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ClassiCube/third_party/bearssl/ec_p256_m64.c

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/*
* Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
*
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
* BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*/
#include "inner.h"
#if BR_INT128 || BR_UMUL128
#if BR_UMUL128
#include <intrin.h>
#endif
static const unsigned char P256_G[] = {
0x04, 0x6B, 0x17, 0xD1, 0xF2, 0xE1, 0x2C, 0x42, 0x47, 0xF8,
0xBC, 0xE6, 0xE5, 0x63, 0xA4, 0x40, 0xF2, 0x77, 0x03, 0x7D,
0x81, 0x2D, 0xEB, 0x33, 0xA0, 0xF4, 0xA1, 0x39, 0x45, 0xD8,
0x98, 0xC2, 0x96, 0x4F, 0xE3, 0x42, 0xE2, 0xFE, 0x1A, 0x7F,
0x9B, 0x8E, 0xE7, 0xEB, 0x4A, 0x7C, 0x0F, 0x9E, 0x16, 0x2B,
0xCE, 0x33, 0x57, 0x6B, 0x31, 0x5E, 0xCE, 0xCB, 0xB6, 0x40,
0x68, 0x37, 0xBF, 0x51, 0xF5
};
static const unsigned char P256_N[] = {
0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0xFF, 0xFF,
0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xBC, 0xE6, 0xFA, 0xAD,
0xA7, 0x17, 0x9E, 0x84, 0xF3, 0xB9, 0xCA, 0xC2, 0xFC, 0x63,
0x25, 0x51
};
static const unsigned char *
api_generator(int curve, size_t *len)
{
(void)curve;
*len = sizeof P256_G;
return P256_G;
}
static const unsigned char *
api_order(int curve, size_t *len)
{
(void)curve;
*len = sizeof P256_N;
return P256_N;
}
static size_t
api_xoff(int curve, size_t *len)
{
(void)curve;
*len = 32;
return 1;
}
/*
* A field element is encoded as four 64-bit integers, in basis 2^64.
* Values may reach up to 2^256-1. Montgomery multiplication is used.
*/
/* R = 2^256 mod p */
static const uint64_t F256_R[] = {
0x0000000000000001, 0xFFFFFFFF00000000,
0xFFFFFFFFFFFFFFFF, 0x00000000FFFFFFFE
};
/* Curve equation is y^2 = x^3 - 3*x + B. This constant is B*R mod p
(Montgomery representation of B). */
static const uint64_t P256_B_MONTY[] = {
0xD89CDF6229C4BDDF, 0xACF005CD78843090,
0xE5A220ABF7212ED6, 0xDC30061D04874834
};
/*
* Addition in the field.
*/
static inline void
f256_add(uint64_t *d, const uint64_t *a, const uint64_t *b)
{
#if BR_INT128
unsigned __int128 w;
uint64_t t;
/*
* Do the addition, with an extra carry in t.
*/
w = (unsigned __int128)a[0] + b[0];
d[0] = (uint64_t)w;
w = (unsigned __int128)a[1] + b[1] + (w >> 64);
d[1] = (uint64_t)w;
w = (unsigned __int128)a[2] + b[2] + (w >> 64);
d[2] = (uint64_t)w;
w = (unsigned __int128)a[3] + b[3] + (w >> 64);
d[3] = (uint64_t)w;
t = (uint64_t)(w >> 64);
/*
* Fold carry t, using: 2^256 = 2^224 - 2^192 - 2^96 + 1 mod p.
*/
w = (unsigned __int128)d[0] + t;
d[0] = (uint64_t)w;
w = (unsigned __int128)d[1] + (w >> 64) - (t << 32);
d[1] = (uint64_t)w;
/* Here, carry "w >> 64" can only be 0 or -1 */
w = (unsigned __int128)d[2] - ((w >> 64) & 1);
d[2] = (uint64_t)w;
/* Again, carry is 0 or -1. But there can be carry only if t = 1,
in which case the addition of (t << 32) - t is positive. */
w = (unsigned __int128)d[3] - ((w >> 64) & 1) + (t << 32) - t;
d[3] = (uint64_t)w;
t = (uint64_t)(w >> 64);
/*
* There can be an extra carry here, which we must fold again.
*/
w = (unsigned __int128)d[0] + t;
d[0] = (uint64_t)w;
w = (unsigned __int128)d[1] + (w >> 64) - (t << 32);
d[1] = (uint64_t)w;
w = (unsigned __int128)d[2] - ((w >> 64) & 1);
d[2] = (uint64_t)w;
d[3] += (t << 32) - t - (uint64_t)((w >> 64) & 1);
#elif BR_UMUL128
unsigned char cc;
uint64_t t;
cc = _addcarry_u64(0, a[0], b[0], &d[0]);
cc = _addcarry_u64(cc, a[1], b[1], &d[1]);
cc = _addcarry_u64(cc, a[2], b[2], &d[2]);
cc = _addcarry_u64(cc, a[3], b[3], &d[3]);
/*
* If there is a carry, then we want to subtract p, which we
* do by adding 2^256 - p.
*/
t = cc;
cc = _addcarry_u64(cc, d[0], 0, &d[0]);
cc = _addcarry_u64(cc, d[1], -(t << 32), &d[1]);
cc = _addcarry_u64(cc, d[2], -t, &d[2]);
cc = _addcarry_u64(cc, d[3], (t << 32) - (t << 1), &d[3]);
/*
* We have to do it again if there still is a carry.
*/
t = cc;
cc = _addcarry_u64(cc, d[0], 0, &d[0]);
cc = _addcarry_u64(cc, d[1], -(t << 32), &d[1]);
cc = _addcarry_u64(cc, d[2], -t, &d[2]);
(void)_addcarry_u64(cc, d[3], (t << 32) - (t << 1), &d[3]);
#endif
}
/*
* Subtraction in the field.
*/
static inline void
f256_sub(uint64_t *d, const uint64_t *a, const uint64_t *b)
{
#if BR_INT128
unsigned __int128 w;
uint64_t t;
w = (unsigned __int128)a[0] - b[0];
d[0] = (uint64_t)w;
w = (unsigned __int128)a[1] - b[1] - ((w >> 64) & 1);
d[1] = (uint64_t)w;
w = (unsigned __int128)a[2] - b[2] - ((w >> 64) & 1);
d[2] = (uint64_t)w;
w = (unsigned __int128)a[3] - b[3] - ((w >> 64) & 1);
d[3] = (uint64_t)w;
t = (uint64_t)(w >> 64) & 1;
/*
* If there is a borrow (t = 1), then we must add the modulus
* p = 2^256 - 2^224 + 2^192 + 2^96 - 1.
*/
w = (unsigned __int128)d[0] - t;
d[0] = (uint64_t)w;
w = (unsigned __int128)d[1] + (t << 32) - ((w >> 64) & 1);
d[1] = (uint64_t)w;
/* Here, carry "w >> 64" can only be 0 or +1 */
w = (unsigned __int128)d[2] + (w >> 64);
d[2] = (uint64_t)w;
/* Again, carry is 0 or +1 */
w = (unsigned __int128)d[3] + (w >> 64) - (t << 32) + t;
d[3] = (uint64_t)w;
t = (uint64_t)(w >> 64) & 1;
/*
* There may be again a borrow, in which case we must add the
* modulus again.
*/
w = (unsigned __int128)d[0] - t;
d[0] = (uint64_t)w;
w = (unsigned __int128)d[1] + (t << 32) - ((w >> 64) & 1);
d[1] = (uint64_t)w;
w = (unsigned __int128)d[2] + (w >> 64);
d[2] = (uint64_t)w;
d[3] += (uint64_t)(w >> 64) - (t << 32) + t;
#elif BR_UMUL128
unsigned char cc;
uint64_t t;
cc = _subborrow_u64(0, a[0], b[0], &d[0]);
cc = _subborrow_u64(cc, a[1], b[1], &d[1]);
cc = _subborrow_u64(cc, a[2], b[2], &d[2]);
cc = _subborrow_u64(cc, a[3], b[3], &d[3]);
/*
* If there is a borrow, then we need to add p. We (virtually)
* add 2^256, then subtract 2^256 - p.
*/
t = cc;
cc = _subborrow_u64(0, d[0], t, &d[0]);
cc = _subborrow_u64(cc, d[1], -(t << 32), &d[1]);
cc = _subborrow_u64(cc, d[2], -t, &d[2]);
cc = _subborrow_u64(cc, d[3], (t << 32) - (t << 1), &d[3]);
/*
* If there still is a borrow, then we need to add p again.
*/
t = cc;
cc = _subborrow_u64(0, d[0], t, &d[0]);
cc = _subborrow_u64(cc, d[1], -(t << 32), &d[1]);
cc = _subborrow_u64(cc, d[2], -t, &d[2]);
(void)_subborrow_u64(cc, d[3], (t << 32) - (t << 1), &d[3]);
#endif
}
/*
* Montgomery multiplication in the field.
*/
static void
f256_montymul(uint64_t *d, const uint64_t *a, const uint64_t *b)
{
#if BR_INT128
uint64_t x, f, t0, t1, t2, t3, t4;
unsigned __int128 z, ff;
int i;
/*
* When computing d <- d + a[u]*b, we also add f*p such
* that d + a[u]*b + f*p is a multiple of 2^64. Since
* p = -1 mod 2^64, we can compute f = d[0] + a[u]*b[0] mod 2^64.
*/
/*
* Step 1: t <- (a[0]*b + f*p) / 2^64
* We have f = a[0]*b[0] mod 2^64. Since p = -1 mod 2^64, this
* ensures that (a[0]*b + f*p) is a multiple of 2^64.
*
* We also have: f*p = f*2^256 - f*2^224 + f*2^192 + f*2^96 - f.
*/
x = a[0];
z = (unsigned __int128)b[0] * x;
f = (uint64_t)z;
z = (unsigned __int128)b[1] * x + (z >> 64) + (uint64_t)(f << 32);
t0 = (uint64_t)z;
z = (unsigned __int128)b[2] * x + (z >> 64) + (uint64_t)(f >> 32);
t1 = (uint64_t)z;
z = (unsigned __int128)b[3] * x + (z >> 64) + f;
t2 = (uint64_t)z;
t3 = (uint64_t)(z >> 64);
ff = ((unsigned __int128)f << 64) - ((unsigned __int128)f << 32);
z = (unsigned __int128)t2 + (uint64_t)ff;
t2 = (uint64_t)z;
z = (unsigned __int128)t3 + (z >> 64) + (ff >> 64);
t3 = (uint64_t)z;
t4 = (uint64_t)(z >> 64);
/*
* Steps 2 to 4: t <- (t + a[i]*b + f*p) / 2^64
*/
for (i = 1; i < 4; i ++) {
x = a[i];
/* t <- (t + x*b - f) / 2^64 */
z = (unsigned __int128)b[0] * x + t0;
f = (uint64_t)z;
z = (unsigned __int128)b[1] * x + t1 + (z >> 64);
t0 = (uint64_t)z;
z = (unsigned __int128)b[2] * x + t2 + (z >> 64);
t1 = (uint64_t)z;
z = (unsigned __int128)b[3] * x + t3 + (z >> 64);
t2 = (uint64_t)z;
z = t4 + (z >> 64);
t3 = (uint64_t)z;
t4 = (uint64_t)(z >> 64);
/* t <- t + f*2^32, carry in the upper half of z */
z = (unsigned __int128)t0 + (uint64_t)(f << 32);
t0 = (uint64_t)z;
z = (z >> 64) + (unsigned __int128)t1 + (uint64_t)(f >> 32);
t1 = (uint64_t)z;
/* t <- t + f*2^192 - f*2^160 + f*2^128 */
ff = ((unsigned __int128)f << 64)
- ((unsigned __int128)f << 32) + f;
z = (z >> 64) + (unsigned __int128)t2 + (uint64_t)ff;
t2 = (uint64_t)z;
z = (unsigned __int128)t3 + (z >> 64) + (ff >> 64);
t3 = (uint64_t)z;
t4 += (uint64_t)(z >> 64);
}
/*
* At that point, we have computed t = (a*b + F*p) / 2^256, where
* F is a 256-bit integer whose limbs are the "f" coefficients
* in the steps above. We have:
* a <= 2^256-1
* b <= 2^256-1
* F <= 2^256-1
* Hence:
* a*b + F*p <= (2^256-1)*(2^256-1) + p*(2^256-1)
* a*b + F*p <= 2^256*(2^256 - 2 + p) + 1 - p
* Therefore:
* t < 2^256 + p - 2
* Since p < 2^256, it follows that:
* t4 can be only 0 or 1
* t - p < 2^256
* We can therefore subtract p from t, conditionally on t4, to
* get a nonnegative result that fits on 256 bits.
*/
z = (unsigned __int128)t0 + t4;
t0 = (uint64_t)z;
z = (unsigned __int128)t1 - (t4 << 32) + (z >> 64);
t1 = (uint64_t)z;
z = (unsigned __int128)t2 - (z >> 127);
t2 = (uint64_t)z;
t3 = t3 - (uint64_t)(z >> 127) - t4 + (t4 << 32);
d[0] = t0;
d[1] = t1;
d[2] = t2;
d[3] = t3;
#elif BR_UMUL128
uint64_t x, f, t0, t1, t2, t3, t4;
uint64_t zl, zh, ffl, ffh;
unsigned char k, m;
int i;
/*
* When computing d <- d + a[u]*b, we also add f*p such
* that d + a[u]*b + f*p is a multiple of 2^64. Since
* p = -1 mod 2^64, we can compute f = d[0] + a[u]*b[0] mod 2^64.
*/
/*
* Step 1: t <- (a[0]*b + f*p) / 2^64
* We have f = a[0]*b[0] mod 2^64. Since p = -1 mod 2^64, this
* ensures that (a[0]*b + f*p) is a multiple of 2^64.
*
* We also have: f*p = f*2^256 - f*2^224 + f*2^192 + f*2^96 - f.
*/
x = a[0];
zl = _umul128(b[0], x, &zh);
f = zl;
t0 = zh;
zl = _umul128(b[1], x, &zh);
k = _addcarry_u64(0, zl, t0, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, f << 32, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
t0 = zl;
t1 = zh;
zl = _umul128(b[2], x, &zh);
k = _addcarry_u64(0, zl, t1, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, f >> 32, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
t1 = zl;
t2 = zh;
zl = _umul128(b[3], x, &zh);
k = _addcarry_u64(0, zl, t2, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, f, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
t2 = zl;
t3 = zh;
t4 = _addcarry_u64(0, t3, f, &t3);
k = _subborrow_u64(0, t2, f << 32, &t2);
k = _subborrow_u64(k, t3, f >> 32, &t3);
(void)_subborrow_u64(k, t4, 0, &t4);
/*
* Steps 2 to 4: t <- (t + a[i]*b + f*p) / 2^64
*/
for (i = 1; i < 4; i ++) {
x = a[i];
/* f = t0 + x * b[0]; -- computed below */
/* t <- (t + x*b - f) / 2^64 */
zl = _umul128(b[0], x, &zh);
k = _addcarry_u64(0, zl, t0, &f);
(void)_addcarry_u64(k, zh, 0, &t0);
zl = _umul128(b[1], x, &zh);
k = _addcarry_u64(0, zl, t0, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, t1, &t0);
(void)_addcarry_u64(k, zh, 0, &t1);
zl = _umul128(b[2], x, &zh);
k = _addcarry_u64(0, zl, t1, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, t2, &t1);
(void)_addcarry_u64(k, zh, 0, &t2);
zl = _umul128(b[3], x, &zh);
k = _addcarry_u64(0, zl, t2, &zl);
(void)_addcarry_u64(k, zh, 0, &zh);
k = _addcarry_u64(0, zl, t3, &t2);
(void)_addcarry_u64(k, zh, 0, &t3);
t4 = _addcarry_u64(0, t3, t4, &t3);
/* t <- t + f*2^32, carry in k */
k = _addcarry_u64(0, t0, f << 32, &t0);
k = _addcarry_u64(k, t1, f >> 32, &t1);
/* t <- t + f*2^192 - f*2^160 + f*2^128 */
m = _subborrow_u64(0, f, f << 32, &ffl);
(void)_subborrow_u64(m, f, f >> 32, &ffh);
k = _addcarry_u64(k, t2, ffl, &t2);
k = _addcarry_u64(k, t3, ffh, &t3);
(void)_addcarry_u64(k, t4, 0, &t4);
}
/*
* At that point, we have computed t = (a*b + F*p) / 2^256, where
* F is a 256-bit integer whose limbs are the "f" coefficients
* in the steps above. We have:
* a <= 2^256-1
* b <= 2^256-1
* F <= 2^256-1
* Hence:
* a*b + F*p <= (2^256-1)*(2^256-1) + p*(2^256-1)
* a*b + F*p <= 2^256*(2^256 - 2 + p) + 1 - p
* Therefore:
* t < 2^256 + p - 2
* Since p < 2^256, it follows that:
* t4 can be only 0 or 1
* t - p < 2^256
* We can therefore subtract p from t, conditionally on t4, to
* get a nonnegative result that fits on 256 bits.
*/
k = _addcarry_u64(0, t0, t4, &t0);
k = _addcarry_u64(k, t1, -(t4 << 32), &t1);
k = _addcarry_u64(k, t2, -t4, &t2);
(void)_addcarry_u64(k, t3, (t4 << 32) - (t4 << 1), &t3);
d[0] = t0;
d[1] = t1;
d[2] = t2;
d[3] = t3;
#endif
}
/*
* Montgomery squaring in the field; currently a basic wrapper around
* multiplication (inline, should be optimized away).
* TODO: see if some extra speed can be gained here.
*/
static inline void
f256_montysquare(uint64_t *d, const uint64_t *a)
{
f256_montymul(d, a, a);
}
/*
* Convert to Montgomery representation.
*/
static void
f256_tomonty(uint64_t *d, const uint64_t *a)
{
/*
* R2 = 2^512 mod p.
* If R = 2^256 mod p, then R2 = R^2 mod p; and the Montgomery
* multiplication of a by R2 is: a*R2/R = a*R mod p, i.e. the
* conversion to Montgomery representation.
*/
static const uint64_t R2[] = {
0x0000000000000003,
0xFFFFFFFBFFFFFFFF,
0xFFFFFFFFFFFFFFFE,
0x00000004FFFFFFFD
};
f256_montymul(d, a, R2);
}
/*
* Convert from Montgomery representation.
*/
static void
f256_frommonty(uint64_t *d, const uint64_t *a)
{
/*
* Montgomery multiplication by 1 is division by 2^256 modulo p.
*/
static const uint64_t one[] = { 1, 0, 0, 0 };
f256_montymul(d, a, one);
}
/*
* Inversion in the field. If the source value is 0 modulo p, then this
* returns 0 or p. This function uses Montgomery representation.
*/
static void
f256_invert(uint64_t *d, const uint64_t *a)
{
/*
* We compute a^(p-2) mod p. The exponent pattern (from high to
* low) is:
* - 32 bits of value 1
* - 31 bits of value 0
* - 1 bit of value 1
* - 96 bits of value 0
* - 94 bits of value 1
* - 1 bit of value 0
* - 1 bit of value 1
* To speed up the square-and-multiply algorithm, we precompute
* a^(2^31-1).
*/
uint64_t r[4], t[4];
int i;
br_memcpy(t, a, sizeof t);
for (i = 0; i < 30; i ++) {
f256_montysquare(t, t);
f256_montymul(t, t, a);
}
br_memcpy(r, t, sizeof t);
for (i = 224; i >= 0; i --) {
f256_montysquare(r, r);
switch (i) {
case 0:
case 2:
case 192:
case 224:
f256_montymul(r, r, a);
break;
case 3:
case 34:
case 65:
f256_montymul(r, r, t);
break;
}
}
br_memcpy(d, r, sizeof r);
}
/*
* Finalize reduction.
* Input value fits on 256 bits. This function subtracts p if and only
* if the input is greater than or equal to p.
*/
static inline void
f256_final_reduce(uint64_t *a)
{
#if BR_INT128
uint64_t t0, t1, t2, t3, cc;
unsigned __int128 z;
/*
* We add 2^224 - 2^192 - 2^96 + 1 to a. If there is no carry,
* then a < p; otherwise, the addition result we computed is
* the value we must return.
*/
z = (unsigned __int128)a[0] + 1;
t0 = (uint64_t)z;
z = (unsigned __int128)a[1] + (z >> 64) - ((uint64_t)1 << 32);
t1 = (uint64_t)z;
z = (unsigned __int128)a[2] - (z >> 127);
t2 = (uint64_t)z;
z = (unsigned __int128)a[3] - (z >> 127) + 0xFFFFFFFF;
t3 = (uint64_t)z;
cc = -(uint64_t)(z >> 64);
a[0] ^= cc & (a[0] ^ t0);
a[1] ^= cc & (a[1] ^ t1);
a[2] ^= cc & (a[2] ^ t2);
a[3] ^= cc & (a[3] ^ t3);
#elif BR_UMUL128
uint64_t t0, t1, t2, t3, m;
unsigned char k;
k = _addcarry_u64(0, a[0], (uint64_t)1, &t0);
k = _addcarry_u64(k, a[1], -((uint64_t)1 << 32), &t1);
k = _addcarry_u64(k, a[2], -(uint64_t)1, &t2);
k = _addcarry_u64(k, a[3], ((uint64_t)1 << 32) - 2, &t3);
m = -(uint64_t)k;
a[0] ^= m & (a[0] ^ t0);
a[1] ^= m & (a[1] ^ t1);
a[2] ^= m & (a[2] ^ t2);
a[3] ^= m & (a[3] ^ t3);
#endif
}
/*
* Points in affine and Jacobian coordinates.
*
* - In affine coordinates, the point-at-infinity cannot be encoded.
* - Jacobian coordinates (X,Y,Z) correspond to affine (X/Z^2,Y/Z^3);
* if Z = 0 then this is the point-at-infinity.
*/
typedef struct {
uint64_t x[4];
uint64_t y[4];
} p256_affine;
typedef struct {
uint64_t x[4];
uint64_t y[4];
uint64_t z[4];
} p256_jacobian;
/*
* Decode a point. The returned point is in Jacobian coordinates, but
* with z = 1. If the encoding is invalid, or encodes a point which is
* not on the curve, or encodes the point at infinity, then this function
* returns 0. Otherwise, 1 is returned.
*
* The buffer is assumed to have length exactly 65 bytes.
*/
static uint32_t
point_decode(p256_jacobian *P, const unsigned char *buf)
{
uint64_t x[4], y[4], t[4], x3[4], tt;
uint32_t r;
/*
* Header byte shall be 0x04.
*/
r = EQ(buf[0], 0x04);
/*
* Decode X and Y coordinates, and convert them into
* Montgomery representation.
*/
x[3] = br_dec64be(buf + 1);
x[2] = br_dec64be(buf + 9);
x[1] = br_dec64be(buf + 17);
x[0] = br_dec64be(buf + 25);
y[3] = br_dec64be(buf + 33);
y[2] = br_dec64be(buf + 41);
y[1] = br_dec64be(buf + 49);
y[0] = br_dec64be(buf + 57);
f256_tomonty(x, x);
f256_tomonty(y, y);
/*
* Verify y^2 = x^3 + A*x + B. In curve P-256, A = -3.
* Note that the Montgomery representation of 0 is 0. We must
* take care to apply the final reduction to make sure we have
* 0 and not p.
*/
f256_montysquare(t, y);
f256_montysquare(x3, x);
f256_montymul(x3, x3, x);
f256_sub(t, t, x3);
f256_add(t, t, x);
f256_add(t, t, x);
f256_add(t, t, x);
f256_sub(t, t, P256_B_MONTY);
f256_final_reduce(t);
tt = t[0] | t[1] | t[2] | t[3];
r &= EQ((uint32_t)(tt | (tt >> 32)), 0);
/*
* Return the point in Jacobian coordinates (and Montgomery
* representation).
*/
br_memcpy(P->x, x, sizeof x);
br_memcpy(P->y, y, sizeof y);
br_memcpy(P->z, F256_R, sizeof F256_R);
return r;
}
/*
* Final conversion for a point:
* - The point is converted back to affine coordinates.
* - Final reduction is performed.
* - The point is encoded into the provided buffer.
*
* If the point is the point-at-infinity, all operations are performed,
* but the buffer contents are indeterminate, and 0 is returned. Otherwise,
* the encoded point is written in the buffer, and 1 is returned.
*/
static uint32_t
point_encode(unsigned char *buf, const p256_jacobian *P)
{
uint64_t t1[4], t2[4], z;
/* Set t1 = 1/z^2 and t2 = 1/z^3. */
f256_invert(t2, P->z);
f256_montysquare(t1, t2);
f256_montymul(t2, t2, t1);
/* Compute affine coordinates x (in t1) and y (in t2). */
f256_montymul(t1, P->x, t1);
f256_montymul(t2, P->y, t2);
/* Convert back from Montgomery representation, and finalize
reductions. */
f256_frommonty(t1, t1);
f256_frommonty(t2, t2);
f256_final_reduce(t1);
f256_final_reduce(t2);
/* Encode. */
buf[0] = 0x04;
br_enc64be(buf + 1, t1[3]);
br_enc64be(buf + 9, t1[2]);
br_enc64be(buf + 17, t1[1]);
br_enc64be(buf + 25, t1[0]);
br_enc64be(buf + 33, t2[3]);
br_enc64be(buf + 41, t2[2]);
br_enc64be(buf + 49, t2[1]);
br_enc64be(buf + 57, t2[0]);
/* Return success if and only if P->z != 0. */
z = P->z[0] | P->z[1] | P->z[2] | P->z[3];
return NEQ((uint32_t)(z | z >> 32), 0);
}
/*
* Point doubling in Jacobian coordinates: point P is doubled.
* Note: if the source point is the point-at-infinity, then the result is
* still the point-at-infinity, which is correct. Moreover, if the three
* coordinates were zero, then they still are zero in the returned value.
*
* (Note: this is true even without the final reduction: if the three
* coordinates are encoded as four words of value zero each, then the
* result will also have all-zero coordinate encodings, not the alternate
* encoding as the integer p.)
*/
static void
p256_double(p256_jacobian *P)
{
/*
* Doubling formulas are:
*
* s = 4*x*y^2
* m = 3*(x + z^2)*(x - z^2)
* x' = m^2 - 2*s
* y' = m*(s - x') - 8*y^4
* z' = 2*y*z
*
* These formulas work for all points, including points of order 2
* and points at infinity:
* - If y = 0 then z' = 0. But there is no such point in P-256
* anyway.
* - If z = 0 then z' = 0.
*/
uint64_t t1[4], t2[4], t3[4], t4[4];
/*
* Compute z^2 in t1.
*/
f256_montysquare(t1, P->z);
/*
* Compute x-z^2 in t2 and x+z^2 in t1.
*/
f256_add(t2, P->x, t1);
f256_sub(t1, P->x, t1);
/*
* Compute 3*(x+z^2)*(x-z^2) in t1.
*/
f256_montymul(t3, t1, t2);
f256_add(t1, t3, t3);
f256_add(t1, t3, t1);
/*
* Compute 4*x*y^2 (in t2) and 2*y^2 (in t3).
*/
f256_montysquare(t3, P->y);
f256_add(t3, t3, t3);
f256_montymul(t2, P->x, t3);
f256_add(t2, t2, t2);
/*
* Compute x' = m^2 - 2*s.
*/
f256_montysquare(P->x, t1);
f256_sub(P->x, P->x, t2);
f256_sub(P->x, P->x, t2);
/*
* Compute z' = 2*y*z.
*/
f256_montymul(t4, P->y, P->z);
f256_add(P->z, t4, t4);
/*
* Compute y' = m*(s - x') - 8*y^4. Note that we already have
* 2*y^2 in t3.
*/
f256_sub(t2, t2, P->x);
f256_montymul(P->y, t1, t2);
f256_montysquare(t4, t3);
f256_add(t4, t4, t4);
f256_sub(P->y, P->y, t4);
}
/*
* Point addition (Jacobian coordinates): P1 is replaced with P1+P2.
* This function computes the wrong result in the following cases:
*
* - If P1 == 0 but P2 != 0
* - If P1 != 0 but P2 == 0
* - If P1 == P2
*
* In all three cases, P1 is set to the point at infinity.
*
* Returned value is 0 if one of the following occurs:
*
* - P1 and P2 have the same Y coordinate.
* - P1 == 0 and P2 == 0.
* - The Y coordinate of one of the points is 0 and the other point is
* the point at infinity.
*
* The third case cannot actually happen with valid points, since a point
* with Y == 0 is a point of order 2, and there is no point of order 2 on
* curve P-256.
*
* Therefore, assuming that P1 != 0 and P2 != 0 on input, then the caller
* can apply the following:
*
* - If the result is not the point at infinity, then it is correct.
* - Otherwise, if the returned value is 1, then this is a case of
* P1+P2 == 0, so the result is indeed the point at infinity.
* - Otherwise, P1 == P2, so a "double" operation should have been
* performed.
*
* Note that you can get a returned value of 0 with a correct result,
* e.g. if P1 and P2 have the same Y coordinate, but distinct X coordinates.
*/
static uint32_t
p256_add(p256_jacobian *P1, const p256_jacobian *P2)
{
/*
* Addtions formulas are:
*
* u1 = x1 * z2^2
* u2 = x2 * z1^2
* s1 = y1 * z2^3
* s2 = y2 * z1^3
* h = u2 - u1
* r = s2 - s1
* x3 = r^2 - h^3 - 2 * u1 * h^2
* y3 = r * (u1 * h^2 - x3) - s1 * h^3
* z3 = h * z1 * z2
*/
uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt;
uint32_t ret;
/*
* Compute u1 = x1*z2^2 (in t1) and s1 = y1*z2^3 (in t3).
*/
f256_montysquare(t3, P2->z);
f256_montymul(t1, P1->x, t3);
f256_montymul(t4, P2->z, t3);
f256_montymul(t3, P1->y, t4);
/*
* Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
*/
f256_montysquare(t4, P1->z);
f256_montymul(t2, P2->x, t4);
f256_montymul(t5, P1->z, t4);
f256_montymul(t4, P2->y, t5);
/*
* Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
* We need to test whether r is zero, so we will do some extra
* reduce.
*/
f256_sub(t2, t2, t1);
f256_sub(t4, t4, t3);
f256_final_reduce(t4);
tt = t4[0] | t4[1] | t4[2] | t4[3];
ret = (uint32_t)(tt | (tt >> 32));
ret = (ret | -ret) >> 31;
/*
* Compute u1*h^2 (in t6) and h^3 (in t5);
*/
f256_montysquare(t7, t2);
f256_montymul(t6, t1, t7);
f256_montymul(t5, t7, t2);
/*
* Compute x3 = r^2 - h^3 - 2*u1*h^2.
*/
f256_montysquare(P1->x, t4);
f256_sub(P1->x, P1->x, t5);
f256_sub(P1->x, P1->x, t6);
f256_sub(P1->x, P1->x, t6);
/*
* Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
*/
f256_sub(t6, t6, P1->x);
f256_montymul(P1->y, t4, t6);
f256_montymul(t1, t5, t3);
f256_sub(P1->y, P1->y, t1);
/*
* Compute z3 = h*z1*z2.
*/
f256_montymul(t1, P1->z, P2->z);
f256_montymul(P1->z, t1, t2);
return ret;
}
/*
* Point addition (mixed coordinates): P1 is replaced with P1+P2.
* This is a specialised function for the case when P2 is a non-zero point
* in affine coordinates.
*
* This function computes the wrong result in the following cases:
*
* - If P1 == 0
* - If P1 == P2
*
* In both cases, P1 is set to the point at infinity.
*
* Returned value is 0 if one of the following occurs:
*
* - P1 and P2 have the same Y (affine) coordinate.
* - The Y coordinate of P2 is 0 and P1 is the point at infinity.
*
* The second case cannot actually happen with valid points, since a point
* with Y == 0 is a point of order 2, and there is no point of order 2 on
* curve P-256.
*
* Therefore, assuming that P1 != 0 on input, then the caller
* can apply the following:
*
* - If the result is not the point at infinity, then it is correct.
* - Otherwise, if the returned value is 1, then this is a case of
* P1+P2 == 0, so the result is indeed the point at infinity.
* - Otherwise, P1 == P2, so a "double" operation should have been
* performed.
*
* Again, a value of 0 may be returned in some cases where the addition
* result is correct.
*/
static uint32_t
p256_add_mixed(p256_jacobian *P1, const p256_affine *P2)
{
/*
* Addtions formulas are:
*
* u1 = x1
* u2 = x2 * z1^2
* s1 = y1
* s2 = y2 * z1^3
* h = u2 - u1
* r = s2 - s1
* x3 = r^2 - h^3 - 2 * u1 * h^2
* y3 = r * (u1 * h^2 - x3) - s1 * h^3
* z3 = h * z1
*/
uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt;
uint32_t ret;
/*
* Compute u1 = x1 (in t1) and s1 = y1 (in t3).
*/
br_memcpy(t1, P1->x, sizeof t1);
br_memcpy(t3, P1->y, sizeof t3);
/*
* Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
*/
f256_montysquare(t4, P1->z);
f256_montymul(t2, P2->x, t4);
f256_montymul(t5, P1->z, t4);
f256_montymul(t4, P2->y, t5);
/*
* Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
* We need to test whether r is zero, so we will do some extra
* reduce.
*/
f256_sub(t2, t2, t1);
f256_sub(t4, t4, t3);
f256_final_reduce(t4);
tt = t4[0] | t4[1] | t4[2] | t4[3];
ret = (uint32_t)(tt | (tt >> 32));
ret = (ret | -ret) >> 31;
/*
* Compute u1*h^2 (in t6) and h^3 (in t5);
*/
f256_montysquare(t7, t2);
f256_montymul(t6, t1, t7);
f256_montymul(t5, t7, t2);
/*
* Compute x3 = r^2 - h^3 - 2*u1*h^2.
*/
f256_montysquare(P1->x, t4);
f256_sub(P1->x, P1->x, t5);
f256_sub(P1->x, P1->x, t6);
f256_sub(P1->x, P1->x, t6);
/*
* Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
*/
f256_sub(t6, t6, P1->x);
f256_montymul(P1->y, t4, t6);
f256_montymul(t1, t5, t3);
f256_sub(P1->y, P1->y, t1);
/*
* Compute z3 = h*z1*z2.
*/
f256_montymul(P1->z, P1->z, t2);
return ret;
}
#if 0
/* unused */
/*
* Point addition (mixed coordinates, complete): P1 is replaced with P1+P2.
* This is a specialised function for the case when P2 is a non-zero point
* in affine coordinates.
*
* This function returns the correct result in all cases.
*/
static uint32_t
p256_add_complete_mixed(p256_jacobian *P1, const p256_affine *P2)
{
/*
* Addtions formulas, in the general case, are:
*
* u1 = x1
* u2 = x2 * z1^2
* s1 = y1
* s2 = y2 * z1^3
* h = u2 - u1
* r = s2 - s1
* x3 = r^2 - h^3 - 2 * u1 * h^2
* y3 = r * (u1 * h^2 - x3) - s1 * h^3
* z3 = h * z1
*
* These formulas mishandle the two following cases:
*
* - If P1 is the point-at-infinity (z1 = 0), then z3 is
* incorrectly set to 0.
*
* - If P1 = P2, then u1 = u2 and s1 = s2, and x3, y3 and z3
* are all set to 0.
*
* However, if P1 + P2 = 0, then u1 = u2 but s1 != s2, and then
* we correctly get z3 = 0 (the point-at-infinity).
*
* To fix the case P1 = 0, we perform at the end a copy of P2
* over P1, conditional to z1 = 0.
*
* For P1 = P2: in that case, both h and r are set to 0, and
* we get x3, y3 and z3 equal to 0. We can test for that
* occurrence to make a mask which will be all-one if P1 = P2,
* or all-zero otherwise; then we can compute the double of P2
* and add it, combined with the mask, to (x3,y3,z3).
*
* Using the doubling formulas in p256_double() on (x2,y2),
* simplifying since P2 is affine (i.e. z2 = 1, implicitly),
* we get:
* s = 4*x2*y2^2
* m = 3*(x2 + 1)*(x2 - 1)
* x' = m^2 - 2*s
* y' = m*(s - x') - 8*y2^4
* z' = 2*y2
* which requires only 6 multiplications. Added to the 11
* multiplications of the normal mixed addition in Jacobian
* coordinates, we get a cost of 17 multiplications in total.
*/
uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt, zz;
int i;
/*
* Set zz to -1 if P1 is the point at infinity, 0 otherwise.
*/
zz = P1->z[0] | P1->z[1] | P1->z[2] | P1->z[3];
zz = ((zz | -zz) >> 63) - (uint64_t)1;
/*
* Compute u1 = x1 (in t1) and s1 = y1 (in t3).
*/
br_memcpy(t1, P1->x, sizeof t1);
br_memcpy(t3, P1->y, sizeof t3);
/*
* Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
*/
f256_montysquare(t4, P1->z);
f256_montymul(t2, P2->x, t4);
f256_montymul(t5, P1->z, t4);
f256_montymul(t4, P2->y, t5);
/*
* Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
* reduce.
*/
f256_sub(t2, t2, t1);
f256_sub(t4, t4, t3);
/*
* If both h = 0 and r = 0, then P1 = P2, and we want to set
* the mask tt to -1; otherwise, the mask will be 0.
*/
f256_final_reduce(t2);
f256_final_reduce(t4);
tt = t2[0] | t2[1] | t2[2] | t2[3] | t4[0] | t4[1] | t4[2] | t4[3];
tt = ((tt | -tt) >> 63) - (uint64_t)1;
/*
* Compute u1*h^2 (in t6) and h^3 (in t5);
*/
f256_montysquare(t7, t2);
f256_montymul(t6, t1, t7);
f256_montymul(t5, t7, t2);
/*
* Compute x3 = r^2 - h^3 - 2*u1*h^2.
*/
f256_montysquare(P1->x, t4);
f256_sub(P1->x, P1->x, t5);
f256_sub(P1->x, P1->x, t6);
f256_sub(P1->x, P1->x, t6);
/*
* Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
*/
f256_sub(t6, t6, P1->x);
f256_montymul(P1->y, t4, t6);
f256_montymul(t1, t5, t3);
f256_sub(P1->y, P1->y, t1);
/*
* Compute z3 = h*z1.
*/
f256_montymul(P1->z, P1->z, t2);
/*
* The "double" result, in case P1 = P2.
*/
/*
* Compute z' = 2*y2 (in t1).
*/
f256_add(t1, P2->y, P2->y);
/*
* Compute 2*(y2^2) (in t2) and s = 4*x2*(y2^2) (in t3).
*/
f256_montysquare(t2, P2->y);
f256_add(t2, t2, t2);
f256_add(t3, t2, t2);
f256_montymul(t3, P2->x, t3);
/*
* Compute m = 3*(x2^2 - 1) (in t4).
*/
f256_montysquare(t4, P2->x);
f256_sub(t4, t4, F256_R);
f256_add(t5, t4, t4);
f256_add(t4, t4, t5);
/*
* Compute x' = m^2 - 2*s (in t5).
*/
f256_montysquare(t5, t4);
f256_sub(t5, t3);
f256_sub(t5, t3);
/*
* Compute y' = m*(s - x') - 8*y2^4 (in t6).
*/
f256_sub(t6, t3, t5);
f256_montymul(t6, t6, t4);
f256_montysquare(t7, t2);
f256_sub(t6, t6, t7);
f256_sub(t6, t6, t7);
/*
* We now have the alternate (doubling) coordinates in (t5,t6,t1).
* We combine them with (x3,y3,z3).
*/
for (i = 0; i < 4; i ++) {
P1->x[i] |= tt & t5[i];
P1->y[i] |= tt & t6[i];
P1->z[i] |= tt & t1[i];
}
/*
* If P1 = 0, then we get z3 = 0 (which is invalid); if z1 is 0,
* then we want to replace the result with a copy of P2. The
* test on z1 was done at the start, in the zz mask.
*/
for (i = 0; i < 4; i ++) {
P1->x[i] ^= zz & (P1->x[i] ^ P2->x[i]);
P1->y[i] ^= zz & (P1->y[i] ^ P2->y[i]);
P1->z[i] ^= zz & (P1->z[i] ^ F256_R[i]);
}
}
#endif
/*
* Inner function for computing a point multiplication. A window is
* provided, with points 1*P to 15*P in affine coordinates.
*
* Assumptions:
* - All provided points are valid points on the curve.
* - Multiplier is non-zero, and smaller than the curve order.
* - Everything is in Montgomery representation.
*/
static void
point_mul_inner(p256_jacobian *R, const p256_affine *W,
const unsigned char *k, size_t klen)
{
p256_jacobian Q;
uint32_t qz;
br_memset(&Q, 0, sizeof Q);
qz = 1;
while (klen -- > 0) {
int i;
unsigned bk;
bk = *k ++;
for (i = 0; i < 2; i ++) {
uint32_t bits;
uint32_t bnz;
p256_affine T;
p256_jacobian U;
uint32_t n;
int j;
uint64_t m;
p256_double(&Q);
p256_double(&Q);
p256_double(&Q);
p256_double(&Q);
bits = (bk >> 4) & 0x0F;
bnz = NEQ(bits, 0);
/*
* Lookup point in window. If the bits are 0,
* we get something invalid, which is not a
* problem because we will use it only if the
* bits are non-zero.
*/
br_memset(&T, 0, sizeof T);
for (n = 0; n < 15; n ++) {
m = -(uint64_t)EQ(bits, n + 1);
T.x[0] |= m & W[n].x[0];
T.x[1] |= m & W[n].x[1];
T.x[2] |= m & W[n].x[2];
T.x[3] |= m & W[n].x[3];
T.y[0] |= m & W[n].y[0];
T.y[1] |= m & W[n].y[1];
T.y[2] |= m & W[n].y[2];
T.y[3] |= m & W[n].y[3];
}
U = Q;
p256_add_mixed(&U, &T);
/*
* If qz is still 1, then Q was all-zeros, and this
* is conserved through p256_double().
*/
m = -(uint64_t)(bnz & qz);
for (j = 0; j < 4; j ++) {
Q.x[j] |= m & T.x[j];
Q.y[j] |= m & T.y[j];
Q.z[j] |= m & F256_R[j];
}
CCOPY(bnz & ~qz, &Q, &U, sizeof Q);
qz &= ~bnz;
bk <<= 4;
}
}
*R = Q;
}
/*
* Convert a window from Jacobian to affine coordinates. A single
* field inversion is used. This function works for windows up to
* 32 elements.
*
* The destination array (aff[]) and the source array (jac[]) may
* overlap, provided that the start of aff[] is not after the start of
* jac[]. Even if the arrays do _not_ overlap, the source array is
* modified.
*/
static void
window_to_affine(p256_affine *aff, p256_jacobian *jac, int num)
{
/*
* Convert the window points to affine coordinates. We use the
* following trick to mutualize the inversion computation: if
* we have z1, z2, z3, and z4, and want to inverse all of them,
* we compute u = 1/(z1*z2*z3*z4), and then we have:
* 1/z1 = u*z2*z3*z4
* 1/z2 = u*z1*z3*z4
* 1/z3 = u*z1*z2*z4
* 1/z4 = u*z1*z2*z3
*
* The partial products are computed recursively:
*
* - on input (z_1,z_2), return (z_2,z_1) and z_1*z_2
* - on input (z_1,z_2,... z_n):
* recurse on (z_1,z_2,... z_(n/2)) -> r1 and m1
* recurse on (z_(n/2+1),z_(n/2+2)... z_n) -> r2 and m2
* multiply elements of r1 by m2 -> s1
* multiply elements of r2 by m1 -> s2
* return r1||r2 and m1*m2
*
* In the example below, we suppose that we have 14 elements.
* Let z1, z2,... zE be the 14 values to invert (index noted in
* hexadecimal, starting at 1).
*
* - Depth 1:
* swap(z1, z2); z12 = z1*z2
* swap(z3, z4); z34 = z3*z4
* swap(z5, z6); z56 = z5*z6
* swap(z7, z8); z78 = z7*z8
* swap(z9, zA); z9A = z9*zA
* swap(zB, zC); zBC = zB*zC
* swap(zD, zE); zDE = zD*zE
*
* - Depth 2:
* z1 <- z1*z34, z2 <- z2*z34, z3 <- z3*z12, z4 <- z4*z12
* z1234 = z12*z34
* z5 <- z5*z78, z6 <- z6*z78, z7 <- z7*z56, z8 <- z8*z56
* z5678 = z56*z78
* z9 <- z9*zBC, zA <- zA*zBC, zB <- zB*z9A, zC <- zC*z9A
* z9ABC = z9A*zBC
*
* - Depth 3:
* z1 <- z1*z5678, z2 <- z2*z5678, z3 <- z3*z5678, z4 <- z4*z5678
* z5 <- z5*z1234, z6 <- z6*z1234, z7 <- z7*z1234, z8 <- z8*z1234
* z12345678 = z1234*z5678
* z9 <- z9*zDE, zA <- zA*zDE, zB <- zB*zDE, zC <- zC*zDE
* zD <- zD*z9ABC, zE*z9ABC
* z9ABCDE = z9ABC*zDE
*
* - Depth 4:
* multiply z1..z8 by z9ABCDE
* multiply z9..zE by z12345678
* final z = z12345678*z9ABCDE
*/
uint64_t z[16][4];
int i, k, s;
#define zt (z[15])
#define zu (z[14])
#define zv (z[13])
/*
* First recursion step (pairwise swapping and multiplication).
* If there is an odd number of elements, then we "invent" an
* extra one with coordinate Z = 1 (in Montgomery representation).
*/
for (i = 0; (i + 1) < num; i += 2) {
br_memcpy(zt, jac[i].z, sizeof zt);
br_memcpy(jac[i].z, jac[i + 1].z, sizeof zt);
br_memcpy(jac[i + 1].z, zt, sizeof zt);
f256_montymul(z[i >> 1], jac[i].z, jac[i + 1].z);
}
if ((num & 1) != 0) {
br_memcpy(z[num >> 1], jac[num - 1].z, sizeof zt);
br_memcpy(jac[num - 1].z, F256_R, sizeof F256_R);
}
/*
* Perform further recursion steps. At the entry of each step,
* the process has been done for groups of 's' points. The
* integer k is the log2 of s.
*/
for (k = 1, s = 2; s < num; k ++, s <<= 1) {
int n;
for (i = 0; i < num; i ++) {
f256_montymul(jac[i].z, jac[i].z, z[(i >> k) ^ 1]);
}
n = (num + s - 1) >> k;
for (i = 0; i < (n >> 1); i ++) {
f256_montymul(z[i], z[i << 1], z[(i << 1) + 1]);
}
if ((n & 1) != 0) {
br_memmove(z[n >> 1], z[n], sizeof zt);
}
}
/*
* Invert the final result, and convert all points.
*/
f256_invert(zt, z[0]);
for (i = 0; i < num; i ++) {
f256_montymul(zv, jac[i].z, zt);
f256_montysquare(zu, zv);
f256_montymul(zv, zv, zu);
f256_montymul(aff[i].x, jac[i].x, zu);
f256_montymul(aff[i].y, jac[i].y, zv);
}
}
/*
* Multiply the provided point by an integer.
* Assumptions:
* - Source point is a valid curve point.
* - Source point is not the point-at-infinity.
* - Integer is not 0, and is lower than the curve order.
* If these conditions are not met, then the result is indeterminate
* (but the process is still constant-time).
*/
static void
p256_mul(p256_jacobian *P, const unsigned char *k, size_t klen)
{
union {
p256_affine aff[15];
p256_jacobian jac[15];
} window;
int i;
/*
* Compute window, in Jacobian coordinates.
*/
window.jac[0] = *P;
for (i = 2; i < 16; i ++) {
window.jac[i - 1] = window.jac[(i >> 1) - 1];
if ((i & 1) == 0) {
p256_double(&window.jac[i - 1]);
} else {
p256_add(&window.jac[i - 1], &window.jac[i >> 1]);
}
}
/*
* Convert the window points to affine coordinates. Point
* window[0] is the source point, already in affine coordinates.
*/
window_to_affine(window.aff, window.jac, 15);
/*
* Perform point multiplication.
*/
point_mul_inner(P, window.aff, k, klen);
}
/*
* Precomputed window for the conventional generator: P256_Gwin[n]
* contains (n+1)*G (affine coordinates, in Montgomery representation).
*/
static const p256_affine P256_Gwin[] = {
{
{ 0x79E730D418A9143C, 0x75BA95FC5FEDB601,
0x79FB732B77622510, 0x18905F76A53755C6 },
{ 0xDDF25357CE95560A, 0x8B4AB8E4BA19E45C,
0xD2E88688DD21F325, 0x8571FF1825885D85 }
},
{
{ 0x850046D410DDD64D, 0xAA6AE3C1A433827D,
0x732205038D1490D9, 0xF6BB32E43DCF3A3B },
{ 0x2F3648D361BEE1A5, 0x152CD7CBEB236FF8,
0x19A8FB0E92042DBE, 0x78C577510A5B8A3B }
},
{
{ 0xFFAC3F904EEBC127, 0xB027F84A087D81FB,
0x66AD77DD87CBBC98, 0x26936A3FB6FF747E },
{ 0xB04C5C1FC983A7EB, 0x583E47AD0861FE1A,
0x788208311A2EE98E, 0xD5F06A29E587CC07 }
},
{
{ 0x74B0B50D46918DCC, 0x4650A6EDC623C173,
0x0CDAACACE8100AF2, 0x577362F541B0176B },
{ 0x2D96F24CE4CBABA6, 0x17628471FAD6F447,
0x6B6C36DEE5DDD22E, 0x84B14C394C5AB863 }
},
{
{ 0xBE1B8AAEC45C61F5, 0x90EC649A94B9537D,
0x941CB5AAD076C20C, 0xC9079605890523C8 },
{ 0xEB309B4AE7BA4F10, 0x73C568EFE5EB882B,
0x3540A9877E7A1F68, 0x73A076BB2DD1E916 }
},
{
{ 0x403947373E77664A, 0x55AE744F346CEE3E,
0xD50A961A5B17A3AD, 0x13074B5954213673 },
{ 0x93D36220D377E44B, 0x299C2B53ADFF14B5,
0xF424D44CEF639F11, 0xA4C9916D4A07F75F }
},
{
{ 0x0746354EA0173B4F, 0x2BD20213D23C00F7,
0xF43EAAB50C23BB08, 0x13BA5119C3123E03 },
{ 0x2847D0303F5B9D4D, 0x6742F2F25DA67BDD,
0xEF933BDC77C94195, 0xEAEDD9156E240867 }
},
{
{ 0x27F14CD19499A78F, 0x462AB5C56F9B3455,
0x8F90F02AF02CFC6B, 0xB763891EB265230D },
{ 0xF59DA3A9532D4977, 0x21E3327DCF9EBA15,
0x123C7B84BE60BBF0, 0x56EC12F27706DF76 }
},
{
{ 0x75C96E8F264E20E8, 0xABE6BFED59A7A841,
0x2CC09C0444C8EB00, 0xE05B3080F0C4E16B },
{ 0x1EB7777AA45F3314, 0x56AF7BEDCE5D45E3,
0x2B6E019A88B12F1A, 0x086659CDFD835F9B }
},
{
{ 0x2C18DBD19DC21EC8, 0x98F9868A0FCF8139,
0x737D2CD648250B49, 0xCC61C94724B3428F },
{ 0x0C2B407880DD9E76, 0xC43A8991383FBE08,
0x5F7D2D65779BE5D2, 0x78719A54EB3B4AB5 }
},
{
{ 0xEA7D260A6245E404, 0x9DE407956E7FDFE0,
0x1FF3A4158DAC1AB5, 0x3E7090F1649C9073 },
{ 0x1A7685612B944E88, 0x250F939EE57F61C8,
0x0C0DAA891EAD643D, 0x68930023E125B88E }
},
{
{ 0x04B71AA7D2697768, 0xABDEDEF5CA345A33,
0x2409D29DEE37385E, 0x4EE1DF77CB83E156 },
{ 0x0CAC12D91CBB5B43, 0x170ED2F6CA895637,
0x28228CFA8ADE6D66, 0x7FF57C9553238ACA }
},
{
{ 0xCCC425634B2ED709, 0x0E356769856FD30D,
0xBCBCD43F559E9811, 0x738477AC5395B759 },
{ 0x35752B90C00EE17F, 0x68748390742ED2E3,
0x7CD06422BD1F5BC1, 0xFBC08769C9E7B797 }
},
{
{ 0xA242A35BB0CF664A, 0x126E48F77F9707E3,
0x1717BF54C6832660, 0xFAAE7332FD12C72E },
{ 0x27B52DB7995D586B, 0xBE29569E832237C2,
0xE8E4193E2A65E7DB, 0x152706DC2EAA1BBB }
},
{
{ 0x72BCD8B7BC60055B, 0x03CC23EE56E27E4B,
0xEE337424E4819370, 0xE2AA0E430AD3DA09 },
{ 0x40B8524F6383C45D, 0xD766355442A41B25,
0x64EFA6DE778A4797, 0x2042170A7079ADF4 }
}
};
/*
* Multiply the conventional generator of the curve by the provided
* integer. Return is written in *P.
*
* Assumptions:
* - Integer is not 0, and is lower than the curve order.
* If this conditions is not met, then the result is indeterminate
* (but the process is still constant-time).
*/
static void
p256_mulgen(p256_jacobian *P, const unsigned char *k, size_t klen)
{
point_mul_inner(P, P256_Gwin, k, klen);
}
/*
* Return 1 if all of the following hold:
* - klen <= 32
* - k != 0
* - k is lower than the curve order
* Otherwise, return 0.
*
* Constant-time behaviour: only klen may be observable.
*/
static uint32_t
check_scalar(const unsigned char *k, size_t klen)
{
uint32_t z;
int32_t c;
size_t u;
if (klen > 32) {
return 0;
}
z = 0;
for (u = 0; u < klen; u ++) {
z |= k[u];
}
if (klen == 32) {
c = 0;
for (u = 0; u < klen; u ++) {
c |= -(int32_t)EQ0(c) & CMP(k[u], P256_N[u]);
}
} else {
c = -1;
}
return NEQ(z, 0) & LT0(c);
}
static uint32_t
api_mul(unsigned char *G, size_t Glen,
const unsigned char *k, size_t klen, int curve)
{
uint32_t r;
p256_jacobian P;
(void)curve;
if (Glen != 65) {
return 0;
}
r = check_scalar(k, klen);
r &= point_decode(&P, G);
p256_mul(&P, k, klen);
r &= point_encode(G, &P);
return r;
}
static size_t
api_mulgen(unsigned char *R,
const unsigned char *k, size_t klen, int curve)
{
p256_jacobian P;
(void)curve;
p256_mulgen(&P, k, klen);
point_encode(R, &P);
return 65;
}
static uint32_t
api_muladd(unsigned char *A, const unsigned char *B, size_t len,
const unsigned char *x, size_t xlen,
const unsigned char *y, size_t ylen, int curve)
{
/*
* We might want to use Shamir's trick here: make a composite
* window of u*P+v*Q points, to merge the two doubling-ladders
* into one. This, however, has some complications:
*
* - During the computation, we may hit the point-at-infinity.
* Thus, we would need p256_add_complete_mixed() (complete
* formulas for point addition), with a higher cost (17 muls
* instead of 11).
*
* - A 4-bit window would be too large, since it would involve
* 16*16-1 = 255 points. For the same window size as in the
* p256_mul() case, we would need to reduce the window size
* to 2 bits, and thus perform twice as many non-doubling
* point additions.
*
* - The window may itself contain the point-at-infinity, and
* thus cannot be in all generality be made of affine points.
* Instead, we would need to make it a window of points in
* Jacobian coordinates. Even p256_add_complete_mixed() would
* be inappropriate.
*
* For these reasons, the code below performs two separate
* point multiplications, then computes the final point addition
* (which is both a "normal" addition, and a doubling, to handle
* all cases).
*/
p256_jacobian P, Q;
uint32_t r, t, s;
uint64_t z;
(void)curve;
if (len != 65) {
return 0;
}
r = point_decode(&P, A);
p256_mul(&P, x, xlen);
if (B == NULL) {
p256_mulgen(&Q, y, ylen);
} else {
r &= point_decode(&Q, B);
p256_mul(&Q, y, ylen);
}
/*
* The final addition may fail in case both points are equal.
*/
t = p256_add(&P, &Q);
f256_final_reduce(P.z);
z = P.z[0] | P.z[1] | P.z[2] | P.z[3];
s = EQ((uint32_t)(z | (z >> 32)), 0);
p256_double(&Q);
/*
* If s is 1 then either P+Q = 0 (t = 1) or P = Q (t = 0). So we
* have the following:
*
* s = 0, t = 0 return P (normal addition)
* s = 0, t = 1 return P (normal addition)
* s = 1, t = 0 return Q (a 'double' case)
* s = 1, t = 1 report an error (P+Q = 0)
*/
CCOPY(s & ~t, &P, &Q, sizeof Q);
point_encode(A, &P);
r &= ~(s & t);
return r;
}
/* see bearssl_ec.h */
const br_ec_impl br_ec_p256_m64 = {
(uint32_t)0x00800000,
&api_generator,
&api_order,
&api_xoff,
&api_mul,
&api_mulgen,
&api_muladd
};
/* see bearssl_ec.h */
const br_ec_impl *
br_ec_p256_m64_get(void)
{
return &br_ec_p256_m64;
}
#else
/* see bearssl_ec.h */
const br_ec_impl *
br_ec_p256_m64_get(void)
{
return 0;
}
#endif
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