u⟫ * ⟪v⟫^7 := by rw [hfe3, hv7, hfe1, hv3, hfe]; ring + have hrval : ⟪r⟫ = ⟪u⟫ * ⟪v⟫^3 * (⟪u⟫ * ⟪v⟫^7)^(2^252-3) := by + rw [hr, hfe2v, hfe4, hfe3v] + have hcheckv : ⟪check⟫ = ⟪v⟫ * ⟪r⟫^2 := by rw [hcheck, hfe5]; ring + have hcore := sqrt_core ⟪u⟫ ⟪v⟫ x hvne hx + rw [← hrval] at hcore + have hrpv : ⟪r_prime⟫ = ⟪im⟫ * ⟪r⟫ := hrp + -- denote transfer along the two selects + have hr1d : (flipped ||| flipped_i).val = 0 ∧ ⟪r1⟫ = ⟪r⟫ ∨ + (flipped ||| flipped_i).val ≠ 0 ∧ ⟪r1⟫ = ⟪r_prime⟫ := by + split at hr1l + · left + refine ⟨by assumption, ?_⟩ + unfold denote + rw [feVal_eq r1 rr0 rr1 rr2 rr3 rr4 (by rw [hr1l]), + feVal_eq r rr0 rr1 rr2 rr3 rr4 hrl] + · right + refine ⟨by assumption, ?_⟩ + unfold denote + rw [feVal_eq r1 rp0 rp1 rp2 rp3 rp4 (by rw [hr1l]), + feVal_eq r_prime rp0 rp1 rp2 rp3 rp4 hrpl] + have hr2d : rneg.val = 0 ∧ ⟪r2⟫ = ⟪r1⟫ ∨ rneg.val ≠ 0 ∧ ⟪r2⟫ = ⟪r_neg⟫ := by + split at hr2l + · left + refine ⟨by assumption, ?_⟩ + unfold denote + rw [feVal_eq r2 q0 q1 q2 q3 q4 (by rw [hr2l]), + feVal_eq r1 q0 q1 q2 q3 q4 hq] + · right + refine ⟨by assumption, ?_⟩ + unfold denote + rw [feVal_eq r2 n0 n1 n2 n3 n4 (by rw [hr2l]), + feVal_eq r_neg n0 n1 n2 n3 n4 hnl] + -- ── choice values from the three checks ────────────────────────────────── + -- the value equation carried by r1 in every case: ⟪r1⟫²·⟪v⟫ = ⟪u⟫ and the + -- flip choice consistent with the branch taken + have hval1 : ⟪r1⟫ ^ 2 * ⟪v⟫ = ⟪u⟫ ∧ (correct ||| flipped).val = 1 := by + haveI : Fact (Nat.Prime P) := ⟨P_prime⟩ + have hc01' := hc01 + have hf01' := hf01 + have hfi01' := hfi01 + by_cases hu0 : ⟪u⟫ = 0 + · -- u = 0: check = v·r² = ±0 = 0; every flag fires; r1 = im·r with r-part 0 + have hchk0 : ⟪check⟫ = 0 := by + rcases hcore with h | h <;> rw [hcheckv] + · rw [show ⟪v⟫ * ⟪r⟫^2 = ⟪v⟫ * (⟪u⟫ * ⟪v⟫^3 * (⟪u⟫*⟪v⟫^7)^(2^252-3))^2 from by rw [hrval]] + rw [hrval] at h + rw [h, hu0] + · rw [hrval] at h ⊢ + rw [h, hu0] + ring + have hr0 : ⟪v⟫ * ⟪r⟫^2 = 0 := by rw [← hcheckv]; exact hchk0 + have hrz : ⟪r⟫ = 0 := by + rcases mul_eq_zero.mp hr0 with h | h + · exact absurd h hvne + · exact pow_eq_zero_iff (n := 2) (by norm_num) |>.mp h + have hcv : correct.val = 1 := hciff.mpr (by rw [hchk0, hu0]) + have hor1 : (correct ||| flipped).val = 1 := by + rcases hf01 with h0 | h1 + · have : flipped = 0#u8 := UScalar.eq_of_val_eq (by simp [h0]) + rw [this] + have : correct = 1#u8 := UScalar.eq_of_val_eq (by simp [hcv]) + rw [this] + rfl + · have : flipped = 1#u8 := UScalar.eq_of_val_eq (by simp [h1]) + rw [this] + have : correct = 1#u8 := UScalar.eq_of_val_eq (by simp [hcv]) + rw [this] + rfl + refine ⟨?_, hor1⟩ + rcases hr1d with ⟨-, hd⟩ | ⟨-, hd⟩ + · rw [hd, hrz, hu0]; ring + · rw [hd, hrpv, hrz, hu0]; ring + · -- u ≠ 0: the disjunct decides everything + rcases hcore with hA | hB + · -- v·r² = u: no flip, correct = 1 + have hcv : correct.val = 1 := hciff.mpr (by rw [hcheckv]; exact hA) + have hfv : flipped.val = 0 := by + rcases hf01 with h | h + · exact h + · exfalso + have := hfiff.mp h + rw [hcheckv, hfe6] at this + rw [hA] at this + exact hu0 ((eq_neg_self_iff_zero ⟪u⟫).mp this) + have hfiv : flipped_i.val = 0 := by + rcases hfi01 with h | h + · exact h + · exfalso + have := hfiiff.mp h + rw [hcheckv, hfe7, hfe6] at this + rw [hA] at this + have hfac : ⟪u⟫ * (1 + ⟪im⟫) = 0 := by linear_combination this + rcases mul_eq_zero.mp hfac with h' | h' + · exact hu0 h' + · exact one_add_i_ne_zero (by rw [← sq]; rw [sq]; exact him) h' + have hflip0 : (flipped ||| flipped_i).val = 0 := by + have h1 : flipped = 0#u8 := UScalar.eq_of_val_eq (by simp [hfv]) + have h2 : flipped_i = 0#u8 := UScalar.eq_of_val_eq (by simp [hfiv]) + rw [h1, h2] + rfl + refine ⟨?_, ?_⟩ + · rcases hr1d with ⟨-, hd⟩ | ⟨hne, -⟩ + · rw [hd]; linear_combination hA + · exact absurd hflip0 hne + · have h1 : correct = 1#u8 := UScalar.eq_of_val_eq (by simp [hcv]) + have h2 : flipped = 0#u8 := UScalar.eq_of_val_eq (by simp [hfv]) + rw [h1, h2] + rfl + · -- v·r² = −u: flip fires, r1 = im·r + have hfv : flipped.val = 1 := hfiff.mpr (by rw [hcheckv, hfe6]; exact hB) + have hflip1 : (flipped ||| flipped_i).val ≠ 0 := by + have h1 : flipped = 1#u8 := UScalar.eq_of_val_eq (by simp [hfv]) + rw [h1] + rcases hfi01 with h | h + · have h2 : flipped_i = 0#u8 := UScalar.eq_of_val_eq (by simp [h]) + rw [h2] + decide + · have h2 : flipped_i = 1#u8 := UScalar.eq_of_val_eq (by simp [h]) + rw [h2] + decide + refine ⟨?_, ?_⟩ + · rcases hr1d with ⟨h0, -⟩ | ⟨-, hd⟩ + · exact absurd h0 hflip1 + · rw [hd, hrpv] + have him2 : ⟪im⟫ ^ 2 = -1 := by rw [sq]; exact him + have : (⟪im⟫ * ⟪r⟫) ^ 2 * ⟪v⟫ = ⟪im⟫^2 * (⟪v⟫ * ⟪r⟫^2) := by ring + rw [this, him2, hB] + ring + · have h2 : flipped = 1#u8 := UScalar.eq_of_val_eq (by simp [hfv]) + rw [h2] + rcases hc01 with h | h + · have h1 : correct = 0#u8 := UScalar.eq_of_val_eq (by simp [h]) + rw [h1] + rfl + · have h1 : correct = 1#u8 := UScalar.eq_of_val_eq (by simp [h]) + rw [h1] + rfl + obtain ⟨hval1', hwas⟩ := hval1 + -- ── parity normalization and the final post ───────────────────────────── + haveI : NeZero P := ⟨by unfold P; norm_num⟩ + have hrnegv : rneg.val = (⟪r1⟫).val % 2 := by + rw [hrneg] + unfold denote + rw [ZMod.val_natCast] + have hbr2 : Bnd r2 (2^52) := by + split at hr2l + · rw [Bnd_eq r2 q0 q1 q2 q3 q4 _ (by rw [hr2l])] + rw [Bnd_eq r1 q0 q1 q2 q3 q4 _ hq] at hbr1 + exact hbr1 + · have hb3 : Bnd r_neg (2^52) := hbrn + rw [Bnd_eq r2 n0 n1 n2 n3 n4 _ (by rw [hr2l])] + rw [Bnd_eq r_neg n0 n1 n2 n3 n4 _ hnl] at hb3 + exact hb3 + refine ⟨hwas, hbr2, ?_, ?_⟩ + · -- the square equation survives the sign normalization + rcases hr2d with ⟨-, hd⟩ | ⟨-, hd⟩ + · rw [hd]; exact hval1' + · rw [hd, hrn] + have : (-⟪r1⟫) ^ 2 * ⟪v⟫ = ⟪r1⟫ ^ 2 * ⟪v⟫ := by ring + rw [this] + exact hval1' + · -- even parity + rcases hr2d with ⟨h0, hd⟩ | ⟨hne, hd⟩ + · rw [hd] + rw [hrnegv] at h0 + exact h0 + · rw [hd, hrn] + have hodd : rneg.val = 1 := by + have := hrnegv + omega + rw [hrnegv] at hodd + have hr1nz : ⟪r1⟫ ≠ 0 := by + intro hz + rw [hz] at hodd + simp at hodd + have hnegval : (-⟪r1⟫).val = P - (⟪r1⟫).val := by + rw [ZMod.neg_val, if_neg hr1nz] + rw [hnegval] + have hlt := ZMod.val_lt ⟪r1⟫ + have hpodd : P % 2 = 1 := by unfold P; norm_num + have hpos : 0 < (⟪r1⟫).val := by + rcases Nat.eq_zero_or_pos (⟪r1⟫).val with h | h + · exact absurd ((ZMod.val_eq_zero _).mp h) hr1nz + · exact h + omega + +end CurveFieldProofs diff --git a/verification/Proofs/FromBytesSpec.lean b/verification/Proofs/FromBytesSpec.lean new file mode 100644 index 0000000..4892036 --- /dev/null +++ b/verification/Proofs/FromBytesSpec.lean @@ -0,0 +1,438 @@ +/- ────────────────────────────────────────────────────────────────────────────── + Proofs/FromBytesSpec.lean — phase 2, decompress step 2: the byte parser. + + `FieldElement51::from_bytes` loads five 64-bit little-endian windows at + byte offsets 0/6/12/19/24, shifts by 0/3/6/1/12, and masks to 51 bits — + the windows tile bits 0..254 exactly, so + + feVal (from_bytes b) = bytesVal b mod 2²⁵⁵ (from_bytes_spec) + + — the top bit is discarded, everything else is exact. This is the y-parse + of decompression: for a canonical encoding (y-residue + sign bit), the + parsed field element denotes exactly the y-residue. + + Structure: a generic 8-byte loader lemma (`load8_at_spec`, the disjoint-OR + idiom), a pure window/digit identity (`limbs_of_bytes`), and the walk. + ────────────────────────────────────────────────────────────────────────────── -/ +import Proofs.DecompressSpec +open Aeneas Aeneas.Std Result +open curve25519 + +set_option maxHeartbeats 8000000 +set_option linter.unusedSimpArgs false +set_option maxRecDepth 8000 + +namespace CurveFieldProofs + +open Aeneas.Std.WP + +/-- Disjoint low-bits OR is addition (product-order-robust form). -/ +theorem or_add_low {a b : ℕ} (k : ℕ) (ha : a < 2^k) : + a ||| b * 2^k = a + b * 2^k := by + have hor := Nat.two_pow_add_eq_or_of_lt (b := a) (i := k) ha b + calc a ||| b * 2^k = a ||| 2^k * b := by rw [Nat.mul_comm b] + _ = 2^k * b ||| a := Nat.lor_comm _ _ + _ = 2^k * b + a := hor.symm + _ = a + b * 2^k := by ring + +/-- Generic 8-byte little-endian loader: given the eight bytes at positions + i..i+7, the loaded word is their LE value. -/ +theorem load8_at_spec (s : Slice Std.U8) (i : Std.Usize) + (c0 c1 c2 c3 c4 c5 c6 c7 : Std.U8) + (hlen : i.val + 7 < s.length) + (h0 : s.val[i.val]! = c0) (h1 : s.val[i.val + 1]! = c1) + (h2 : s.val[i.val + 2]! = c2) (h3 : s.val[i.val + 3]! = c3) + (h4 : s.val[i.val + 4]! = c4) (h5 : s.val[i.val + 5]! = c5) + (h6 : s.val[i.val + 6]! = c6) (h7 : s.val[i.val + 7]! = c7) : + backend.serial.u64.field.FieldElement51.from_bytes.load8_at s i ⦃ w => + w.val = c0.val + c1.val * 2^8 + c2.val * 2^16 + c3.val * 2^24 + + c4.val * 2^32 + c5.val * 2^40 + c6.val * 2^48 + c7.val * 2^56 ⦄ := by + unfold backend.serial.u64.field.FieldElement51.from_bytes.load8_at + step as ⟨x0, hx0⟩ + rw [← getElem!_pos (↑s : List Std.U8) i.val (by scalar_tac)] at hx0 + rw [h0] at hx0 + step as ⟨w0, hw0⟩ + have hw0v : w0.val = c0.val := by + rw [hw0, UScalar.cast_val_eq, hx0] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨p1, hp1⟩ + have hp1v : p1.val = i.val + 1 := by scalar_tac + step as ⟨x1, hx1⟩ + rw [← getElem!_pos (↑s : List Std.U8) p1.val (by scalar_tac)] at hx1 + rw [hp1v, h1] at hx1 + step as ⟨y1, hy1⟩ + have hy1v : y1.val = c1.val := by + rw [hy1, UScalar.cast_val_eq, hx1] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t1, ht1⟩ + have ht1v : t1.val = c1.val * 2^8 := by + rw [ht1] + simp only [Nat.shiftLeft_eq, hy1v] + rw [Nat.mod_eq_of_lt (show c1.val * 2^8 < U64.size by scalar_tac)] + step as ⟨w1, hw1⟩ + have hw1v : w1.val = c0.val + c1.val * 2^8 := by + rw [hw1, UScalar.val_or, hw0v, ht1v] + rw [or_add_low 8 (by scalar_tac)] + try ring + clear hp1 hp1v hx1 hy1 hy1v ht1 ht1v hw0 hw0v + step as ⟨p2, hp2⟩ + have hp2v : p2.val = i.val + 2 := by scalar_tac + step as ⟨x2, hx2⟩ + rw [← getElem!_pos (↑s : List Std.U8) p2.val (by scalar_tac)] at hx2 + rw [hp2v, h2] at hx2 + step as ⟨y2, hy2⟩ + have hy2v : y2.val = c2.val := by + rw [hy2, UScalar.cast_val_eq, hx2] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t2, ht2⟩ + have ht2v : t2.val = c2.val * 2^16 := by + rw [ht2] + simp only [Nat.shiftLeft_eq, hy2v] + rw [Nat.mod_eq_of_lt (show c2.val * 2^16 < U64.size by scalar_tac)] + step as ⟨w2, hw2⟩ + have hw2v : w2.val = c0.val + c1.val * 2^8 + c2.val * 2^16 := by + rw [hw2, UScalar.val_or, hw1v, ht2v] + rw [or_add_low 16 (by scalar_tac)] + try ring + clear hp2 hp2v hx2 hy2 hy2v ht2 ht2v hw1 hw1v + step as ⟨p3, hp3⟩ + have hp3v : p3.val = i.val + 3 := by scalar_tac + step as ⟨x3, hx3⟩ + rw [← getElem!_pos (↑s : List Std.U8) p3.val (by scalar_tac)] at hx3 + rw [hp3v, h3] at hx3 + step as ⟨y3, hy3⟩ + have hy3v : y3.val = c3.val := by + rw [hy3, UScalar.cast_val_eq, hx3] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t3, ht3⟩ + have ht3v : t3.val = c3.val * 2^24 := by + rw [ht3] + simp only [Nat.shiftLeft_eq, hy3v] + rw [Nat.mod_eq_of_lt (show c3.val * 2^24 < U64.size by scalar_tac)] + step as ⟨w3, hw3⟩ + have hw3v : w3.val = c0.val + c1.val * 2^8 + c2.val * 2^16 + c3.val * 2^24 := by + rw [hw3, UScalar.val_or, hw2v, ht3v] + rw [or_add_low 24 (by scalar_tac)] + try ring + clear hp3 hp3v hx3 hy3 hy3v ht3 ht3v hw2 hw2v + step as ⟨p4, hp4⟩ + have hp4v : p4.val = i.val + 4 := by scalar_tac + step as ⟨x4, hx4⟩ + rw [← getElem!_pos (↑s : List Std.U8) p4.val (by scalar_tac)] at hx4 + rw [hp4v, h4] at hx4 + step as ⟨y4, hy4⟩ + have hy4v : y4.val = c4.val := by + rw [hy4, UScalar.cast_val_eq, hx4] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t4, ht4⟩ + have ht4v : t4.val = c4.val * 2^32 := by + rw [ht4] + simp only [Nat.shiftLeft_eq, hy4v] + rw [Nat.mod_eq_of_lt (show c4.val * 2^32 < U64.size by scalar_tac)] + step as ⟨w4, hw4⟩ + have hw4v : w4.val = c0.val + c1.val * 2^8 + c2.val * 2^16 + c3.val * 2^24 + c4.val * 2^32 := by + rw [hw4, UScalar.val_or, hw3v, ht4v] + rw [or_add_low 32 (by scalar_tac)] + try ring + clear hp4 hp4v hx4 hy4 hy4v ht4 ht4v hw3 hw3v + step as ⟨p5, hp5⟩ + have hp5v : p5.val = i.val + 5 := by scalar_tac + step as ⟨x5, hx5⟩ + rw [← getElem!_pos (↑s : List Std.U8) p5.val (by scalar_tac)] at hx5 + rw [hp5v, h5] at hx5 + step as ⟨y5, hy5⟩ + have hy5v : y5.val = c5.val := by + rw [hy5, UScalar.cast_val_eq, hx5] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t5, ht5⟩ + have ht5v : t5.val = c5.val * 2^40 := by + rw [ht5] + simp only [Nat.shiftLeft_eq, hy5v] + rw [Nat.mod_eq_of_lt (show c5.val * 2^40 < U64.size by scalar_tac)] + step as ⟨w5, hw5⟩ + have hw5v : w5.val = c0.val + c1.val * 2^8 + c2.val * 2^16 + c3.val * 2^24 + c4.val * 2^32 + c5.val * 2^40 := by + rw [hw5, UScalar.val_or, hw4v, ht5v] + rw [or_add_low 40 (by scalar_tac)] + try ring + clear hp5 hp5v hx5 hy5 hy5v ht5 ht5v hw4 hw4v + step as ⟨p6, hp6⟩ + have hp6v : p6.val = i.val + 6 := by scalar_tac + step as ⟨x6, hx6⟩ + rw [← getElem!_pos (↑s : List Std.U8) p6.val (by scalar_tac)] at hx6 + rw [hp6v, h6] at hx6 + step as ⟨y6, hy6⟩ + have hy6v : y6.val = c6.val := by + rw [hy6, UScalar.cast_val_eq, hx6] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t6, ht6⟩ + have ht6v : t6.val = c6.val * 2^48 := by + rw [ht6] + simp only [Nat.shiftLeft_eq, hy6v] + rw [Nat.mod_eq_of_lt (show c6.val * 2^48 < U64.size by scalar_tac)] + step as ⟨w6, hw6⟩ + have hw6v : w6.val = c0.val + c1.val * 2^8 + c2.val * 2^16 + c3.val * 2^24 + c4.val * 2^32 + c5.val * 2^40 + c6.val * 2^48 := by + rw [hw6, UScalar.val_or, hw5v, ht6v] + rw [or_add_low 48 (by scalar_tac)] + try ring + clear hp6 hp6v hx6 hy6 hy6v ht6 ht6v hw5 hw5v + step as ⟨p7, hp7⟩ + have hp7v : p7.val = i.val + 7 := by scalar_tac + step as ⟨x7, hx7⟩ + rw [← getElem!_pos (↑s : List Std.U8) p7.val (by scalar_tac)] at hx7 + rw [hp7v, h7] at hx7 + step as ⟨y7, hy7⟩ + have hy7v : y7.val = c7.val := by + rw [hy7, UScalar.cast_val_eq, hx7] + norm_num [UScalarTy.numBits] + scalar_tac + step as ⟨t7, ht7⟩ + have ht7v : t7.val = c7.val * 2^56 := by + rw [ht7] + simp only [Nat.shiftLeft_eq, hy7v] + rw [Nat.mod_eq_of_lt (show c7.val * 2^56 < U64.size by scalar_tac)] + try simp only [spec_ok] + rw [UScalar.val_or, hw6v, ht7v] + rw [or_add_low 56 (by scalar_tac)] + try ring + + +/-- Extract a 64-bit window: with the value decomposed as low + 2^m·window + + 2^m·2^64·high (window < 2^64, low < 2^m), division and mod recover the + window. -/ +theorem window_extract (Lo M H : ℕ) (m : ℕ) (hL : Lo < 2^m) (hM : M < 2^64) : + (Lo + 2^m * M + 2^m * 2^64 * H) / 2^m % 2^64 = M := by + have h1 : (Lo + 2^m * M + 2^m * 2^64 * H) / 2^m = M + 2^64 * H := by + rw [show Lo + 2^m * M + 2^m * 2^64 * H = Lo + 2^m * (M + 2^64 * H) by ring] + rw [Nat.add_mul_div_left _ _ (Nat.two_pow_pos m), Nat.div_eq_of_lt hL] + omega + rw [h1] + omega + +/-- Shift inside a 64-bit window: for sh + 51 ≤ 64, + ((B/2^(8o)) mod 2^64 / 2^sh) mod 2^51 = (B / 2^(8o+sh)) mod 2^51. -/ +theorem window_shift (B o sh : ℕ) (hsh : sh + 51 ≤ 64) : + (B / 2^(8*o) % 2^64 / 2^sh) % 2^51 = B / 2^(8*o + sh) % 2^51 := by + have h1 : B / 2^(8*o) % 2^64 / 2^sh = B / 2^(8*o) / 2^sh % 2^(64 - sh) := by + have hsplit : (2:ℕ)^64 = 2^sh * 2^(64 - sh) := by + rw [← pow_add] + congr 1 + omega + rw [hsplit, Nat.mod_mul_right_div_self] + rw [h1, Nat.div_div_eq_div_mul, ← pow_add] + have h2 : B / 2^(8*o + sh) % 2^(64 - sh) % 2^51 = B / 2^(8*o + sh) % 2^51 := by + apply Nat.mod_mod_of_dvd + exact pow_dvd_pow 2 (by omega) + rw [h2] + +/-- Base-2⁵¹ five-digit tiling: the masked digits reassemble the value + mod 2²⁵⁵. -/ +theorem digits_tile (B : ℕ) : + B % 2^51 + + (B / 2^51 % 2^51) * 2^51 + + (B / 2^102 % 2^51) * 2^102 + + (B / 2^153 % 2^51) * 2^153 + + (B / 2^204 % 2^51) * 2^204 = B % 2^255 := by + have e1 : B / 2^51 / 2^51 = B / 2^102 := by rw [Nat.div_div_eq_div_mul]; norm_num + have e2 : B / 2^102 / 2^51 = B / 2^153 := by rw [Nat.div_div_eq_div_mul]; norm_num + have e3 : B / 2^153 / 2^51 = B / 2^204 := by rw [Nat.div_div_eq_div_mul]; norm_num + have e4 : B / 2^204 / 2^51 = B / 2^255 := by rw [Nat.div_div_eq_div_mul]; norm_num + have d0 := Nat.div_add_mod B (2^51) + have d1 := Nat.div_add_mod (B / 2^51) (2^51) + have d2 := Nat.div_add_mod (B / 2^102) (2^51) + have d3 := Nat.div_add_mod (B / 2^153) (2^51) + have d4 := Nat.div_add_mod (B / 2^204) (2^51) + have dT := Nat.div_add_mod B (2^255) + omega + +/-- **The byte parser is exact below bit 255**: for any 32 input bytes, + `from_bytes` succeeds with 51-bit limbs denoting `bytesVal b mod 2²⁵⁵` + (the sign bit is discarded, the rest is the little-endian value). -/ +theorem from_bytes_spec (bytes : Std.Array Std.U8 32#usize) + (b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 : Std.U8) + (hbl : (↑bytes : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31]) : + backend.serial.u64.field.FieldElement51.from_bytes bytes ⦃ r => + Bnd r (2^51) ∧ feVal r = bytesVal bytes % 2^255 ⦄ := by + unfold backend.serial.u64.field.FieldElement51.from_bytes + step as ⟨msk0, hmsk0⟩ + step as ⟨mask, hmask⟩ + have hmaskv : mask.val = 2251799813685247 := by + rw [hmask, hmsk0] + simp [Nat.shiftLeft_eq] + scalar_tac + -- window 0: bytes 0..7, shift 0 + step as ⟨s0, hs0⟩ + have hs0v : s0.val = (↑bytes : List Std.U8) := by rw [hs0]; rfl + step with (load8_at_spec s0 0#usize b0 b1 b2 b3 b4 b5 b6 b7 + (by rw [Slice.length]; simp [hs0v, hbl]) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + (by simp only [hs0v, hbl]; rfl) + ) as ⟨w0, hw0⟩ + step as ⟨l0, hl0⟩ + have hl0v : l0.val = w0.val % 2^51 := by + rw [hl0, UScalar.val_and, hmaskv, nat_and_mask] + norm_num + -- window 1: bytes 6..13, shift 3 + step as ⟨s1, hs1⟩ + have hs1v : s1.val = (↑bytes : List Std.U8) := by rw [hs1]; rfl + step with (load8_at_spec s1 6#usize b6 b7 b8 b9 b10 b11 b12 b13 + (by rw [Slice.length]; simp [hs1v, hbl]) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + (by simp only [hs1v, hbl]; rfl) + ) as ⟨w1, hw1⟩ + step as ⟨sh1, hsh1⟩ + step as ⟨l1, hl1⟩ + have hl1v : l1.val = (w1.val / 2^3) % 2^51 := by + rw [hl1, UScalar.val_and, hmaskv, nat_and_mask, hsh1, nat_shr] + norm_num + -- window 2: bytes 12..19, shift 6 + step as ⟨s2, hs2⟩ + have hs2v : s2.val = (↑bytes : List Std.U8) := by rw [hs2]; rfl + step with (load8_at_spec s2 12#usize b12 b13 b14 b15 b16 b17 b18 b19 + (by rw [Slice.length]; simp [hs2v, hbl]) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + (by simp only [hs2v, hbl]; rfl) + ) as ⟨w2, hw2⟩ + step as ⟨sh2, hsh2⟩ + step as ⟨l2, hl2⟩ + have hl2v : l2.val = (w2.val / 2^6) % 2^51 := by + rw [hl2, UScalar.val_and, hmaskv, nat_and_mask, hsh2, nat_shr] + norm_num + -- window 3: bytes 19..26, shift 1 + step as ⟨s3, hs3⟩ + have hs3v : s3.val = (↑bytes : List Std.U8) := by rw [hs3]; rfl + step with (load8_at_spec s3 19#usize b19 b20 b21 b22 b23 b24 b25 b26 + (by rw [Slice.length]; simp [hs3v, hbl]) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + (by simp only [hs3v, hbl]; rfl) + ) as ⟨w3, hw3⟩ + step as ⟨sh3, hsh3⟩ + step as ⟨l3, hl3⟩ + have hl3v : l3.val = (w3.val / 2^1) % 2^51 := by + rw [hl3, UScalar.val_and, hmaskv, nat_and_mask, hsh3, nat_shr] + norm_num + -- window 4: bytes 24..31, shift 12 + step as ⟨s4, hs4⟩ + have hs4v : s4.val = (↑bytes : List Std.U8) := by rw [hs4]; rfl + step with (load8_at_spec s4 24#usize b24 b25 b26 b27 b28 b29 b30 b31 + (by rw [Slice.length]; simp [hs4v, hbl]) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + (by simp only [hs4v, hbl]; rfl) + ) as ⟨w4, hw4⟩ + step as ⟨sh4, hsh4⟩ + step as ⟨l4, hl4⟩ + have hl4v : l4.val = (w4.val / 2^12) % 2^51 := by + rw [hl4, UScalar.val_and, hmaskv, nat_and_mask, hsh4, nat_shr] + norm_num + try simp only [spec_ok] + constructor + · rw [Bnd_eq _ l0 l1 l2 l3 l4 _ rfl] + refine ⟨?_, ?_, ?_, ?_, ?_⟩ <;> · first + | (rw [hl0v]; exact Nat.mod_lt _ (by norm_num)) + | (rw [hl1v]; exact Nat.mod_lt _ (by norm_num)) + | (rw [hl2v]; exact Nat.mod_lt _ (by norm_num)) + | (rw [hl3v]; exact Nat.mod_lt _ (by norm_num)) + | (rw [hl4v]; exact Nat.mod_lt _ (by norm_num)) + · rw [feVal_eq _ l0 l1 l2 l3 l4 rfl] + unfold limbsVal + have hB : bytesVal bytes = b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248 := by + simp only [bytesVal, hbl] + have hwin0 : w0.val = bytesVal bytes / 2^0 % 2^64 := by + have hdecomp : (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248) + = (0) + 2^0 * (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56) + 2^0 * 2^64 * (b8.val + b9.val * 2^8 + b10.val * 2^16 + b11.val * 2^24 + b12.val * 2^32 + b13.val * 2^40 + b14.val * 2^48 + b15.val * 2^56 + b16.val * 2^64 + b17.val * 2^72 + b18.val * 2^80 + b19.val * 2^88 + b20.val * 2^96 + b21.val * 2^104 + b22.val * 2^112 + b23.val * 2^120 + b24.val * 2^128 + b25.val * 2^136 + b26.val * 2^144 + b27.val * 2^152 + b28.val * 2^160 + b29.val * 2^168 + b30.val * 2^176 + b31.val * 2^184) := by + ring + rw [hw0, hB, hdecomp] + exact (window_extract (0) (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56) (b8.val + b9.val * 2^8 + b10.val * 2^16 + b11.val * 2^24 + b12.val * 2^32 + b13.val * 2^40 + b14.val * 2^48 + b15.val * 2^56 + b16.val * 2^64 + b17.val * 2^72 + b18.val * 2^80 + b19.val * 2^88 + b20.val * 2^96 + b21.val * 2^104 + b22.val * 2^112 + b23.val * 2^120 + b24.val * 2^128 + b25.val * 2^136 + b26.val * 2^144 + b27.val * 2^152 + b28.val * 2^160 + b29.val * 2^168 + b30.val * 2^176 + b31.val * 2^184) 0 + (by scalar_tac) (by scalar_tac)).symm + have hlimb0 : l0.val = bytesVal bytes / 2^0 % 2^51 := by + rw [hl0v, hwin0] + exact Nat.mod_mod_of_dvd _ (pow_dvd_pow 2 (by norm_num)) + have hwin1 : w1.val = bytesVal bytes / 2^48 % 2^64 := by + have hdecomp : (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248) + = (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40) + 2^48 * (b6.val + b7.val * 2^8 + b8.val * 2^16 + b9.val * 2^24 + b10.val * 2^32 + b11.val * 2^40 + b12.val * 2^48 + b13.val * 2^56) + 2^48 * 2^64 * (b14.val + b15.val * 2^8 + b16.val * 2^16 + b17.val * 2^24 + b18.val * 2^32 + b19.val * 2^40 + b20.val * 2^48 + b21.val * 2^56 + b22.val * 2^64 + b23.val * 2^72 + b24.val * 2^80 + b25.val * 2^88 + b26.val * 2^96 + b27.val * 2^104 + b28.val * 2^112 + b29.val * 2^120 + b30.val * 2^128 + b31.val * 2^136) := by + ring + rw [hw1, hB, hdecomp] + exact (window_extract (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40) (b6.val + b7.val * 2^8 + b8.val * 2^16 + b9.val * 2^24 + b10.val * 2^32 + b11.val * 2^40 + b12.val * 2^48 + b13.val * 2^56) (b14.val + b15.val * 2^8 + b16.val * 2^16 + b17.val * 2^24 + b18.val * 2^32 + b19.val * 2^40 + b20.val * 2^48 + b21.val * 2^56 + b22.val * 2^64 + b23.val * 2^72 + b24.val * 2^80 + b25.val * 2^88 + b26.val * 2^96 + b27.val * 2^104 + b28.val * 2^112 + b29.val * 2^120 + b30.val * 2^128 + b31.val * 2^136) 48 + (by scalar_tac) (by scalar_tac)).symm + have hlimb1 : l1.val = bytesVal bytes / 2^51 % 2^51 := by + rw [hl1v, hwin1] + have := window_shift (bytesVal bytes) 6 3 (by norm_num) + norm_num at this ⊢ + rw [this] + have hwin2 : w2.val = bytesVal bytes / 2^96 % 2^64 := by + have hdecomp : (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248) + = (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88) + 2^96 * (b12.val + b13.val * 2^8 + b14.val * 2^16 + b15.val * 2^24 + b16.val * 2^32 + b17.val * 2^40 + b18.val * 2^48 + b19.val * 2^56) + 2^96 * 2^64 * (b20.val + b21.val * 2^8 + b22.val * 2^16 + b23.val * 2^24 + b24.val * 2^32 + b25.val * 2^40 + b26.val * 2^48 + b27.val * 2^56 + b28.val * 2^64 + b29.val * 2^72 + b30.val * 2^80 + b31.val * 2^88) := by + ring + rw [hw2, hB, hdecomp] + exact (window_extract (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88) (b12.val + b13.val * 2^8 + b14.val * 2^16 + b15.val * 2^24 + b16.val * 2^32 + b17.val * 2^40 + b18.val * 2^48 + b19.val * 2^56) (b20.val + b21.val * 2^8 + b22.val * 2^16 + b23.val * 2^24 + b24.val * 2^32 + b25.val * 2^40 + b26.val * 2^48 + b27.val * 2^56 + b28.val * 2^64 + b29.val * 2^72 + b30.val * 2^80 + b31.val * 2^88) 96 + (by scalar_tac) (by scalar_tac)).symm + have hlimb2 : l2.val = bytesVal bytes / 2^102 % 2^51 := by + rw [hl2v, hwin2] + have := window_shift (bytesVal bytes) 12 6 (by norm_num) + norm_num at this ⊢ + rw [this] + have hwin3 : w3.val = bytesVal bytes / 2^152 % 2^64 := by + have hdecomp : (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248) + = (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144) + 2^152 * (b19.val + b20.val * 2^8 + b21.val * 2^16 + b22.val * 2^24 + b23.val * 2^32 + b24.val * 2^40 + b25.val * 2^48 + b26.val * 2^56) + 2^152 * 2^64 * (b27.val + b28.val * 2^8 + b29.val * 2^16 + b30.val * 2^24 + b31.val * 2^32) := by + ring + rw [hw3, hB, hdecomp] + exact (window_extract (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144) (b19.val + b20.val * 2^8 + b21.val * 2^16 + b22.val * 2^24 + b23.val * 2^32 + b24.val * 2^40 + b25.val * 2^48 + b26.val * 2^56) (b27.val + b28.val * 2^8 + b29.val * 2^16 + b30.val * 2^24 + b31.val * 2^32) 152 + (by scalar_tac) (by scalar_tac)).symm + have hlimb3 : l3.val = bytesVal bytes / 2^153 % 2^51 := by + rw [hl3v, hwin3] + have := window_shift (bytesVal bytes) 19 1 (by norm_num) + norm_num at this ⊢ + rw [this] + have hwin4 : w4.val = bytesVal bytes / 2^192 % 2^64 := by + have hdecomp : (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248) + = (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184) + 2^192 * (b24.val + b25.val * 2^8 + b26.val * 2^16 + b27.val * 2^24 + b28.val * 2^32 + b29.val * 2^40 + b30.val * 2^48 + b31.val * 2^56) + 2^192 * 2^64 * 0 := by + ring + rw [hw4, hB, hdecomp] + exact (window_extract (b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184) (b24.val + b25.val * 2^8 + b26.val * 2^16 + b27.val * 2^24 + b28.val * 2^32 + b29.val * 2^40 + b30.val * 2^48 + b31.val * 2^56) 0 192 + (by scalar_tac) (by scalar_tac)).symm + have hlimb4 : l4.val = bytesVal bytes / 2^204 % 2^51 := by + rw [hl4v, hwin4] + have := window_shift (bytesVal bytes) 24 12 (by norm_num) + norm_num at this ⊢ + rw [this] + have ht := digits_tile (bytesVal bytes) + norm_num at ht hlimb0 hlimb1 hlimb2 hlimb3 hlimb4 ⊢ + omega + +end CurveFieldProofs diff --git a/verification/check.sh b/verification/check.sh index 6a69e45..b969b67 100755 --- a/verification/check.sh +++ b/verification/check.sh @@ -67,6 +67,9 @@ PROOFS=( SigApexSpec PointLiftSpec PointEqSpec + DecompressSpec + FromBytesSpec + DecompressMain ) # Fully-qualified certificate names; each must be axiom-clean. CERTS=( @@ -91,6 +94,12 @@ CERTS=( ScalarProofs.from_bytes_mod_order_wide_spec CurveFieldProofs.vartime_dsm_basepoint_spec CurveFieldProofs.enc_point_inj + CurveFieldProofs.pow_p58_spec + CurveFieldProofs.fe_ct_eq_spec + CurveFieldProofs.sqrt_core + CurveFieldProofs.sqrt_ratio_i_sq_spec + CurveFieldProofs.from_bytes_spec + CurveFieldProofs.decompress_of_canonical ) # Imports needed so every certificate in CERTS is in scope for the audit. AUDIT_IMPORTS=( @@ -107,6 +116,9 @@ AUDIT_IMPORTS=( Proofs.ScalarPackSpec Proofs.PointLiftSpec Proofs.PointEqSpec + Proofs.DecompressSpec + Proofs.FromBytesSpec + Proofs.DecompressMain ) # ── Phase 0: resource + integrity guards ──────────────────────────────────── @@ -198,17 +210,18 @@ lake env bash -c " cd '$HERE' ALLOWED='[propext, Classical.choice, Quot.sound, ed25519.Signature, ed_sigs.sha512_hash3, ed25519.Signature.r_bytes, ed25519.Signature.s_bytes]' AUD=\$(mktemp '$HERE/.apex-XXXX.lean') - { echo 'import Proofs.SigApexSpec'; echo 'import Proofs.PointLiftSpec'; echo 'import Proofs.PointEqSpec'; echo '#print axioms CurveFieldProofs.verify_accepts_iff'; echo '#print axioms CurveFieldProofs.verify_accepts_iff_point'; echo '#print axioms CurveFieldProofs.verify_accepts_iff_point_eq'; } > \"\$AUD\" + { echo 'import Proofs.SigApexSpec'; echo 'import Proofs.PointLiftSpec'; echo 'import Proofs.PointEqSpec'; echo 'import Proofs.DecompressMain'; echo '#print axioms CurveFieldProofs.verify_accepts_iff'; echo '#print axioms CurveFieldProofs.verify_accepts_iff_point'; echo '#print axioms CurveFieldProofs.verify_accepts_iff_point_eq'; echo '#print axioms CurveFieldProofs.verify_accepts_iff_decompress'; } > \"\$AUD\" OUT=\$(LEAN_TIMEOUT=$TIMEOUT LEAN_MEM_MB=4096 '$HERE/lean-guard' \"\$AUD\" 2>&1) echo \"\$OUT\" rm -f \"\$AUD\" FLAT=\$(echo \"\$OUT\" | tr '\\n' ' ' | tr -s ' ') if echo \"\$FLAT\" | grep -qF \"'CurveFieldProofs.verify_accepts_iff' depends on axioms: \$ALLOWED\" \ && echo \"\$FLAT\" | grep -qF \"'CurveFieldProofs.verify_accepts_iff_point' depends on axioms: \$ALLOWED\" \ - && echo \"\$FLAT\" | grep -qF \"'CurveFieldProofs.verify_accepts_iff_point_eq' depends on axioms: \$ALLOWED\"; then - echo ' apex axiom cone = exactly the SHA-512 + wire-format boundary (no curve/scalar/backend axioms)' + && echo \"\$FLAT\" | grep -qF \"'CurveFieldProofs.verify_accepts_iff_point_eq' depends on axioms: \$ALLOWED\" \ + && echo \"\$FLAT\" | grep -qF \"'CurveFieldProofs.verify_accepts_iff_decompress' depends on axioms: \$ALLOWED\"; then + echo ' apex + full-lift axiom cones = exactly the SHA-512 + wire-format boundary (no curve/scalar/backend axioms)' else - echo 'APEX AUDIT FAILED: verify_accepts_iff cone is not the documented boundary'; exit 1 + echo 'APEX AUDIT FAILED: apex/full-lift cone is not the documented boundary'; exit 1 fi " diff --git a/verification/extract.sh b/verification/extract.sh index 18ffa3c..67e08d0 100755 --- a/verification/extract.sh +++ b/verification/extract.sh @@ -5,9 +5,11 @@ # roots: crate::field, crate::backend::serial::u64::field, # crate::backend::serial::curve_models, crate::edwards # (same widening the reference solution used for its Tier-1 addition-law -# theorem; scalar-mul backends and decompress internals stay opaque — -# upstream Aeneas cannot translate them; they are modeled/axiomatized in -# gen/CurveField/FunsExternal.lean OUTSIDE every certificate's cone). +# theorem; scalar-mul backends stay opaque — upstream Aeneas cannot +# translate them; they are modeled/axiomatized in +# gen/CurveField/FunsExternal.lean OUTSIDE every certificate's cone. +# decompress IS extracted since the phase-2 full lift — the source's +# step_2 uses the documented negate-then-conditional-assign rewrite). # # Rust --charon--> CurveField.llbc --aeneas--> gen/CurveField/*.lean # @@ -59,7 +61,6 @@ charon cargo --preset=aeneas \ --opaque 'crate::backend::serial::scalar_mul::pippenger' \ --opaque 'crate::backend::vector' \ --opaque 'crate::backend::scalar_fits_in_128_bits' \ - --opaque 'crate::edwards::decompress' \ --opaque 'crate::edwards::_::sum' \ --opaque 'crate::edwards::_::from_slice' \ --dest-file "$HERE/CurveField.llbc" \ diff --git a/verification/gen/CurveField/Funs.lean b/verification/gen/CurveField/Funs.lean index dfe5918..ba75db1 100644 --- a/verification/gen/CurveField/Funs.lean +++ b/verification/gen/CurveField/Funs.lean @@ -2098,7 +2098,7 @@ def backend.serial.scalar_mul.vartime_double_base.dsm_loop table_a table_b r j /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::as_projective]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 541:4-547:5 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 546:4-552:5 -/ def edwards.EdwardsPoint.as_projective (self : edwards.EdwardsPoint) : Result backend.serial.curve_models.ProjectivePoint @@ -2106,7 +2106,7 @@ def edwards.EdwardsPoint.as_projective ok { X := self.X, Y := self.Y, Z := self.Z } /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::double]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 774:4-776:5 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 779:4-781:5 -/ def edwards.EdwardsPoint.double (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do let pp ← edwards.EdwardsPoint.as_projective self @@ -2125,7 +2125,7 @@ def backend.serial.u64.constants.EDWARDS_D2 ]) /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::as_projective_niels]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 528:4-535:5 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 533:4-540:5 -/ def edwards.EdwardsPoint.as_projective_niels (self : edwards.EdwardsPoint) : Result backend.serial.curve_models.ProjectiveNielsPoint @@ -3764,7 +3764,7 @@ def backend.vartime_double_base_mul backend.serial.scalar_mul.vartime_double_base.mul a A b /-- [curve25519::edwards::{impl core::ops::arith::Add<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}::add]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 785:4-787:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 790:4-792:5 Visibility: public -/ def Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint.add @@ -3788,7 +3788,7 @@ def edwards.EdwardsPoint.Insts.CoreOpsArithAddEdwardsPointEdwardsPoint.add self rhs /-- [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::scalar::Scalar, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}::mul]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 884:4-886:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 889:4-891:5 Visibility: public -/ def Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul (self : edwards.EdwardsPoint) (scalar : scalar.Scalar) : @@ -3797,7 +3797,7 @@ def Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul backend.variable_base_mul self scalar /-- [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::scalar::Scalar}::mul]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 896:4-898:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 901:4-903:5 Visibility: public -/ def Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint.mul (self : scalar.Scalar) (point : edwards.EdwardsPoint) : @@ -3817,7 +3817,7 @@ def SharedAScalar.Insts.CoreOpsArithMulEdwardsPointEdwardsPoint.mul rhs /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_base]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 906:4-916:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 911:4-921:5 Visibility: public -/ def edwards.EdwardsPoint.mul_base (scalar : scalar.Scalar) : Result edwards.EdwardsPoint := do @@ -4190,7 +4190,7 @@ def scalar.Scalar.from_bytes_mod_order_wide scalar.Scalar52.pack unpacked /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::vartime_double_scalar_mul_basepoint]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1068:4-1074:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1073:4-1079:5 Visibility: public -/ def edwards.EdwardsPoint.vartime_double_scalar_mul_basepoint (a : scalar.Scalar) (A : edwards.EdwardsPoint) (b : scalar.Scalar) : @@ -4260,7 +4260,7 @@ def field.FieldElement51.invert t20 t3 /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::to_affine]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 564:4-569:5 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 569:4-574:5 -/ def edwards.EdwardsPoint.to_affine (self : edwards.EdwardsPoint) : Result edwards.affine.AffinePoint := do let recip ← field.FieldElement51.invert self.Z @@ -4296,7 +4296,7 @@ def edwards.affine.AffinePoint.compress ok s1 /-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::compress]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 615:4-617:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 620:4-622:5 Visibility: public -/ def edwards.EdwardsPoint.compress (self : edwards.EdwardsPoint) : Result edwards.CompressedEdwardsY := do @@ -4326,7 +4326,7 @@ def ed_sigs.verification_key.VerificationKey.recompute_r_sha512 edwards.EdwardsPoint.compress ep /-- [curve25519::edwards::{curve25519::edwards::CompressedEdwardsY}::as_bytes]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 198:4-200:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 197:4-199:5 Visibility: public -/ def edwards.CompressedEdwardsY.as_bytes (self : edwards.CompressedEdwardsY) : Result (Array Std.U8 32#usize) := do @@ -4625,14 +4625,14 @@ def scalar.Scalar.Insts.CoreOpsArithMulShared0AffinePointEdwardsPoint : } /-- [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::CompressedEdwardsY}::clone]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:15-174:20 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 173:15-173:20 Visibility: public -/ def edwards.CompressedEdwardsY.Insts.CoreCloneClone.clone (self : edwards.CompressedEdwardsY) : Result edwards.CompressedEdwardsY := do ok self /-- Trait implementation: [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:15-174:20 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 173:15-173:20 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreCloneClone : core.clone.Clone edwards.CompressedEdwardsY := { @@ -4640,7 +4640,7 @@ def edwards.CompressedEdwardsY.Insts.CoreCloneClone : core.clone.Clone } /-- Trait implementation: [curve25519::edwards::{impl core::marker::Copy for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:9-174:13 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 173:9-173:13 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreMarkerCopy : core.marker.Copy edwards.CompressedEdwardsY := { @@ -4648,7 +4648,7 @@ def edwards.CompressedEdwardsY.Insts.CoreMarkerCopy : core.marker.Copy } /-- [curve25519::edwards::{impl core::hash::Hash for curve25519::edwards::CompressedEdwardsY}::hash]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:22-174:26 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 173:22-173:26 Visibility: public -/ def edwards.CompressedEdwardsY.Insts.CoreHashHash.hash {__H : Type} (corehashHasherInst : core.hash.Hasher __H) @@ -4659,7 +4659,7 @@ def edwards.CompressedEdwardsY.Insts.CoreHashHash.hash state /-- Trait implementation: [curve25519::edwards::{impl core::hash::Hash for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:22-174:26 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 173:22-173:26 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreHashHash : core.hash.Hash edwards.CompressedEdwardsY := { @@ -4668,7 +4668,7 @@ def edwards.CompressedEdwardsY.Insts.CoreHashHash : core.hash.Hash } /-- [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::CompressedEdwardsY}::ct_eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 178:4-180:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 177:4-179:5 Visibility: public -/ def edwards.CompressedEdwardsY.Insts.SubtleConstantTimeEq.ct_eq (self : edwards.CompressedEdwardsY) (other : edwards.CompressedEdwardsY) : @@ -4681,7 +4681,7 @@ def edwards.CompressedEdwardsY.Insts.SubtleConstantTimeEq.ct_eq Slice.Insts.SubtleConstantTimeEq.ct_eq U8.Insts.SubtleConstantTimeEq s s1 /-- Trait implementation: [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 177:0-181:1 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 176:0-180:1 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.SubtleConstantTimeEq : subtle.ConstantTimeEq edwards.CompressedEdwardsY := { @@ -4689,7 +4689,7 @@ def edwards.CompressedEdwardsY.Insts.SubtleConstantTimeEq : } /-- [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::CompressedEdwardsY}::eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 185:4-187:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 184:4-186:5 Visibility: public -/ def edwards.CompressedEdwardsY.Insts.CoreCmpPartialEqCompressedEdwardsY.eq (self : edwards.CompressedEdwardsY) (other : edwards.CompressedEdwardsY) : @@ -4700,7 +4700,7 @@ def edwards.CompressedEdwardsY.Insts.CoreCmpPartialEqCompressedEdwardsY.eq core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice c /-- Trait implementation: [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 184:0-188:1 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 183:0-187:1 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreCmpPartialEqCompressedEdwardsY : core.cmp.PartialEq edwards.CompressedEdwardsY edwards.CompressedEdwardsY := { @@ -4708,7 +4708,7 @@ def edwards.CompressedEdwardsY.Insts.CoreCmpPartialEqCompressedEdwardsY : } /-- Trait implementation: [curve25519::edwards::{impl core::cmp::Eq for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 183:0-183:33 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 182:0-182:33 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreCmpEq : core.cmp.Eq edwards.CompressedEdwardsY := { @@ -4719,7 +4719,7 @@ def edwards.CompressedEdwardsY.Insts.CoreCmpEq : core.cmp.Eq } /-- [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::CompressedEdwardsY}::fmt]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 191:4-193:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 190:4-192:5 Visibility: public -/ def edwards.CompressedEdwardsY.Insts.CoreFmtDebug.fmt (self : edwards.CompressedEdwardsY) (f : core.fmt.Formatter) : @@ -4739,7 +4739,7 @@ def edwards.CompressedEdwardsY.Insts.CoreFmtDebug.fmt core.fmt.Formatter.write_fmt f a1 /-- Trait implementation: [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 190:0-194:1 -/ + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 189:0-193:1 -/ @[reducible] def edwards.CompressedEdwardsY.Insts.CoreFmtDebug : core.fmt.Debug edwards.CompressedEdwardsY := { @@ -4747,717 +4747,33 @@ def edwards.CompressedEdwardsY.Insts.CoreFmtDebug : core.fmt.Debug } /-- [curve25519::edwards::{curve25519::edwards::CompressedEdwardsY}::to_bytes]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 203:4-205:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 202:4-204:5 Visibility: public -/ def edwards.CompressedEdwardsY.to_bytes (self : edwards.CompressedEdwardsY) : Result (Array Std.U8 32#usize) := do ok self -/-- [curve25519::edwards::{curve25519::edwards::CompressedEdwardsY}::decompress]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 211:4-219:5 - Visibility: public -/ -def edwards.CompressedEdwardsY.decompress - (self : edwards.CompressedEdwardsY) : - Result (Option edwards.EdwardsPoint) +/-- [curve25519::edwards::decompress::step_2]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 239:4-262:5 -/ +def edwards.decompress.step_2 + (repr : edwards.CompressedEdwardsY) + (X : backend.serial.u64.field.FieldElement51) + (Y : backend.serial.u64.field.FieldElement51) + (Z : backend.serial.u64.field.FieldElement51) : + Result edwards.EdwardsPoint := do - let (is_valid_y_coord, X, Y, Z) ← edwards.decompress.step_1 self - let b ← - core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice - is_valid_y_coord - if b - then let ep ← edwards.decompress.step_2 self X Y Z - ok (some ep) - else ok none - -/-- [curve25519::edwards::{impl core::convert::TryFrom<&'_0 [u8], core::array::TryFromSliceError> for curve25519::edwards::CompressedEdwardsY}::try_from]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 263:4-265:5 - Visibility: public -/ -def - edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError.try_from - (slice : Slice Std.U8) : - Result (core.result.Result edwards.CompressedEdwardsY - core.array.TryFromSliceError) - := do - edwards.CompressedEdwardsY.from_slice slice - -/-- Trait implementation: [curve25519::edwards::{impl core::convert::TryFrom<&'_0 [u8], core::array::TryFromSliceError> for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 260:0-266:1 -/ -@[reducible] -def - edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError - : core.convert.TryFrom edwards.CompressedEdwardsY (Slice Std.U8) - core.array.TryFromSliceError := { - try_from := - edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError.try_from -} - -/-- [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::EdwardsPoint}::clone]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 388:15-388:20 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreCloneClone.clone - (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do - ok self - -/-- Trait implementation: [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 388:15-388:20 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreCloneClone : core.clone.Clone - edwards.EdwardsPoint := { - clone := edwards.EdwardsPoint.Insts.CoreCloneClone.clone -} - -/-- Trait implementation: [curve25519::edwards::{impl core::marker::Copy for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 388:9-388:13 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreMarkerCopy : core.marker.Copy - edwards.EdwardsPoint := { - cloneInst := edwards.EdwardsPoint.Insts.CoreCloneClone -} - -/-- [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::CompressedEdwardsY}::identity]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 402:4-407:5 - Visibility: public -/ -def edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity - : Result edwards.CompressedEdwardsY := do - ok - (Array.make 32#usize [ - 1#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, - 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, - 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8 - ]) - -/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 401:0-408:1 -/ -@[reducible] -def edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity : traits.Identity - edwards.CompressedEdwardsY := { - identity := - edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity -} - -/-- [curve25519::edwards::{impl core::default::Default for curve25519::edwards::CompressedEdwardsY}::default]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 411:4-413:5 - Visibility: public -/ -def edwards.CompressedEdwardsY.Insts.CoreDefaultDefault.default - : Result edwards.CompressedEdwardsY := do - edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity - -/-- Trait implementation: [curve25519::edwards::{impl core::default::Default for curve25519::edwards::CompressedEdwardsY}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 410:0-414:1 -/ -@[reducible] -def edwards.CompressedEdwardsY.Insts.CoreDefaultDefault : core.default.Default - edwards.CompressedEdwardsY := { - default := edwards.CompressedEdwardsY.Insts.CoreDefaultDefault.default -} - -/-- [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::EdwardsPoint}::identity]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 429:4-436:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity - : Result edwards.EdwardsPoint := do - let fe ← backend.serial.u64.field.FieldElement51.ZERO - let fe1 ← backend.serial.u64.field.FieldElement51.ONE - ok { X := fe, Y := fe1, Z := fe1, T := fe } - -/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 428:0-437:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity : traits.Identity - edwards.EdwardsPoint := { - identity := edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity -} - -/-- [curve25519::edwards::{impl core::default::Default for curve25519::edwards::EdwardsPoint}::default]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 440:4-442:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreDefaultDefault.default - : Result edwards.EdwardsPoint := do - edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity - -/-- Trait implementation: [curve25519::edwards::{impl core::default::Default for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 439:0-443:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreDefaultDefault : core.default.Default - edwards.EdwardsPoint := { - default := edwards.EdwardsPoint.Insts.CoreDefaultDefault.default -} - -/-- [curve25519::edwards::{impl curve25519::traits::ValidityCheck for curve25519::edwards::EdwardsPoint}::is_valid]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 474:4-479:5 -/ -def edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck.is_valid - (self : edwards.EdwardsPoint) : Result Bool := do - let pp ← edwards.EdwardsPoint.as_projective self - let point_on_curve ← - backend.serial.curve_models.ProjectivePoint.Insts.Curve25519TraitsValidityCheck.is_valid - pp + let a ← edwards.CompressedEdwardsY.as_bytes repr + let i ← Array.index_usize a 31#usize + let i1 ← i >>> 7#i32 + let compressed_sign_bit ← subtle.Choice.Insts.CoreConvertFromU8.from i1 + let X_neg ← Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg X + let X1 ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_assign + X X_neg compressed_sign_bit let fe ← Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.X self.Y - let fe1 ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.Z self.T - let on_segre_image ← - backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51.eq - fe fe1 - if point_on_curve - then ok on_segre_image - else ok false - -/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::ValidityCheck for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 473:0-480:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck : - traits.ValidityCheck edwards.EdwardsPoint := { - is_valid := edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck.is_valid -} - -/-- [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}::conditional_select]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 487:4-494:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_select - (a : edwards.EdwardsPoint) (b : edwards.EdwardsPoint) - (choice : subtle.Choice) : - Result edwards.EdwardsPoint - := do - let fe ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select - a.X b.X choice - let fe1 ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select - a.Y b.Y choice - let fe2 ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select - a.Z b.Z choice - let fe3 ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select - a.T b.T choice - ok { X := fe, Y := fe1, Z := fe2, T := fe3 } - -/-- Trait implementation: [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 486:0-495:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.SubtleConditionallySelectable : - subtle.ConditionallySelectable edwards.EdwardsPoint := { - coremarkerCopyInst := edwards.EdwardsPoint.Insts.CoreMarkerCopy - conditional_select := - edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_select - conditional_assign := - edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_assign - conditional_swap := - edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_swap -} - -/-- [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::EdwardsPoint}::ct_eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 502:4-511:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq - (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : - Result subtle.Choice - := do - let fe ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.X other.Z - let fe1 ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - other.X self.Z - let c ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq fe - fe1 - let fe2 ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.Y other.Z - let fe3 ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - other.Y self.Z - let c1 ← - backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq - fe2 fe3 - subtle.Choice.Insts.CoreOpsBitBitAndChoiceChoice.bitand c c1 - -/-- Trait implementation: [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 501:0-512:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.SubtleConstantTimeEq : subtle.ConstantTimeEq - edwards.EdwardsPoint := { - ct_eq := edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq -} - -/-- [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::EdwardsPoint}::eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 515:4-517:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint.eq - (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : - Result Bool - := do - let c ← edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq self other - core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice c - -/-- Trait implementation: [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 514:0-518:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint : - core.cmp.PartialEq edwards.EdwardsPoint edwards.EdwardsPoint := { - eq := edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint.eq -} - -/-- Trait implementation: [curve25519::edwards::{impl core::cmp::Eq for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 520:0-520:27 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreCmpEq : core.cmp.Eq edwards.EdwardsPoint - := { - partialEqInst := edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint - assert_fields_are_eq := - edwards.EdwardsPoint.Insts.CoreCmpEq.assert_fields_are_eq -} - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::as_affine_niels]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 551:4-561:5 -/ -def edwards.EdwardsPoint.as_affine_niels - (self : edwards.EdwardsPoint) : - Result backend.serial.curve_models.AffineNielsPoint - := do - let recip ← field.FieldElement51.invert self.Z - let x ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.X recip - let y ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - self.Y recip - let fe ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - x y - let fe1 ← backend.serial.u64.constants.EDWARDS_D2 - let xy2d ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - fe fe1 - let fe2 ← - Shared0FieldElement51.Insts.CoreOpsArithAddSharedAFieldElement51FieldElement51.add - y x - let fe3 ← - Shared0FieldElement51.Insts.CoreOpsArithSubSharedAFieldElement51FieldElement51.sub - y x - ok { y_plus_x := fe2, y_minus_x := fe3, xy2d } - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::to_montgomery]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 580:4-590:5 - Visibility: public -/ -def edwards.EdwardsPoint.to_montgomery - (self : edwards.EdwardsPoint) : Result montgomery.MontgomeryPoint := do - let U ← - Shared0FieldElement51.Insts.CoreOpsArithAddSharedAFieldElement51FieldElement51.add - self.Z self.Y - let W ← - Shared0FieldElement51.Insts.CoreOpsArithSubSharedAFieldElement51FieldElement51.sub - self.Z self.Y - let fe ← field.FieldElement51.invert W - let u ← - Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul - U fe - let a ← backend.serial.u64.field.FieldElement51.to_bytes u - ok a - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Add<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 783:0-788:1 -/ -@[reducible] -def Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint : - core.ops.arith.Add edwards.EdwardsPoint edwards.EdwardsPoint - edwards.EdwardsPoint := { - add := - Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint.add -} - -/-- [curve25519::edwards::{impl core::ops::arith::AddAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}::add_assign]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 797:4-799:5 - Visibility: public -/ -def - edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint.add_assign - (self : edwards.EdwardsPoint) (_rhs : edwards.EdwardsPoint) : - Result edwards.EdwardsPoint - := do - Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint.add - self _rhs - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::AddAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 796:0-800:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint : - core.ops.arith.AddAssign edwards.EdwardsPoint edwards.EdwardsPoint := { - add_assign := - edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint.add_assign -} - -/-- [curve25519::edwards::{impl core::ops::arith::Sub<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}::sub]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 806:4-808:5 - Visibility: public -/ -def - Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub - (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : - Result edwards.EdwardsPoint - := do - let pnp ← edwards.EdwardsPoint.as_projective_niels other - let cp ← - Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAProjectiveNielsPointCompletedPoint.sub - self pnp - backend.serial.curve_models.CompletedPoint.as_extended cp - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Sub<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 804:0-809:1 -/ -@[reducible] -def Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint : - core.ops.arith.Sub edwards.EdwardsPoint edwards.EdwardsPoint - edwards.EdwardsPoint := { - sub := - Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub -} - -/-- [curve25519::edwards::{impl core::ops::arith::SubAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}::sub_assign]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 818:4-820:5 - Visibility: public -/ -def - edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint.sub_assign - (self : edwards.EdwardsPoint) (_rhs : edwards.EdwardsPoint) : - Result edwards.EdwardsPoint - := do - Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub - self _rhs - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::SubAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 817:0-821:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint : - core.ops.arith.SubAssign edwards.EdwardsPoint edwards.EdwardsPoint := { - sub_assign := - edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint.sub_assign -} - -/-- Trait implementation: [curve25519::edwards::{impl core::iter::traits::accum::Sum for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 825:0-835:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreIterTraitsAccumSum {T : Type} - (coreborrowBorrowTEdwardsPointInst : core.borrow.Borrow T - edwards.EdwardsPoint) : core.iter.traits.accum.Sum edwards.EdwardsPoint T - := { - sum := fun {I : Type} (coreitertraitsiteratorIteratorInst : - core.iter.traits.iterator.Iterator I T) => - edwards.EdwardsPoint.Insts.CoreIterTraitsAccumSum.sum - coreborrowBorrowTEdwardsPointInst coreitertraitsiteratorIteratorInst -} - -/-- [curve25519::edwards::{impl core::ops::arith::Neg for &'_0 curve25519::edwards::EdwardsPoint}::neg]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 844:4-851:5 - Visibility: public -/ -def Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg - (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do - let fe ← - Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg self.X - let fe1 ← - Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg self.T - ok { self with X := fe, T := fe1 } - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Neg for &'_0 curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 841:0-852:1 -/ -@[reducible] -def Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint : core.ops.arith.Neg - edwards.EdwardsPoint edwards.EdwardsPoint := { - neg := Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg -} - -/-- [curve25519::edwards::{impl core::ops::arith::Neg for curve25519::edwards::EdwardsPoint}::neg]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 857:4-859:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg - (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do - Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg self - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Neg for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 854:0-860:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint : core.ops.arith.Neg - edwards.EdwardsPoint edwards.EdwardsPoint := { - neg := edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg -} - -/-- [curve25519::edwards::{impl core::ops::arith::MulAssign<&'a curve25519::scalar::Scalar> for curve25519::edwards::EdwardsPoint}::mul_assign]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 867:4-870:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar.mul_assign - (self : edwards.EdwardsPoint) (scalar : scalar.Scalar) : - Result edwards.EdwardsPoint - := do - Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul self - scalar - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::MulAssign<&'a curve25519::scalar::Scalar> for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 866:0-871:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar : - core.ops.arith.MulAssign edwards.EdwardsPoint scalar.Scalar := { - mul_assign := - edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar.mul_assign -} - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::scalar::Scalar, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 878:0-887:1 -/ -@[reducible] -def Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint : - core.ops.arith.Mul edwards.EdwardsPoint scalar.Scalar edwards.EdwardsPoint - := { - mul := Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul -} - -/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::scalar::Scalar}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 889:0-899:1 -/ -@[reducible] -def Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint : - core.ops.arith.Mul scalar.Scalar edwards.EdwardsPoint edwards.EdwardsPoint - := { - mul := Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint.mul -} - -/-- [curve25519::scalar::clamp_integer]: - Source: 'curve25519/solana-ed25519/src/scalar.rs', lines 1610:0-1615:1 - Visibility: public -/ -def scalar.clamp_integer - (bytes : Array Std.U8 32#usize) : Result (Array Std.U8 32#usize) := do - let i ← Array.index_usize bytes 0#usize - let i1 ← lift (i &&& 248#u8) - let bytes1 ← Array.update bytes 0#usize i1 - let i2 ← Array.index_usize bytes1 31#usize - let i3 ← lift (i2 &&& 127#u8) - let bytes2 ← Array.update bytes1 31#usize i3 - let i4 ← Array.index_usize bytes2 31#usize - let i5 ← lift (i4 ||| 64#u8) - Array.update bytes2 31#usize i5 - -/-- [curve25519::edwards::{impl core::ops::arith::Mul for curve25519::scalar::Scalar}::mul]: - Source: 'curve25519/solana-ed25519/src/macros.rs', lines 107:12-109:13 - Visibility: public -/ -def scalar.Scalar.Insts.CoreOpsArithMulEdwardsPointEdwardsPoint.mul - (self : scalar.Scalar) (rhs : edwards.EdwardsPoint) : - Result edwards.EdwardsPoint - := do - Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint.mul self - rhs - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_clamped]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 920:4-932:5 - Visibility: public -/ -def edwards.EdwardsPoint.mul_clamped - (self : edwards.EdwardsPoint) (bytes : Array Std.U8 32#usize) : - Result edwards.EdwardsPoint - := do - let a ← scalar.clamp_integer bytes - scalar.Scalar.Insts.CoreOpsArithMulEdwardsPointEdwardsPoint.mul - { bytes := a } self - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_base_clamped]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 936:4-944:5 - Visibility: public -/ -def edwards.EdwardsPoint.mul_base_clamped - (bytes : Array Std.U8 32#usize) : Result edwards.EdwardsPoint := do - let a ← scalar.clamp_integer bytes - edwards.EdwardsPoint.mul_base { bytes := a } - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::vartime_triple_scalar_mul_basepoint]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1099:4-1107:5 - Visibility: public -/ -def edwards.EdwardsPoint.vartime_triple_scalar_mul_basepoint - (a1 : scalar.Scalar) (A1 : edwards.EdwardsPoint) (a2 : scalar.Scalar) - (A2 : edwards.EdwardsPoint) (b : scalar.Scalar) : - Result edwards.EdwardsPoint - := do - backend.vartime_triple_base_mul_128_128_256 a1 A1 a2 A2 b - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: loop body 0: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1395:8-1398:9 -/ -@[rust_loop_body] -def edwards.EdwardsPoint.mul_by_pow_2_loop.body - (iter : core.ops.range.Range Std.U32) - (s : backend.serial.curve_models.ProjectivePoint) : - Result (ControlFlow ((core.ops.range.Range Std.U32) × - backend.serial.curve_models.ProjectivePoint) - backend.serial.curve_models.ProjectivePoint) - := do - let (o, iter1) ← - core.iter.range.IteratorRange.next U32.Insts.CoreIterRangeStep iter - match o with - | none => ok (done s) - | some _ => - let r ← backend.serial.curve_models.ProjectivePoint.double s - let s1 ← backend.serial.curve_models.CompletedPoint.as_projective r - ok (cont (iter1, s1)) - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: loop 0: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1395:8-1398:9 -/ -@[rust_loop] -def edwards.EdwardsPoint.mul_by_pow_2_loop - (iter : core.ops.range.Range Std.U32) - (s : backend.serial.curve_models.ProjectivePoint) : - Result backend.serial.curve_models.ProjectivePoint - := do - loop - (fun (iter1, s1) => edwards.EdwardsPoint.mul_by_pow_2_loop.body iter1 s1) - (iter, s) - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1391:4-1401:5 -/ -def edwards.EdwardsPoint.mul_by_pow_2 - (self : edwards.EdwardsPoint) (k : Std.U32) : - Result edwards.EdwardsPoint - := do - massert (k > 0#u32) - let s ← edwards.EdwardsPoint.as_projective self - let i ← k - 1#u32 - let s1 ← - edwards.EdwardsPoint.mul_by_pow_2_loop { start := 0#u32, «end» := i } s - let cp ← backend.serial.curve_models.ProjectivePoint.double s1 - backend.serial.curve_models.CompletedPoint.as_extended cp - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_cofactor]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1386:4-1388:5 - Visibility: public -/ -def edwards.EdwardsPoint.mul_by_cofactor - (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do - edwards.EdwardsPoint.mul_by_pow_2 self 3#u32 - -/-- [curve25519::traits::{impl curve25519::traits::IsIdentity for T}::is_identity]: - Source: 'curve25519/solana-ed25519/src/traits.rs', lines 51:4-53:5 - Visibility: public -/ -def traits.IsIdentity.Blanket.is_identity - {T : Type} (subtleConstantTimeEqInst : subtle.ConstantTimeEq T) (IdentityInst - : traits.Identity T) (self : T) : - Result Bool - := do - let t ← IdentityInst.identity - let c ← subtleConstantTimeEqInst.ct_eq self t - core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice c - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::is_small_order]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1426:4-1428:5 - Visibility: public -/ -def edwards.EdwardsPoint.is_small_order - (self : edwards.EdwardsPoint) : Result Bool := do - let ep ← edwards.EdwardsPoint.mul_by_cofactor self - traits.IsIdentity.Blanket.is_identity - edwards.EdwardsPoint.Insts.SubtleConstantTimeEq - edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity ep - -/-- [curve25519::edwards::{impl core::ops::arith::Mul for &'a curve25519::edwards::EdwardsPoint}::mul]: - Source: 'curve25519/solana-ed25519/src/macros.rs', lines 100:12-102:13 - Visibility: public -/ -def SharedAEdwardsPoint.Insts.CoreOpsArithMulScalarEdwardsPoint.mul - (self : edwards.EdwardsPoint) (rhs : scalar.Scalar) : - Result edwards.EdwardsPoint - := do - Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul self - rhs - -/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::is_torsion_free]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1456:4-1458:5 - Visibility: public -/ -def edwards.EdwardsPoint.is_torsion_free - (self : edwards.EdwardsPoint) : Result Bool := do - let ep ← - SharedAEdwardsPoint.Insts.CoreOpsArithMulScalarEdwardsPoint.mul self - constants.BASEPOINT_ORDER - traits.IsIdentity.Blanket.is_identity - edwards.EdwardsPoint.Insts.SubtleConstantTimeEq - edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity ep - -/-- [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::EdwardsPoint}::fmt]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1466:4-1472:5 - Visibility: public -/ -def edwards.EdwardsPoint.Insts.CoreFmtDebug.fmt - (self : edwards.EdwardsPoint) (f : core.fmt.Formatter) : - Result ((core.result.Result Unit core.fmt.Error) × core.fmt.Formatter) - := do - let a ← - core.fmt.rt.Argument.new_debug (core.fmt.DebugShared - backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.X - let a1 ← - core.fmt.rt.Argument.new_debug (core.fmt.DebugShared - backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.Y - let a2 ← - core.fmt.rt.Argument.new_debug (core.fmt.DebugShared - backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.Z - let a3 ← - core.fmt.rt.Argument.new_debug (core.fmt.DebugShared - backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.T - let a4 ← - core.fmt.Arguments.new - (Array.make 48#usize [ - 18#u8, 69#u8, 100#u8, 119#u8, 97#u8, 114#u8, 100#u8, 115#u8, 80#u8, - 111#u8, 105#u8, 110#u8, 116#u8, 123#u8, 10#u8, 9#u8, 88#u8, 58#u8, - 32#u8, 192#u8, 6#u8, 44#u8, 10#u8, 9#u8, 89#u8, 58#u8, 32#u8, 192#u8, - 6#u8, 44#u8, 10#u8, 9#u8, 90#u8, 58#u8, 32#u8, 192#u8, 6#u8, 44#u8, - 10#u8, 9#u8, 84#u8, 58#u8, 32#u8, 192#u8, 2#u8, 10#u8, 125#u8, 0#u8 - ]) (Array.make 4#usize [ a, a1, a2, a3 ]) - core.fmt.Formatter.write_fmt f a4 - -/-- Trait implementation: [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::EdwardsPoint}] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1465:0-1473:1 -/ -@[reducible] -def edwards.EdwardsPoint.Insts.CoreFmtDebug : core.fmt.Debug - edwards.EdwardsPoint := { - fmt := edwards.EdwardsPoint.Insts.CoreFmtDebug.fmt -} - -/-- Trait implementation: [curve25519::field::{impl core::cmp::PartialEq for curve25519::backend::serial::u64::field::FieldElement51}] - Source: 'curve25519/solana-ed25519/src/field.rs', lines 54:0-58:1 -/ -@[reducible] -def - backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51 - : core.cmp.PartialEq backend.serial.u64.field.FieldElement51 - backend.serial.u64.field.FieldElement51 := { - eq := - backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51.eq -} - -/-- Trait implementation: [curve25519::field::{impl core::cmp::Eq for curve25519::backend::serial::u64::field::FieldElement51}] - Source: 'curve25519/solana-ed25519/src/field.rs', lines 52:0-52:27 -/ -@[reducible] -def backend.serial.u64.field.FieldElement51.Insts.CoreCmpEq : core.cmp.Eq - backend.serial.u64.field.FieldElement51 := { - partialEqInst := - backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51 - assert_fields_are_eq := - backend.serial.u64.field.FieldElement51.Insts.CoreCmpEq.assert_fields_are_eq -} - -/-- Trait implementation: [curve25519::field::{impl subtle::ConstantTimeEq for curve25519::backend::serial::u64::field::FieldElement51}] - Source: 'curve25519/solana-ed25519/src/field.rs', lines 60:0-67:1 -/ -@[reducible] -def backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq : - subtle.ConstantTimeEq backend.serial.u64.field.FieldElement51 := { - ct_eq := - backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq -} - -/-- [curve25519::field::{impl core::default::Default for curve25519::backend::serial::u64::field::FieldElement51}::default]: - Source: 'curve25519/solana-ed25519/src/field.rs', lines 70:4-72:5 - Visibility: public -/ -def backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault.default - : Result backend.serial.u64.field.FieldElement51 := do - backend.serial.u64.field.FieldElement51.ZERO - -/-- Trait implementation: [curve25519::field::{impl core::default::Default for curve25519::backend::serial::u64::field::FieldElement51}] - Source: 'curve25519/solana-ed25519/src/field.rs', lines 69:0-73:1 -/ -@[reducible] -def backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault : - core.default.Default backend.serial.u64.field.FieldElement51 := { - default := - backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault.default -} - -/-- [curve25519::field::{curve25519::backend::serial::u64::field::FieldElement51}::is_zero]: - Source: 'curve25519/solana-ed25519/src/field.rs', lines 149:4-154:5 -/ -def field.FieldElement51.is_zero - (self : backend.serial.u64.field.FieldElement51) : Result subtle.Choice := do - let zero := Array.repeat 32#usize 0#u8 - let bytes ← backend.serial.u64.field.FieldElement51.to_bytes self - let s ← lift (Array.to_slice bytes) - let s1 ← lift (Array.to_slice zero) - Slice.Insts.SubtleConstantTimeEq.ct_eq U8.Insts.SubtleConstantTimeEq s s1 + X1 Y + ok { X := X1, Y, Z, T := fe } /-- [curve25519::field::{curve25519::backend::serial::u64::field::FieldElement51}::pow_p58]: Source: 'curve25519/solana-ed25519/src/field.rs', lines 294:4-303:5 -/ @@ -5533,6 +4849,737 @@ def field.FieldElement51.sqrt_ratio_i flipped_sign_sqrt ok (was_nonzero_square, r2) +/-- [curve25519::edwards::decompress::step_1]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 225:4-236:5 -/ +def edwards.decompress.step_1 + (repr : edwards.CompressedEdwardsY) : + Result (subtle.Choice × backend.serial.u64.field.FieldElement51 × + backend.serial.u64.field.FieldElement51 × + backend.serial.u64.field.FieldElement51) + := do + let a ← edwards.CompressedEdwardsY.as_bytes repr + let Y ← backend.serial.u64.field.FieldElement51.from_bytes a + let Z ← backend.serial.u64.field.FieldElement51.ONE + let YY ← backend.serial.u64.field.FieldElement51.square Y + let u ← + Shared0FieldElement51.Insts.CoreOpsArithSubSharedAFieldElement51FieldElement51.sub + YY Z + let fe ← backend.serial.u64.constants.EDWARDS_D + let fe1 ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + YY fe + let v ← + Shared0FieldElement51.Insts.CoreOpsArithAddSharedAFieldElement51FieldElement51.add + fe1 Z + let (is_valid_y_coord, X) ← field.FieldElement51.sqrt_ratio_i u v + ok (is_valid_y_coord, X, Y, Z) + +/-- [curve25519::edwards::{curve25519::edwards::CompressedEdwardsY}::decompress]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 210:4-218:5 + Visibility: public -/ +def edwards.CompressedEdwardsY.decompress + (self : edwards.CompressedEdwardsY) : + Result (Option edwards.EdwardsPoint) + := do + let (is_valid_y_coord, X, Y, Z) ← edwards.decompress.step_1 self + let b ← + core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice + is_valid_y_coord + if b + then let ep ← edwards.decompress.step_2 self X Y Z + ok (some ep) + else ok none + +/-- [curve25519::edwards::{impl core::convert::TryFrom<&'_0 [u8], core::array::TryFromSliceError> for curve25519::edwards::CompressedEdwardsY}::try_from]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 268:4-270:5 + Visibility: public -/ +def + edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError.try_from + (slice : Slice Std.U8) : + Result (core.result.Result edwards.CompressedEdwardsY + core.array.TryFromSliceError) + := do + edwards.CompressedEdwardsY.from_slice slice + +/-- Trait implementation: [curve25519::edwards::{impl core::convert::TryFrom<&'_0 [u8], core::array::TryFromSliceError> for curve25519::edwards::CompressedEdwardsY}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 265:0-271:1 -/ +@[reducible] +def + edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError + : core.convert.TryFrom edwards.CompressedEdwardsY (Slice Std.U8) + core.array.TryFromSliceError := { + try_from := + edwards.CompressedEdwardsY.Insts.CoreConvertTryFromShared0SliceU8TryFromSliceError.try_from +} + +/-- [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::EdwardsPoint}::clone]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 393:15-393:20 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreCloneClone.clone + (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do + ok self + +/-- Trait implementation: [curve25519::edwards::{impl core::clone::Clone for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 393:15-393:20 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreCloneClone : core.clone.Clone + edwards.EdwardsPoint := { + clone := edwards.EdwardsPoint.Insts.CoreCloneClone.clone +} + +/-- Trait implementation: [curve25519::edwards::{impl core::marker::Copy for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 393:9-393:13 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreMarkerCopy : core.marker.Copy + edwards.EdwardsPoint := { + cloneInst := edwards.EdwardsPoint.Insts.CoreCloneClone +} + +/-- [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::CompressedEdwardsY}::identity]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 407:4-412:5 + Visibility: public -/ +def edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity + : Result edwards.CompressedEdwardsY := do + ok + (Array.make 32#usize [ + 1#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, + 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, + 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8 + ]) + +/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::CompressedEdwardsY}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 406:0-413:1 -/ +@[reducible] +def edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity : traits.Identity + edwards.CompressedEdwardsY := { + identity := + edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity +} + +/-- [curve25519::edwards::{impl core::default::Default for curve25519::edwards::CompressedEdwardsY}::default]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 416:4-418:5 + Visibility: public -/ +def edwards.CompressedEdwardsY.Insts.CoreDefaultDefault.default + : Result edwards.CompressedEdwardsY := do + edwards.CompressedEdwardsY.Insts.Curve25519TraitsIdentity.identity + +/-- Trait implementation: [curve25519::edwards::{impl core::default::Default for curve25519::edwards::CompressedEdwardsY}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 415:0-419:1 -/ +@[reducible] +def edwards.CompressedEdwardsY.Insts.CoreDefaultDefault : core.default.Default + edwards.CompressedEdwardsY := { + default := edwards.CompressedEdwardsY.Insts.CoreDefaultDefault.default +} + +/-- [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::EdwardsPoint}::identity]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 434:4-441:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity + : Result edwards.EdwardsPoint := do + let fe ← backend.serial.u64.field.FieldElement51.ZERO + let fe1 ← backend.serial.u64.field.FieldElement51.ONE + ok { X := fe, Y := fe1, Z := fe1, T := fe } + +/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::Identity for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 433:0-442:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity : traits.Identity + edwards.EdwardsPoint := { + identity := edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity +} + +/-- [curve25519::edwards::{impl core::default::Default for curve25519::edwards::EdwardsPoint}::default]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 445:4-447:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreDefaultDefault.default + : Result edwards.EdwardsPoint := do + edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity.identity + +/-- Trait implementation: [curve25519::edwards::{impl core::default::Default for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 444:0-448:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreDefaultDefault : core.default.Default + edwards.EdwardsPoint := { + default := edwards.EdwardsPoint.Insts.CoreDefaultDefault.default +} + +/-- [curve25519::edwards::{impl curve25519::traits::ValidityCheck for curve25519::edwards::EdwardsPoint}::is_valid]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 479:4-484:5 -/ +def edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck.is_valid + (self : edwards.EdwardsPoint) : Result Bool := do + let pp ← edwards.EdwardsPoint.as_projective self + let point_on_curve ← + backend.serial.curve_models.ProjectivePoint.Insts.Curve25519TraitsValidityCheck.is_valid + pp + let fe ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.X self.Y + let fe1 ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.Z self.T + let on_segre_image ← + backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51.eq + fe fe1 + if point_on_curve + then ok on_segre_image + else ok false + +/-- Trait implementation: [curve25519::edwards::{impl curve25519::traits::ValidityCheck for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 478:0-485:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck : + traits.ValidityCheck edwards.EdwardsPoint := { + is_valid := edwards.EdwardsPoint.Insts.Curve25519TraitsValidityCheck.is_valid +} + +/-- [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}::conditional_select]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 492:4-499:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_select + (a : edwards.EdwardsPoint) (b : edwards.EdwardsPoint) + (choice : subtle.Choice) : + Result edwards.EdwardsPoint + := do + let fe ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select + a.X b.X choice + let fe1 ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select + a.Y b.Y choice + let fe2 ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select + a.Z b.Z choice + let fe3 ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConditionallySelectable.conditional_select + a.T b.T choice + ok { X := fe, Y := fe1, Z := fe2, T := fe3 } + +/-- Trait implementation: [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 491:0-500:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.SubtleConditionallySelectable : + subtle.ConditionallySelectable edwards.EdwardsPoint := { + coremarkerCopyInst := edwards.EdwardsPoint.Insts.CoreMarkerCopy + conditional_select := + edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_select + conditional_assign := + edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_assign + conditional_swap := + edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_swap +} + +/-- [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::EdwardsPoint}::ct_eq]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 507:4-516:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq + (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : + Result subtle.Choice + := do + let fe ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.X other.Z + let fe1 ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + other.X self.Z + let c ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq fe + fe1 + let fe2 ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.Y other.Z + let fe3 ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + other.Y self.Z + let c1 ← + backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq + fe2 fe3 + subtle.Choice.Insts.CoreOpsBitBitAndChoiceChoice.bitand c c1 + +/-- Trait implementation: [curve25519::edwards::{impl subtle::ConstantTimeEq for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 506:0-517:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.SubtleConstantTimeEq : subtle.ConstantTimeEq + edwards.EdwardsPoint := { + ct_eq := edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq +} + +/-- [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::EdwardsPoint}::eq]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 520:4-522:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint.eq + (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : + Result Bool + := do + let c ← edwards.EdwardsPoint.Insts.SubtleConstantTimeEq.ct_eq self other + core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice c + +/-- Trait implementation: [curve25519::edwards::{impl core::cmp::PartialEq for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 519:0-523:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint : + core.cmp.PartialEq edwards.EdwardsPoint edwards.EdwardsPoint := { + eq := edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint.eq +} + +/-- Trait implementation: [curve25519::edwards::{impl core::cmp::Eq for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 525:0-525:27 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreCmpEq : core.cmp.Eq edwards.EdwardsPoint + := { + partialEqInst := edwards.EdwardsPoint.Insts.CoreCmpPartialEqEdwardsPoint + assert_fields_are_eq := + edwards.EdwardsPoint.Insts.CoreCmpEq.assert_fields_are_eq +} + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::as_affine_niels]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 556:4-566:5 -/ +def edwards.EdwardsPoint.as_affine_niels + (self : edwards.EdwardsPoint) : + Result backend.serial.curve_models.AffineNielsPoint + := do + let recip ← field.FieldElement51.invert self.Z + let x ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.X recip + let y ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + self.Y recip + let fe ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + x y + let fe1 ← backend.serial.u64.constants.EDWARDS_D2 + let xy2d ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + fe fe1 + let fe2 ← + Shared0FieldElement51.Insts.CoreOpsArithAddSharedAFieldElement51FieldElement51.add + y x + let fe3 ← + Shared0FieldElement51.Insts.CoreOpsArithSubSharedAFieldElement51FieldElement51.sub + y x + ok { y_plus_x := fe2, y_minus_x := fe3, xy2d } + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::to_montgomery]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 585:4-595:5 + Visibility: public -/ +def edwards.EdwardsPoint.to_montgomery + (self : edwards.EdwardsPoint) : Result montgomery.MontgomeryPoint := do + let U ← + Shared0FieldElement51.Insts.CoreOpsArithAddSharedAFieldElement51FieldElement51.add + self.Z self.Y + let W ← + Shared0FieldElement51.Insts.CoreOpsArithSubSharedAFieldElement51FieldElement51.sub + self.Z self.Y + let fe ← field.FieldElement51.invert W + let u ← + Shared0FieldElement51.Insts.CoreOpsArithMulSharedAFieldElement51FieldElement51.mul + U fe + let a ← backend.serial.u64.field.FieldElement51.to_bytes u + ok a + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Add<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 788:0-793:1 -/ +@[reducible] +def Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint : + core.ops.arith.Add edwards.EdwardsPoint edwards.EdwardsPoint + edwards.EdwardsPoint := { + add := + Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint.add +} + +/-- [curve25519::edwards::{impl core::ops::arith::AddAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}::add_assign]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 802:4-804:5 + Visibility: public -/ +def + edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint.add_assign + (self : edwards.EdwardsPoint) (_rhs : edwards.EdwardsPoint) : + Result edwards.EdwardsPoint + := do + Shared0EdwardsPoint.Insts.CoreOpsArithAddSharedAEdwardsPointEdwardsPoint.add + self _rhs + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::AddAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 801:0-805:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint : + core.ops.arith.AddAssign edwards.EdwardsPoint edwards.EdwardsPoint := { + add_assign := + edwards.EdwardsPoint.Insts.CoreOpsArithAddAssignSharedAEdwardsPoint.add_assign +} + +/-- [curve25519::edwards::{impl core::ops::arith::Sub<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}::sub]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 811:4-813:5 + Visibility: public -/ +def + Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub + (self : edwards.EdwardsPoint) (other : edwards.EdwardsPoint) : + Result edwards.EdwardsPoint + := do + let pnp ← edwards.EdwardsPoint.as_projective_niels other + let cp ← + Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAProjectiveNielsPointCompletedPoint.sub + self pnp + backend.serial.curve_models.CompletedPoint.as_extended cp + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Sub<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 809:0-814:1 -/ +@[reducible] +def Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint : + core.ops.arith.Sub edwards.EdwardsPoint edwards.EdwardsPoint + edwards.EdwardsPoint := { + sub := + Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub +} + +/-- [curve25519::edwards::{impl core::ops::arith::SubAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}::sub_assign]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 823:4-825:5 + Visibility: public -/ +def + edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint.sub_assign + (self : edwards.EdwardsPoint) (_rhs : edwards.EdwardsPoint) : + Result edwards.EdwardsPoint + := do + Shared0EdwardsPoint.Insts.CoreOpsArithSubSharedAEdwardsPointEdwardsPoint.sub + self _rhs + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::SubAssign<&'a curve25519::edwards::EdwardsPoint> for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 822:0-826:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint : + core.ops.arith.SubAssign edwards.EdwardsPoint edwards.EdwardsPoint := { + sub_assign := + edwards.EdwardsPoint.Insts.CoreOpsArithSubAssignSharedAEdwardsPoint.sub_assign +} + +/-- Trait implementation: [curve25519::edwards::{impl core::iter::traits::accum::Sum for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 830:0-840:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreIterTraitsAccumSum {T : Type} + (coreborrowBorrowTEdwardsPointInst : core.borrow.Borrow T + edwards.EdwardsPoint) : core.iter.traits.accum.Sum edwards.EdwardsPoint T + := { + sum := fun {I : Type} (coreitertraitsiteratorIteratorInst : + core.iter.traits.iterator.Iterator I T) => + edwards.EdwardsPoint.Insts.CoreIterTraitsAccumSum.sum + coreborrowBorrowTEdwardsPointInst coreitertraitsiteratorIteratorInst +} + +/-- [curve25519::edwards::{impl core::ops::arith::Neg for &'_0 curve25519::edwards::EdwardsPoint}::neg]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 849:4-856:5 + Visibility: public -/ +def Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg + (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do + let fe ← + Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg self.X + let fe1 ← + Shared0FieldElement51.Insts.CoreOpsArithNegFieldElement51.neg self.T + ok { self with X := fe, T := fe1 } + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Neg for &'_0 curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 846:0-857:1 -/ +@[reducible] +def Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint : core.ops.arith.Neg + edwards.EdwardsPoint edwards.EdwardsPoint := { + neg := Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg +} + +/-- [curve25519::edwards::{impl core::ops::arith::Neg for curve25519::edwards::EdwardsPoint}::neg]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 862:4-864:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg + (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do + Shared0EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg self + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Neg for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 859:0-865:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint : core.ops.arith.Neg + edwards.EdwardsPoint edwards.EdwardsPoint := { + neg := edwards.EdwardsPoint.Insts.CoreOpsArithNegEdwardsPoint.neg +} + +/-- [curve25519::edwards::{impl core::ops::arith::MulAssign<&'a curve25519::scalar::Scalar> for curve25519::edwards::EdwardsPoint}::mul_assign]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 872:4-875:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar.mul_assign + (self : edwards.EdwardsPoint) (scalar : scalar.Scalar) : + Result edwards.EdwardsPoint + := do + Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul self + scalar + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::MulAssign<&'a curve25519::scalar::Scalar> for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 871:0-876:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar : + core.ops.arith.MulAssign edwards.EdwardsPoint scalar.Scalar := { + mul_assign := + edwards.EdwardsPoint.Insts.CoreOpsArithMulAssignSharedAScalar.mul_assign +} + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::scalar::Scalar, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 883:0-892:1 -/ +@[reducible] +def Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint : + core.ops.arith.Mul edwards.EdwardsPoint scalar.Scalar edwards.EdwardsPoint + := { + mul := Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul +} + +/-- Trait implementation: [curve25519::edwards::{impl core::ops::arith::Mul<&'a curve25519::edwards::EdwardsPoint, curve25519::edwards::EdwardsPoint> for &'_1 curve25519::scalar::Scalar}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 894:0-904:1 -/ +@[reducible] +def Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint : + core.ops.arith.Mul scalar.Scalar edwards.EdwardsPoint edwards.EdwardsPoint + := { + mul := Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint.mul +} + +/-- [curve25519::scalar::clamp_integer]: + Source: 'curve25519/solana-ed25519/src/scalar.rs', lines 1610:0-1615:1 + Visibility: public -/ +def scalar.clamp_integer + (bytes : Array Std.U8 32#usize) : Result (Array Std.U8 32#usize) := do + let i ← Array.index_usize bytes 0#usize + let i1 ← lift (i &&& 248#u8) + let bytes1 ← Array.update bytes 0#usize i1 + let i2 ← Array.index_usize bytes1 31#usize + let i3 ← lift (i2 &&& 127#u8) + let bytes2 ← Array.update bytes1 31#usize i3 + let i4 ← Array.index_usize bytes2 31#usize + let i5 ← lift (i4 ||| 64#u8) + Array.update bytes2 31#usize i5 + +/-- [curve25519::edwards::{impl core::ops::arith::Mul for curve25519::scalar::Scalar}::mul]: + Source: 'curve25519/solana-ed25519/src/macros.rs', lines 107:12-109:13 + Visibility: public -/ +def scalar.Scalar.Insts.CoreOpsArithMulEdwardsPointEdwardsPoint.mul + (self : scalar.Scalar) (rhs : edwards.EdwardsPoint) : + Result edwards.EdwardsPoint + := do + Shared0Scalar.Insts.CoreOpsArithMulSharedAEdwardsPointEdwardsPoint.mul self + rhs + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_clamped]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 925:4-937:5 + Visibility: public -/ +def edwards.EdwardsPoint.mul_clamped + (self : edwards.EdwardsPoint) (bytes : Array Std.U8 32#usize) : + Result edwards.EdwardsPoint + := do + let a ← scalar.clamp_integer bytes + scalar.Scalar.Insts.CoreOpsArithMulEdwardsPointEdwardsPoint.mul + { bytes := a } self + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_base_clamped]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 941:4-949:5 + Visibility: public -/ +def edwards.EdwardsPoint.mul_base_clamped + (bytes : Array Std.U8 32#usize) : Result edwards.EdwardsPoint := do + let a ← scalar.clamp_integer bytes + edwards.EdwardsPoint.mul_base { bytes := a } + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::vartime_triple_scalar_mul_basepoint]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1104:4-1112:5 + Visibility: public -/ +def edwards.EdwardsPoint.vartime_triple_scalar_mul_basepoint + (a1 : scalar.Scalar) (A1 : edwards.EdwardsPoint) (a2 : scalar.Scalar) + (A2 : edwards.EdwardsPoint) (b : scalar.Scalar) : + Result edwards.EdwardsPoint + := do + backend.vartime_triple_base_mul_128_128_256 a1 A1 a2 A2 b + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: loop body 0: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1400:8-1403:9 -/ +@[rust_loop_body] +def edwards.EdwardsPoint.mul_by_pow_2_loop.body + (iter : core.ops.range.Range Std.U32) + (s : backend.serial.curve_models.ProjectivePoint) : + Result (ControlFlow ((core.ops.range.Range Std.U32) × + backend.serial.curve_models.ProjectivePoint) + backend.serial.curve_models.ProjectivePoint) + := do + let (o, iter1) ← + core.iter.range.IteratorRange.next U32.Insts.CoreIterRangeStep iter + match o with + | none => ok (done s) + | some _ => + let r ← backend.serial.curve_models.ProjectivePoint.double s + let s1 ← backend.serial.curve_models.CompletedPoint.as_projective r + ok (cont (iter1, s1)) + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: loop 0: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1400:8-1403:9 -/ +@[rust_loop] +def edwards.EdwardsPoint.mul_by_pow_2_loop + (iter : core.ops.range.Range Std.U32) + (s : backend.serial.curve_models.ProjectivePoint) : + Result backend.serial.curve_models.ProjectivePoint + := do + loop + (fun (iter1, s1) => edwards.EdwardsPoint.mul_by_pow_2_loop.body iter1 s1) + (iter, s) + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_pow_2]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1396:4-1406:5 -/ +def edwards.EdwardsPoint.mul_by_pow_2 + (self : edwards.EdwardsPoint) (k : Std.U32) : + Result edwards.EdwardsPoint + := do + massert (k > 0#u32) + let s ← edwards.EdwardsPoint.as_projective self + let i ← k - 1#u32 + let s1 ← + edwards.EdwardsPoint.mul_by_pow_2_loop { start := 0#u32, «end» := i } s + let cp ← backend.serial.curve_models.ProjectivePoint.double s1 + backend.serial.curve_models.CompletedPoint.as_extended cp + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::mul_by_cofactor]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1391:4-1393:5 + Visibility: public -/ +def edwards.EdwardsPoint.mul_by_cofactor + (self : edwards.EdwardsPoint) : Result edwards.EdwardsPoint := do + edwards.EdwardsPoint.mul_by_pow_2 self 3#u32 + +/-- [curve25519::traits::{impl curve25519::traits::IsIdentity for T}::is_identity]: + Source: 'curve25519/solana-ed25519/src/traits.rs', lines 51:4-53:5 + Visibility: public -/ +def traits.IsIdentity.Blanket.is_identity + {T : Type} (subtleConstantTimeEqInst : subtle.ConstantTimeEq T) (IdentityInst + : traits.Identity T) (self : T) : + Result Bool + := do + let t ← IdentityInst.identity + let c ← subtleConstantTimeEqInst.ct_eq self t + core.convert.IntoFrom.into Bool.Insts.CoreConvertFromChoice c + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::is_small_order]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1431:4-1433:5 + Visibility: public -/ +def edwards.EdwardsPoint.is_small_order + (self : edwards.EdwardsPoint) : Result Bool := do + let ep ← edwards.EdwardsPoint.mul_by_cofactor self + traits.IsIdentity.Blanket.is_identity + edwards.EdwardsPoint.Insts.SubtleConstantTimeEq + edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity ep + +/-- [curve25519::edwards::{impl core::ops::arith::Mul for &'a curve25519::edwards::EdwardsPoint}::mul]: + Source: 'curve25519/solana-ed25519/src/macros.rs', lines 100:12-102:13 + Visibility: public -/ +def SharedAEdwardsPoint.Insts.CoreOpsArithMulScalarEdwardsPoint.mul + (self : edwards.EdwardsPoint) (rhs : scalar.Scalar) : + Result edwards.EdwardsPoint + := do + Shared0EdwardsPoint.Insts.CoreOpsArithMulSharedAScalarEdwardsPoint.mul self + rhs + +/-- [curve25519::edwards::{curve25519::edwards::EdwardsPoint}::is_torsion_free]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1461:4-1463:5 + Visibility: public -/ +def edwards.EdwardsPoint.is_torsion_free + (self : edwards.EdwardsPoint) : Result Bool := do + let ep ← + SharedAEdwardsPoint.Insts.CoreOpsArithMulScalarEdwardsPoint.mul self + constants.BASEPOINT_ORDER + traits.IsIdentity.Blanket.is_identity + edwards.EdwardsPoint.Insts.SubtleConstantTimeEq + edwards.EdwardsPoint.Insts.Curve25519TraitsIdentity ep + +/-- [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::EdwardsPoint}::fmt]: + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1471:4-1477:5 + Visibility: public -/ +def edwards.EdwardsPoint.Insts.CoreFmtDebug.fmt + (self : edwards.EdwardsPoint) (f : core.fmt.Formatter) : + Result ((core.result.Result Unit core.fmt.Error) × core.fmt.Formatter) + := do + let a ← + core.fmt.rt.Argument.new_debug (core.fmt.DebugShared + backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.X + let a1 ← + core.fmt.rt.Argument.new_debug (core.fmt.DebugShared + backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.Y + let a2 ← + core.fmt.rt.Argument.new_debug (core.fmt.DebugShared + backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.Z + let a3 ← + core.fmt.rt.Argument.new_debug (core.fmt.DebugShared + backend.serial.u64.field.FieldElement51.Insts.CoreFmtDebug) self.T + let a4 ← + core.fmt.Arguments.new + (Array.make 48#usize [ + 18#u8, 69#u8, 100#u8, 119#u8, 97#u8, 114#u8, 100#u8, 115#u8, 80#u8, + 111#u8, 105#u8, 110#u8, 116#u8, 123#u8, 10#u8, 9#u8, 88#u8, 58#u8, + 32#u8, 192#u8, 6#u8, 44#u8, 10#u8, 9#u8, 89#u8, 58#u8, 32#u8, 192#u8, + 6#u8, 44#u8, 10#u8, 9#u8, 90#u8, 58#u8, 32#u8, 192#u8, 6#u8, 44#u8, + 10#u8, 9#u8, 84#u8, 58#u8, 32#u8, 192#u8, 2#u8, 10#u8, 125#u8, 0#u8 + ]) (Array.make 4#usize [ a, a1, a2, a3 ]) + core.fmt.Formatter.write_fmt f a4 + +/-- Trait implementation: [curve25519::edwards::{impl core::fmt::Debug for curve25519::edwards::EdwardsPoint}] + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 1470:0-1478:1 -/ +@[reducible] +def edwards.EdwardsPoint.Insts.CoreFmtDebug : core.fmt.Debug + edwards.EdwardsPoint := { + fmt := edwards.EdwardsPoint.Insts.CoreFmtDebug.fmt +} + +/-- Trait implementation: [curve25519::field::{impl core::cmp::PartialEq for curve25519::backend::serial::u64::field::FieldElement51}] + Source: 'curve25519/solana-ed25519/src/field.rs', lines 54:0-58:1 -/ +@[reducible] +def + backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51 + : core.cmp.PartialEq backend.serial.u64.field.FieldElement51 + backend.serial.u64.field.FieldElement51 := { + eq := + backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51.eq +} + +/-- Trait implementation: [curve25519::field::{impl core::cmp::Eq for curve25519::backend::serial::u64::field::FieldElement51}] + Source: 'curve25519/solana-ed25519/src/field.rs', lines 52:0-52:27 -/ +@[reducible] +def backend.serial.u64.field.FieldElement51.Insts.CoreCmpEq : core.cmp.Eq + backend.serial.u64.field.FieldElement51 := { + partialEqInst := + backend.serial.u64.field.FieldElement51.Insts.CoreCmpPartialEqFieldElement51 + assert_fields_are_eq := + backend.serial.u64.field.FieldElement51.Insts.CoreCmpEq.assert_fields_are_eq +} + +/-- Trait implementation: [curve25519::field::{impl subtle::ConstantTimeEq for curve25519::backend::serial::u64::field::FieldElement51}] + Source: 'curve25519/solana-ed25519/src/field.rs', lines 60:0-67:1 -/ +@[reducible] +def backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq : + subtle.ConstantTimeEq backend.serial.u64.field.FieldElement51 := { + ct_eq := + backend.serial.u64.field.FieldElement51.Insts.SubtleConstantTimeEq.ct_eq +} + +/-- [curve25519::field::{impl core::default::Default for curve25519::backend::serial::u64::field::FieldElement51}::default]: + Source: 'curve25519/solana-ed25519/src/field.rs', lines 70:4-72:5 + Visibility: public -/ +def backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault.default + : Result backend.serial.u64.field.FieldElement51 := do + backend.serial.u64.field.FieldElement51.ZERO + +/-- Trait implementation: [curve25519::field::{impl core::default::Default for curve25519::backend::serial::u64::field::FieldElement51}] + Source: 'curve25519/solana-ed25519/src/field.rs', lines 69:0-73:1 -/ +@[reducible] +def backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault : + core.default.Default backend.serial.u64.field.FieldElement51 := { + default := + backend.serial.u64.field.FieldElement51.Insts.CoreDefaultDefault.default +} + +/-- [curve25519::field::{curve25519::backend::serial::u64::field::FieldElement51}::is_zero]: + Source: 'curve25519/solana-ed25519/src/field.rs', lines 149:4-154:5 -/ +def field.FieldElement51.is_zero + (self : backend.serial.u64.field.FieldElement51) : Result subtle.Choice := do + let zero := Array.repeat 32#usize 0#u8 + let bytes ← backend.serial.u64.field.FieldElement51.to_bytes self + let s ← lift (Array.to_slice bytes) + let s1 ← lift (Array.to_slice zero) + Slice.Insts.SubtleConstantTimeEq.ct_eq U8.Insts.SubtleConstantTimeEq s s1 + /-- [curve25519::field::{curve25519::backend::serial::u64::field::FieldElement51}::invsqrt]: Source: 'curve25519/solana-ed25519/src/field.rs', lines 384:4-386:5 -/ def field.FieldElement51.invsqrt diff --git a/verification/gen/CurveField/FunsExternal_Template.lean b/verification/gen/CurveField/FunsExternal_Template.lean index 9806919..1073412 100644 --- a/verification/gen/CurveField/FunsExternal_Template.lean +++ b/verification/gen/CurveField/FunsExternal_Template.lean @@ -358,30 +358,13 @@ axiom edwards.affine.AffinePoint.Insts.CoreCmpEq.assert_fields_are_eq : edwards.affine.AffinePoint → Result Unit /-- [curve25519::edwards::{impl core::cmp::Eq for curve25519::edwards::CompressedEdwardsY}::assert_fields_are_eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 183:0-183:33 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 182:0-182:33 Visibility: public -/ axiom edwards.CompressedEdwardsY.Insts.CoreCmpEq.assert_fields_are_eq : edwards.CompressedEdwardsY → Result Unit -/-- [curve25519::edwards::decompress::step_2]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 240:4-257:5 -/ -axiom edwards.decompress.step_2 - : - edwards.CompressedEdwardsY → backend.serial.u64.field.FieldElement51 → - backend.serial.u64.field.FieldElement51 → - backend.serial.u64.field.FieldElement51 → Result edwards.EdwardsPoint - -/-- [curve25519::edwards::decompress::step_1]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 226:4-237:5 -/ -axiom edwards.decompress.step_1 - : - edwards.CompressedEdwardsY → Result (subtle.Choice × - backend.serial.u64.field.FieldElement51 × - backend.serial.u64.field.FieldElement51 × - backend.serial.u64.field.FieldElement51) - /-- [curve25519::edwards::{curve25519::edwards::CompressedEdwardsY}::from_slice]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 423:4-425:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 428:4-430:5 Visibility: public -/ axiom edwards.CompressedEdwardsY.from_slice : @@ -389,7 +372,7 @@ axiom edwards.CompressedEdwardsY.from_slice core.array.TryFromSliceError) /-- [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}::conditional_swap]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 486:0-495:1 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 491:0-500:1 Visibility: public -/ axiom edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_swap : @@ -397,7 +380,7 @@ axiom edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_swap (edwards.EdwardsPoint × edwards.EdwardsPoint) /-- [curve25519::edwards::{impl subtle::ConditionallySelectable for curve25519::edwards::EdwardsPoint}::conditional_assign]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 486:0-495:1 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 491:0-500:1 Visibility: public -/ axiom edwards.EdwardsPoint.Insts.SubtleConditionallySelectable.conditional_assign @@ -406,13 +389,13 @@ axiom edwards.EdwardsPoint /-- [curve25519::edwards::{impl core::cmp::Eq for curve25519::edwards::EdwardsPoint}::assert_fields_are_eq]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 520:0-520:27 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 525:0-525:27 Visibility: public -/ axiom edwards.EdwardsPoint.Insts.CoreCmpEq.assert_fields_are_eq : edwards.EdwardsPoint → Result Unit /-- [curve25519::edwards::{impl core::iter::traits::accum::Sum for curve25519::edwards::EdwardsPoint}::sum]: - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 829:4-834:5 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 834:4-839:5 Visibility: public -/ axiom edwards.EdwardsPoint.Insts.CoreIterTraitsAccumSum.sum {T : Type} {I : Type} (coreborrowBorrowTEdwardsPointInst : core.borrow.Borrow diff --git a/verification/gen/CurveField/Types.lean b/verification/gen/CurveField/Types.lean index b982d62..1c1682f 100644 --- a/verification/gen/CurveField/Types.lean +++ b/verification/gen/CurveField/Types.lean @@ -161,7 +161,7 @@ structure traits.ValidityCheck (Self : Type) where is_valid : Self → Result Bool /-- [curve25519::edwards::EdwardsPoint] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 390:0-395:1 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 395:0-400:1 Visibility: public -/ structure edwards.EdwardsPoint where X : backend.serial.u64.field.FieldElement51 @@ -223,7 +223,7 @@ structure edwards.affine.AffinePoint where y : backend.serial.u64.field.FieldElement51 /-- [curve25519::edwards::CompressedEdwardsY] - Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 175:0-175:44 + Source: 'curve25519/solana-ed25519/src/edwards.rs', lines 174:0-174:44 Visibility: public -/ @[reducible] def edwards.CompressedEdwardsY := Array Std.U8 32#usize