vector/avx2/field.rs

// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2021 isis lovecruft
// Copyright (c) 2016-2019 Henry de Valence
// See LICENSE for licensing information.
//
// Authors:
// - isis agora lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>

//! An implementation of 4-way vectorized 32bit field arithmetic using
//! AVX2.
//!
//! The `FieldElement2625x4` struct provides a vector of four field
//! elements, implemented using AVX2 operations.  Its API is designed
//! to abstract away the platform-dependent details, so that point
//! arithmetic can be implemented only in terms of a vector of field
//! elements.
//!
//! At this level, the API is optimized for speed and not safety.  The
//! `FieldElement2625x4` does not always perform reductions.  The pre-
//! and post-conditions on the bounds of the coefficients are
//! documented for each method, but it is the caller's responsibility
//! to ensure that there are no overflows.

#![allow(non_snake_case)]

const A_LANES: u8 = 0b0000_0101;
const B_LANES: u8 = 0b0000_1010;
const C_LANES: u8 = 0b0101_0000;
const D_LANES: u8 = 0b1010_0000;

#[allow(unused)]
const A_LANES64: u8 = 0b00_00_00_11;
#[allow(unused)]
const B_LANES64: u8 = 0b00_00_11_00;
#[allow(unused)]
const C_LANES64: u8 = 0b00_11_00_00;
#[allow(unused)]
const D_LANES64: u8 = 0b11_00_00_00;

use core::ops::{Add, Mul, Neg};
use packed_simd::{i32x8, u32x8, u64x4, IntoBits};

use backend::vector::avx2::constants::{P_TIMES_16_HI, P_TIMES_16_LO, P_TIMES_2_HI, P_TIMES_2_LO};
use backend::serial::u64::field::FieldElement51;

/// Unpack 32-bit lanes into 64-bit lanes:
/// ```ascii,no_run
/// (a0, b0, a1, b1, c0, d0, c1, d1)
/// ```
/// into
/// ```ascii,no_run
/// (a0, 0, b0, 0, c0, 0, d0, 0)
/// (a1, 0, b1, 0, c1, 0, d1, 0)
/// ```
#[inline(always)]
fn unpack_pair(src: u32x8) -> (u32x8, u32x8) {
    let a: u32x8;
    let b: u32x8;
    let zero = i32x8::new(0, 0, 0, 0, 0, 0, 0, 0);
    unsafe {
        use core::arch::x86_64::_mm256_unpackhi_epi32;
        use core::arch::x86_64::_mm256_unpacklo_epi32;
        a = _mm256_unpacklo_epi32(src.into_bits(), zero.into_bits()).into_bits();
        b = _mm256_unpackhi_epi32(src.into_bits(), zero.into_bits()).into_bits();
    }
    (a, b)
}

/// Repack 64-bit lanes into 32-bit lanes:
/// ```ascii,no_run
/// (a0, 0, b0, 0, c0, 0, d0, 0)
/// (a1, 0, b1, 0, c1, 0, d1, 0)
/// ```
/// into
/// ```ascii,no_run
/// (a0, b0, a1, b1, c0, d0, c1, d1)
/// ```
#[inline(always)]
fn repack_pair(x: u32x8, y: u32x8) -> u32x8 {
    unsafe {
        use core::arch::x86_64::_mm256_blend_epi32;
        use core::arch::x86_64::_mm256_shuffle_epi32;

        // Input: x = (a0, 0, b0, 0, c0, 0, d0, 0)
        // Input: y = (a1, 0, b1, 0, c1, 0, d1, 0)

        let x_shuffled = _mm256_shuffle_epi32(x.into_bits(), 0b11_01_10_00);
        let y_shuffled = _mm256_shuffle_epi32(y.into_bits(), 0b10_00_11_01);

        // x' = (a0, b0,  0,  0, c0, d0,  0,  0)
        // y' = ( 0,  0, a1, b1,  0,  0, c1, d1)

        return _mm256_blend_epi32(x_shuffled, y_shuffled, 0b11001100).into_bits();
    }
}

/// The `Lanes` enum represents a subset of the lanes `A,B,C,D` of a
/// `FieldElement2625x4`.
///
/// It's used to specify blend operations without
/// having to know details about the data layout of the
/// `FieldElement2625x4`.
#[derive(Copy, Clone, Debug)]
pub enum Lanes {
    C,
    D,
    AB,
    AC,
    CD,
    AD,
    BC,
    ABCD,
}

/// The `Shuffle` enum represents a shuffle of a `FieldElement2625x4`.
///
/// The enum variants are named by what they do to a vector \\(
/// (A,B,C,D) \\); for instance, `Shuffle::BADC` turns \\( (A, B, C,
/// D) \\) into \\( (B, A, D, C) \\).
#[derive(Copy, Clone, Debug)]
pub enum Shuffle {
    AAAA,
    BBBB,
    CACA,
    DBBD,
    ADDA,
    CBCB,
    ABAB,
    BADC,
    BACD,
    ABDC,
}

/// A vector of four field elements.
///
/// Each operation on a `FieldElement2625x4` has documented effects on
/// the bounds of the coefficients.  This API is designed for speed
/// and not safety; it is the caller's responsibility to ensure that
/// the post-conditions of one operation are compatible with the
/// pre-conditions of the next.
#[derive(Clone, Copy, Debug)]
pub struct FieldElement2625x4(pub(crate) [u32x8; 5]);

use subtle::Choice;
use subtle::ConditionallySelectable;

impl ConditionallySelectable for FieldElement2625x4 {
    fn conditional_select(
        a: &FieldElement2625x4,
        b: &FieldElement2625x4,
        choice: Choice,
    ) -> FieldElement2625x4 {
        let mask = (-(choice.unwrap_u8() as i32)) as u32;
        let mask_vec = u32x8::splat(mask);
        FieldElement2625x4([
            a.0[0] ^ (mask_vec & (a.0[0] ^ b.0[0])),
            a.0[1] ^ (mask_vec & (a.0[1] ^ b.0[1])),
            a.0[2] ^ (mask_vec & (a.0[2] ^ b.0[2])),
            a.0[3] ^ (mask_vec & (a.0[3] ^ b.0[3])),
            a.0[4] ^ (mask_vec & (a.0[4] ^ b.0[4])),
        ])
    }

    fn conditional_assign(
        &mut self,
        other: &FieldElement2625x4,
        choice: Choice,
    ) {
        let mask = (-(choice.unwrap_u8() as i32)) as u32;
        let mask_vec = u32x8::splat(mask);
        self.0[0] ^= mask_vec & (self.0[0] ^ other.0[0]);
        self.0[1] ^= mask_vec & (self.0[1] ^ other.0[1]);
        self.0[2] ^= mask_vec & (self.0[2] ^ other.0[2]);
        self.0[3] ^= mask_vec & (self.0[3] ^ other.0[3]);
        self.0[4] ^= mask_vec & (self.0[4] ^ other.0[4]);
    }
}

impl FieldElement2625x4 {
    /// Split this vector into an array of four (serial) field
    /// elements.
    pub fn split(&self) -> [FieldElement51; 4] {
        let mut out = [FieldElement51::zero(); 4];
        for i in 0..5 {
            let a_2i   = self.0[i].extract(0) as u64; //
            let b_2i   = self.0[i].extract(1) as u64; //
            let a_2i_1 = self.0[i].extract(2) as u64; // `.
            let b_2i_1 = self.0[i].extract(3) as u64; //  | pre-swapped to avoid
            let c_2i   = self.0[i].extract(4) as u64; //  | a cross lane shuffle
            let d_2i   = self.0[i].extract(5) as u64; // .'
            let c_2i_1 = self.0[i].extract(6) as u64; //
            let d_2i_1 = self.0[i].extract(7) as u64; //

            out[0].0[i] = a_2i + (a_2i_1 << 26);
            out[1].0[i] = b_2i + (b_2i_1 << 26);
            out[2].0[i] = c_2i + (c_2i_1 << 26);
            out[3].0[i] = d_2i + (d_2i_1 << 26);
        }

        out
    }

    /// Rearrange the elements of this vector according to `control`.
    ///
    /// The `control` parameter should be a compile-time constant, so
    /// that when this function is inlined, LLVM is able to lower the
    /// shuffle using an immediate.
    #[inline]
    pub fn shuffle(&self, control: Shuffle) -> FieldElement2625x4 {
        #[inline(always)]
        fn shuffle_lanes(x: u32x8, control: Shuffle) -> u32x8 {
            unsafe {
                use core::arch::x86_64::_mm256_permutevar8x32_epi32;

                let c: u32x8 = match control {
                    Shuffle::AAAA => u32x8::new(0, 0, 2, 2, 0, 0, 2, 2),
                    Shuffle::BBBB => u32x8::new(1, 1, 3, 3, 1, 1, 3, 3),
                    Shuffle::CACA => u32x8::new(4, 0, 6, 2, 4, 0, 6, 2),
                    Shuffle::DBBD => u32x8::new(5, 1, 7, 3, 1, 5, 3, 7),
                    Shuffle::ADDA => u32x8::new(0, 5, 2, 7, 5, 0, 7, 2),
                    Shuffle::CBCB => u32x8::new(4, 1, 6, 3, 4, 1, 6, 3),
                    Shuffle::ABAB => u32x8::new(0, 1, 2, 3, 0, 1, 2, 3),
                    Shuffle::BADC => u32x8::new(1, 0, 3, 2, 5, 4, 7, 6),
                    Shuffle::BACD => u32x8::new(1, 0, 3, 2, 4, 5, 6, 7),
                    Shuffle::ABDC => u32x8::new(0, 1, 2, 3, 5, 4, 7, 6),
                };
                // Note that this gets turned into a generic LLVM
                // shuffle-by-constants, which can be lowered to a simpler
                // instruction than a generic permute.
                _mm256_permutevar8x32_epi32(x.into_bits(), c.into_bits()).into_bits()
            }
        }

        FieldElement2625x4([
            shuffle_lanes(self.0[0], control),
            shuffle_lanes(self.0[1], control),
            shuffle_lanes(self.0[2], control),
            shuffle_lanes(self.0[3], control),
            shuffle_lanes(self.0[4], control),
        ])
    }

    /// Blend `self` with `other`, taking lanes specified in `control` from `other`.
    ///
    /// The `control` parameter should be a compile-time constant, so
    /// that this function can be inlined and LLVM can lower it to a
    /// blend instruction using an immediate.
    #[inline]
    pub fn blend(&self, other: FieldElement2625x4, control: Lanes) -> FieldElement2625x4 {
        #[inline(always)]
        fn blend_lanes(x: u32x8, y: u32x8, control: Lanes) -> u32x8 {
            unsafe {
                use core::arch::x86_64::_mm256_blend_epi32;

                // This would be much cleaner if we could factor out the match
                // statement on the control. Unfortunately, rustc forgets
                // constant-info very quickly, so we can't even write
                // ```
                // match control {
                //     Lanes::C => {
                //         let imm = C_LANES as i32;
                //         _mm256_blend_epi32(..., imm)
                // ```
                // let alone
                // ```
                // let imm = match control {
                //     Lanes::C => C_LANES as i32,
                // }
                // _mm256_blend_epi32(..., imm)
                // ```
                // even though both of these would be constant-folded by LLVM
                // at a lower level (as happens in the shuffle implementation,
                // which does not require a shuffle immediate but *is* lowered
                // to immediate shuffles anyways).
                match control {
                    Lanes::C => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), C_LANES as i32).into_bits()
                    }
                    Lanes::D => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), D_LANES as i32).into_bits()
                    }
                    Lanes::AD => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), (A_LANES | D_LANES) as i32)
                            .into_bits()
                    }
                    Lanes::AB => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), (A_LANES | B_LANES) as i32)
                            .into_bits()
                    }
                    Lanes::AC => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), (A_LANES | C_LANES) as i32)
                            .into_bits()
                    }
                    Lanes::CD => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), (C_LANES | D_LANES) as i32)
                            .into_bits()
                    }
                    Lanes::BC => {
                        _mm256_blend_epi32(x.into_bits(), y.into_bits(), (B_LANES | C_LANES) as i32)
                            .into_bits()
                    }
                    Lanes::ABCD => _mm256_blend_epi32(
                        x.into_bits(),
                        y.into_bits(),
                        (A_LANES | B_LANES | C_LANES | D_LANES) as i32,
                    ).into_bits(),
                }
            }
        }

        FieldElement2625x4([
            blend_lanes(self.0[0], other.0[0], control),
            blend_lanes(self.0[1], other.0[1], control),
            blend_lanes(self.0[2], other.0[2], control),
            blend_lanes(self.0[3], other.0[3], control),
            blend_lanes(self.0[4], other.0[4], control),
        ])
    }

    /// Construct a vector of zeros.
    pub fn zero() -> FieldElement2625x4 {
        FieldElement2625x4([u32x8::splat(0); 5])
    }

    /// Convenience wrapper around `new(x,x,x,x)`.
    pub fn splat(x: &FieldElement51) -> FieldElement2625x4 {
        FieldElement2625x4::new(x, x, x, x)
    }

    /// Create a `FieldElement2625x4` from four `FieldElement51`s.
    ///
    /// # Postconditions
    ///
    /// The resulting `FieldElement2625x4` is bounded with \\( b < 0.0002 \\).
    pub fn new(
        x0: &FieldElement51,
        x1: &FieldElement51,
        x2: &FieldElement51,
        x3: &FieldElement51,
    ) -> FieldElement2625x4 {
        let mut buf = [u32x8::splat(0); 5];
        let low_26_bits = (1 << 26) - 1;
        for i in 0..5 {
            let a_2i   = (x0.0[i] & low_26_bits) as u32;
            let a_2i_1 = (x0.0[i] >> 26) as u32;
            let b_2i   = (x1.0[i] & low_26_bits) as u32;
            let b_2i_1 = (x1.0[i] >> 26) as u32;
            let c_2i   = (x2.0[i] & low_26_bits) as u32;
            let c_2i_1 = (x2.0[i] >> 26) as u32;
            let d_2i   = (x3.0[i] & low_26_bits) as u32;
            let d_2i_1 = (x3.0[i] >> 26) as u32;

            buf[i] = u32x8::new(a_2i, b_2i, a_2i_1, b_2i_1, c_2i, d_2i, c_2i_1, d_2i_1);
        }

        // We don't know that the original `FieldElement51`s were
        // fully reduced, so the odd limbs may exceed 2^25.
        // Reduce them to be sure.
        FieldElement2625x4(buf).reduce()
    }

    /// Given \\((A,B,C,D)\\), compute \\((-A,-B,-C,-D)\\), without
    /// performing a reduction.
    ///
    /// # Preconditions
    ///
    /// The coefficients of `self` must be bounded with \\( b < 0.999 \\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 1 \\).
    #[inline]
    pub fn negate_lazy(&self) -> FieldElement2625x4 {
        // The limbs of self are bounded with b < 0.999, while the
        // smallest limb of 2*p is 67108845 > 2^{26+0.9999}, so
        // underflows are not possible.
        FieldElement2625x4([
            P_TIMES_2_LO - self.0[0],
            P_TIMES_2_HI - self.0[1],
            P_TIMES_2_HI - self.0[2],
            P_TIMES_2_HI - self.0[3],
            P_TIMES_2_HI - self.0[4],
        ])
    }

    /// Given `self = (A,B,C,D)`, compute `(B - A, B + A, D - C, D + C)`.
    ///
    /// # Preconditions
    ///
    /// The coefficients of `self` must be bounded with \\( b < 0.01 \\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 1.6 \\).
    #[inline]
    pub fn diff_sum(&self) -> FieldElement2625x4 {
        // tmp1 = (B, A, D, C)
        let tmp1 = self.shuffle(Shuffle::BADC);
        // tmp2 = (-A, B, -C, D)
        let tmp2 = self.blend(self.negate_lazy(), Lanes::AC);
        // (B - A, B + A, D - C, D + C) bounded with b < 1.6
        tmp1 + tmp2
    }

    /// Reduce this vector of field elements \\(\mathrm{mod} p\\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.0002 \\).
    #[inline]
    pub fn reduce(&self) -> FieldElement2625x4 {
        let shifts = i32x8::new(26, 26, 25, 25, 26, 26, 25, 25);
        let masks = u32x8::new(
            (1 << 26) - 1,
            (1 << 26) - 1,
            (1 << 25) - 1,
            (1 << 25) - 1,
            (1 << 26) - 1,
            (1 << 26) - 1,
            (1 << 25) - 1,
            (1 << 25) - 1,
        );

        // Let c(x) denote the carryout of the coefficient x.
        //
        // Given    (   x0,    y0,    x1,    y1,    z0,    w0,    z1,    w1),
        // compute  (c(x1), c(y1), c(x0), c(y0), c(z1), c(w1), c(z0), c(w0)).
        //
        // The carryouts are bounded by 2^(32 - 25) = 2^7.
        let rotated_carryout = |v: u32x8| -> u32x8 {
            unsafe {
                use core::arch::x86_64::_mm256_srlv_epi32;
                use core::arch::x86_64::_mm256_shuffle_epi32;

                let c = _mm256_srlv_epi32(v.into_bits(), shifts.into_bits());
                _mm256_shuffle_epi32(c, 0b01_00_11_10).into_bits()
            }
        };

        // Combine (lo, lo, lo, lo, lo, lo, lo, lo)
        //    with (hi, hi, hi, hi, hi, hi, hi, hi)
        //      to (lo, lo, hi, hi, lo, lo, hi, hi)
        //
        // This allows combining carryouts, e.g.,
        //
        // lo  (c(x1), c(y1), c(x0), c(y0), c(z1), c(w1), c(z0), c(w0))
        // hi  (c(x3), c(y3), c(x2), c(y2), c(z3), c(w3), c(z2), c(w2))
        // ->  (c(x1), c(y1), c(x2), c(y2), c(z1), c(w1), c(z2), c(w2))
        //
        // which is exactly the vector of carryins for
        //
        //     (   x2,    y2,    x3,    y3,    z2,    w2,    z3,    w3).
        //
        let combine = |v_lo: u32x8, v_hi: u32x8| -> u32x8 {
            unsafe {
                use core::arch::x86_64::_mm256_blend_epi32;
                _mm256_blend_epi32(v_lo.into_bits(), v_hi.into_bits(), 0b11_00_11_00).into_bits()
            }
        };

        let mut v = self.0;

        let c10 = rotated_carryout(v[0]);
        v[0] = (v[0] & masks) + combine(u32x8::splat(0), c10);

        let c32 = rotated_carryout(v[1]);
        v[1] = (v[1] & masks) + combine(c10, c32);

        let c54 = rotated_carryout(v[2]);
        v[2] = (v[2] & masks) + combine(c32, c54);

        let c76 = rotated_carryout(v[3]);
        v[3] = (v[3] & masks) + combine(c54, c76);

        let c98 = rotated_carryout(v[4]);
        v[4] = (v[4] & masks) + combine(c76, c98);

        let c9_19: u32x8 = unsafe {
            use core::arch::x86_64::_mm256_mul_epu32;
            use core::arch::x86_64::_mm256_shuffle_epi32;

            // Need to rearrange c98, since vpmuludq uses the low
            // 32-bits of each 64-bit lane to compute the product:
            //
            // c98       = (c(x9), c(y9), c(x8), c(y8), c(z9), c(w9), c(z8), c(w8));
            // c9_spread = (c(x9), c(x8), c(y9), c(y8), c(z9), c(z8), c(w9), c(w8)).
            let c9_spread = _mm256_shuffle_epi32(c98.into_bits(), 0b11_01_10_00);

            // Since the carryouts are bounded by 2^7, their products with 19
            // are bounded by 2^11.25.  This means that
            //
            // c9_19_spread = (19*c(x9), 0, 19*c(y9), 0, 19*c(z9), 0, 19*c(w9), 0).
            let c9_19_spread = _mm256_mul_epu32(c9_spread, u64x4::splat(19).into_bits());

            // Unshuffle:
            // c9_19 = (19*c(x9), 19*c(y9), 0, 0, 19*c(z9), 19*c(w9), 0, 0).
            _mm256_shuffle_epi32(c9_19_spread, 0b11_01_10_00).into_bits()
        };

        // Add the final carryin.
        v[0] = v[0] + c9_19;

        // Each output coefficient has exactly one carryin, which is
        // bounded by 2^11.25, so they are bounded as
        //
        // c_even < 2^26 + 2^11.25 < 26.00006 < 2^{26+b}
        // c_odd  < 2^25 + 2^11.25 < 25.0001  < 2^{25+b}
        //
        // where b = 0.0002.
        FieldElement2625x4(v)
    }

    /// Given an array of wide coefficients, reduce them to a `FieldElement2625x4`.
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.007 \\).
    #[inline]
    fn reduce64(mut z: [u64x4; 10]) -> FieldElement2625x4 {
        // These aren't const because splat isn't a const fn
        let LOW_25_BITS: u64x4 = u64x4::splat((1 << 25) - 1);
        let LOW_26_BITS: u64x4 = u64x4::splat((1 << 26) - 1);

        // Carry the value from limb i = 0..8 to limb i+1
        let carry = |z: &mut [u64x4; 10], i: usize| {
            debug_assert!(i < 9);
            if i % 2 == 0 {
                // Even limbs have 26 bits
                z[i + 1] = z[i + 1] + (z[i] >> 26);
                z[i] = z[i] & LOW_26_BITS;
            } else {
                // Odd limbs have 25 bits
                z[i + 1] = z[i + 1] + (z[i] >> 25);
                z[i] = z[i] & LOW_25_BITS;
            }
        };

        // Perform two halves of the carry chain in parallel.
        carry(&mut z, 0); carry(&mut z, 4);
        carry(&mut z, 1); carry(&mut z, 5);
        carry(&mut z, 2); carry(&mut z, 6);
        carry(&mut z, 3); carry(&mut z, 7);
        // Since z[3] < 2^64, c < 2^(64-25) = 2^39,
        // so    z[4] < 2^26 + 2^39 < 2^39.0002
        carry(&mut z, 4); carry(&mut z, 8);
        // Now z[4] < 2^26
        // and z[5] < 2^25 + 2^13.0002 < 2^25.0004 (good enough)

        // Last carry has a multiplication by 19.  In the serial case we
        // do a 64-bit multiplication by 19, but here we want to do a
        // 32-bit multiplication.  However, if we only know z[9] < 2^64,
        // the carry is bounded as c < 2^(64-25) = 2^39, which is too
        // big.  To ensure c < 2^32, we would need z[9] < 2^57.
        // Instead, we split the carry in two, with c = c_0 + c_1*2^26.

        let c = z[9] >> 25;
        z[9] = z[9] & LOW_25_BITS;
        let mut c0: u64x4 = c & LOW_26_BITS; // c0 < 2^26;
        let mut c1: u64x4 = c >> 26;         // c1 < 2^(39-26) = 2^13;

        unsafe {
            use core::arch::x86_64::_mm256_mul_epu32;
            let x19 = u64x4::splat(19);
            c0 = _mm256_mul_epu32(c0.into_bits(), x19.into_bits()).into_bits(); // c0 < 2^30.25
            c1 = _mm256_mul_epu32(c1.into_bits(), x19.into_bits()).into_bits(); // c1 < 2^17.25
        }

        z[0] = z[0] + c0; // z0 < 2^26 + 2^30.25 < 2^30.33
        z[1] = z[1] + c1; // z1 < 2^25 + 2^17.25 < 2^25.0067
        carry(&mut z, 0); // z0 < 2^26, z1 < 2^25.0067 + 2^4.33 = 2^25.007

        // The output coefficients are bounded with
        //
        // b = 0.007  for z[1]
        // b = 0.0004 for z[5]
        // b = 0      for other z[i].
        //
        // So the packed result is bounded with b = 0.007.
        FieldElement2625x4([
            repack_pair(z[0].into_bits(), z[1].into_bits()),
            repack_pair(z[2].into_bits(), z[3].into_bits()),
            repack_pair(z[4].into_bits(), z[5].into_bits()),
            repack_pair(z[6].into_bits(), z[7].into_bits()),
            repack_pair(z[8].into_bits(), z[9].into_bits()),
        ])
    }

    /// Square this field element, and negate the result's \\(D\\) value.
    ///
    /// # Preconditions
    ///
    /// The coefficients of `self` must be bounded with \\( b < 1.5 \\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.007 \\).
    pub fn square_and_negate_D(&self) -> FieldElement2625x4 {
        #[inline(always)]
        fn m(x: u32x8, y: u32x8) -> u64x4 {
            use core::arch::x86_64::_mm256_mul_epu32;
            unsafe { _mm256_mul_epu32(x.into_bits(), y.into_bits()).into_bits() }
        }

        #[inline(always)]
        fn m_lo(x: u32x8, y: u32x8) -> u32x8 {
            use core::arch::x86_64::_mm256_mul_epu32;
            unsafe { _mm256_mul_epu32(x.into_bits(), y.into_bits()).into_bits() }
        }

        let v19 = u32x8::new(19, 0, 19, 0, 19, 0, 19, 0);

        let (x0, x1) = unpack_pair(self.0[0]);
        let (x2, x3) = unpack_pair(self.0[1]);
        let (x4, x5) = unpack_pair(self.0[2]);
        let (x6, x7) = unpack_pair(self.0[3]);
        let (x8, x9) = unpack_pair(self.0[4]);

        let x0_2   = x0 << 1;
        let x1_2   = x1 << 1;
        let x2_2   = x2 << 1;
        let x3_2   = x3 << 1;
        let x4_2   = x4 << 1;
        let x5_2   = x5 << 1;
        let x6_2   = x6 << 1;
        let x7_2   = x7 << 1;

        let x5_19  = m_lo(v19, x5);
        let x6_19  = m_lo(v19, x6);
        let x7_19  = m_lo(v19, x7);
        let x8_19  = m_lo(v19, x8);
        let x9_19  = m_lo(v19, x9);

        let mut z0 = m(x0,  x0) + m(x2_2,x8_19) + m(x4_2,x6_19) + ((m(x1_2,x9_19) +  m(x3_2,x7_19) +    m(x5,x5_19)) << 1);
        let mut z1 = m(x0_2,x1) + m(x3_2,x8_19) + m(x5_2,x6_19) +                  ((m(x2,x9_19)   +    m(x4,x7_19)) << 1);
        let mut z2 = m(x0_2,x2) + m(x1_2,x1)    + m(x4_2,x8_19) + m(x6,x6_19)    + ((m(x3_2,x9_19) +  m(x5_2,x7_19)) << 1);
        let mut z3 = m(x0_2,x3) + m(x1_2,x2)    + m(x5_2,x8_19) +                  ((m(x4,x9_19)   +    m(x6,x7_19)) << 1);
        let mut z4 = m(x0_2,x4) + m(x1_2,x3_2)  + m(x2,  x2)    + m(x6_2,x8_19)  + ((m(x5_2,x9_19) +    m(x7,x7_19)) << 1);
        let mut z5 = m(x0_2,x5) + m(x1_2,x4)    + m(x2_2,x3)    + m(x7_2,x8_19)                    +  ((m(x6,x9_19)) << 1);
        let mut z6 = m(x0_2,x6) + m(x1_2,x5_2)  + m(x2_2,x4)    + m(x3_2,x3) + m(x8,x8_19)        + ((m(x7_2,x9_19)) << 1);
        let mut z7 = m(x0_2,x7) + m(x1_2,x6)    + m(x2_2,x5)    + m(x3_2,x4)                      +   ((m(x8,x9_19)) << 1);
        let mut z8 = m(x0_2,x8) + m(x1_2,x7_2)  + m(x2_2,x6)    + m(x3_2,x5_2) + m(x4,x4)         +   ((m(x9,x9_19)) << 1);
        let mut z9 = m(x0_2,x9) + m(x1_2,x8)    + m(x2_2,x7)    + m(x3_2,x6) + m(x4_2,x5);

        // The biggest z_i is bounded as z_i < 249*2^(51 + 2*b);
        // if b < 1.5 we get z_i < 4485585228861014016.
        //
        // The limbs of the multiples of p are bounded above by
        //
        // 0x3fffffff << 37 = 9223371899415822336 < 2^63
        //
        // and below by
        //
        // 0x1fffffff << 37 = 4611685880988434432
        //                  > 4485585228861014016
        //
        // So these multiples of p are big enough to avoid underflow
        // in subtraction, and small enough to fit within u64
        // with room for a carry.

        let low__p37 = u64x4::splat(0x3ffffed << 37);
        let even_p37 = u64x4::splat(0x3ffffff << 37);
        let odd__p37 = u64x4::splat(0x1ffffff << 37);

        let negate_D = |x: u64x4, p: u64x4| -> u64x4 {
            unsafe {
                use core::arch::x86_64::_mm256_blend_epi32;
                _mm256_blend_epi32(x.into_bits(), (p - x).into_bits(), D_LANES64 as i32).into_bits()
            }
        };

        z0 = negate_D(z0, low__p37);
        z1 = negate_D(z1, odd__p37);
        z2 = negate_D(z2, even_p37);
        z3 = negate_D(z3, odd__p37);
        z4 = negate_D(z4, even_p37);
        z5 = negate_D(z5, odd__p37);
        z6 = negate_D(z6, even_p37);
        z7 = negate_D(z7, odd__p37);
        z8 = negate_D(z8, even_p37);
        z9 = negate_D(z9, odd__p37);

        FieldElement2625x4::reduce64([z0, z1, z2, z3, z4, z5, z6, z7, z8, z9])
    }
}

impl Neg for FieldElement2625x4 {
    type Output = FieldElement2625x4;

    /// Negate this field element, performing a reduction.
    ///
    /// If the coefficients are known to be small, use `negate_lazy`
    /// to avoid performing a reduction.
    ///
    /// # Preconditions
    ///
    /// The coefficients of `self` must be bounded with \\( b < 4.0 \\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.0002 \\).
    #[inline]
    fn neg(self) -> FieldElement2625x4 {
        FieldElement2625x4([
            P_TIMES_16_LO - self.0[0],
            P_TIMES_16_HI - self.0[1],
            P_TIMES_16_HI - self.0[2],
            P_TIMES_16_HI - self.0[3],
            P_TIMES_16_HI - self.0[4],
        ]).reduce()
    }
}

impl Add<FieldElement2625x4> for FieldElement2625x4 {
    type Output = FieldElement2625x4;
    /// Add two `FieldElement2625x4`s, without performing a reduction.
    #[inline]
    fn add(self, rhs: FieldElement2625x4) -> FieldElement2625x4 {
        FieldElement2625x4([
            self.0[0] + rhs.0[0],
            self.0[1] + rhs.0[1],
            self.0[2] + rhs.0[2],
            self.0[3] + rhs.0[3],
            self.0[4] + rhs.0[4],
        ])
    }
}

impl Mul<(u32, u32, u32, u32)> for FieldElement2625x4 {
    type Output = FieldElement2625x4;
    /// Perform a multiplication by a vector of small constants.
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.007 \\).
    #[inline]
    fn mul(self, scalars: (u32, u32, u32, u32)) -> FieldElement2625x4 {
        unsafe {
            use core::arch::x86_64::_mm256_mul_epu32;

            let consts = u32x8::new(scalars.0, 0, scalars.1, 0, scalars.2, 0, scalars.3, 0);

            let (b0, b1) = unpack_pair(self.0[0]);
            let (b2, b3) = unpack_pair(self.0[1]);
            let (b4, b5) = unpack_pair(self.0[2]);
            let (b6, b7) = unpack_pair(self.0[3]);
            let (b8, b9) = unpack_pair(self.0[4]);

            FieldElement2625x4::reduce64([
                _mm256_mul_epu32(b0.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b1.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b2.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b3.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b4.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b5.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b6.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b7.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b8.into_bits(), consts.into_bits()).into_bits(),
                _mm256_mul_epu32(b9.into_bits(), consts.into_bits()).into_bits(),
            ])
        }
    }
}

impl<'a, 'b> Mul<&'b FieldElement2625x4> for &'a FieldElement2625x4 {
    type Output = FieldElement2625x4;
    /// Multiply `self` by `rhs`.
    ///
    /// # Preconditions
    ///
    /// The coefficients of `self` must be bounded with \\( b < 2.5 \\).
    ///
    /// The coefficients of `rhs` must be bounded with \\( b < 1.75 \\).
    ///
    /// # Postconditions
    ///
    /// The coefficients of the result are bounded with \\( b < 0.007 \\).
    ///
    fn mul(self, rhs: &'b FieldElement2625x4) -> FieldElement2625x4 {
        #[inline(always)]
        fn m(x: u32x8, y: u32x8) -> u64x4 {
            use core::arch::x86_64::_mm256_mul_epu32;
            unsafe { _mm256_mul_epu32(x.into_bits(), y.into_bits()).into_bits() }
        }

        #[inline(always)]
        fn m_lo(x: u32x8, y: u32x8) -> u32x8 {
            use core::arch::x86_64::_mm256_mul_epu32;
            unsafe { _mm256_mul_epu32(x.into_bits(), y.into_bits()).into_bits() }
        }

        let (x0, x1) = unpack_pair(self.0[0]);
        let (x2, x3) = unpack_pair(self.0[1]);
        let (x4, x5) = unpack_pair(self.0[2]);
        let (x6, x7) = unpack_pair(self.0[3]);
        let (x8, x9) = unpack_pair(self.0[4]);

        let (y0, y1) = unpack_pair(rhs.0[0]);
        let (y2, y3) = unpack_pair(rhs.0[1]);
        let (y4, y5) = unpack_pair(rhs.0[2]);
        let (y6, y7) = unpack_pair(rhs.0[3]);
        let (y8, y9) = unpack_pair(rhs.0[4]);

        let v19 = u32x8::new(19, 0, 19, 0, 19, 0, 19, 0);

        let y1_19 = m_lo(v19, y1); // This fits in a u32
        let y2_19 = m_lo(v19, y2); // iff 26 + b + lg(19) < 32
        let y3_19 = m_lo(v19, y3); // if  b < 32 - 26 - 4.248 = 1.752
        let y4_19 = m_lo(v19, y4);
        let y5_19 = m_lo(v19, y5);
        let y6_19 = m_lo(v19, y6);
        let y7_19 = m_lo(v19, y7);
        let y8_19 = m_lo(v19, y8);
        let y9_19 = m_lo(v19, y9);

        let x1_2 = x1 + x1; // This fits in a u32 iff 25 + b + 1 < 32
        let x3_2 = x3 + x3; //                    iff b < 6
        let x5_2 = x5 + x5;
        let x7_2 = x7 + x7;
        let x9_2 = x9 + x9;

        let z0 = m(x0,y0) + m(x1_2,y9_19) + m(x2,y8_19) + m(x3_2,y7_19) + m(x4,y6_19) + m(x5_2,y5_19) + m(x6,y4_19) + m(x7_2,y3_19) + m(x8,y2_19) + m(x9_2,y1_19);
        let z1 = m(x0,y1) +   m(x1,y0)    + m(x2,y9_19) +   m(x3,y8_19) + m(x4,y7_19) +   m(x5,y6_19) + m(x6,y5_19) +   m(x7,y4_19) + m(x8,y3_19) + m(x9,y2_19);
        let z2 = m(x0,y2) + m(x1_2,y1)    + m(x2,y0)    + m(x3_2,y9_19) + m(x4,y8_19) + m(x5_2,y7_19) + m(x6,y6_19) + m(x7_2,y5_19) + m(x8,y4_19) + m(x9_2,y3_19);
        let z3 = m(x0,y3) +   m(x1,y2)    + m(x2,y1)    +   m(x3,y0)    + m(x4,y9_19) +   m(x5,y8_19) + m(x6,y7_19) +   m(x7,y6_19) + m(x8,y5_19) + m(x9,y4_19);
        let z4 = m(x0,y4) + m(x1_2,y3)    + m(x2,y2)    + m(x3_2,y1)    + m(x4,y0)    + m(x5_2,y9_19) + m(x6,y8_19) + m(x7_2,y7_19) + m(x8,y6_19) + m(x9_2,y5_19);
        let z5 = m(x0,y5) +   m(x1,y4)    + m(x2,y3)    +   m(x3,y2)    + m(x4,y1)    +   m(x5,y0)    + m(x6,y9_19) +   m(x7,y8_19) + m(x8,y7_19) + m(x9,y6_19);
        let z6 = m(x0,y6) + m(x1_2,y5)    + m(x2,y4)    + m(x3_2,y3)    + m(x4,y2)    + m(x5_2,y1)    + m(x6,y0)    + m(x7_2,y9_19) + m(x8,y8_19) + m(x9_2,y7_19);
        let z7 = m(x0,y7) +   m(x1,y6)    + m(x2,y5)    +   m(x3,y4)    + m(x4,y3)    +   m(x5,y2)    + m(x6,y1)    +   m(x7,y0)    + m(x8,y9_19) + m(x9,y8_19);
        let z8 = m(x0,y8) + m(x1_2,y7)    + m(x2,y6)    + m(x3_2,y5)    + m(x4,y4)    + m(x5_2,y3)    + m(x6,y2)    + m(x7_2,y1)    + m(x8,y0)    + m(x9_2,y9_19);
        let z9 = m(x0,y9) +   m(x1,y8)    + m(x2,y7)    +   m(x3,y6)    + m(x4,y5)    +   m(x5,y4)    + m(x6,y3)    +   m(x7,y2)    + m(x8,y1)    + m(x9,y0);

        // The bounds on z[i] are the same as in the serial 32-bit code
        // and the comment below is copied from there:

        // How big is the contribution to z[i+j] from x[i], y[j]?
        //
        // Using the bounds above, we get:
        //
        // i even, j even:   x[i]*y[j] <   2^(26+b)*2^(26+b) = 2*2^(51+2*b)
        // i  odd, j even:   x[i]*y[j] <   2^(25+b)*2^(26+b) = 1*2^(51+2*b)
        // i even, j  odd:   x[i]*y[j] <   2^(26+b)*2^(25+b) = 1*2^(51+2*b)
        // i  odd, j  odd: 2*x[i]*y[j] < 2*2^(25+b)*2^(25+b) = 1*2^(51+2*b)
        //
        // We perform inline reduction mod p by replacing 2^255 by 19
        // (since 2^255 - 19 = 0 mod p).  This adds a factor of 19, so
        // we get the bounds (z0 is the biggest one, but calculated for
        // posterity here in case finer estimation is needed later):
        //
        //  z0 < ( 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 249*2^(51 + 2*b)
        //  z1 < ( 1 +  1   + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 154*2^(51 + 2*b)
        //  z2 < ( 2 +  1   +  2   + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 195*2^(51 + 2*b)
        //  z3 < ( 1 +  1   +  1   +  1   + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 118*2^(51 + 2*b)
        //  z4 < ( 2 +  1   +  2   +  1   +  2   + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 141*2^(51 + 2*b)
        //  z5 < ( 1 +  1   +  1   +  1   +  1   +  1   + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) =  82*2^(51 + 2*b)
        //  z6 < ( 2 +  1   +  2   +  1   +  2   +  1   +  2   + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) =  87*2^(51 + 2*b)
        //  z7 < ( 1 +  1   +  1   +  1   +  1   +  1   +  1   +  1   + 1*19 + 1*19 )*2^(51 + 2b) =  46*2^(51 + 2*b)
        //  z8 < ( 2 +  1   +  2   +  1   +  2   +  1   +  2   +  1   +  2   + 1*19 )*2^(51 + 2b) =  33*2^(51 + 2*b)
        //  z9 < ( 1 +  1   +  1   +  1   +  1   +  1   +  1   +  1   +  1   +  1   )*2^(51 + 2b) =  10*2^(51 + 2*b)
        //
        // So z[0] fits into a u64 if 51 + 2*b + lg(249) < 64
        //                         if b < 2.5.

        // In fact this bound is slightly sloppy, since it treats both
        // inputs x and y as being bounded by the same parameter b,
        // while they are in fact bounded by b_x and b_y, and we
        // already require that b_y < 1.75 in order to fit the
        // multiplications by 19 into a u32.  The tighter bound on b_y
        // means we could get a tighter bound on the outputs, or a
        // looser bound on b_x.
        FieldElement2625x4::reduce64([z0, z1, z2, z3, z4, z5, z6, z7, z8, z9])
    }
}

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

    #[test]
    fn scale_by_curve_constants() {
        let mut x = FieldElement2625x4::splat(&FieldElement51::one());

        x = x * (121666, 121666, 2*121666, 2*121665);

        let xs = x.split();
        assert_eq!(xs[0], FieldElement51([121666, 0, 0, 0, 0]));
        assert_eq!(xs[1], FieldElement51([121666, 0, 0, 0, 0]));
        assert_eq!(xs[2], FieldElement51([2 * 121666, 0, 0, 0, 0]));
        assert_eq!(xs[3], FieldElement51([2 * 121665, 0, 0, 0, 0]));
    }

    #[test]
    fn diff_sum_vs_serial() {
        let x0 = FieldElement51([10000, 10001, 10002, 10003, 10004]);
        let x1 = FieldElement51([10100, 10101, 10102, 10103, 10104]);
        let x2 = FieldElement51([10200, 10201, 10202, 10203, 10204]);
        let x3 = FieldElement51([10300, 10301, 10302, 10303, 10304]);

        let vec = FieldElement2625x4::new(&x0, &x1, &x2, &x3).diff_sum();

        let result = vec.split();

        assert_eq!(result[0], &x1 - &x0);
        assert_eq!(result[1], &x1 + &x0);
        assert_eq!(result[2], &x3 - &x2);
        assert_eq!(result[3], &x3 + &x2);
    }

    #[test]
    fn square_vs_serial() {
        let x0 = FieldElement51([10000, 10001, 10002, 10003, 10004]);
        let x1 = FieldElement51([10100, 10101, 10102, 10103, 10104]);
        let x2 = FieldElement51([10200, 10201, 10202, 10203, 10204]);
        let x3 = FieldElement51([10300, 10301, 10302, 10303, 10304]);

        let vec = FieldElement2625x4::new(&x0, &x1, &x2, &x3);

        let result = vec.square_and_negate_D().split();

        assert_eq!(result[0], &x0 * &x0);
        assert_eq!(result[1], &x1 * &x1);
        assert_eq!(result[2], &x2 * &x2);
        assert_eq!(result[3], -&(&x3 * &x3));
    }

    #[test]
    fn multiply_vs_serial() {
        let x0 = FieldElement51([10000, 10001, 10002, 10003, 10004]);
        let x1 = FieldElement51([10100, 10101, 10102, 10103, 10104]);
        let x2 = FieldElement51([10200, 10201, 10202, 10203, 10204]);
        let x3 = FieldElement51([10300, 10301, 10302, 10303, 10304]);

        let vec = FieldElement2625x4::new(&x0, &x1, &x2, &x3);
        let vecprime = vec.clone();

        let result = (&vec * &vecprime).split();

        assert_eq!(result[0], &x0 * &x0);
        assert_eq!(result[1], &x1 * &x1);
        assert_eq!(result[2], &x2 * &x2);
        assert_eq!(result[3], &x3 * &x3);
    }

    #[test]
    fn test_unpack_repack_pair() {
        let x0 = FieldElement51([10000 + (10001 << 26), 0, 0, 0, 0]);
        let x1 = FieldElement51([10100 + (10101 << 26), 0, 0, 0, 0]);
        let x2 = FieldElement51([10200 + (10201 << 26), 0, 0, 0, 0]);
        let x3 = FieldElement51([10300 + (10301 << 26), 0, 0, 0, 0]);

        let vec = FieldElement2625x4::new(&x0, &x1, &x2, &x3);

        let src = vec.0[0];

        let (a, b) = unpack_pair(src);

        let expected_a = u32x8::new(10000, 0, 10100, 0, 10200, 0, 10300, 0);
        let expected_b = u32x8::new(10001, 0, 10101, 0, 10201, 0, 10301, 0);

        assert_eq!(a, expected_a);
        assert_eq!(b, expected_b);

        let expected_src = repack_pair(a, b);

        assert_eq!(src, expected_src);
    }

    #[test]
    fn new_split_roundtrips() {
        let x0 = FieldElement51::from_bytes(&[0x10; 32]);
        let x1 = FieldElement51::from_bytes(&[0x11; 32]);
        let x2 = FieldElement51::from_bytes(&[0x12; 32]);
        let x3 = FieldElement51::from_bytes(&[0x13; 32]);

        let vec = FieldElement2625x4::new(&x0, &x1, &x2, &x3);

        let splits = vec.split();

        assert_eq!(x0, splits[0]);
        assert_eq!(x1, splits[1]);
        assert_eq!(x2, splits[2]);
        assert_eq!(x3, splits[3]);
    }
}