564 lines
20 KiB
Rust
564 lines
20 KiB
Rust
// -*- mode: rust; coding: utf-8; -*-
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//
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// This file is part of curve25519-dalek.
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// Copyright (c) 2016-2019 Isis Lovecruft, Henry de Valence
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// See LICENSE for licensing information.
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//
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// Authors:
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// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
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// - Henry de Valence <hdevalence@hdevalence.ca>
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//! Field arithmetic modulo \\(p = 2\^{255} - 19\\), using \\(64\\)-bit
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//! limbs with \\(128\\)-bit products.
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use core::fmt::Debug;
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use core::ops::Neg;
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use core::ops::{Add, AddAssign};
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use core::ops::{Mul, MulAssign};
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use core::ops::{Sub, SubAssign};
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use subtle::Choice;
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use subtle::ConditionallySelectable;
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use zeroize::Zeroize;
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/// A `FieldElement51` represents an element of the field
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/// \\( \mathbb Z / (2\^{255} - 19)\\).
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///
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/// In the 64-bit implementation, a `FieldElement` is represented in
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/// radix \\(2\^{51}\\) as five `u64`s; the coefficients are allowed to
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/// grow up to \\(2\^{54}\\) between reductions modulo \\(p\\).
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///
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/// # Note
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///
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/// The `curve25519_dalek::field` module provides a type alias
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/// `curve25519_dalek::field::FieldElement` to either `FieldElement51`
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/// or `FieldElement2625`.
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///
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/// The backend-specific type `FieldElement51` should not be used
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/// outside of the `curve25519_dalek::field` module.
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#[derive(Copy, Clone)]
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pub struct FieldElement51(pub (crate) [u64; 5]);
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impl Debug for FieldElement51 {
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fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
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write!(f, "FieldElement51({:?})", &self.0[..])
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}
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}
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impl Zeroize for FieldElement51 {
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fn zeroize(&mut self) {
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self.0.zeroize();
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}
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}
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impl<'b> AddAssign<&'b FieldElement51> for FieldElement51 {
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fn add_assign(&mut self, _rhs: &'b FieldElement51) {
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for i in 0..5 {
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self.0[i] += _rhs.0[i];
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}
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}
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}
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impl<'a, 'b> Add<&'b FieldElement51> for &'a FieldElement51 {
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type Output = FieldElement51;
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fn add(self, _rhs: &'b FieldElement51) -> FieldElement51 {
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let mut output = *self;
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output += _rhs;
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output
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}
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}
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impl<'b> SubAssign<&'b FieldElement51> for FieldElement51 {
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fn sub_assign(&mut self, _rhs: &'b FieldElement51) {
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let result = (self as &FieldElement51) - _rhs;
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self.0 = result.0;
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}
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}
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impl<'a, 'b> Sub<&'b FieldElement51> for &'a FieldElement51 {
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type Output = FieldElement51;
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fn sub(self, _rhs: &'b FieldElement51) -> FieldElement51 {
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// To avoid underflow, first add a multiple of p.
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// Choose 16*p = p << 4 to be larger than 54-bit _rhs.
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//
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// If we could statically track the bitlengths of the limbs
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// of every FieldElement51, we could choose a multiple of p
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// just bigger than _rhs and avoid having to do a reduction.
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//
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// Since we don't yet have type-level integers to do this, we
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// have to add an explicit reduction call here.
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FieldElement51::reduce([
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(self.0[0] + 36028797018963664u64) - _rhs.0[0],
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(self.0[1] + 36028797018963952u64) - _rhs.0[1],
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(self.0[2] + 36028797018963952u64) - _rhs.0[2],
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(self.0[3] + 36028797018963952u64) - _rhs.0[3],
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(self.0[4] + 36028797018963952u64) - _rhs.0[4],
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])
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}
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}
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impl<'b> MulAssign<&'b FieldElement51> for FieldElement51 {
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fn mul_assign(&mut self, _rhs: &'b FieldElement51) {
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let result = (self as &FieldElement51) * _rhs;
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self.0 = result.0;
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}
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}
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impl<'a, 'b> Mul<&'b FieldElement51> for &'a FieldElement51 {
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type Output = FieldElement51;
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fn mul(self, _rhs: &'b FieldElement51) -> FieldElement51 {
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/// Helper function to multiply two 64-bit integers with 128
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/// bits of output.
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#[inline(always)]
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fn m(x: u64, y: u64) -> u128 { (x as u128) * (y as u128) }
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// Alias self, _rhs for more readable formulas
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let a: &[u64; 5] = &self.0;
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let b: &[u64; 5] = &_rhs.0;
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// Precondition: assume input limbs a[i], b[i] are bounded as
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//
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// a[i], b[i] < 2^(51 + b)
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//
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// where b is a real parameter measuring the "bit excess" of the limbs.
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// 64-bit precomputations to avoid 128-bit multiplications.
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//
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// This fits into a u64 whenever 51 + b + lg(19) < 64.
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//
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// Since 51 + b + lg(19) < 51 + 4.25 + b
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// = 55.25 + b,
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// this fits if b < 8.75.
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let b1_19 = b[1] * 19;
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let b2_19 = b[2] * 19;
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let b3_19 = b[3] * 19;
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let b4_19 = b[4] * 19;
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// Multiply to get 128-bit coefficients of output
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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);
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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);
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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);
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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);
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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]);
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// How big are the c[i]? We have
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//
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// c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)
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// < 2^(102 + lg(1 + 4*19) + 2*b)
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// < 2^(108.27 + 2*b)
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//
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// The carry (c[i] >> 51) fits into a u64 when
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// 108.27 + 2*b - 51 < 64
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// 2*b < 6.73
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// b < 3.365.
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//
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// So we require b < 3 to ensure this fits.
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debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));
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debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));
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debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));
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debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));
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debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));
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// Casting to u64 and back tells the compiler that the carry is
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// bounded by 2^64, so that the addition is a u128 + u64 rather
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// than u128 + u128.
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const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;
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let mut out = [0u64; 5];
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c1 += ((c0 >> 51) as u64) as u128;
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out[0] = (c0 as u64) & LOW_51_BIT_MASK;
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c2 += ((c1 >> 51) as u64) as u128;
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out[1] = (c1 as u64) & LOW_51_BIT_MASK;
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c3 += ((c2 >> 51) as u64) as u128;
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out[2] = (c2 as u64) & LOW_51_BIT_MASK;
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c4 += ((c3 >> 51) as u64) as u128;
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out[3] = (c3 as u64) & LOW_51_BIT_MASK;
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let carry: u64 = (c4 >> 51) as u64;
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out[4] = (c4 as u64) & LOW_51_BIT_MASK;
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// To see that this does not overflow, we need out[0] + carry * 19 < 2^64.
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//
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// c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)
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// < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)
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// < 2^(102 + 2*b + lg(5)) + 2^64.
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//
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// When b < 3 we get
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//
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// c4 < 2^110.33 so that carry < 2^59.33
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//
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// so that
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//
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// out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58
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//
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// and there is no overflow.
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out[0] = out[0] + carry * 19;
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// Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).
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out[1] += out[0] >> 51;
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out[0] &= LOW_51_BIT_MASK;
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// Now out[i] < 2^(51 + epsilon) for all i.
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FieldElement51(out)
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}
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}
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impl<'a> Neg for &'a FieldElement51 {
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type Output = FieldElement51;
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fn neg(self) -> FieldElement51 {
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let mut output = *self;
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output.negate();
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output
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}
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}
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impl ConditionallySelectable for FieldElement51 {
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fn conditional_select(
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a: &FieldElement51,
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b: &FieldElement51,
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choice: Choice,
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) -> FieldElement51 {
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FieldElement51([
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u64::conditional_select(&a.0[0], &b.0[0], choice),
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u64::conditional_select(&a.0[1], &b.0[1], choice),
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u64::conditional_select(&a.0[2], &b.0[2], choice),
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u64::conditional_select(&a.0[3], &b.0[3], choice),
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u64::conditional_select(&a.0[4], &b.0[4], choice),
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])
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}
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fn conditional_swap(a: &mut FieldElement51, b: &mut FieldElement51, choice: Choice) {
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u64::conditional_swap(&mut a.0[0], &mut b.0[0], choice);
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u64::conditional_swap(&mut a.0[1], &mut b.0[1], choice);
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u64::conditional_swap(&mut a.0[2], &mut b.0[2], choice);
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u64::conditional_swap(&mut a.0[3], &mut b.0[3], choice);
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u64::conditional_swap(&mut a.0[4], &mut b.0[4], choice);
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}
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fn conditional_assign(&mut self, other: &FieldElement51, choice: Choice) {
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self.0[0].conditional_assign(&other.0[0], choice);
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self.0[1].conditional_assign(&other.0[1], choice);
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self.0[2].conditional_assign(&other.0[2], choice);
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self.0[3].conditional_assign(&other.0[3], choice);
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self.0[4].conditional_assign(&other.0[4], choice);
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}
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}
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impl FieldElement51 {
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/// Invert the sign of this field element
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pub fn negate(&mut self) {
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// See commentary in the Sub impl
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let neg = FieldElement51::reduce([
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36028797018963664u64 - self.0[0],
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36028797018963952u64 - self.0[1],
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36028797018963952u64 - self.0[2],
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36028797018963952u64 - self.0[3],
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36028797018963952u64 - self.0[4],
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]);
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self.0 = neg.0;
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}
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/// Construct zero.
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pub fn zero() -> FieldElement51 {
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FieldElement51([ 0, 0, 0, 0, 0 ])
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}
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/// Construct one.
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pub fn one() -> FieldElement51 {
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FieldElement51([ 1, 0, 0, 0, 0 ])
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}
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/// Construct -1.
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pub fn minus_one() -> FieldElement51 {
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FieldElement51([2251799813685228, 2251799813685247, 2251799813685247, 2251799813685247, 2251799813685247])
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}
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/// Given 64-bit input limbs, reduce to enforce the bound 2^(51 + epsilon).
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#[inline(always)]
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fn reduce(mut limbs: [u64; 5]) -> FieldElement51 {
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const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;
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// Since the input limbs are bounded by 2^64, the biggest
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// carry-out is bounded by 2^13.
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//
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// The biggest carry-in is c4 * 19, resulting in
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//
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// 2^51 + 19*2^13 < 2^51.0000000001
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//
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// Because we don't need to canonicalize, only to reduce the
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// limb sizes, it's OK to do a "weak reduction", where we
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// compute the carry-outs in parallel.
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let c0 = limbs[0] >> 51;
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let c1 = limbs[1] >> 51;
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let c2 = limbs[2] >> 51;
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let c3 = limbs[3] >> 51;
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let c4 = limbs[4] >> 51;
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limbs[0] &= LOW_51_BIT_MASK;
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limbs[1] &= LOW_51_BIT_MASK;
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limbs[2] &= LOW_51_BIT_MASK;
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limbs[3] &= LOW_51_BIT_MASK;
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limbs[4] &= LOW_51_BIT_MASK;
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limbs[0] += c4 * 19;
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limbs[1] += c0;
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limbs[2] += c1;
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limbs[3] += c2;
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limbs[4] += c3;
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FieldElement51(limbs)
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}
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/// Load a `FieldElement51` from the low 255 bits of a 256-bit
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/// input.
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///
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/// # Warning
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///
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/// This function does not check that the input used the canonical
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/// representative. It masks the high bit, but it will happily
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/// decode 2^255 - 18 to 1. Applications that require a canonical
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/// encoding of every field element should decode, re-encode to
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/// the canonical encoding, and check that the input was
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/// canonical.
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///
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pub fn from_bytes(bytes: &[u8; 32]) -> FieldElement51 {
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let load8 = |input: &[u8]| -> u64 {
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(input[0] as u64)
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| ((input[1] as u64) << 8)
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| ((input[2] as u64) << 16)
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| ((input[3] as u64) << 24)
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| ((input[4] as u64) << 32)
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| ((input[5] as u64) << 40)
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| ((input[6] as u64) << 48)
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| ((input[7] as u64) << 56)
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};
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let low_51_bit_mask = (1u64 << 51) - 1;
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FieldElement51(
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// load bits [ 0, 64), no shift
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[ load8(&bytes[ 0..]) & low_51_bit_mask
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// load bits [ 48,112), shift to [ 51,112)
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, (load8(&bytes[ 6..]) >> 3) & low_51_bit_mask
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// load bits [ 96,160), shift to [102,160)
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, (load8(&bytes[12..]) >> 6) & low_51_bit_mask
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// load bits [152,216), shift to [153,216)
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, (load8(&bytes[19..]) >> 1) & low_51_bit_mask
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// load bits [192,256), shift to [204,112)
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, (load8(&bytes[24..]) >> 12) & low_51_bit_mask
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])
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}
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/// Serialize this `FieldElement51` to a 32-byte array. The
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/// encoding is canonical.
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pub fn to_bytes(&self) -> [u8; 32] {
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// Let h = limbs[0] + limbs[1]*2^51 + ... + limbs[4]*2^204.
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//
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// Write h = pq + r with 0 <= r < p.
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//
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// We want to compute r = h mod p.
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//
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// If h < 2*p = 2^256 - 38,
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// then q = 0 or 1,
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//
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// with q = 0 when h < p
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// and q = 1 when h >= p.
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//
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// Notice that h >= p <==> h + 19 >= p + 19 <==> h + 19 >= 2^255.
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// Therefore q can be computed as the carry bit of h + 19.
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// First, reduce the limbs to ensure h < 2*p.
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let mut limbs = FieldElement51::reduce(self.0).0;
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let mut q = (limbs[0] + 19) >> 51;
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q = (limbs[1] + q) >> 51;
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q = (limbs[2] + q) >> 51;
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q = (limbs[3] + q) >> 51;
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q = (limbs[4] + q) >> 51;
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// Now we can compute r as r = h - pq = r - (2^255-19)q = r + 19q - 2^255q
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limbs[0] += 19*q;
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// Now carry the result to compute r + 19q ...
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let low_51_bit_mask = (1u64 << 51) - 1;
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limbs[1] += limbs[0] >> 51;
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limbs[0] = limbs[0] & low_51_bit_mask;
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limbs[2] += limbs[1] >> 51;
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limbs[1] = limbs[1] & low_51_bit_mask;
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limbs[3] += limbs[2] >> 51;
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limbs[2] = limbs[2] & low_51_bit_mask;
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limbs[4] += limbs[3] >> 51;
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limbs[3] = limbs[3] & low_51_bit_mask;
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// ... but instead of carrying (limbs[4] >> 51) = 2^255q
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// into another limb, discard it, subtracting the value
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limbs[4] = limbs[4] & low_51_bit_mask;
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// Now arrange the bits of the limbs.
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let mut s = [0u8;32];
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s[ 0] = limbs[0] as u8;
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s[ 1] = (limbs[0] >> 8) as u8;
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s[ 2] = (limbs[0] >> 16) as u8;
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s[ 3] = (limbs[0] >> 24) as u8;
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s[ 4] = (limbs[0] >> 32) as u8;
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s[ 5] = (limbs[0] >> 40) as u8;
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s[ 6] = ((limbs[0] >> 48) | (limbs[1] << 3)) as u8;
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s[ 7] = (limbs[1] >> 5) as u8;
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s[ 8] = (limbs[1] >> 13) as u8;
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s[ 9] = (limbs[1] >> 21) as u8;
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s[10] = (limbs[1] >> 29) as u8;
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s[11] = (limbs[1] >> 37) as u8;
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s[12] = ((limbs[1] >> 45) | (limbs[2] << 6)) as u8;
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s[13] = (limbs[2] >> 2) as u8;
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s[14] = (limbs[2] >> 10) as u8;
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s[15] = (limbs[2] >> 18) as u8;
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s[16] = (limbs[2] >> 26) as u8;
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s[17] = (limbs[2] >> 34) as u8;
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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
|
|
}
|
|
|
|
/// Given `k > 0`, return `self^(2^k)`.
|
|
pub fn pow2k(&self, mut k: u32) -> FieldElement51 {
|
|
|
|
debug_assert!( k > 0 );
|
|
|
|
/// 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) }
|
|
|
|
let mut a: [u64; 5] = self.0;
|
|
|
|
loop {
|
|
// Precondition: assume input limbs a[i] are bounded as
|
|
//
|
|
// a[i] < 2^(51 + b)
|
|
//
|
|
// where b is a real parameter measuring the "bit excess" of the limbs.
|
|
|
|
// Precomputation: 64-bit multiply by 19.
|
|
//
|
|
// This fits into a u64 whenever 51 + b + lg(19) < 64.
|
|
//
|
|
// Since 51 + b + lg(19) < 51 + 4.25 + b
|
|
// = 55.25 + b,
|
|
// this fits if b < 8.75.
|
|
let a3_19 = 19 * a[3];
|
|
let a4_19 = 19 * a[4];
|
|
|
|
// Multiply to get 128-bit coefficients of output.
|
|
//
|
|
// The 128-bit multiplications by 2 turn into 1 slr + 1 slrd each,
|
|
// which doesn't seem any better or worse than doing them as precomputations
|
|
// on the 64-bit inputs.
|
|
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^(102 + 2*b) * (1+i + (4-i)*19)
|
|
// < 2^(102 + lg(1 + 4*19) + 2*b)
|
|
// < 2^(108.27 + 2*b)
|
|
//
|
|
// The carry (c[i] >> 51) fits into a u64 when
|
|
// 108.27 + 2*b - 51 < 64
|
|
// 2*b < 6.73
|
|
// b < 3.365.
|
|
//
|
|
// So we require b < 3 to ensure this fits.
|
|
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));
|
|
|
|
const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;
|
|
|
|
// Casting to u64 and back tells the compiler that the carry is bounded by 2^64, so
|
|
// that the addition is a u128 + u64 rather than u128 + u128.
|
|
c1 += ((c0 >> 51) as u64) as u128;
|
|
a[0] = (c0 as u64) & LOW_51_BIT_MASK;
|
|
|
|
c2 += ((c1 >> 51) as u64) as u128;
|
|
a[1] = (c1 as u64) & LOW_51_BIT_MASK;
|
|
|
|
c3 += ((c2 >> 51) as u64) as u128;
|
|
a[2] = (c2 as u64) & LOW_51_BIT_MASK;
|
|
|
|
c4 += ((c3 >> 51) as u64) as u128;
|
|
a[3] = (c3 as u64) & LOW_51_BIT_MASK;
|
|
|
|
let carry: u64 = (c4 >> 51) as u64;
|
|
a[4] = (c4 as u64) & LOW_51_BIT_MASK;
|
|
|
|
// To see that this does not overflow, we need a[0] + carry * 19 < 2^64.
|
|
//
|
|
// c4 < a2^2 + 2*a0*a4 + 2*a1*a3 + (carry from c3)
|
|
// < 2^(102 + 2*b + lg(5)) + 2^64.
|
|
//
|
|
// When b < 3 we get
|
|
//
|
|
// c4 < 2^110.33 so that carry < 2^59.33
|
|
//
|
|
// so that
|
|
//
|
|
// a[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58
|
|
//
|
|
// and there is no overflow.
|
|
a[0] = a[0] + carry * 19;
|
|
|
|
// Now a[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + epsilon).
|
|
a[1] += a[0] >> 51;
|
|
a[0] &= LOW_51_BIT_MASK;
|
|
|
|
// Now all a[i] < 2^(51 + epsilon) and a = self^(2^k).
|
|
|
|
k = k - 1;
|
|
if k == 0 {
|
|
break;
|
|
}
|
|
}
|
|
|
|
FieldElement51(a)
|
|
}
|
|
|
|
/// Returns the square of this field element.
|
|
pub fn square(&self) -> FieldElement51 {
|
|
self.pow2k(1)
|
|
}
|
|
|
|
/// Returns 2 times the square of this field element.
|
|
pub fn square2(&self) -> FieldElement51 {
|
|
let mut square = self.pow2k(1);
|
|
for i in 0..5 {
|
|
square.0[i] *= 2;
|
|
}
|
|
|
|
square
|
|
}
|
|
}
|