3c849bc901
Move base 2^64 code to own #if section. It was nested in base 2^51 section, which arguably might have been tricky to follow. Reviewed-by: Rich Salz <rsalz@openssl.org> (Merged from https://github.com/openssl/openssl/pull/6699)
5457 lines
190 KiB
C
5457 lines
190 KiB
C
/*
|
|
* Copyright 2016-2018 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_lcl.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(__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 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) & 8388607) << 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},
|
|
{-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
|
|
285431, 2763829, 15736322, 4143876},
|
|
},
|
|
{
|
|
{2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
|
|
-14594663, 23527084, -16458268},
|
|
{33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
|
|
-32625340, 4087881, -15188911, -14416214},
|
|
{1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
|
|
4357868, -4774191, -16323038},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
|
|
23365147, -3949732, 7390890, 2759800},
|
|
{4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
|
|
-4264057, 1244380, -12919645},
|
|
{-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
|
|
9208236, 15886429, 16489664},
|
|
},
|
|
{
|
|
{1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
|
|
-13693198, 398369, -30606455, -712933},
|
|
{-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
|
|
13348553, 12076947, -30836462, 5113182},
|
|
{-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
|
|
-30341101, -7336386, 13847711, 5387222},
|
|
},
|
|
{
|
|
{-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
|
|
8763061, 3617786, -19600662, 10370991},
|
|
{20246567, -14369378, 22358229, -543712, 18507283, -10413996,
|
|
14554437, -8746092, 32232924, 16763880},
|
|
{9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
|
|
11094161, 15689506, 3140038, -16510092},
|
|
},
|
|
{
|
|
{-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
|
|
-27224800, 9448613, -28774454, 366295},
|
|
{19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
|
|
28344573, 8041113, 719605, 11671788},
|
|
{8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
|
|
-15266516, 27000813, -10195553},
|
|
},
|
|
{
|
|
{-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
|
|
5336097, 6750977, -14521026},
|
|
{11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
|
|
1695823, -8819122, 8169720, 16220347},
|
|
{-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
|
|
-11144307, -2627664, -5990708, -14166033},
|
|
},
|
|
{
|
|
{-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
|
|
27884329, 2847284, 2655861, 1738395},
|
|
{-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
|
|
21651608, -3239336, -19087449, -11005278},
|
|
{1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
|
|
5821408, 10478196, 8544890},
|
|
},
|
|
{
|
|
{32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
|
|
19270449, 12217473, 17789017, -3395995},
|
|
{-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
|
|
15479401, -3853233, 30460520, 1052596},
|
|
{-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
|
|
27491595, -4612359, 3179268, -9478891},
|
|
},
|
|
{
|
|
{31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
|
|
-14756777, -16411740, 19072640, -9511060},
|
|
{11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
|
|
5977896, -5215017, 473099, 5040608},
|
|
{-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
|
|
28326862, 1721092, -19558642, -3131606},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
|
|
8076149, -27868496, 11538389},
|
|
{-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
|
|
8754525, 7446702, -5676054, 5797016},
|
|
{-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
|
|
2014099, -9050574, -2369172, -5877341},
|
|
},
|
|
{
|
|
{-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
|
|
1192730, -3714199, 15123619, 10811505},
|
|
{14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
|
|
15776356, -28886779, -11974553},
|
|
{-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
|
|
-20654173, -16484855, 4714547, -9600655},
|
|
},
|
|
{
|
|
{15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
|
|
24611599, -4543832, -11745876, 12340220},
|
|
{12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
|
|
9613953, 8241152, 15370987, 9608631},
|
|
{-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
|
|
15866074, -28210621, -8814099},
|
|
},
|
|
{
|
|
{26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
|
|
858697, 20571223, 8420556},
|
|
{14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
|
|
33531827, 12516406, -21574435, -12476749},
|
|
{236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
|
|
7256740, 8791136, 15069930},
|
|
},
|
|
{
|
|
{1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
|
|
14711875, 4874229, -30663140, -2331391},
|
|
{5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
|
|
-7912378, -33069337, 9234253},
|
|
{20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
|
|
31559055, -11609587, 18979186, 13396066},
|
|
},
|
|
{
|
|
{24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
|
|
-18816887, 13594782, 33514650, 7021958},
|
|
{-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
|
|
-25948728, -3916677, -21480480, 12868082},
|
|
{-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
|
|
-21446107, 2244500, -12455797, -8089383},
|
|
},
|
|
{
|
|
{-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
|
|
33086546, 8957937, -15233648, 5540521},
|
|
{-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
|
|
-23710744, -1568984, -16128528, -14962807},
|
|
{23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
|
|
892185, -11513277, -15205948},
|
|
},
|
|
{
|
|
{9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
|
|
4763127, -19179614, 5867134},
|
|
{-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
|
|
27846559, 5931263, -29749703, -16108455},
|
|
{27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
|
|
7283490, -15148073, -19526700, 7734629},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
|
|
7585295, -3176626, 18549497, 15302069},
|
|
{-32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
|
|
10458790, -6418461, -8872242, 8424746},
|
|
{24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
|
|
19206234, 7134917, -11284482, -828919},
|
|
},
|
|
{
|
|
{11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
|
|
10243738, -14685461, -5066034, 16498837},
|
|
{8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
|
|
-14124238, 6536641, 10543906},
|
|
{-28946384, 15479763, -17466835, 568876, -1497683, 11223454,
|
|
-2669190, -16625574, -27235709, 8876771},
|
|
},
|
|
{
|
|
{-25742899, -12566864, -15649966, -846607, -33026686, -796288,
|
|
-33481822, 15824474, -604426, -9039817},
|
|
{10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697,
|
|
-4890037, 1657394, 3084098},
|
|
{10477963, -7470260, 12119566, -13250805, 29016247, -5365589,
|
|
31280319, 14396151, -30233575, 15272409},
|
|
},
|
|
{
|
|
{-12288309, 3169463, 28813183, 16658753, 25116432, -5630466,
|
|
-25173957, -12636138, -25014757, 1950504},
|
|
{-26180358, 9489187, 11053416, -14746161, -31053720, 5825630,
|
|
-8384306, -8767532, 15341279, 8373727},
|
|
{28685821, 7759505, -14378516, -12002860, -31971820, 4079242,
|
|
298136, -10232602, -2878207, 15190420},
|
|
},
|
|
{
|
|
{-32932876, 13806336, -14337485, -15794431, -24004620, 10940928,
|
|
8669718, 2742393, -26033313, -6875003},
|
|
{-1580388, -11729417, -25979658, -11445023, -17411874, -10912854,
|
|
9291594, -16247779, -12154742, 6048605},
|
|
{-30305315, 14843444, 1539301, 11864366, 20201677, 1900163,
|
|
13934231, 5128323, 11213262, 9168384},
|
|
},
|
|
{
|
|
{-26280513, 11007847, 19408960, -940758, -18592965, -4328580,
|
|
-5088060, -11105150, 20470157, -16398701},
|
|
{-23136053, 9282192, 14855179, -15390078, -7362815, -14408560,
|
|
-22783952, 14461608, 14042978, 5230683},
|
|
{29969567, -2741594, -16711867, -8552442, 9175486, -2468974,
|
|
21556951, 3506042, -5933891, -12449708},
|
|
},
|
|
{
|
|
{-3144746, 8744661, 19704003, 4581278, -20430686, 6830683,
|
|
-21284170, 8971513, -28539189, 15326563},
|
|
{-19464629, 10110288, -17262528, -3503892, -23500387, 1355669,
|
|
-15523050, 15300988, -20514118, 9168260},
|
|
{-5353335, 4488613, -23803248, 16314347, 7780487, -15638939,
|
|
-28948358, 9601605, 33087103, -9011387},
|
|
},
|
|
{
|
|
{-19443170, -15512900, -20797467, -12445323, -29824447, 10229461,
|
|
-27444329, -15000531, -5996870, 15664672},
|
|
{23294591, -16632613, -22650781, -8470978, 27844204, 11461195,
|
|
13099750, -2460356, 18151676, 13417686},
|
|
{-24722913, -4176517, -31150679, 5988919, -26858785, 6685065,
|
|
1661597, -12551441, 15271676, -15452665},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{11433042, -13228665, 8239631, -5279517, -1985436, -725718,
|
|
-18698764, 2167544, -6921301, -13440182},
|
|
{-31436171, 15575146, 30436815, 12192228, -22463353, 9395379,
|
|
-9917708, -8638997, 12215110, 12028277},
|
|
{14098400, 6555944, 23007258, 5757252, -15427832, -12950502,
|
|
30123440, 4617780, -16900089, -655628},
|
|
},
|
|
{
|
|
{-4026201, -15240835, 11893168, 13718664, -14809462, 1847385,
|
|
-15819999, 10154009, 23973261, -12684474},
|
|
{-26531820, -3695990, -1908898, 2534301, -31870557, -16550355,
|
|
18341390, -11419951, 32013174, -10103539},
|
|
{-25479301, 10876443, -11771086, -14625140, -12369567, 1838104,
|
|
21911214, 6354752, 4425632, -837822},
|
|
},
|
|
{
|
|
{-10433389, -14612966, 22229858, -3091047, -13191166, 776729,
|
|
-17415375, -12020462, 4725005, 14044970},
|
|
{19268650, -7304421, 1555349, 8692754, -21474059, -9910664, 6347390,
|
|
-1411784, -19522291, -16109756},
|
|
{-24864089, 12986008, -10898878, -5558584, -11312371, -148526,
|
|
19541418, 8180106, 9282262, 10282508},
|
|
},
|
|
{
|
|
{-26205082, 4428547, -8661196, -13194263, 4098402, -14165257,
|
|
15522535, 8372215, 5542595, -10702683},
|
|
{-10562541, 14895633, 26814552, -16673850, -17480754, -2489360,
|
|
-2781891, 6993761, -18093885, 10114655},
|
|
{-20107055, -929418, 31422704, 10427861, -7110749, 6150669,
|
|
-29091755, -11529146, 25953725, -106158},
|
|
},
|
|
{
|
|
{-4234397, -8039292, -9119125, 3046000, 2101609, -12607294,
|
|
19390020, 6094296, -3315279, 12831125},
|
|
{-15998678, 7578152, 5310217, 14408357, -33548620, -224739,
|
|
31575954, 6326196, 7381791, -2421839},
|
|
{-20902779, 3296811, 24736065, -16328389, 18374254, 7318640,
|
|
6295303, 8082724, -15362489, 12339664},
|
|
},
|
|
{
|
|
{27724736, 2291157, 6088201, -14184798, 1792727, 5857634, 13848414,
|
|
15768922, 25091167, 14856294},
|
|
{-18866652, 8331043, 24373479, 8541013, -701998, -9269457, 12927300,
|
|
-12695493, -22182473, -9012899},
|
|
{-11423429, -5421590, 11632845, 3405020, 30536730, -11674039,
|
|
-27260765, 13866390, 30146206, 9142070},
|
|
},
|
|
{
|
|
{3924129, -15307516, -13817122, -10054960, 12291820, -668366,
|
|
-27702774, 9326384, -8237858, 4171294},
|
|
{-15921940, 16037937, 6713787, 16606682, -21612135, 2790944,
|
|
26396185, 3731949, 345228, -5462949},
|
|
{-21327538, 13448259, 25284571, 1143661, 20614966, -8849387,
|
|
2031539, -12391231, -16253183, -13582083},
|
|
},
|
|
{
|
|
{31016211, -16722429, 26371392, -14451233, -5027349, 14854137,
|
|
17477601, 3842657, 28012650, -16405420},
|
|
{-5075835, 9368966, -8562079, -4600902, -15249953, 6970560,
|
|
-9189873, 16292057, -8867157, 3507940},
|
|
{29439664, 3537914, 23333589, 6997794, -17555561, -11018068,
|
|
-15209202, -15051267, -9164929, 6580396},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-12185861, -7679788, 16438269, 10826160, -8696817, -6235611,
|
|
17860444, -9273846, -2095802, 9304567},
|
|
{20714564, -4336911, 29088195, 7406487, 11426967, -5095705,
|
|
14792667, -14608617, 5289421, -477127},
|
|
{-16665533, -10650790, -6160345, -13305760, 9192020, -1802462,
|
|
17271490, 12349094, 26939669, -3752294},
|
|
},
|
|
{
|
|
{-12889898, 9373458, 31595848, 16374215, 21471720, 13221525,
|
|
-27283495, -12348559, -3698806, 117887},
|
|
{22263325, -6560050, 3984570, -11174646, -15114008, -566785,
|
|
28311253, 5358056, -23319780, 541964},
|
|
{16259219, 3261970, 2309254, -15534474, -16885711, -4581916,
|
|
24134070, -16705829, -13337066, -13552195},
|
|
},
|
|
{
|
|
{9378160, -13140186, -22845982, -12745264, 28198281, -7244098,
|
|
-2399684, -717351, 690426, 14876244},
|
|
{24977353, -314384, -8223969, -13465086, 28432343, -1176353,
|
|
-13068804, -12297348, -22380984, 6618999},
|
|
{-1538174, 11685646, 12944378, 13682314, -24389511, -14413193,
|
|
8044829, -13817328, 32239829, -5652762},
|
|
},
|
|
{
|
|
{-18603066, 4762990, -926250, 8885304, -28412480, -3187315, 9781647,
|
|
-10350059, 32779359, 5095274},
|
|
{-33008130, -5214506, -32264887, -3685216, 9460461, -9327423,
|
|
-24601656, 14506724, 21639561, -2630236},
|
|
{-16400943, -13112215, 25239338, 15531969, 3987758, -4499318,
|
|
-1289502, -6863535, 17874574, 558605},
|
|
},
|
|
{
|
|
{-13600129, 10240081, 9171883, 16131053, -20869254, 9599700,
|
|
33499487, 5080151, 2085892, 5119761},
|
|
{-22205145, -2519528, -16381601, 414691, -25019550, 2170430,
|
|
30634760, -8363614, -31999993, -5759884},
|
|
{-6845704, 15791202, 8550074, -1312654, 29928809, -12092256,
|
|
27534430, -7192145, -22351378, 12961482},
|
|
},
|
|
{
|
|
{-24492060, -9570771, 10368194, 11582341, -23397293, -2245287,
|
|
16533930, 8206996, -30194652, -5159638},
|
|
{-11121496, -3382234, 2307366, 6362031, -135455, 8868177, -16835630,
|
|
7031275, 7589640, 8945490},
|
|
{-32152748, 8917967, 6661220, -11677616, -1192060, -15793393,
|
|
7251489, -11182180, 24099109, -14456170},
|
|
},
|
|
{
|
|
{5019558, -7907470, 4244127, -14714356, -26933272, 6453165,
|
|
-19118182, -13289025, -6231896, -10280736},
|
|
{10853594, 10721687, 26480089, 5861829, -22995819, 1972175,
|
|
-1866647, -10557898, -3363451, -6441124},
|
|
{-17002408, 5906790, 221599, -6563147, 7828208, -13248918, 24362661,
|
|
-2008168, -13866408, 7421392},
|
|
},
|
|
{
|
|
{8139927, -6546497, 32257646, -5890546, 30375719, 1886181,
|
|
-21175108, 15441252, 28826358, -4123029},
|
|
{6267086, 9695052, 7709135, -16603597, -32869068, -1886135,
|
|
14795160, -7840124, 13746021, -1742048},
|
|
{28584902, 7787108, -6732942, -15050729, 22846041, -7571236,
|
|
-3181936, -363524, 4771362, -8419958},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{24949256, 6376279, -27466481, -8174608, -18646154, -9930606,
|
|
33543569, -12141695, 3569627, 11342593},
|
|
{26514989, 4740088, 27912651, 3697550, 19331575, -11472339, 6809886,
|
|
4608608, 7325975, -14801071},
|
|
{-11618399, -14554430, -24321212, 7655128, -1369274, 5214312,
|
|
-27400540, 10258390, -17646694, -8186692},
|
|
},
|
|
{
|
|
{11431204, 15823007, 26570245, 14329124, 18029990, 4796082,
|
|
-31446179, 15580664, 9280358, -3973687},
|
|
{-160783, -10326257, -22855316, -4304997, -20861367, -13621002,
|
|
-32810901, -11181622, -15545091, 4387441},
|
|
{-20799378, 12194512, 3937617, -5805892, -27154820, 9340370,
|
|
-24513992, 8548137, 20617071, -7482001},
|
|
},
|
|
{
|
|
{-938825, -3930586, -8714311, 16124718, 24603125, -6225393,
|
|
-13775352, -11875822, 24345683, 10325460},
|
|
{-19855277, -1568885, -22202708, 8714034, 14007766, 6928528,
|
|
16318175, -1010689, 4766743, 3552007},
|
|
{-21751364, -16730916, 1351763, -803421, -4009670, 3950935, 3217514,
|
|
14481909, 10988822, -3994762},
|
|
},
|
|
{
|
|
{15564307, -14311570, 3101243, 5684148, 30446780, -8051356,
|
|
12677127, -6505343, -8295852, 13296005},
|
|
{-9442290, 6624296, -30298964, -11913677, -4670981, -2057379,
|
|
31521204, 9614054, -30000824, 12074674},
|
|
{4771191, -135239, 14290749, -13089852, 27992298, 14998318,
|
|
-1413936, -1556716, 29832613, -16391035},
|
|
},
|
|
{
|
|
{7064884, -7541174, -19161962, -5067537, -18891269, -2912736,
|
|
25825242, 5293297, -27122660, 13101590},
|
|
{-2298563, 2439670, -7466610, 1719965, -27267541, -16328445,
|
|
32512469, -5317593, -30356070, -4190957},
|
|
{-30006540, 10162316, -33180176, 3981723, -16482138, -13070044,
|
|
14413974, 9515896, 19568978, 9628812},
|
|
},
|
|
{
|
|
{33053803, 199357, 15894591, 1583059, 27380243, -4580435, -17838894,
|
|
-6106839, -6291786, 3437740},
|
|
{-18978877, 3884493, 19469877, 12726490, 15913552, 13614290,
|
|
-22961733, 70104, 7463304, 4176122},
|
|
{-27124001, 10659917, 11482427, -16070381, 12771467, -6635117,
|
|
-32719404, -5322751, 24216882, 5944158},
|
|
},
|
|
{
|
|
{8894125, 7450974, -2664149, -9765752, -28080517, -12389115,
|
|
19345746, 14680796, 11632993, 5847885},
|
|
{26942781, -2315317, 9129564, -4906607, 26024105, 11769399,
|
|
-11518837, 6367194, -9727230, 4782140},
|
|
{19916461, -4828410, -22910704, -11414391, 25606324, -5972441,
|
|
33253853, 8220911, 6358847, -1873857},
|
|
},
|
|
{
|
|
{801428, -2081702, 16569428, 11065167, 29875704, 96627, 7908388,
|
|
-4480480, -13538503, 1387155},
|
|
{19646058, 5720633, -11416706, 12814209, 11607948, 12749789,
|
|
14147075, 15156355, -21866831, 11835260},
|
|
{19299512, 1155910, 28703737, 14890794, 2925026, 7269399, 26121523,
|
|
15467869, -26560550, 5052483},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-3017432, 10058206, 1980837, 3964243, 22160966, 12322533, -6431123,
|
|
-12618185, 12228557, -7003677},
|
|
{32944382, 14922211, -22844894, 5188528, 21913450, -8719943,
|
|
4001465, 13238564, -6114803, 8653815},
|
|
{22865569, -4652735, 27603668, -12545395, 14348958, 8234005,
|
|
24808405, 5719875, 28483275, 2841751},
|
|
},
|
|
{
|
|
{-16420968, -1113305, -327719, -12107856, 21886282, -15552774,
|
|
-1887966, -315658, 19932058, -12739203},
|
|
{-11656086, 10087521, -8864888, -5536143, -19278573, -3055912,
|
|
3999228, 13239134, -4777469, -13910208},
|
|
{1382174, -11694719, 17266790, 9194690, -13324356, 9720081,
|
|
20403944, 11284705, -14013818, 3093230},
|
|
},
|
|
{
|
|
{16650921, -11037932, -1064178, 1570629, -8329746, 7352753, -302424,
|
|
16271225, -24049421, -6691850},
|
|
{-21911077, -5927941, -4611316, -5560156, -31744103, -10785293,
|
|
24123614, 15193618, -21652117, -16739389},
|
|
{-9935934, -4289447, -25279823, 4372842, 2087473, 10399484,
|
|
31870908, 14690798, 17361620, 11864968},
|
|
},
|
|
{
|
|
{-11307610, 6210372, 13206574, 5806320, -29017692, -13967200,
|
|
-12331205, -7486601, -25578460, -16240689},
|
|
{14668462, -12270235, 26039039, 15305210, 25515617, 4542480,
|
|
10453892, 6577524, 9145645, -6443880},
|
|
{5974874, 3053895, -9433049, -10385191, -31865124, 3225009,
|
|
-7972642, 3936128, -5652273, -3050304},
|
|
},
|
|
{
|
|
{30625386, -4729400, -25555961, -12792866, -20484575, 7695099,
|
|
17097188, -16303496, -27999779, 1803632},
|
|
{-3553091, 9865099, -5228566, 4272701, -5673832, -16689700,
|
|
14911344, 12196514, -21405489, 7047412},
|
|
{20093277, 9920966, -11138194, -5343857, 13161587, 12044805,
|
|
-32856851, 4124601, -32343828, -10257566},
|
|
},
|
|
{
|
|
{-20788824, 14084654, -13531713, 7842147, 19119038, -13822605,
|
|
4752377, -8714640, -21679658, 2288038},
|
|
{-26819236, -3283715, 29965059, 3039786, -14473765, 2540457,
|
|
29457502, 14625692, -24819617, 12570232},
|
|
{-1063558, -11551823, 16920318, 12494842, 1278292, -5869109,
|
|
-21159943, -3498680, -11974704, 4724943},
|
|
},
|
|
{
|
|
{17960970, -11775534, -4140968, -9702530, -8876562, -1410617,
|
|
-12907383, -8659932, -29576300, 1903856},
|
|
{23134274, -14279132, -10681997, -1611936, 20684485, 15770816,
|
|
-12989750, 3190296, 26955097, 14109738},
|
|
{15308788, 5320727, -30113809, -14318877, 22902008, 7767164,
|
|
29425325, -11277562, 31960942, 11934971},
|
|
},
|
|
{
|
|
{-27395711, 8435796, 4109644, 12222639, -24627868, 14818669,
|
|
20638173, 4875028, 10491392, 1379718},
|
|
{-13159415, 9197841, 3875503, -8936108, -1383712, -5879801,
|
|
33518459, 16176658, 21432314, 12180697},
|
|
{-11787308, 11500838, 13787581, -13832590, -22430679, 10140205,
|
|
1465425, 12689540, -10301319, -13872883},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{5414091, -15386041, -21007664, 9643570, 12834970, 1186149,
|
|
-2622916, -1342231, 26128231, 6032912},
|
|
{-26337395, -13766162, 32496025, -13653919, 17847801, -12669156,
|
|
3604025, 8316894, -25875034, -10437358},
|
|
{3296484, 6223048, 24680646, -12246460, -23052020, 5903205,
|
|
-8862297, -4639164, 12376617, 3188849},
|
|
},
|
|
{
|
|
{29190488, -14659046, 27549113, -1183516, 3520066, -10697301,
|
|
32049515, -7309113, -16109234, -9852307},
|
|
{-14744486, -9309156, 735818, -598978, -20407687, -5057904,
|
|
25246078, -15795669, 18640741, -960977},
|
|
{-6928835, -16430795, 10361374, 5642961, 4910474, 12345252,
|
|
-31638386, -494430, 10530747, 1053335},
|
|
},
|
|
{
|
|
{-29265967, -14186805, -13538216, -12117373, -19457059, -10655384,
|
|
-31462369, -2948985, 24018831, 15026644},
|
|
{-22592535, -3145277, -2289276, 5953843, -13440189, 9425631,
|
|
25310643, 13003497, -2314791, -15145616},
|
|
{-27419985, -603321, -8043984, -1669117, -26092265, 13987819,
|
|
-27297622, 187899, -23166419, -2531735},
|
|
},
|
|
{
|
|
{-21744398, -13810475, 1844840, 5021428, -10434399, -15911473,
|
|
9716667, 16266922, -5070217, 726099},
|
|
{29370922, -6053998, 7334071, -15342259, 9385287, 2247707,
|
|
-13661962, -4839461, 30007388, -15823341},
|
|
{-936379, 16086691, 23751945, -543318, -1167538, -5189036, 9137109,
|
|
730663, 9835848, 4555336},
|
|
},
|
|
{
|
|
{-23376435, 1410446, -22253753, -12899614, 30867635, 15826977,
|
|
17693930, 544696, -11985298, 12422646},
|
|
{31117226, -12215734, -13502838, 6561947, -9876867, -12757670,
|
|
-5118685, -4096706, 29120153, 13924425},
|
|
{-17400879, -14233209, 19675799, -2734756, -11006962, -5858820,
|
|
-9383939, -11317700, 7240931, -237388},
|
|
},
|
|
{
|
|
{-31361739, -11346780, -15007447, -5856218, -22453340, -12152771,
|
|
1222336, 4389483, 3293637, -15551743},
|
|
{-16684801, -14444245, 11038544, 11054958, -13801175, -3338533,
|
|
-24319580, 7733547, 12796905, -6335822},
|
|
{-8759414, -10817836, -25418864, 10783769, -30615557, -9746811,
|
|
-28253339, 3647836, 3222231, -11160462},
|
|
},
|
|
{
|
|
{18606113, 1693100, -25448386, -15170272, 4112353, 10045021,
|
|
23603893, -2048234, -7550776, 2484985},
|
|
{9255317, -3131197, -12156162, -1004256, 13098013, -9214866,
|
|
16377220, -2102812, -19802075, -3034702},
|
|
{-22729289, 7496160, -5742199, 11329249, 19991973, -3347502,
|
|
-31718148, 9936966, -30097688, -10618797},
|
|
},
|
|
{
|
|
{21878590, -5001297, 4338336, 13643897, -3036865, 13160960,
|
|
19708896, 5415497, -7360503, -4109293},
|
|
{27736861, 10103576, 12500508, 8502413, -3413016, -9633558,
|
|
10436918, -1550276, -23659143, -8132100},
|
|
{19492550, -12104365, -29681976, -852630, -3208171, 12403437,
|
|
30066266, 8367329, 13243957, 8709688},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{12015105, 2801261, 28198131, 10151021, 24818120, -4743133,
|
|
-11194191, -5645734, 5150968, 7274186},
|
|
{2831366, -12492146, 1478975, 6122054, 23825128, -12733586,
|
|
31097299, 6083058, 31021603, -9793610},
|
|
{-2529932, -2229646, 445613, 10720828, -13849527, -11505937,
|
|
-23507731, 16354465, 15067285, -14147707},
|
|
},
|
|
{
|
|
{7840942, 14037873, -33364863, 15934016, -728213, -3642706,
|
|
21403988, 1057586, -19379462, -12403220},
|
|
{915865, -16469274, 15608285, -8789130, -24357026, 6060030,
|
|
-17371319, 8410997, -7220461, 16527025},
|
|
{32922597, -556987, 20336074, -16184568, 10903705, -5384487,
|
|
16957574, 52992, 23834301, 6588044},
|
|
},
|
|
{
|
|
{32752030, 11232950, 3381995, -8714866, 22652988, -10744103,
|
|
17159699, 16689107, -20314580, -1305992},
|
|
{-4689649, 9166776, -25710296, -10847306, 11576752, 12733943,
|
|
7924251, -2752281, 1976123, -7249027},
|
|
{21251222, 16309901, -2983015, -6783122, 30810597, 12967303, 156041,
|
|
-3371252, 12331345, -8237197},
|
|
},
|
|
{
|
|
{8651614, -4477032, -16085636, -4996994, 13002507, 2950805,
|
|
29054427, -5106970, 10008136, -4667901},
|
|
{31486080, 15114593, -14261250, 12951354, 14369431, -7387845,
|
|
16347321, -13662089, 8684155, -10532952},
|
|
{19443825, 11385320, 24468943, -9659068, -23919258, 2187569,
|
|
-26263207, -6086921, 31316348, 14219878},
|
|
},
|
|
{
|
|
{-28594490, 1193785, 32245219, 11392485, 31092169, 15722801,
|
|
27146014, 6992409, 29126555, 9207390},
|
|
{32382935, 1110093, 18477781, 11028262, -27411763, -7548111,
|
|
-4980517, 10843782, -7957600, -14435730},
|
|
{2814918, 7836403, 27519878, -7868156, -20894015, -11553689,
|
|
-21494559, 8550130, 28346258, 1994730},
|
|
},
|
|
{
|
|
{-19578299, 8085545, -14000519, -3948622, 2785838, -16231307,
|
|
-19516951, 7174894, 22628102, 8115180},
|
|
{-30405132, 955511, -11133838, -15078069, -32447087, -13278079,
|
|
-25651578, 3317160, -9943017, 930272},
|
|
{-15303681, -6833769, 28856490, 1357446, 23421993, 1057177,
|
|
24091212, -1388970, -22765376, -10650715},
|
|
},
|
|
{
|
|
{-22751231, -5303997, -12907607, -12768866, -15811511, -7797053,
|
|
-14839018, -16554220, -1867018, 8398970},
|
|
{-31969310, 2106403, -4736360, 1362501, 12813763, 16200670,
|
|
22981545, -6291273, 18009408, -15772772},
|
|
{-17220923, -9545221, -27784654, 14166835, 29815394, 7444469,
|
|
29551787, -3727419, 19288549, 1325865},
|
|
},
|
|
{
|
|
{15100157, -15835752, -23923978, -1005098, -26450192, 15509408,
|
|
12376730, -3479146, 33166107, -8042750},
|
|
{20909231, 13023121, -9209752, 16251778, -5778415, -8094914,
|
|
12412151, 10018715, 2213263, -13878373},
|
|
{32529814, -11074689, 30361439, -16689753, -9135940, 1513226,
|
|
22922121, 6382134, -5766928, 8371348},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{9923462, 11271500, 12616794, 3544722, -29998368, -1721626,
|
|
12891687, -8193132, -26442943, 10486144},
|
|
{-22597207, -7012665, 8587003, -8257861, 4084309, -12970062, 361726,
|
|
2610596, -23921530, -11455195},
|
|
{5408411, -1136691, -4969122, 10561668, 24145918, 14240566,
|
|
31319731, -4235541, 19985175, -3436086},
|
|
},
|
|
{
|
|
{-13994457, 16616821, 14549246, 3341099, 32155958, 13648976,
|
|
-17577068, 8849297, 65030, 8370684},
|
|
{-8320926, -12049626, 31204563, 5839400, -20627288, -1057277,
|
|
-19442942, 6922164, 12743482, -9800518},
|
|
{-2361371, 12678785, 28815050, 4759974, -23893047, 4884717,
|
|
23783145, 11038569, 18800704, 255233},
|
|
},
|
|
{
|
|
{-5269658, -1773886, 13957886, 7990715, 23132995, 728773, 13393847,
|
|
9066957, 19258688, -14753793},
|
|
{-2936654, -10827535, -10432089, 14516793, -3640786, 4372541,
|
|
-31934921, 2209390, -1524053, 2055794},
|
|
{580882, 16705327, 5468415, -2683018, -30926419, -14696000,
|
|
-7203346, -8994389, -30021019, 7394435},
|
|
},
|
|
{
|
|
{23838809, 1822728, -15738443, 15242727, 8318092, -3733104,
|
|
-21672180, -3492205, -4821741, 14799921},
|
|
{13345610, 9759151, 3371034, -16137791, 16353039, 8577942, 31129804,
|
|
13496856, -9056018, 7402518},
|
|
{2286874, -4435931, -20042458, -2008336, -13696227, 5038122,
|
|
11006906, -15760352, 8205061, 1607563},
|
|
},
|
|
{
|
|
{14414086, -8002132, 3331830, -3208217, 22249151, -5594188,
|
|
18364661, -2906958, 30019587, -9029278},
|
|
{-27688051, 1585953, -10775053, 931069, -29120221, -11002319,
|
|
-14410829, 12029093, 9944378, 8024},
|
|
{4368715, -3709630, 29874200, -15022983, -20230386, -11410704,
|
|
-16114594, -999085, -8142388, 5640030},
|
|
},
|
|
{
|
|
{10299610, 13746483, 11661824, 16234854, 7630238, 5998374, 9809887,
|
|
-16694564, 15219798, -14327783},
|
|
{27425505, -5719081, 3055006, 10660664, 23458024, 595578, -15398605,
|
|
-1173195, -18342183, 9742717},
|
|
{6744077, 2427284, 26042789, 2720740, -847906, 1118974, 32324614,
|
|
7406442, 12420155, 1994844},
|
|
},
|
|
{
|
|
{14012521, -5024720, -18384453, -9578469, -26485342, -3936439,
|
|
-13033478, -10909803, 24319929, -6446333},
|
|
{16412690, -4507367, 10772641, 15929391, -17068788, -4658621,
|
|
10555945, -10484049, -30102368, -4739048},
|
|
{22397382, -7767684, -9293161, -12792868, 17166287, -9755136,
|
|
-27333065, 6199366, 21880021, -12250760},
|
|
},
|
|
{
|
|
{-4283307, 5368523, -31117018, 8163389, -30323063, 3209128,
|
|
16557151, 8890729, 8840445, 4957760},
|
|
{-15447727, 709327, -6919446, -10870178, -29777922, 6522332,
|
|
-21720181, 12130072, -14796503, 5005757},
|
|
{-2114751, -14308128, 23019042, 15765735, -25269683, 6002752,
|
|
10183197, -13239326, -16395286, -2176112},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-19025756, 1632005, 13466291, -7995100, -23640451, 16573537,
|
|
-32013908, -3057104, 22208662, 2000468},
|
|
{3065073, -1412761, -25598674, -361432, -17683065, -5703415,
|
|
-8164212, 11248527, -3691214, -7414184},
|
|
{10379208, -6045554, 8877319, 1473647, -29291284, -12507580,
|
|
16690915, 2553332, -3132688, 16400289},
|
|
},
|
|
{
|
|
{15716668, 1254266, -18472690, 7446274, -8448918, 6344164,
|
|
-22097271, -7285580, 26894937, 9132066},
|
|
{24158887, 12938817, 11085297, -8177598, -28063478, -4457083,
|
|
-30576463, 64452, -6817084, -2692882},
|
|
{13488534, 7794716, 22236231, 5989356, 25426474, -12578208, 2350710,
|
|
-3418511, -4688006, 2364226},
|
|
},
|
|
{
|
|
{16335052, 9132434, 25640582, 6678888, 1725628, 8517937, -11807024,
|
|
-11697457, 15445875, -7798101},
|
|
{29004207, -7867081, 28661402, -640412, -12794003, -7943086,
|
|
31863255, -4135540, -278050, -15759279},
|
|
{-6122061, -14866665, -28614905, 14569919, -10857999, -3591829,
|
|
10343412, -6976290, -29828287, -10815811},
|
|
},
|
|
{
|
|
{27081650, 3463984, 14099042, -4517604, 1616303, -6205604, 29542636,
|
|
15372179, 17293797, 960709},
|
|
{20263915, 11434237, -5765435, 11236810, 13505955, -10857102,
|
|
-16111345, 6493122, -19384511, 7639714},
|
|
{-2830798, -14839232, 25403038, -8215196, -8317012, -16173699,
|
|
18006287, -16043750, 29994677, -15808121},
|
|
},
|
|
{
|
|
{9769828, 5202651, -24157398, -13631392, -28051003, -11561624,
|
|
-24613141, -13860782, -31184575, 709464},
|
|
{12286395, 13076066, -21775189, -1176622, -25003198, 4057652,
|
|
-32018128, -8890874, 16102007, 13205847},
|
|
{13733362, 5599946, 10557076, 3195751, -5557991, 8536970, -25540170,
|
|
8525972, 10151379, 10394400},
|
|
},
|
|
{
|
|
{4024660, -16137551, 22436262, 12276534, -9099015, -2686099,
|
|
19698229, 11743039, -33302334, 8934414},
|
|
{-15879800, -4525240, -8580747, -2934061, 14634845, -698278,
|
|
-9449077, 3137094, -11536886, 11721158},
|
|
{17555939, -5013938, 8268606, 2331751, -22738815, 9761013, 9319229,
|
|
8835153, -9205489, -1280045},
|
|
},
|
|
{
|
|
{-461409, -7830014, 20614118, 16688288, -7514766, -4807119,
|
|
22300304, 505429, 6108462, -6183415},
|
|
{-5070281, 12367917, -30663534, 3234473, 32617080, -8422642,
|
|
29880583, -13483331, -26898490, -7867459},
|
|
{-31975283, 5726539, 26934134, 10237677, -3173717, -605053,
|
|
24199304, 3795095, 7592688, -14992079},
|
|
},
|
|
{
|
|
{21594432, -14964228, 17466408, -4077222, 32537084, 2739898,
|
|
6407723, 12018833, -28256052, 4298412},
|
|
{-20650503, -11961496, -27236275, 570498, 3767144, -1717540,
|
|
13891942, -1569194, 13717174, 10805743},
|
|
{-14676630, -15644296, 15287174, 11927123, 24177847, -8175568,
|
|
-796431, 14860609, -26938930, -5863836},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{12962541, 5311799, -10060768, 11658280, 18855286, -7954201,
|
|
13286263, -12808704, -4381056, 9882022},
|
|
{18512079, 11319350, -20123124, 15090309, 18818594, 5271736,
|
|
-22727904, 3666879, -23967430, -3299429},
|
|
{-6789020, -3146043, 16192429, 13241070, 15898607, -14206114,
|
|
-10084880, -6661110, -2403099, 5276065},
|
|
},
|
|
{
|
|
{30169808, -5317648, 26306206, -11750859, 27814964, 7069267,
|
|
7152851, 3684982, 1449224, 13082861},
|
|
{10342826, 3098505, 2119311, 193222, 25702612, 12233820, 23697382,
|
|
15056736, -21016438, -8202000},
|
|
{-33150110, 3261608, 22745853, 7948688, 19370557, -15177665,
|
|
-26171976, 6482814, -10300080, -11060101},
|
|
},
|
|
{
|
|
{32869458, -5408545, 25609743, 15678670, -10687769, -15471071,
|
|
26112421, 2521008, -22664288, 6904815},
|
|
{29506923, 4457497, 3377935, -9796444, -30510046, 12935080, 1561737,
|
|
3841096, -29003639, -6657642},
|
|
{10340844, -6630377, -18656632, -2278430, 12621151, -13339055,
|
|
30878497, -11824370, -25584551, 5181966},
|
|
},
|
|
{
|
|
{25940115, -12658025, 17324188, -10307374, -8671468, 15029094,
|
|
24396252, -16450922, -2322852, -12388574},
|
|
{-21765684, 9916823, -1300409, 4079498, -1028346, 11909559, 1782390,
|
|
12641087, 20603771, -6561742},
|
|
{-18882287, -11673380, 24849422, 11501709, 13161720, -4768874,
|
|
1925523, 11914390, 4662781, 7820689},
|
|
},
|
|
{
|
|
{12241050, -425982, 8132691, 9393934, 32846760, -1599620, 29749456,
|
|
12172924, 16136752, 15264020},
|
|
{-10349955, -14680563, -8211979, 2330220, -17662549, -14545780,
|
|
10658213, 6671822, 19012087, 3772772},
|
|
{3753511, -3421066, 10617074, 2028709, 14841030, -6721664, 28718732,
|
|
-15762884, 20527771, 12988982},
|
|
},
|
|
{
|
|
{-14822485, -5797269, -3707987, 12689773, -898983, -10914866,
|
|
-24183046, -10564943, 3299665, -12424953},
|
|
{-16777703, -15253301, -9642417, 4978983, 3308785, 8755439, 6943197,
|
|
6461331, -25583147, 8991218},
|
|
{-17226263, 1816362, -1673288, -6086439, 31783888, -8175991,
|
|
-32948145, 7417950, -30242287, 1507265},
|
|
},
|
|
{
|
|
{29692663, 6829891, -10498800, 4334896, 20945975, -11906496,
|
|
-28887608, 8209391, 14606362, -10647073},
|
|
{-3481570, 8707081, 32188102, 5672294, 22096700, 1711240, -33020695,
|
|
9761487, 4170404, -2085325},
|
|
{-11587470, 14855945, -4127778, -1531857, -26649089, 15084046,
|
|
22186522, 16002000, -14276837, -8400798},
|
|
},
|
|
{
|
|
{-4811456, 13761029, -31703877, -2483919, -3312471, 7869047,
|
|
-7113572, -9620092, 13240845, 10965870},
|
|
{-7742563, -8256762, -14768334, -13656260, -23232383, 12387166,
|
|
4498947, 14147411, 29514390, 4302863},
|
|
{-13413405, -12407859, 20757302, -13801832, 14785143, 8976368,
|
|
-5061276, -2144373, 17846988, -13971927},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-2244452, -754728, -4597030, -1066309, -6247172, 1455299,
|
|
-21647728, -9214789, -5222701, 12650267},
|
|
{-9906797, -16070310, 21134160, 12198166, -27064575, 708126, 387813,
|
|
13770293, -19134326, 10958663},
|
|
{22470984, 12369526, 23446014, -5441109, -21520802, -9698723,
|
|
-11772496, -11574455, -25083830, 4271862},
|
|
},
|
|
{
|
|
{-25169565, -10053642, -19909332, 15361595, -5984358, 2159192,
|
|
75375, -4278529, -32526221, 8469673},
|
|
{15854970, 4148314, -8893890, 7259002, 11666551, 13824734,
|
|
-30531198, 2697372, 24154791, -9460943},
|
|
{15446137, -15806644, 29759747, 14019369, 30811221, -9610191,
|
|
-31582008, 12840104, 24913809, 9815020},
|
|
},
|
|
{
|
|
{-4709286, -5614269, -31841498, -12288893, -14443537, 10799414,
|
|
-9103676, 13438769, 18735128, 9466238},
|
|
{11933045, 9281483, 5081055, -5183824, -2628162, -4905629, -7727821,
|
|
-10896103, -22728655, 16199064},
|
|
{14576810, 379472, -26786533, -8317236, -29426508, -10812974,
|
|
-102766, 1876699, 30801119, 2164795},
|
|
},
|
|
{
|
|
{15995086, 3199873, 13672555, 13712240, -19378835, -4647646,
|
|
-13081610, -15496269, -13492807, 1268052},
|
|
{-10290614, -3659039, -3286592, 10948818, 23037027, 3794475,
|
|
-3470338, -12600221, -17055369, 3565904},
|
|
{29210088, -9419337, -5919792, -4952785, 10834811, -13327726,
|
|
-16512102, -10820713, -27162222, -14030531},
|
|
},
|
|
{
|
|
{-13161890, 15508588, 16663704, -8156150, -28349942, 9019123,
|
|
-29183421, -3769423, 2244111, -14001979},
|
|
{-5152875, -3800936, -9306475, -6071583, 16243069, 14684434,
|
|
-25673088, -16180800, 13491506, 4641841},
|
|
{10813417, 643330, -19188515, -728916, 30292062, -16600078,
|
|
27548447, -7721242, 14476989, -12767431},
|
|
},
|
|
{
|
|
{10292079, 9984945, 6481436, 8279905, -7251514, 7032743, 27282937,
|
|
-1644259, -27912810, 12651324},
|
|
{-31185513, -813383, 22271204, 11835308, 10201545, 15351028,
|
|
17099662, 3988035, 21721536, -3148940},
|
|
{10202177, -6545839, -31373232, -9574638, -32150642, -8119683,
|
|
-12906320, 3852694, 13216206, 14842320},
|
|
},
|
|
{
|
|
{-15815640, -10601066, -6538952, -7258995, -6984659, -6581778,
|
|
-31500847, 13765824, -27434397, 9900184},
|
|
{14465505, -13833331, -32133984, -14738873, -27443187, 12990492,
|
|
33046193, 15796406, -7051866, -8040114},
|
|
{30924417, -8279620, 6359016, -12816335, 16508377, 9071735,
|
|
-25488601, 15413635, 9524356, -7018878},
|
|
},
|
|
{
|
|
{12274201, -13175547, 32627641, -1785326, 6736625, 13267305,
|
|
5237659, -5109483, 15663516, 4035784},
|
|
{-2951309, 8903985, 17349946, 601635, -16432815, -4612556,
|
|
-13732739, -15889334, -22258478, 4659091},
|
|
{-16916263, -4952973, -30393711, -15158821, 20774812, 15897498,
|
|
5736189, 15026997, -2178256, -13455585},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-8858980, -2219056, 28571666, -10155518, -474467, -10105698,
|
|
-3801496, 278095, 23440562, -290208},
|
|
{10226241, -5928702, 15139956, 120818, -14867693, 5218603, 32937275,
|
|
11551483, -16571960, -7442864},
|
|
{17932739, -12437276, -24039557, 10749060, 11316803, 7535897,
|
|
22503767, 5561594, -3646624, 3898661},
|
|
},
|
|
{
|
|
{7749907, -969567, -16339731, -16464, -25018111, 15122143, -1573531,
|
|
7152530, 21831162, 1245233},
|
|
{26958459, -14658026, 4314586, 8346991, -5677764, 11960072,
|
|
-32589295, -620035, -30402091, -16716212},
|
|
{-12165896, 9166947, 33491384, 13673479, 29787085, 13096535,
|
|
6280834, 14587357, -22338025, 13987525},
|
|
},
|
|
{
|
|
{-24349909, 7778775, 21116000, 15572597, -4833266, -5357778,
|
|
-4300898, -5124639, -7469781, -2858068},
|
|
{9681908, -6737123, -31951644, 13591838, -6883821, 386950, 31622781,
|
|
6439245, -14581012, 4091397},
|
|
{-8426427, 1470727, -28109679, -1596990, 3978627, -5123623,
|
|
-19622683, 12092163, 29077877, -14741988},
|
|
},
|
|
{
|
|
{5269168, -6859726, -13230211, -8020715, 25932563, 1763552,
|
|
-5606110, -5505881, -20017847, 2357889},
|
|
{32264008, -15407652, -5387735, -1160093, -2091322, -3946900,
|
|
23104804, -12869908, 5727338, 189038},
|
|
{14609123, -8954470, -6000566, -16622781, -14577387, -7743898,
|
|
-26745169, 10942115, -25888931, -14884697},
|
|
},
|
|
{
|
|
{20513500, 5557931, -15604613, 7829531, 26413943, -2019404,
|
|
-21378968, 7471781, 13913677, -5137875},
|
|
{-25574376, 11967826, 29233242, 12948236, -6754465, 4713227,
|
|
-8940970, 14059180, 12878652, 8511905},
|
|
{-25656801, 3393631, -2955415, -7075526, -2250709, 9366908,
|
|
-30223418, 6812974, 5568676, -3127656},
|
|
},
|
|
{
|
|
{11630004, 12144454, 2116339, 13606037, 27378885, 15676917,
|
|
-17408753, -13504373, -14395196, 8070818},
|
|
{27117696, -10007378, -31282771, -5570088, 1127282, 12772488,
|
|
-29845906, 10483306, -11552749, -1028714},
|
|
{10637467, -5688064, 5674781, 1072708, -26343588, -6982302,
|
|
-1683975, 9177853, -27493162, 15431203},
|
|
},
|
|
{
|
|
{20525145, 10892566, -12742472, 12779443, -29493034, 16150075,
|
|
-28240519, 14943142, -15056790, -7935931},
|
|
{-30024462, 5626926, -551567, -9981087, 753598, 11981191, 25244767,
|
|
-3239766, -3356550, 9594024},
|
|
{-23752644, 2636870, -5163910, -10103818, 585134, 7877383, 11345683,
|
|
-6492290, 13352335, -10977084},
|
|
},
|
|
{
|
|
{-1931799, -5407458, 3304649, -12884869, 17015806, -4877091,
|
|
-29783850, -7752482, -13215537, -319204},
|
|
{20239939, 6607058, 6203985, 3483793, -18386976, -779229, -20723742,
|
|
15077870, -22750759, 14523817},
|
|
{27406042, -6041657, 27423596, -4497394, 4996214, 10002360,
|
|
-28842031, -4545494, -30172742, -4805667},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{11374242, 12660715, 17861383, -12540833, 10935568, 1099227,
|
|
-13886076, -9091740, -27727044, 11358504},
|
|
{-12730809, 10311867, 1510375, 10778093, -2119455, -9145702,
|
|
32676003, 11149336, -26123651, 4985768},
|
|
{-19096303, 341147, -6197485, -239033, 15756973, -8796662, -983043,
|
|
13794114, -19414307, -15621255},
|
|
},
|
|
{
|
|
{6490081, 11940286, 25495923, -7726360, 8668373, -8751316, 3367603,
|
|
6970005, -1691065, -9004790},
|
|
{1656497, 13457317, 15370807, 6364910, 13605745, 8362338, -19174622,
|
|
-5475723, -16796596, -5031438},
|
|
{-22273315, -13524424, -64685, -4334223, -18605636, -10921968,
|
|
-20571065, -7007978, -99853, -10237333},
|
|
},
|
|
{
|
|
{17747465, 10039260, 19368299, -4050591, -20630635, -16041286,
|
|
31992683, -15857976, -29260363, -5511971},
|
|
{31932027, -4986141, -19612382, 16366580, 22023614, 88450, 11371999,
|
|
-3744247, 4882242, -10626905},
|
|
{29796507, 37186, 19818052, 10115756, -11829032, 3352736, 18551198,
|
|
3272828, -5190932, -4162409},
|
|
},
|
|
{
|
|
{12501286, 4044383, -8612957, -13392385, -32430052, 5136599,
|
|
-19230378, -3529697, 330070, -3659409},
|
|
{6384877, 2899513, 17807477, 7663917, -2358888, 12363165, 25366522,
|
|
-8573892, -271295, 12071499},
|
|
{-8365515, -4042521, 25133448, -4517355, -6211027, 2265927,
|
|
-32769618, 1936675, -5159697, 3829363},
|
|
},
|
|
{
|
|
{28425966, -5835433, -577090, -4697198, -14217555, 6870930, 7921550,
|
|
-6567787, 26333140, 14267664},
|
|
{-11067219, 11871231, 27385719, -10559544, -4585914, -11189312,
|
|
10004786, -8709488, -21761224, 8930324},
|
|
{-21197785, -16396035, 25654216, -1725397, 12282012, 11008919,
|
|
1541940, 4757911, -26491501, -16408940},
|
|
},
|
|
{
|
|
{13537262, -7759490, -20604840, 10961927, -5922820, -13218065,
|
|
-13156584, 6217254, -15943699, 13814990},
|
|
{-17422573, 15157790, 18705543, 29619, 24409717, -260476, 27361681,
|
|
9257833, -1956526, -1776914},
|
|
{-25045300, -10191966, 15366585, 15166509, -13105086, 8423556,
|
|
-29171540, 12361135, -18685978, 4578290},
|
|
},
|
|
{
|
|
{24579768, 3711570, 1342322, -11180126, -27005135, 14124956,
|
|
-22544529, 14074919, 21964432, 8235257},
|
|
{-6528613, -2411497, 9442966, -5925588, 12025640, -1487420,
|
|
-2981514, -1669206, 13006806, 2355433},
|
|
{-16304899, -13605259, -6632427, -5142349, 16974359, -10911083,
|
|
27202044, 1719366, 1141648, -12796236},
|
|
},
|
|
{
|
|
{-12863944, -13219986, -8318266, -11018091, -6810145, -4843894,
|
|
13475066, -3133972, 32674895, 13715045},
|
|
{11423335, -5468059, 32344216, 8962751, 24989809, 9241752,
|
|
-13265253, 16086212, -28740881, -15642093},
|
|
{-1409668, 12530728, -6368726, 10847387, 19531186, -14132160,
|
|
-11709148, 7791794, -27245943, 4383347},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-28970898, 5271447, -1266009, -9736989, -12455236, 16732599,
|
|
-4862407, -4906449, 27193557, 6245191},
|
|
{-15193956, 5362278, -1783893, 2695834, 4960227, 12840725, 23061898,
|
|
3260492, 22510453, 8577507},
|
|
{-12632451, 11257346, -32692994, 13548177, -721004, 10879011,
|
|
31168030, 13952092, -29571492, -3635906},
|
|
},
|
|
{
|
|
{3877321, -9572739, 32416692, 5405324, -11004407, -13656635,
|
|
3759769, 11935320, 5611860, 8164018},
|
|
{-16275802, 14667797, 15906460, 12155291, -22111149, -9039718,
|
|
32003002, -8832289, 5773085, -8422109},
|
|
{-23788118, -8254300, 1950875, 8937633, 18686727, 16459170, -905725,
|
|
12376320, 31632953, 190926},
|
|
},
|
|
{
|
|
{-24593607, -16138885, -8423991, 13378746, 14162407, 6901328,
|
|
-8288749, 4508564, -25341555, -3627528},
|
|
{8884438, -5884009, 6023974, 10104341, -6881569, -4941533, 18722941,
|
|
-14786005, -1672488, 827625},
|
|
{-32720583, -16289296, -32503547, 7101210, 13354605, 2659080,
|
|
-1800575, -14108036, -24878478, 1541286},
|
|
},
|
|
{
|
|
{2901347, -1117687, 3880376, -10059388, -17620940, -3612781,
|
|
-21802117, -3567481, 20456845, -1885033},
|
|
{27019610, 12299467, -13658288, -1603234, -12861660, -4861471,
|
|
-19540150, -5016058, 29439641, 15138866},
|
|
{21536104, -6626420, -32447818, -10690208, -22408077, 5175814,
|
|
-5420040, -16361163, 7779328, 109896},
|
|
},
|
|
{
|
|
{30279744, 14648750, -8044871, 6425558, 13639621, -743509, 28698390,
|
|
12180118, 23177719, -554075},
|
|
{26572847, 3405927, -31701700, 12890905, -19265668, 5335866,
|
|
-6493768, 2378492, 4439158, -13279347},
|
|
{-22716706, 3489070, -9225266, -332753, 18875722, -1140095,
|
|
14819434, -12731527, -17717757, -5461437},
|
|
},
|
|
{
|
|
{-5056483, 16566551, 15953661, 3767752, -10436499, 15627060,
|
|
-820954, 2177225, 8550082, -15114165},
|
|
{-18473302, 16596775, -381660, 15663611, 22860960, 15585581,
|
|
-27844109, -3582739, -23260460, -8428588},
|
|
{-32480551, 15707275, -8205912, -5652081, 29464558, 2713815,
|
|
-22725137, 15860482, -21902570, 1494193},
|
|
},
|
|
{
|
|
{-19562091, -14087393, -25583872, -9299552, 13127842, 759709,
|
|
21923482, 16529112, 8742704, 12967017},
|
|
{-28464899, 1553205, 32536856, -10473729, -24691605, -406174,
|
|
-8914625, -2933896, -29903758, 15553883},
|
|
{21877909, 3230008, 9881174, 10539357, -4797115, 2841332, 11543572,
|
|
14513274, 19375923, -12647961},
|
|
},
|
|
{
|
|
{8832269, -14495485, 13253511, 5137575, 5037871, 4078777, 24880818,
|
|
-6222716, 2862653, 9455043},
|
|
{29306751, 5123106, 20245049, -14149889, 9592566, 8447059, -2077124,
|
|
-2990080, 15511449, 4789663},
|
|
{-20679756, 7004547, 8824831, -9434977, -4045704, -3750736,
|
|
-5754762, 108893, 23513200, 16652362},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-33256173, 4144782, -4476029, -6579123, 10770039, -7155542,
|
|
-6650416, -12936300, -18319198, 10212860},
|
|
{2756081, 8598110, 7383731, -6859892, 22312759, -1105012, 21179801,
|
|
2600940, -9988298, -12506466},
|
|
{-24645692, 13317462, -30449259, -15653928, 21365574, -10869657,
|
|
11344424, 864440, -2499677, -16710063},
|
|
},
|
|
{
|
|
{-26432803, 6148329, -17184412, -14474154, 18782929, -275997,
|
|
-22561534, 211300, 2719757, 4940997},
|
|
{-1323882, 3911313, -6948744, 14759765, -30027150, 7851207,
|
|
21690126, 8518463, 26699843, 5276295},
|
|
{-13149873, -6429067, 9396249, 365013, 24703301, -10488939, 1321586,
|
|
149635, -15452774, 7159369},
|
|
},
|
|
{
|
|
{9987780, -3404759, 17507962, 9505530, 9731535, -2165514, 22356009,
|
|
8312176, 22477218, -8403385},
|
|
{18155857, -16504990, 19744716, 9006923, 15154154, -10538976,
|
|
24256460, -4864995, -22548173, 9334109},
|
|
{2986088, -4911893, 10776628, -3473844, 10620590, -7083203,
|
|
-21413845, 14253545, -22587149, 536906},
|
|
},
|
|
{
|
|
{4377756, 8115836, 24567078, 15495314, 11625074, 13064599, 7390551,
|
|
10589625, 10838060, -15420424},
|
|
{-19342404, 867880, 9277171, -3218459, -14431572, -1986443,
|
|
19295826, -15796950, 6378260, 699185},
|
|
{7895026, 4057113, -7081772, -13077756, -17886831, -323126, -716039,
|
|
15693155, -5045064, -13373962},
|
|
},
|
|
{
|
|
{-7737563, -5869402, -14566319, -7406919, 11385654, 13201616,
|
|
31730678, -10962840, -3918636, -9669325},
|
|
{10188286, -15770834, -7336361, 13427543, 22223443, 14896287,
|
|
30743455, 7116568, -21786507, 5427593},
|
|
{696102, 13206899, 27047647, -10632082, 15285305, -9853179,
|
|
10798490, -4578720, 19236243, 12477404},
|
|
},
|
|
{
|
|
{-11229439, 11243796, -17054270, -8040865, -788228, -8167967,
|
|
-3897669, 11180504, -23169516, 7733644},
|
|
{17800790, -14036179, -27000429, -11766671, 23887827, 3149671,
|
|
23466177, -10538171, 10322027, 15313801},
|
|
{26246234, 11968874, 32263343, -5468728, 6830755, -13323031,
|
|
-15794704, -101982, -24449242, 10890804},
|
|
},
|
|
{
|
|
{-31365647, 10271363, -12660625, -6267268, 16690207, -13062544,
|
|
-14982212, 16484931, 25180797, -5334884},
|
|
{-586574, 10376444, -32586414, -11286356, 19801893, 10997610,
|
|
2276632, 9482883, 316878, 13820577},
|
|
{-9882808, -4510367, -2115506, 16457136, -11100081, 11674996,
|
|
30756178, -7515054, 30696930, -3712849},
|
|
},
|
|
{
|
|
{32988917, -9603412, 12499366, 7910787, -10617257, -11931514,
|
|
-7342816, -9985397, -32349517, 7392473},
|
|
{-8855661, 15927861, 9866406, -3649411, -2396914, -16655781,
|
|
-30409476, -9134995, 25112947, -2926644},
|
|
{-2504044, -436966, 25621774, -5678772, 15085042, -5479877,
|
|
-24884878, -13526194, 5537438, -13914319},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-11225584, 2320285, -9584280, 10149187, -33444663, 5808648,
|
|
-14876251, -1729667, 31234590, 6090599},
|
|
{-9633316, 116426, 26083934, 2897444, -6364437, -2688086, 609721,
|
|
15878753, -6970405, -9034768},
|
|
{-27757857, 247744, -15194774, -9002551, 23288161, -10011936,
|
|
-23869595, 6503646, 20650474, 1804084},
|
|
},
|
|
{
|
|
{-27589786, 15456424, 8972517, 8469608, 15640622, 4439847, 3121995,
|
|
-10329713, 27842616, -202328},
|
|
{-15306973, 2839644, 22530074, 10026331, 4602058, 5048462, 28248656,
|
|
5031932, -11375082, 12714369},
|
|
{20807691, -7270825, 29286141, 11421711, -27876523, -13868230,
|
|
-21227475, 1035546, -19733229, 12796920},
|
|
},
|
|
{
|
|
{12076899, -14301286, -8785001, -11848922, -25012791, 16400684,
|
|
-17591495, -12899438, 3480665, -15182815},
|
|
{-32361549, 5457597, 28548107, 7833186, 7303070, -11953545,
|
|
-24363064, -15921875, -33374054, 2771025},
|
|
{-21389266, 421932, 26597266, 6860826, 22486084, -6737172,
|
|
-17137485, -4210226, -24552282, 15673397},
|
|
},
|
|
{
|
|
{-20184622, 2338216, 19788685, -9620956, -4001265, -8740893,
|
|
-20271184, 4733254, 3727144, -12934448},
|
|
{6120119, 814863, -11794402, -622716, 6812205, -15747771, 2019594,
|
|
7975683, 31123697, -10958981},
|
|
{30069250, -11435332, 30434654, 2958439, 18399564, -976289,
|
|
12296869, 9204260, -16432438, 9648165},
|
|
},
|
|
{
|
|
{32705432, -1550977, 30705658, 7451065, -11805606, 9631813, 3305266,
|
|
5248604, -26008332, -11377501},
|
|
{17219865, 2375039, -31570947, -5575615, -19459679, 9219903, 294711,
|
|
15298639, 2662509, -16297073},
|
|
{-1172927, -7558695, -4366770, -4287744, -21346413, -8434326,
|
|
32087529, -1222777, 32247248, -14389861},
|
|
},
|
|
{
|
|
{14312628, 1221556, 17395390, -8700143, -4945741, -8684635,
|
|
-28197744, -9637817, -16027623, -13378845},
|
|
{-1428825, -9678990, -9235681, 6549687, -7383069, -468664, 23046502,
|
|
9803137, 17597934, 2346211},
|
|
{18510800, 15337574, 26171504, 981392, -22241552, 7827556,
|
|
-23491134, -11323352, 3059833, -11782870},
|
|
},
|
|
{
|
|
{10141598, 6082907, 17829293, -1947643, 9830092, 13613136,
|
|
-25556636, -5544586, -33502212, 3592096},
|
|
{33114168, -15889352, -26525686, -13343397, 33076705, 8716171,
|
|
1151462, 1521897, -982665, -6837803},
|
|
{-32939165, -4255815, 23947181, -324178, -33072974, -12305637,
|
|
-16637686, 3891704, 26353178, 693168},
|
|
},
|
|
{
|
|
{30374239, 1595580, -16884039, 13186931, 4600344, 406904, 9585294,
|
|
-400668, 31375464, 14369965},
|
|
{-14370654, -7772529, 1510301, 6434173, -18784789, -6262728,
|
|
32732230, -13108839, 17901441, 16011505},
|
|
{18171223, -11934626, -12500402, 15197122, -11038147, -15230035,
|
|
-19172240, -16046376, 8764035, 12309598},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{5975908, -5243188, -19459362, -9681747, -11541277, 14015782,
|
|
-23665757, 1228319, 17544096, -10593782},
|
|
{5811932, -1715293, 3442887, -2269310, -18367348, -8359541,
|
|
-18044043, -15410127, -5565381, 12348900},
|
|
{-31399660, 11407555, 25755363, 6891399, -3256938, 14872274,
|
|
-24849353, 8141295, -10632534, -585479},
|
|
},
|
|
{
|
|
{-12675304, 694026, -5076145, 13300344, 14015258, -14451394,
|
|
-9698672, -11329050, 30944593, 1130208},
|
|
{8247766, -6710942, -26562381, -7709309, -14401939, -14648910,
|
|
4652152, 2488540, 23550156, -271232},
|
|
{17294316, -3788438, 7026748, 15626851, 22990044, 113481, 2267737,
|
|
-5908146, -408818, -137719},
|
|
},
|
|
{
|
|
{16091085, -16253926, 18599252, 7340678, 2137637, -1221657,
|
|
-3364161, 14550936, 3260525, -7166271},
|
|
{-4910104, -13332887, 18550887, 10864893, -16459325, -7291596,
|
|
-23028869, -13204905, -12748722, 2701326},
|
|
{-8574695, 16099415, 4629974, -16340524, -20786213, -6005432,
|
|
-10018363, 9276971, 11329923, 1862132},
|
|
},
|
|
{
|
|
{14763076, -15903608, -30918270, 3689867, 3511892, 10313526,
|
|
-21951088, 12219231, -9037963, -940300},
|
|
{8894987, -3446094, 6150753, 3013931, 301220, 15693451, -31981216,
|
|
-2909717, -15438168, 11595570},
|
|
{15214962, 3537601, -26238722, -14058872, 4418657, -15230761,
|
|
13947276, 10730794, -13489462, -4363670},
|
|
},
|
|
{
|
|
{-2538306, 7682793, 32759013, 263109, -29984731, -7955452,
|
|
-22332124, -10188635, 977108, 699994},
|
|
{-12466472, 4195084, -9211532, 550904, -15565337, 12917920,
|
|
19118110, -439841, -30534533, -14337913},
|
|
{31788461, -14507657, 4799989, 7372237, 8808585, -14747943, 9408237,
|
|
-10051775, 12493932, -5409317},
|
|
},
|
|
{
|
|
{-25680606, 5260744, -19235809, -6284470, -3695942, 16566087,
|
|
27218280, 2607121, 29375955, 6024730},
|
|
{842132, -2794693, -4763381, -8722815, 26332018, -12405641,
|
|
11831880, 6985184, -9940361, 2854096},
|
|
{-4847262, -7969331, 2516242, -5847713, 9695691, -7221186, 16512645,
|
|
960770, 12121869, 16648078},
|
|
},
|
|
{
|
|
{-15218652, 14667096, -13336229, 2013717, 30598287, -464137,
|
|
-31504922, -7882064, 20237806, 2838411},
|
|
{-19288047, 4453152, 15298546, -16178388, 22115043, -15972604,
|
|
12544294, -13470457, 1068881, -12499905},
|
|
{-9558883, -16518835, 33238498, 13506958, 30505848, -1114596,
|
|
-8486907, -2630053, 12521378, 4845654},
|
|
},
|
|
{
|
|
{-28198521, 10744108, -2958380, 10199664, 7759311, -13088600,
|
|
3409348, -873400, -6482306, -12885870},
|
|
{-23561822, 6230156, -20382013, 10655314, -24040585, -11621172,
|
|
10477734, -1240216, -3113227, 13974498},
|
|
{12966261, 15550616, -32038948, -1615346, 21025980, -629444,
|
|
5642325, 7188737, 18895762, 12629579},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{14741879, -14946887, 22177208, -11721237, 1279741, 8058600,
|
|
11758140, 789443, 32195181, 3895677},
|
|
{10758205, 15755439, -4509950, 9243698, -4879422, 6879879, -2204575,
|
|
-3566119, -8982069, 4429647},
|
|
{-2453894, 15725973, -20436342, -10410672, -5803908, -11040220,
|
|
-7135870, -11642895, 18047436, -15281743},
|
|
},
|
|
{
|
|
{-25173001, -11307165, 29759956, 11776784, -22262383, -15820455,
|
|
10993114, -12850837, -17620701, -9408468},
|
|
{21987233, 700364, -24505048, 14972008, -7774265, -5718395,
|
|
32155026, 2581431, -29958985, 8773375},
|
|
{-25568350, 454463, -13211935, 16126715, 25240068, 8594567,
|
|
20656846, 12017935, -7874389, -13920155},
|
|
},
|
|
{
|
|
{6028182, 6263078, -31011806, -11301710, -818919, 2461772,
|
|
-31841174, -5468042, -1721788, -2776725},
|
|
{-12278994, 16624277, 987579, -5922598, 32908203, 1248608, 7719845,
|
|
-4166698, 28408820, 6816612},
|
|
{-10358094, -8237829, 19549651, -12169222, 22082623, 16147817,
|
|
20613181, 13982702, -10339570, 5067943},
|
|
},
|
|
{
|
|
{-30505967, -3821767, 12074681, 13582412, -19877972, 2443951,
|
|
-19719286, 12746132, 5331210, -10105944},
|
|
{30528811, 3601899, -1957090, 4619785, -27361822, -15436388,
|
|
24180793, -12570394, 27679908, -1648928},
|
|
{9402404, -13957065, 32834043, 10838634, -26580150, -13237195,
|
|
26653274, -8685565, 22611444, -12715406},
|
|
},
|
|
{
|
|
{22190590, 1118029, 22736441, 15130463, -30460692, -5991321,
|
|
19189625, -4648942, 4854859, 6622139},
|
|
{-8310738, -2953450, -8262579, -3388049, -10401731, -271929,
|
|
13424426, -3567227, 26404409, 13001963},
|
|
{-31241838, -15415700, -2994250, 8939346, 11562230, -12840670,
|
|
-26064365, -11621720, -15405155, 11020693},
|
|
},
|
|
{
|
|
{1866042, -7949489, -7898649, -10301010, 12483315, 13477547,
|
|
3175636, -12424163, 28761762, 1406734},
|
|
{-448555, -1777666, 13018551, 3194501, -9580420, -11161737,
|
|
24760585, -4347088, 25577411, -13378680},
|
|
{-24290378, 4759345, -690653, -1852816, 2066747, 10693769,
|
|
-29595790, 9884936, -9368926, 4745410},
|
|
},
|
|
{
|
|
{-9141284, 6049714, -19531061, -4341411, -31260798, 9944276,
|
|
-15462008, -11311852, 10931924, -11931931},
|
|
{-16561513, 14112680, -8012645, 4817318, -8040464, -11414606,
|
|
-22853429, 10856641, -20470770, 13434654},
|
|
{22759489, -10073434, -16766264, -1871422, 13637442, -10168091,
|
|
1765144, -12654326, 28445307, -5364710},
|
|
},
|
|
{
|
|
{29875063, 12493613, 2795536, -3786330, 1710620, 15181182,
|
|
-10195717, -8788675, 9074234, 1167180},
|
|
{-26205683, 11014233, -9842651, -2635485, -26908120, 7532294,
|
|
-18716888, -9535498, 3843903, 9367684},
|
|
{-10969595, -6403711, 9591134, 9582310, 11349256, 108879, 16235123,
|
|
8601684, -139197, 4242895},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{22092954, -13191123, -2042793, -11968512, 32186753, -11517388,
|
|
-6574341, 2470660, -27417366, 16625501},
|
|
{-11057722, 3042016, 13770083, -9257922, 584236, -544855, -7770857,
|
|
2602725, -27351616, 14247413},
|
|
{6314175, -10264892, -32772502, 15957557, -10157730, 168750,
|
|
-8618807, 14290061, 27108877, -1180880},
|
|
},
|
|
{
|
|
{-8586597, -7170966, 13241782, 10960156, -32991015, -13794596,
|
|
33547976, -11058889, -27148451, 981874},
|
|
{22833440, 9293594, -32649448, -13618667, -9136966, 14756819,
|
|
-22928859, -13970780, -10479804, -16197962},
|
|
{-7768587, 3326786, -28111797, 10783824, 19178761, 14905060,
|
|
22680049, 13906969, -15933690, 3797899},
|
|
},
|
|
{
|
|
{21721356, -4212746, -12206123, 9310182, -3882239, -13653110,
|
|
23740224, -2709232, 20491983, -8042152},
|
|
{9209270, -15135055, -13256557, -6167798, -731016, 15289673,
|
|
25947805, 15286587, 30997318, -6703063},
|
|
{7392032, 16618386, 23946583, -8039892, -13265164, -1533858,
|
|
-14197445, -2321576, 17649998, -250080},
|
|
},
|
|
{
|
|
{-9301088, -14193827, 30609526, -3049543, -25175069, -1283752,
|
|
-15241566, -9525724, -2233253, 7662146},
|
|
{-17558673, 1763594, -33114336, 15908610, -30040870, -12174295,
|
|
7335080, -8472199, -3174674, 3440183},
|
|
{-19889700, -5977008, -24111293, -9688870, 10799743, -16571957,
|
|
40450, -4431835, 4862400, 1133},
|
|
},
|
|
{
|
|
{-32856209, -7873957, -5422389, 14860950, -16319031, 7956142,
|
|
7258061, 311861, -30594991, -7379421},
|
|
{-3773428, -1565936, 28985340, 7499440, 24445838, 9325937, 29727763,
|
|
16527196, 18278453, 15405622},
|
|
{-4381906, 8508652, -19898366, -3674424, -5984453, 15149970,
|
|
-13313598, 843523, -21875062, 13626197},
|
|
},
|
|
{
|
|
{2281448, -13487055, -10915418, -2609910, 1879358, 16164207,
|
|
-10783882, 3953792, 13340839, 15928663},
|
|
{31727126, -7179855, -18437503, -8283652, 2875793, -16390330,
|
|
-25269894, -7014826, -23452306, 5964753},
|
|
{4100420, -5959452, -17179337, 6017714, -18705837, 12227141,
|
|
-26684835, 11344144, 2538215, -7570755},
|
|
},
|
|
{
|
|
{-9433605, 6123113, 11159803, -2156608, 30016280, 14966241,
|
|
-20474983, 1485421, -629256, -15958862},
|
|
{-26804558, 4260919, 11851389, 9658551, -32017107, 16367492,
|
|
-20205425, -13191288, 11659922, -11115118},
|
|
{26180396, 10015009, -30844224, -8581293, 5418197, 9480663, 2231568,
|
|
-10170080, 33100372, -1306171},
|
|
},
|
|
{
|
|
{15121113, -5201871, -10389905, 15427821, -27509937, -15992507,
|
|
21670947, 4486675, -5931810, -14466380},
|
|
{16166486, -9483733, -11104130, 6023908, -31926798, -1364923,
|
|
2340060, -16254968, -10735770, -10039824},
|
|
{28042865, -3557089, -12126526, 12259706, -3717498, -6945899,
|
|
6766453, -8689599, 18036436, 5803270},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-817581, 6763912, 11803561, 1585585, 10958447, -2671165, 23855391,
|
|
4598332, -6159431, -14117438},
|
|
{-31031306, -14256194, 17332029, -2383520, 31312682, -5967183,
|
|
696309, 50292, -20095739, 11763584},
|
|
{-594563, -2514283, -32234153, 12643980, 12650761, 14811489, 665117,
|
|
-12613632, -19773211, -10713562},
|
|
},
|
|
{
|
|
{30464590, -11262872, -4127476, -12734478, 19835327, -7105613,
|
|
-24396175, 2075773, -17020157, 992471},
|
|
{18357185, -6994433, 7766382, 16342475, -29324918, 411174, 14578841,
|
|
8080033, -11574335, -10601610},
|
|
{19598397, 10334610, 12555054, 2555664, 18821899, -10339780,
|
|
21873263, 16014234, 26224780, 16452269},
|
|
},
|
|
{
|
|
{-30223925, 5145196, 5944548, 16385966, 3976735, 2009897, -11377804,
|
|
-7618186, -20533829, 3698650},
|
|
{14187449, 3448569, -10636236, -10810935, -22663880, -3433596,
|
|
7268410, -10890444, 27394301, 12015369},
|
|
{19695761, 16087646, 28032085, 12999827, 6817792, 11427614,
|
|
20244189, -1312777, -13259127, -3402461},
|
|
},
|
|
{
|
|
{30860103, 12735208, -1888245, -4699734, -16974906, 2256940,
|
|
-8166013, 12298312, -8550524, -10393462},
|
|
{-5719826, -11245325, -1910649, 15569035, 26642876, -7587760,
|
|
-5789354, -15118654, -4976164, 12651793},
|
|
{-2848395, 9953421, 11531313, -5282879, 26895123, -12697089,
|
|
-13118820, -16517902, 9768698, -2533218},
|
|
},
|
|
{
|
|
{-24719459, 1894651, -287698, -4704085, 15348719, -8156530,
|
|
32767513, 12765450, 4940095, 10678226},
|
|
{18860224, 15980149, -18987240, -1562570, -26233012, -11071856,
|
|
-7843882, 13944024, -24372348, 16582019},
|
|
{-15504260, 4970268, -29893044, 4175593, -20993212, -2199756,
|
|
-11704054, 15444560, -11003761, 7989037},
|
|
},
|
|
{
|
|
{31490452, 5568061, -2412803, 2182383, -32336847, 4531686,
|
|
-32078269, 6200206, -19686113, -14800171},
|
|
{-17308668, -15879940, -31522777, -2831, -32887382, 16375549,
|
|
8680158, -16371713, 28550068, -6857132},
|
|
{-28126887, -5688091, 16837845, -1820458, -6850681, 12700016,
|
|
-30039981, 4364038, 1155602, 5988841},
|
|
},
|
|
{
|
|
{21890435, -13272907, -12624011, 12154349, -7831873, 15300496,
|
|
23148983, -4470481, 24618407, 8283181},
|
|
{-33136107, -10512751, 9975416, 6841041, -31559793, 16356536,
|
|
3070187, -7025928, 1466169, 10740210},
|
|
{-1509399, -15488185, -13503385, -10655916, 32799044, 909394,
|
|
-13938903, -5779719, -32164649, -15327040},
|
|
},
|
|
{
|
|
{3960823, -14267803, -28026090, -15918051, -19404858, 13146868,
|
|
15567327, 951507, -3260321, -573935},
|
|
{24740841, 5052253, -30094131, 8961361, 25877428, 6165135,
|
|
-24368180, 14397372, -7380369, -6144105},
|
|
{-28888365, 3510803, -28103278, -1158478, -11238128, -10631454,
|
|
-15441463, -14453128, -1625486, -6494814},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{793299, -9230478, 8836302, -6235707, -27360908, -2369593, 33152843,
|
|
-4885251, -9906200, -621852},
|
|
{5666233, 525582, 20782575, -8038419, -24538499, 14657740, 16099374,
|
|
1468826, -6171428, -15186581},
|
|
{-4859255, -3779343, -2917758, -6748019, 7778750, 11688288,
|
|
-30404353, -9871238, -1558923, -9863646},
|
|
},
|
|
{
|
|
{10896332, -7719704, 824275, 472601, -19460308, 3009587, 25248958,
|
|
14783338, -30581476, -15757844},
|
|
{10566929, 12612572, -31944212, 11118703, -12633376, 12362879,
|
|
21752402, 8822496, 24003793, 14264025},
|
|
{27713862, -7355973, -11008240, 9227530, 27050101, 2504721,
|
|
23886875, -13117525, 13958495, -5732453},
|
|
},
|
|
{
|
|
{-23481610, 4867226, -27247128, 3900521, 29838369, -8212291,
|
|
-31889399, -10041781, 7340521, -15410068},
|
|
{4646514, -8011124, -22766023, -11532654, 23184553, 8566613,
|
|
31366726, -1381061, -15066784, -10375192},
|
|
{-17270517, 12723032, -16993061, 14878794, 21619651, -6197576,
|
|
27584817, 3093888, -8843694, 3849921},
|
|
},
|
|
{
|
|
{-9064912, 2103172, 25561640, -15125738, -5239824, 9582958,
|
|
32477045, -9017955, 5002294, -15550259},
|
|
{-12057553, -11177906, 21115585, -13365155, 8808712, -12030708,
|
|
16489530, 13378448, -25845716, 12741426},
|
|
{-5946367, 10645103, -30911586, 15390284, -3286982, -7118677,
|
|
24306472, 15852464, 28834118, -7646072},
|
|
},
|
|
{
|
|
{-17335748, -9107057, -24531279, 9434953, -8472084, -583362,
|
|
-13090771, 455841, 20461858, 5491305},
|
|
{13669248, -16095482, -12481974, -10203039, -14569770, -11893198,
|
|
-24995986, 11293807, -28588204, -9421832},
|
|
{28497928, 6272777, -33022994, 14470570, 8906179, -1225630,
|
|
18504674, -14165166, 29867745, -8795943},
|
|
},
|
|
{
|
|
{-16207023, 13517196, -27799630, -13697798, 24009064, -6373891,
|
|
-6367600, -13175392, 22853429, -4012011},
|
|
{24191378, 16712145, -13931797, 15217831, 14542237, 1646131,
|
|
18603514, -11037887, 12876623, -2112447},
|
|
{17902668, 4518229, -411702, -2829247, 26878217, 5258055, -12860753,
|
|
608397, 16031844, 3723494},
|
|
},
|
|
{
|
|
{-28632773, 12763728, -20446446, 7577504, 33001348, -13017745,
|
|
17558842, -7872890, 23896954, -4314245},
|
|
{-20005381, -12011952, 31520464, 605201, 2543521, 5991821, -2945064,
|
|
7229064, -9919646, -8826859},
|
|
{28816045, 298879, -28165016, -15920938, 19000928, -1665890,
|
|
-12680833, -2949325, -18051778, -2082915},
|
|
},
|
|
{
|
|
{16000882, -344896, 3493092, -11447198, -29504595, -13159789,
|
|
12577740, 16041268, -19715240, 7847707},
|
|
{10151868, 10572098, 27312476, 7922682, 14825339, 4723128,
|
|
-32855931, -6519018, -10020567, 3852848},
|
|
{-11430470, 15697596, -21121557, -4420647, 5386314, 15063598,
|
|
16514493, -15932110, 29330899, -15076224},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-25499735, -4378794, -15222908, -6901211, 16615731, 2051784,
|
|
3303702, 15490, -27548796, 12314391},
|
|
{15683520, -6003043, 18109120, -9980648, 15337968, -5997823,
|
|
-16717435, 15921866, 16103996, -3731215},
|
|
{-23169824, -10781249, 13588192, -1628807, -3798557, -1074929,
|
|
-19273607, 5402699, -29815713, -9841101},
|
|
},
|
|
{
|
|
{23190676, 2384583, -32714340, 3462154, -29903655, -1529132,
|
|
-11266856, 8911517, -25205859, 2739713},
|
|
{21374101, -3554250, -33524649, 9874411, 15377179, 11831242,
|
|
-33529904, 6134907, 4931255, 11987849},
|
|
{-7732, -2978858, -16223486, 7277597, 105524, -322051, -31480539,
|
|
13861388, -30076310, 10117930},
|
|
},
|
|
{
|
|
{-29501170, -10744872, -26163768, 13051539, -25625564, 5089643,
|
|
-6325503, 6704079, 12890019, 15728940},
|
|
{-21972360, -11771379, -951059, -4418840, 14704840, 2695116, 903376,
|
|
-10428139, 12885167, 8311031},
|
|
{-17516482, 5352194, 10384213, -13811658, 7506451, 13453191,
|
|
26423267, 4384730, 1888765, -5435404},
|
|
},
|
|
{
|
|
{-25817338, -3107312, -13494599, -3182506, 30896459, -13921729,
|
|
-32251644, -12707869, -19464434, -3340243},
|
|
{-23607977, -2665774, -526091, 4651136, 5765089, 4618330, 6092245,
|
|
14845197, 17151279, -9854116},
|
|
{-24830458, -12733720, -15165978, 10367250, -29530908, -265356,
|
|
22825805, -7087279, -16866484, 16176525},
|
|
},
|
|
{
|
|
{-23583256, 6564961, 20063689, 3798228, -4740178, 7359225, 2006182,
|
|
-10363426, -28746253, -10197509},
|
|
{-10626600, -4486402, -13320562, -5125317, 3432136, -6393229,
|
|
23632037, -1940610, 32808310, 1099883},
|
|
{15030977, 5768825, -27451236, -2887299, -6427378, -15361371,
|
|
-15277896, -6809350, 2051441, -15225865},
|
|
},
|
|
{
|
|
{-3362323, -7239372, 7517890, 9824992, 23555850, 295369, 5148398,
|
|
-14154188, -22686354, 16633660},
|
|
{4577086, -16752288, 13249841, -15304328, 19958763, -14537274,
|
|
18559670, -10759549, 8402478, -9864273},
|
|
{-28406330, -1051581, -26790155, -907698, -17212414, -11030789,
|
|
9453451, -14980072, 17983010, 9967138},
|
|
},
|
|
{
|
|
{-25762494, 6524722, 26585488, 9969270, 24709298, 1220360, -1677990,
|
|
7806337, 17507396, 3651560},
|
|
{-10420457, -4118111, 14584639, 15971087, -15768321, 8861010,
|
|
26556809, -5574557, -18553322, -11357135},
|
|
{2839101, 14284142, 4029895, 3472686, 14402957, 12689363, -26642121,
|
|
8459447, -5605463, -7621941},
|
|
},
|
|
{
|
|
{-4839289, -3535444, 9744961, 2871048, 25113978, 3187018, -25110813,
|
|
-849066, 17258084, -7977739},
|
|
{18164541, -10595176, -17154882, -1542417, 19237078, -9745295,
|
|
23357533, -15217008, 26908270, 12150756},
|
|
{-30264870, -7647865, 5112249, -7036672, -1499807, -6974257, 43168,
|
|
-5537701, -32302074, 16215819},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-6898905, 9824394, -12304779, -4401089, -31397141, -6276835,
|
|
32574489, 12532905, -7503072, -8675347},
|
|
{-27343522, -16515468, -27151524, -10722951, 946346, 16291093,
|
|
254968, 7168080, 21676107, -1943028},
|
|
{21260961, -8424752, -16831886, -11920822, -23677961, 3968121,
|
|
-3651949, -6215466, -3556191, -7913075},
|
|
},
|
|
{
|
|
{16544754, 13250366, -16804428, 15546242, -4583003, 12757258,
|
|
-2462308, -8680336, -18907032, -9662799},
|
|
{-2415239, -15577728, 18312303, 4964443, -15272530, -12653564,
|
|
26820651, 16690659, 25459437, -4564609},
|
|
{-25144690, 11425020, 28423002, -11020557, -6144921, -15826224,
|
|
9142795, -2391602, -6432418, -1644817},
|
|
},
|
|
{
|
|
{-23104652, 6253476, 16964147, -3768872, -25113972, -12296437,
|
|
-27457225, -16344658, 6335692, 7249989},
|
|
{-30333227, 13979675, 7503222, -12368314, -11956721, -4621693,
|
|
-30272269, 2682242, 25993170, -12478523},
|
|
{4364628, 5930691, 32304656, -10044554, -8054781, 15091131,
|
|
22857016, -10598955, 31820368, 15075278},
|
|
},
|
|
{
|
|
{31879134, -8918693, 17258761, 90626, -8041836, -4917709, 24162788,
|
|
-9650886, -17970238, 12833045},
|
|
{19073683, 14851414, -24403169, -11860168, 7625278, 11091125,
|
|
-19619190, 2074449, -9413939, 14905377},
|
|
{24483667, -11935567, -2518866, -11547418, -1553130, 15355506,
|
|
-25282080, 9253129, 27628530, -7555480},
|
|
},
|
|
{
|
|
{17597607, 8340603, 19355617, 552187, 26198470, -3176583, 4593324,
|
|
-9157582, -14110875, 15297016},
|
|
{510886, 14337390, -31785257, 16638632, 6328095, 2713355, -20217417,
|
|
-11864220, 8683221, 2921426},
|
|
{18606791, 11874196, 27155355, -5281482, -24031742, 6265446,
|
|
-25178240, -1278924, 4674690, 13890525},
|
|
},
|
|
{
|
|
{13609624, 13069022, -27372361, -13055908, 24360586, 9592974,
|
|
14977157, 9835105, 4389687, 288396},
|
|
{9922506, -519394, 13613107, 5883594, -18758345, -434263, -12304062,
|
|
8317628, 23388070, 16052080},
|
|
{12720016, 11937594, -31970060, -5028689, 26900120, 8561328,
|
|
-20155687, -11632979, -14754271, -10812892},
|
|
},
|
|
{
|
|
{15961858, 14150409, 26716931, -665832, -22794328, 13603569,
|
|
11829573, 7467844, -28822128, 929275},
|
|
{11038231, -11582396, -27310482, -7316562, -10498527, -16307831,
|
|
-23479533, -9371869, -21393143, 2465074},
|
|
{20017163, -4323226, 27915242, 1529148, 12396362, 15675764,
|
|
13817261, -9658066, 2463391, -4622140},
|
|
},
|
|
{
|
|
{-16358878, -12663911, -12065183, 4996454, -1256422, 1073572,
|
|
9583558, 12851107, 4003896, 12673717},
|
|
{-1731589, -15155870, -3262930, 16143082, 19294135, 13385325,
|
|
14741514, -9103726, 7903886, 2348101},
|
|
{24536016, -16515207, 12715592, -3862155, 1511293, 10047386,
|
|
-3842346, -7129159, -28377538, 10048127},
|
|
},
|
|
},
|
|
{
|
|
{
|
|
{-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 = 2097151 & load_3(s);
|
|
int64_t s1 = 2097151 & (load_4(s + 2) >> 5);
|
|
int64_t s2 = 2097151 & (load_3(s + 5) >> 2);
|
|
int64_t s3 = 2097151 & (load_4(s + 7) >> 7);
|
|
int64_t s4 = 2097151 & (load_4(s + 10) >> 4);
|
|
int64_t s5 = 2097151 & (load_3(s + 13) >> 1);
|
|
int64_t s6 = 2097151 & (load_4(s + 15) >> 6);
|
|
int64_t s7 = 2097151 & (load_3(s + 18) >> 3);
|
|
int64_t s8 = 2097151 & load_3(s + 21);
|
|
int64_t s9 = 2097151 & (load_4(s + 23) >> 5);
|
|
int64_t s10 = 2097151 & (load_3(s + 26) >> 2);
|
|
int64_t s11 = 2097151 & (load_4(s + 28) >> 7);
|
|
int64_t s12 = 2097151 & (load_4(s + 31) >> 4);
|
|
int64_t s13 = 2097151 & (load_3(s + 34) >> 1);
|
|
int64_t s14 = 2097151 & (load_4(s + 36) >> 6);
|
|
int64_t s15 = 2097151 & (load_3(s + 39) >> 3);
|
|
int64_t s16 = 2097151 & load_3(s + 42);
|
|
int64_t s17 = 2097151 & (load_4(s + 44) >> 5);
|
|
int64_t s18 = 2097151 & (load_3(s + 47) >> 2);
|
|
int64_t s19 = 2097151 & (load_4(s + 49) >> 7);
|
|
int64_t s20 = 2097151 & (load_4(s + 52) >> 4);
|
|
int64_t s21 = 2097151 & (load_3(s + 55) >> 1);
|
|
int64_t s22 = 2097151 & (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 = 2097151 & load_3(a);
|
|
int64_t a1 = 2097151 & (load_4(a + 2) >> 5);
|
|
int64_t a2 = 2097151 & (load_3(a + 5) >> 2);
|
|
int64_t a3 = 2097151 & (load_4(a + 7) >> 7);
|
|
int64_t a4 = 2097151 & (load_4(a + 10) >> 4);
|
|
int64_t a5 = 2097151 & (load_3(a + 13) >> 1);
|
|
int64_t a6 = 2097151 & (load_4(a + 15) >> 6);
|
|
int64_t a7 = 2097151 & (load_3(a + 18) >> 3);
|
|
int64_t a8 = 2097151 & load_3(a + 21);
|
|
int64_t a9 = 2097151 & (load_4(a + 23) >> 5);
|
|
int64_t a10 = 2097151 & (load_3(a + 26) >> 2);
|
|
int64_t a11 = (load_4(a + 28) >> 7);
|
|
int64_t b0 = 2097151 & load_3(b);
|
|
int64_t b1 = 2097151 & (load_4(b + 2) >> 5);
|
|
int64_t b2 = 2097151 & (load_3(b + 5) >> 2);
|
|
int64_t b3 = 2097151 & (load_4(b + 7) >> 7);
|
|
int64_t b4 = 2097151 & (load_4(b + 10) >> 4);
|
|
int64_t b5 = 2097151 & (load_3(b + 13) >> 1);
|
|
int64_t b6 = 2097151 & (load_4(b + 15) >> 6);
|
|
int64_t b7 = 2097151 & (load_3(b + 18) >> 3);
|
|
int64_t b8 = 2097151 & load_3(b + 21);
|
|
int64_t b9 = 2097151 & (load_4(b + 23) >> 5);
|
|
int64_t b10 = 2097151 & (load_3(b + 26) >> 2);
|
|
int64_t b11 = (load_4(b + 28) >> 7);
|
|
int64_t c0 = 2097151 & load_3(c);
|
|
int64_t c1 = 2097151 & (load_4(c + 2) >> 5);
|
|
int64_t c2 = 2097151 & (load_3(c + 5) >> 2);
|
|
int64_t c3 = 2097151 & (load_4(c + 7) >> 7);
|
|
int64_t c4 = 2097151 & (load_4(c + 10) >> 4);
|
|
int64_t c5 = 2097151 & (load_3(c + 13) >> 1);
|
|
int64_t c6 = 2097151 & (load_4(c + 15) >> 6);
|
|
int64_t c7 = 2097151 & (load_3(c + 18) >> 3);
|
|
int64_t c8 = 2097151 & load_3(c + 21);
|
|
int64_t c9 = 2097151 & (load_4(c + 23) >> 5);
|
|
int64_t c10 = 2097151 & (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;
|
|
}
|
|
|
|
int ED25519_verify(const uint8_t *message, size_t message_len,
|
|
const uint8_t signature[64], const uint8_t public_key[32]) {
|
|
ge_p3 A;
|
|
uint8_t rcopy[32];
|
|
uint8_t scopy[32];
|
|
SHA512_CTX hash_ctx;
|
|
ge_p2 R;
|
|
uint8_t rcheck[32];
|
|
uint8_t h[SHA512_DIGEST_LENGTH];
|
|
|
|
if ((signature[63] & 224) != 0 ||
|
|
ge_frombytes_vartime(&A, public_key) != 0) {
|
|
return 0;
|
|
}
|
|
|
|
fe_neg(A.X, A.X);
|
|
fe_neg(A.T, A.T);
|
|
|
|
memcpy(rcopy, signature, 32);
|
|
memcpy(scopy, signature + 32, 32);
|
|
|
|
SHA512_Init(&hash_ctx);
|
|
SHA512_Update(&hash_ctx, signature, 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, scopy);
|
|
|
|
ge_tobytes(rcheck, &R);
|
|
|
|
return CRYPTO_memcmp(rcheck, rcopy, 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));
|
|
}
|