k256 0.13.4

secp256k1 elliptic curve library written in pure Rust with support for ECDSA signing/verification/public-key recovery, Taproot Schnorr signatures (BIP340), Elliptic Curve Diffie-Hellman (ECDH), and general-purpose secp256k1 elliptic curve group operations which can be used to implement arbitrary protocols
Documentation 
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//! Wide scalar (64-bit limbs)

use super::{Scalar, MODULUS};
use crate::ORDER;
use elliptic_curve::{
    bigint::{Limb, U256, U512},
    subtle::{Choice, ConditionallySelectable},
};

/// Limbs of 2^256 minus the secp256k1 order.
const NEG_MODULUS: [u64; 4] = [!MODULUS[0] + 1, !MODULUS[1], !MODULUS[2], !MODULUS[3]];

#[derive(Clone, Copy, Debug, Default)]
pub(crate) struct WideScalar(pub(super) U512);

impl WideScalar {
    pub const fn from_bytes(bytes: &[u8; 64]) -> Self {
        Self(U512::from_be_slice(bytes))
    }

    /// Multiplies two scalars without modulo reduction, producing up to a 512-bit scalar.
    #[inline(always)] // only used in Scalar::mul(), so won't cause binary bloat
    pub fn mul_wide(a: &Scalar, b: &Scalar) -> Self {
        let a = a.0.to_words();
        let b = b.0.to_words();

        // 160 bit accumulator.
        let c0 = 0;
        let c1 = 0;
        let c2 = 0;

        // l[0..7] = a[0..3] * b[0..3].
        let (c0, c1) = muladd_fast(a[0], b[0], c0, c1);
        let (l0, c0, c1) = (c0, c1, 0);
        let (c0, c1, c2) = muladd(a[0], b[1], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[1], b[0], c0, c1, c2);
        let (l1, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = muladd(a[0], b[2], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[1], b[1], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[2], b[0], c0, c1, c2);
        let (l2, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = muladd(a[0], b[3], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[1], b[2], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[2], b[1], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[3], b[0], c0, c1, c2);
        let (l3, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = muladd(a[1], b[3], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[2], b[2], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[3], b[1], c0, c1, c2);
        let (l4, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = muladd(a[2], b[3], c0, c1, c2);
        let (c0, c1, c2) = muladd(a[3], b[2], c0, c1, c2);
        let (l5, c0, c1, _c2) = (c0, c1, c2, 0);
        let (c0, c1) = muladd_fast(a[3], b[3], c0, c1);
        let (l6, c0, _c1) = (c0, c1, 0);
        let l7 = c0;

        Self(U512::from_words([l0, l1, l2, l3, l4, l5, l6, l7]))
    }

    /// Multiplies `a` by `b` (without modulo reduction) divide the result by `2^shift`
    /// (rounding to the nearest integer).
    /// Variable time in `shift`.
    pub(crate) fn mul_shift_vartime(a: &Scalar, b: &Scalar, shift: usize) -> Scalar {
        debug_assert!(shift >= 256);

        let l = Self::mul_wide(a, b).0.to_words();
        let shiftlimbs = shift >> 6;
        let shiftlow = shift & 0x3F;
        let shifthigh = 64 - shiftlow;

        let r0 = if shift < 512 {
            let lo = l[shiftlimbs] >> shiftlow;
            let hi = if shift < 448 && shiftlow != 0 {
                l[1 + shiftlimbs] << shifthigh
            } else {
                0
            };
            hi | lo
        } else {
            0
        };

        let r1 = if shift < 448 {
            let lo = l[1 + shiftlimbs] >> shiftlow;
            let hi = if shift < 384 && shiftlow != 0 {
                l[2 + shiftlimbs] << shifthigh
            } else {
                0
            };
            hi | lo
        } else {
            0
        };

        let r2 = if shift < 384 {
            let lo = l[2 + shiftlimbs] >> shiftlow;
            let hi = if shift < 320 && shiftlow != 0 {
                l[3 + shiftlimbs] << shifthigh
            } else {
                0
            };
            hi | lo
        } else {
            0
        };

        let r3 = if shift < 320 {
            l[3 + shiftlimbs] >> shiftlow
        } else {
            0
        };

        let res = Scalar(U256::from_words([r0, r1, r2, r3]));

        // Check the highmost discarded bit and round up if it is set.
        let c = (l[(shift - 1) >> 6] >> ((shift - 1) & 0x3f)) & 1;
        Scalar::conditional_select(&res, &res.add(&Scalar::ONE), Choice::from(c as u8))
    }

    fn reduce_impl(&self, modulus_minus_one: bool) -> Scalar {
        let neg_modulus0 = if modulus_minus_one {
            NEG_MODULUS[0] + 1
        } else {
            NEG_MODULUS[0]
        };
        let modulus = if modulus_minus_one {
            ORDER.wrapping_sub(&U256::ONE)
        } else {
            ORDER
        };

        let w = self.0.to_words();
        let n0 = w[4];
        let n1 = w[5];
        let n2 = w[6];
        let n3 = w[7];

        // Reduce 512 bits into 385.
        // m[0..6] = self[0..3] + n[0..3] * neg_modulus.
        let c0 = w[0];
        let c1 = 0;
        let c2 = 0;
        let (c0, c1) = muladd_fast(n0, neg_modulus0, c0, c1);
        let (m0, c0, c1) = (c0, c1, 0);
        let (c0, c1) = sumadd_fast(w[1], c0, c1);
        let (c0, c1, c2) = muladd(n1, neg_modulus0, c0, c1, c2);
        let (c0, c1, c2) = muladd(n0, NEG_MODULUS[1], c0, c1, c2);
        let (m1, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = sumadd(w[2], c0, c1, c2);
        let (c0, c1, c2) = muladd(n2, neg_modulus0, c0, c1, c2);
        let (c0, c1, c2) = muladd(n1, NEG_MODULUS[1], c0, c1, c2);
        let (c0, c1, c2) = sumadd(n0, c0, c1, c2);
        let (m2, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = sumadd(w[3], c0, c1, c2);
        let (c0, c1, c2) = muladd(n3, neg_modulus0, c0, c1, c2);
        let (c0, c1, c2) = muladd(n2, NEG_MODULUS[1], c0, c1, c2);
        let (c0, c1, c2) = sumadd(n1, c0, c1, c2);
        let (m3, c0, c1, c2) = (c0, c1, c2, 0);
        let (c0, c1, c2) = muladd(n3, NEG_MODULUS[1], c0, c1, c2);
        let (c0, c1, c2) = sumadd(n2, c0, c1, c2);
        let (m4, c0, c1, _c2) = (c0, c1, c2, 0);
        let (c0, c1) = sumadd_fast(n3, c0, c1);
        let (m5, c0, _c1) = (c0, c1, 0);
        debug_assert!(c0 <= 1);
        let m6 = c0;

        // Reduce 385 bits into 258.
        // p[0..4] = m[0..3] + m[4..6] * neg_modulus.
        let c0 = m0;
        let c1 = 0;
        let c2 = 0;
        let (c0, c1) = muladd_fast(m4, neg_modulus0, c0, c1);
        let (p0, c0, c1) = (c0, c1, 0);
        let (c0, c1) = sumadd_fast(m1, c0, c1);
        let (c0, c1, c2) = muladd(m5, neg_modulus0, c0, c1, c2);
        let (c0, c1, c2) = muladd(m4, NEG_MODULUS[1], c0, c1, c2);
        let (p1, c0, c1) = (c0, c1, 0);
        let (c0, c1, c2) = sumadd(m2, c0, c1, c2);
        let (c0, c1, c2) = muladd(m6, neg_modulus0, c0, c1, c2);
        let (c0, c1, c2) = muladd(m5, NEG_MODULUS[1], c0, c1, c2);
        let (c0, c1, c2) = sumadd(m4, c0, c1, c2);
        let (p2, c0, c1, _c2) = (c0, c1, c2, 0);
        let (c0, c1) = sumadd_fast(m3, c0, c1);
        let (c0, c1) = muladd_fast(m6, NEG_MODULUS[1], c0, c1);
        let (c0, c1) = sumadd_fast(m5, c0, c1);
        let (p3, c0, _c1) = (c0, c1, 0);
        let p4 = c0 + m6;
        debug_assert!(p4 <= 2);

        // Reduce 258 bits into 256.
        // r[0..3] = p[0..3] + p[4] * neg_modulus.
        let mut c = (p0 as u128) + (neg_modulus0 as u128) * (p4 as u128);
        let r0 = (c & 0xFFFFFFFFFFFFFFFFu128) as u64;
        c >>= 64;
        c += (p1 as u128) + (NEG_MODULUS[1] as u128) * (p4 as u128);
        let r1 = (c & 0xFFFFFFFFFFFFFFFFu128) as u64;
        c >>= 64;
        c += (p2 as u128) + (p4 as u128);
        let r2 = (c & 0xFFFFFFFFFFFFFFFFu128) as u64;
        c >>= 64;
        c += p3 as u128;
        let r3 = (c & 0xFFFFFFFFFFFFFFFFu128) as u64;
        c >>= 64;

        // Final reduction of r.
        let r = U256::from([r0, r1, r2, r3]);
        let (r2, underflow) = r.sbb(&modulus, Limb::ZERO);
        let high_bit = Choice::from(c as u8);
        let underflow = Choice::from((underflow.0 >> 63) as u8);
        Scalar(U256::conditional_select(&r, &r2, !underflow | high_bit))
    }

    #[inline(always)] // only used in Scalar::mul(), so won't cause binary bloat
    pub(super) fn reduce(&self) -> Scalar {
        self.reduce_impl(false)
    }

    pub(super) fn reduce_nonzero(&self) -> Scalar {
        self.reduce_impl(true) + Scalar::ONE
    }
}

/// Add a to the number defined by (c0,c1,c2). c2 must never overflow.
fn sumadd(a: u64, c0: u64, c1: u64, c2: u64) -> (u64, u64, u64) {
    let (new_c0, carry0) = c0.overflowing_add(a);
    let (new_c1, carry1) = c1.overflowing_add(carry0 as u64);
    let new_c2 = c2 + (carry1 as u64);
    (new_c0, new_c1, new_c2)
}

/// Add a to the number defined by (c0,c1). c1 must never overflow.
fn sumadd_fast(a: u64, c0: u64, c1: u64) -> (u64, u64) {
    let (new_c0, carry0) = c0.overflowing_add(a);
    let new_c1 = c1 + (carry0 as u64);
    (new_c0, new_c1)
}

/// Add a*b to the number defined by (c0,c1,c2). c2 must never overflow.
fn muladd(a: u64, b: u64, c0: u64, c1: u64, c2: u64) -> (u64, u64, u64) {
    let t = (a as u128) * (b as u128);
    let th = (t >> 64) as u64; // at most 0xFFFFFFFFFFFFFFFE
    let tl = t as u64;

    let (new_c0, carry0) = c0.overflowing_add(tl);
    let new_th = th.wrapping_add(carry0 as u64); // at most 0xFFFFFFFFFFFFFFFF
    let (new_c1, carry1) = c1.overflowing_add(new_th);
    let new_c2 = c2 + (carry1 as u64);

    (new_c0, new_c1, new_c2)
}

/// Add a*b to the number defined by (c0,c1). c1 must never overflow.
fn muladd_fast(a: u64, b: u64, c0: u64, c1: u64) -> (u64, u64) {
    let t = (a as u128) * (b as u128);
    let th = (t >> 64) as u64; // at most 0xFFFFFFFFFFFFFFFE
    let tl = t as u64;

    let (new_c0, carry0) = c0.overflowing_add(tl);
    let new_th = th.wrapping_add(carry0 as u64); // at most 0xFFFFFFFFFFFFFFFF
    let new_c1 = c1 + new_th;

    (new_c0, new_c1)
}