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initial work on rewriting 32bit arithmetic

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Henry de Valence 2017-11-16 11:14:49 -08:00
parent 399a9db478
commit 8952c7f476
3 changed files with 282 additions and 321 deletions
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4
src/constants.rs
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@ -37,8 +37,8 @@ use scalar::Scalar;
#[cfg(feature="radix_51")]
pub use constants_64bit::*;
#[cfg(not(feature="radix_51"))]
pub use constants_32bit::*;
//#[cfg(not(feature="radix_51"))]
//pub use constants_32bit::*;
/// Basepoint has y = 4/5.
///

596
src/field_32bit.rs
View file

@ -11,8 +11,8 @@
//! Field arithmetic for ℤ/(2²⁵⁵-19), using 32-bit arithmetic with
//! 64-bit products.
//!
//! Based on Adam Langley's curve25519-donna and (Golang) ed25519
//! implementations.
//! This code was originally derived from Adam Langley's
//! curve25519-donna and (Golang) ed25519 implementations.
//!
//! This implementation is intended for platforms that can multiply
//! 32-bit inputs to produce 64-bit outputs.
@ -30,14 +30,11 @@ use core::ops::Neg;
use subtle::ConditionallyAssignable;
use utils::{load3, load4};
/// A `FieldElement32` represents an element of the field GF(2^255 - 19).
///
/// In the 32-bit implementation, a `FieldElement32` is represented in
/// radix 2^25.5 as ten `i32`s, so that 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].
/// radix 2^25.5 as ten `u32`s, so that an element t, entries
/// t[0],...,t[9], represents `sum(t[i]*2^ceil(i*51/2))`.
///
/// The coefficients t[i] are allowed to grow between multiplications.
///
@ -56,7 +53,7 @@ use utils::{load3, load4};
/// faster. However, the `FieldElement64` implementation requires Rust's
/// `u128`, which is not yet stable.
#[derive(Copy, Clone)]
pub struct FieldElement32(pub (crate) [i32; 10]);
pub struct FieldElement32(pub (crate) [u32; 10]);
impl Debug for FieldElement32 {
fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
@ -83,9 +80,22 @@ impl<'a, 'b> Add<&'b FieldElement32> for &'a FieldElement32 {
impl<'b> SubAssign<&'b FieldElement32> for FieldElement32 {
fn sub_assign(&mut self, _rhs: &'b FieldElement32) {
for i in 0..10 {
self.0[i] -= _rhs.0[i];
}
// See comment in FieldElement64::Sub
//
// Compute a - b as ((a + 2^4 * p) - b) to avoid underflow.
let b = &_rhs.0;
self.0 = FieldElement32::reduce([
((self.0[0] + (0x3ffffed << 4)) - b[0]) as u64,
((self.0[1] + (0x1ffffff << 4)) - b[1]) as u64,
((self.0[2] + (0x3ffffff << 4)) - b[2]) as u64,
((self.0[3] + (0x1ffffff << 4)) - b[3]) as u64,
((self.0[4] + (0x3ffffff << 4)) - b[4]) as u64,
((self.0[5] + (0x1ffffff << 4)) - b[5]) as u64,
((self.0[6] + (0x3ffffff << 4)) - b[6]) as u64,
((self.0[7] + (0x1ffffff << 4)) - b[7]) as u64,
((self.0[8] + (0x3ffffff << 4)) - b[8]) as u64,
((self.0[9] + (0x1ffffff << 4)) - b[9]) as u64,
]).0;
}
}
@ -108,86 +118,101 @@ impl<'b> MulAssign<&'b FieldElement32> for FieldElement32 {
impl<'a, 'b> Mul<&'b FieldElement32> for &'a FieldElement32 {
type Output = FieldElement32;
fn mul(self, _rhs: &'b FieldElement32) -> FieldElement32 {
// Notes preserved from ed25519.go (presumably originally from ref10):
//
// 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.
let f0 = self.0[0] as i64;
let f1 = self.0[1] as i64;
let f2 = self.0[2] as i64;
let f3 = self.0[3] as i64;
let f4 = self.0[4] as i64;
let f5 = self.0[5] as i64;
let f6 = self.0[6] as i64;
let f7 = self.0[7] as i64;
let f8 = self.0[8] as i64;
let f9 = self.0[9] as i64;
/// Helper function to multiply two 32-bit integers with 64 bits
/// of output.
#[inline(always)]
fn m(x: u32, y: u32) -> u64 { (x as u64) * (y as u64) }
let f1_2 = (2 * self.0[1]) as i64;
let f3_2 = (2 * self.0[3]) as i64;
let f5_2 = (2 * self.0[5]) as i64;
let f7_2 = (2 * self.0[7]) as i64;
let f9_2 = (2 * self.0[9]) as i64;
// Alias self, _rhs for more readable formulas
let x: &[u32;10] = &self.0; let y: &[u32;10] = &_rhs.0;
let g0 = _rhs.0[0] as i64;
let g1 = _rhs.0[1] as i64;
let g2 = _rhs.0[2] as i64;
let g3 = _rhs.0[3] as i64;
let g4 = _rhs.0[4] as i64;
let g5 = _rhs.0[5] as i64;
let g6 = _rhs.0[6] as i64;
let g7 = _rhs.0[7] as i64;
let g8 = _rhs.0[8] as i64;
let g9 = _rhs.0[9] as i64;
// We assume that the input limbs x[i], y[i] are bounded by:
//
// x[i], y[i] < 2^(26 + b) if i even
// x[i], y[i] < 2^(25 + b) if i odd
//
// where b is a (real) parameter representing the excess bits of
// the limbs. We track the bitsizes of all variables through
// the computation and solve at the end for the allowable
// headroom bitsize b (which determines how many additions we
// can perform between reductions or multiplications).
let g1_19 = (19 * _rhs.0[1]) as i64; /* 1.4*2^29 */
let g2_19 = (19 * _rhs.0[2]) as i64; /* 1.4*2^30; still ok */
let g3_19 = (19 * _rhs.0[3]) as i64;
let g4_19 = (19 * _rhs.0[4]) as i64;
let g5_19 = (19 * _rhs.0[5]) as i64;
let g6_19 = (19 * _rhs.0[6]) as i64;
let g7_19 = (19 * _rhs.0[7]) as i64;
let g8_19 = (19 * _rhs.0[8]) as i64;
let g9_19 = (19 * _rhs.0[9]) as i64;
let y1_19 = 19 * y[1]; // This fits in a u32
let y2_19 = 19 * y[2]; // iff 26 + b + lg(19) < 32
let y3_19 = 19 * y[3]; // if b < 32 - 26 - 4.248 = 1.752
let y4_19 = 19 * y[4];
let y5_19 = 19 * y[5]; // below, b<2.5: this is a bottleneck,
let y6_19 = 19 * y[6]; // could be avoided by promoting to
let y7_19 = 19 * y[7]; // u64 here instead of in m()
let y8_19 = 19 * y[8];
let y9_19 = 19 * y[9];
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;
// What happens when we multiply x[i] with y[j] and place the
// result into the (i+j)-th limb?
//
// x[i] represents the value x[i]*2^ceil(i*51/2)
// y[j] represents the value y[j]*2^ceil(j*51/2)
// z[i+j] represents the value z[i+j]*2^ceil((i+j)*51/2)
// x[i]*y[j] represents the value x[i]*y[i]*2^(ceil(i*51/2)+ceil(j*51/2))
//
// Since the radix is already accounted for, the result placed
// into the (i+j)-th limb should be
//
// x[i]*y[i]*2^(ceil(i*51/2)+ceil(j*51/2) - ceil((i+j)*51/2)).
//
// The value of ceil(i*51/2)+ceil(j*51/2) - ceil((i+j)*51/2) is
// 1 when both i and j are odd, and 0 otherwise. So we add
//
// x[i]*y[j] if either i or j is even
// 2*x[i]*y[j] if i and j are both odd
//
// by using precomputed multiples of x[i] for odd i:
FieldElement32::reduce([h0, h1, h2, h3, h4, h5, h6, h7, h8, h9])
let x1_2 = 2 * x[1]; // This fits in a u32 iff 25 + b + 1 < 32
let x3_2 = 2 * x[3]; // iff b < 6
let x5_2 = 2 * x[5];
let x7_2 = 2 * x[7];
let x9_2 = 2 * x[9];
let z0 = m(x[0],y[0]) + m(x1_2,y9_19) + m(x[2],y8_19) + m(x3_2,y7_19) + m(x[4],y6_19) + m(x5_2,y5_19) + m(x[6],y4_19) + m(x7_2,y3_19) + m(x[8],y2_19) + m(x9_2,y1_19);
let z1 = m(x[0],y[1]) + m(x[1],y[0]) + m(x[2],y9_19) + m(x[3],y8_19) + m(x[4],y7_19) + m(x[5],y6_19) + m(x[6],y5_19) + m(x[7],y4_19) + m(x[8],y3_19) + m(x[9],y2_19);
let z2 = m(x[0],y[2]) + m(x1_2,y[1]) + m(x[2],y[0]) + m(x3_2,y9_19) + m(x[4],y8_19) + m(x5_2,y7_19) + m(x[6],y6_19) + m(x7_2,y5_19) + m(x[8],y4_19) + m(x9_2,y3_19);
let z3 = m(x[0],y[3]) + m(x[1],y[2]) + m(x[2],y[1]) + m(x[3],y[0]) + m(x[4],y9_19) + m(x[5],y8_19) + m(x[6],y7_19) + m(x[7],y6_19) + m(x[8],y5_19) + m(x[9],y4_19);
let z4 = m(x[0],y[4]) + m(x1_2,y[3]) + m(x[2],y[2]) + m(x3_2,y[1]) + m(x[4],y[0]) + m(x5_2,y9_19) + m(x[6],y8_19) + m(x7_2,y7_19) + m(x[8],y6_19) + m(x9_2,y5_19);
let z5 = m(x[0],y[5]) + m(x[1],y[4]) + m(x[2],y[3]) + m(x[3],y[2]) + m(x[4],y[1]) + m(x[5],y[0]) + m(x[6],y9_19) + m(x[7],y8_19) + m(x[8],y7_19) + m(x[9],y6_19);
let z6 = m(x[0],y[6]) + m(x1_2,y[5]) + m(x[2],y[4]) + m(x3_2,y[3]) + m(x[4],y[2]) + m(x5_2,y[1]) + m(x[6],y[0]) + m(x7_2,y9_19) + m(x[8],y8_19) + m(x9_2,y7_19);
let z7 = m(x[0],y[7]) + m(x[1],y[6]) + m(x[2],y[5]) + m(x[3],y[4]) + m(x[4],y[3]) + m(x[5],y[2]) + m(x[6],y[1]) + m(x[7],y[0]) + m(x[8],y9_19) + m(x[9],y8_19);
let z8 = m(x[0],y[8]) + m(x1_2,y[7]) + m(x[2],y[6]) + m(x3_2,y[5]) + m(x[4],y[4]) + m(x5_2,y[3]) + m(x[6],y[2]) + m(x7_2,y[1]) + m(x[8],y[0]) + m(x9_2,y9_19);
let z9 = m(x[0],y[9]) + m(x[1],y[8]) + m(x[2],y[7]) + m(x[3],y[6]) + m(x[4],y[5]) + m(x[5],y[4]) + m(x[6],y[3]) + m(x[7],y[2]) + m(x[8],y[1]) + m(x[9],y[0]);
// How big is the contribution to z[i+j] from x[i], y[j]?
//
// Using the bounds above, we get:
//
// i even, j even: x[i]*y[j] < 2^(26+b)*2^(26+b) = 2*2^(51+2*b)
// i odd, j even: x[i]*y[j] < 2^(25+b)*2^(26+b) = 1*2^(51+2*b)
// i even, j odd: x[i]*y[j] < 2^(26+b)*2^(25+b) = 1*2^(51+2*b)
// i odd, j odd: 2*x[i]*y[j] < 2*2^(25+b)*2^(25+b) = 1*2^(51+2*b)
//
// We perform inline reduction mod p by replacing 2^255 by 19
// (since 2^255 - 19 = 0 mod p). This adds a factor of 19, so
// we get the bounds (z0 is the biggest one, but calculated for
// posterity here in case finer estimation is needed later):
//
// z0 < ( 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 249*2^(51 + 2*b)
// z1 < ( 1 + 1 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 154*2^(51 + 2*b)
// z2 < ( 2 + 1 + 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 195*2^(51 + 2*b)
// z3 < ( 1 + 1 + 1 + 1 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 118*2^(51 + 2*b)
// z4 < ( 2 + 1 + 2 + 1 + 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 141*2^(51 + 2*b)
// z5 < ( 1 + 1 + 1 + 1 + 1 + 1 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 82*2^(51 + 2*b)
// z6 < ( 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 87*2^(51 + 2*b)
// z7 < ( 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1*19 + 1*19 )*2^(51 + 2b) = 46*2^(51 + 2*b)
// z6 < ( 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1*19 )*2^(51 + 2b) = 33*2^(51 + 2*b)
// z7 < ( 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 )*2^(51 + 2b) = 10*2^(51 + 2*b)
//
// So z[0] fits into a u64 if 51 + 2*b + lg(249) < 64
// if b < 2.5.
FieldElement32::reduce([z0, z1, z2, z3, z4, z5, z6, z7, z8, z9])
}
}
@ -202,7 +227,7 @@ impl<'a> Neg for &'a FieldElement32 {
impl ConditionallyAssignable for FieldElement32 {
fn conditional_assign(&mut self, f: &FieldElement32, choice: u8) {
let mask = -(choice as i32);
let mask = (-(choice as i32)) as u32;
for i in 0..10 {
self.0[i] ^= mask & (self.0[i] ^ f.0[i]);
}
@ -212,9 +237,20 @@ impl ConditionallyAssignable for FieldElement32 {
impl FieldElement32 {
/// Invert the sign of this field element
pub fn negate(&mut self) {
for i in 0..10 {
self.0[i] = -self.0[i];
}
// Compute -b as ((2^4 * p) - b) to avoid underflow.
let neg = FieldElement32::reduce([
((0x3ffffed << 4) - self.0[0]) as u64,
((0x1ffffff << 4) - self.0[1]) as u64,
((0x3ffffff << 4) - self.0[2]) as u64,
((0x1ffffff << 4) - self.0[3]) as u64,
((0x3ffffff << 4) - self.0[4]) as u64,
((0x1ffffff << 4) - self.0[5]) as u64,
((0x3ffffff << 4) - self.0[6]) as u64,
((0x1ffffff << 4) - self.0[7]) as u64,
((0x3ffffff << 4) - self.0[8]) as u64,
((0x1ffffff << 4) - self.0[9]) as u64,
]);
self.0 = neg.0;
}
/// Construct zero.
@ -229,98 +265,64 @@ impl FieldElement32 {
/// Construct -1.
pub fn minus_one() -> FieldElement32 {
FieldElement32([-1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ])
FieldElement32([
0x3ffffec, 0x1ffffff, 0x3ffffff, 0x1ffffff, 0x3ffffff,
0x1ffffff, 0x3ffffff, 0x1ffffff, 0x3ffffff, 0x1ffffff,
])
}
fn reduce(mut h: [i64; 10]) -> FieldElement32 { //FeCombine
let mut c = [0i64; 10];
/// Given unreduced coefficients `z[0], ..., z[9]` of any size,
/// carry and reduce them mod p to obtain a `FieldElement32`
/// whose coefficients have excess `b < 0.007`.
///
/// In other words, each coefficient of the result is bounded by
/// either `2^(25 + 0.007)` or `2^(26 + 0.007)`, as appropriate.
fn reduce(mut z: [u64; 10]) -> FieldElement32 {
/*
|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]
*/
const LOW_25_BITS: u64 = (1 << 25) - 1;
const LOW_26_BITS: u64 = (1 << 26) - 1;
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 */
/// Carry the value from limb i = 0..8 to limb i+1
#[inline(always)]
fn carry(z: &mut [u64; 10], i: usize) {
debug_assert!(i < 9);
if i % 2 == 0 {
// Even limbs have 26 bits
z[i+1] += z[i] >> 26;
z[i] &= LOW_26_BITS;
} else {
// Odd limbs have 25 bits
z[i+1] += z[i] >> 25;
z[i] &= LOW_25_BITS;
}
}
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 */
// Perform two halves of the carry chain in parallel.
carry(&mut z, 0); carry(&mut z, 4);
carry(&mut z, 1); carry(&mut z, 5);
carry(&mut z, 2); carry(&mut z, 6);
carry(&mut z, 3); carry(&mut z, 7);
// Since z[3] < 2^64, c < 2^(64-25) = 2^39,
// so z[4] < 2^26 + 2^39 < 2^39.0002
carry(&mut z, 4); carry(&mut z, 8);
// Now z[4] < 2^26
// and z[5] < 2^25 + 2^13.0002 < 2^25.0004 (good enough)
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 */
// Last carry has a multiplication by 19:
z[0] += 19*(z[9] >> 25);
z[9] &= LOW_25_BITS;
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 */
// Since z[9] < 2^64, c < 2^(64-25) = 2^39,
// so z[0] + 19*c < 2^26 + 2^43.248 < 2^43.249.
carry(&mut z, 0);
// Now z[1] < 2^25 - 2^(43.249 - 26)
// < 2^25.007 (good enough)
// and we're done.
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 = FieldElement32([0i32; 10]);
output.0[0] = h[0] as i32;
output.0[1] = h[1] as i32;
output.0[2] = h[2] as i32;
output.0[3] = h[3] as i32;
output.0[4] = h[4] as i32;
output.0[5] = h[5] as i32;
output.0[6] = h[6] as i32;
output.0[7] = h[7] as i32;
output.0[8] = h[8] as i32;
output.0[9] = h[9] as i32;
output
FieldElement32([
z[0] as u32, z[1] as u32, z[2] as u32, z[3] as u32, z[4] as u32,
z[5] as u32, z[6] as u32, z[7] as u32, z[8] as u32, z[9] as u32,
])
}
/// Load a `FieldElement64` from the low 255 bits of a 256-bit
@ -334,11 +336,19 @@ impl FieldElement32 {
/// encoding of every field element should decode, re-encode to
/// the canonical encoding, and check that the input was
/// canonical.
///
/// XXX the above applies to the 64-bit implementation; check that
/// it applies here too.
pub fn from_bytes(data: &[u8; 32]) -> FieldElement32 { //FeFromBytes
let mut h = [0i64;10];
#[inline]
fn load3(b: &[u8]) -> u64 {
(b[0] as u64) | ((b[1] as u64) << 8) | ((b[2] as u64) << 16)
}
#[inline]
fn load4(b: &[u8]) -> u64 {
(b[0] as u64) | ((b[1] as u64) << 8) | ((b[2] as u64) << 16) | ((b[3] as u64) << 24)
}
let mut h = [0u64;10];
const LOW_23_BITS: u64 = (1 << 23) - 1;
h[0] = load4(&data[ 0..]);
h[1] = load3(&data[ 4..]) << 6;
h[2] = load3(&data[ 7..]) << 5;
@ -348,52 +358,33 @@ impl FieldElement32 {
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;
h[9] = (load3(&data[29..]) & LOW_23_BITS) << 2;
FieldElement32::reduce(h)
}
/// Serialize this `FieldElement64` to a 32-byte array. The
/// encoding is canonical.
pub fn to_bytes(&self) -> [u8; 32] { //FeToBytes
// Comment preserved from ed25519.go (presumably originally from ref10):
//
// # 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<y<1.
//
// Write r = h - pq.
//
// Have 0 <= r< = p-1 = 2^255 - 20.
//
// Thus 0 <= r + 19 * 2^-255 * r < r + 19 * 2^-255 * 2^255 <= 2^255 - 1.
//
// Write x = r + 19 * 2^-255 * r + y.
//
// Then 0 < x < 2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
//
// Have q+2^(-255)x = 2^-255 * (h + 19 * 2^-25 * h9 + 2^-1),
// so floor(2^-255 * (h + 19 * 2^-25 * h9 + 2^-1)) = q.
//
let mut carry = [0i32; 10];
let mut h: [i32; 10] = self.0;
pub fn to_bytes(&self) -> [u8; 32] {
let mut q:i32 = (19*h[9] + (1 << 24)) >> 25;
q = (h[0] + q) >> 26;
let inp = &self.0;
// Reduce the value represented by `in` to the range [0,2*p)
let mut h: [u32; 10] = FieldElement32::reduce([
// XXX this cast is annoying
inp[0] as u64, inp[1] as u64, inp[2] as u64, inp[3] as u64, inp[4] as u64,
inp[5] as u64, inp[6] as u64, inp[7] as u64, inp[8] as u64, inp[9] as u64,
]).0;
// Let h be the value to encode.
//
// Write h = pq + r with 0 <= r < p. We want to compute r = h mod p.
//
// Since h < 2*p, q = 0 or 1, with q = 0 when h < p and q = 1 when h >= p.
//
// Notice that h >= p <==> h + 19 >= p + 19 <==> h + 19 >= 2^255.
// Therefore q can be computed as the carry bit of h + 19.
let mut q: u32 = (h[0] + 19) >> 26;
q = (h[1] + q) >> 25;
q = (h[2] + q) >> 26;
q = (h[3] + q) >> 25;
@ -404,45 +395,40 @@ impl FieldElement32 {
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.
debug_assert!( q == 0 || q == 1 );
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
// Now we can compute r as r = h - pq = r - (2^255-19)q = r + 19q - 2^255q
// 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].
const LOW_25_BITS: u32 = (1 << 25) - 1;
const LOW_26_BITS: u32 = (1 << 26) - 1;
h[0] += 19*q;
// Now carry the result to compute r + 19q...
h[1] += h[0] >> 26;
h[0] = h[0] & LOW_26_BITS;
h[2] += h[1] >> 25;
h[1] = h[1] & LOW_25_BITS;
h[3] += h[2] >> 26;
h[2] = h[2] & LOW_26_BITS;
h[4] += h[3] >> 25;
h[3] = h[3] & LOW_25_BITS;
h[5] += h[4] >> 26;
h[4] = h[4] & LOW_26_BITS;
h[6] += h[5] >> 25;
h[5] = h[5] & LOW_25_BITS;
h[7] += h[6] >> 26;
h[6] = h[6] & LOW_26_BITS;
h[8] += h[7] >> 25;
h[7] = h[7] & LOW_25_BITS;
h[9] += h[8] >> 26;
h[8] = h[8] & LOW_26_BITS;
// ... but instead of carrying the value
// (h[9] >> 25) = q*2^255 into another limb,
// discard it, subtracting the value from h.
debug_assert!( (h[9] >> 25) == 0 || (h[9] >> 25) == 1);
h[9] = h[9] & LOW_25_BITS;
let mut s = [0u8; 32];
s[0] = (h[0] >> 0) as u8;
@ -484,77 +470,53 @@ impl FieldElement32 {
s
}
fn square_inner(&self) -> [i64; 10] {
let f0 = self.0[0] as i64;
let f1 = self.0[1] as i64;
let f2 = self.0[2] as i64;
let f3 = self.0[3] as i64;
let f4 = self.0[4] as i64;
let f5 = self.0[5] as i64;
let f6 = self.0[6] as i64;
let f7 = self.0[7] as i64;
let f8 = self.0[8] as i64;
let f9 = self.0[9] as i64;
let f0_2 = (2 * self.0[0]) as i64;
let f1_2 = (2 * self.0[1]) as i64;
let f2_2 = (2 * self.0[2]) as i64;
let f3_2 = (2 * self.0[3]) as i64;
let f4_2 = (2 * self.0[4]) as i64;
let f5_2 = (2 * self.0[5]) as i64;
let f6_2 = (2 * self.0[6]) as i64;
let f7_2 = (2 * self.0[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
fn square_inner(&self) -> [u64; 10] {
// Optimized version of multiplication for the case of squaring.
// Pre- and post- conditions identical to multiplication function.
let x = &self.0;
let x0_2 = 2 * x[0];
let x1_2 = 2 * x[1];
let x2_2 = 2 * x[2];
let x3_2 = 2 * x[3];
let x4_2 = 2 * x[4];
let x5_2 = 2 * x[5];
let x6_2 = 2 * x[6];
let x7_2 = 2 * x[7];
let x5_19 = 19 * x[5];
let x6_19 = 19 * x[6];
let x7_19 = 19 * x[7];
let x8_19 = 19 * x[8];
let x9_19 = 19 * x[9];
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;
/// Helper function to multiply two 32-bit integers with 64 bits
/// of output.
#[inline(always)]
fn m(x: u32, y: u32) -> u64 { (x as u64) * (y as u64) }
h
// This block is rearranged so that instead of doing a 32-bit multiplication by 38, we do a
// 64-bit multiplication by 2 on the results. This is because lg(38) is too big: we would
// have less than 1 bit of headroom left, which is too little.
let mut z = [0u64;10];
z[0] = m(x[0],x[0]) + m(x2_2,x8_19) + m(x4_2,x6_19) + (m(x1_2,x9_19) + m(x3_2,x7_19) + m(x[5],x5_19))*2;
z[1] = m(x0_2,x[1]) + m(x3_2,x8_19) + m(x5_2,x6_19) + (m(x[2],x9_19) + m(x[4],x7_19))*2;
z[2] = m(x0_2,x[2]) + m(x1_2,x[1]) + m(x4_2,x8_19) + m(x[6],x6_19) + (m(x3_2,x9_19) + m(x5_2,x7_19))*2;
z[3] = m(x0_2,x[3]) + m(x1_2,x[2]) + m(x5_2,x8_19) + (m(x[4],x9_19) + m(x[6],x7_19))*2;
z[4] = m(x0_2,x[4]) + m(x1_2,x3_2) + m(x[2],x[2]) + m(x6_2,x8_19) + (m(x5_2,x9_19) + m(x[7],x7_19))*2;
z[5] = m(x0_2,x[5]) + m(x1_2,x[4]) + m(x2_2,x[3]) + m(x7_2,x8_19) + m(x[6],x9_19)*2;
z[6] = m(x0_2,x[6]) + m(x1_2,x5_2) + m(x2_2,x[4]) + m(x3_2,x[3]) + m(x[8],x8_19) + m(x7_2,x9_19)*2;
z[7] = m(x0_2,x[7]) + m(x1_2,x[6]) + m(x2_2,x[5]) + m(x3_2,x[4]) + m(x[8],x9_19)*2;
z[8] = m(x0_2,x[8]) + m(x1_2,x7_2) + m(x2_2,x[6]) + m(x3_2,x5_2) + m(x[4],x[4]) + m(x[9],x9_19)*2;
z[9] = m(x0_2,x[9]) + m(x1_2,x[8]) + m(x2_2,x[7]) + m(x3_2,x[6]) + m(x4_2,x[5]) ;
z
}
/// Calculates h = f*f. Can overlap h with f.
///
/// XXX limbs: better to talk about headroom?
///
/// # 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.
/// Compute `self^2`.
pub fn square(&self) -> FieldElement32 {
FieldElement32::reduce(self.square_inner())
}
/// Square this field element and multiply the result by 2.
///
/// XXX explain why square2 exists vs square (overflow)
///
/// # 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.
/// Compute `2*self^2`.
pub fn square2(&self) -> FieldElement32 {
let mut coeffs = self.square_inner();
for i in 0..self.0.len() {

3
src/lib.rs
View file

@ -82,7 +82,7 @@ pub mod ristretto;
// Other miscelaneous utilities.
pub mod utils;
mod utils;
// Low-level curve and point constants, as well as pre-computed curve group elements.
@ -91,4 +91,3 @@ pub mod constants;
mod constants_32bit;
#[cfg(feature="radix_51")]
mod constants_64bit;