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openssl/crypto/ec/curve25519.c

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/*
* Copyright 2016-2019 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (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
*/
#include <string.h>
#include "ec_local.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},
},
{
{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},
},
{
{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
-19188627, -15224819, -9818940, -12085777},
{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
-15689887, 1762328, 14866737},
{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
-28236412, 3959421, 27914454, 4383652},
},
{
{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},
},
{
{14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
30290735, 10876454, -33154098, 2381726},
{-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
-6236617, 3696005, -32300832, 15351955},
{27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
-32961401, -2970515, 29551813, 10109425},
},
},
{
{
{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
-2378284, -1627556, 10092783, -4764171},
{27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
-32508754, 12005538, -17810127, 12803510},
{17228999, -15661624, -1233527, 300140, -1224870, -11714777,
30364213, -9038194, 18016357, 4397660},
},
{
{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
-26619106, 14544525, -17477504, 982639},
{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
-4120128, -21047696, 9934963},
{5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
-14747930, 4559895, -30123922, -10897950},
},
{
{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
24191034, 4541697, -13338309, 5500568},
{12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
-17430592, 12264343, 10874051, 13524335},
{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
5080568, -22528059, 5376628},
},
{
{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
-22321305, -9447443, 4535768, 1569007},
{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
-30494562, 3044290, 31848280, 12543772},
{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
-27377195, -2062731, 7718482, 14474653},
},
{
{2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
-7236665, 24316168, -5253567},
{13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
33040651, -13424532, -20729456, 8321686},
{21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
23845965, -11874838, -9984458, 608372},
},
{
{-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
1123968, -6780577, 27229399, 23887},
{-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
12797024, -6440308, -1633405, 16678954},
{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
-1508144, -4795045, -17169265, 4904953},
},
{
{24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
5169211, 16191880, 2128236, -4326833},
{-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
-29806336, 916033, -6882542, -2986532},
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{-12622226, -6204820, 30718825, 2591312, -10617028, 12192840,
18873298, -7297090, -32297756, 15221632},
{-26478122, -11103864, 11546244, -1852483, 9180880, 7656409,
-21343950, 2095755, 29769758, 6593415},
{-31994208, -2907461, 4176912, 3264766, 12538965, -868111, 26312345,
-6118678, 30958054, 8292160},
},
{
{31429822, -13959116, 29173532, 15632448, 12174511, -2760094,
32808831, 3977186, 26143136, -3148876},
{22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633,
-1674433, -3758243, -2304625},
{-15491917, 8012313, -2514730, -12702462, -23965846, -10254029,
-1612713, -1535569, -16664475, 8194478},
},
{
{27338066, -7507420, -7414224, 10140405, -19026427, -6589889,
27277191, 8855376, 28572286, 3005164},
{26287124, 4821776, 25476601, -4145903, -3764513, -15788984,
-18008582, 1182479, -26094821, -13079595},
{-7171154, 3178080, 23970071, 6201893, -17195577, -4489192,
-21876275, -13982627, 32208683, -1198248},
},
{
{-16657702, 2817643, -10286362, 14811298, 6024667, 13349505,
-27315504, -10497842, -27672585, -11539858},
{15941029, -9405932, -21367050, 8062055, 31876073, -238629,
-15278393, -1444429, 15397331, -4130193},
{8934485, -13485467, -23286397, -13423241, -32446090, 14047986,
31170398, -1441021, -27505566, 15087184},
},
{
{-18357243, -2156491, 24524913, -16677868, 15520427, -6360776,
-15502406, 11461896, 16788528, -5868942},
{-1947386, 16013773, 21750665, 3714552, -17401782, -16055433,
-3770287, -10323320, 31322514, -11615635},
{21426655, -5650218, -13648287, -5347537, -28812189, -4920970,
-18275391, -14621414, 13040862, -12112948},
},
{
{11293895, 12478086, -27136401, 15083750, -29307421, 14748872,
14555558, -13417103, 1613711, 4896935},
{-25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
2825960, -4897045, -23971776, -11267415},
{-15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
20615400, 12405433, -23753030, -8436416},
},
{
{-7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
4378436, 2432030, 23097949, -566018},
{4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
10103221, -18512313, 2424778},
{366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
1344109, -3642553, 12412659},
},
{
{-24001791, 7690286, 14929416, -168257, -32210835, -13412986,
24162697, -15326504, -3141501, 11179385},
{18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
-6001441, -1486897, -18684645, -11443503},
{476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
13403813, 11052904, 5219329},
},
},
{
{
{20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
31186971, -3973730, 9014762, -8579056},
{-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
-33102500, 9160280, 8473550, -3256838},
{24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
-7689309, -16335821, -24568481, 11788948},
},
{
{-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
-20037437, 10410733, -24568470, -1458691},
{-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
11871841, -12505194, -18513325, 8464118},
{-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
14325289, 8628612, 33313881, -8370517},
},
{
{-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
-24805667, -10236854, -8940735, -5818269},
{-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
15989197, -12838188, 28358192, -4253904},
{-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
-16637684, 4072016, -5351664, 5596589},
},
{
{-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
29266239, 2557221, 1768301, 15373193},
{-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
-4504991, -24660491, 3442910},
{-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
22597931, 7176455, -18585478, 13365930},
},
{
{-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
-8570186, -9689599, -3031667},
{25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
683793, -11823784, 15723479, -15163481},
{-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
11879682, 5400171, 519526, -1235876},
},
{
{22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
-20353881, 7315967, 16648397, 7605640},
{-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
23994942, -5281555, -9468848, 4763278},
{-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
31088447, -7764523, -11356529, 728112},
},
{
{26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
-4273545, -12555558, -29365436, -5498272},
{17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
12327945, 10750447, 10014012},
{-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
-27481051, -666732, 3424691, 7540221},
},
{
{30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
-16317219, -9244265, 15258046},
{13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
2711395, 1062915, -5136345},
{-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
-6066489, 12194497, 32960380, 1459310},
},
},
{
{
{19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
-6101885, 18638003, -11174937},
{31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
9012486, -7584354, -6643087, -5442636},
{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
9677543, -32294889, -6456008},
},
{
{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
-7839692, -7852844, -8138429},
{-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
-27333451, 10754588, -9431476, 5203576},
{31834314, 14135496, -770007, 5159118, 20917671, -16768096,
-7467973, -7337524, 31809243, 7347066},
},
{
{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
19797970, -12211255, 15192876, -2087490},
{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
10609330, 12694420, 33473243, -13382104},
{33184999, 11180355, 15832085, -11385430, -1633671, 225884,
15089336, -11023903, -6135662, 14480053},
},
{
{31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
5496208, 13685227, 27595050, 8737275},
{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
-31008351, -12610604, 26498114, 66511},
{22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
30540766, -4286747, -13327787, -7515095},
},
{
{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
8205540, 13585437, -17127465, 15115439},
{23711543, -672915, 31206561, -8362711, 6164647, -9709987,
-33535882, -1426096, 8236921, 16492939},
{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
19574902, 10071562, 6708380, -6222424},
},
{
{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
9328700, 29955601, -11678310},
{3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
-25892142, -12635595, -9917575, 6216608},
{-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
24822830, -6146567, -26767480, 7525079},
},
{
{-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
-910336, -2782495, -19386633, 11994101},
{21691500, -13624626, -641331, -14367021, 3285881, -3483596,
-25064666, 9718258, -7477437, 13381418},
{18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
23111648, -6375247, 28535282, 15779576},
},
{
{30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
-14068454, 12021730, 9955285, -16303356},
{9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
-18426029, 9924399, 20194861, 13380996},
{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
-1984914, 15707771, 26342023, 10146099},
},
},
{
{
{-26016874, -219943, 21339191, -41388, 19745256, -2878700,
-29637280, 2227040, 21612326, -545728},
{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
25764461, 12243797, -20856566, 11649658},
{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
6114064, 33514190, 2333242},
},
{
{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
-6679750, -12670638, 24350578, -13450001},
{-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
-31536088, -10406836, 8317860, 12352766},
{18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
-23552096, -2287550, 20712163, 6719373},
},
{
{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
-3763210, 26224235, -3297458},
{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
21728352, 9493610, 18620611, -16428628},
{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
-5269471, -9725556, -30701573, -16479657},
},
{
{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
12248509, -5240639, 13735342, 1934062},
{25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
-15136294, -3765346, -21277997, 5473616},
{31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
-7125085, 12469656, 29111212, -5451014},
},
{
{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])
{
uint8_t az[SHA512_DIGEST_LENGTH];
uint8_t nonce[SHA512_DIGEST_LENGTH];
ge_p3 R;
uint8_t hram[SHA512_DIGEST_LENGTH];
SHA512_CTX hash_ctx;
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, private_key, 32);
SHA512_Final(az, &hash_ctx);
az[0] &= 248;
az[31] &= 63;
az[31] |= 64;
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, az + 32, 32);
SHA512_Update(&hash_ctx, message, message_len);
SHA512_Final(nonce, &hash_ctx);
x25519_sc_reduce(nonce);
ge_scalarmult_base(&R, nonce);
ge_p3_tobytes(out_sig, &R);
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, out_sig, 32);
SHA512_Update(&hash_ctx, public_key, 32);
SHA512_Update(&hash_ctx, message, message_len);
SHA512_Final(hram, &hash_ctx);
x25519_sc_reduce(hram);
sc_muladd(out_sig + 32, hram, az, nonce);
OPENSSL_cleanse(&hash_ctx, sizeof(hash_ctx));
OPENSSL_cleanse(nonce, sizeof(nonce));
OPENSSL_cleanse(az, sizeof(az));
return 1;
}
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])
{
int i;
ge_p3 A;
const uint8_t *r, *s;
SHA512_CTX hash_ctx;
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_Init(&hash_ctx);
SHA512_Update(&hash_ctx, r, 32);
SHA512_Update(&hash_ctx, public_key, 32);
SHA512_Update(&hash_ctx, message, message_len);
SHA512_Final(h, &hash_ctx);
x25519_sc_reduce(h);
ge_double_scalarmult_vartime(&R, h, &A, s);
ge_tobytes(rcheck, &R);
return CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
}
void ED25519_public_from_private(uint8_t out_public_key[32],
const uint8_t private_key[32])
{
uint8_t az[SHA512_DIGEST_LENGTH];
ge_p3 A;
SHA512(private_key, 32, az);
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));
}
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));
}
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