commit 387a56fe2ca976476458ad1330cb1faaa6ed3095 Author: Isis Lovecruft Date: Thu Dec 8 05:12:00 2016 +0000 Initial commit. diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..7acc1af --- /dev/null +++ b/.gitignore @@ -0,0 +1,11 @@ +target +Cargo.lock + +*~ +\#* +.\#* +*.swp +*.orig +*.bak + +*.s diff --git a/Cargo.toml b/Cargo.toml new file mode 100644 index 0000000..f7f6686 --- /dev/null +++ b/Cargo.toml @@ -0,0 +1,69 @@ +[package] +name = "curve25519-dalek" +version = "0.1.0" +authors = ["Isis Lovecruft ", + "Henry de Valence "] +readme = "README.md" +license-file = "LICENSE" +repository = "https://code.ciph.re/isis/curve25519-dalek" +homepage = "https://code.ciph.re/isis/curve25519-dalek" +documentation = "https://docs.rs/curve25519-dalek" +keywords = ["cryptography", "curve25519", "elliptic", "curve", "ECC"] +description = "A low-level cryptographic library for point, group, field, and scalar operations on a curve isomorphic to the twisted Edwards curve defined by x²+y² = 121665/121666 x²y² over GF(2²⁵⁵ - 19)." +exclude = [ + ".gitignore" +] + +[dependencies] +arrayref = "0.3.2" + +# The development profile, used for `cargo build`. +[profile.dev] +opt-level = 0 # controls the `--opt-level` the compiler builds with +debug = true # controls whether the compiler passes `-g` +rpath = false # controls whether the compiler passes `-C rpath` +lto = false # controls `-C lto` for binaries and staticlibs +debug-assertions = true # controls whether debug assertions are enabled +codegen-units = 1 # controls whether the compiler passes `-C codegen-units` + # `codegen-units` is ignored when `lto = true` + panic = 'unwind' # panic strategy (`-C panic=...`), can also be 'abort' + +# The release profile, used for `cargo build --release`. +[profile.release] +opt-level = 3 +debug = false +rpath = false +lto = false +debug-assertions = false +codegen-units = 1 +panic = 'unwind' + +# The testing profile, used for `cargo test`. +[profile.test] +opt-level = 0 +debug = true +rpath = false +lto = false +debug-assertions = true +codegen-units = 1 +panic = 'unwind' + +# The benchmarking profile, used for `cargo bench`. +[profile.bench] +opt-level = 3 +debug = false +rpath = false +lto = false +debug-assertions = false +codegen-units = 1 +panic = 'unwind' + +# The documentation profile, used for `cargo doc`. +[profile.doc] +opt-level = 0 +debug = true +rpath = false +lto = false +debug-assertions = true +codegen-units = 1 +panic = 'unwind' diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..f89d2cc --- /dev/null +++ b/LICENSE @@ -0,0 +1,129 @@ +To the extent possible under law, the author(s) have waived all copyright and related or +neighboring rights to curve25519-dalek, using the Creative Commons "CC0" public domain dedication. + + + +Creative Commons CC0 1.0 Universal + +CREATIVE COMMONS CORPORATION IS NOT A LAW FIRM AND DOES NOT PROVIDE LEGAL +SERVICES. 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Affirmer understands and acknowledges that Creative Commons is not a + party to this document and has no duty or obligation with respect to + this CC0 or use of the Work. diff --git a/README.md b/README.md new file mode 100644 index 0000000..91e7605 --- /dev/null +++ b/README.md @@ -0,0 +1,28 @@ + +# curve25519-dalek [](https://docs.rs/curve25519-dalek/badge.svg) + +*A low-level cryptographic library for point, group, field, and scalar +operations on a curve isomorphic to the twisted Edwards curve defined by x²+y² += 121665/121666 x²y² over GF(2²⁵⁵ - 19).* + +*SPOILER ALERT:* **The Twelfth Doctor's first encounter with the Daleks is in +his second full episode, "Into the Dalek". A beleaguered ship of the "Combined +Galactic Resistance" has discovered a broken Dalek that has turned "good", +desiring to kill all other Daleks. The Doctor, Clara and a team of soldiers +are miniaturized and enter the Dalek, which the Doctor names Rusty. They +repair the damage, but accidentally restore it to its original nature, causing +it to go on the rampage and alert the Dalek fleet to the whereabouts of the +rebel ship. However, the Doctor manages to return Rusty to its previous state +by linking his mind with the Dalek's: Rusty shares the Doctor's view of the +universe's beauty, but also his deep hatred of the Daleks. Rusty destroys the +other Daleks and departs the ship, determined to track down and bring an end +to the Dalek race.** + +Significant portions of this code are ported from +[Adam Langley's Golang ed25519 library](https://github.com/agl/ed25519), along +with referencing the ref10 implementation. + +## TODO + + * Implement hashing to a point on the curve. + * Maybe implement Mike Hamburg's Decaf point compression format. diff --git a/src/curve.rs b/src/curve.rs new file mode 100644 index 0000000..da84984 --- /dev/null +++ b/src/curve.rs @@ -0,0 +1,1121 @@ +// -*- mode: rust; -*- +// +// To the extent possible under law, the authors have waived all copyright and +// related or neighboring rights to curve25519-dalek, using the Creative +// Commons "CC0" public domain dedication. See +// for full details. +// +// Authors: +// - Isis Agora Lovecruft +// - Henry de Valence + +//! Group operations for Curve25519, in the form of the twisted +//! Edwards curve -x²+y²=1+dx²y² modulo p=2²⁵⁵-19 with +//! parameter d=-121665/121666. +//! +//! # Curve representations +//! +//! Internally, we use several different models for the curve. Here +//! is a sketch of the relationship between the models, following [a +//! post](https://moderncrypto.org/mail-archive/curves/2016/000807.html) +//! by Ben Smith on the moderncrypto mailing list. +//! +//! Begin with the affine equation for the curve, +//! +//!     -x² + y² = 1 + dx²y².       (1) +//! +//! Next, pass to the projective closure 𝗣^1 x 𝗣^1 by setting x=X/Z, +//! y=Y/T. Clearing denominators gives the model +//! +//!     -X²T² + Y²Z² = Z²T² + dX²Y². (2) +//! +//! To map from 𝗣^1 x 𝗣^1, a product of two lines, to 𝗣^3, we use the +//! Segre embedding, +//! +//!     σ : ((X:Z),(Y:T)) ↦ (XY:XT:ZY:ZT).  (3) +//! +//! Using coordinates (W₀:W₁:W₂:W₃) for 𝗣^3, the image of σ(𝗣^1 x 𝗣^1) +//! is the surface defined by W₀W₃=W₁W₂, and under σ, equation (2) +//! becomes +//! +//!     -W₁² + W₂² = W₃² + dW₀².   (4) +//! +//! Up to variable naming, this is exactly the curve model introduced +//! in ["Twisted Edwards Curves +//! Revisited"](iacr.org/archive/asiacrypt2008/53500329/53500329.pdf) +//! by Hisil, Wong, Carter, and Dawson. We can map from 𝗣^3 to 𝗣² by +//! sending (W₀:W₁:W₂:W₃) to (W₁:W₂:W₃). Notice that +//! +//!     W₁/W₃ = XT/ZT = X/Z = x    (5) +//! +//!     W₂/W₃ = ZY/ZT = Y/T = y,   (6) +//! +//! so this is the same as if we had started with the affine model (1) +//! and passed to 𝗣^2 by setting `x = W₁/W₃`, `y = W₂/W₃`. Up to +//! variable naming, this is the projective representation introduced +//! in ["Twisted Edwards Curves"](https://eprint.iacr.org/2008/013). +//! +//! Following the implementation strategy in the ref10 reference +//! implementation for Ed25519, we use several different models for +//! curve points: +//! +//! * CompletedPoint: points in 𝗣^1 x 𝗣^1; +//! * ExtendedPoint: points in 𝗣^3; +//! * ProjectivePoint: points in 𝗣^2. +//! +//! Finally, to accelerate additions, we use two cached point formats, +//! one for the affine model and one for the 𝗣^3 model: +//! +//! * PreComputedPoint: `(y+x, y-x, 2dxy)` +//! * CachedPoint: `(Y+X, Y-X, Z, 2dXY)` +//! +//! [1]: https://moderncrypto.org/mail-archive/curves/2016/000807.html + +// We allow non snake_case names because coordinates in projective space are +// traditionally denoted by the capitalisation of their respective +// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my +// affine and projective cakes and eat both of them too. +#![allow(non_snake_case)] + +use std::fmt::Debug; +use std::iter::Iterator; +use std::ops::{Add, Sub, Neg, Index}; +use std::cmp::{PartialEq, Eq}; + +use constants; +use field::FieldElement; +use scalar::Scalar; +use util::bytes_equal_ct; + +// ------------------------------------------------------------------------ +// Compressed points +// ------------------------------------------------------------------------ + +/// An affine point `(x,y)` on the curve is determined by the +/// `y`-coordinate and the sign of `x`, marshalled into a 32-byte array. +/// +/// The first 255 bits of a CompressedPoint represent the +/// y-coordinate. The high bit of the 32nd byte gives the sign of `x`. +#[derive(Copy, Clone)] +pub struct CompressedPoint(pub [u8; 32]); + +impl Debug for CompressedPoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "CompressedPoint: {:?}", &self.0[..]) + } +} + +impl Eq for CompressedPoint {} +impl PartialEq for CompressedPoint { + /// Determine if this `CompressedPoint` is equal to another. + /// + /// # Warning + /// + /// This function is NOT constant time. + fn eq(&self, other: &CompressedPoint) -> bool { + return self.0 == other.0; + } +} + +impl Index for CompressedPoint { + type Output = u8; + + fn index<'a>(&'a self, _index: usize) -> &'a u8 { + let ret: &'a u8 = &(self.0[_index]); + ret + } +} + +impl CompressedPoint { + /// View this `CompressedPoint` as an array of bytes. + pub fn to_bytes(&self) -> [u8;32] { + self.0 + } + + /// Attempt to decompress to an `ExtendedPoint`. + /// + /// # Warning + /// + /// This function will fail and return None if both vx²-u=0 and vx²+u=0. + pub fn decompress(&self) -> Option { // FromBytes() + let mut u: FieldElement; + let mut v: FieldElement; + let v3: FieldElement; + let vxx: FieldElement; + + let mut X: FieldElement; + let Y: FieldElement; + let Z: FieldElement; + let T: FieldElement; + + Y = FieldElement::from_bytes(&self.0); + Z = FieldElement::one(); + + u = Y.square(); + v = &u * &constants::d; + u -= &Z; // u = y²-1 + v += &Z; // v = dy²+1 + v3 = &v.square() * &v; // v3 = v³ + X = (&v3.square() * &(&v * &u)).pow_p58(); // x = (uv⁷)^((q-5)/8) + X *= &(&u * &v3); // x = (uv³)(uv⁷)^((q-5)/8) + + vxx = &v * &X.square(); + if (&vxx - &u).is_nonzero() == 1 { // vx²-u + if (&vxx + &u).is_nonzero() == 1 { // vx²+u + return None; + } + X *= &constants::SQRT_M1; + } + + if X.is_negative() != (self[31] >> 7) as i32 { + X = X.neg(); + } + T = &X * &Y; + + Some(ExtendedPoint{ X: X, Y: Y, Z: Z, T: T }) + } +} + +// ------------------------------------------------------------------------ +// Internal point representations +// ------------------------------------------------------------------------ + +/// An `ExtendedPoint` is a point on the curve in 𝗣³(𝔽ₚ). +/// A point (x,y) in the affine model corresponds to (x:y:1:xy). +#[derive(Copy, Clone)] +pub struct ExtendedPoint { + X: FieldElement, + Y: FieldElement, + Z: FieldElement, + T: FieldElement, +} + +/// A `ProjectivePoint` is a point on the curve in 𝗣²(𝔽ₚ). +/// A point (x,y) in the affine model corresponds to (x:y:1). +#[derive(Copy, Clone)] +pub struct ProjectivePoint { + X: FieldElement, + Y: FieldElement, + Z: FieldElement, +} + +/// A CompletedPoint is a point ((X:Z), (Y:T)) in 𝗣¹(𝔽ₚ)×𝗣¹(𝔽ₚ). +/// A point (x,y) in the affine model corresponds to ((x:1),(y:1)). +#[derive(Copy, Clone)] +pub struct CompletedPoint { + X: FieldElement, + Y: FieldElement, + Z: FieldElement, + T: FieldElement, +} + +/// A pre-computed point in the affine model for the curve, +/// represented as (y+x, y-x, 2dxy). These precomputations +/// accelerate addition and subtraction. +#[derive(Copy, Clone)] +#[allow(missing_docs)] +pub struct PreComputedPoint { + pub y_plus_x: FieldElement, + pub y_minus_x: FieldElement, + pub xy2d: FieldElement, +} + +/// A pre-computed point in the P³(𝔽ₚ) model for the curve, +/// represented as (Y+X, Y-X, Z, 2dXY). These precomputations +/// accelerate addition and subtraction. +#[derive(Copy, Clone)] +pub struct CachedPoint { + Y_plus_X: FieldElement, + Y_minus_X: FieldElement, + Z: FieldElement, + T2d: FieldElement, +} + +// ------------------------------------------------------------------------ +// Constructors +// ------------------------------------------------------------------------ + +/// Trait for curve point types that have an identity constructor. +pub trait Identity { + /// Returns the identity element of the curve. + /// Can be used as a constructor. + fn identity() -> Self; +} + +impl Identity for ExtendedPoint { + fn identity() -> ExtendedPoint { + ExtendedPoint{ X: FieldElement::zero(), + Y: FieldElement::one(), + Z: FieldElement::one(), + T: FieldElement::zero() } + } +} + +impl Identity for ProjectivePoint { + fn identity() -> ProjectivePoint { + ProjectivePoint{ X: FieldElement::zero(), + Y: FieldElement::one(), + Z: FieldElement::one() } + } +} + +impl Identity for CachedPoint { + fn identity() -> CachedPoint { + CachedPoint{ Y_plus_X: FieldElement::one(), + Y_minus_X: FieldElement::one(), + Z: FieldElement::one(), + T2d: FieldElement::zero() } + } +} + +impl Identity for PreComputedPoint { + fn identity() -> PreComputedPoint { + PreComputedPoint{ + y_plus_x: FieldElement::one(), + y_minus_x: FieldElement::one(), + xy2d: FieldElement::zero(), + } + } +} + +// ------------------------------------------------------------------------ +// Constant-time assignment +// ------------------------------------------------------------------------ + +/// Trait for items which can be conditionally assigned in constant time. +pub trait CTAssignable { + /// If `choice == 1u8`, assign `other` to `self`. + /// Otherwise, leave `self` unchanged. + /// Executes in constant time. + // XXX this trait should be extracted? + fn conditional_assign(&mut self, other: &Self, choice: u8); +} + +impl CTAssignable for CachedPoint { + fn conditional_assign(&mut self, other: &CachedPoint, choice: u8) { + self.Y_plus_X.conditional_assign(&other.Y_plus_X, choice); + self.Y_minus_X.conditional_assign(&other.Y_minus_X, choice); + self.Z.conditional_assign(&other.Z, choice); + self.T2d.conditional_assign(&other.T2d, choice); + } +} + +impl CTAssignable for PreComputedPoint { + fn conditional_assign(&mut self, other: &PreComputedPoint, choice: u8) { + // PreComputedGroupElementCMove() + self.y_plus_x.conditional_assign(&other.y_plus_x, choice); + self.y_minus_x.conditional_assign(&other.y_minus_x, choice); + self.xy2d.conditional_assign(&other.xy2d, choice); + } +} + +// ------------------------------------------------------------------------ +// Point conversions +// ------------------------------------------------------------------------ + +impl ProjectivePoint { + /// Convert to the extended twisted Edwards representation of this + /// point. + /// + /// From §3 in [0]: + /// + /// Given (X:Y:Z) in Ɛ, passing to Ɛₑ can be performed in 3M+1S by + /// computing (XZ,YZ,XY,Z²). (Note that in that paper, points are + /// (X:Y:T:Z) so this really does match the code below). + #[allow(dead_code)] // rustc complains this is unused even when it's used + fn to_extended(&self) -> ExtendedPoint { + ExtendedPoint{ + X: &self.X * &self.Z, + Y: &self.Y * &self.Z, + Z: self.Z.square(), + T: &self.X * &self.Y, + } + } + + /// Convert this point to a `CompressedPoint` + pub fn compress(&self) -> CompressedPoint { + let recip = self.Z.invert(); + let x = &self.X * &recip; + let y = &self.Y * &recip; + let mut s: [u8; 32]; + + s = y.to_bytes(); + s[31] ^= (x.is_negative() << 7) as u8; + CompressedPoint(s) + } +} + +impl ExtendedPoint { + fn to_cached(&self) -> CachedPoint { + CachedPoint{ + Y_plus_X: &self.Y + &self.X, + Y_minus_X: &self.Y - &self.X, + Z: self.Z, + T2d: &self.T * &constants::d2, + } + } + + /// Convert the representation of this point from extended Twisted Edwards + /// coodinates to projective coordinates. + /// + /// Given a point in Ɛₑ, we can convert to projective coordinates + /// cost-free by simply ignoring T. + fn to_projective(&self) -> ProjectivePoint { + ProjectivePoint{ + X: self.X, + Y: self.Y, + Z: self.Z, + } + } + + /// Convert this point to a `CompressedPoint` + pub fn compress(&self) -> CompressedPoint { + self.to_projective().compress() + } + + /// XXX rewrite + /// We only need the x-coordinate of the curve25519 point, which I'll + /// call u. The isomorphism is u=(y+1)/(1-y), since y=Y/Z, this gives + /// u=(Y+Z)/(Z-Y). We know that Z=1, thus u=(Y+1)/(1-Y). + pub fn edwards_to_montgomery_x(&self) -> FieldElement { // edwardsToMontgomeryX + let one = FieldElement::one(); + &((&one - &self.Y).invert()) * &(&self.Y + &one) + } +} + +impl CompletedPoint { + fn to_projective(&self) -> ProjectivePoint { + ProjectivePoint{ + X: &self.X * &self.T, + Y: &self.Y * &self.Z, + Z: &self.Z * &self.T, + } + } + + fn to_extended(&self) -> ExtendedPoint { + ExtendedPoint{ + X: &self.X * &self.T, + Y: &self.Y * &self.Z, + Z: &self.Z * &self.T, + T: &self.X * &self.Y, + } + } +} + +// ------------------------------------------------------------------------ +// Doubling +// ------------------------------------------------------------------------ + +impl ProjectivePoint { + /// Double this point: return self + self + fn double(&self) -> CompletedPoint { // Double() + let XX = self.X.square(); + let YY = self.Y.square(); + let ZZ2 = self.Z.square2(); + let X_plus_Y = &self.X + &self.Y; + let X_plus_Y_sq = X_plus_Y.square(); + let YY_plus_XX = &YY + &XX; + let YY_minus_XX = &YY - &XX; + + CompletedPoint{ + X: &X_plus_Y_sq - &YY_plus_XX, + Y: YY_plus_XX, + Z: YY_minus_XX, + T: &ZZ2 - &YY_minus_XX + } + } +} + +impl ExtendedPoint { + /// Add this point to itself. + fn double(&self) -> ExtendedPoint { + self.to_projective().double().to_extended() + } +} + +// ------------------------------------------------------------------------ +// Addition and Subtraction +// ------------------------------------------------------------------------ + +impl<'a,'b> Add<&'b CachedPoint> for &'a ExtendedPoint { + type Output = CompletedPoint; + + fn add(self, other: &'b CachedPoint) -> CompletedPoint { + let Y_plus_X = &self.Y + &self.X; + let Y_minus_X = &self.Y - &self.X; + let PP = &Y_plus_X * &other.Y_plus_X; + let MM = &Y_minus_X * &other.Y_minus_X; + let TT2d = &self.T * &other.T2d; + let ZZ = &self.Z * &other.Z; + let ZZ2 = &ZZ + &ZZ; + + CompletedPoint{ + X: &PP - &MM, + Y: &PP + &MM, + Z: &ZZ2 + &TT2d, + T: &ZZ2 - &TT2d + } + } +} + +impl<'a,'b> Sub<&'b CachedPoint> for &'a ExtendedPoint { + type Output = CompletedPoint; + + fn sub(self, other: &'b CachedPoint) -> CompletedPoint { + let Y_plus_X = &self.Y + &self.X; + let Y_minus_X = &self.Y - &self.X; + let PM = &Y_plus_X * &other.Y_minus_X; + let MP = &Y_minus_X * &other.Y_plus_X; + let TT2d = &self.T * &other.T2d; + let ZZ = &self.Z * &other.Z; + let ZZ2 = &ZZ + &ZZ; + + CompletedPoint{ + X: &PM - &MP, + Y: &PM + &MP, + Z: &ZZ2 - &TT2d, + T: &ZZ2 + &TT2d + } + } +} + +impl<'a,'b> Add<&'b PreComputedPoint> for &'a ExtendedPoint { + type Output = CompletedPoint; + + fn add(self, other: &'b PreComputedPoint) -> CompletedPoint { + let Y_plus_X = &self.Y + &self.X; + let Y_minus_X = &self.Y - &self.X; + let PP = &Y_plus_X * &other.y_plus_x; + let MM = &Y_minus_X * &other.y_minus_x; + let Txy2d = &self.T * &other.xy2d; + let Z2 = &self.Z + &self.Z; + + CompletedPoint{ + X: &PP - &MM, + Y: &PP + &MM, + Z: &Z2 + &Txy2d, + T: &Z2 - &Txy2d + } + } +} + +impl<'a,'b> Sub<&'b PreComputedPoint> for &'a ExtendedPoint { + type Output = CompletedPoint; + + fn sub(self, other: &'b PreComputedPoint) -> CompletedPoint { + let Y_plus_X = &self.Y + &self.X; + let Y_minus_X = &self.Y - &self.X; + let PM = &Y_plus_X * &other.y_minus_x; + let MP = &Y_minus_X * &other.y_plus_x; + let Txy2d = &self.T * &other.xy2d; + let Z2 = &self.Z + &self.Z; + + CompletedPoint{ + X: &PM - &MP, + Y: &PM + &MP, + Z: &Z2 - &Txy2d, + T: &Z2 + &Txy2d + } + } +} + +impl<'a,'b> Add<&'b ExtendedPoint> for &'a ExtendedPoint { + type Output = ExtendedPoint; + fn add(self, other: &'b ExtendedPoint) -> ExtendedPoint { + (self + &other.to_cached()).to_extended() + } +} + +impl<'a,'b> Sub<&'b ExtendedPoint> for &'a ExtendedPoint { + type Output = ExtendedPoint; + fn sub(self, other: &'b ExtendedPoint) -> ExtendedPoint { + (self - &other.to_cached()).to_extended() + } +} + +impl<'a> Neg for &'a ExtendedPoint { + type Output = ExtendedPoint; + + fn neg(self) -> ExtendedPoint { + ExtendedPoint{ + X: -(&self.X), + Y: self.Y, + Z: self.Z, + T: -(&self.T), + } + } +} + +impl<'a> Neg for &'a CachedPoint { + type Output = CachedPoint; + + fn neg(self) -> CachedPoint { + CachedPoint{ + Y_plus_X: self.Y_minus_X, + Y_minus_X: self.Y_plus_X, + Z: self.Z, + T2d: -(&self.T2d), + } + } +} + + +impl<'a> Neg for &'a PreComputedPoint { + type Output = PreComputedPoint; + + fn neg(self) -> PreComputedPoint { + PreComputedPoint{ + y_plus_x: self.y_minus_x, + y_minus_x: self.y_plus_x, + xy2d: -(&self.xy2d) + } + } +} + +// ------------------------------------------------------------------------ +// Scalar multiplication +// ------------------------------------------------------------------------ + +impl ExtendedPoint { + /// Scalar multiplication: compute `a * self`. + /// + /// Uses a window of size 4. Note: for scalar multiplication of + /// the basepoint, `basepoint_mult` is approximately 4x faster. + pub fn scalar_mult(&self, a: &Scalar) -> ExtendedPoint { + let A = self.to_cached(); + let mut As: [CachedPoint; 8] = [A; 8]; + for i in 0..7 { + As[i+1] = (self + &As[i]).to_extended().to_cached(); + } + let e = a.to_radix_16(); + let mut h = ExtendedPoint::identity(); + let mut t: CompletedPoint; + for i in (0..64).rev() { + h = h.mult_by_pow_2(4); + t = &h + &select_precomputed_point(e[i], &As); + h = t.to_extended(); + } + h + } + + /// Construct an `ExtendedPoint` from a `Scalar`, `a`, by + /// computing the multiple `aB` of the basepoint `B`. + /// + /// Precondition: the scalar must be reduced. + /// + /// The computation proceeds as follows, as described on page 13 + /// of the Ed25519 paper. Write the scalar `a` in radix 16 with + /// coefficients in [-8,8), i.e., + /// + /// a = a_0 + a_1*16^1 + ... + a_63*16^63, + /// + /// with -8 ≤ a_i < 8. Then + /// + /// a*B = a_0*B + a_1*16^1*B + ... + a_63*16^63*B. + /// + /// Grouping even and odd coefficients gives + /// + /// a*B = a_0*16^0*B + a_2*16^2*B + ... + a_62*16^62*B + /// + a_1*16^1*B + a_3*16^3*B + ... + a_63*16^63*B + /// = (a_0*16^0*B + a_2*16^2*B + ... + a_62*16^62*B) + /// + 16*(a_1*16^0*B + a_3*16^2*B + ... + a_63*16^62*B). + /// + /// We then use the `select_precomputed_point` function, which + /// takes `-8 ≤ x < 8` and `[16^2i * B, ..., 8 * 16^2i * B]`, + /// and returns `x * 16^2i * B` in constant time. + pub fn basepoint_mult(a: &Scalar) -> ExtendedPoint { //GeScalarMultBase + let e = a.to_radix_16(); + let mut h = ExtendedPoint::identity(); + let mut t: CompletedPoint; + + for i in (0..64).filter(|x| x % 2 == 1) { + t = &h + &select_precomputed_point(e[i], &constants::base[i/2]); + h = t.to_extended(); + } + + h = h.mult_by_pow_2(4); + + for i in (0..64).filter(|x| x % 2 == 0) { + t = &h + &select_precomputed_point(e[i], &constants::base[i/2]); + h = t.to_extended(); + } + + h + } + + /// Compute `2^k * self` by successive doublings. + /// Requires `k > 0`. + #[inline] + pub fn mult_by_pow_2(&self, k: u32) -> ExtendedPoint { + let mut r: CompletedPoint; + let mut s = self.to_projective(); + for _ in 0..(k-1) { + r = s.double(); s = r.to_projective(); + } + // Unroll last iteration so we can go directly to_extended() + r = s.double(); + return r.to_extended(); + } +} + +/// Given a point `A` and scalars `a` and `b`, compute the point +/// `aA+bB`, where `B` is the Ed25519 basepoint (i.e., `B = (x,4/5)` +/// with x positive). +/// +/// # Warning +/// +/// This function is *not* constant time, hence its name. +// XXX should return ExtendedPoint? +pub fn double_scalar_mult_vartime(a: &Scalar, A: &ExtendedPoint, b: &Scalar) -> ProjectivePoint { + let a_naf = a.non_adjacent_form(); + let b_naf = b.non_adjacent_form(); + + // Build a lookup table of odd multiples of A + let mut Ai = [CachedPoint::identity(); 8]; + let A2 = A.double(); + Ai[0] = A.to_cached(); + for i in 0..7 { + Ai[i+1] = (&A2 + &Ai[i]).to_extended().to_cached(); + } + // Now Ai = [A, 3A, 5A, 7A, 9A, 11A, 13A, 15A] + + // Find starting index + let mut i: usize = 255; + for j in (0..255).rev() { + i = j; + if a_naf[i] != 0 || b_naf[i] != 0 { + break; + } + } + + let mut r = ProjectivePoint::identity(); + loop { + let mut t = r.double(); + + if a_naf[i] > 0 { + t = &t.to_extended() + &Ai[( a_naf[i]/2) as usize]; + } else if a_naf[i] < 0 { + t = &t.to_extended() - &Ai[(-a_naf[i]/2) as usize]; + } + + if b_naf[i] > 0 { + t = &t.to_extended() + &constants::bi[( b_naf[i]/2) as usize]; + } else if b_naf[i] < 0 { + t = &t.to_extended() - &constants::bi[(-b_naf[i]/2) as usize]; + } + + r = t.to_projective(); + + if i == 0 { + break; + } + i -= 1; + } + + r +} + +/// Given precomputed points `[P, 2P, 3P, ..., 8P]`, as well as `-8 ≤ +/// x ≤ 8`, compute `x * B` in constant time, i.e., without branching +/// on x or using it as an array index. +fn select_precomputed_point(x: i8, points: &[T; 8]) -> T + where T: Identity + CTAssignable, for<'a> &'a T: Neg +{ + debug_assert!(x >= -8); debug_assert!(x <= 8); + + // Compute xabs = |x| + let xmask = x >> 7; + let xabs = (x + xmask) ^ xmask; + + // Set t = 0 * P = identity + let mut t = T::identity(); + for j in 1..9 { + // Copy `points[j-1] == j*P` onto `t` in constant time if `|x| == j`. + t.conditional_assign(&points[j-1], + bytes_equal_ct(xabs as u8, j as u8)); + } + // Now t == |x| * P. + + let minus_t = -(&t); + let neg_mask = (xmask & 1) as u8; + t.conditional_assign(&minus_t, neg_mask); + // Now t == x * P. + + t +} + +// ------------------------------------------------------------------------ +// Elligator2 (uniform encoding/decoding of curve points) +// ------------------------------------------------------------------------ + +impl ExtendedPoint { + /// Use Elligator2 to try to convert `self` to a uniformly random + /// string. + /// + /// Returns `Some<[u8;32]>` if `self` is in the image of the + /// Elligator2 map. For a random point on the curve, this happens + /// with probability 1/2. Otherwise, returns `None`. + pub fn to_uniform_representative(&self) -> Option<[u8;32]> { + unimplemented!(); + } + + /// Use Elligator2 to convert a uniformly random string to a curve + /// point. + #[allow(unused_variables)] // REMOVE WHEN IMPLEMENTED + pub fn from_uniform_representative(bytes: &[u8;32]) -> ExtendedPoint { + unimplemented!(); + } +} + +// ------------------------------------------------------------------------ +// Debug traits +// ------------------------------------------------------------------------ + +impl Debug for ExtendedPoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "ExtendedPoint(\n\tX: {:?},\n\tY: {:?},\n\tZ: {:?},\n\tT: {:?}\n)", + &self.X, &self.Y, &self.Z, &self.T) + } +} + +impl Debug for ProjectivePoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "ProjectivePoint(\n\tX: {:?},\n\tY: {:?},\n\tZ: {:?}\n)", + &self.X, &self.Y, &self.Z) + } +} + +impl Debug for CompletedPoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "CompletedPoint(\n\tX: {:?},\n\tY: {:?},\n\tZ: {:?},\n\tT: {:?}\n)", + &self.X, &self.Y, &self.Z, &self.T) + } +} + +impl Debug for PreComputedPoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "PreComputedPoint(\n\ty_plus_x: {:?},\n\ty_minus_x: {:?},\n\txy2d: {:?}\n)", + &self.y_plus_x, &self.y_minus_x, &self.xy2d) + } +} + +impl Debug for CachedPoint { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "CachedPoint(\n\tY_plus_X: {:?},\n\tY_minus_X: {:?},\n\tZ: {:?},\n\tT2d: {:?}\n)", + &self.Y_plus_X, &self.Y_minus_X, &self.Z, &self.T2d) + } +} + +// ------------------------------------------------------------------------ +// Tests +// ------------------------------------------------------------------------ + +#[cfg(test)] +mod test { + use test::Bencher; + use field::FieldElement; + use scalar::Scalar; + use constants; + use super::*; + use super::select_precomputed_point; + + /// Basepoint has y = 4/5. + /// + /// Generated with Sage: these are the bytes of 4/5 in 𝔽_p. The + /// sign bit is 0 since the basepoint has x chosen to be positive. + static BASE_CMPRSSD: CompressedPoint = + CompressedPoint([0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66]); + + /// X coordinate of the basepoint. + /// = 15112221349535400772501151409588531511454012693041857206046113283949847762202 + static BASE_X_COORD_BYTES: [u8; 32] = + [0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, + 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21]; + + static BASE2_CMPRSSD: CompressedPoint = + CompressedPoint([0xc9, 0xa3, 0xf8, 0x6a, 0xae, 0x46, 0x5f, 0xe, + 0x56, 0x51, 0x38, 0x64, 0x51, 0x0f, 0x39, 0x97, + 0x56, 0x1f, 0xa2, 0xc9, 0xe8, 0x5e, 0xa2, 0x1d, + 0xc2, 0x29, 0x23, 0x09, 0xf3, 0xcd, 0x60, 0x22]); + + static BASE16_CMPRSSD: CompressedPoint = + CompressedPoint([0xeb, 0x27, 0x67, 0xc1, 0x37, 0xab, 0x7a, 0xd8, + 0x27, 0x9c, 0x07, 0x8e, 0xff, 0x11, 0x6a, 0xb0, + 0x78, 0x6e, 0xad, 0x3a, 0x2e, 0x0f, 0x98, 0x9f, + 0x72, 0xc3, 0x7f, 0x82, 0xf2, 0x96, 0x96, 0x70]); + + /// 4493907448824000747700850167940867464579944529806937181821189941592931634714 + static A_SCALAR: Scalar = Scalar([ + 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, + 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d, + 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, + 0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09]); + + /// 2506056684125797857694181776241676200180934651973138769173342316833279714961 + static B_SCALAR: Scalar = Scalar([ + 0x91, 0x26, 0x7a, 0xcf, 0x25, 0xc2, 0x09, 0x1b, + 0xa2, 0x17, 0x74, 0x7b, 0x66, 0xf0, 0xb3, 0x2e, + 0x9d, 0xf2, 0xa5, 0x67, 0x41, 0xcf, 0xda, 0xc4, + 0x56, 0xa7, 0xd4, 0xaa, 0xb8, 0x60, 0x8a, 0x05]); + + /// A_SCALAR * basepoint, computed with ed25519.py + static A_TIMES_BASEPOINT: CompressedPoint = CompressedPoint([ + 0xea, 0x27, 0xe2, 0x60, 0x53, 0xdf, 0x1b, 0x59, + 0x56, 0xf1, 0x4d, 0x5d, 0xec, 0x3c, 0x34, 0xc3, + 0x84, 0xa2, 0x69, 0xb7, 0x4c, 0xc3, 0x80, 0x3e, + 0xa8, 0xe2, 0xe7, 0xc9, 0x42, 0x5e, 0x40, 0xa5]); + + /// A_SCALAR * (A_TIMES_BASEPOINT) + B_SCALAR * BASEPOINT + static DOUBLE_SCALAR_MULT_RESULT: CompressedPoint = CompressedPoint([ + 0x7d, 0xfd, 0x6c, 0x45, 0xaf, 0x6d, 0x6e, 0x0e, + 0xba, 0x20, 0x37, 0x1a, 0x23, 0x64, 0x59, 0xc4, + 0xc0, 0x46, 0x83, 0x43, 0xde, 0x70, 0x4b, 0x85, + 0x09, 0x6f, 0xfe, 0x35, 0x4f, 0x13, 0x2b, 0x42]); + + /// Test round-trip decompression for the basepoint. + #[test] + fn test_basepoint_decompression_compression() { + let base_X = FieldElement::from_bytes(&BASE_X_COORD_BYTES); + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp2 = BASE2_CMPRSSD.decompress().unwrap(); + let compressed = bp.compress(); + let compressed2 = bp2.compress(); + // Check that decompression actually gives the correct X coordinate + assert_eq!(base_X, bp.X); + assert_eq!(compressed, BASE_CMPRSSD); + assert_eq!(compressed2, BASE2_CMPRSSD); + } + + /// Test sign handling in decompression + #[test] + fn test_decompression_sign_handling() { + let mut m_bp_bytes: [u8;32] = BASE_CMPRSSD.to_bytes().clone(); + // Set the high bit of the last byte to flip the sign + m_bp_bytes[31] |= 1 << 7; + let m_bp = CompressedPoint(m_bp_bytes).decompress().unwrap(); + let bp = BASE_CMPRSSD.decompress().unwrap(); + assert_eq!(m_bp.X, -(&bp.X)); + assert_eq!(m_bp.Y, bp.Y); + assert_eq!(m_bp.Z, bp.Z); + assert_eq!(m_bp.T, -(&bp.T)); + } + + /// Test that computing 1*basepoint gives the correct basepoint. + #[test] + fn test_basepoint_mult_one_vs_basepoint() { + let bp = ExtendedPoint::basepoint_mult(&Scalar::one()); + let compressed = bp.compress(); + assert_eq!(compressed, BASE_CMPRSSD); + } + + /// Test `impl Add for ExtendedPoint` + /// using basepoint + basepoint versus the 2*basepoint constant. + #[test] + fn test_basepoint_plus_basepoint() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp_added = &bp + &bp; + assert_eq!( bp_added.compress(), BASE2_CMPRSSD); + } + + /// Test `impl Add for ExtendedPoint` + /// using the basepoint, basepoint2 constants + #[test] + fn test_basepoint_plus_basepoint_cached() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp_added = (&bp + &bp.to_cached()).to_extended(); + assert_eq!( bp_added.compress(), BASE2_CMPRSSD); + } + + /// Test `impl Add for ExtendedPoint` + /// using the basepoint, basepoint2 constants + #[test] + fn test_basepoint_plus_basepoint_precomputed() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + // on decode, Z =1, so x = X/Z = X, y = Y/Z = Y, xy = T + let bp_precomputed = PreComputedPoint{ + y_plus_x: &bp.Y + &bp.X, + y_minus_x: &bp.Y - &bp.X, + xy2d: &bp.T * &constants::d2, + }; + let bp_added = (&bp + &bp_precomputed).to_extended(); + assert_eq!( bp_added.compress(), BASE2_CMPRSSD); + } + + /// Test basepoint_mult versus a known scalar multiple from ed25519.py + #[test] + fn test_basepoint_mult() { + let aB = ExtendedPoint::basepoint_mult(&A_SCALAR); + assert_eq!(aB.compress(), A_TIMES_BASEPOINT); + } + + /// Test scalar_mult versus a known scalar multiple from ed25519.py + #[test] + fn test_scalar_mult() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let aB = bp.scalar_mult(&A_SCALAR); + assert_eq!(aB.compress(), A_TIMES_BASEPOINT); + } + + /// Test double_scalar_mult_vartime vs ed25519.py + #[test] + fn test_double_scalar_mult_vartime() { + let A = A_TIMES_BASEPOINT.decompress().unwrap(); + let result = double_scalar_mult_vartime(&A_SCALAR, &A, &B_SCALAR); + assert_eq!(result.compress(), DOUBLE_SCALAR_MULT_RESULT); + } + + /// Test basepoint.double() versus the 2*basepoint constant. + #[test] + fn test_basepoint_double() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp_doubled = bp.double(); + assert_eq!(bp_doubled.compress(), BASE2_CMPRSSD); + } + + /// Test that computing 2*basepoint is the same as basepoint.double() + #[test] + fn test_scalar_mult_two_vs_double() { + // XXX this seems like a pain point: better way to construct small + // scalars? + let two = Scalar([ 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, 0, 0, 0 ]); + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp_doubled = bp.double(); + let bp2 = ExtendedPoint::basepoint_mult(&two); + assert_eq!(bp_doubled.compress(), bp2.compress()); + } + + #[test] + fn test_basepoint_projective_extended_round_trip() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp_roundtrip = bp.to_projective().to_extended(); + + assert_eq!(BASE_CMPRSSD, bp_roundtrip.compress()); + } + + /// Test computing 16*basepoint vs mult_by_pow_2 + #[test] + fn test_mult_by_pow_2() { + let bp = BASE_CMPRSSD.decompress().unwrap(); + let bp16 = bp.mult_by_pow_2(4); + assert_eq!(bp16.compress(), BASE16_CMPRSSD); + } + + /// The basepoint, doubled, minus the basepoint should equal the basepoint. + #[test] + fn test_ge_sub() { + let p1: ExtendedPoint = BASE_CMPRSSD.decompress().unwrap(); + let p2: ExtendedPoint = BASE2_CMPRSSD.decompress().unwrap(); + let p3: ExtendedPoint = (&p2 - &p1.to_cached()).to_extended(); + + assert_eq!(p1.compress(), p3.compress()); + } + + /// The basepoint plus the identity should equal the basepoint. + #[test] + fn test_ge_add() { + let p1: ExtendedPoint = BASE_CMPRSSD.decompress().unwrap(); + let p2: ExtendedPoint = ExtendedPoint::identity(); + let p3: ExtendedPoint = (&p1 + &p2.to_cached()).to_extended(); + + assert_eq!(p1.compress(), p3.compress()); + } + + #[test] + fn test_PreComputedPoint_conditional_assign() { + let id = PreComputedPoint::identity(); + let mut p1 = PreComputedPoint::identity(); + let p2: PreComputedPoint = PreComputedPoint{ + y_plus_x: FieldElement([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]), + y_minus_x: FieldElement([11, 22, 33, 44, 55, 66, 77, 88, 99, 100]), + xy2d: FieldElement([10, 20, 30, 40, 50, 60, 70, 80, 90, 101]), + }; + + p1.conditional_assign(&p2, 0); + assert_eq!(p1.y_plus_x, id.y_plus_x); + assert_eq!(p1.y_minus_x, id.y_minus_x); + assert_eq!(p1.xy2d, id.xy2d); + p1.conditional_assign(&p2, 1); + assert_eq!(p1.y_plus_x, p2.y_plus_x); + assert_eq!(p1.y_minus_x, p2.y_minus_x); + assert_eq!(p1.xy2d, p2.xy2d); + } + + #[bench] + fn bench_basepoint_mult(b: &mut Bencher) { + b.iter(|| ExtendedPoint::basepoint_mult(&A_SCALAR)); + } + + #[bench] + fn bench_scalar_mult(b: &mut Bencher) { + let bp = BASE_CMPRSSD.decompress().unwrap(); + b.iter(|| bp.scalar_mult(&A_SCALAR)); + } + + #[bench] + fn bench_select_precomputed_point(b: &mut Bencher) { + b.iter(|| select_precomputed_point(0, &constants::base[12])); + } + + #[bench] + fn bench_double_scalar_mult_vartime(bench: &mut Bencher) { + let A = A_TIMES_BASEPOINT.decompress().unwrap(); + bench.iter(|| double_scalar_mult_vartime(&A_SCALAR, &A, &B_SCALAR)); + } + + #[bench] + fn bench_extended_add_cached(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap(); + let p2 = BASE2_CMPRSSD.decompress().unwrap().to_cached(); + + b.iter(| | &p1 + &p2); + } + + #[bench] + fn bench_extended_add_cached_to_extended(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap(); + let p2 = BASE2_CMPRSSD.decompress().unwrap().to_cached(); + + b.iter(| | (&p1 + &p2).to_extended()); + } + + #[bench] + fn bench_extended_add_precomputed(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap(); + let p2 = select_precomputed_point(6, &constants::base[27]); + + b.iter(| | &p1 + &p2); + } + + #[bench] + fn bench_extended_add_precomputed_to_extended(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap(); + let p2 = select_precomputed_point(6, &constants::base[27]); + + b.iter(| | (&p1 + &p2).to_extended()); + } + + #[bench] + fn bench_double(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap().to_projective(); + + b.iter(| | p1.double() ); + } + + #[bench] + fn bench_double_to_extended(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap().to_projective(); + + b.iter(| | p1.double().to_extended() ); + } + + #[bench] + fn bench_mult_by_pow2_4(b: &mut Bencher) { + let p1 = BASE_CMPRSSD.decompress().unwrap(); + + b.iter(| | p1.mult_by_pow_2(4) ); + } +} diff --git a/src/field.rs b/src/field.rs new file mode 100644 index 0000000..2f1301f --- /dev/null +++ b/src/field.rs @@ -0,0 +1,972 @@ +// -*- mode: rust; coding: utf-8; -*- +// +// To the extent possible under law, the authors have waived all +// copyright and related or neighboring rights to curve25519-dalek, +// using the Creative Commons "CC0" public domain dedication. See +// for full +// details. +// +// Authors: +// - Isis Agora Lovecruft +// - Henry de Valence + +//! Field arithmetic for ℤ/(2²⁵⁵-19). +//! +//! Based on Adam Langley's curve25519-donna and (Golang) ed25519 +//! implementations. + +use std::clone::Clone; +use std::fmt::Debug; +use std::ops::{Add, AddAssign}; +use std::ops::{Sub, SubAssign}; +use std::ops::{Mul, MulAssign}; +use std::ops::{Index, IndexMut}; +use std::cmp::{Eq, PartialEq}; +use std::ops::Neg; + +use util::byte_is_nonzero; + +/// FieldElements are represented as an array of ten "Limbs", which are radix +/// 25.5, that is, each Limb of a FieldElement alternates between being +/// represented as a factor of 2^25 or 2^26 more than the last corresponding +/// integer. +pub type Limb = i32; + +/// FieldElement represents an element of the field GF(2^255 - 19). An element +/// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 +/// t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on +/// context. +#[derive(Copy, Clone)] +pub struct FieldElement(pub [Limb; 10]); + +impl PartialEq for FieldElement { + /// Test equality between two FieldElements by converting them to bytes. + /// + /// # Warning + /// + /// This comparison is *not* constant time. It could easily be + /// made to be, but the main use of an `Eq` implementation is for + /// branching, so it seems pointless. + /// + /// XXX it would be good to encode constant-time considerations + /// (no data flow from secret information) into Rust's type + /// system. + fn eq(&self, other: &FieldElement) -> bool { + let self_bytes = self.to_bytes(); + let other_bytes = other.to_bytes(); + let mut are_equal: bool = true; + for i in 0..32 { + are_equal &= self_bytes[i] == other_bytes[i]; + } + return are_equal; + } +} + +impl Eq for FieldElement {} + +impl Debug for FieldElement { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + write!(f, "FieldElement: {:?}", &self.0[..]) + } +} + +impl Index for FieldElement { + type Output = Limb; + + fn index<'a>(&'a self, _index: usize) -> &'a Limb { + let ret: &'a Limb = &(self.0[_index]); + ret + } +} + +impl IndexMut for FieldElement { + fn index_mut<'a>(&'a mut self, _index: usize) -> &'a mut Limb { + let ret: &'a mut Limb = &mut(self.0[_index]); + ret + } +} + +impl<'b> AddAssign<&'b FieldElement> for FieldElement { + fn add_assign(&mut self, _rhs: &'b FieldElement) { // fsum() + for i in 0..10 { + self[i] += _rhs[i]; + } + } +} + +impl<'a, 'b> Add<&'b FieldElement> for &'a FieldElement { + type Output = FieldElement; + fn add(self, _rhs: &'b FieldElement) -> FieldElement { + let mut output = self.clone(); + output += _rhs; + output + } +} + +impl<'b> SubAssign<&'b FieldElement> for FieldElement { + fn sub_assign(&mut self, _rhs: &'b FieldElement) { // fdifference() + for i in 0..10 { + self[i] -= _rhs[i]; + } + } +} + +impl<'a, 'b> Sub<&'b FieldElement> for &'a FieldElement { + type Output = FieldElement; + fn sub(self, _rhs: &'b FieldElement) -> FieldElement { + let mut output = self.clone(); + output -= _rhs; + output + } +} + +impl<'b> MulAssign<&'b FieldElement> for FieldElement { + fn mul_assign(&mut self, _rhs: &'b FieldElement) { + self.0 = self.multiply(_rhs).0; + } +} + +impl<'a, 'b> Mul<&'b FieldElement> for &'a FieldElement { + type Output = FieldElement; + fn mul(self, _rhs: &'b FieldElement) -> FieldElement { + self.multiply(_rhs) + } +} + +impl<'a> Neg for &'a FieldElement { + type Output = FieldElement; + fn neg(self) -> FieldElement { + let mut output = self.clone(); + output.negate(); + output + } +} + +/// Convert an array of (at least) three bytes into an i64. +#[inline] +#[allow(dead_code)] +pub fn load3(input: &[u8]) -> i64 { + (input[0] as i64) + | ((input[1] as i64) << 8) + | ((input[2] as i64) << 16) +} + +/// Convert an array of (at least) four bytes into an i64. +#[inline] +#[allow(dead_code)] +pub fn load4(input: &[u8]) -> i64 { + (input[0] as i64) + | ((input[1] as i64) << 8) + | ((input[2] as i64) << 16) + | ((input[3] as i64) << 24) +} + +impl FieldElement { + /// Invert the sign of this field element + pub fn negate(&mut self) { + for i in 0..10 { + self[i] = -self[i]; + } + } + + /// Construct the additive identity + pub fn zero() -> FieldElement { + FieldElement([ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]) + } + + /// Construct the multiplicative identity + pub fn one() -> FieldElement { + FieldElement([ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]) + } + + /// Overwrite this FieldElement with one of the inputs without branching. + /// Like `conditional_assign`, but chooses between two inputs instead of + /// one input and the original value. + /// + /// If `choice == 0`, replace `self` with `f`: + /// + /// ``` + /// # use curve25519_dalek::field::FieldElement; + /// let f = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// let g = FieldElement([2,2,2,2,2,2,2,2,2,2]); + /// let mut h = FieldElement([0,0,0,0,0,0,0,0,0,0]); + /// h.conditional_choose(&f, &g, 0); + /// assert!(h == f); + /// ``` + /// + /// If `choice == 1`, replace `self` with `g`: + /// + /// ``` + /// # use curve25519_dalek::field::FieldElement; + /// # let f = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// # let g = FieldElement([2,2,2,2,2,2,2,2,2,2]); + /// # let mut h = FieldElement([0,0,0,0,0,0,0,0,0,0]); + /// h.conditional_choose(&f, &g, 1); + /// assert!(h == g); + /// ``` + /// + /// # Preconditions + /// + /// * `b` in {0,1} + pub fn conditional_choose(&mut self, + f: &FieldElement, + g: &FieldElement, + choice: u8) + { + let mask = -(choice as Limb); + for i in 0..10 { + self[i] = f[i] ^ (mask & (f[i] ^ g[i])); + } + } + + /// Conditionally assign the Limbs of another FieldElement to this + /// one. Like `conditional_choose`, but choosing between one + /// input and the original value. + /// + /// If `choice == 0`, replace `self` with `self`: + /// + /// ``` + /// # use curve25519_dalek::field::FieldElement; + /// let f = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// let g = FieldElement([2,2,2,2,2,2,2,2,2,2]); + /// let mut h = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// h.conditional_assign(&g, 0); + /// assert!(h == f); + /// ``` + /// + /// If `choice == 1`, replace `self` with `f`: + /// + /// ``` + /// # use curve25519_dalek::field::FieldElement; + /// # let f = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// # let g = FieldElement([2,2,2,2,2,2,2,2,2,2]); + /// # let mut h = FieldElement([1,1,1,1,1,1,1,1,1,1]); + /// h.conditional_assign(&g, 1); + /// assert!(h == g); + /// ``` + /// + /// # Preconditions + /// + /// * `choice` in {0,1} + pub fn conditional_assign(&mut self, f: &FieldElement, choice: u8) { + let mask = -(choice as Limb); + for i in 0..10 { + self[i] ^= mask & (self[i] ^ f[i]); + } + } + + fn combine_coeffs(input: &[i64;10]) -> FieldElement { //FeCombine + let mut c = [0i64;10]; + let mut h = input.clone(); + + /* + |h[0]| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) + i.e. |h[0]| <= 1.2*2^59; narrower ranges for h[2], h[4], h[6], h[8] + |h[1]| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) + i.e. |h[1]| <= 1.5*2^58; narrower ranges for h[3], h[5], h[7], h[9] + */ + + c[0] = (h[0] + (1 << 25)) >> 26; + h[1] += c[0]; + h[0] -= c[0] << 26; + c[4] = (h[4] + (1 << 25)) >> 26; + h[5] += c[4]; + h[4] -= c[4] << 26; + /* |h[0]| <= 2^25 */ + /* |h[4]| <= 2^25 */ + /* |h[1]| <= 1.51*2^58 */ + /* |h[5]| <= 1.51*2^58 */ + + c[1] = (h[1] + (1 << 24)) >> 25; + h[2] += c[1]; + h[1] -= c[1] << 25; + c[5] = (h[5] + (1 << 24)) >> 25; + h[6] += c[5]; + h[5] -= c[5] << 25; + /* |h[1]| <= 2^24; from now on fits into int32 */ + /* |h[5]| <= 2^24; from now on fits into int32 */ + /* |h[2]| <= 1.21*2^59 */ + /* |h[6]| <= 1.21*2^59 */ + + c[2] = (h[2] + (1 << 25)) >> 26; + h[3] += c[2]; + h[2] -= c[2] << 26; + c[6] = (h[6] + (1 << 25)) >> 26; + h[7] += c[6]; + h[6] -= c[6] << 26; + /* |h[2]| <= 2^25; from now on fits into int32 unchanged */ + /* |h[6]| <= 2^25; from now on fits into int32 unchanged */ + /* |h[3]| <= 1.51*2^58 */ + /* |h[7]| <= 1.51*2^58 */ + + c[3] = (h[3] + (1 << 24)) >> 25; + h[4] += c[3]; + h[3] -= c[3] << 25; + c[7] = (h[7] + (1 << 24)) >> 25; + h[8] += c[7]; + h[7] -= c[7] << 25; + /* |h[3]| <= 2^24; from now on fits into int32 unchanged */ + /* |h[7]| <= 2^24; from now on fits into int32 unchanged */ + /* |h[4]| <= 1.52*2^33 */ + /* |h[8]| <= 1.52*2^33 */ + + c[4] = (h[4] + (1 << 25)) >> 26; + h[5] += c[4]; + h[4] -= c[4] << 26; + c[8] = (h[8] + (1 << 25)) >> 26; + h[9] += c[8]; + h[8] -= c[8] << 26; + /* |h[4]| <= 2^25; from now on fits into int32 unchanged */ + /* |h[8]| <= 2^25; from now on fits into int32 unchanged */ + /* |h[5]| <= 1.01*2^24 */ + /* |h[9]| <= 1.51*2^58 */ + + c[9] = (h[9] + (1 << 24)) >> 25; + h[0] += c[9] * 19; + h[9] -= c[9] << 25; + /* |h[9]| <= 2^24; from now on fits into int32 unchanged */ + /* |h[0]| <= 1.8*2^37 */ + + c[0] = (h[0] + (1 << 25)) >> 26; + h[1] += c[0]; + h[0] -= c[0] << 26; + /* |h[0]| <= 2^25; from now on fits into int32 unchanged */ + /* |h[1]| <= 1.01*2^24 */ + + let mut output = FieldElement([0i32;10]); + output[0] = h[0] as i32; + output[1] = h[1] as i32; + output[2] = h[2] as i32; + output[3] = h[3] as i32; + output[4] = h[4] as i32; + output[5] = h[5] as i32; + output[6] = h[6] as i32; + output[7] = h[7] as i32; + output[8] = h[8] as i32; + output[9] = h[9] as i32; + output + } + + /// Create a FieldElement by demarshalling an array of 32 bytes. + /// + /// # Example + /// + /// ``` + /// # use curve25519_dalek::field::FieldElement; + /// let data: [u8; 32] = [ 1, 2, 3, 4, 5, 6, 7, 8, + /// 9, 10, 11, 12, 13, 14, 15, 16, + /// 17, 18, 19, 20, 21, 22, 23, 24, + /// 25, 26, 27, 28, 29, 30, 31, 32 ]; + /// let fe: FieldElement = FieldElement::from_bytes(&data); + /// assert_eq!(fe, + /// FieldElement([ 197121, -4095679, 21045505, 6840408, 4209720, + /// 1249809, -7665014, -12377341, 30523826, 8420472])) + /// ``` + /// + /// # Return + /// + /// Returns a new FieldElement. + pub fn from_bytes(data: &[u8;32]) -> FieldElement { //FeFromBytes + let mut h = [0i64;10]; + h[0] = load4(&data[ 0..]); + h[1] = load3(&data[ 4..]) << 6; + h[2] = load3(&data[ 7..]) << 5; + h[3] = load3(&data[10..]) << 3; + h[4] = load3(&data[13..]) << 2; + h[5] = load4(&data[16..]); + h[6] = load3(&data[20..]) << 7; + h[7] = load3(&data[23..]) << 5; + h[8] = load3(&data[26..]) << 4; + h[9] = (load3(&data[29..]) & 8388607) << 2; + + FieldElement::combine_coeffs(&h) + } + + /// Marshal this FieldElement into a 32-byte array. + /// + /// # Preconditions + /// + /// * `|h[i]|` bounded by 1.1*2^25, 1.1*2^24, 1.1*2^25, 1.1*2^24, etc. + /// + /// # Lemma + /// + /// Write p = 2^255 - 19 and q = floor(h/p). + /// + /// Basic claim: q = floor(2^(-255)(h + 19 * 2^-25 h9 + 2^-1)). + /// + /// # Proof + /// + /// Have |h|<=p so |q|<=1 so |19^2 * 2^-255 * q| < 1/4. + /// + /// Also have |h-2^230 * h9| < 2^230 so |19 * 2^-255 * (h-2^230 * h9)| < 1/4. + /// + /// Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9), then 0 [u8;32] { //FeToBytes + let mut carry = [0i32; 10]; + let mut h = self.clone(); + + let mut q:i32 = (19*h[9] + (1 << 24)) >> 25; + q = (h[0] + q) >> 26; + q = (h[1] + q) >> 25; + q = (h[2] + q) >> 26; + q = (h[3] + q) >> 25; + q = (h[4] + q) >> 26; + q = (h[5] + q) >> 25; + q = (h[6] + q) >> 26; + q = (h[7] + q) >> 25; + q = (h[8] + q) >> 26; + q = (h[9] + q) >> 25; + + // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. + h[0] += 19 * q; + // Goal: Output h-2^255 q, which is between 0 and 2^255-20. + + carry[0] = h[0] >> 26; + h[1] += carry[0]; + h[0] -= carry[0] << 26; + carry[1] = h[1] >> 25; + h[2] += carry[1]; + h[1] -= carry[1] << 25; + carry[2] = h[2] >> 26; + h[3] += carry[2]; + h[2] -= carry[2] << 26; + carry[3] = h[3] >> 25; + h[4] += carry[3]; + h[3] -= carry[3] << 25; + carry[4] = h[4] >> 26; + h[5] += carry[4]; + h[4] -= carry[4] << 26; + carry[5] = h[5] >> 25; + h[6] += carry[5]; + h[5] -= carry[5] << 25; + carry[6] = h[6] >> 26; + h[7] += carry[6]; + h[6] -= carry[6] << 26; + carry[7] = h[7] >> 25; + h[8] += carry[7]; + h[7] -= carry[7] << 25; + carry[8] = h[8] >> 26; + h[9] += carry[8]; + h[8] -= carry[8] << 26; + carry[9] = h[9] >> 25; + h[9] -= carry[9] << 25; + // h10 = carry9 + + // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. + // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; + // evidently 2^255 h10-2^255 q = 0. + // Goal: Output h[0]+...+2^230 h[9]. + + let mut s = [0u8;32]; + s[0] = (h[0] >> 0) as u8; + s[1] = (h[0] >> 8) as u8; + s[2] = (h[0] >> 16) as u8; + s[3] = ((h[0] >> 24) | (h[1] << 2)) as u8; + s[4] = (h[1] >> 6) as u8; + s[5] = (h[1] >> 14) as u8; + s[6] = ((h[1] >> 22) | (h[2] << 3)) as u8; + s[7] = (h[2] >> 5) as u8; + s[8] = (h[2] >> 13) as u8; + s[9] = ((h[2] >> 21) | (h[3] << 5)) as u8; + s[10] = (h[3] >> 3) as u8; + s[11] = (h[3] >> 11) as u8; + s[12] = ((h[3] >> 19) | (h[4] << 6)) as u8; + s[13] = (h[4] >> 2) as u8; + s[14] = (h[4] >> 10) as u8; + s[15] = (h[4] >> 18) as u8; + s[16] = (h[5] >> 0) as u8; + s[17] = (h[5] >> 8) as u8; + s[18] = (h[5] >> 16) as u8; + s[19] = ((h[5] >> 24) | (h[6] << 1)) as u8; + s[20] = (h[6] >> 7) as u8; + s[21] = (h[6] >> 15) as u8; + s[22] = ((h[6] >> 23) | (h[7] << 3)) as u8; + s[23] = (h[7] >> 5) as u8; + s[24] = (h[7] >> 13) as u8; + s[25] = ((h[7] >> 21) | (h[8] << 4)) as u8; + s[26] = (h[8] >> 4) as u8; + s[27] = (h[8] >> 12) as u8; + s[28] = ((h[8] >> 20) | (h[9] << 6)) as u8; + s[29] = (h[9] >> 2) as u8; + s[30] = (h[9] >> 10) as u8; + s[31] = (h[9] >> 18) as u8; + + //Clear high bit + s[31] &= 127u8; + + s + } + + /// XXX clarify documentation + /// Determine if this field element, represented as a byte array, + /// is less than or equal to another field element represented as + /// a byte array. + /// + /// # Returns + /// + /// Returns `1u8` if `self.to_bytes() <= other.to_bytes()`, and `0u8` otherwise. + pub fn bytes_equal_less_than(&self, other: &[u8; 32]) -> u8 { // feBytesLess + // XXX cleanup + let mut equal_so_far: i32 = -1i32; + let mut greater: i32 = 0i32; + + let this: [u8; 32] = self.to_bytes(); + + for i in 32 .. 0 { + let x: i32 = this[i-1] as i32; + let y: i32 = other[i-1] as i32; + + greater = (!equal_so_far & greater) | (equal_so_far & ((x - y) >> 31)); + equal_so_far = equal_so_far & (((x ^ y) - 1) >> 31); + } + + (!equal_so_far & 1 & greater) as u8 + } + + /// Determine if this `FieldElement` is negative. + /// + /// # Return + /// + /// If negative, return `1i32`. Otherwise, return `0i32`. + // XXX should return u8 + pub fn is_negative(&self) -> i32 { //FeIsNegative + let bytes = self.to_bytes(); + (bytes[0] & 1) as i32 + } + + /// Determine if this `FieldElement` is non-zero. + /// + /// # Return + /// + /// If non-zero, return `1u8`. Otherwise, return `0u8`. + pub fn is_nonzero(&self) -> u8 { //FeIsNonZero + let bytes = self.to_bytes(); + let mut x = 0u8; + for b in &bytes { + x |= *b; + } + return byte_is_nonzero(x); + } + + /// Calculates h = f * g. Can overlap h with f or g. + /// + /// # Preconditions + /// + /// * |f[i]| bounded by 1.1*2^26, 1.1*2^25, 1.1*2^26, 1.1*2^25, etc. + /// * |g[i]| bounded by 1.1*2^26, 1.1*2^25, 1.1*2^26, 1.1*2^25, etc. + /// + /// # Postconditions + /// + /// * |h| bounded by 1.1*2^25, 1.1*2^24, 1.1*2^25, 1.1*2^24, etc. + /// + /// ## Notes on implementation strategy + /// + /// * Using schoolbook multiplication. + /// * Karatsuba would save a little in some cost models. + /// + /// * Most multiplications by 2 and 19 are 32-bit precomputations; + /// cheaper than 64-bit postcomputations. + /// + /// * There is one remaining multiplication by 19 in the carry chain; + /// one *19 precomputation can be merged into this, + /// but the resulting data flow is considerably less clean. + /// + /// * There are 12 carries below. + /// 10 of them are 2-way parallelizable and vectorizable. + /// Can get away with 11 carries, but then data flow is much deeper. + /// + /// * With tighter constraints on inputs can squeeze carries into int32. + pub fn multiply(&self, _rhs: &FieldElement) -> FieldElement { + let f0 = self[0] as i64; + let f1 = self[1] as i64; + let f2 = self[2] as i64; + let f3 = self[3] as i64; + let f4 = self[4] as i64; + let f5 = self[5] as i64; + let f6 = self[6] as i64; + let f7 = self[7] as i64; + let f8 = self[8] as i64; + let f9 = self[9] as i64; + + let f1_2 = (2 * self[1]) as i64; + let f3_2 = (2 * self[3]) as i64; + let f5_2 = (2 * self[5]) as i64; + let f7_2 = (2 * self[7]) as i64; + let f9_2 = (2 * self[9]) as i64; + + let g0 = _rhs[0] as i64; + let g1 = _rhs[1] as i64; + let g2 = _rhs[2] as i64; + let g3 = _rhs[3] as i64; + let g4 = _rhs[4] as i64; + let g5 = _rhs[5] as i64; + let g6 = _rhs[6] as i64; + let g7 = _rhs[7] as i64; + let g8 = _rhs[8] as i64; + let g9 = _rhs[9] as i64; + + let g1_19 = (19 * _rhs[1]) as i64; /* 1.4*2^29 */ + let g2_19 = (19 * _rhs[2]) as i64; /* 1.4*2^30; still ok */ + let g3_19 = (19 * _rhs[3]) as i64; + let g4_19 = (19 * _rhs[4]) as i64; + let g5_19 = (19 * _rhs[5]) as i64; + let g6_19 = (19 * _rhs[6]) as i64; + let g7_19 = (19 * _rhs[7]) as i64; + let g8_19 = (19 * _rhs[8]) as i64; + let g9_19 = (19 * _rhs[9]) as i64; + + let h0 = f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19; + let h1 = f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19; + let h2 = f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19; + let h3 = f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19; + let h4 = f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19; + let h5 = f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19; + let h6 = f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19; + let h7 = f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19; + let h8 = f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19; + let h9 = f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0; + + FieldElement::combine_coeffs(&[h0, h1, h2, h3, h4, h5, h6, h7, h8, h9]) + } + + fn square_inner(&self) -> [i64;10] { + let f0 = self[0] as i64; + let f1 = self[1] as i64; + let f2 = self[2] as i64; + let f3 = self[3] as i64; + let f4 = self[4] as i64; + let f5 = self[5] as i64; + let f6 = self[6] as i64; + let f7 = self[7] as i64; + let f8 = self[8] as i64; + let f9 = self[9] as i64; + let f0_2 = (2 * self[0]) as i64; + let f1_2 = (2 * self[1]) as i64; + let f2_2 = (2 * self[2]) as i64; + let f3_2 = (2 * self[3]) as i64; + let f4_2 = (2 * self[4]) as i64; + let f5_2 = (2 * self[5]) as i64; + let f6_2 = (2 * self[6]) as i64; + let f7_2 = (2 * self[7]) as i64; + let f5_38 = 38 * f5; // 1.31*2^30 + let f6_19 = 19 * f6; // 1.31*2^30 + let f7_38 = 38 * f7; // 1.31*2^30 + let f8_19 = 19 * f8; // 1.31*2^30 + let f9_38 = 38 * f9; // 1.31*2^30 + + let mut h = [0i64;10]; + h[0] = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38; + h[1] = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19; + h[2] = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19; + h[3] = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38; + h[4] = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38; + h[5] = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19; + h[6] = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19; + h[7] = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38; + h[8] = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38; + h[9] = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5; + + h + } + + /// Calculates h = f*f. Can overlap h with f. + /// + /// # Preconditions + /// + /// * |f[i]| bounded by 1.1*2^26, 1.1*2^25, 1.1*2^26, 1.1*2^25, etc. + /// + /// # Postconditions + /// + /// * |h[i]| bounded by 1.1*2^25, 1.1*2^24, 1.1*2^25, 1.1*2^24, etc. + pub fn square(&self) -> FieldElement { + FieldElement::combine_coeffs(&self.square_inner()) + } + + /// Square this field element and multiply the result by 2. + /// + /// # Preconditions + /// + /// * |f[i]| bounded by 1.65*2^26, 1.65*2^25, 1.65*2^26, 1.65*2^25, etc. + /// + /// # Postconditions + /// + /// * |h[i]| bounded by 1.01*2^25, 1.01*2^24, 1.01*2^25, 1.01*2^24, etc. + /// + /// # Notes + /// + /// See fe_mul.c in ref10 implementation for discussion of implementation + /// strategy. + pub fn square2(&self) -> FieldElement { + let mut coeffs = self.square_inner(); + for i in 0..10 { + coeffs[i] += coeffs[i]; + } + FieldElement::combine_coeffs(&coeffs) + } + + #[inline] + #[allow(dead_code)] + /// Requires k > 0; raise self to the 2^(2^k)-th power. + fn pow2k(&self, k: u32) -> FieldElement { + let mut z = self.square(); + for _ in 1..k { z = z.square(); } + z + } + + /// Compute (self^(2^250-1), self^11), used as a helper function + /// within invert() and pow22523(). + /// + /// XXX This returns an extra intermediate to save computation in + /// finding inverses, at the cost of an extra copy when it's not + /// used (e.g., when raising to (p-1)/2 or (p-5)/8). Good idea? + fn pow22501(&self) -> (FieldElement,FieldElement) { + // Instead of managing which temporary variables are used + // for what, we define as many as we need and trust the + // compiler to reuse stack space as appropriate. + // + // XXX testing some examples suggests that this does happen, + // but it would be good to check asm for this function. + // + // 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 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). + pub 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. + /// Used in decoding. + pub 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 + } + + /// chi calculates `self^((p-1)/2)`. + /// + /// # Return + /// + /// * If this element is a non-zero square, returns `1`. + /// * If it is zero, returns `0`. + /// * If it is non-square, returns `-1`. + pub fn chi(&self) -> FieldElement { // extra25519.chi + // The bits of (p-1)/2 = 2^254 -10 are 0110111111...11. + // + // nonzero bits of exponent + let (t19, _) = self.pow22501(); // 249..0 + let t20 = t19.pow2k(4); // 253..4 + let t21 = self.square(); // 1 + let t22 = t21.square(); // 2 + let t23 = &t22 * &t21; // 2,1 + let t24 = &t20 * &t23; // 253..4,2,1 + + t24 + } +} + +#[cfg(test)] +mod test { + use field::*; + use test::Bencher; + + #[bench] + fn bench_fieldelement_a_mul_a(b: &mut Bencher) { + let a = FieldElement::from_bytes(&A_BYTES); + b.iter(|| &a*&a); + } + + #[bench] + fn bench_fieldelement_a_sq(b: &mut Bencher) { + let a = FieldElement::from_bytes(&A_BYTES); + b.iter(|| a.square()); + } + + #[bench] + fn bench_fieldelement_a_inv(b: &mut Bencher) { + let a = FieldElement::from_bytes(&A_BYTES); + b.iter(|| a.invert()); + } + + /// 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 test_fieldelement_a_mul_a() { + let a = FieldElement::from_bytes(&A_BYTES); + let asq = FieldElement::from_bytes(&ASQ_BYTES); + assert_eq!(asq, &a*&a); + assert_eq!(asq, a.square()); + } + + #[test] + fn test_fieldelement_a_square2() { + let a = FieldElement::from_bytes(&A_BYTES); + let asq = FieldElement::from_bytes(&ASQ_BYTES); + assert_eq!(a.square2(), &asq+&asq); + } + + #[test] + fn test_fieldelement_a_inv() { + let a = FieldElement::from_bytes(&A_BYTES); + let ainv = FieldElement::from_bytes(&AINV_BYTES); + assert_eq!(ainv, a.invert()); + } + + #[test] + fn test_fieldelement_a_p58() { + let a = FieldElement::from_bytes(&A_BYTES); + let ap58 = FieldElement::from_bytes(&AP58_BYTES); + assert_eq!(ap58, a.pow_p58()); + } + + #[test] + fn test_fieldelement_a_chi() { + let a = FieldElement::from_bytes(&A_BYTES); + // a is square + assert_eq!(a.chi(), FieldElement::one()); + } + + #[test] + fn test_fieldelement_eq() { + 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]; + + static B_LIMBS: FieldElement = FieldElement( + [-5652623, 8034020, 8266223, -13556020, -5672552, -5582839, -12603138, 15161929, -16418207, 13296296]); + + #[test] + fn test_fieldelement_frombytes_highbit_is_ignored() { + let mut cleared_bytes = B_BYTES.clone(); + cleared_bytes[31] &= 127u8; + let orig_elt = FieldElement::from_bytes(&B_BYTES); + let cleared_elt = FieldElement::from_bytes(&cleared_bytes); + for i in 0..10 { + assert!(orig_elt[i] == cleared_elt[i]); + } + } + + #[test] + fn test_fieldelement_to_bytes() { + let test_elt = FieldElement::from_bytes(&B_BYTES); + for i in 0..10 { + assert!(test_elt[i] == B_LIMBS[i]); + } + } + + #[test] + fn test_fieldelement_from_bytes() { + let test_bytes = B_LIMBS.to_bytes(); + for i in 0..31 { + assert!(test_bytes[i] == B_BYTES[i]); + } + // high bit is set to zero in to_bytes + assert!(test_bytes[31] == (B_BYTES[31] & 127u8)); + } +} diff --git a/src/lib.rs b/src/lib.rs new file mode 100644 index 0000000..1a6d498 --- /dev/null +++ b/src/lib.rs @@ -0,0 +1,1481 @@ +// -*- mode: rust; coding: utf-8; -*- +// +// To the extent possible under law, the authors have waived all copyright and +// related or neighboring rights to curve25519-dalek, using the Creative +// Commons "CC0" public domain dedication. +// See for full details. +// +// Authors: +// - Isis Agora Lovecruft +// - Henry de Valence + +#![allow(unused_features)] +#![feature(test)] +#![deny(missing_docs)] // refuse to compile if documentation is missing + +//! # curve25519-dalek +//! +//! **A Rust implementation of field and group operations on an Edwards curve +//! over F(2^255 - 19).** +//! +//! **[SPOILER ALERT]** The Twelfth Doctor's first encounter with the Daleks is +//! in his second full episode, "Into the Dalek". A beleaguered ship of the +//! "Combined Galactic Resistance" has discovered a broken Dalek that has +//! turned "good", desiring to kill all other Daleks. The Doctor, Clara and a +//! team of soldiers are miniaturized and enter the Dalek, which the Doctor +//! names Rusty. They repair the damage, but accidentally restore it to its +//! original nature, causing it to go on the rampage and alert the Dalek fleet +//! to the whereabouts of the rebel ship. However, the Doctor manages to +//! return Rusty to its previous state by linking his mind with the Dalek's: +//! Rusty shares the Doctor's view of the universe's beauty, but also his deep +//! hatred of the Daleks. Rusty destroys the other Daleks and departs the +//! ship, determined to track down and bring an end to the Dalek race. + +#[cfg(test)] +extern crate test; + +#[macro_use] +extern crate arrayref; + +// Modules for low-level operations directly on field elements and curve points. + +pub mod field; +pub mod curve; +pub mod scalar; + +// Utilities module. + +pub mod util; + +// Low-level curve and point constants, as well as pre-computed curve group elements. + +mod constants { + #![allow(dead_code)] + #![allow(non_upper_case_globals)] + + use field::FieldElement; + use curve::PreComputedPoint; + + pub const a: FieldElement = FieldElement([ + 121665, 0, 0, 0, 0, 0, 0, 0, 0, 0, ]); + pub const d: FieldElement = FieldElement([ + -10913610, 13857413, -15372611, 6949391, 114729, + -8787816, -6275908, -3247719, -18696448, -12055116, ]); + pub const d2: FieldElement = FieldElement([ + -21827239, -5839606, -30745221, 13898782, 229458, + 15978800, -12551817, -6495438, 29715968, 9444199, ]); + pub const SQRT_M1: FieldElement = FieldElement([ + -32595792, -7943725, 9377950, 3500415, 12389472, + -272473, -25146209, -2005654, 326686, 11406482, ]); + + /// When we define the curve according to the form y² = x³+Ax²+Bx, + /// we get A=486662 and B=1. + pub const A: FieldElement = FieldElement([ + 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, ]); + + /// SQRT_MINUS_A is sqrt(-486662) + pub const SQRT_MINUS_A: FieldElement = FieldElement([ // sqrtMinusA + 12222970, 8312128, 11511410, -9067497, 15300785, + 241793, -25456130, -14121551, 12187136, -3972024, ]); + + /// SQRT_MINUS_HALF is sqrt(-1/2) + pub const SQRT_MINUS_HALF: FieldElement = FieldElement([ // sqrtMinusHalf + -17256545, 3971863, 28865457, -1750208, 27359696, + -16640980, 12573105, 1002827, -163343, 11073975, ]); + + /// HALF_Q_MINUS_1_BYTES is (2^255-20)/2 expressed in little endian form. + pub const HALF_Q_MINUS_1_BYTES: [u8; 32] = [ // halfQMinus1Bytes + 0xf6, 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, 0x3f, ]; + + pub const bi: [PreComputedPoint; 8] = [ + PreComputedPoint{ + y_plus_x: FieldElement([25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605]), + y_minus_x: FieldElement([-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378]), + xy2d: FieldElement([-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024]), + y_minus_x: FieldElement([16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574]), + xy2d: FieldElement([30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380]), + y_minus_x: FieldElement([4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306]), + xy2d: FieldElement([19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766]), + y_minus_x: FieldElement([-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701]), + xy2d: FieldElement([28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877]), + y_minus_x: FieldElement([-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951]), + xy2d: FieldElement([4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436]), + y_minus_x: FieldElement([25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918]), + xy2d: FieldElement([23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800]), + y_minus_x: FieldElement([-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305]), + xy2d: FieldElement([-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876]), + y_minus_x: FieldElement([-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619]), + xy2d: FieldElement([-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683]), + }, + ]; + + pub const base: [[PreComputedPoint; 8]; 32] = [ + [ + PreComputedPoint{ + y_plus_x: FieldElement([25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605]), + y_minus_x: FieldElement([-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378]), + xy2d: FieldElement([-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303]), + y_minus_x: FieldElement([-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081]), + xy2d: FieldElement([26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024]), + y_minus_x: FieldElement([16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574]), + xy2d: FieldElement([30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540]), + y_minus_x: FieldElement([23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397]), + xy2d: FieldElement([7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380]), + y_minus_x: FieldElement([4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306]), + xy2d: FieldElement([19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777]), + y_minus_x: FieldElement([-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737]), + xy2d: FieldElement([-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766]), + y_minus_x: FieldElement([-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701]), + xy2d: FieldElement([28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726]), + y_minus_x: FieldElement([-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955]), + xy2d: FieldElement([27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425]), + }, + ], + [ + PreComputedPoint{ + y_plus_x: FieldElement([-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171]), + y_minus_x: FieldElement([27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510]), + xy2d: FieldElement([17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639]), + y_minus_x: FieldElement([29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963]), + xy2d: FieldElement([5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568]), + y_minus_x: FieldElement([12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335]), + xy2d: FieldElement([25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007]), + y_minus_x: FieldElement([-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772]), + xy2d: FieldElement([-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, -7236665, 24316168, -5253567]), + y_minus_x: FieldElement([13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 33040651, -13424532, -20729456, 8321686]), + xy2d: FieldElement([21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 23845965, -11874838, -9984458, 608372]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 1123968, -6780577, 27229399, 23887]), + y_minus_x: FieldElement([-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 12797024, -6440308, -1633405, 16678954]), + xy2d: FieldElement([-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, -1508144, -4795045, -17169265, 4904953]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 5169211, 16191880, 2128236, -4326833]), + y_minus_x: FieldElement([-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, -29806336, 916033, -6882542, -2986532]), + xy2d: FieldElement([-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 285431, 2763829, 15736322, 4143876]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, -14594663, 23527084, -16458268]), + y_minus_x: FieldElement([33431127, -11130478, -17838966, -15626900, 8909499, 8376530, -32625340, 4087881, -15188911, -14416214]), + xy2d: FieldElement([1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 4357868, -4774191, -16323038]), + }, + ], + [ + PreComputedPoint{ + y_plus_x: FieldElement([6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 23365147, -3949732, 7390890, 2759800]), + y_minus_x: FieldElement([4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, -4264057, 1244380, -12919645]), + xy2d: FieldElement([-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 9208236, 15886429, 16489664]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([1996075, 10375649, 14346367, 13311202, -6874135, -16438411, -13693198, 398369, -30606455, -712933]), + y_minus_x: FieldElement([-25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 13348553, 12076947, -30836462, 5113182]), + xy2d: FieldElement([-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, -30341101, -7336386, 13847711, 5387222]), + }, + PreComputedPoint{ + y_plus_x: FieldElement([-18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 8763061, 3617786, 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4642684]), + xy2d: FieldElement([-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476]), + }, + ]]; +} diff --git a/src/scalar.rs b/src/scalar.rs new file mode 100644 index 0000000..610c563 --- /dev/null +++ b/src/scalar.rs @@ -0,0 +1,527 @@ +// -*- mode: rust; -*- +// +// To the extent possible under law, the authors have waived all +// copyright and related or neighboring rights to curve25519-dalek, +// using the Creative Commons "CC0" public domain dedication. See +// for full +// details. +// +// Authors: +// - Isis Agora Lovecruft +// - Henry de Valence + +//! Arithmetic for scalar multiplication. +//! +//! The Ed25519 basepoint P has prime order +//! +//! l = 2^252 + 27742317777372353535851937790883648493. +//! +//! Thus a multiple `aP` of the basepoint (with a ∈ ℤ) depends only +//! on the value of `a (mod l)`, or equivalently, the image of `a` in +//! the quotient ℤ/lℤ. +//! +//! The `Scalar` struct represents an element in ℤ/lℤ. +//! +//! Arithmetic operations on `Scalar`s are done using 12 21-bit limbs. +//! However, in contrast to `FieldElement`s, `Scalar`s are stored in +//! memory as bytes, allowing easy access to the bits of the `Scalar`. + +use std::clone::Clone; +use std::ops::{Index, IndexMut}; + +use field::{load3, load4}; + +/// The `Scalar` struct represents an element in ℤ/lℤ, where +/// +/// l = 2^252 + 27742317777372353535851937790883648493 +/// +/// is the order of the basepoint. +#[derive(Copy)] +pub struct Scalar(pub [u8; 32]); + +impl Clone for Scalar { + fn clone(&self) -> Scalar { *self } +} + +impl Index for Scalar { + type Output = u8; + + fn index<'a>(&'a self, _index: usize) -> &'a u8 { + let ret: &'a u8 = &(self.0[_index]); + ret + } +} + +impl IndexMut for Scalar { + fn index_mut<'a>(&'a mut self, _index: usize) -> &'a mut u8 { + let ret: &'a mut u8 = &mut(self.0[_index]); + ret + } +} + +impl Scalar { + /// Construct the additive identity + pub fn zero() -> Self { + Scalar([0u8; 32]) + } + + /// Construct the multiplicative identity + pub fn one() -> Self { + Scalar([ 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, 0, 0, 0 ]) + } + + /// Compute a width-5 "Non-Adjacent Form" of this scalar. + /// + /// A width-`w` NAF of a positive integer `k` is an expression + /// `k = sum(k[i]*2^i for i in range(l))`, where each nonzero + /// coefficient `k[i]` is odd and bounded by `|k[i]| < 2^(w-1)`, + /// `k[l-1]` is nonzero, and at most one of any `w` consecutive + /// coefficients is nonzero. (Hankerson, Menezes, Vanstone; def 3.32). + /// + /// Intuitively, this is like a binary expansion, except that we + /// allow some coefficients to grow up to `2^(w-1)` so that the + /// nonzero coefficients are as sparse as possible. + pub fn non_adjacent_form(&self) -> [i8;256] { + // Step 1: write out bits of the scalar + let mut naf = [0i8; 256]; + for i in 0..256 { + // As i runs from 0..256, the bottom 3 bits index the bit, + // while the upper bits index the byte. + naf[i] = ((self.0[i>>3] >> (i&7)) & 1u8) as i8; + } + + // Step 2: zero coefficients by carrying them upwards or downwards + 'bits: for i in 0..256 { + if naf[i] == 0 { continue 'bits; } + 'window: for b in 1..6 { + if i+b >= 256 { break 'window; } + if naf[i+b] == 0 { continue 'window; } + let potential_carry = naf[i+b] << b; + if naf[i+b] + potential_carry <= 15 { + // Eliminate naf[i+b] by carrying its value onto naf[i] + naf[i] += potential_carry; + naf[i+b] = 0; + } else if naf[i+b] - potential_carry >= -15 { + // Eliminate naf[i+b] by carrying its value upwards. + naf[i] -= potential_carry; // Subtract 2^(i+b) + 'carry: for k in i+b..256 { + if naf[k] != 0 { + // Since naf[k] = 0 or 1 for k > i, naf[k] == 1. + naf[k] = 0; // Subtract 2^k + } else { + // By now we have subtracted 2^k = + // 2^(i+b) + 2^(i+b) + 2^(i+b+1) + ... + 2^(k-1). + naf[k] = 1; // Add back 2^k. + break 'carry; + } + } + } + } + } + + naf + } + + /// Create a scalar by packing 12 21-bit limbs into bytes. + fn pack_limbs(limbs: &[i64;12]) -> Scalar { + let mut s = Scalar::zero(); + s[0] = (limbs[ 0] >> 0) as u8; + s[1] = (limbs[ 0] >> 8) as u8; + s[2] = ((limbs[ 0] >> 16) | (limbs[ 1] << 5)) as u8; + s[3] = (limbs[ 1] >> 3) as u8; + s[4] = (limbs[ 1] >> 11) as u8; + s[5] = ((limbs[ 1] >> 19) | (limbs[ 2] << 2)) as u8; + s[6] = (limbs[ 2] >> 6) as u8; + s[7] = ((limbs[ 2] >> 14) | (limbs[ 3] << 7)) as u8; + s[8] = (limbs[ 3] >> 1) as u8; + s[9] = (limbs[ 3] >> 9) as u8; + s[10] = ((limbs[ 3] >> 17) | (limbs[ 4] << 4)) as u8; + s[11] = (limbs[ 4] >> 4) as u8; + s[12] = (limbs[ 4] >> 12) as u8; + s[13] = ((limbs[ 4] >> 20) | (limbs[ 5] << 1)) as u8; + s[14] = (limbs[ 5] >> 7) as u8; + s[15] = ((limbs[ 5] >> 15) | (limbs[ 6] << 6)) as u8; + s[16] = (limbs[ 6] >> 2) as u8; + s[17] = (limbs[ 6] >> 10) as u8; + s[18] = ((limbs[ 6] >> 18) | (limbs[ 7] << 3)) as u8; + s[19] = (limbs[ 7] >> 5) as u8; + s[20] = (limbs[ 7] >> 13) as u8; + s[21] = (limbs[ 8] >> 0) as u8; + s[22] = (limbs[ 8] >> 8) as u8; + s[23] = ((limbs[ 8] >> 16) | (limbs[ 9] << 5)) as u8; + s[24] = (limbs[ 9] >> 3) as u8; + s[25] = (limbs[ 9] >> 11) as u8; + s[26] = ((limbs[ 9] >> 19) | (limbs[10] << 2)) as u8; + s[27] = (limbs[10] >> 6) as u8; + s[28] = ((limbs[10] >> 14) | (limbs[11] << 7)) as u8; + s[29] = (limbs[11] >> 1) as u8; + s[30] = (limbs[11] >> 9) as u8; + s[31] = (limbs[11] >> 17) as u8; + + s + } + + // Unpack a scalar into 12 21-bit limbs. + fn unpack_limbs(&self) -> [i64;12] { + let mask_21bits: i64 = (1 << 21) -1; + let mut a = [0i64;12]; + a[ 0] = mask_21bits & load3(&self.0[ 0..]) ; + a[ 1] = mask_21bits & (load4(&self.0[ 2..]) >> 5); + a[ 2] = mask_21bits & (load3(&self.0[ 5..]) >> 2); + a[ 3] = mask_21bits & (load4(&self.0[ 7..]) >> 7); + a[ 4] = mask_21bits & (load4(&self.0[10..]) >> 4); + a[ 5] = mask_21bits & (load3(&self.0[13..]) >> 1); + a[ 6] = mask_21bits & (load4(&self.0[15..]) >> 6); + a[ 7] = mask_21bits & (load3(&self.0[18..]) >> 3); + a[ 8] = mask_21bits & load3(&self.0[21..]) ; + a[ 9] = mask_21bits & (load4(&self.0[23..]) >> 5); + a[10] = mask_21bits & (load3(&self.0[26..]) >> 2); + a[11] = load4(&self.0[28..]) >> 7 ; + + a + } + + /// Write this scalar in radix 16, with coefficients in `[-8,8)`, + /// i.e., compute `a_i` such that + /// + /// a = a_0 + a_1*16^1 + ... + a_63*16^63, + /// + /// with `-8 ≤ a_i < 8` for `0 ≤ i < 63` and `-8 ≤ a_63 ≤ 8`. + /// + /// Precondition: self[31] <= 127. This is the case whenever + /// `self` is reduced. + pub fn to_radix_16(&self) -> [i8;64] { + debug_assert!(self[31] <= 127); + let mut output = [0i8; 64]; + + // Step 1: change radix. + // Convert from radix 256 (bytes) to radix 16 (nibbles) + #[inline(always)] + fn bot_half(x: u8) -> u8 { (x >> 0) & 15 } + #[inline(always)] + fn top_half(x: u8) -> u8 { (x >> 4) & 15 } + + for i in 0..32 { + output[2*i ] = bot_half(self[i]) as i8; + output[2*i+1] = top_half(self[i]) as i8; + } + // Precondition note: since self[31] <= 127, output[63] <= 7 + + // Step 2: recenter coefficients from [0,16) to [-8,8) + for i in 0..63 { + let carry = (output[i] + 8) >> 4; + output[i ] -= carry << 4; + output[i+1] += carry; + } + // Precondition note: output[63] is not recentered. It + // increases by carry <= 1. Thus output[63] <= 8. + + output + } + + /// Reduce limbs in-place. Reduction is mod + /// + /// l = 2^252 + 27742317777372353535851937790883648493, + /// + /// so + /// + /// 2^252 = -27742317777372353535851937790883648493 (mod l). + /// + /// We can write the right-hand side in 21-bit limbs as + /// + /// rhs = 666643 * 2^0 + /// + 470296 * 2^21 + /// + 654183 * 2^42 + /// - 997805 * 2^63 + /// + 136657 * 2^84 + /// - 683901 * 2^105 + /// + /// The (12+k)-th limb of `limbs` is the coefficient of + /// + /// 2^(252 + 21*k) + /// + /// since 12*21 = 252. By the above, we have that + /// + /// c * 2^(252 + 21*k) = c * 666643 * 2^(21*k) + /// + c * 470296 * 2^(42*k) + ... + /// + /// so we can eliminate it by adding those values to the lower + /// limbs. Reduction mod l amounts to eliminating all of the + /// high limbs while carrying as appropriate to prevent + /// overflows in the lower limbs. + fn reduce_limbs(mut limbs: &mut [i64;24]) { + #[inline] + #[allow(dead_code)] + fn do_reduction(limbs: &mut [i64;24], i:usize) { + limbs[i - 12] += limbs[i] * 666643; + limbs[i - 11] += limbs[i] * 470296; + limbs[i - 10] += limbs[i] * 654183; + limbs[i - 9] -= limbs[i] * 997805; + limbs[i - 8] += limbs[i] * 136657; + limbs[i - 7] -= limbs[i] * 683901; + limbs[i] = 0; + } + /// Carry excess from the `i`-th limb into the `(i+1)`-th limb. + /// Postcondition: `0 <= limbs[i] < 2^21`. + #[inline] + #[allow(dead_code)] + fn do_carry_uncentered(limbs: &mut [i64; 24], i: usize) { + let carry: i64 = limbs[i] >> 21; + limbs[i+1] += carry; + limbs[i ] -= carry << 21; + } + #[inline] + #[allow(dead_code)] + /// Carry excess from the `i`-th limb into the `(i+1)`-th limb. + /// Postcondition: `-2^20 <= limbs[i] < 2^20`. + fn do_carry_centered(limbs: &mut [i64;24], i:usize) { + let carry: i64 = (limbs[i] + (1<<20)) >> 21; + limbs[i+1] += carry; + limbs[i ] -= carry << 21; + } + + for i in 0..23 { + do_carry_centered(&mut limbs, i); + } + for i in (0..23).filter(|x| x % 2 == 1) { + do_carry_centered(&mut limbs, i); + } + + do_reduction(&mut limbs, 23); + do_reduction(&mut limbs, 22); + do_reduction(&mut limbs, 21); + do_reduction(&mut limbs, 20); + do_reduction(&mut limbs, 19); + do_reduction(&mut limbs, 18); + + for i in (6..18).filter(|x| x % 2 == 0) { + do_carry_centered(&mut limbs, i); + } + for i in (6..16).filter(|x| x % 2 == 1) { + do_carry_centered(&mut limbs, i); + } + + do_reduction(&mut limbs, 17); + do_reduction(&mut limbs, 16); + do_reduction(&mut limbs, 15); + do_reduction(&mut limbs, 14); + do_reduction(&mut limbs, 13); + do_reduction(&mut limbs, 12); + + for i in (0..12).filter(|x| x % 2 == 0) { + do_carry_centered(&mut limbs, i); + } + for i in (0..12).filter(|x| x % 2 == 1) { + do_carry_centered(&mut limbs, i); + } + + do_reduction(&mut limbs, 12); + + for i in 0..12 { + do_carry_uncentered(&mut limbs, i); + } + + do_reduction(&mut limbs, 12); + + for i in 0..11 { + do_carry_uncentered(&mut limbs, i); + } + } + + /// Compute `ab+c (mod l)`. + pub fn multiply_add(a: &Scalar, b: &Scalar, c: &Scalar) -> Scalar { + // Unpack scalars into limbs + let al = a.unpack_limbs(); + let bl = b.unpack_limbs(); + let cl = c.unpack_limbs(); + + let mut result = [0i64;24]; + + // Multiply a and b, and add c + result[0] = cl[0] + al[0]*bl[0]; + result[1] = cl[1] + al[0]*bl[1] + al[1]*bl[0]; + result[2] = cl[2] + al[0]*bl[2] + al[1]*bl[1] + al[2]*bl[0]; + result[3] = cl[3] + al[0]*bl[3] + al[1]*bl[2] + al[2]*bl[1] + al[3]*bl[0]; + result[4] = cl[4] + al[0]*bl[4] + al[1]*bl[3] + al[2]*bl[2] + al[3]*bl[1] + al[4]*bl[0]; + result[5] = cl[5] + al[0]*bl[5] + al[1]*bl[4] + al[2]*bl[3] + al[3]*bl[2] + al[4]*bl[1] + al[5]*bl[0]; + result[6] = cl[6] + al[0]*bl[6] + al[1]*bl[5] + al[2]*bl[4] + al[3]*bl[3] + al[4]*bl[2] + al[5]*bl[1] + al[6]*bl[0]; + result[7] = cl[7] + al[0]*bl[7] + al[1]*bl[6] + al[2]*bl[5] + al[3]*bl[4] + al[4]*bl[3] + al[5]*bl[2] + al[6]*bl[1] + al[7]*bl[0]; + result[8] = cl[8] + al[0]*bl[8] + al[1]*bl[7] + al[2]*bl[6] + al[3]*bl[5] + al[4]*bl[4] + al[5]*bl[3] + al[6]*bl[2] + al[7]*bl[1] + al[8]*bl[0]; + result[9] = cl[9] + al[0]*bl[9] + al[1]*bl[8] + al[2]*bl[7] + al[3]*bl[6] + al[4]*bl[5] + al[5]*bl[4] + al[6]*bl[3] + al[7]*bl[2] + al[8]*bl[1] + al[9]*bl[0]; + result[10] = cl[10] + al[0]*bl[10] + al[1]*bl[9] + al[2]*bl[8] + al[3]*bl[7] + al[4]*bl[6] + al[5]*bl[5] + al[6]*bl[4] + al[7]*bl[3] + al[8]*bl[2] + al[9]*bl[1] + al[10]*bl[0]; + result[11] = cl[11] + al[0]*bl[11] + al[1]*bl[10] + al[2]*bl[9] + al[3]*bl[8] + al[4]*bl[7] + al[5]*bl[6] + al[6]*bl[5] + al[7]*bl[4] + al[8]*bl[3] + al[9]*bl[2] + al[10]*bl[1] + al[11]*bl[0]; + result[12] = al[1]*bl[11] + al[2]*bl[10] + al[3]*bl[9] + al[4]*bl[8] + al[5]*bl[7] + al[6]*bl[6] + al[7]*bl[5] + al[8]*bl[4] + al[9]*bl[3] + al[10]*bl[2] + al[11]*bl[1]; + result[13] = al[2]*bl[11] + al[3]*bl[10] + al[4]*bl[9] + al[5]*bl[8] + al[6]*bl[7] + al[7]*bl[6] + al[8]*bl[5] + al[9]*bl[4] + al[10]*bl[3] + al[11]*bl[2]; + result[14] = al[3]*bl[11] + al[4]*bl[10] + al[5]*bl[9] + al[6]*bl[8] + al[7]*bl[7] + al[8]*bl[6] + al[9]*bl[5] + al[10]*bl[4] + al[11]*bl[3]; + result[15] = al[4]*bl[11] + al[5]*bl[10] + al[6]*bl[9] + al[7]*bl[8] + al[8]*bl[7] + al[9]*bl[6] + al[10]*bl[5] + al[11]*bl[4]; + result[16] = al[5]*bl[11] + al[6]*bl[10] + al[7]*bl[9] + al[8]*bl[8] + al[9]*bl[7] + al[10]*bl[6] + al[11]*bl[5]; + result[17] = al[6]*bl[11] + al[7]*bl[10] + al[8]*bl[9] + al[9]*bl[8] + al[10]*bl[7] + al[11]*bl[6]; + result[18] = al[7]*bl[11] + al[8]*bl[10] + al[9]*bl[9] + al[10]*bl[8] + al[11]*bl[7]; + result[19] = al[8]*bl[11] + al[9]*bl[10] + al[10]*bl[9] + al[11]*bl[8]; + result[20] = al[9]*bl[11] + al[10]*bl[10] + al[11]*bl[9]; + result[21] = al[10]*bl[11] + al[11]*bl[10]; + result[22] = al[11]*bl[11]; + result[23] = 0i64; + + // reduce limbs and pack into output + Scalar::reduce_limbs(&mut result); + Scalar::pack_limbs(array_ref!(result, 0, 12)) + } + + /// Reduce a 512-bit little endian number mod l + pub fn reduce(input: &[u8;64]) -> Scalar { + let mut s = [0i64;24]; + + // XXX express this as two unpack_limbs + // some issues re: masking with the top byte of the 32byte input + let mask_21bits: i64 = (1 << 21) -1; + s[0] = mask_21bits & load3(&input[ 0..]) ; + s[1] = mask_21bits & (load4(&input[ 2..]) >> 5); + s[2] = mask_21bits & (load3(&input[ 5..]) >> 2); + s[3] = mask_21bits & (load4(&input[ 7..]) >> 7); + s[4] = mask_21bits & (load4(&input[10..]) >> 4); + s[5] = mask_21bits & (load3(&input[13..]) >> 1); + s[6] = mask_21bits & (load4(&input[15..]) >> 6); + s[7] = mask_21bits & (load3(&input[18..]) >> 3); + s[8] = mask_21bits & load3(&input[21..]) ; + s[9] = mask_21bits & (load4(&input[23..]) >> 5); + s[10] = mask_21bits & (load3(&input[26..]) >> 2); + s[11] = mask_21bits & (load4(&input[28..]) >> 7); + s[12] = mask_21bits & (load4(&input[31..]) >> 4); + s[13] = mask_21bits & (load3(&input[34..]) >> 1); + s[14] = mask_21bits & (load4(&input[36..]) >> 6); + s[15] = mask_21bits & (load3(&input[39..]) >> 3); + s[16] = mask_21bits & load3(&input[42..]) ; + s[17] = mask_21bits & (load4(&input[44..]) >> 5); + s[18] = mask_21bits & (load3(&input[47..]) >> 2); + s[19] = mask_21bits & (load4(&input[49..]) >> 7); + s[20] = mask_21bits & (load4(&input[52..]) >> 4); + s[21] = mask_21bits & (load3(&input[55..]) >> 1); + s[22] = mask_21bits & (load4(&input[57..]) >> 6); + s[23] = load4(&input[60..]) >> 3 ; + + // XXX replacing the previous code in this function with the + // call to reduce_limbs adds two extra carry passes (the ones + // at the top of the reduce_limbs function). Otherwise they + // are identical. The test seems to work OK but it would be + // good to check that this really is OK to add. + Scalar::reduce_limbs(&mut s); + + Scalar::pack_limbs(array_ref!(s,0,12)) + } +} + +#[cfg(test)] +mod test { + use super::*; + use test::Bencher; + + #[bench] + fn bench_scalar_multiply_add(b: &mut Bencher) { + b.iter(|| Scalar::multiply_add(&X, &Y, &Z) ); + } + + /// x = 2238329342913194256032495932344128051776374960164957527413114840482143558222 + static X: Scalar = Scalar( + [0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84, + 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52, + 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44, + 0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04]); + /// y = 2592331292931086675770238855846338635550719849568364935475441891787804997264 + static Y: Scalar = Scalar( + [0x90, 0x76, 0x33, 0xfe, 0x1c, 0x4b, 0x66, 0xa4, + 0xa2, 0x8d, 0x2d, 0xd7, 0x67, 0x83, 0x86, 0xc3, + 0x53, 0xd0, 0xde, 0x54, 0x55, 0xd4, 0xfc, 0x9d, + 0xe8, 0xef, 0x7a, 0xc3, 0x1f, 0x35, 0xbb, 0x05]); + /// z = 5033871415930814945849241457262266927579821285980625165479289807629491019013 + static Z: Scalar = Scalar( + [0x05, 0x9d, 0x3e, 0x0b, 0x09, 0x26, 0x50, 0x3d, + 0xa3, 0x84, 0xa1, 0x3c, 0x92, 0x7a, 0xc2, 0x06, + 0x41, 0x98, 0xcf, 0x34, 0x3a, 0x24, 0xd5, 0xb7, + 0xeb, 0x33, 0x6a, 0x2d, 0xfc, 0x11, 0x21, 0x0b]); + /// w = 3486911242272497535104403593250518247409663771668155364040899665266216860804 + static W: Scalar = Scalar( + [0x84, 0xfc, 0xbc, 0x4f, 0x78, 0x12, 0xa0, 0x06, + 0xd7, 0x91, 0xd9, 0x7a, 0x3a, 0x27, 0xdd, 0x1e, + 0x21, 0x43, 0x45, 0xf7, 0xb1, 0xb9, 0x56, 0x7a, + 0x81, 0x30, 0x73, 0x44, 0x96, 0x85, 0xb5, 0x07]); + + /// x*y = 5690045403673944803228348699031245560686958845067437804563560795922180092780 + static X_TIMES_Y: Scalar = Scalar( + [0x6c, 0x33, 0x74, 0xa1, 0x89, 0x4f, 0x62, 0x21, + 0x0a, 0xaa, 0x2f, 0xe1, 0x86, 0xa6, 0xf9, 0x2c, + 0xe0, 0xaa, 0x75, 0xc2, 0x77, 0x95, 0x81, 0xc2, + 0x95, 0xfc, 0x08, 0x17, 0x9a, 0x73, 0x94, 0x0c]); + + static A_SCALAR: Scalar = Scalar([ + 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, + 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d, + 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, + 0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09]); + + static A_NAF: [i8;256] = + [0,13,0,0,0,0,0,0,0,7,0,0,0,0,0,0,-9,0,0,0,0,-11,0,0,0,0,3,0,0,0,0,1, + 0,0,0,0,9,0,0,0,0,-5,0,0,0,0,0,0,3,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0, + -9,0,0,0,0,0,-3,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,9,0, + 0,0,0,-15,0,0,0,0,-7,0,0,0,0,-9,0,0,0,0,0,5,0,0,0,0,13,0,0,0,0,0,-3,0, + 0,0,0,-11,0,0,0,0,-7,0,0,0,0,-13,0,0,0,0,11,0,0,0,0,-9,0,0,0,0,0,1,0,0, + 0,0,0,-15,0,0,0,0,1,0,0,0,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,0,13,0,0,0, + 0,0,0,11,0,0,0,0,0,15,0,0,0,0,0,-9,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,7, + 0,0,0,0,0,-15,0,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0]; + + #[test] + fn test_non_adjacent_form() { + let naf = A_SCALAR.non_adjacent_form(); + for i in 0..256 { + assert_eq!(naf[i], A_NAF[i]); + } + } + + #[test] + fn test_scalar_multiply_by_one() { + let one = Scalar::one(); + let zero = Scalar::zero(); + let test_scalar = Scalar::multiply_add(&X, &one, &zero); + for i in 0..32 { + assert!(test_scalar[i] == X[i]); + } + } + + #[test] + fn test_scalar_multiply_only() { + let zero = Scalar::zero(); + let test_scalar = Scalar::multiply_add(&X, &Y, &zero); + for i in 0..32 { + assert!(test_scalar[i] == X_TIMES_Y[i]); + } + } + + #[test] + fn test_scalar_multiply_add() { + let test_scalar = Scalar::multiply_add(&X, &Y, &Z); + for i in 0..32 { + assert!(test_scalar[i] == W[i]); + } + } + + #[test] + fn test_scalar_reduce() { + let mut bignum = [0u8;64]; + // set bignum = x + 2^256x + for i in 0..32 { + bignum[ i] = X[i]; + bignum[32+i] = X[i]; + } + // 3958878930004874126169954872055634648693766179881526445624823978500314864344 + // = x + 2^256x (mod l) + let reduced = Scalar([216, 154, 179, 139, 210, 121, 2, 71, + 69, 99, 158, 216, 23, 173, 63, 100, + 204, 0, 91, 50, 219, 153, 57, 249, + 28, 82, 31, 197, 100, 165, 192, 8]); + let test_red = Scalar::reduce(&bignum); + for i in 0..32 { + assert!(test_red[i] == reduced[i]); + } + } +} diff --git a/src/util.rs b/src/util.rs new file mode 100644 index 0000000..d94a853 --- /dev/null +++ b/src/util.rs @@ -0,0 +1,68 @@ +// -*- mode: rust; -*- +// +// To the extent possible under law, the authors have waived all copyright and +// related or neighboring rights to curve25519-dalek, using the Creative +// Commons "CC0" public domain dedication. See +// for full details. +// +// Authors: +// - Isis Agora Lovecruft +// - Henry de Valence + +//! Utility functions and tools for constant-time comparisons. + +/// Check equality of two bytes in constant time. +/// +/// # Return +/// +/// Returns 1 if `a == b` and 0 otherwise. +#[inline(always)] +pub fn bytes_equal_ct(a: u8, b: u8) -> u8 { + let mut x: u8; + + x = !(a ^ b); + x &= x >> 4; + x &= x >> 2; + x &= x >> 1; + x +} + +/// Test if a byte is non-zero in constant time. +/// +/// ```rust,ignore +/// let mut x: u8; +/// x = 0; +/// assert!(byte_is_nonzero(x)); +/// x = 3; +/// assert!(byte_is_nonzero(x) == 1); +/// ``` +/// +/// # Return +/// +/// * If b != 0, returns 1u8. +/// * If b == 0, returns 0u8. +#[inline(always)] +pub fn byte_is_nonzero(b: u8) -> u8 { + let mut x = b; + x |= x >> 4; + x |= x >> 2; + x |= x >> 1; + (x & 1) +} + +/// Check equality of two 32-byte arrays in constant time. +/// +/// # Return +/// +/// Returns 1 if `a == b` and 0 otherwise. +#[inline(always)] +// We don't use this in curve25519-dalek, but it's useful for e.g. an ed25519 implementation. +#[allow(dead_code)] +pub fn arrays_equal_ct(a: &[u8; 32], b: &[u8; 32]) -> u8 { + let mut x: u8 = 0; + + for i in 0..32 { + x |= a[i] ^ b[i]; + } + bytes_equal_ct(x, 0) +}