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//! Field arithmetic modulo p = 2^{384} − 2^{128} − 2^{96} + 2^{32} − 1
//!
//! Arithmetic implementations are extracted Rust code from the Coq fiat-crypto
//! libraries.
//!
//! # License
//!
//! Copyright (c) 2015-2020 the fiat-crypto authors
//!
//! fiat-crypto is distributed under the terms of the MIT License, the
//! Apache License (Version 2.0), and the BSD 1-Clause License;
//! users may pick which license to apply.
#![allow(
clippy::should_implement_trait,
clippy::suspicious_op_assign_impl,
clippy::unused_unit,
clippy::unnecessary_cast,
clippy::too_many_arguments,
clippy::identity_op
)]
#[cfg_attr(target_pointer_width = "32", path = "field/p384_32.rs")]
#[cfg_attr(target_pointer_width = "64", path = "field/p384_64.rs")]
mod field_impl;
use self::field_impl::*;
use crate::{FieldBytes, NistP384};
use core::{
iter::{Product, Sum},
ops::{AddAssign, MulAssign, Neg, SubAssign},
};
use elliptic_curve::{
bigint::{self, Limb, U384},
ff::PrimeField,
subtle::{Choice, ConstantTimeEq, CtOption},
};
/// Constant representing the modulus
/// p = 2^{384} − 2^{128} − 2^{96} + 2^{32} − 1
pub(crate) const MODULUS: U384 = U384::from_be_hex(FieldElement::MODULUS);
/// Element of the secp384r1 base field used for curve coordinates.
#[derive(Clone, Copy, Debug)]
pub struct FieldElement(pub(super) U384);
primeorder::impl_mont_field_element!(
NistP384,
FieldElement,
FieldBytes,
U384,
MODULUS,
fiat_p384_montgomery_domain_field_element,
fiat_p384_from_montgomery,
fiat_p384_to_montgomery,
fiat_p384_add,
fiat_p384_sub,
fiat_p384_mul,
fiat_p384_opp,
fiat_p384_square
);
impl FieldElement {
/// Compute [`FieldElement`] inversion: `1 / self`.
pub fn invert(&self) -> CtOption<Self> {
CtOption::new(self.invert_unchecked(), !self.is_zero())
}
/// Returns the multiplicative inverse of self.
///
/// Does not check that self is non-zero.
const fn invert_unchecked(&self) -> Self {
let words = impl_field_invert!(
self.to_canonical().as_words(),
Self::ONE.0.to_words(),
Limb::BITS,
bigint::nlimbs!(U384::BITS),
fiat_p384_mul,
fiat_p384_opp,
fiat_p384_divstep_precomp,
fiat_p384_divstep,
fiat_p384_msat,
fiat_p384_selectznz,
);
Self(U384::from_words(words))
}
/// Returns the square root of self mod p, or `None` if no square root
/// exists.
pub fn sqrt(&self) -> CtOption<Self> {
// p mod 4 = 3 -> compute sqrt(x) using x^((p+1)/4) =
// x^9850501549098619803069760025035903451269934817616361666987073351061430442874217582261816522064734500465401743278080
let t1 = *self;
let t10 = t1.square();
let t11 = t1 * t10;
let t110 = t11.square();
let t111 = t1 * t110;
let t111000 = t111.sqn(3);
let t111111 = t111 * t111000;
let t1111110 = t111111.square();
let t1111111 = t1 * t1111110;
let x12 = t1111110.sqn(5) * t111111;
let x24 = x12.sqn(12) * x12;
let x31 = x24.sqn(7) * t1111111;
let x32 = x31.square() * t1;
let x63 = x32.sqn(31) * x31;
let x126 = x63.sqn(63) * x63;
let x252 = x126.sqn(126) * x126;
let x255 = x252.sqn(3) * t111;
let x = ((x255.sqn(33) * x32).sqn(64) * t1).sqn(30);
CtOption::new(x, x.square().ct_eq(&t1))
}
/// Returns self^(2^n) mod p.
fn sqn(&self, n: usize) -> Self {
let mut x = *self;
for _ in 0..n {
x = x.square();
}
x
}
}
impl PrimeField for FieldElement {
type Repr = FieldBytes;
const MODULUS: &'static str = "fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff";
const NUM_BITS: u32 = 384;
const CAPACITY: u32 = 383;
const TWO_INV: Self = Self::from_u64(2).invert_unchecked();
const MULTIPLICATIVE_GENERATOR: Self = Self::from_u64(19);
const S: u32 = 1;
const ROOT_OF_UNITY: Self = Self::from_hex("fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000fffffffe");
const ROOT_OF_UNITY_INV: Self = Self::ROOT_OF_UNITY.invert_unchecked();
const DELTA: Self = Self::from_u64(49);
#[inline]
fn from_repr(bytes: FieldBytes) -> CtOption<Self> {
Self::from_bytes(&bytes)
}
#[inline]
fn to_repr(&self) -> FieldBytes {
self.to_bytes()
}
#[inline]
fn is_odd(&self) -> Choice {
self.is_odd()
}
}
#[cfg(test)]
mod tests {
use super::FieldElement;
use elliptic_curve::ff::PrimeField;
use primeorder::impl_primefield_tests;
/// t = (modulus - 1) >> S
const T: [u64; 6] = [
0x000000007fffffff,
0x7fffffff80000000,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0x7fffffffffffffff,
];
impl_primefield_tests!(FieldElement, T);
/// Basic tests that field inversion works.
#[test]
fn invert() {
let one = FieldElement::ONE;
assert_eq!(one.invert().unwrap(), one);
let three = one + &one + &one;
let inv_three = three.invert().unwrap();
assert_eq!(three * &inv_three, one);
let minus_three = -three;
let inv_minus_three = minus_three.invert().unwrap();
assert_eq!(inv_minus_three, -inv_three);
assert_eq!(three * &inv_minus_three, -one);
}
#[test]
fn sqrt() {
let one = FieldElement::ONE;
let two = one + &one;
let four = two.square();
assert_eq!(four.sqrt().unwrap(), two);
}
}
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