/*
* Copyright (c) 2017 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_INT128
/*
* Compute x*y+v1+v2. Operands are 64-bit, and result is 128-bit, with
* high word in "hi" and low word in "lo".
*/
#define FMA1(hi, lo, x, y, v1, v2) do { \
unsigned __int128 fmaz; \
fmaz = (unsigned __int128)(x) * (unsigned __int128)(y) \
+ (unsigned __int128)(v1) + (unsigned __int128)(v2); \
(hi) = (uint64_t)(fmaz >> 64); \
(lo) = (uint64_t)fmaz; \
} while (0)
/*
* Compute x1*y1+x2*y2+v1+v2. Operands are 64-bit, and result is 128-bit,
* with high word in "hi" and low word in "lo".
*
* Callers should ensure that the two inner products, and the v1 and v2
* operands, are multiple of 4 (this is not used by this specific definition
* but may help other implementations).
*/
#define FMA2(hi, lo, x1, y1, x2, y2, v1, v2) do { \
unsigned __int128 fmaz; \
fmaz = (unsigned __int128)(x1) * (unsigned __int128)(y1) \
+ (unsigned __int128)(x2) * (unsigned __int128)(y2) \
+ (unsigned __int128)(v1) + (unsigned __int128)(v2); \
(hi) = (uint64_t)(fmaz >> 64); \
(lo) = (uint64_t)fmaz; \
} while (0)
#elif BR_UMUL128
#include <intrin.h>
#define FMA1(hi, lo, x, y, v1, v2) do { \
uint64_t fmahi, fmalo; \
unsigned char fmacc; \
fmalo = _umul128((x), (y), &fmahi); \
fmacc = _addcarry_u64(0, fmalo, (v1), &fmalo); \
_addcarry_u64(fmacc, fmahi, 0, &fmahi); \
fmacc = _addcarry_u64(0, fmalo, (v2), &(lo)); \
_addcarry_u64(fmacc, fmahi, 0, &(hi)); \
} while (0)
/*
* Normally we should use _addcarry_u64() for FMA2 too, but it makes
* Visual Studio crash. Instead we use this version, which leverages
* the fact that the vx operands, and the products, are multiple of 4.
* This is unfortunately slower.
*/
#define FMA2(hi, lo, x1, y1, x2, y2, v1, v2) do { \
uint64_t fma1hi, fma1lo; \
uint64_t fma2hi, fma2lo; \
uint64_t fmatt; \
fma1lo = _umul128((x1), (y1), &fma1hi); \
fma2lo = _umul128((x2), (y2), &fma2hi); \
fmatt = (fma1lo >> 2) + (fma2lo >> 2) \
+ ((v1) >> 2) + ((v2) >> 2); \
(lo) = fmatt << 2; \
(hi) = fma1hi + fma2hi + (fmatt >> 62); \
} while (0)
/*
* The FMA2 macro definition we would prefer to use, but it triggers
* an internal compiler error in Visual Studio 2015.
*
#define FMA2(hi, lo, x1, y1, x2, y2, v1, v2) do { \
uint64_t fma1hi, fma1lo; \
uint64_t fma2hi, fma2lo; \
unsigned char fmacc; \
fma1lo = _umul128((x1), (y1), &fma1hi); \
fma2lo = _umul128((x2), (y2), &fma2hi); \
fmacc = _addcarry_u64(0, fma1lo, (v1), &fma1lo); \
_addcarry_u64(fmacc, fma1hi, 0, &fma1hi); \
fmacc = _addcarry_u64(0, fma2lo, (v2), &fma2lo); \
_addcarry_u64(fmacc, fma2hi, 0, &fma2hi); \
fmacc = _addcarry_u64(0, fma1lo, fma2lo, &(lo)); \
_addcarry_u64(fmacc, fma1hi, fma2hi, &(hi)); \
} while (0)
*/
#endif
#define MASK62 ((uint64_t)0x3FFFFFFFFFFFFFFF)
#define MUL62_lo(x, y) (((uint64_t)(x) * (uint64_t)(y)) & MASK62)
/*
* Subtract b from a, and return the final carry. If 'ctl32' is 0, then
* a[] is kept unmodified, but the final carry is still computed and
* returned.
*/
static uint32_t
i62_sub(uint64_t *a, const uint64_t *b, size_t num, uint32_t ctl32)
{
uint64_t cc, mask;
size_t u;
cc = 0;
ctl32 = -ctl32;
mask = (uint64_t)ctl32 | ((uint64_t)ctl32 << 32);
for (u = 0; u < num; u ++) {
uint64_t aw, bw, dw;
aw = a[u];
bw = b[u];
dw = aw - bw - cc;
cc = dw >> 63;
dw &= MASK62;
a[u] = aw ^ (mask & (dw ^ aw));
}
return (uint32_t)cc;
}
/*
* Montgomery multiplication, over arrays of 62-bit values. The
* destination array (d) must be distinct from the other operands
* (x, y and m). All arrays are in little-endian format (least
* significant word comes first) over 'num' words.
*/
static void
montymul(uint64_t *d, const uint64_t *x, const uint64_t *y,
const uint64_t *m, size_t num, uint64_t m0i)
{
uint64_t dh;
size_t u, num4;
num4 = 1 + ((num - 1) & ~(size_t)3);
memset(d, 0, num * sizeof *d);
dh = 0;
for (u = 0; u < num; u ++) {
size_t v;
uint64_t f, xu;
uint64_t r, zh;
uint64_t hi, lo;
xu = x[u] << 2;
f = MUL62_lo(d[0] + MUL62_lo(x[u], y[0]), m0i) << 2;
FMA2(hi, lo, xu, y[0], f, m[0], d[0] << 2, 0);
r = hi;
for (v = 1; v < num4; v += 4) {
FMA2(hi, lo, xu, y[v + 0],
f, m[v + 0], d[v + 0] << 2, r << 2);
r = hi + (r >> 62);
d[v - 1] = lo >> 2;
FMA2(hi, lo, xu, y[v + 1],
f, m[v + 1], d[v + 1] << 2, r << 2);
r = hi + (r >> 62);
d[v + 0] = lo >> 2;
FMA2(hi, lo, xu, y[v + 2],
f, m[v + 2], d[v + 2] << 2, r << 2);
r = hi + (r >> 62);
d[v + 1] = lo >> 2;
FMA2(hi, lo, xu, y[v + 3],
f, m[v + 3], d[v + 3] << 2, r << 2);
r = hi + (r >> 62);
d[v + 2] = lo >> 2;
}
for (; v < num; v ++) {
FMA2(hi, lo, xu, y[v], f, m[v], d[v] << 2, r << 2);
r = hi + (r >> 62);
d[v - 1] = lo >> 2;
}
zh = dh + r;
d[num - 1] = zh & MASK62;
dh = zh >> 62;
}
i62_sub(d, m, num, (uint32_t)dh | NOT(i62_sub(d, m, num, 0)));
}
/*
* Conversion back from Montgomery representation.
*/
static void
frommonty(uint64_t *x, const uint64_t *m, size_t num, uint64_t m0i)
{
size_t u, v;
for (u = 0; u < num; u ++) {
uint64_t f, cc;
f = MUL62_lo(x[0], m0i) << 2;
cc = 0;
for (v = 0; v < num; v ++) {
uint64_t hi, lo;
FMA1(hi, lo, f, m[v], x[v] << 2, cc);
cc = hi << 2;
if (v != 0) {
x[v - 1] = lo >> 2;
}
}
x[num - 1] = cc >> 2;
}
i62_sub(x, m, num, NOT(i62_sub(x, m, num, 0)));
}
/* see inner.h */
uint32_t
br_i62_modpow_opt(uint32_t *x31, const unsigned char *e, size_t elen,
const uint32_t *m31, uint32_t m0i31, uint64_t *tmp, size_t twlen)
{
size_t u, mw31num, mw62num;
uint64_t *x, *m, *t1, *t2;
uint64_t m0i;
uint32_t acc;
int win_len, acc_len;
/*
* Get modulus size, in words.
*/
mw31num = (m31[0] + 31) >> 5;
mw62num = (mw31num + 1) >> 1;
/*
* In order to apply this function, we must have enough room to
* copy the operand and modulus into the temporary array, along
* with at least two temporaries. If there is not enough room,
* switch to br_i31_modpow(). We also use br_i31_modpow() if the
* modulus length is not at least four words (94 bits or more).
*/
if (mw31num < 4 || (mw62num << 2) > twlen) {
/*
* We assume here that we can split an aligned uint64_t
* into two properly aligned uint32_t. Since both types
* are supposed to have an exact width with no padding,
* then this property must hold.
*/
size_t txlen;
txlen = mw31num + 1;
if (twlen < txlen) {
return 0;
}
br_i31_modpow(x31, e, elen, m31, m0i31,
(uint32_t *)tmp, (uint32_t *)tmp + txlen);
return 1;
}
/*
* Convert x to Montgomery representation: this means that
* we replace x with x*2^z mod m, where z is the smallest multiple
* of the word size such that 2^z >= m. We want to reuse the 31-bit
* functions here (for constant-time operation), but we need z
* for a 62-bit word size.
*/
for (u = 0; u < mw62num; u ++) {
br_i31_muladd_small(x31, 0, m31);
br_i31_muladd_small(x31, 0, m31);
}
/*
* Assemble operands into arrays of 62-bit words. Note that
* all the arrays of 62-bit words that we will handle here
* are without any leading size word.
*
* We also adjust tmp and twlen to account for the words used
* for these extra arrays.
*/
m = tmp;
x = tmp + mw62num;
tmp += (mw62num << 1);
twlen -= (mw62num << 1);
for (u = 0; u < mw31num; u += 2) {
size_t v;
v = u >> 1;
if ((u + 1) == mw31num) {
m[v] = (uint64_t)m31[u + 1];
x[v] = (uint64_t)x31[u + 1];
} else {
m[v] = (uint64_t)m31[u + 1]
+ ((uint64_t)m31[u + 2] << 31);
x[v] = (uint64_t)x31[u + 1]
+ ((uint64_t)x31[u + 2] << 31);
}
}
/*
* Compute window size. We support windows up to 5 bits; for a
* window of size k bits, we need 2^k+1 temporaries (for k = 1,
* we use special code that uses only 2 temporaries).
*/
for (win_len = 5; win_len > 1; win_len --) {
if ((((uint32_t)1 << win_len) + 1) * mw62num <= twlen) {
break;
}
}
t1 = tmp;
t2 = tmp + mw62num;
/*
* Compute m0i, which is equal to -(1/m0) mod 2^62. We were
* provided with m0i31, which already fulfills this property
* modulo 2^31; the single expression below is then sufficient.
*/
m0i = (uint64_t)m0i31;
m0i = MUL62_lo(m0i, (uint64_t)2 + MUL62_lo(m0i, m[0]));
/*
* Compute window contents. If the window has size one bit only,
* then t2 is set to x; otherwise, t2[0] is left untouched, and
* t2[k] is set to x^k (for k >= 1).
*/
if (win_len == 1) {
memcpy(t2, x, mw62num * sizeof *x);
} else {
uint64_t *base;
memcpy(t2 + mw62num, x, mw62num * sizeof *x);
base = t2 + mw62num;
for (u = 2; u < ((unsigned)1 << win_len); u ++) {
montymul(base + mw62num, base, x, m, mw62num, m0i);
base += mw62num;
}
}
/*
* Set x to 1, in Montgomery representation. We again use the
* 31-bit code.
*/
br_i31_zero(x31, m31[0]);
x31[(m31[0] + 31) >> 5] = 1;
br_i31_muladd_small(x31, 0, m31);
if (mw31num & 1) {
br_i31_muladd_small(x31, 0, m31);
}
for (u = 0; u < mw31num; u += 2) {
size_t v;
v = u >> 1;
if ((u + 1) == mw31num) {
x[v] = (uint64_t)x31[u + 1];
} else {
x[v] = (uint64_t)x31[u + 1]
+ ((uint64_t)x31[u + 2] << 31);
}
}
/*
* We process bits from most to least significant. At each
* loop iteration, we have acc_len bits in acc.
*/
acc = 0;
acc_len = 0;
while (acc_len > 0 || elen > 0) {
int i, k;
uint32_t bits;
uint64_t mask1, mask2;
/*
* Get the next bits.
*/
k = win_len;
if (acc_len < win_len) {
if (elen > 0) {
acc = (acc << 8) | *e ++;
elen --;
acc_len += 8;
} else {
k = acc_len;
}
}
bits = (acc >> (acc_len - k)) & (((uint32_t)1 << k) - 1);
acc_len -= k;
/*
* We could get exactly k bits. Compute k squarings.
*/
for (i = 0; i < k; i ++) {
montymul(t1, x, x, m, mw62num, m0i);
memcpy(x, t1, mw62num * sizeof *x);
}
/*
* Window lookup: we want to set t2 to the window
* lookup value, assuming the bits are non-zero. If
* the window length is 1 bit only, then t2 is
* already set; otherwise, we do a constant-time lookup.
*/
if (win_len > 1) {
uint64_t *base;
memset(t2, 0, mw62num * sizeof *t2);
base = t2 + mw62num;
for (u = 1; u < ((uint32_t)1 << k); u ++) {
uint64_t mask;
size_t v;
mask = -(uint64_t)EQ(u, bits);
for (v = 0; v < mw62num; v ++) {
t2[v] |= mask & base[v];
}
base += mw62num;
}
}
/*
* Multiply with the looked-up value. We keep the product
* only if the exponent bits are not all-zero.
*/
montymul(t1, x, t2, m, mw62num, m0i);
mask1 = -(uint64_t)EQ(bits, 0);
mask2 = ~mask1;
for (u = 0; u < mw62num; u ++) {
x[u] = (mask1 & x[u]) | (mask2 & t1[u]);
}
}
/*
* Convert back from Montgomery representation.
*/
frommonty(x, m, mw62num, m0i);
/*
* Convert result into 31-bit words.
*/
for (u = 0; u < mw31num; u += 2) {
uint64_t zw;
zw = x[u >> 1];
x31[u + 1] = (uint32_t)zw & 0x7FFFFFFF;
if ((u + 1) < mw31num) {
x31[u + 2] = (uint32_t)(zw >> 31);
}
}
return 1;
}
#else
/* see inner.h */
uint32_t
br_i62_modpow_opt(uint32_t *x31, const unsigned char *e, size_t elen,
const uint32_t *m31, uint32_t m0i31, uint64_t *tmp, size_t twlen)
{
size_t mwlen;
mwlen = (m31[0] + 63) >> 5;
if (twlen < mwlen) {
return 0;
}
return br_i31_modpow_opt(x31, e, elen, m31, m0i31,
(uint32_t *)tmp, twlen << 1);
}
#endif
/* see inner.h */
uint32_t
br_i62_modpow_opt_as_i31(uint32_t *x31, const unsigned char *e, size_t elen,
const uint32_t *m31, uint32_t m0i31, uint32_t *tmp, size_t twlen)
{
/*
* As documented, this function expects the 'tmp' argument to be
* 64-bit aligned. This is OK since this function is internal (it
* is not part of BearSSL's public API).
*/
return br_i62_modpow_opt(x31, e, elen, m31, m0i31,
(uint64_t *)tmp, twlen >> 1);
}