// -*- 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), using 64-bit arithmetic wuth
//! 128-bit products.
//!
//! On x86_64, the multiplications lower to `MUL` instructions taking
//! 64-bit inputs and producing 128-bit outputs. On other platforms,
//! this implementation is not recommended. On Haswell and newer, the
//! BMI2 instruction set provides `MULX` and friends, which gives even
//! better performance.
use core::fmt::Debug;
use core::ops::{Add, AddAssign};
use core::ops::{Sub, SubAssign};
use core::ops::{Mul, MulAssign};
use core::ops::Neg;
use subtle::ConditionallyAssignable;
use utils::load8;
/// In the 64-bit implementation, field elements are represented in
/// radix 2^51 as five `u64`s.
pub type Limb = u64;
/// A `FieldElement64` represents an element of the field GF(2^255 - 19).
///
/// In the 64-bit implementation, a `FieldElement` is represented in
/// radix 2^51 as five `u64`s; the coefficients are allowed to grow up
/// to 2^54 between reductions mod `p`.
///
/// # Warning
///
/// You almost certainly do not want to use `FieldElement64` directly. Consider
/// using `curve25519_dalek::field::FieldElement`, which will automatically
/// select between `FieldElement32` and `FieldElement64` depending on whether
/// curve25519-dalek was compiled with `--features="nightly"`.
///
/// This implementation, `FieldElement64`, is intended for x64_64 platforms,
/// which have the `MUL` instructions taking 64-bit inputs and producing 128-bit
/// outputs. On other platforms, this implementation is not recommended. On
/// Haswell and newer, the BMI2 instruction set provides `MULX` and friends,
/// which gives even better performance. This implementation requires Rust's
/// `u128`, which is not yet stable.
#[derive(Copy, Clone)]
pub struct FieldElement64(pub [u64; 5]);
impl Debug for FieldElement64 {
fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
write!(f, "FieldElement64: {:?}", &self.0[..])
}
}
impl<'b> AddAssign<&'b FieldElement64> for FieldElement64 {
fn add_assign(&mut self, _rhs: &'b FieldElement64) {
for i in 0..5 {
self.0[i] += _rhs.0[i];
}
}
}
impl<'a, 'b> Add<&'b FieldElement64> for &'a FieldElement64 {
type Output = FieldElement64;
fn add(self, _rhs: &'b FieldElement64) -> FieldElement64 {
let mut output = *self;
output += _rhs;
output
}
}
impl<'b> SubAssign<&'b FieldElement64> for FieldElement64 {
fn sub_assign(&mut self, _rhs: &'b FieldElement64) {
let result = (self as &FieldElement64) - _rhs;
self.0 = result.0;
}
}
impl<'a, 'b> Sub<&'b FieldElement64> for &'a FieldElement64 {
type Output = FieldElement64;
fn sub(self, _rhs: &'b FieldElement64) -> FieldElement64 {
// To avoid underflow, first add a multiple of p.
// Choose 16*p = p << 4 to be larger than 54-bit _rhs.
//
// If we could statically track the bitlengths of the limbs
// of every FieldElement64, we could choose a multiple of p
// just bigger than _rhs and avoid having to do a reduction.
//
// Since we don't yet have type-level integers to do this, we
// have to add an explicit reduction call here, which is a
// somewhat significant cost.
FieldElement64::reduce([
(self.0[0] + 36028797018963664u64) - _rhs.0[0],
(self.0[1] + 36028797018963952u64) - _rhs.0[1],
(self.0[2] + 36028797018963952u64) - _rhs.0[2],
(self.0[3] + 36028797018963952u64) - _rhs.0[3],
(self.0[4] + 36028797018963952u64) - _rhs.0[4],
])
}
}
impl<'b> MulAssign<&'b FieldElement64> for FieldElement64 {
fn mul_assign(&mut self, _rhs: &'b FieldElement64) {
let result = (self as &FieldElement64) * _rhs;
self.0 = result.0;
}
}
impl<'a, 'b> Mul<&'b FieldElement64> for &'a FieldElement64 {
type Output = FieldElement64;
fn mul(self, _rhs: &'b FieldElement64) -> FieldElement64 {
/// Helper function to multiply two 64-bit integers with 128
/// bits of output.
#[inline(always)]
fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }
// Alias self, _rhs for more readable formulas
let a: &[u64; 5] = &self.0;
let b: &[u64; 5] = &_rhs.0;
// 64-bit precomputations to avoid 128-bit multiplications
let b1_19 = b[1] * 19;
let b2_19 = b[2] * 19;
let b3_19 = b[3] * 19;
let b4_19 = b[4] * 19;
// Multiply to get 128-bit coefficients of output
let c0: u128 = m(a[0],b[0]) + m(a[4],b1_19) + m(a[3],b2_19) + m(a[2],b3_19) + m(a[1],b4_19);
let mut c1: u128 = m(a[1],b[0]) + m(a[0],b[1]) + m(a[4],b2_19) + m(a[3],b3_19) + m(a[2],b4_19);
let mut c2: u128 = m(a[2],b[0]) + m(a[1],b[1]) + m(a[0],b[2]) + m(a[4],b3_19) + m(a[3],b4_19);
let mut c3: u128 = m(a[3],b[0]) + m(a[2],b[1]) + m(a[1],b[2]) + m(a[0],b[3]) + m(a[4],b4_19);
let mut c4: u128 = m(a[4],b[0]) + m(a[3],b[1]) + m(a[2],b[2]) + m(a[1],b[3]) + m(a[0],b[4]);
// Now c[i] < 2^2b * (1+i + (4-i)*19) < 2^(2b + lg(1+4*19)) < 2^(2b + 6.27)
// where b is the bitlength of the input limbs.
// The carry (c[i] >> 51) fits into a u64 iff 2b+6.27 < 64+51 iff b <= 54.
// After the first carry pass, all c[i] fit into u64.
debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));
debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));
debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));
debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));
debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));
// The 128-bit output limbs are stored in two 64-bit registers
// (low/high part). By rebinding the names after carrying, we
// inform LLVM that the values have shrunk, so it can
// efficiently allocate registers.
let low_51_bit_mask = (1u64 << 51) - 1;
c1 += (c0 >> 51) as u128;
let mut c0: u64 = (c0 as u64) & low_51_bit_mask;
c2 += (c1 >> 51) as u128;
let c1: u64 = (c1 as u64) & low_51_bit_mask;
c3 += (c2 >> 51) as u128;
let c2: u64 = (c2 as u64) & low_51_bit_mask;
c4 += (c3 >> 51) as u128;
let c3: u64 = (c3 as u64) & low_51_bit_mask;
c0 += ((c4 >> 51) as u64) * 19;
let c4: u64 = (c4 as u64) & low_51_bit_mask;
FieldElement64::reduce([c0,c1,c2,c3,c4])
}
}
impl<'a> Neg for &'a FieldElement64 {
type Output = FieldElement64;
fn neg(self) -> FieldElement64 {
let mut output = *self;
output.negate();
output
}
}
impl ConditionallyAssignable for FieldElement64 {
fn conditional_assign(&mut self, f: &FieldElement64, choice: u8) {
let mask = (-(choice as i64)) as u64;
for i in 0..5 {
self.0[i] ^= mask & (self.0[i] ^ f.0[i]);
}
}
}
impl FieldElement64 {
/// Invert the sign of this field element
pub fn negate(&mut self) {
// See commentary in the Sub impl
let neg = FieldElement64::reduce([
36028797018963664u64 - self.0[0],
36028797018963952u64 - self.0[1],
36028797018963952u64 - self.0[2],
36028797018963952u64 - self.0[3],
36028797018963952u64 - self.0[4],
]);
self.0 = neg.0;
}
/// Construct zero.
pub fn zero() -> FieldElement64 {
FieldElement64([ 0, 0, 0, 0, 0 ])
}
/// Construct one.
pub fn one() -> FieldElement64 {
FieldElement64([ 1, 0, 0, 0, 0 ])
}
/// Construct -1.
pub fn minus_one() -> FieldElement64 {
FieldElement64([2251799813685228, 2251799813685247, 2251799813685247, 2251799813685247, 2251799813685247])
}
/// Given 64-bit limbs, reduce to enforce the bound c_i < 2^51.
#[inline(always)]
fn reduce(mut limbs: [u64; 5]) -> FieldElement64 {
let low_51_bit_mask = (1u64 << 51) - 1;
limbs[1] += limbs[0] >> 51;
limbs[0] = limbs[0] & low_51_bit_mask;
limbs[2] += limbs[1] >> 51;
limbs[1] = limbs[1] & low_51_bit_mask;
limbs[3] += limbs[2] >> 51;
limbs[2] = limbs[2] & low_51_bit_mask;
limbs[4] += limbs[3] >> 51;
limbs[3] = limbs[3] & low_51_bit_mask;
limbs[0] += (limbs[4] >> 51) * 19;
limbs[4] = limbs[4] & low_51_bit_mask;
FieldElement64(limbs)
}
/// Load a `FieldElement64` from the low 255 bits of a 256-bit
/// input.
///
/// # Warning
///
/// This function does not check that the input used the canonical
/// representative. It masks the high bit, but it will happily
/// decode 2^255 - 18 to 1. Applications that require a canonical
/// encoding of every field element should decode, re-encode to
/// the canonical encoding, and check that the input was
/// canonical.
///
pub fn from_bytes(bytes: &[u8; 32]) -> FieldElement64 {
let low_51_bit_mask = (1u64 << 51) - 1;
FieldElement64(
// load bits [ 0, 64), no shift
[ load8(&bytes[ 0..]) & low_51_bit_mask
// load bits [ 48,112), shift to [ 51,112)
, (load8(&bytes[ 6..]) >> 3) & low_51_bit_mask
// load bits [ 96,160), shift to [102,160)
, (load8(&bytes[12..]) >> 6) & low_51_bit_mask
// load bits [152,216), shift to [153,216)
, (load8(&bytes[19..]) >> 1) & low_51_bit_mask
// load bits [192,256), shift to [204,112)
, (load8(&bytes[24..]) >> 12) & low_51_bit_mask
])
}
/// Serialize this `FieldElement64` to a 32-byte array. The
/// encoding is canonical.
pub fn to_bytes(&self) -> [u8; 32] {
// This reduces to the range [0,2^255), but we need [0,2^255-19).
let mut limbs = FieldElement64::reduce(self.0).0;
// Let h = limbs[0] + limbs[1]*2^51 + ... + limbs[4]*2^204.
//
// Write h = pq + r with 0 <= r < p. We want to compute r = h mod p.
//
// Since h < 2^255, 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 = (limbs[0] + 19) >> 51;
q = (limbs[1] + q) >> 51;
q = (limbs[2] + q) >> 51;
q = (limbs[3] + q) >> 51;
q = (limbs[4] + q) >> 51;
// Now we can compute r as r = h - pq = r - (2^255-19)q = r + 19q - 2^255q
limbs[0] += 19*q;
// Now carry the result to compute r + 19q ...
let low_51_bit_mask = (1u64 << 51) - 1;
limbs[1] += limbs[0] >> 51;
limbs[0] = limbs[0] & low_51_bit_mask;
limbs[2] += limbs[1] >> 51;
limbs[1] = limbs[1] & low_51_bit_mask;
limbs[3] += limbs[2] >> 51;
limbs[2] = limbs[2] & low_51_bit_mask;
limbs[4] += limbs[3] >> 51;
limbs[3] = limbs[3] & low_51_bit_mask;
// ... but instead of carrying (limbs[4] >> 51) = 2^255q
// into another limb, discard it, subtracting the value
limbs[4] = limbs[4] & low_51_bit_mask;
// Now arrange the bits of the limbs.
let mut s = [0u8;32];
s[ 0] = limbs[0] as u8;
s[ 1] = (limbs[0] >> 8) as u8;
s[ 2] = (limbs[0] >> 16) as u8;
s[ 3] = (limbs[0] >> 24) as u8;
s[ 4] = (limbs[0] >> 32) as u8;
s[ 5] = (limbs[0] >> 40) as u8;
s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;
s[ 7] = (limbs[1] >> 5) as u8;
s[ 8] = (limbs[1] >> 13) as u8;
s[ 9] = (limbs[1] >> 21) as u8;
s[10] = (limbs[1] >> 29) as u8;
s[11] = (limbs[1] >> 37) as u8;
s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;
s[13] = (limbs[2] >> 2) as u8;
s[14] = (limbs[2] >> 10) as u8;
s[15] = (limbs[2] >> 18) as u8;
s[16] = (limbs[2] >> 26) as u8;
s[17] = (limbs[2] >> 34) as u8;
s[18] = (limbs[2] >> 42) as u8;
s[19] = ((limbs[2] >> 50) | (limbs[3] << 1)) as u8;
s[20] = (limbs[3] >> 7) as u8;
s[21] = (limbs[3] >> 15) as u8;
s[22] = (limbs[3] >> 23) as u8;
s[23] = (limbs[3] >> 31) as u8;
s[24] = (limbs[3] >> 39) as u8;
s[25] = ((limbs[3] >> 47) | (limbs[4] << 4)) as u8;
s[26] = (limbs[4] >> 4) as u8;
s[27] = (limbs[4] >> 12) as u8;
s[28] = (limbs[4] >> 20) as u8;
s[29] = (limbs[4] >> 28) as u8;
s[30] = (limbs[4] >> 36) as u8;
s[31] = (limbs[4] >> 44) as u8;
// High bit should be zero.
debug_assert!((s[31] & 0b1000_0000u8) == 0u8);
s
}
#[inline(always)]
fn square_inner(&self) -> [u64; 5] {
/// Multiply two 64-bit integers with 128 bits of output.
#[inline(always)]
fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }
// Alias self, _rhs for more readable formulas
let a: &[u64; 5] = &self.0;
// Precomputation: 64-bit multiply by 19
let a3_19 = 19 * a[3];
let a4_19 = 19 * a[4];
// Multiply to get 128-bit coefficients of output
let c0: u128 = m(a[0], a[0]) + 2*( m(a[1], a4_19) + m(a[2], a3_19) );
let mut c1: u128 = m(a[3], a3_19) + 2*( m(a[0], a[1]) + m(a[2], a4_19) );
let mut c2: u128 = m(a[1], a[1]) + 2*( m(a[0], a[2]) + m(a[4], a3_19) );
let mut c3: u128 = m(a[4], a4_19) + 2*( m(a[0], a[3]) + m(a[1], a[2]) );
let mut c4: u128 = m(a[2], a[2]) + 2*( m(a[0], a[4]) + m(a[1], a[3]) );
// Same bound as in multiply:
// c[i] < 2^2b * (1+i + (4-i)*19) < 2^(2b + lg(1+4*19)) < 2^(2b + 6.27)
// where b is the bitlength of the input limbs.
//
// The carry (c[i] >> 51) fits into a u64 iff 2b+6.27 < 64+51 iff b <= 54.
// After the first carry pass, all c[i] fit into u64.
debug_assert!(a[0] < (1 << 54));
debug_assert!(a[1] < (1 << 54));
debug_assert!(a[2] < (1 << 54));
debug_assert!(a[3] < (1 << 54));
debug_assert!(a[4] < (1 << 54));
// The 128-bit output limbs are stored in two 64-bit registers (low/high part).
// By rebinding the names after carrying, we free the upper registers for reuse.
let low_51_bit_mask = (1u64 << 51) - 1;
c1 += (c0 >> 51) as u128;
let mut c0: u64 = (c0 as u64) & low_51_bit_mask;
c2 += (c1 >> 51) as u128;
let c1: u64 = (c1 as u64) & low_51_bit_mask;
c3 += (c2 >> 51) as u128;
let c2: u64 = (c2 as u64) & low_51_bit_mask;
c4 += (c3 >> 51) as u128;
let c3: u64 = (c3 as u64) & low_51_bit_mask;
c0 += ((c4 >> 51) as u64) * 19;
let c4: u64 = (c4 as u64) & low_51_bit_mask;
// Now c_i all fit into u64, but are not yet bounded by 2^51.
[c0,c1,c2,c3,c4]
}
/// Returns the square of this field element.
pub fn square(&self) -> FieldElement64 {
FieldElement64::reduce(self.square_inner())
}
/// Returns 2 times the square of this field element.
pub fn square2(&self) -> FieldElement64 {
let mut limbs = self.square_inner();
// For this to work, need to have 1 extra bit of headroom after carry
// --> max 53 bit inputs, not 54
//
// XXX check that this is correct; I think it isn't -- hdevalence
limbs[0] *= 2;
limbs[1] *= 2;
limbs[2] *= 2;
limbs[3] *= 2;
limbs[4] *= 2;
FieldElement64::reduce(limbs)
}
}