field.rs

//! 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;
use core::ops::{AddAssign, MulAssign, Neg, SubAssign};
use elliptic_curve::{
    bigint::{ArrayEncoding, Encoding, Integer, U384},
    subtle::{Choice, ConstantTimeEq, ConstantTimeLess, CtOption},
};

/// Constant representing the modulus
/// p = 2^{384} − 2^{128} − 2^{96} + 2^{32} − 1
pub(crate) const MODULUS: U384 = U384::from_be_hex("fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff");

/// Element of the secp384r1 base field used for curve coordinates.
#[derive(Clone, Copy, Debug)]
pub struct FieldElement(pub(super) U384);

impl_field_element!(
    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,
    fiat_p384_divstep_precomp,
    fiat_p384_divstep,
    fiat_p384_msat,
);

impl FieldElement {
    /// Parse the given byte array as an SEC1-encoded field element.
    ///
    /// Returns `None` if the byte array does not contain a big-endian integer in
    /// the range `[0, p)`.
    pub fn from_sec1(bytes: FieldBytes) -> CtOption<Self> {
        Self::from_be_bytes(bytes)
    }

    /// Returns the SEC1 encoding of this field element.
    pub fn to_sec1(self) -> FieldBytes {
        self.to_be_bytes()
    }

    /// 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
    }
}

#[cfg(test)]
mod tests {
    use super::FieldElement;

    /// 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);
    }
}