openssl/crypto/ec/curve25519.c

/*
 * Copyright 2016-2020 The OpenSSL Project Authors. All Rights Reserved.
 *
 * Licensed under the Apache License 2.0 (the "License").  You may not use
 * this file except in compliance with the License.  You can obtain a copy
 * in the file LICENSE in the source distribution or at
 * https://www.openssl.org/source/license.html
 */

/*
 * ECDSA low level APIs are deprecated for public use, but still ok for
 * internal use.
 */
#include "internal/deprecated.h"

#include <string.h>
#include "crypto/ecx.h"
#include "ec_local.h"
#include <openssl/evp.h>
#include <openssl/sha.h>

#if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
                            defined(_M_AMD64) || defined(_M_X64))

# define BASE_2_64_IMPLEMENTED

typedef uint64_t fe64[4];

int x25519_fe64_eligible(void);

/*
 * Following subroutines perform corresponding operations modulo
 * 2^256-38, i.e. double the curve modulus. However, inputs and
 * outputs are permitted to be partially reduced, i.e. to remain
 * in [0..2^256) range. It's all tied up in final fe64_tobytes
 * that performs full reduction modulo 2^255-19.
 *
 * There are no reference C implementations for these.
 */
void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
void x25519_fe64_sqr(fe64 h, const fe64 f);
void x25519_fe64_mul121666(fe64 h, fe64 f);
void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
# define fe64_mul x25519_fe64_mul
# define fe64_sqr x25519_fe64_sqr
# define fe64_mul121666 x25519_fe64_mul121666
# define fe64_add x25519_fe64_add
# define fe64_sub x25519_fe64_sub
# define fe64_tobytes x25519_fe64_tobytes

static uint64_t load_8(const uint8_t *in)
{
    uint64_t result;

    result = in[0];
    result |= ((uint64_t)in[1]) << 8;
    result |= ((uint64_t)in[2]) << 16;
    result |= ((uint64_t)in[3]) << 24;
    result |= ((uint64_t)in[4]) << 32;
    result |= ((uint64_t)in[5]) << 40;
    result |= ((uint64_t)in[6]) << 48;
    result |= ((uint64_t)in[7]) << 56;

    return result;
}

static void fe64_frombytes(fe64 h, const uint8_t *s)
{
    h[0] = load_8(s);
    h[1] = load_8(s + 8);
    h[2] = load_8(s + 16);
    h[3] = load_8(s + 24) & 0x7fffffffffffffff;
}

static void fe64_0(fe64 h)
{
    h[0] = 0;
    h[1] = 0;
    h[2] = 0;
    h[3] = 0;
}

static void fe64_1(fe64 h)
{
    h[0] = 1;
    h[1] = 0;
    h[2] = 0;
    h[3] = 0;
}

static void fe64_copy(fe64 h, const fe64 f)
{
    h[0] = f[0];
    h[1] = f[1];
    h[2] = f[2];
    h[3] = f[3];
}

static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
{
    int i;
    uint64_t mask = 0 - (uint64_t)b;

    for (i = 0; i < 4; i++) {
        uint64_t x = f[i] ^ g[i];
        x &= mask;
        f[i] ^= x;
        g[i] ^= x;
    }
}

static void fe64_invert(fe64 out, const fe64 z)
{
    fe64 t0;
    fe64 t1;
    fe64 t2;
    fe64 t3;
    int i;

    /*
     * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
     * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
     */

    /* t0 = z ** 2 */
    fe64_sqr(t0, z);

    /* t1 = t0 ** (2 ** 2) = z ** 8 */
    fe64_sqr(t1, t0);
    fe64_sqr(t1, t1);

    /* t1 = z * t1 = z ** 9 */
    fe64_mul(t1, z, t1);
    /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
    fe64_mul(t0, t0, t1);

    /* t2 = t0 ** 2 = z ** 22 */
    fe64_sqr(t2, t0);

    /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
    fe64_mul(t1, t1, t2);

    /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
    fe64_sqr(t2, t1);
    for (i = 1; i < 5; ++i)
        fe64_sqr(t2, t2);

    /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
    fe64_mul(t1, t2, t1);

    /* Continuing similarly... */

    /* t2 = z ** (2 ** 20 - 1) */
    fe64_sqr(t2, t1);
    for (i = 1; i < 10; ++i)
        fe64_sqr(t2, t2);

    fe64_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 40 - 1) */
    fe64_sqr(t3, t2);
    for (i = 1; i < 20; ++i)
        fe64_sqr(t3, t3);

    fe64_mul(t2, t3, t2);

    /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
    for (i = 0; i < 10; ++i)
        fe64_sqr(t2, t2);

    /* t1 = z ** (2 ** 50 - 1) */
    fe64_mul(t1, t2, t1);

    /* t2 = z ** (2 ** 100 - 1) */
    fe64_sqr(t2, t1);
    for (i = 1; i < 50; ++i)
        fe64_sqr(t2, t2);

    fe64_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 200 - 1) */
    fe64_sqr(t3, t2);
    for (i = 1; i < 100; ++i)
        fe64_sqr(t3, t3);

    fe64_mul(t2, t3, t2);

    /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
    for (i = 0; i < 50; ++i)
        fe64_sqr(t2, t2);

    /* t1 = z ** (2 ** 250 - 1) */
    fe64_mul(t1, t2, t1);

    /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
    for (i = 0; i < 5; ++i)
        fe64_sqr(t1, t1);

    /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
    fe64_mul(out, t1, t0);
}

/*
 * Duplicate of original x25519_scalar_mult_generic, but using
 * fe64_* subroutines.
 */
static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
                               const uint8_t point[32])
{
    fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
    uint8_t e[32];
    unsigned swap = 0;
    int pos;

    memcpy(e, scalar, 32);
    e[0]  &= 0xf8;
    e[31] &= 0x7f;
    e[31] |= 0x40;
    fe64_frombytes(x1, point);
    fe64_1(x2);
    fe64_0(z2);
    fe64_copy(x3, x1);
    fe64_1(z3);

    for (pos = 254; pos >= 0; --pos) {
        unsigned int b = 1 & (e[pos / 8] >> (pos & 7));

        swap ^= b;
        fe64_cswap(x2, x3, swap);
        fe64_cswap(z2, z3, swap);
        swap = b;
        fe64_sub(tmp0, x3, z3);
        fe64_sub(tmp1, x2, z2);
        fe64_add(x2, x2, z2);
        fe64_add(z2, x3, z3);
        fe64_mul(z3, x2, tmp0);
        fe64_mul(z2, z2, tmp1);
        fe64_sqr(tmp0, tmp1);
        fe64_sqr(tmp1, x2);
        fe64_add(x3, z3, z2);
        fe64_sub(z2, z3, z2);
        fe64_mul(x2, tmp1, tmp0);
        fe64_sub(tmp1, tmp1, tmp0);
        fe64_sqr(z2, z2);
        fe64_mul121666(z3, tmp1);
        fe64_sqr(x3, x3);
        fe64_add(tmp0, tmp0, z3);
        fe64_mul(z3, x1, z2);
        fe64_mul(z2, tmp1, tmp0);
    }

    fe64_invert(z2, z2);
    fe64_mul(x2, x2, z2);
    fe64_tobytes(out, x2);

    OPENSSL_cleanse(e, sizeof(e));
}
#endif

#if defined(X25519_ASM) \
    || ( (defined(__SIZEOF_INT128__) && __SIZEOF_INT128__ == 16) \
         && !defined(__sparc__) \
         && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
         && !(defined(__ANDROID__) && !defined(__clang__)) )
/*
 * Base 2^51 implementation. It's virtually no different from reference
 * base 2^25.5 implementation in respect to lax boundary conditions for
 * intermediate values and even individual limbs. So that whatever you
 * know about the reference, applies even here...
 */
# define BASE_2_51_IMPLEMENTED

typedef uint64_t fe51[5];

static const uint64_t MASK51 = 0x7ffffffffffff;

static uint64_t load_7(const uint8_t *in)
{
    uint64_t result;

    result = in[0];
    result |= ((uint64_t)in[1]) << 8;
    result |= ((uint64_t)in[2]) << 16;
    result |= ((uint64_t)in[3]) << 24;
    result |= ((uint64_t)in[4]) << 32;
    result |= ((uint64_t)in[5]) << 40;
    result |= ((uint64_t)in[6]) << 48;

    return result;
}

static uint64_t load_6(const uint8_t *in)
{
    uint64_t result;

    result = in[0];
    result |= ((uint64_t)in[1]) << 8;
    result |= ((uint64_t)in[2]) << 16;
    result |= ((uint64_t)in[3]) << 24;
    result |= ((uint64_t)in[4]) << 32;
    result |= ((uint64_t)in[5]) << 40;

    return result;
}

static void fe51_frombytes(fe51 h, const uint8_t *s)
{
    uint64_t h0 = load_7(s);                                /* 56 bits */
    uint64_t h1 = load_6(s + 7) << 5;                       /* 53 bits */
    uint64_t h2 = load_7(s + 13) << 2;                      /* 58 bits */
    uint64_t h3 = load_6(s + 20) << 7;                      /* 55 bits */
    uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4;   /* 51 bits */

    h1 |= h0 >> 51; h0 &= MASK51;
    h2 |= h1 >> 51; h1 &= MASK51;
    h3 |= h2 >> 51; h2 &= MASK51;
    h4 |= h3 >> 51; h3 &= MASK51;

    h[0] = h0;
    h[1] = h1;
    h[2] = h2;
    h[3] = h3;
    h[4] = h4;
}

static void fe51_tobytes(uint8_t *s, const fe51 h)
{
    uint64_t h0 = h[0];
    uint64_t h1 = h[1];
    uint64_t h2 = h[2];
    uint64_t h3 = h[3];
    uint64_t h4 = h[4];
    uint64_t q;

    /* compare to modulus */
    q = (h0 + 19) >> 51;
    q = (h1 + q) >> 51;
    q = (h2 + q) >> 51;
    q = (h3 + q) >> 51;
    q = (h4 + q) >> 51;

    /* full reduce */
    h0 += 19 * q;
    h1 += h0 >> 51; h0 &= MASK51;
    h2 += h1 >> 51; h1 &= MASK51;
    h3 += h2 >> 51; h2 &= MASK51;
    h4 += h3 >> 51; h3 &= MASK51;
                    h4 &= MASK51;

    /* smash */
    s[0] = (uint8_t)(h0 >> 0);
    s[1] = (uint8_t)(h0 >> 8);
    s[2] = (uint8_t)(h0 >> 16);
    s[3] = (uint8_t)(h0 >> 24);
    s[4] = (uint8_t)(h0 >> 32);
    s[5] = (uint8_t)(h0 >> 40);
    s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
    s[7] = (uint8_t)(h1 >> 5);
    s[8] = (uint8_t)(h1 >> 13);
    s[9] = (uint8_t)(h1 >> 21);
    s[10] = (uint8_t)(h1 >> 29);
    s[11] = (uint8_t)(h1 >> 37);
    s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
    s[13] = (uint8_t)(h2 >> 2);
    s[14] = (uint8_t)(h2 >> 10);
    s[15] = (uint8_t)(h2 >> 18);
    s[16] = (uint8_t)(h2 >> 26);
    s[17] = (uint8_t)(h2 >> 34);
    s[18] = (uint8_t)(h2 >> 42);
    s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
    s[20] = (uint8_t)(h3 >> 7);
    s[21] = (uint8_t)(h3 >> 15);
    s[22] = (uint8_t)(h3 >> 23);
    s[23] = (uint8_t)(h3 >> 31);
    s[24] = (uint8_t)(h3 >> 39);
    s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
    s[26] = (uint8_t)(h4 >> 4);
    s[27] = (uint8_t)(h4 >> 12);
    s[28] = (uint8_t)(h4 >> 20);
    s[29] = (uint8_t)(h4 >> 28);
    s[30] = (uint8_t)(h4 >> 36);
    s[31] = (uint8_t)(h4 >> 44);
}

# if defined(X25519_ASM)
void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
void x25519_fe51_sqr(fe51 h, const fe51 f);
void x25519_fe51_mul121666(fe51 h, fe51 f);
#  define fe51_mul x25519_fe51_mul
#  define fe51_sq  x25519_fe51_sqr
#  define fe51_mul121666 x25519_fe51_mul121666
# else

typedef __uint128_t u128;

static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
{
    u128 h0, h1, h2, h3, h4;
    uint64_t f_i, g0, g1, g2, g3, g4;

    f_i = f[0];
    h0 = (u128)f_i * (g0 = g[0]);
    h1 = (u128)f_i * (g1 = g[1]);
    h2 = (u128)f_i * (g2 = g[2]);
    h3 = (u128)f_i * (g3 = g[3]);
    h4 = (u128)f_i * (g4 = g[4]);

    f_i = f[1];
    h0 += (u128)f_i * (g4 *= 19);
    h1 += (u128)f_i * g0;
    h2 += (u128)f_i * g1;
    h3 += (u128)f_i * g2;
    h4 += (u128)f_i * g3;

    f_i = f[2];
    h0 += (u128)f_i * (g3 *= 19);
    h1 += (u128)f_i * g4;
    h2 += (u128)f_i * g0;
    h3 += (u128)f_i * g1;
    h4 += (u128)f_i * g2;

    f_i = f[3];
    h0 += (u128)f_i * (g2 *= 19);
    h1 += (u128)f_i * g3;
    h2 += (u128)f_i * g4;
    h3 += (u128)f_i * g0;
    h4 += (u128)f_i * g1;

    f_i = f[4];
    h0 += (u128)f_i * (g1 *= 19);
    h1 += (u128)f_i * g2;
    h2 += (u128)f_i * g3;
    h3 += (u128)f_i * g4;
    h4 += (u128)f_i * g0;

    /* partial [lazy] reduction */
    h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
    h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;

    h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
    g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;

    g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
    g3 += g2 >> 51; g2 &= MASK51;
    g1 += g0 >> 51; g0 &= MASK51;

    h[0] = g0;
    h[1] = g1;
    h[2] = g2;
    h[3] = g3;
    h[4] = g4;
}

static void fe51_sq(fe51 h, const fe51 f)
{
#  if defined(OPENSSL_SMALL_FOOTPRINT)
    fe51_mul(h, f, f);
#  else
    /* dedicated squaring gives 16-25% overall improvement */
    uint64_t g0 = f[0];
    uint64_t g1 = f[1];
    uint64_t g2 = f[2];
    uint64_t g3 = f[3];
    uint64_t g4 = f[4];
    u128 h0, h1, h2, h3, h4;

    h0 = (u128)g0 * g0;     g0 *= 2;
    h1 = (u128)g0 * g1;
    h2 = (u128)g0 * g2;
    h3 = (u128)g0 * g3;
    h4 = (u128)g0 * g4;

    g0 = g4;                /* borrow g0 */
    h3 += (u128)g0 * (g4 *= 19);

    h2 += (u128)g1 * g1;    g1 *= 2;
    h3 += (u128)g1 * g2;
    h4 += (u128)g1 * g3;
    h0 += (u128)g1 * g4;

    g0 = g3;                /* borrow g0 */
    h1 += (u128)g0 * (g3 *= 19);
    h2 += (u128)(g0 * 2) * g4;

    h4 += (u128)g2 * g2;    g2 *= 2;
    h0 += (u128)g2 * g3;
    h1 += (u128)g2 * g4;

    /* partial [lazy] reduction */
    h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
    h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;

    h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
    g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;

    g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
    g3 += g2 >> 51; g2 &= MASK51;
    g1 += g0 >> 51; g0 &= MASK51;

    h[0] = g0;
    h[1] = g1;
    h[2] = g2;
    h[3] = g3;
    h[4] = g4;
#  endif
}

static void fe51_mul121666(fe51 h, fe51 f)
{
    u128 h0 = f[0] * (u128)121666;
    u128 h1 = f[1] * (u128)121666;
    u128 h2 = f[2] * (u128)121666;
    u128 h3 = f[3] * (u128)121666;
    u128 h4 = f[4] * (u128)121666;
    uint64_t g0, g1, g2, g3, g4;

    h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
    h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;

    h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
    g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;

    g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
    g3 += g2 >> 51; g2 &= MASK51;
    g1 += g0 >> 51; g0 &= MASK51;

    h[0] = g0;
    h[1] = g1;
    h[2] = g2;
    h[3] = g3;
    h[4] = g4;
}
# endif

static void fe51_add(fe51 h, const fe51 f, const fe51 g)
{
    h[0] = f[0] + g[0];
    h[1] = f[1] + g[1];
    h[2] = f[2] + g[2];
    h[3] = f[3] + g[3];
    h[4] = f[4] + g[4];
}

static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
{
    /*
     * Add 2*modulus to ensure that result remains positive
     * even if subtrahend is partially reduced.
     */
    h[0] = (f[0] + 0xfffffffffffda) - g[0];
    h[1] = (f[1] + 0xffffffffffffe) - g[1];
    h[2] = (f[2] + 0xffffffffffffe) - g[2];
    h[3] = (f[3] + 0xffffffffffffe) - g[3];
    h[4] = (f[4] + 0xffffffffffffe) - g[4];
}

static void fe51_0(fe51 h)
{
    h[0] = 0;
    h[1] = 0;
    h[2] = 0;
    h[3] = 0;
    h[4] = 0;
}

static void fe51_1(fe51 h)
{
    h[0] = 1;
    h[1] = 0;
    h[2] = 0;
    h[3] = 0;
    h[4] = 0;
}

static void fe51_copy(fe51 h, const fe51 f)
{
    h[0] = f[0];
    h[1] = f[1];
    h[2] = f[2];
    h[3] = f[3];
    h[4] = f[4];
}

static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
{
    int i;
    uint64_t mask = 0 - (uint64_t)b;

    for (i = 0; i < 5; i++) {
        int64_t x = f[i] ^ g[i];
        x &= mask;
        f[i] ^= x;
        g[i] ^= x;
    }
}

static void fe51_invert(fe51 out, const fe51 z)
{
    fe51 t0;
    fe51 t1;
    fe51 t2;
    fe51 t3;
    int i;

    /*
     * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
     * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
     */

    /* t0 = z ** 2 */
    fe51_sq(t0, z);

    /* t1 = t0 ** (2 ** 2) = z ** 8 */
    fe51_sq(t1, t0);
    fe51_sq(t1, t1);

    /* t1 = z * t1 = z ** 9 */
    fe51_mul(t1, z, t1);
    /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
    fe51_mul(t0, t0, t1);

    /* t2 = t0 ** 2 = z ** 22 */
    fe51_sq(t2, t0);

    /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
    fe51_mul(t1, t1, t2);

    /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
    fe51_sq(t2, t1);
    for (i = 1; i < 5; ++i)
        fe51_sq(t2, t2);

    /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
    fe51_mul(t1, t2, t1);

    /* Continuing similarly... */

    /* t2 = z ** (2 ** 20 - 1) */
    fe51_sq(t2, t1);
    for (i = 1; i < 10; ++i)
        fe51_sq(t2, t2);

    fe51_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 40 - 1) */
    fe51_sq(t3, t2);
    for (i = 1; i < 20; ++i)
        fe51_sq(t3, t3);

    fe51_mul(t2, t3, t2);

    /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
    for (i = 0; i < 10; ++i)
        fe51_sq(t2, t2);

    /* t1 = z ** (2 ** 50 - 1) */
    fe51_mul(t1, t2, t1);

    /* t2 = z ** (2 ** 100 - 1) */
    fe51_sq(t2, t1);
    for (i = 1; i < 50; ++i)
        fe51_sq(t2, t2);

    fe51_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 200 - 1) */
    fe51_sq(t3, t2);
    for (i = 1; i < 100; ++i)
        fe51_sq(t3, t3);

    fe51_mul(t2, t3, t2);

    /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
    for (i = 0; i < 50; ++i)
        fe51_sq(t2, t2);

    /* t1 = z ** (2 ** 250 - 1) */
    fe51_mul(t1, t2, t1);

    /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
    for (i = 0; i < 5; ++i)
        fe51_sq(t1, t1);

    /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
    fe51_mul(out, t1, t0);
}

/*
 * Duplicate of original x25519_scalar_mult_generic, but using
 * fe51_* subroutines.
 */
static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
                               const uint8_t point[32])
{
    fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
    uint8_t e[32];
    unsigned swap = 0;
    int pos;

# ifdef BASE_2_64_IMPLEMENTED
    if (x25519_fe64_eligible()) {
        x25519_scalar_mulx(out, scalar, point);
        return;
    }
# endif

    memcpy(e, scalar, 32);
    e[0]  &= 0xf8;
    e[31] &= 0x7f;
    e[31] |= 0x40;
    fe51_frombytes(x1, point);
    fe51_1(x2);
    fe51_0(z2);
    fe51_copy(x3, x1);
    fe51_1(z3);

    for (pos = 254; pos >= 0; --pos) {
        unsigned int b = 1 & (e[pos / 8] >> (pos & 7));

        swap ^= b;
        fe51_cswap(x2, x3, swap);
        fe51_cswap(z2, z3, swap);
        swap = b;
        fe51_sub(tmp0, x3, z3);
        fe51_sub(tmp1, x2, z2);
        fe51_add(x2, x2, z2);
        fe51_add(z2, x3, z3);
        fe51_mul(z3, tmp0, x2);
        fe51_mul(z2, z2, tmp1);
        fe51_sq(tmp0, tmp1);
        fe51_sq(tmp1, x2);
        fe51_add(x3, z3, z2);
        fe51_sub(z2, z3, z2);
        fe51_mul(x2, tmp1, tmp0);
        fe51_sub(tmp1, tmp1, tmp0);
        fe51_sq(z2, z2);
        fe51_mul121666(z3, tmp1);
        fe51_sq(x3, x3);
        fe51_add(tmp0, tmp0, z3);
        fe51_mul(z3, x1, z2);
        fe51_mul(z2, tmp1, tmp0);
    }

    fe51_invert(z2, z2);
    fe51_mul(x2, x2, z2);
    fe51_tobytes(out, x2);

    OPENSSL_cleanse(e, sizeof(e));
}
#endif

/*
 * Reference base 2^25.5 implementation.
 *
 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
 * 20141124 (http://bench.cr.yp.to/supercop.html).
 *
 * The field functions are shared by Ed25519 and X25519 where possible.
 */

/*
 * fe means field element. Here the field is \Z/(2^255-19). An element t,
 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
 * context.
 */
typedef int32_t fe[10];

static const int64_t kBottom21Bits =  0x1fffffLL;
static const int64_t kBottom25Bits = 0x1ffffffLL;
static const int64_t kBottom26Bits = 0x3ffffffLL;
static const int64_t kTop39Bits = 0xfffffffffe000000LL;
static const int64_t kTop38Bits = 0xfffffffffc000000LL;

static uint64_t load_3(const uint8_t *in)
{
    uint64_t result;

    result  = ((uint64_t)in[0]);
    result |= ((uint64_t)in[1]) << 8;
    result |= ((uint64_t)in[2]) << 16;
    return result;
}

static uint64_t load_4(const uint8_t *in)
{
    uint64_t result;

    result  = ((uint64_t)in[0]);
    result |= ((uint64_t)in[1]) << 8;
    result |= ((uint64_t)in[2]) << 16;
    result |= ((uint64_t)in[3]) << 24;
    return result;
}

static void fe_frombytes(fe h, const uint8_t *s)
{
    /* Ignores top bit of h. */
    int64_t h0 =  load_4(s);
    int64_t h1 =  load_3(s +  4) << 6;
    int64_t h2 =  load_3(s +  7) << 5;
    int64_t h3 =  load_3(s + 10) << 3;
    int64_t h4 =  load_3(s + 13) << 2;
    int64_t h5 =  load_4(s + 16);
    int64_t h6 =  load_3(s + 20) << 7;
    int64_t h7 =  load_3(s + 23) << 5;
    int64_t h8 =  load_3(s + 26) << 4;
    int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;

    carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
    carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
    carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
    carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
    carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
    carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

    h[0] = (int32_t)h0;
    h[1] = (int32_t)h1;
    h[2] = (int32_t)h2;
    h[3] = (int32_t)h3;
    h[4] = (int32_t)h4;
    h[5] = (int32_t)h5;
    h[6] = (int32_t)h6;
    h[7] = (int32_t)h7;
    h[8] = (int32_t)h8;
    h[9] = (int32_t)h9;
}

/*
 * Preconditions:
 *   |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 *
 * Write p=2^255-19; q=floor(h/p).
 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
 *
 * Proof:
 *   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
 *   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
 *
 *   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
 *   Then 0<y<1.
 *
 *   Write r=h-pq.
 *   Have 0<=r<=p-1=2^255-20.
 *   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
 *
 *   Write x=r+19(2^-255)r+y.
 *   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
 *
 *   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
 *   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
 */
static void fe_tobytes(uint8_t *s, const fe h)
{
    int32_t h0 = h[0];
    int32_t h1 = h[1];
    int32_t h2 = h[2];
    int32_t h3 = h[3];
    int32_t h4 = h[4];
    int32_t h5 = h[5];
    int32_t h6 = h[6];
    int32_t h7 = h[7];
    int32_t h8 = h[8];
    int32_t h9 = h[9];
    int32_t q;

    q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
    q = (h0 + q) >> 26;
    q = (h1 + q) >> 25;
    q = (h2 + q) >> 26;
    q = (h3 + q) >> 25;
    q = (h4 + q) >> 26;
    q = (h5 + q) >> 25;
    q = (h6 + q) >> 26;
    q = (h7 + q) >> 25;
    q = (h8 + q) >> 26;
    q = (h9 + q) >> 25;

    /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
    h0 += 19 * q;
    /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */

    h1 += h0 >> 26; h0 &= kBottom26Bits;
    h2 += h1 >> 25; h1 &= kBottom25Bits;
    h3 += h2 >> 26; h2 &= kBottom26Bits;
    h4 += h3 >> 25; h3 &= kBottom25Bits;
    h5 += h4 >> 26; h4 &= kBottom26Bits;
    h6 += h5 >> 25; h5 &= kBottom25Bits;
    h7 += h6 >> 26; h6 &= kBottom26Bits;
    h8 += h7 >> 25; h7 &= kBottom25Bits;
    h9 += h8 >> 26; h8 &= kBottom26Bits;
                    h9 &= kBottom25Bits;
    /* h10 = carry9 */

    /*
     * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
     * Have h0+...+2^230 h9 between 0 and 2^255-1;
     * evidently 2^255 h10-2^255 q = 0.
     * Goal: Output h0+...+2^230 h9.
     */
    s[ 0] = (uint8_t) (h0 >>  0);
    s[ 1] = (uint8_t) (h0 >>  8);
    s[ 2] = (uint8_t) (h0 >> 16);
    s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
    s[ 4] = (uint8_t) (h1 >>  6);
    s[ 5] = (uint8_t) (h1 >> 14);
    s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
    s[ 7] = (uint8_t) (h2 >>  5);
    s[ 8] = (uint8_t) (h2 >> 13);
    s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
    s[10] = (uint8_t) (h3 >>  3);
    s[11] = (uint8_t) (h3 >> 11);
    s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
    s[13] = (uint8_t) (h4 >>  2);
    s[14] = (uint8_t) (h4 >> 10);
    s[15] = (uint8_t) (h4 >> 18);
    s[16] = (uint8_t) (h5 >>  0);
    s[17] = (uint8_t) (h5 >>  8);
    s[18] = (uint8_t) (h5 >> 16);
    s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
    s[20] = (uint8_t) (h6 >>  7);
    s[21] = (uint8_t) (h6 >> 15);
    s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
    s[23] = (uint8_t) (h7 >>  5);
    s[24] = (uint8_t) (h7 >> 13);
    s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
    s[26] = (uint8_t) (h8 >>  4);
    s[27] = (uint8_t) (h8 >> 12);
    s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
    s[29] = (uint8_t) (h9 >>  2);
    s[30] = (uint8_t) (h9 >> 10);
    s[31] = (uint8_t) (h9 >> 18);
}

/* h = f */
static void fe_copy(fe h, const fe f)
{
    memmove(h, f, sizeof(int32_t) * 10);
}

/* h = 0 */
static void fe_0(fe h)
{
    memset(h, 0, sizeof(int32_t) * 10);
}

/* h = 1 */
static void fe_1(fe h)
{
    memset(h, 0, sizeof(int32_t) * 10);
    h[0] = 1;
}

/*
 * h = f + g
 *
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 */
static void fe_add(fe h, const fe f, const fe g)
{
    unsigned i;

    for (i = 0; i < 10; i++) {
        h[i] = f[i] + g[i];
    }
}

/*
 * h = f - g
 *
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 */
static void fe_sub(fe h, const fe f, const fe g)
{
    unsigned i;

    for (i = 0; i < 10; i++) {
        h[i] = f[i] - g[i];
    }
}

/*
 * h = f * g
 *
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *    |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * Notes on implementation strategy:
 *
 * Using schoolbook multiplication.
 * Karatsuba would save a little in some cost models.
 *
 * Most multiplications by 2 and 19 are 32-bit precomputations;
 * cheaper than 64-bit postcomputations.
 *
 * There is one remaining multiplication by 19 in the carry chain;
 * one *19 precomputation can be merged into this,
 * but the resulting data flow is considerably less clean.
 *
 * There are 12 carries below.
 * 10 of them are 2-way parallelizable and vectorizable.
 * Can get away with 11 carries, but then data flow is much deeper.
 *
 * With tighter constraints on inputs can squeeze carries into int32.
 */
static void fe_mul(fe h, const fe f, const fe g)
{
    int32_t f0 = f[0];
    int32_t f1 = f[1];
    int32_t f2 = f[2];
    int32_t f3 = f[3];
    int32_t f4 = f[4];
    int32_t f5 = f[5];
    int32_t f6 = f[6];
    int32_t f7 = f[7];
    int32_t f8 = f[8];
    int32_t f9 = f[9];
    int32_t g0 = g[0];
    int32_t g1 = g[1];
    int32_t g2 = g[2];
    int32_t g3 = g[3];
    int32_t g4 = g[4];
    int32_t g5 = g[5];
    int32_t g6 = g[6];
    int32_t g7 = g[7];
    int32_t g8 = g[8];
    int32_t g9 = g[9];
    int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
    int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
    int32_t g3_19 = 19 * g3;
    int32_t g4_19 = 19 * g4;
    int32_t g5_19 = 19 * g5;
    int32_t g6_19 = 19 * g6;
    int32_t g7_19 = 19 * g7;
    int32_t g8_19 = 19 * g8;
    int32_t g9_19 = 19 * g9;
    int32_t f1_2 = 2 * f1;
    int32_t f3_2 = 2 * f3;
    int32_t f5_2 = 2 * f5;
    int32_t f7_2 = 2 * f7;
    int32_t f9_2 = 2 * f9;
    int64_t f0g0    = f0   * (int64_t) g0;
    int64_t f0g1    = f0   * (int64_t) g1;
    int64_t f0g2    = f0   * (int64_t) g2;
    int64_t f0g3    = f0   * (int64_t) g3;
    int64_t f0g4    = f0   * (int64_t) g4;
    int64_t f0g5    = f0   * (int64_t) g5;
    int64_t f0g6    = f0   * (int64_t) g6;
    int64_t f0g7    = f0   * (int64_t) g7;
    int64_t f0g8    = f0   * (int64_t) g8;
    int64_t f0g9    = f0   * (int64_t) g9;
    int64_t f1g0    = f1   * (int64_t) g0;
    int64_t f1g1_2  = f1_2 * (int64_t) g1;
    int64_t f1g2    = f1   * (int64_t) g2;
    int64_t f1g3_2  = f1_2 * (int64_t) g3;
    int64_t f1g4    = f1   * (int64_t) g4;
    int64_t f1g5_2  = f1_2 * (int64_t) g5;
    int64_t f1g6    = f1   * (int64_t) g6;
    int64_t f1g7_2  = f1_2 * (int64_t) g7;
    int64_t f1g8    = f1   * (int64_t) g8;
    int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
    int64_t f2g0    = f2   * (int64_t) g0;
    int64_t f2g1    = f2   * (int64_t) g1;
    int64_t f2g2    = f2   * (int64_t) g2;
    int64_t f2g3    = f2   * (int64_t) g3;
    int64_t f2g4    = f2   * (int64_t) g4;
    int64_t f2g5    = f2   * (int64_t) g5;
    int64_t f2g6    = f2   * (int64_t) g6;
    int64_t f2g7    = f2   * (int64_t) g7;
    int64_t f2g8_19 = f2   * (int64_t) g8_19;
    int64_t f2g9_19 = f2   * (int64_t) g9_19;
    int64_t f3g0    = f3   * (int64_t) g0;
    int64_t f3g1_2  = f3_2 * (int64_t) g1;
    int64_t f3g2    = f3   * (int64_t) g2;
    int64_t f3g3_2  = f3_2 * (int64_t) g3;
    int64_t f3g4    = f3   * (int64_t) g4;
    int64_t f3g5_2  = f3_2 * (int64_t) g5;
    int64_t f3g6    = f3   * (int64_t) g6;
    int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
    int64_t f3g8_19 = f3   * (int64_t) g8_19;
    int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
    int64_t f4g0    = f4   * (int64_t) g0;
    int64_t f4g1    = f4   * (int64_t) g1;
    int64_t f4g2    = f4   * (int64_t) g2;
    int64_t f4g3    = f4   * (int64_t) g3;
    int64_t f4g4    = f4   * (int64_t) g4;
    int64_t f4g5    = f4   * (int64_t) g5;
    int64_t f4g6_19 = f4   * (int64_t) g6_19;
    int64_t f4g7_19 = f4   * (int64_t) g7_19;
    int64_t f4g8_19 = f4   * (int64_t) g8_19;
    int64_t f4g9_19 = f4   * (int64_t) g9_19;
    int64_t f5g0    = f5   * (int64_t) g0;
    int64_t f5g1_2  = f5_2 * (int64_t) g1;
    int64_t f5g2    = f5   * (int64_t) g2;
    int64_t f5g3_2  = f5_2 * (int64_t) g3;
    int64_t f5g4    = f5   * (int64_t) g4;
    int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
    int64_t f5g6_19 = f5   * (int64_t) g6_19;
    int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
    int64_t f5g8_19 = f5   * (int64_t) g8_19;
    int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
    int64_t f6g0    = f6   * (int64_t) g0;
    int64_t f6g1    = f6   * (int64_t) g1;
    int64_t f6g2    = f6   * (int64_t) g2;
    int64_t f6g3    = f6   * (int64_t) g3;
    int64_t f6g4_19 = f6   * (int64_t) g4_19;
    int64_t f6g5_19 = f6   * (int64_t) g5_19;
    int64_t f6g6_19 = f6   * (int64_t) g6_19;
    int64_t f6g7_19 = f6   * (int64_t) g7_19;
    int64_t f6g8_19 = f6   * (int64_t) g8_19;
    int64_t f6g9_19 = f6   * (int64_t) g9_19;
    int64_t f7g0    = f7   * (int64_t) g0;
    int64_t f7g1_2  = f7_2 * (int64_t) g1;
    int64_t f7g2    = f7   * (int64_t) g2;
    int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
    int64_t f7g4_19 = f7   * (int64_t) g4_19;
    int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
    int64_t f7g6_19 = f7   * (int64_t) g6_19;
    int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
    int64_t f7g8_19 = f7   * (int64_t) g8_19;
    int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
    int64_t f8g0    = f8   * (int64_t) g0;
    int64_t f8g1    = f8   * (int64_t) g1;
    int64_t f8g2_19 = f8   * (int64_t) g2_19;
    int64_t f8g3_19 = f8   * (int64_t) g3_19;
    int64_t f8g4_19 = f8   * (int64_t) g4_19;
    int64_t f8g5_19 = f8   * (int64_t) g5_19;
    int64_t f8g6_19 = f8   * (int64_t) g6_19;
    int64_t f8g7_19 = f8   * (int64_t) g7_19;
    int64_t f8g8_19 = f8   * (int64_t) g8_19;
    int64_t f8g9_19 = f8   * (int64_t) g9_19;
    int64_t f9g0    = f9   * (int64_t) g0;
    int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
    int64_t f9g2_19 = f9   * (int64_t) g2_19;
    int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
    int64_t f9g4_19 = f9   * (int64_t) g4_19;
    int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
    int64_t f9g6_19 = f9   * (int64_t) g6_19;
    int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
    int64_t f9g8_19 = f9   * (int64_t) g8_19;
    int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
    int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
    int64_t h1 = f0g1 + f1g0    + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
    int64_t h2 = f0g2 + f1g1_2  + f2g0    + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
    int64_t h3 = f0g3 + f1g2    + f2g1    + f3g0    + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
    int64_t h4 = f0g4 + f1g3_2  + f2g2    + f3g1_2  + f4g0    + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
    int64_t h5 = f0g5 + f1g4    + f2g3    + f3g2    + f4g1    + f5g0    + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
    int64_t h6 = f0g6 + f1g5_2  + f2g4    + f3g3_2  + f4g2    + f5g1_2  + f6g0    + f7g9_38 + f8g8_19 + f9g7_38;
    int64_t h7 = f0g7 + f1g6    + f2g5    + f3g4    + f4g3    + f5g2    + f6g1    + f7g0    + f8g9_19 + f9g8_19;
    int64_t h8 = f0g8 + f1g7_2  + f2g6    + f3g5_2  + f4g4    + f5g3_2  + f6g2    + f7g1_2  + f8g0    + f9g9_38;
    int64_t h9 = f0g9 + f1g8    + f2g7    + f3g6    + f4g5    + f5g4    + f6g3    + f7g2    + f8g1    + f9g0   ;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;

    /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
     *   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
     * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
     *   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    /* |h0| <= 2^25 */
    /* |h4| <= 2^25 */
    /* |h1| <= 1.71*2^59 */
    /* |h5| <= 1.71*2^59 */

    carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
    carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
    /* |h1| <= 2^24; from now on fits into int32 */
    /* |h5| <= 2^24; from now on fits into int32 */
    /* |h2| <= 1.41*2^60 */
    /* |h6| <= 1.41*2^60 */

    carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
    carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
    /* |h2| <= 2^25; from now on fits into int32 unchanged */
    /* |h6| <= 2^25; from now on fits into int32 unchanged */
    /* |h3| <= 1.71*2^59 */
    /* |h7| <= 1.71*2^59 */

    carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
    carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
    /* |h3| <= 2^24; from now on fits into int32 unchanged */
    /* |h7| <= 2^24; from now on fits into int32 unchanged */
    /* |h4| <= 1.72*2^34 */
    /* |h8| <= 1.41*2^60 */

    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
    /* |h4| <= 2^25; from now on fits into int32 unchanged */
    /* |h8| <= 2^25; from now on fits into int32 unchanged */
    /* |h5| <= 1.01*2^24 */
    /* |h9| <= 1.71*2^59 */

    carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
    /* |h9| <= 2^24; from now on fits into int32 unchanged */
    /* |h0| <= 1.1*2^39 */

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    /* |h0| <= 2^25; from now on fits into int32 unchanged */
    /* |h1| <= 1.01*2^24 */

    h[0] = (int32_t)h0;
    h[1] = (int32_t)h1;
    h[2] = (int32_t)h2;
    h[3] = (int32_t)h3;
    h[4] = (int32_t)h4;
    h[5] = (int32_t)h5;
    h[6] = (int32_t)h6;
    h[7] = (int32_t)h7;
    h[8] = (int32_t)h8;
    h[9] = (int32_t)h9;
}

/*
 * h = f * f
 *
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * See fe_mul.c for discussion of implementation strategy.
 */
static void fe_sq(fe h, const fe f)
{
    int32_t f0 = f[0];
    int32_t f1 = f[1];
    int32_t f2 = f[2];
    int32_t f3 = f[3];
    int32_t f4 = f[4];
    int32_t f5 = f[5];
    int32_t f6 = f[6];
    int32_t f7 = f[7];
    int32_t f8 = f[8];
    int32_t f9 = f[9];
    int32_t f0_2 = 2 * f0;
    int32_t f1_2 = 2 * f1;
    int32_t f2_2 = 2 * f2;
    int32_t f3_2 = 2 * f3;
    int32_t f4_2 = 2 * f4;
    int32_t f5_2 = 2 * f5;
    int32_t f6_2 = 2 * f6;
    int32_t f7_2 = 2 * f7;
    int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
    int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
    int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
    int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
    int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
    int64_t f0f0    = f0   * (int64_t) f0;
    int64_t f0f1_2  = f0_2 * (int64_t) f1;
    int64_t f0f2_2  = f0_2 * (int64_t) f2;
    int64_t f0f3_2  = f0_2 * (int64_t) f3;
    int64_t f0f4_2  = f0_2 * (int64_t) f4;
    int64_t f0f5_2  = f0_2 * (int64_t) f5;
    int64_t f0f6_2  = f0_2 * (int64_t) f6;
    int64_t f0f7_2  = f0_2 * (int64_t) f7;
    int64_t f0f8_2  = f0_2 * (int64_t) f8;
    int64_t f0f9_2  = f0_2 * (int64_t) f9;
    int64_t f1f1_2  = f1_2 * (int64_t) f1;
    int64_t f1f2_2  = f1_2 * (int64_t) f2;
    int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
    int64_t f1f4_2  = f1_2 * (int64_t) f4;
    int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
    int64_t f1f6_2  = f1_2 * (int64_t) f6;
    int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
    int64_t f1f8_2  = f1_2 * (int64_t) f8;
    int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
    int64_t f2f2    = f2   * (int64_t) f2;
    int64_t f2f3_2  = f2_2 * (int64_t) f3;
    int64_t f2f4_2  = f2_2 * (int64_t) f4;
    int64_t f2f5_2  = f2_2 * (int64_t) f5;
    int64_t f2f6_2  = f2_2 * (int64_t) f6;
    int64_t f2f7_2  = f2_2 * (int64_t) f7;
    int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
    int64_t f2f9_38 = f2   * (int64_t) f9_38;
    int64_t f3f3_2  = f3_2 * (int64_t) f3;
    int64_t f3f4_2  = f3_2 * (int64_t) f4;
    int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
    int64_t f3f6_2  = f3_2 * (int64_t) f6;
    int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
    int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
    int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
    int64_t f4f4    = f4   * (int64_t) f4;
    int64_t f4f5_2  = f4_2 * (int64_t) f5;
    int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
    int64_t f4f7_38 = f4   * (int64_t) f7_38;
    int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
    int64_t f4f9_38 = f4   * (int64_t) f9_38;
    int64_t f5f5_38 = f5   * (int64_t) f5_38;
    int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
    int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
    int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
    int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
    int64_t f6f6_19 = f6   * (int64_t) f6_19;
    int64_t f6f7_38 = f6   * (int64_t) f7_38;
    int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
    int64_t f6f9_38 = f6   * (int64_t) f9_38;
    int64_t f7f7_38 = f7   * (int64_t) f7_38;
    int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
    int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
    int64_t f8f8_19 = f8   * (int64_t) f8_19;
    int64_t f8f9_38 = f8   * (int64_t) f9_38;
    int64_t f9f9_38 = f9   * (int64_t) f9_38;
    int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
    int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
    int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
    int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
    int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
    int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
    int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
    int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
    int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
    int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;

    carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
    carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;

    carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
    carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;

    carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
    carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

    carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;

    h[0] = (int32_t)h0;
    h[1] = (int32_t)h1;
    h[2] = (int32_t)h2;
    h[3] = (int32_t)h3;
    h[4] = (int32_t)h4;
    h[5] = (int32_t)h5;
    h[6] = (int32_t)h6;
    h[7] = (int32_t)h7;
    h[8] = (int32_t)h8;
    h[9] = (int32_t)h9;
}

static void fe_invert(fe out, const fe z)
{
    fe t0;
    fe t1;
    fe t2;
    fe t3;
    int i;

    /*
     * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
     * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
     */

    /* t0 = z ** 2 */
    fe_sq(t0, z);

    /* t1 = t0 ** (2 ** 2) = z ** 8 */
    fe_sq(t1, t0);
    fe_sq(t1, t1);

    /* t1 = z * t1 = z ** 9 */
    fe_mul(t1, z, t1);
    /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
    fe_mul(t0, t0, t1);

    /* t2 = t0 ** 2 = z ** 22 */
    fe_sq(t2, t0);

    /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
    fe_mul(t1, t1, t2);

    /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
    fe_sq(t2, t1);
    for (i = 1; i < 5; ++i) {
        fe_sq(t2, t2);
    }

    /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
    fe_mul(t1, t2, t1);

    /* Continuing similarly... */

    /* t2 = z ** (2 ** 20 - 1) */
    fe_sq(t2, t1);
    for (i = 1; i < 10; ++i) {
        fe_sq(t2, t2);
    }
    fe_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 40 - 1) */
    fe_sq(t3, t2);
    for (i = 1; i < 20; ++i) {
        fe_sq(t3, t3);
    }
    fe_mul(t2, t3, t2);

    /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
    for (i = 0; i < 10; ++i) {
        fe_sq(t2, t2);
    }
    /* t1 = z ** (2 ** 50 - 1) */
    fe_mul(t1, t2, t1);

    /* t2 = z ** (2 ** 100 - 1) */
    fe_sq(t2, t1);
    for (i = 1; i < 50; ++i) {
        fe_sq(t2, t2);
    }
    fe_mul(t2, t2, t1);

    /* t2 = z ** (2 ** 200 - 1) */
    fe_sq(t3, t2);
    for (i = 1; i < 100; ++i) {
        fe_sq(t3, t3);
    }
    fe_mul(t2, t3, t2);

    /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
    fe_sq(t2, t2);
    for (i = 1; i < 50; ++i) {
        fe_sq(t2, t2);
    }

    /* t1 = z ** (2 ** 250 - 1) */
    fe_mul(t1, t2, t1);

    /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
    fe_sq(t1, t1);
    for (i = 1; i < 5; ++i) {
        fe_sq(t1, t1);
    }

    /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
    fe_mul(out, t1, t0);
}

/*
 * h = -f
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 */
static void fe_neg(fe h, const fe f)
{
    unsigned i;

    for (i = 0; i < 10; i++) {
        h[i] = -f[i];
    }
}

/*
 * Replace (f,g) with (g,g) if b == 1;
 * replace (f,g) with (f,g) if b == 0.
 *
 * Preconditions: b in {0,1}.
 */
static void fe_cmov(fe f, const fe g, unsigned b)
{
    size_t i;

    b = 0-b;
    for (i = 0; i < 10; i++) {
        int32_t x = f[i] ^ g[i];
        x &= b;
        f[i] ^= x;
    }
}

/*
 * return 0 if f == 0
 * return 1 if f != 0
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 */
static int fe_isnonzero(const fe f)
{
    uint8_t s[32];
    static const uint8_t zero[32] = {0};

    fe_tobytes(s, f);

    return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
}

/*
 * return 1 if f is in {1,3,5,...,q-2}
 * return 0 if f is in {0,2,4,...,q-1}
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 */
static int fe_isnegative(const fe f)
{
    uint8_t s[32];

    fe_tobytes(s, f);
    return s[0] & 1;
}

/*
 * h = 2 * f * f
 *
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * See fe_mul.c for discussion of implementation strategy.
 */
static void fe_sq2(fe h, const fe f)
{
    int32_t f0 = f[0];
    int32_t f1 = f[1];
    int32_t f2 = f[2];
    int32_t f3 = f[3];
    int32_t f4 = f[4];
    int32_t f5 = f[5];
    int32_t f6 = f[6];
    int32_t f7 = f[7];
    int32_t f8 = f[8];
    int32_t f9 = f[9];
    int32_t f0_2 = 2 * f0;
    int32_t f1_2 = 2 * f1;
    int32_t f2_2 = 2 * f2;
    int32_t f3_2 = 2 * f3;
    int32_t f4_2 = 2 * f4;
    int32_t f5_2 = 2 * f5;
    int32_t f6_2 = 2 * f6;
    int32_t f7_2 = 2 * f7;
    int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
    int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
    int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
    int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
    int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
    int64_t f0f0    = f0   * (int64_t) f0;
    int64_t f0f1_2  = f0_2 * (int64_t) f1;
    int64_t f0f2_2  = f0_2 * (int64_t) f2;
    int64_t f0f3_2  = f0_2 * (int64_t) f3;
    int64_t f0f4_2  = f0_2 * (int64_t) f4;
    int64_t f0f5_2  = f0_2 * (int64_t) f5;
    int64_t f0f6_2  = f0_2 * (int64_t) f6;
    int64_t f0f7_2  = f0_2 * (int64_t) f7;
    int64_t f0f8_2  = f0_2 * (int64_t) f8;
    int64_t f0f9_2  = f0_2 * (int64_t) f9;
    int64_t f1f1_2  = f1_2 * (int64_t) f1;
    int64_t f1f2_2  = f1_2 * (int64_t) f2;
    int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
    int64_t f1f4_2  = f1_2 * (int64_t) f4;
    int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
    int64_t f1f6_2  = f1_2 * (int64_t) f6;
    int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
    int64_t f1f8_2  = f1_2 * (int64_t) f8;
    int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
    int64_t f2f2    = f2   * (int64_t) f2;
    int64_t f2f3_2  = f2_2 * (int64_t) f3;
    int64_t f2f4_2  = f2_2 * (int64_t) f4;
    int64_t f2f5_2  = f2_2 * (int64_t) f5;
    int64_t f2f6_2  = f2_2 * (int64_t) f6;
    int64_t f2f7_2  = f2_2 * (int64_t) f7;
    int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
    int64_t f2f9_38 = f2   * (int64_t) f9_38;
    int64_t f3f3_2  = f3_2 * (int64_t) f3;
    int64_t f3f4_2  = f3_2 * (int64_t) f4;
    int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
    int64_t f3f6_2  = f3_2 * (int64_t) f6;
    int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
    int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
    int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
    int64_t f4f4    = f4   * (int64_t) f4;
    int64_t f4f5_2  = f4_2 * (int64_t) f5;
    int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
    int64_t f4f7_38 = f4   * (int64_t) f7_38;
    int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
    int64_t f4f9_38 = f4   * (int64_t) f9_38;
    int64_t f5f5_38 = f5   * (int64_t) f5_38;
    int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
    int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
    int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
    int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
    int64_t f6f6_19 = f6   * (int64_t) f6_19;
    int64_t f6f7_38 = f6   * (int64_t) f7_38;
    int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
    int64_t f6f9_38 = f6   * (int64_t) f9_38;
    int64_t f7f7_38 = f7   * (int64_t) f7_38;
    int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
    int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
    int64_t f8f8_19 = f8   * (int64_t) f8_19;
    int64_t f8f9_38 = f8   * (int64_t) f9_38;
    int64_t f9f9_38 = f9   * (int64_t) f9_38;
    int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
    int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
    int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
    int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
    int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
    int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
    int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
    int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
    int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
    int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;

    h0 += h0;
    h1 += h1;
    h2 += h2;
    h3 += h3;
    h4 += h4;
    h5 += h5;
    h6 += h6;
    h7 += h7;
    h8 += h8;
    h9 += h9;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;

    carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
    carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;

    carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
    carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;

    carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
    carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

    carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;

    h[0] = (int32_t)h0;
    h[1] = (int32_t)h1;
    h[2] = (int32_t)h2;
    h[3] = (int32_t)h3;
    h[4] = (int32_t)h4;
    h[5] = (int32_t)h5;
    h[6] = (int32_t)h6;
    h[7] = (int32_t)h7;
    h[8] = (int32_t)h8;
    h[9] = (int32_t)h9;
}

static void fe_pow22523(fe out, const fe z)
{
    fe t0;
    fe t1;
    fe t2;
    int i;

    fe_sq(t0, z);
    fe_sq(t1, t0);
    for (i = 1; i < 2; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t1, z, t1);
    fe_mul(t0, t0, t1);
    fe_sq(t0, t0);
    fe_mul(t0, t1, t0);
    fe_sq(t1, t0);
    for (i = 1; i < 5; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t0, t1, t0);
    fe_sq(t1, t0);
    for (i = 1; i < 10; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t1, t1, t0);
    fe_sq(t2, t1);
    for (i = 1; i < 20; ++i) {
        fe_sq(t2, t2);
    }
    fe_mul(t1, t2, t1);
    fe_sq(t1, t1);
    for (i = 1; i < 10; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t0, t1, t0);
    fe_sq(t1, t0);
    for (i = 1; i < 50; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t1, t1, t0);
    fe_sq(t2, t1);
    for (i = 1; i < 100; ++i) {
        fe_sq(t2, t2);
    }
    fe_mul(t1, t2, t1);
    fe_sq(t1, t1);
    for (i = 1; i < 50; ++i) {
        fe_sq(t1, t1);
    }
    fe_mul(t0, t1, t0);
    fe_sq(t0, t0);
    for (i = 1; i < 2; ++i) {
        fe_sq(t0, t0);
    }
    fe_mul(out, t0, z);
}

/*
 * ge means group element.
 *
 * Here the group is the set of pairs (x,y) of field elements (see fe.h)
 * satisfying -x^2 + y^2 = 1 + d x^2y^2
 * where d = -121665/121666.
 *
 * Representations:
 *   ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
 *   ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
 *   ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
 *   ge_precomp (Duif): (y+x,y-x,2dxy)
 */
typedef struct {
    fe X;
    fe Y;
    fe Z;
} ge_p2;

typedef struct {
    fe X;
    fe Y;
    fe Z;
    fe T;
} ge_p3;

typedef struct {
    fe X;
    fe Y;
    fe Z;
    fe T;
} ge_p1p1;

typedef struct {
    fe yplusx;
    fe yminusx;
    fe xy2d;
} ge_precomp;

typedef struct {
    fe YplusX;
    fe YminusX;
    fe Z;
    fe T2d;
} ge_cached;

static void ge_tobytes(uint8_t *s, const ge_p2 *h)
{
    fe recip;
    fe x;
    fe y;

    fe_invert(recip, h->Z);
    fe_mul(x, h->X, recip);
    fe_mul(y, h->Y, recip);
    fe_tobytes(s, y);
    s[31] ^= fe_isnegative(x) << 7;
}

static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
{
    fe recip;
    fe x;
    fe y;

    fe_invert(recip, h->Z);
    fe_mul(x, h->X, recip);
    fe_mul(y, h->Y, recip);
    fe_tobytes(s, y);
    s[31] ^= fe_isnegative(x) << 7;
}

static const fe d = {
    -10913610, 13857413, -15372611, 6949391,   114729,
    -8787816,  -6275908, -3247719,  -18696448, -12055116
};

static const fe sqrtm1 = {
    -32595792, -7943725,  9377950,  3500415, 12389472,
    -272473,   -25146209, -2005654, 326686,  11406482
};

static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
{
    fe u;
    fe v;
    fe v3;
    fe vxx;
    fe check;

    fe_frombytes(h->Y, s);
    fe_1(h->Z);
    fe_sq(u, h->Y);
    fe_mul(v, u, d);
    fe_sub(u, u, h->Z); /* u = y^2-1 */
    fe_add(v, v, h->Z); /* v = dy^2+1 */

    fe_sq(v3, v);
    fe_mul(v3, v3, v); /* v3 = v^3 */
    fe_sq(h->X, v3);
    fe_mul(h->X, h->X, v);
    fe_mul(h->X, h->X, u); /* x = uv^7 */

    fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
    fe_mul(h->X, h->X, v3);
    fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */

    fe_sq(vxx, h->X);
    fe_mul(vxx, vxx, v);
    fe_sub(check, vxx, u); /* vx^2-u */
    if (fe_isnonzero(check)) {
        fe_add(check, vxx, u); /* vx^2+u */
        if (fe_isnonzero(check)) {
            return -1;
        }
        fe_mul(h->X, h->X, sqrtm1);
    }

    if (fe_isnegative(h->X) != (s[31] >> 7)) {
        fe_neg(h->X, h->X);
    }

    fe_mul(h->T, h->X, h->Y);
    return 0;
}

static void ge_p2_0(ge_p2 *h)
{
    fe_0(h->X);
    fe_1(h->Y);
    fe_1(h->Z);
}

static void ge_p3_0(ge_p3 *h)
{
    fe_0(h->X);
    fe_1(h->Y);
    fe_1(h->Z);
    fe_0(h->T);
}

static void ge_precomp_0(ge_precomp *h)
{
    fe_1(h->yplusx);
    fe_1(h->yminusx);
    fe_0(h->xy2d);
}

/* r = p */
static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
{
    fe_copy(r->X, p->X);
    fe_copy(r->Y, p->Y);
    fe_copy(r->Z, p->Z);
}

static const fe d2 = {
    -21827239, -5839606,  -30745221, 13898782, 229458,
    15978800,  -12551817, -6495438,  29715968, 9444199
};

/* r = p */
static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
{
    fe_add(r->YplusX, p->Y, p->X);
    fe_sub(r->YminusX, p->Y, p->X);
    fe_copy(r->Z, p->Z);
    fe_mul(r->T2d, p->T, d2);
}

/* r = p */
static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
{
    fe_mul(r->X, p->X, p->T);
    fe_mul(r->Y, p->Y, p->Z);
    fe_mul(r->Z, p->Z, p->T);
}

/* r = p */
static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
{
    fe_mul(r->X, p->X, p->T);
    fe_mul(r->Y, p->Y, p->Z);
    fe_mul(r->Z, p->Z, p->T);
    fe_mul(r->T, p->X, p->Y);
}

/* r = 2 * p */
static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
{
    fe t0;

    fe_sq(r->X, p->X);
    fe_sq(r->Z, p->Y);
    fe_sq2(r->T, p->Z);
    fe_add(r->Y, p->X, p->Y);
    fe_sq(t0, r->Y);
    fe_add(r->Y, r->Z, r->X);
    fe_sub(r->Z, r->Z, r->X);
    fe_sub(r->X, t0, r->Y);
    fe_sub(r->T, r->T, r->Z);
}

/* r = 2 * p */
static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
{
    ge_p2 q;
    ge_p3_to_p2(&q, p);
    ge_p2_dbl(r, &q);
}

/* r = p + q */
static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
{
    fe t0;

    fe_add(r->X, p->Y, p->X);
    fe_sub(r->Y, p->Y, p->X);
    fe_mul(r->Z, r->X, q->yplusx);
    fe_mul(r->Y, r->Y, q->yminusx);
    fe_mul(r->T, q->xy2d, p->T);
    fe_add(t0, p->Z, p->Z);
    fe_sub(r->X, r->Z, r->Y);
    fe_add(r->Y, r->Z, r->Y);
    fe_add(r->Z, t0, r->T);
    fe_sub(r->T, t0, r->T);
}

/* r = p - q */
static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
{
    fe t0;

    fe_add(r->X, p->Y, p->X);
    fe_sub(r->Y, p->Y, p->X);
    fe_mul(r->Z, r->X, q->yminusx);
    fe_mul(r->Y, r->Y, q->yplusx);
    fe_mul(r->T, q->xy2d, p->T);
    fe_add(t0, p->Z, p->Z);
    fe_sub(r->X, r->Z, r->Y);
    fe_add(r->Y, r->Z, r->Y);
    fe_sub(r->Z, t0, r->T);
    fe_add(r->T, t0, r->T);
}

/* r = p + q */
static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
{
    fe t0;

    fe_add(r->X, p->Y, p->X);
    fe_sub(r->Y, p->Y, p->X);
    fe_mul(r->Z, r->X, q->YplusX);
    fe_mul(r->Y, r->Y, q->YminusX);
    fe_mul(r->T, q->T2d, p->T);
    fe_mul(r->X, p->Z, q->Z);
    fe_add(t0, r->X, r->X);
    fe_sub(r->X, r->Z, r->Y);
    fe_add(r->Y, r->Z, r->Y);
    fe_add(r->Z, t0, r->T);
    fe_sub(r->T, t0, r->T);
}

/* r = p - q */
static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
{
    fe t0;

    fe_add(r->X, p->Y, p->X);
    fe_sub(r->Y, p->Y, p->X);
    fe_mul(r->Z, r->X, q->YminusX);
    fe_mul(r->Y, r->Y, q->YplusX);
    fe_mul(r->T, q->T2d, p->T);
    fe_mul(r->X, p->Z, q->Z);
    fe_add(t0, r->X, r->X);
    fe_sub(r->X, r->Z, r->Y);
    fe_add(r->Y, r->Z, r->Y);
    fe_sub(r->Z, t0, r->T);
    fe_add(r->T, t0, r->T);
}

static uint8_t equal(signed char b, signed char c)
{
    uint8_t ub = b;
    uint8_t uc = c;
    uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
    uint32_t y = x;      /* 0: yes; 1..255: no */
    y -= 1;              /* 4294967295: yes; 0..254: no */
    y >>= 31;            /* 1: yes; 0: no */
    return y;
}

static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
{
    fe_cmov(t->yplusx, u->yplusx, b);
    fe_cmov(t->yminusx, u->yminusx, b);
    fe_cmov(t->xy2d, u->xy2d, b);
}

/* k25519Precomp[i][j] = (j+1)*256^i*B */
static const ge_precomp k25519Precomp[32][8] = {
    {
        {
            {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
             27544626, -11754271, -6079156, 2047605},
            {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
             5043384, 19500929, -15469378},
            {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
             29287919, 11864899, -24514362, -4438546},
        },
        {
            {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
             -11717903, -3814571, -358445, -10211303},
            {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
             -15616551, 11189268, -26829678, -5319081},
            {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
             -3128826, -9541118, -15472047, -4166697},
        },
        {
            {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
             27787600, -14772189, 28944400, -1550024},
            {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
             16354577, -11775962, 7689662, 11199574},
            {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
             7512774, 10017326, -17749093, -9920357},
        },
        {
            {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
             -28926210, 15006023, 3284568, -6276540},
            {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
             7464579, 9656445, 13059162, 10374397},
            {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
             -3839045, -641708, -101325},
        },
        {
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        },
        {
            {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
             24367466, 6388839, -10295587, 452383},
            {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
             -24236251, -5915248, 15766062, 8407814},
            {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
             -8917023, -4388953, -8067909, 2276718},
        },
        {
            {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
             -23827587, 5096219, 22740376, -7303417},
            {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
             24051124, 13742383, -15637599, 13295222},
            {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
             -17720195, -4612972, -4451357, -14669444},
        },
        {
            {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
             -2469266, -4141880, 7770569, 9620597},
            {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
             -1694323, -33502340, -14767970},
            {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
             1220118, 30494170, -11440799},
        },
        {
            {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
             -26739026, 926050, -1684339, -13333647},
            {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
             9021034, 9078865, 3353509, 4033511},
            {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
             23161163, 8839127, 27485041, 7356032},
        },
    },
    {
        {
            {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
             2625015, 28431036, -16771834},
            {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
             -22545972, 14150565, 15970762, 4099461},
            {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
             13617293, -9937143, 11465739, 8317062},
        },
        {
            {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
             14898637, 3848455, 20969334, -5157516},
            {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
             -21610826, -3649888, 11177095, 14989547},
            {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
             13515641, 2581286, -28487508, 9930240},
        },
        {
            {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
             18345767, -13403753, 16291481, -5314038},
            {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
             6957617, 4368891, 9788741},
            {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
             -21722536, -8613148, 16250552, -11111103},
        },
        {
            {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
             10604807, -30190403, 4782747},
            {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
             -9981571, 4383045, 22546403, 437323},
            {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
             27343084, 2786261, -30633590, -14097016},
        },
        {
            {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
             -19690631, 2355319, -19284671, -6114373},
            {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
             18952177, 15496498, -29380133, 11754228},
            {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
             7141596, 11724556, 22761615, -10134141},
        },
        {
            {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
             -28741185, -12227393, 32851222, 11717399},
            {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
             31474879, 3483633, -1193175, -4030831},
            {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
             33142652, 6546660, -19985279, -3948376},
        },
        {
            {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
             -8537131, -12833048, -30772034, -15486313},
            {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
             -31135347, -16049879, 10928917, 3011958},
            {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
             -30831056, -12805180, 18008031, 10258577},
        },
        {
            {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
             -1853465, 1367120, 25127874, 6671743},
            {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
             21382910, 11042292, 25838796, 4642684},
            {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
             30468147, -13900640, 18423289, 4177476},
        },
    },
};

static uint8_t negative(signed char b)
{
    uint32_t x = b;

    x >>= 31; /* 1: yes; 0: no */
    return x;
}

static void table_select(ge_precomp *t, int pos, signed char b)
{
    ge_precomp minust;
    uint8_t bnegative = negative(b);
    uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);

    ge_precomp_0(t);
    cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
    cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
    cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
    cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
    cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
    cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
    cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
    cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
    fe_copy(minust.yplusx, t->yminusx);
    fe_copy(minust.yminusx, t->yplusx);
    fe_neg(minust.xy2d, t->xy2d);
    cmov(t, &minust, bnegative);
}

/*
 * h = a * B
 *
 * where a = a[0]+256*a[1]+...+256^31 a[31]
 * B is the Ed25519 base point (x,4/5) with x positive.
 *
 * Preconditions:
 *   a[31] <= 127
 */
static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
{
    signed char e[64];
    signed char carry;
    ge_p1p1 r;
    ge_p2 s;
    ge_precomp t;
    int i;

    for (i = 0; i < 32; ++i) {
        e[2 * i + 0] = (a[i] >> 0) & 15;
        e[2 * i + 1] = (a[i] >> 4) & 15;
    }
    /* each e[i] is between 0 and 15 */
    /* e[63] is between 0 and 7 */

    carry = 0;
    for (i = 0; i < 63; ++i) {
        e[i] += carry;
        carry = e[i] + 8;
        carry >>= 4;
        e[i] -= carry << 4;
    }
    e[63] += carry;
    /* each e[i] is between -8 and 8 */

    ge_p3_0(h);
    for (i = 1; i < 64; i += 2) {
        table_select(&t, i / 2, e[i]);
        ge_madd(&r, h, &t);
        ge_p1p1_to_p3(h, &r);
    }

    ge_p3_dbl(&r, h);
    ge_p1p1_to_p2(&s, &r);
    ge_p2_dbl(&r, &s);
    ge_p1p1_to_p2(&s, &r);
    ge_p2_dbl(&r, &s);
    ge_p1p1_to_p2(&s, &r);
    ge_p2_dbl(&r, &s);
    ge_p1p1_to_p3(h, &r);

    for (i = 0; i < 64; i += 2) {
        table_select(&t, i / 2, e[i]);
        ge_madd(&r, h, &t);
        ge_p1p1_to_p3(h, &r);
    }

    OPENSSL_cleanse(e, sizeof(e));
}

#if !defined(BASE_2_51_IMPLEMENTED)
/*
 * Replace (f,g) with (g,f) if b == 1;
 * replace (f,g) with (f,g) if b == 0.
 *
 * Preconditions: b in {0,1}.
 */
static void fe_cswap(fe f, fe g, unsigned int b)
{
    size_t i;

    b = 0-b;
    for (i = 0; i < 10; i++) {
        int32_t x = f[i] ^ g[i];
        x &= b;
        f[i] ^= x;
        g[i] ^= x;
    }
}

/*
 * h = f * 121666
 *
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 */
static void fe_mul121666(fe h, fe f)
{
    int32_t f0 = f[0];
    int32_t f1 = f[1];
    int32_t f2 = f[2];
    int32_t f3 = f[3];
    int32_t f4 = f[4];
    int32_t f5 = f[5];
    int32_t f6 = f[6];
    int32_t f7 = f[7];
    int32_t f8 = f[8];
    int32_t f9 = f[9];
    int64_t h0 = f0 * (int64_t) 121666;
    int64_t h1 = f1 * (int64_t) 121666;
    int64_t h2 = f2 * (int64_t) 121666;
    int64_t h3 = f3 * (int64_t) 121666;
    int64_t h4 = f4 * (int64_t) 121666;
    int64_t h5 = f5 * (int64_t) 121666;
    int64_t h6 = f6 * (int64_t) 121666;
    int64_t h7 = f7 * (int64_t) 121666;
    int64_t h8 = f8 * (int64_t) 121666;
    int64_t h9 = f9 * (int64_t) 121666;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;

    carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
    carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
    carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
    carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
    carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

    carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
    carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
    carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
    carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
    carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

    h[0] = (int32_t)h0;
    h[1] = (int32_t)h1;
    h[2] = (int32_t)h2;
    h[3] = (int32_t)h3;
    h[4] = (int32_t)h4;
    h[5] = (int32_t)h5;
    h[6] = (int32_t)h6;
    h[7] = (int32_t)h7;
    h[8] = (int32_t)h8;
    h[9] = (int32_t)h9;
}

static void x25519_scalar_mult_generic(uint8_t out[32],
                                       const uint8_t scalar[32],
                                       const uint8_t point[32]) {
    fe x1, x2, z2, x3, z3, tmp0, tmp1;
    uint8_t e[32];
    unsigned swap = 0;
    int pos;

    memcpy(e, scalar, 32);
    e[0] &= 248;
    e[31] &= 127;
    e[31] |= 64;
    fe_frombytes(x1, point);
    fe_1(x2);
    fe_0(z2);
    fe_copy(x3, x1);
    fe_1(z3);

    for (pos = 254; pos >= 0; --pos) {
        unsigned b = 1 & (e[pos / 8] >> (pos & 7));
        swap ^= b;
        fe_cswap(x2, x3, swap);
        fe_cswap(z2, z3, swap);
        swap = b;
        fe_sub(tmp0, x3, z3);
        fe_sub(tmp1, x2, z2);
        fe_add(x2, x2, z2);
        fe_add(z2, x3, z3);
        fe_mul(z3, tmp0, x2);
        fe_mul(z2, z2, tmp1);
        fe_sq(tmp0, tmp1);
        fe_sq(tmp1, x2);
        fe_add(x3, z3, z2);
        fe_sub(z2, z3, z2);
        fe_mul(x2, tmp1, tmp0);
        fe_sub(tmp1, tmp1, tmp0);
        fe_sq(z2, z2);
        fe_mul121666(z3, tmp1);
        fe_sq(x3, x3);
        fe_add(tmp0, tmp0, z3);
        fe_mul(z3, x1, z2);
        fe_mul(z2, tmp1, tmp0);
    }

    fe_invert(z2, z2);
    fe_mul(x2, x2, z2);
    fe_tobytes(out, x2);

    OPENSSL_cleanse(e, sizeof(e));
}

static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
                               const uint8_t point[32]) {
    x25519_scalar_mult_generic(out, scalar, point);
}
#endif

static void slide(signed char *r, const uint8_t *a)
{
    int i;
    int b;
    int k;

    for (i = 0; i < 256; ++i) {
        r[i] = 1 & (a[i >> 3] >> (i & 7));
    }

    for (i = 0; i < 256; ++i) {
        if (r[i]) {
            for (b = 1; b <= 6 && i + b < 256; ++b) {
                if (r[i + b]) {
                    if (r[i] + (r[i + b] << b) <= 15) {
                        r[i] += r[i + b] << b;
                        r[i + b] = 0;
                    } else if (r[i] - (r[i + b] << b) >= -15) {
                        r[i] -= r[i + b] << b;
                        for (k = i + b; k < 256; ++k) {
                            if (!r[k]) {
                                r[k] = 1;
                                break;
                            }
                            r[k] = 0;
                        }
                    } else {
                        break;
                    }
                }
            }
        }
    }
}

static const ge_precomp Bi[8] = {
    {
        {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
         -11754271, -6079156, 2047605},
        {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
         5043384, 19500929, -15469378},
        {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
         11864899, -24514362, -4438546},
    },
    {
        {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
         -14772189, 28944400, -1550024},
        {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
         -11775962, 7689662, 11199574},
        {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
         10017326, -17749093, -9920357},
    },
    {
        {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
         14515107, -15438304, 10819380},
        {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
         12483688, -12668491, 5581306},
        {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
         13850243, -23678021, -15815942},
    },
    {
        {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
         5230134, -23952439, -15175766},
        {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
         16520125, 30598449, 7715701},
        {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
         1370708, 29794553, -1409300},
    },
    {
        {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
         -1361450, -13062696, 13821877},
        {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
         -7212327, 18853322, -14220951},
        {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
         -10431137, 2207753, -3209784},
    },
    {
        {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
         -663000, -31111463, -16132436},
        {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
         15725684, 171356, 6466918},
        {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
         -14088058, -30714912, 16193877},
    },
    {
        {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
         4729455, -18074513, 9256800},
        {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
         9761698, -19827198, 630305},
        {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
         -15960994, -2449256, -14291300},
    },
    {
        {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
         15033784, 25105118, -7894876},
        {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
         1573892, -2625887, 2198790, -15804619},
        {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
         -16236442, -32461234, -12290683},
    },
};

/*
 * r = a * A + b * B
 *
 * where a = a[0]+256*a[1]+...+256^31 a[31].
 * and b = b[0]+256*b[1]+...+256^31 b[31].
 * B is the Ed25519 base point (x,4/5) with x positive.
 */
static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
                                         const ge_p3 *A, const uint8_t *b)
{
    signed char aslide[256];
    signed char bslide[256];
    ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
    ge_p1p1 t;
    ge_p3 u;
    ge_p3 A2;
    int i;

    slide(aslide, a);
    slide(bslide, b);

    ge_p3_to_cached(&Ai[0], A);
    ge_p3_dbl(&t, A);
    ge_p1p1_to_p3(&A2, &t);
    ge_add(&t, &A2, &Ai[0]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[1], &u);
    ge_add(&t, &A2, &Ai[1]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[2], &u);
    ge_add(&t, &A2, &Ai[2]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[3], &u);
    ge_add(&t, &A2, &Ai[3]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[4], &u);
    ge_add(&t, &A2, &Ai[4]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[5], &u);
    ge_add(&t, &A2, &Ai[5]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[6], &u);
    ge_add(&t, &A2, &Ai[6]);
    ge_p1p1_to_p3(&u, &t);
    ge_p3_to_cached(&Ai[7], &u);

    ge_p2_0(r);

    for (i = 255; i >= 0; --i) {
        if (aslide[i] || bslide[i]) {
            break;
        }
    }

    for (; i >= 0; --i) {
        ge_p2_dbl(&t, r);

        if (aslide[i] > 0) {
            ge_p1p1_to_p3(&u, &t);
            ge_add(&t, &u, &Ai[aslide[i] / 2]);
        } else if (aslide[i] < 0) {
            ge_p1p1_to_p3(&u, &t);
            ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
        }

        if (bslide[i] > 0) {
            ge_p1p1_to_p3(&u, &t);
            ge_madd(&t, &u, &Bi[bslide[i] / 2]);
        } else if (bslide[i] < 0) {
            ge_p1p1_to_p3(&u, &t);
            ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
        }

        ge_p1p1_to_p2(r, &t);
    }
}

/*
 * The set of scalars is \Z/l
 * where l = 2^252 + 27742317777372353535851937790883648493.
 *
 * Input:
 *   s[0]+256*s[1]+...+256^63*s[63] = s
 *
 * Output:
 *   s[0]+256*s[1]+...+256^31*s[31] = s mod l
 *   where l = 2^252 + 27742317777372353535851937790883648493.
 *   Overwrites s in place.
*/
static void x25519_sc_reduce(uint8_t *s)
{
    int64_t s0  = kBottom21Bits &  load_3(s);
    int64_t s1  = kBottom21Bits & (load_4(s +  2) >> 5);
    int64_t s2  = kBottom21Bits & (load_3(s +  5) >> 2);
    int64_t s3  = kBottom21Bits & (load_4(s +  7) >> 7);
    int64_t s4  = kBottom21Bits & (load_4(s + 10) >> 4);
    int64_t s5  = kBottom21Bits & (load_3(s + 13) >> 1);
    int64_t s6  = kBottom21Bits & (load_4(s + 15) >> 6);
    int64_t s7  = kBottom21Bits & (load_3(s + 18) >> 3);
    int64_t s8  = kBottom21Bits &  load_3(s + 21);
    int64_t s9  = kBottom21Bits & (load_4(s + 23) >> 5);
    int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
    int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
    int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
    int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
    int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
    int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
    int64_t s16 = kBottom21Bits &  load_3(s + 42);
    int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
    int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
    int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
    int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
    int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
    int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
    int64_t s23 =                 (load_4(s + 60) >> 3);
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;
    int64_t carry10;
    int64_t carry11;
    int64_t carry12;
    int64_t carry13;
    int64_t carry14;
    int64_t carry15;
    int64_t carry16;

    s11 += s23 * 666643;
    s12 += s23 * 470296;
    s13 += s23 * 654183;
    s14 -= s23 * 997805;
    s15 += s23 * 136657;
    s16 -= s23 * 683901;
    s23  = 0;

    s10 += s22 * 666643;
    s11 += s22 * 470296;
    s12 += s22 * 654183;
    s13 -= s22 * 997805;
    s14 += s22 * 136657;
    s15 -= s22 * 683901;
    s22  = 0;

    s9  += s21 * 666643;
    s10 += s21 * 470296;
    s11 += s21 * 654183;
    s12 -= s21 * 997805;
    s13 += s21 * 136657;
    s14 -= s21 * 683901;
    s21  = 0;

    s8  += s20 * 666643;
    s9  += s20 * 470296;
    s10 += s20 * 654183;
    s11 -= s20 * 997805;
    s12 += s20 * 136657;
    s13 -= s20 * 683901;
    s20  = 0;

    s7  += s19 * 666643;
    s8  += s19 * 470296;
    s9  += s19 * 654183;
    s10 -= s19 * 997805;
    s11 += s19 * 136657;
    s12 -= s19 * 683901;
    s19  = 0;

    s6  += s18 * 666643;
    s7  += s18 * 470296;
    s8  += s18 * 654183;
    s9  -= s18 * 997805;
    s10 += s18 * 136657;
    s11 -= s18 * 683901;
    s18  = 0;

    carry6 = (s6 + (1 << 20)) >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry8 = (s8 + (1 << 20)) >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry10 = (s10 + (1 << 20)) >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);
    carry12 = (s12 + (1 << 20)) >> 21;
    s13 += carry12;
    s12 -= carry12 * (1 << 21);
    carry14 = (s14 + (1 << 20)) >> 21;
    s15 += carry14;
    s14 -= carry14 * (1 << 21);
    carry16 = (s16 + (1 << 20)) >> 21;
    s17 += carry16;
    s16 -= carry16 * (1 << 21);

    carry7 = (s7 + (1 << 20)) >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry9 = (s9 + (1 << 20)) >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry11 = (s11 + (1 << 20)) >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);
    carry13 = (s13 + (1 << 20)) >> 21;
    s14 += carry13;
    s13 -= carry13 * (1 << 21);
    carry15 = (s15 + (1 << 20)) >> 21;
    s16 += carry15;
    s15 -= carry15 * (1 << 21);

    s5  += s17 * 666643;
    s6  += s17 * 470296;
    s7  += s17 * 654183;
    s8  -= s17 * 997805;
    s9  += s17 * 136657;
    s10 -= s17 * 683901;
    s17  = 0;

    s4  += s16 * 666643;
    s5  += s16 * 470296;
    s6  += s16 * 654183;
    s7  -= s16 * 997805;
    s8  += s16 * 136657;
    s9  -= s16 * 683901;
    s16  = 0;

    s3  += s15 * 666643;
    s4  += s15 * 470296;
    s5  += s15 * 654183;
    s6  -= s15 * 997805;
    s7  += s15 * 136657;
    s8  -= s15 * 683901;
    s15  = 0;

    s2  += s14 * 666643;
    s3  += s14 * 470296;
    s4  += s14 * 654183;
    s5  -= s14 * 997805;
    s6  += s14 * 136657;
    s7  -= s14 * 683901;
    s14  = 0;

    s1  += s13 * 666643;
    s2  += s13 * 470296;
    s3  += s13 * 654183;
    s4  -= s13 * 997805;
    s5  += s13 * 136657;
    s6  -= s13 * 683901;
    s13  = 0;

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12  = 0;

    carry0 = (s0 + (1 << 20)) >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry2 = (s2 + (1 << 20)) >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry4 = (s4 + (1 << 20)) >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry6 = (s6 + (1 << 20)) >> 21;
    s7 += carry6;
    s6 -= carry6 * (1 << 21);
    carry8 = (s8 + (1 << 20)) >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry10 = (s10 + (1 << 20)) >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);

    carry1 = (s1 + (1 << 20)) >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry3 = (s3 + (1 << 20)) >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry5 = (s5 + (1 << 20)) >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry7 = (s7 + (1 << 20)) >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry9 = (s9 + (1 << 20)) >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry11 = (s11 + (1 << 20)) >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12  = 0;

    carry0 = s0 >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry1 = s1 >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry2 = s2 >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry3 = s3 >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry4 = s4 >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry5 = s5 >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry6 = s6 >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry7 = s7 >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry8 = s8 >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry9 = s9 >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry10 = s10 >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);
    carry11 = s11 >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12  = 0;

    carry0 = s0 >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry1 = s1 >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry2 = s2 >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry3 = s3 >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry4 = s4 >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry5 = s5 >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry6 = s6 >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry7 = s7 >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry8 = s8 >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry9 = s9 >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry10 = s10 >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);

    s[ 0] = (uint8_t) (s0  >>  0);
    s[ 1] = (uint8_t) (s0  >>  8);
    s[ 2] = (uint8_t)((s0  >> 16) | (s1  <<  5));
    s[ 3] = (uint8_t) (s1  >>  3);
    s[ 4] = (uint8_t) (s1  >> 11);
    s[ 5] = (uint8_t)((s1  >> 19) | (s2  <<  2));
    s[ 6] = (uint8_t) (s2  >>  6);
    s[ 7] = (uint8_t)((s2  >> 14) | (s3  <<  7));
    s[ 8] = (uint8_t) (s3  >>  1);
    s[ 9] = (uint8_t) (s3  >>  9);
    s[10] = (uint8_t)((s3  >> 17) | (s4  <<  4));
    s[11] = (uint8_t) (s4  >>  4);
    s[12] = (uint8_t) (s4  >> 12);
    s[13] = (uint8_t)((s4  >> 20) | (s5  <<  1));
    s[14] = (uint8_t) (s5  >>  7);
    s[15] = (uint8_t)((s5  >> 15) | (s6  <<  6));
    s[16] = (uint8_t) (s6  >>  2);
    s[17] = (uint8_t) (s6  >> 10);
    s[18] = (uint8_t)((s6  >> 18) | (s7  <<  3));
    s[19] = (uint8_t) (s7  >>  5);
    s[20] = (uint8_t) (s7  >> 13);
    s[21] = (uint8_t) (s8  >>  0);
    s[22] = (uint8_t) (s8  >>  8);
    s[23] = (uint8_t)((s8  >> 16) | (s9  <<  5));
    s[24] = (uint8_t) (s9  >>  3);
    s[25] = (uint8_t) (s9  >> 11);
    s[26] = (uint8_t)((s9  >> 19) | (s10 <<  2));
    s[27] = (uint8_t) (s10 >>  6);
    s[28] = (uint8_t)((s10 >> 14) | (s11 <<  7));
    s[29] = (uint8_t) (s11 >>  1);
    s[30] = (uint8_t) (s11 >>  9);
    s[31] = (uint8_t) (s11 >> 17);
}

/*
 * Input:
 *   a[0]+256*a[1]+...+256^31*a[31] = a
 *   b[0]+256*b[1]+...+256^31*b[31] = b
 *   c[0]+256*c[1]+...+256^31*c[31] = c
 *
 * Output:
 *   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
 *   where l = 2^252 + 27742317777372353535851937790883648493.
 */
static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
                      const uint8_t *c)
{
    int64_t a0  = kBottom21Bits &  load_3(a);
    int64_t a1  = kBottom21Bits & (load_4(a +  2) >> 5);
    int64_t a2  = kBottom21Bits & (load_3(a +  5) >> 2);
    int64_t a3  = kBottom21Bits & (load_4(a +  7) >> 7);
    int64_t a4  = kBottom21Bits & (load_4(a + 10) >> 4);
    int64_t a5  = kBottom21Bits & (load_3(a + 13) >> 1);
    int64_t a6  = kBottom21Bits & (load_4(a + 15) >> 6);
    int64_t a7  = kBottom21Bits & (load_3(a + 18) >> 3);
    int64_t a8  = kBottom21Bits &  load_3(a + 21);
    int64_t a9  = kBottom21Bits & (load_4(a + 23) >> 5);
    int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
    int64_t a11 =                 (load_4(a + 28) >> 7);
    int64_t b0  = kBottom21Bits &  load_3(b);
    int64_t b1  = kBottom21Bits & (load_4(b +  2) >> 5);
    int64_t b2  = kBottom21Bits & (load_3(b +  5) >> 2);
    int64_t b3  = kBottom21Bits & (load_4(b +  7) >> 7);
    int64_t b4  = kBottom21Bits & (load_4(b + 10) >> 4);
    int64_t b5  = kBottom21Bits & (load_3(b + 13) >> 1);
    int64_t b6  = kBottom21Bits & (load_4(b + 15) >> 6);
    int64_t b7  = kBottom21Bits & (load_3(b + 18) >> 3);
    int64_t b8  = kBottom21Bits &  load_3(b + 21);
    int64_t b9  = kBottom21Bits & (load_4(b + 23) >> 5);
    int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
    int64_t b11 =                 (load_4(b + 28) >> 7);
    int64_t c0  = kBottom21Bits &  load_3(c);
    int64_t c1  = kBottom21Bits & (load_4(c +  2) >> 5);
    int64_t c2  = kBottom21Bits & (load_3(c +  5) >> 2);
    int64_t c3  = kBottom21Bits & (load_4(c +  7) >> 7);
    int64_t c4  = kBottom21Bits & (load_4(c + 10) >> 4);
    int64_t c5  = kBottom21Bits & (load_3(c + 13) >> 1);
    int64_t c6  = kBottom21Bits & (load_4(c + 15) >> 6);
    int64_t c7  = kBottom21Bits & (load_3(c + 18) >> 3);
    int64_t c8  = kBottom21Bits &  load_3(c + 21);
    int64_t c9  = kBottom21Bits & (load_4(c + 23) >> 5);
    int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
    int64_t c11 =                 (load_4(c + 28) >> 7);
    int64_t s0;
    int64_t s1;
    int64_t s2;
    int64_t s3;
    int64_t s4;
    int64_t s5;
    int64_t s6;
    int64_t s7;
    int64_t s8;
    int64_t s9;
    int64_t s10;
    int64_t s11;
    int64_t s12;
    int64_t s13;
    int64_t s14;
    int64_t s15;
    int64_t s16;
    int64_t s17;
    int64_t s18;
    int64_t s19;
    int64_t s20;
    int64_t s21;
    int64_t s22;
    int64_t s23;
    int64_t carry0;
    int64_t carry1;
    int64_t carry2;
    int64_t carry3;
    int64_t carry4;
    int64_t carry5;
    int64_t carry6;
    int64_t carry7;
    int64_t carry8;
    int64_t carry9;
    int64_t carry10;
    int64_t carry11;
    int64_t carry12;
    int64_t carry13;
    int64_t carry14;
    int64_t carry15;
    int64_t carry16;
    int64_t carry17;
    int64_t carry18;
    int64_t carry19;
    int64_t carry20;
    int64_t carry21;
    int64_t carry22;

    s0  = c0   +   a0 * b0;
    s1  = c1   +   a0 * b1   +   a1 * b0;
    s2  = c2   +   a0 * b2   +   a1 * b1   +   a2 * b0;
    s3  = c3   +   a0 * b3   +   a1 * b2   +   a2 * b1  +   a3 * b0;
    s4  = c4   +   a0 * b4   +   a1 * b3   +   a2 * b2  +   a3 * b1  +   a4 * b0;
    s5  = c5   +   a0 * b5   +   a1 * b4   +   a2 * b3  +   a3 * b2  +   a4 * b1  +   a5 * b0;
    s6  = c6   +   a0 * b6   +   a1 * b5   +   a2 * b4  +   a3 * b3  +   a4 * b2  +   a5 * b1 +   a6 * b0;
    s7  = c7   +   a0 * b7   +   a1 * b6   +   a2 * b5  +   a3 * b4  +   a4 * b3  +   a5 * b2 +   a6 * b1   +   a7 * b0;
    s8  = c8   +   a0 * b8   +   a1 * b7   +   a2 * b6  +   a3 * b5  +   a4 * b4  +   a5 * b3 +   a6 * b2   +   a7 * b1   +   a8 * b0;
    s9  = c9   +   a0 * b9   +   a1 * b8   +   a2 * b7  +   a3 * b6  +   a4 * b5  +   a5 * b4 +   a6 * b3   +   a7 * b2   +   a8 * b1  +   a9 * b0;
    s10 = c10  +   a0 * b10  +   a1 * b9   +   a2 * b8  +   a3 * b7  +   a4 * b6  +   a5 * b5 +   a6 * b4   +   a7 * b3   +   a8 * b2  +   a9 * b1  +  a10 * b0;
    s11 = c11  +   a0 * b11  +   a1 * b10  +   a2 * b9  +   a3 * b8  +   a4 * b7  +   a5 * b6 +   a6 * b5   +   a7 * b4   +   a8 * b3  +   a9 * b2  +  a10 * b1  +  a11 * b0;
    s12 =          a1 * b11  +   a2 * b10  +   a3 * b9  +   a4 * b8  +   a5 * b7  +   a6 * b6 +   a7 * b5   +   a8 * b4   +   a9 * b3  +  a10 * b2  +  a11 * b1;
    s13 =          a2 * b11  +   a3 * b10  +   a4 * b9  +   a5 * b8  +   a6 * b7  +   a7 * b6 +   a8 * b5   +   a9 * b4   +  a10 * b3  +  a11 * b2;
    s14 =          a3 * b11  +   a4 * b10  +   a5 * b9  +   a6 * b8  +   a7 * b7  +   a8 * b6 +   a9 * b5   +  a10 * b4   +  a11 * b3;
    s15 =          a4 * b11  +   a5 * b10  +   a6 * b9  +   a7 * b8  +   a8 * b7  +   a9 * b6 +  a10 * b5   +  a11 * b4;
    s16 =          a5 * b11  +   a6 * b10  +   a7 * b9  +   a8 * b8  +   a9 * b7  +  a10 * b6 +  a11 * b5;
    s17 =          a6 * b11  +   a7 * b10  +   a8 * b9  +   a9 * b8  +  a10 * b7  +  a11 * b6;
    s18 =          a7 * b11  +   a8 * b10  +   a9 * b9  +  a10 * b8  +  a11 * b7;
    s19 =          a8 * b11  +   a9 * b10  +  a10 * b9  +  a11 * b8;
    s20 =          a9 * b11  +  a10 * b10  +  a11 * b9;
    s21 =         a10 * b11  +  a11 * b10;
    s22 =         a11 * b11;
    s23 =         0;

    carry0 = (s0 + (1 << 20)) >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry2 = (s2 + (1 << 20)) >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry4 = (s4 + (1 << 20)) >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry6 = (s6 + (1 << 20)) >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry8 = (s8 + (1 << 20)) >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry10 = (s10 + (1 << 20)) >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);
    carry12 = (s12 + (1 << 20)) >> 21;
    s13 += carry12;
    s12 -= carry12 * (1 << 21);
    carry14 = (s14 + (1 << 20)) >> 21;
    s15 += carry14;
    s14 -= carry14 * (1 << 21);
    carry16 = (s16 + (1 << 20)) >> 21;
    s17 += carry16;
    s16 -= carry16 * (1 << 21);
    carry18 = (s18 + (1 << 20)) >> 21;
    s19 += carry18;
    s18 -= carry18 * (1 << 21);
    carry20 = (s20 + (1 << 20)) >> 21;
    s21 += carry20;
    s20 -= carry20 * (1 << 21);
    carry22 = (s22 + (1 << 20)) >> 21;
    s23 += carry22;
    s22 -= carry22 * (1 << 21);

    carry1 = (s1 + (1 << 20)) >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry3 = (s3 + (1 << 20)) >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry5 = (s5 + (1 << 20)) >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry7 = (s7 + (1 << 20)) >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry9 = (s9 + (1 << 20)) >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry11 = (s11 + (1 << 20)) >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);
    carry13 = (s13 + (1 << 20)) >> 21;
    s14 += carry13;
    s13 -= carry13 * (1 << 21);
    carry15 = (s15 + (1 << 20)) >> 21;
    s16 += carry15;
    s15 -= carry15 * (1 << 21);
    carry17 = (s17 + (1 << 20)) >> 21;
    s18 += carry17;
    s17 -= carry17 * (1 << 21);
    carry19 = (s19 + (1 << 20)) >> 21;
    s20 += carry19;
    s19 -= carry19 * (1 << 21);
    carry21 = (s21 + (1 << 20)) >> 21;
    s22 += carry21;
    s21 -= carry21 * (1 << 21);

    s11 += s23 * 666643;
    s12 += s23 * 470296;
    s13 += s23 * 654183;
    s14 -= s23 * 997805;
    s15 += s23 * 136657;
    s16 -= s23 * 683901;
    s23  = 0;

    s10 += s22 * 666643;
    s11 += s22 * 470296;
    s12 += s22 * 654183;
    s13 -= s22 * 997805;
    s14 += s22 * 136657;
    s15 -= s22 * 683901;
    s22  = 0;

    s9  += s21 * 666643;
    s10 += s21 * 470296;
    s11 += s21 * 654183;
    s12 -= s21 * 997805;
    s13 += s21 * 136657;
    s14 -= s21 * 683901;
    s21  = 0;

    s8  += s20 * 666643;
    s9  += s20 * 470296;
    s10 += s20 * 654183;
    s11 -= s20 * 997805;
    s12 += s20 * 136657;
    s13 -= s20 * 683901;
    s20  = 0;

    s7  += s19 * 666643;
    s8  += s19 * 470296;
    s9  += s19 * 654183;
    s10 -= s19 * 997805;
    s11 += s19 * 136657;
    s12 -= s19 * 683901;
    s19  = 0;

    s6  += s18 * 666643;
    s7  += s18 * 470296;
    s8  += s18 * 654183;
    s9  -= s18 * 997805;
    s10 += s18 * 136657;
    s11 -= s18 * 683901;
    s18  = 0;

    carry6 = (s6 + (1 << 20)) >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry8 = (s8 + (1 << 20)) >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry10 = (s10 + (1 << 20)) >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);
    carry12 = (s12 + (1 << 20)) >> 21;
    s13 += carry12;
    s12 -= carry12 * (1 << 21);
    carry14 = (s14 + (1 << 20)) >> 21;
    s15 += carry14;
    s14 -= carry14 * (1 << 21);
    carry16 = (s16 + (1 << 20)) >> 21;
    s17 += carry16;
    s16 -= carry16 * (1 << 21);

    carry7 = (s7 + (1 << 20)) >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry9 = (s9 + (1 << 20)) >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry11 = (s11 + (1 << 20)) >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);
    carry13 = (s13 + (1 << 20)) >> 21;
    s14 += carry13;
    s13 -= carry13 * (1 << 21);
    carry15 = (s15 + (1 << 20)) >> 21;
    s16 += carry15;
    s15 -= carry15 * (1 << 21);

    s5  += s17 * 666643;
    s6  += s17 * 470296;
    s7  += s17 * 654183;
    s8  -= s17 * 997805;
    s9  += s17 * 136657;
    s10 -= s17 * 683901;
    s17  = 0;

    s4  += s16 * 666643;
    s5  += s16 * 470296;
    s6  += s16 * 654183;
    s7  -= s16 * 997805;
    s8  += s16 * 136657;
    s9  -= s16 * 683901;
    s16  = 0;

    s3  += s15 * 666643;
    s4  += s15 * 470296;
    s5  += s15 * 654183;
    s6  -= s15 * 997805;
    s7  += s15 * 136657;
    s8  -= s15 * 683901;
    s15  = 0;

    s2  += s14 * 666643;
    s3  += s14 * 470296;
    s4  += s14 * 654183;
    s5  -= s14 * 997805;
    s6  += s14 * 136657;
    s7  -= s14 * 683901;
    s14  = 0;

    s1  += s13 * 666643;
    s2  += s13 * 470296;
    s3  += s13 * 654183;
    s4  -= s13 * 997805;
    s5  += s13 * 136657;
    s6  -= s13 * 683901;
    s13  = 0;

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12 = 0;

    carry0 = (s0 + (1 << 20)) >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry2 = (s2 + (1 << 20)) >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry4 = (s4 + (1 << 20)) >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry6 = (s6 + (1 << 20)) >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry8 = (s8 + (1 << 20)) >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry10 = (s10 + (1 << 20)) >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);

    carry1 = (s1 + (1 << 20)) >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry3 = (s3 + (1 << 20)) >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry5 = (s5 + (1 << 20)) >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry7 = (s7 + (1 << 20)) >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry9 = (s9 + (1 << 20)) >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry11 = (s11 + (1 << 20)) >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12  = 0;

    carry0 = s0 >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry1 = s1 >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry2 = s2 >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry3 = s3 >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry4 = s4 >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry5 = s5 >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry6 = s6 >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry7 = s7 >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry8 = s8 >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry9 = s9 >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry10 = s10 >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);
    carry11 = s11 >> 21;
    s12 += carry11;
    s11 -= carry11 * (1 << 21);

    s0  += s12 * 666643;
    s1  += s12 * 470296;
    s2  += s12 * 654183;
    s3  -= s12 * 997805;
    s4  += s12 * 136657;
    s5  -= s12 * 683901;
    s12  = 0;

    carry0 = s0 >> 21;
    s1  += carry0;
    s0  -= carry0 * (1 << 21);
    carry1 = s1 >> 21;
    s2  += carry1;
    s1  -= carry1 * (1 << 21);
    carry2 = s2 >> 21;
    s3  += carry2;
    s2  -= carry2 * (1 << 21);
    carry3 = s3 >> 21;
    s4  += carry3;
    s3  -= carry3 * (1 << 21);
    carry4 = s4 >> 21;
    s5  += carry4;
    s4  -= carry4 * (1 << 21);
    carry5 = s5 >> 21;
    s6  += carry5;
    s5  -= carry5 * (1 << 21);
    carry6 = s6 >> 21;
    s7  += carry6;
    s6  -= carry6 * (1 << 21);
    carry7 = s7 >> 21;
    s8  += carry7;
    s7  -= carry7 * (1 << 21);
    carry8 = s8 >> 21;
    s9  += carry8;
    s8  -= carry8 * (1 << 21);
    carry9 = s9 >> 21;
    s10 += carry9;
    s9  -= carry9 * (1 << 21);
    carry10 = s10 >> 21;
    s11 += carry10;
    s10 -= carry10 * (1 << 21);

    s[ 0] = (uint8_t) (s0  >>  0);
    s[ 1] = (uint8_t) (s0  >>  8);
    s[ 2] = (uint8_t)((s0  >> 16) | (s1 << 5));
    s[ 3] = (uint8_t) (s1  >>  3);
    s[ 4] = (uint8_t) (s1  >> 11);
    s[ 5] = (uint8_t)((s1  >> 19) | (s2 << 2));
    s[ 6] = (uint8_t) (s2  >>  6);
    s[ 7] = (uint8_t)((s2  >> 14) | (s3 << 7));
    s[ 8] = (uint8_t) (s3  >>  1);
    s[ 9] = (uint8_t) (s3  >>  9);
    s[10] = (uint8_t)((s3  >> 17) | (s4 << 4));
    s[11] = (uint8_t) (s4  >>  4);
    s[12] = (uint8_t) (s4  >> 12);
    s[13] = (uint8_t)((s4  >> 20) | (s5 << 1));
    s[14] = (uint8_t) (s5  >>  7);
    s[15] = (uint8_t)((s5  >> 15) | (s6 << 6));
    s[16] = (uint8_t) (s6  >>  2);
    s[17] = (uint8_t) (s6  >> 10);
    s[18] = (uint8_t)((s6  >> 18) | (s7 << 3));
    s[19] = (uint8_t) (s7  >>  5);
    s[20] = (uint8_t) (s7  >> 13);
    s[21] = (uint8_t) (s8  >>  0);
    s[22] = (uint8_t) (s8  >>  8);
    s[23] = (uint8_t)((s8  >> 16) | (s9 << 5));
    s[24] = (uint8_t) (s9  >>  3);
    s[25] = (uint8_t) (s9  >> 11);
    s[26] = (uint8_t)((s9  >> 19) | (s10 << 2));
    s[27] = (uint8_t) (s10 >>  6);
    s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
    s[29] = (uint8_t) (s11 >>  1);
    s[30] = (uint8_t) (s11 >>  9);
    s[31] = (uint8_t) (s11 >> 17);
}

int ED25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
                 const uint8_t public_key[32], const uint8_t private_key[32],
                 OSSL_LIB_CTX *libctx, const char *propq)
{
    uint8_t az[SHA512_DIGEST_LENGTH];
    uint8_t nonce[SHA512_DIGEST_LENGTH];
    ge_p3 R;
    uint8_t hram[SHA512_DIGEST_LENGTH];
    EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
    EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new();
    unsigned int sz;
    int res = 0;

    if (sha512 == NULL || hash_ctx == NULL)
        goto err;

    if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
        || !EVP_DigestUpdate(hash_ctx, private_key, 32)
        || !EVP_DigestFinal_ex(hash_ctx, az, &sz))
        goto err;

    az[0] &= 248;
    az[31] &= 63;
    az[31] |= 64;

    if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
        || !EVP_DigestUpdate(hash_ctx, az + 32, 32)
        || !EVP_DigestUpdate(hash_ctx, message, message_len)
        || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz))
        goto err;

    x25519_sc_reduce(nonce);
    ge_scalarmult_base(&R, nonce);
    ge_p3_tobytes(out_sig, &R);

    if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
        || !EVP_DigestUpdate(hash_ctx, out_sig, 32)
        || !EVP_DigestUpdate(hash_ctx, public_key, 32)
        || !EVP_DigestUpdate(hash_ctx, message, message_len)
        || !EVP_DigestFinal_ex(hash_ctx, hram, &sz))
        goto err;

    x25519_sc_reduce(hram);
    sc_muladd(out_sig + 32, hram, az, nonce);

    res = 1;
err:
    OPENSSL_cleanse(nonce, sizeof(nonce));
    OPENSSL_cleanse(az, sizeof(az));
    EVP_MD_free(sha512);
    EVP_MD_CTX_free(hash_ctx);
    return res;
}

static const char allzeroes[15];

int ED25519_verify(const uint8_t *message, size_t message_len,
                   const uint8_t signature[64], const uint8_t public_key[32],
                   OSSL_LIB_CTX *libctx, const char *propq)
{
    int i;
    ge_p3 A;
    const uint8_t *r, *s;
    EVP_MD *sha512;
    EVP_MD_CTX *hash_ctx = NULL;
    unsigned int sz;
    int res = 0;
    ge_p2 R;
    uint8_t rcheck[32];
    uint8_t h[SHA512_DIGEST_LENGTH];
    /* 27742317777372353535851937790883648493 in little endian format */
    const uint8_t l_low[16] = {
        0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
        0xDE, 0xF9, 0xDE, 0x14
    };

    r = signature;
    s = signature + 32;

    /*
     * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
     *
     * If not the signature is publicly invalid. Since it's public we can do the
     * check in variable time.
     *
     * First check the most significant byte
     */
    if (s[31] > 0x10)
        return 0;
    if (s[31] == 0x10) {
        /*
         * Most significant byte indicates a value close to 2^252 so check the
         * rest
         */
        if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
            return 0;
        for (i = 15; i >= 0; i--) {
            if (s[i] < l_low[i])
                break;
            if (s[i] > l_low[i])
                return 0;
        }
        if (i < 0)
            return 0;
    }

    if (ge_frombytes_vartime(&A, public_key) != 0) {
        return 0;
    }

    fe_neg(A.X, A.X);
    fe_neg(A.T, A.T);

    sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
    if (sha512 == NULL)
        return 0;
    hash_ctx = EVP_MD_CTX_new();
    if (hash_ctx == NULL)
        goto err;

    if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
        || !EVP_DigestUpdate(hash_ctx, r, 32)
        || !EVP_DigestUpdate(hash_ctx, public_key, 32)
        || !EVP_DigestUpdate(hash_ctx, message, message_len)
        || !EVP_DigestFinal_ex(hash_ctx, h, &sz))
        goto err;

    x25519_sc_reduce(h);

    ge_double_scalarmult_vartime(&R, h, &A, s);

    ge_tobytes(rcheck, &R);

    res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
err:
    EVP_MD_free(sha512);
    EVP_MD_CTX_free(hash_ctx);
    return res;
}

int ED25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32],
                                const uint8_t private_key[32], const char *propq)
{
    uint8_t az[SHA512_DIGEST_LENGTH];
    ge_p3 A;
    int r;
    EVP_MD *sha512 = NULL;

    sha512 = EVP_MD_fetch(ctx, SN_sha512, propq);
    if (sha512 == NULL)
        return 0;
    r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL);
    EVP_MD_free(sha512);
    if (!r) {
        OPENSSL_cleanse(az, sizeof(az));
        return 0;
    }

    az[0] &= 248;
    az[31] &= 63;
    az[31] |= 64;

    ge_scalarmult_base(&A, az);
    ge_p3_tobytes(out_public_key, &A);

    OPENSSL_cleanse(az, sizeof(az));
    return 1;
}

int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
           const uint8_t peer_public_value[32])
{
    static const uint8_t kZeros[32] = {0};
    x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
    /* The all-zero output results when the input is a point of small order. */
    return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
}

void X25519_public_from_private(uint8_t out_public_value[32],
                                const uint8_t private_key[32])
{
    uint8_t e[32];
    ge_p3 A;
    fe zplusy, zminusy, zminusy_inv;

    memcpy(e, private_key, 32);
    e[0] &= 248;
    e[31] &= 127;
    e[31] |= 64;

    ge_scalarmult_base(&A, e);

    /*
     * We only need the u-coordinate of the curve25519 point.
     * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
     * u=(Z+Y)/(Z-Y).
     */
    fe_add(zplusy, A.Z, A.Y);
    fe_sub(zminusy, A.Z, A.Y);
    fe_invert(zminusy_inv, zminusy);
    fe_mul(zplusy, zplusy, zminusy_inv);
    fe_tobytes(out_public_value, zplusy);

    OPENSSL_cleanse(e, sizeof(e));
}