// -*- mode: rust; -*- // // This file is part of curve25519-dalek. // Copyright (c) 2016-2021 isis agora lovecruft // Copyright (c) 2016-2019 Henry de Valence // See LICENSE for licensing information. // // Authors: // - Isis Agora Lovecruft // - Henry de Valence //! Field arithmetic modulo \\(p = 2\^{255} - 19\\). //! //! The `curve25519_dalek::field` module provides a type alias //! `curve25519_dalek::field::FieldElement` to a field element type //! defined in the `backend` module; either `FieldElement51` or //! `FieldElement2625`. //! //! Field operations defined in terms of machine //! operations, such as field multiplication or squaring, are defined in //! the backend implementation. //! //! Field operations defined in terms of other field operations, such as //! field inversion or square roots, are defined here. #![allow(unused_qualifications)] use cfg_if::cfg_if; use subtle::Choice; use subtle::ConditionallyNegatable; use subtle::ConditionallySelectable; use subtle::ConstantTimeEq; use crate::backend; use crate::constants; #[cfg(feature = "digest")] use digest::{ Digest, FixedOutput, HashMarker, array::{Array, typenum::U64}, block_api::BlockSizeUser, typenum::{IsGreater, True}, }; cfg_if! { if #[cfg(curve25519_dalek_backend = "fiat")] { /// A `FieldElement` represents an element of the field /// \\( \mathbb Z / (2\^{255} - 19)\\). /// /// The `FieldElement` type is an alias for one of the platform-specific /// implementations. /// /// Using formally-verified field arithmetic from fiat-crypto. #[cfg(curve25519_dalek_bits = "32")] pub(crate) type FieldElement = backend::serial::fiat_u32::field::FieldElement2625; /// A `FieldElement` represents an element of the field /// \\( \mathbb Z / (2\^{255} - 19)\\). /// /// The `FieldElement` type is an alias for one of the platform-specific /// implementations. /// /// Using formally-verified field arithmetic from fiat-crypto. #[cfg(curve25519_dalek_bits = "64")] pub(crate) type FieldElement = backend::serial::fiat_u64::field::FieldElement51; } else if #[cfg(curve25519_dalek_bits = "64")] { /// A `FieldElement` represents an element of the field /// \\( \mathbb Z / (2\^{255} - 19)\\). /// /// The `FieldElement` type is an alias for one of the platform-specific /// implementations. pub(crate) type FieldElement = backend::serial::u64::field::FieldElement51; } else { /// A `FieldElement` represents an element of the field /// \\( \mathbb Z / (2\^{255} - 19)\\). /// /// The `FieldElement` type is an alias for one of the platform-specific /// implementations. pub(crate) type FieldElement = backend::serial::u32::field::FieldElement2625; } } impl Eq for FieldElement {} impl PartialEq for FieldElement { fn eq(&self, other: &FieldElement) -> bool { self.ct_eq(other).into() } } impl ConstantTimeEq for FieldElement { /// Test equality between two `FieldElement`s. Since the /// internal representation is not canonical, the field elements /// are normalized to wire format before comparison. fn ct_eq(&self, other: &FieldElement) -> Choice { self.to_bytes().ct_eq(&other.to_bytes()) } } impl Default for FieldElement { fn default() -> Self { FieldElement::ZERO } } impl FieldElement { /// Load a `FieldElement` from 64 bytes, by reducing modulo q. #[cfg(feature = "digest")] pub(crate) fn from_bytes_wide(bytes: &[u8; 64]) -> Self { let mut fl = [0u8; 32]; let mut gl = [0u8; 32]; fl.copy_from_slice(&bytes[..32]); gl.copy_from_slice(&bytes[32..]); // Mask off the top bits of both halves, since from_bytes masks them off anyway. We'll add // them back in later. let fl_top_bit = (fl[31] >> 7) as u16; let gl_top_bit = (gl[31] >> 7) as u16; fl[31] &= 0x7f; gl[31] &= 0x7f; // Interpret both sides as field elements let mut fe_f = Self::from_bytes(&fl); let fe_g = Self::from_bytes(&gl); // The full field elem is now fe_f + 2²⁵⁵ fl_top_bit + 2²⁵⁶ fe_g + 2⁵¹¹ gl_top_bit // Add the masked off bits back to fe_f. fl_top_bit, if set, is 2^255 ≡ 19 (mod q). // gl_top_bit, if set, is 2^511 ≡ 722 (mod q) let top_bits_sum = { // This only need to be a u16 because the max value is 741 let addend: u16 = fl_top_bit * 19 + gl_top_bit * 722; let mut addend_bytes = [0u8; 32]; addend_bytes[..2].copy_from_slice(&addend.to_le_bytes()); Self::from_bytes(&addend_bytes) }; fe_f += &top_bits_sum; // Now add the high half into fe_f. The RHS is multiplied by 2^256 ≡ 38 (mod q) const THIRTY_EIGHT: FieldElement = FieldElement::from_bytes(&[ 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, 0, 0, 0, ]); fe_f += &(&THIRTY_EIGHT * &fe_g); fe_f } /// Determine if this `FieldElement` is negative, in the sense /// used in the ed25519 paper: `x` is negative if the low bit is /// set. /// /// # Return /// /// If negative, return `Choice(1)`. Otherwise, return `Choice(0)`. pub(crate) fn is_negative(&self) -> Choice { let bytes = self.to_bytes(); (bytes[0] & 1).into() } /// Determine if this `FieldElement` is zero. /// /// # Return /// /// If zero, return `Choice(1)`. Otherwise, return `Choice(0)`. pub(crate) fn is_zero(&self) -> Choice { let zero = [0u8; 32]; let bytes = self.to_bytes(); bytes.ct_eq(&zero) } /// Compute (self^(2^250-1), self^11), used as a helper function /// within invert() and pow22523(). #[rustfmt::skip] // keep alignment of explanatory comments fn pow22501(&self) -> (FieldElement, FieldElement) { // Instead of managing which temporary variables are used // for what, we define as many as we need and leave stack // allocation to the compiler // // Each temporary variable t_i is of the form (self)^e_i. // Squaring t_i corresponds to multiplying e_i by 2, // so the pow2k function shifts e_i left by k places. // Multiplying t_i and t_j corresponds to adding e_i + e_j. // // Temporary t_i Nonzero bits of e_i // let t0 = self.square(); // 1 e_0 = 2^1 let t1 = t0.square().square(); // 3 e_1 = 2^3 let t2 = self * &t1; // 3,0 e_2 = 2^3 + 2^0 let t3 = &t0 * &t2; // 3,1,0 let t4 = t3.square(); // 4,2,1 let t5 = &t2 * &t4; // 4,3,2,1,0 let t6 = t5.pow2k(5); // 9,8,7,6,5 let t7 = &t6 * &t5; // 9,8,7,6,5,4,3,2,1,0 let t8 = t7.pow2k(10); // 19..10 let t9 = &t8 * &t7; // 19..0 let t10 = t9.pow2k(20); // 39..20 let t11 = &t10 * &t9; // 39..0 let t12 = t11.pow2k(10); // 49..10 let t13 = &t12 * &t7; // 49..0 let t14 = t13.pow2k(50); // 99..50 let t15 = &t14 * &t13; // 99..0 let t16 = t15.pow2k(100); // 199..100 let t17 = &t16 * &t15; // 199..0 let t18 = t17.pow2k(50); // 249..50 let t19 = &t18 * &t13; // 249..0 (t19, t3) } /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse. /// /// When an input `FieldElement` is zero, its value is unchanged. pub(crate) fn invert_batch(inputs: &mut [FieldElement; N]) { let mut scratch = [FieldElement::ONE; N]; Self::internal_invert_batch(inputs, &mut scratch); } /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse. /// /// When an input `FieldElement` is zero, its value is unchanged. #[cfg(feature = "alloc")] pub(crate) fn invert_batch_alloc(inputs: &mut [FieldElement]) { let n = inputs.len(); let mut scratch = vec![FieldElement::ONE; n]; Self::internal_invert_batch(inputs, &mut scratch); } /// Given a slice of pub(crate)lic `FieldElements`, replace each with its inverse. `scratch` can /// contain anything, so long as its length is the same as `inputs`. /// /// When an input `FieldElement` is zero, its value is unchanged. /// /// # Panics /// Panics when `scratch.len() != inputs.len()` fn internal_invert_batch(inputs: &mut [FieldElement], scratch: &mut [FieldElement]) { // Montgomery’s Trick and Fast Implementation of Masked AES // Genelle, Prouff and Quisquater // Section 3.2 debug_assert_eq!(inputs.len(), scratch.len()); // Keep an accumulator of all of the previous products let mut acc = FieldElement::ONE; // Pass through the input vector, recording the previous // products in the scratch space for (input, scratch) in inputs.iter().zip(scratch.iter_mut()) { *scratch = acc; // acc <- acc * input, but skipping zeros (constant-time) acc.conditional_assign(&(&acc * input), !input.is_zero()); } // acc is nonzero because we skipped zeros in inputs assert!(bool::from(!acc.is_zero())); // Compute the inverse of all products acc = acc.invert(); // Pass through the vector backwards to compute the inverses // in place for (input, scratch) in inputs.iter_mut().rev().zip(scratch.iter_mut().rev()) { let tmp = &acc * input; // input <- acc * scratch, then acc <- tmp // Again, we skip zeros in a constant-time way let nz = !input.is_zero(); input.conditional_assign(&(&acc * scratch), nz); acc.conditional_assign(&tmp, nz); } } /// Given a nonzero field element, compute its inverse. /// /// The inverse is computed as self^(p-2), since /// x^(p-2)x = x^(p-1) = 1 (mod p). /// /// This function returns zero on input zero. #[rustfmt::skip] // keep alignment of explanatory comments #[allow(clippy::let_and_return)] pub(crate) fn invert(&self) -> FieldElement { // The bits of p-2 = 2^255 -19 -2 are 11010111111...11. // // nonzero bits of exponent let (t19, t3) = self.pow22501(); // t19: 249..0 ; t3: 3,1,0 let t20 = t19.pow2k(5); // 254..5 let t21 = &t20 * &t3; // 254..5,3,1,0 t21 } /// Raise this field element to the power (p-5)/8 = 2^252 -3. #[rustfmt::skip] // keep alignment of explanatory comments #[allow(clippy::let_and_return)] pub(crate) fn pow_p58(&self) -> FieldElement { // The bits of (p-5)/8 are 101111.....11. // // nonzero bits of exponent let (t19, _) = self.pow22501(); // 249..0 let t20 = t19.pow2k(2); // 251..2 let t21 = self * &t20; // 251..2,0 t21 } /// Given `FieldElements` `u` and `v`, compute either `sqrt(u/v)` /// or `sqrt(i*u/v)` in constant time. /// /// This function always returns the nonnegative square root. /// /// # Return /// /// - `(Choice(1), +sqrt(u/v)) ` if `v` is nonzero and `u/v` is square; /// - `(Choice(1), zero) ` if `u` is zero; /// - `(Choice(0), zero) ` if `v` is zero and `u` is nonzero; /// - `(Choice(0), +sqrt(i*u/v))` if `u/v` is nonsquare (so `i*u/v` is square). /// pub(crate) fn sqrt_ratio_i(u: &FieldElement, v: &FieldElement) -> (Choice, FieldElement) { // Using the same trick as in ed25519 decoding, we merge the // inversion, the square root, and the square test as follows. // // To compute sqrt(α), we can compute β = α^((p+3)/8). // Then β^2 = ±α, so multiplying β by sqrt(-1) if necessary // gives sqrt(α). // // To compute 1/sqrt(α), we observe that // 1/β = α^(p-1 - (p+3)/8) = α^((7p-11)/8) // = α^3 * (α^7)^((p-5)/8). // // We can therefore compute sqrt(u/v) = sqrt(u)/sqrt(v) // by first computing // r = u^((p+3)/8) v^(p-1-(p+3)/8) // = u u^((p-5)/8) v^3 (v^7)^((p-5)/8) // = (uv^3) (uv^7)^((p-5)/8). // // If v is nonzero and u/v is square, then r^2 = ±u/v, // so vr^2 = ±u. // If vr^2 = u, then sqrt(u/v) = r. // If vr^2 = -u, then sqrt(u/v) = r*sqrt(-1). // // If v is zero, r is also zero. let v3 = &v.square() * v; let v7 = &v3.square() * v; let mut r = &(u * &v3) * &(u * &v7).pow_p58(); let check = v * &r.square(); let i = &constants::SQRT_M1; let correct_sign_sqrt = check.ct_eq(u); let flipped_sign_sqrt = check.ct_eq(&(-u)); let flipped_sign_sqrt_i = check.ct_eq(&(&(-u) * i)); let r_prime = &constants::SQRT_M1 * &r; r.conditional_assign(&r_prime, flipped_sign_sqrt | flipped_sign_sqrt_i); // Choose the nonnegative square root. let r_is_negative = r.is_negative(); r.conditional_negate(r_is_negative); let was_nonzero_square = correct_sign_sqrt | flipped_sign_sqrt; (was_nonzero_square, r) } /// Attempt to compute `sqrt(1/self)` in constant time. /// /// Convenience wrapper around `sqrt_ratio_i`. /// /// This function always returns the nonnegative square root. /// /// # Return /// /// - `(Choice(1), +sqrt(1/self)) ` if `self` is a nonzero square; /// - `(Choice(0), zero) ` if `self` is zero; /// - `(Choice(0), +sqrt(i/self)) ` if `self` is a nonzero nonsquare; /// pub(crate) fn invsqrt(&self) -> (Choice, FieldElement) { FieldElement::sqrt_ratio_i(&FieldElement::ONE, self) } #[cfg(feature = "digest")] /// Hashes the given message and domain separator to produce `COUNT` [`FieldElement`]s, per RFC /// 9380. The hash's input is the concatenation of the elements of `msg`. Likewise for the /// domain separator with `domain_sep`. At least one element of `domain_sep`, MUST be nonempty, /// its concatenation MUST NOT exceed 255 bytes, and `COUNT` MUST be 1 or 2. /// /// The specification names SHA-512 as an example of a secure hash to use with this function, /// but you may use any 512-bit hash within reason (see the /// [`spec`](https://www.rfc-editor.org/rfc/rfc9380.html#section-5.2) for details). /// /// # Panics /// Panics if `domain_sep.collect().len() == 0` or `> 255`. Also panics if `COUNT > 2` or /// `COUNT == 0`. pub fn hash_to_field( msg: &[&[u8]], domain_sep: &[&[u8]], ) -> [Self; COUNT] where D: BlockSizeUser + Default + FixedOutput + HashMarker, D::BlockSize: IsGreater, { // We only use `hash_to_field` for Elligator2, which uses 0 < COUNT <= 2 assert!(COUNT == 1 || COUNT == 2); // §5.2, we generate count * m * L = COUNT * 1 * (256 + 128)/8 bytes let len_in_bytes = COUNT * 48; // Buffer should hold however many digests we need to produce len_in_bytes bytes. // len_in_bytes is at most 2*48=96, and the digest is 64B. So that's 2 digests, or 128B let mut buf = [0u8; 128]; // We can call this without worrying about panics: len_in_bytes is at most 96, and buf is // bigger than it. Further, if `domain_sep` is too large, that's also a panic condition of // this function so it's fine let uniform_bytes = expand_msg_xmd::(msg, domain_sep, &mut buf, len_in_bytes); let mut result = [FieldElement::ONE; COUNT]; for i in 0..COUNT { let mut bytes_wide = [0u8; 64]; bytes_wide[..48].copy_from_slice(&uniform_bytes[48 * i..48 * (i + 1)]); bytes_wide[..48].reverse(); result[i] = FieldElement::from_bytes_wide(&bytes_wide); } result } } /// Hashes the concatenation of the elements of `msg` with domain separator equal to the /// concatenation of `domain_sep`. The output is an `outlen`-length slice into `buf`. Follows /// https://www.rfc-editor.org/rfc/rfc9380.html#section-5.3.1 /// /// # Panics /// Panics if the domain separator is empty, if the total length of the domain separator is more /// than 255 bytes, if `outlen` is more than `255 * D::output_size()`, if `outlen is more than /// 65535, or if `dst.len() < outlen`. #[cfg(feature = "digest")] fn expand_msg_xmd<'a, D>( msg: &[&[u8]], domain_sep: &[&[u8]], buf: &'a mut [u8], outlen: usize, ) -> &'a [u8] where D: BlockSizeUser + Default + FixedOutput + HashMarker, { use core::iter::once; // The notation we use in this function is the same as in the spec let len_in_bytes = u16::try_from(outlen).expect("outlen must not exceed 65535"); assert!(buf.len() >= outlen); let ell = u8::try_from((len_in_bytes as usize).div_ceil(D::output_size())) .expect("output length cannot exceed 255 times digest size"); let domain_sep_len = u8::try_from(domain_sep.iter().map(|c| c.len()).sum::()) .expect("unexpected overflow from domain separator's size."); assert_ne!( domain_sep_len, 0, "domain separator MUST have nonzero length." ); let domain_sep_len_slice = &[domain_sep_len][..]; let dst_prime = domain_sep.iter().copied().chain(once(domain_sep_len_slice)); let z_pad = Array::::default(); let l_i_b_str = len_in_bytes.to_be_bytes(); // Collect the components of msg_prime let msg_prime = once(z_pad.as_slice()) .chain(msg.iter().copied()) .chain(once(l_i_b_str.as_slice())) .chain(once(&[0u8][..])) .chain(dst_prime.clone()); // Hash all of msg_prime let b_0 = msg_prime .fold(D::new(), |h, slice| h.chain_update(slice)) .finalize(); // Collect the input components for the b_1 hash let b_1_input = once(b_0.as_slice()) .chain(once(&[1u8][..])) .chain(dst_prime.clone()); let b_1 = b_1_input .fold(D::new(), |h, slice| h.chain_update(slice)) .finalize(); // Write b_1 to the output buffer buf[..D::output_size()].copy_from_slice(&b_1); for i in 2..=ell { // Get the last digest we produced let i_ = i as usize; let last_b = &buf[(i_ - 2) * D::output_size()..(i_ - 1) * D::output_size()]; // XOR it with b_0 let mut xor_bs = b_0.clone(); xor_bs .as_mut_slice() .iter_mut() .zip(last_b) .for_each(|(l, r)| *l ^= *r); // Hash (b_0 ^ b_{i-1}) || i || dst_prime let i_slice = &[i][..]; let b_i_input = once(xor_bs.as_slice()) .chain(once(i_slice)) .chain(dst_prime.clone()); let b_i = b_i_input .fold(D::new(), |h, slice| h.chain_update(slice)) .finalize(); // Write b_i to the output buffer buf[(i_ - 1) * D::output_size()..i_ * D::output_size()].copy_from_slice(&b_i); } &buf[..outlen] } #[cfg(test)] mod test { use crate::field::*; /// Random element a of GF(2^255-19), from Sage /// a = 1070314506888354081329385823235218444233221\ /// 2228051251926706380353716438957572 static A_BYTES: [u8; 32] = [ 0x04, 0xfe, 0xdf, 0x98, 0xa7, 0xfa, 0x0a, 0x68, 0x84, 0x92, 0xbd, 0x59, 0x08, 0x07, 0xa7, 0x03, 0x9e, 0xd1, 0xf6, 0xf2, 0xe1, 0xd9, 0xe2, 0xa4, 0xa4, 0x51, 0x47, 0x36, 0xf3, 0xc3, 0xa9, 0x17, ]; /// Byte representation of a**2 static ASQ_BYTES: [u8; 32] = [ 0x75, 0x97, 0x24, 0x9e, 0xe6, 0x06, 0xfe, 0xab, 0x24, 0x04, 0x56, 0x68, 0x07, 0x91, 0x2d, 0x5d, 0x0b, 0x0f, 0x3f, 0x1c, 0xb2, 0x6e, 0xf2, 0xe2, 0x63, 0x9c, 0x12, 0xba, 0x73, 0x0b, 0xe3, 0x62, ]; /// Byte representation of 1/a static AINV_BYTES: [u8; 32] = [ 0x96, 0x1b, 0xcd, 0x8d, 0x4d, 0x5e, 0xa2, 0x3a, 0xe9, 0x36, 0x37, 0x93, 0xdb, 0x7b, 0x4d, 0x70, 0xb8, 0x0d, 0xc0, 0x55, 0xd0, 0x4c, 0x1d, 0x7b, 0x90, 0x71, 0xd8, 0xe9, 0xb6, 0x18, 0xe6, 0x30, ]; /// Byte representation of a^((p-5)/8) static AP58_BYTES: [u8; 32] = [ 0x6a, 0x4f, 0x24, 0x89, 0x1f, 0x57, 0x60, 0x36, 0xd0, 0xbe, 0x12, 0x3c, 0x8f, 0xf5, 0xb1, 0x59, 0xe0, 0xf0, 0xb8, 0x1b, 0x20, 0xd2, 0xb5, 0x1f, 0x15, 0x21, 0xf9, 0xe3, 0xe1, 0x61, 0x21, 0x55, ]; #[test] fn a_mul_a_vs_a_squared_constant() { let a = FieldElement::from_bytes(&A_BYTES); let asq = FieldElement::from_bytes(&ASQ_BYTES); assert_eq!(asq, &a * &a); } #[test] fn a_square_vs_a_squared_constant() { let a = FieldElement::from_bytes(&A_BYTES); let asq = FieldElement::from_bytes(&ASQ_BYTES); assert_eq!(asq, a.square()); } #[test] fn a_square2_vs_a_squared_constant() { let a = FieldElement::from_bytes(&A_BYTES); let asq = FieldElement::from_bytes(&ASQ_BYTES); assert_eq!(a.square2(), &asq + &asq); } #[test] fn a_invert_vs_inverse_of_a_constant() { let a = FieldElement::from_bytes(&A_BYTES); let ainv = FieldElement::from_bytes(&AINV_BYTES); let should_be_inverse = a.invert(); assert_eq!(ainv, should_be_inverse); assert_eq!(FieldElement::ONE, &a * &should_be_inverse); } #[test] #[cfg(feature = "alloc")] fn invert_batch_a_matches_nonbatched() { let a = FieldElement::from_bytes(&A_BYTES); let ap58 = FieldElement::from_bytes(&AP58_BYTES); let asq = FieldElement::from_bytes(&ASQ_BYTES); let ainv = FieldElement::from_bytes(&AINV_BYTES); let a0 = &a - &a; let a2 = &a + &a; let a_list = vec![a, ap58, asq, ainv, a0, a2]; let mut ainv_list = a_list.clone(); FieldElement::invert_batch_alloc(&mut ainv_list[..]); for i in 0..6 { assert_eq!(a_list[i].invert(), ainv_list[i]); } } #[test] fn sqrt_ratio_behavior() { let zero = FieldElement::ZERO; let one = FieldElement::ONE; let i = constants::SQRT_M1; let two = &one + &one; // 2 is nonsquare mod p. let four = &two + &two; // 4 is square mod p. // 0/0 should return (1, 0) since u is 0 let (choice, sqrt) = FieldElement::sqrt_ratio_i(&zero, &zero); assert!(bool::from(choice)); assert_eq!(sqrt, zero); assert!(bool::from(!sqrt.is_negative())); // 1/0 should return (0, 0) since v is 0, u is nonzero let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &zero); assert!(bool::from(!choice)); assert_eq!(sqrt, zero); assert!(bool::from(!sqrt.is_negative())); // 2/1 is nonsquare, so we expect (0, sqrt(i*2)) let (choice, sqrt) = FieldElement::sqrt_ratio_i(&two, &one); assert!(bool::from(!choice)); assert_eq!(sqrt.square(), &two * &i); assert!(bool::from(!sqrt.is_negative())); // 4/1 is square, so we expect (1, sqrt(4)) let (choice, sqrt) = FieldElement::sqrt_ratio_i(&four, &one); assert!(bool::from(choice)); assert_eq!(sqrt.square(), four); assert!(bool::from(!sqrt.is_negative())); // 1/4 is square, so we expect (1, 1/sqrt(4)) let (choice, sqrt) = FieldElement::sqrt_ratio_i(&one, &four); assert!(bool::from(choice)); assert_eq!(&sqrt.square() * &four, one); assert!(bool::from(!sqrt.is_negative())); } #[test] fn a_p58_vs_ap58_constant() { let a = FieldElement::from_bytes(&A_BYTES); let ap58 = FieldElement::from_bytes(&AP58_BYTES); assert_eq!(ap58, a.pow_p58()); } #[test] fn equality() { let a = FieldElement::from_bytes(&A_BYTES); let ainv = FieldElement::from_bytes(&AINV_BYTES); assert!(a == a); assert!(a != ainv); } /// Notice that the last element has the high bit set, which /// should be ignored static B_BYTES: [u8; 32] = [ 113, 191, 169, 143, 91, 234, 121, 15, 241, 131, 217, 36, 230, 101, 92, 234, 8, 208, 170, 251, 97, 127, 70, 210, 58, 23, 166, 87, 240, 169, 184, 178, ]; #[test] fn from_bytes_highbit_is_ignored() { let mut cleared_bytes = B_BYTES; cleared_bytes[31] &= 127u8; let with_highbit_set = FieldElement::from_bytes(&B_BYTES); let without_highbit_set = FieldElement::from_bytes(&cleared_bytes); assert_eq!(without_highbit_set, with_highbit_set); } #[test] fn conditional_negate() { let one = FieldElement::ONE; let minus_one = FieldElement::MINUS_ONE; let mut x = one; x.conditional_negate(Choice::from(1)); assert_eq!(x, minus_one); x.conditional_negate(Choice::from(0)); assert_eq!(x, minus_one); x.conditional_negate(Choice::from(1)); assert_eq!(x, one); } #[test] fn encoding_is_canonical() { // Encode 1 wrongly as 1 + (2^255 - 19) = 2^255 - 18 let one_encoded_wrongly_bytes: [u8; 32] = [ 0xee, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f, ]; // Decode to a field element let one = FieldElement::from_bytes(&one_encoded_wrongly_bytes); // .. then check that the encoding is correct let one_bytes = one.to_bytes(); assert_eq!(one_bytes[0], 1); for byte in &one_bytes[1..] { assert_eq!(*byte, 0); } } #[test] #[cfg(feature = "alloc")] fn invert_batch_empty() { FieldElement::invert_batch_alloc(&mut []); } // The following two consts were generated with the following sage script: // // import random // // F = GF(2**255 - 19) // # Use a seed to make sure we produce the same test vectors every time // random.seed("Ozamataz Buckshank") // // # Generates test vectors, each of the form (input_bytes, reduced_field_elem_bytes), // # where input_bytes is length input_bytes_len // def gen_example(input_bytes_len): // # Generate random bytes // input_bytes = [random.randint(0, 255) for _ in range(input_bytes_len)] // // # Now convert to a field element and get the reduced byte representation // elem = F(int.from_bytes(input_bytes, byteorder='little')) // reduced_bytes = list(int(elem).to_bytes(32, byteorder='little')) // // # Format input and output as hex strings // input_bytes_hex = ''.join(f'{byte:02x}' for byte in input_bytes) // reduced_bytes_hex = ''.join(f'{byte:02x}' for byte in reduced_bytes) // return f"(\"{input_bytes_hex}\", \"{reduced_bytes_hex}\")" // // print("SET 1: Input bytes are length 64") // for _ in range(5): // print(gen_example(64)) // // print("SET 2: Input bytes are length 48") // for _ in range(5): // print(gen_example(48)) /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-64 bytestring, /// reduced field element) #[cfg(feature = "digest")] const FROM_BYTES_WIDE_KAT_BIG: &[(&str, &str)] = &[ ( "77b663085cac0e916f40dbeea5116f201816406e68ccf01b32a97162ae1d5bf95d0d01c2c72fbeeb27a63\ 5b85b715d5ce6f74118a60a7aec53c798ad648a482f", "62b38bd402c4498f5cead14643e54dd649e20a0810610e36a73f1f27a0a81f7e", ), ( "d437c75ec79886650243a79c62933bb307eb12ff16d05db4a6a8a877f4a91abb6eeb64d2e20519c021799\ 3a1dc5639283a06639985a2c892208171503335afb5", "3d2ec29972783de9043e8b982278beaba9d7c5c3ebef257e7cd38168928f1c33", ), ( "6daa9e1abe6c604fb6e841c04bf90a6ef88aef6b1eab17dd44f7207ef472cd2d54bac849f703e64f36e56\ 77e7e86b82be7d26aa220daf1f208bb36dcc1a12338", "28546a0e7303852bc6eead8312f06eeb48d9ca87f60bfeec98ba402ebb751703", ), ( "c3920e326dbf806a50105be78263c1dc9390fb4741587b250cd758c2bfa3ed70faedbbc5f9b1d024e00fe\ 7d7daf796866853f42e72d638e6533c5eb5b7caf3c6", "40eaf38b802a7be1956ba7f3fe2d2ad717f23f40342deb5180cb55ae04bb1d79", ), ( "23f143c72ead6c0f336b4e746a06921f0eb180002e8ce916d196de16216788617c6aeb90a074a85196f03\ 81375011248927c1215e9ec65b382a6ec556fb3f504", "b1bf354a04fd6d2e8321c24ecb3d3ed2c42e3f21c7b60ab8374effd7a709011e", ), ]; /// Test vectors for FieldElement::from_bytes_wide. Elements are of the form (len-48 bytestring, /// reduced field element) #[cfg(feature = "digest")] const FROM_BYTES_WIDE_KAT_MEDIUM: &[(&str, &str)] = &[ ( "82e9cbe4928e3d0bbf1f91824a91acfb30d929f7a2fa5cbcc967c63ea0f3357c29c19f1bc9dcad69d85c1\ c6265970685", "989582fe6c540cbbdee7c612570aa7ba44d929f7a2fa5cbcc967c63ea0f3357c", ), ( "5480494df4fb3a3b19da17e1c8b9192ccb09ec76720321977079300c42c17b9e95b01eb37ffe7048fcd1c\ 9e6094da6c4", "85b6d7e3e8c200fc8b050d234129c95ce809ec76720321977079300c42c17b1e", ), ( "93ec8a480dde098f74bcd341ef4f248f6440cc6e631d7000784f66975a4fd628438bb1350ba4c1421fec3\ 670decced06", "8598e540b737c87718c9fae9f3b870966540cc6e631d7000784f66975a4fd628", ), ( "fd0154ff9a5c4c9ee4e8183c23db97018e0e6201a812f6d4faedda50652d51f65c110b9a1a100a3fc3ff1\ c4ea3cf22e4", "b895f8dc8dc0caf9dfdf66d460adc2deaf0e6201a812f6d4faedda50652d5176", ), ( "0e829dc955e0a1e0dbda9849cb2022b295275782348bd6308b3d0c5836f3ca0130911a17fd54054c3a0f8\ b2486f8ce85", "2e0f8f37e77d6c29831d3db6b404db8ea9275782348bd6308b3d0c5836f3ca01", ), ]; #[cfg(feature = "digest")] #[test] fn from_bytes_wide() { // Do the 64-byte input ones first for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_BIG { let reduce_fe = FieldElement::from_bytes_wide( &hex::decode(input_bytes) .unwrap() .as_slice() .try_into() .unwrap(), ); assert_eq!( &reduce_fe.to_bytes(), hex::decode(expected_reduced).unwrap().as_slice() ); } // Now do the 48-byte inputs for (input_bytes, expected_reduced) in FROM_BYTES_WIDE_KAT_MEDIUM { let mut padded_input_bytes = [0u8; 64]; padded_input_bytes[..48].copy_from_slice(&hex::decode(input_bytes).unwrap()); let reduce_fe = FieldElement::from_bytes_wide(&padded_input_bytes); assert_eq!( &reduce_fe.to_bytes(), hex::decode(expected_reduced).unwrap().as_slice() ); } } #[cfg(feature = "digest")] fn fe_from_test_vector(expected_hex: &str) -> FieldElement { let mut expected_hash = hex::decode(expected_hex).unwrap(); expected_hash.reverse(); FieldElement::from_bytes(&expected_hash.try_into().unwrap()) } /// Hash to field test vectors from /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.2 /// These are of the form (input_msg, output_field_elem) #[cfg(feature = "digest")] const RFC_HASH_TO_FIELD_KAT: &[(&[u8], &str)] = &[ ( b"", "7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0374a30641d" ), ( b"abc", "09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd201a556e253b" ), ( b"abcdef0123456789", "475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75d33edba941", ), ( b"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq", "049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf6929814ac95d03d" ), ( b"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", "3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f0750d24fa0" ) ]; #[test] #[cfg(feature = "digest")] fn hash_to_field() { use sha2::Sha512; let dst = "QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_"; for (msg, expected_hash_hex) in RFC_HASH_TO_FIELD_KAT { let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()])[0]; let expected_fe = fe_from_test_vector(expected_hash_hex); assert_eq!(fe, expected_fe); } } /// Hash to field test vectors from /// https://www.rfc-editor.org/rfc/rfc9380.html#appendix-J.5.1 /// These are of the form (input_msg, output_field_elem, output_field_elem) #[cfg(feature = "digest")] const RFC_HASH_TO_FIELD_KAT_2: &[(&[u8], &str, &str)] = &[ ( b"", "03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee712d203f3a", "780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae793e6a60c75" ), ( b"abc", "5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78f9bd57c227", "005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119aa0b925b76" ), ( b"abcdef0123456789", "285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642636a78d5b3", "2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b369760b81b31" ), ( b"q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq", "4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c533bf321af", "02f20716a5801b843987097a8276b6d869295b2e11253751ca72c109d37485a9" ), ( b"a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", "6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06df4a4ea33b", "1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e84068f5fd96" ) ]; #[test] #[cfg(feature = "digest")] fn hash_to_field_2() { use sha2::Sha512; let dst = "QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_"; for (msg, expected_hash_hex_1, expected_hash_hex_2) in crate::field::test::RFC_HASH_TO_FIELD_KAT_2 { let fe = FieldElement::hash_to_field::(&[msg], &[dst.as_bytes()]); let expected_fe_1 = fe_from_test_vector(expected_hash_hex_1); assert_eq!(fe[0], expected_fe_1); let expected_fe_2 = fe_from_test_vector(expected_hash_hex_2); assert_eq!(fe[1], expected_fe_2); } } }