p256 0.13.2

Pure Rust implementation of the NIST P-256 (a.k.a. secp256r1, prime256v1) elliptic curve as defined in SP 800-186, with support for ECDH, ECDSA signing/verification, and general purpose curve arithmetic
Documentation 
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//! 32-bit secp256r1 scalar field algorithms.

// TODO(tarcieri): adapt 64-bit arithmetic to proper 32-bit arithmetic

use super::{MODULUS, MU};
use crate::{
    arithmetic::util::{adc, mac, sbb},
    U256,
};

/// Barrett Reduction
///
/// The general algorithm is:
/// ```text
/// p = n = order of group
/// b = 2^64 = 64bit machine word
/// k = 4
/// a \in [0, 2^512]
/// mu := floor(b^{2k} / p)
/// q1 := floor(a / b^{k - 1})
/// q2 := q1 * mu
/// q3 := <- floor(a / b^{k - 1})
/// r1 := a mod b^{k + 1}
/// r2 := q3 * m mod b^{k + 1}
/// r := r1 - r2
///
/// if r < 0: r := r + b^{k + 1}
/// while r >= p: do r := r - p (at most twice)
/// ```
///
/// References:
/// - Handbook of Applied Cryptography, Chapter 14
///   Algorithm 14.42
///   http://cacr.uwaterloo.ca/hac/about/chap14.pdf
///
/// - Efficient and Secure Elliptic Curve Cryptography Implementation of Curve P-256
///   Algorithm 6) Barrett Reduction modulo p
///   https://csrc.nist.gov/csrc/media/events/workshop-on-elliptic-curve-cryptography-standards/documents/papers/session6-adalier-mehmet.pdf
#[inline]
#[allow(clippy::too_many_arguments)]
pub(super) const fn barrett_reduce(lo: U256, hi: U256) -> U256 {
    let lo = u256_to_u64x4(lo);
    let hi = u256_to_u64x4(hi);
    let a0 = lo[0];
    let a1 = lo[1];
    let a2 = lo[2];
    let a3 = lo[3];
    let a4 = hi[0];
    let a5 = hi[1];
    let a6 = hi[2];
    let a7 = hi[3];
    let q1: [u64; 5] = [a3, a4, a5, a6, a7];
    let q3 = q1_times_mu_shift_five(&q1);

    let r1: [u64; 5] = [a0, a1, a2, a3, a4];
    let r2: [u64; 5] = q3_times_n_keep_five(&q3);
    let r: [u64; 5] = sub_inner_five(r1, r2);

    // Result is in range (0, 3*n - 1),
    // and 90% of the time, no subtraction will be needed.
    let r = subtract_n_if_necessary(r[0], r[1], r[2], r[3], r[4]);
    let r = subtract_n_if_necessary(r[0], r[1], r[2], r[3], r[4]);

    U256::from_words([
        (r[0] & 0xFFFFFFFF) as u32,
        (r[0] >> 32) as u32,
        (r[1] & 0xFFFFFFFF) as u32,
        (r[1] >> 32) as u32,
        (r[2] & 0xFFFFFFFF) as u32,
        (r[2] >> 32) as u32,
        (r[3] & 0xFFFFFFFF) as u32,
        (r[3] >> 32) as u32,
    ])
}

const fn q1_times_mu_shift_five(q1: &[u64; 5]) -> [u64; 5] {
    // Schoolbook multiplication.

    let (_w0, carry) = mac(0, q1[0], MU[0], 0);
    let (w1, carry) = mac(0, q1[0], MU[1], carry);
    let (w2, carry) = mac(0, q1[0], MU[2], carry);
    let (w3, carry) = mac(0, q1[0], MU[3], carry);
    let (w4, w5) = mac(0, q1[0], MU[4], carry);

    let (_w1, carry) = mac(w1, q1[1], MU[0], 0);
    let (w2, carry) = mac(w2, q1[1], MU[1], carry);
    let (w3, carry) = mac(w3, q1[1], MU[2], carry);
    let (w4, carry) = mac(w4, q1[1], MU[3], carry);
    let (w5, w6) = mac(w5, q1[1], MU[4], carry);

    let (_w2, carry) = mac(w2, q1[2], MU[0], 0);
    let (w3, carry) = mac(w3, q1[2], MU[1], carry);
    let (w4, carry) = mac(w4, q1[2], MU[2], carry);
    let (w5, carry) = mac(w5, q1[2], MU[3], carry);
    let (w6, w7) = mac(w6, q1[2], MU[4], carry);

    let (_w3, carry) = mac(w3, q1[3], MU[0], 0);
    let (w4, carry) = mac(w4, q1[3], MU[1], carry);
    let (w5, carry) = mac(w5, q1[3], MU[2], carry);
    let (w6, carry) = mac(w6, q1[3], MU[3], carry);
    let (w7, w8) = mac(w7, q1[3], MU[4], carry);

    let (_w4, carry) = mac(w4, q1[4], MU[0], 0);
    let (w5, carry) = mac(w5, q1[4], MU[1], carry);
    let (w6, carry) = mac(w6, q1[4], MU[2], carry);
    let (w7, carry) = mac(w7, q1[4], MU[3], carry);
    let (w8, w9) = mac(w8, q1[4], MU[4], carry);

    // let q2 = [_w0, _w1, _w2, _w3, _w4, w5, w6, w7, w8, w9];
    [w5, w6, w7, w8, w9]
}

const fn q3_times_n_keep_five(q3: &[u64; 5]) -> [u64; 5] {
    // Schoolbook multiplication.

    let modulus = u256_to_u64x4(MODULUS);

    let (w0, carry) = mac(0, q3[0], modulus[0], 0);
    let (w1, carry) = mac(0, q3[0], modulus[1], carry);
    let (w2, carry) = mac(0, q3[0], modulus[2], carry);
    let (w3, carry) = mac(0, q3[0], modulus[3], carry);
    let (w4, _) = mac(0, q3[0], 0, carry);

    let (w1, carry) = mac(w1, q3[1], modulus[0], 0);
    let (w2, carry) = mac(w2, q3[1], modulus[1], carry);
    let (w3, carry) = mac(w3, q3[1], modulus[2], carry);
    let (w4, _) = mac(w4, q3[1], modulus[3], carry);

    let (w2, carry) = mac(w2, q3[2], modulus[0], 0);
    let (w3, carry) = mac(w3, q3[2], modulus[1], carry);
    let (w4, _) = mac(w4, q3[2], modulus[2], carry);

    let (w3, carry) = mac(w3, q3[3], modulus[0], 0);
    let (w4, _) = mac(w4, q3[3], modulus[1], carry);

    let (w4, _) = mac(w4, q3[4], modulus[0], 0);

    [w0, w1, w2, w3, w4]
}

#[inline]
#[allow(clippy::too_many_arguments)]
const fn sub_inner_five(l: [u64; 5], r: [u64; 5]) -> [u64; 5] {
    let (w0, borrow) = sbb(l[0], r[0], 0);
    let (w1, borrow) = sbb(l[1], r[1], borrow);
    let (w2, borrow) = sbb(l[2], r[2], borrow);
    let (w3, borrow) = sbb(l[3], r[3], borrow);
    let (w4, _borrow) = sbb(l[4], r[4], borrow);

    // If underflow occurred on the final limb - don't care (= add b^{k+1}).
    [w0, w1, w2, w3, w4]
}

#[inline]
#[allow(clippy::too_many_arguments)]
const fn subtract_n_if_necessary(r0: u64, r1: u64, r2: u64, r3: u64, r4: u64) -> [u64; 5] {
    let modulus = u256_to_u64x4(MODULUS);

    let (w0, borrow) = sbb(r0, modulus[0], 0);
    let (w1, borrow) = sbb(r1, modulus[1], borrow);
    let (w2, borrow) = sbb(r2, modulus[2], borrow);
    let (w3, borrow) = sbb(r3, modulus[3], borrow);
    let (w4, borrow) = sbb(r4, 0, borrow);

    // If underflow occurred on the final limb, borrow = 0xfff...fff, otherwise
    // borrow = 0x000...000. Thus, we use it as a mask to conditionally add the
    // modulus.
    let (w0, carry) = adc(w0, modulus[0] & borrow, 0);
    let (w1, carry) = adc(w1, modulus[1] & borrow, carry);
    let (w2, carry) = adc(w2, modulus[2] & borrow, carry);
    let (w3, carry) = adc(w3, modulus[3] & borrow, carry);
    let (w4, _carry) = adc(w4, 0, carry);

    [w0, w1, w2, w3, w4]
}

// TODO(tarcieri): replace this with proper 32-bit arithmetic
#[inline]
const fn u256_to_u64x4(u256: U256) -> [u64; 4] {
    let words = u256.as_words();

    [
        (words[0] as u64) | ((words[1] as u64) << 32),
        (words[2] as u64) | ((words[3] as u64) << 32),
        (words[4] as u64) | ((words[5] as u64) << 32),
        (words[6] as u64) | ((words[7] as u64) << 32),
    ]
}