You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
362 lines
13 KiB
362 lines
13 KiB
/*
|
|
Public domain by Andrew M. <liquidsun@gmail.com>
|
|
*/
|
|
|
|
|
|
/*
|
|
Arithmetic modulo the group order n = 2^252 + 27742317777372353535851937790883648493 = 7237005577332262213973186563042994240857116359379907606001950938285454250989
|
|
|
|
k = 32
|
|
b = 1 << 8 = 256
|
|
m = 2^252 + 27742317777372353535851937790883648493 = 0x1000000000000000000000000000000014def9dea2f79cd65812631a5cf5d3ed
|
|
mu = floor( b^(k*2) / m ) = 0xfffffffffffffffffffffffffffffffeb2106215d086329a7ed9ce5a30a2c131b
|
|
*/
|
|
|
|
#define bignum256modm_bits_per_limb 56
|
|
#define bignum256modm_limb_size 5
|
|
|
|
typedef uint64_t bignum256modm_element_t;
|
|
typedef bignum256modm_element_t bignum256modm[5];
|
|
|
|
static const bignum256modm modm_m = {
|
|
0x12631a5cf5d3ed,
|
|
0xf9dea2f79cd658,
|
|
0x000000000014de,
|
|
0x00000000000000,
|
|
0x00000010000000
|
|
};
|
|
|
|
static const bignum256modm modm_mu = {
|
|
0x9ce5a30a2c131b,
|
|
0x215d086329a7ed,
|
|
0xffffffffeb2106,
|
|
0xffffffffffffff,
|
|
0x00000fffffffff
|
|
};
|
|
|
|
static bignum256modm_element_t
|
|
lt_modm(bignum256modm_element_t a, bignum256modm_element_t b) {
|
|
return (a - b) >> 63;
|
|
}
|
|
|
|
static void
|
|
reduce256_modm(bignum256modm r) {
|
|
bignum256modm t;
|
|
bignum256modm_element_t b = 0, pb, mask;
|
|
|
|
/* t = r - m */
|
|
pb = 0;
|
|
pb += modm_m[0]; b = lt_modm(r[0], pb); t[0] = (r[0] - pb + (b << 56)); pb = b;
|
|
pb += modm_m[1]; b = lt_modm(r[1], pb); t[1] = (r[1] - pb + (b << 56)); pb = b;
|
|
pb += modm_m[2]; b = lt_modm(r[2], pb); t[2] = (r[2] - pb + (b << 56)); pb = b;
|
|
pb += modm_m[3]; b = lt_modm(r[3], pb); t[3] = (r[3] - pb + (b << 56)); pb = b;
|
|
pb += modm_m[4]; b = lt_modm(r[4], pb); t[4] = (r[4] - pb + (b << 32));
|
|
|
|
/* keep r if r was smaller than m */
|
|
mask = b - 1;
|
|
|
|
r[0] ^= mask & (r[0] ^ t[0]);
|
|
r[1] ^= mask & (r[1] ^ t[1]);
|
|
r[2] ^= mask & (r[2] ^ t[2]);
|
|
r[3] ^= mask & (r[3] ^ t[3]);
|
|
r[4] ^= mask & (r[4] ^ t[4]);
|
|
}
|
|
|
|
static void
|
|
barrett_reduce256_modm(bignum256modm r, const bignum256modm q1, const bignum256modm r1) {
|
|
bignum256modm q3, r2;
|
|
uint128_t c, mul;
|
|
bignum256modm_element_t f, b, pb;
|
|
|
|
/* q1 = x >> 248 = 264 bits = 5 56 bit elements
|
|
q2 = mu * q1
|
|
q3 = (q2 / 256(32+1)) = q2 / (2^8)^(32+1) = q2 >> 264 */
|
|
mul64x64_128(c, modm_mu[0], q1[3]) mul64x64_128(mul, modm_mu[3], q1[0]) add128(c, mul) mul64x64_128(mul, modm_mu[1], q1[2]) add128(c, mul) mul64x64_128(mul, modm_mu[2], q1[1]) add128(c, mul) shr128(f, c, 56);
|
|
mul64x64_128(c, modm_mu[0], q1[4]) add128_64(c, f) mul64x64_128(mul, modm_mu[4], q1[0]) add128(c, mul) mul64x64_128(mul, modm_mu[3], q1[1]) add128(c, mul) mul64x64_128(mul, modm_mu[1], q1[3]) add128(c, mul) mul64x64_128(mul, modm_mu[2], q1[2]) add128(c, mul)
|
|
f = lo128(c); q3[0] = (f >> 40) & 0xffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_mu[4], q1[1]) add128_64(c, f) mul64x64_128(mul, modm_mu[1], q1[4]) add128(c, mul) mul64x64_128(mul, modm_mu[2], q1[3]) add128(c, mul) mul64x64_128(mul, modm_mu[3], q1[2]) add128(c, mul)
|
|
f = lo128(c); q3[0] |= (f << 16) & 0xffffffffffffff; q3[1] = (f >> 40) & 0xffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_mu[4], q1[2]) add128_64(c, f) mul64x64_128(mul, modm_mu[2], q1[4]) add128(c, mul) mul64x64_128(mul, modm_mu[3], q1[3]) add128(c, mul)
|
|
f = lo128(c); q3[1] |= (f << 16) & 0xffffffffffffff; q3[2] = (f >> 40) & 0xffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_mu[4], q1[3]) add128_64(c, f) mul64x64_128(mul, modm_mu[3], q1[4]) add128(c, mul)
|
|
f = lo128(c); q3[2] |= (f << 16) & 0xffffffffffffff; q3[3] = (f >> 40) & 0xffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_mu[4], q1[4]) add128_64(c, f)
|
|
f = lo128(c); q3[3] |= (f << 16) & 0xffffffffffffff; q3[4] = (f >> 40) & 0xffff; shr128(f, c, 56);
|
|
q3[4] |= (f << 16);
|
|
|
|
mul64x64_128(c, modm_m[0], q3[0])
|
|
r2[0] = lo128(c) & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_m[0], q3[1]) add128_64(c, f) mul64x64_128(mul, modm_m[1], q3[0]) add128(c, mul)
|
|
r2[1] = lo128(c) & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_m[0], q3[2]) add128_64(c, f) mul64x64_128(mul, modm_m[2], q3[0]) add128(c, mul) mul64x64_128(mul, modm_m[1], q3[1]) add128(c, mul)
|
|
r2[2] = lo128(c) & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_m[0], q3[3]) add128_64(c, f) mul64x64_128(mul, modm_m[3], q3[0]) add128(c, mul) mul64x64_128(mul, modm_m[1], q3[2]) add128(c, mul) mul64x64_128(mul, modm_m[2], q3[1]) add128(c, mul)
|
|
r2[3] = lo128(c) & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, modm_m[0], q3[4]) add128_64(c, f) mul64x64_128(mul, modm_m[4], q3[0]) add128(c, mul) mul64x64_128(mul, modm_m[3], q3[1]) add128(c, mul) mul64x64_128(mul, modm_m[1], q3[3]) add128(c, mul) mul64x64_128(mul, modm_m[2], q3[2]) add128(c, mul)
|
|
r2[4] = lo128(c) & 0x0000ffffffffff;
|
|
|
|
pb = 0;
|
|
pb += r2[0]; b = lt_modm(r1[0], pb); r[0] = (r1[0] - pb + (b << 56)); pb = b;
|
|
pb += r2[1]; b = lt_modm(r1[1], pb); r[1] = (r1[1] - pb + (b << 56)); pb = b;
|
|
pb += r2[2]; b = lt_modm(r1[2], pb); r[2] = (r1[2] - pb + (b << 56)); pb = b;
|
|
pb += r2[3]; b = lt_modm(r1[3], pb); r[3] = (r1[3] - pb + (b << 56)); pb = b;
|
|
pb += r2[4]; b = lt_modm(r1[4], pb); r[4] = (r1[4] - pb + (b << 40));
|
|
|
|
reduce256_modm(r);
|
|
reduce256_modm(r);
|
|
}
|
|
|
|
|
|
static void
|
|
add256_modm(bignum256modm r, const bignum256modm x, const bignum256modm y) {
|
|
bignum256modm_element_t c;
|
|
|
|
c = x[0] + y[0]; r[0] = c & 0xffffffffffffff; c >>= 56;
|
|
c += x[1] + y[1]; r[1] = c & 0xffffffffffffff; c >>= 56;
|
|
c += x[2] + y[2]; r[2] = c & 0xffffffffffffff; c >>= 56;
|
|
c += x[3] + y[3]; r[3] = c & 0xffffffffffffff; c >>= 56;
|
|
c += x[4] + y[4]; r[4] = c;
|
|
|
|
reduce256_modm(r);
|
|
}
|
|
|
|
static void
|
|
mul256_modm(bignum256modm r, const bignum256modm x, const bignum256modm y) {
|
|
bignum256modm q1, r1;
|
|
uint128_t c, mul;
|
|
bignum256modm_element_t f;
|
|
|
|
mul64x64_128(c, x[0], y[0])
|
|
f = lo128(c); r1[0] = f & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[0], y[1]) add128_64(c, f) mul64x64_128(mul, x[1], y[0]) add128(c, mul)
|
|
f = lo128(c); r1[1] = f & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[0], y[2]) add128_64(c, f) mul64x64_128(mul, x[2], y[0]) add128(c, mul) mul64x64_128(mul, x[1], y[1]) add128(c, mul)
|
|
f = lo128(c); r1[2] = f & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[0], y[3]) add128_64(c, f) mul64x64_128(mul, x[3], y[0]) add128(c, mul) mul64x64_128(mul, x[1], y[2]) add128(c, mul) mul64x64_128(mul, x[2], y[1]) add128(c, mul)
|
|
f = lo128(c); r1[3] = f & 0xffffffffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[0], y[4]) add128_64(c, f) mul64x64_128(mul, x[4], y[0]) add128(c, mul) mul64x64_128(mul, x[3], y[1]) add128(c, mul) mul64x64_128(mul, x[1], y[3]) add128(c, mul) mul64x64_128(mul, x[2], y[2]) add128(c, mul)
|
|
f = lo128(c); r1[4] = f & 0x0000ffffffffff; q1[0] = (f >> 24) & 0xffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[4], y[1]) add128_64(c, f) mul64x64_128(mul, x[1], y[4]) add128(c, mul) mul64x64_128(mul, x[2], y[3]) add128(c, mul) mul64x64_128(mul, x[3], y[2]) add128(c, mul)
|
|
f = lo128(c); q1[0] |= (f << 32) & 0xffffffffffffff; q1[1] = (f >> 24) & 0xffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[4], y[2]) add128_64(c, f) mul64x64_128(mul, x[2], y[4]) add128(c, mul) mul64x64_128(mul, x[3], y[3]) add128(c, mul)
|
|
f = lo128(c); q1[1] |= (f << 32) & 0xffffffffffffff; q1[2] = (f >> 24) & 0xffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[4], y[3]) add128_64(c, f) mul64x64_128(mul, x[3], y[4]) add128(c, mul)
|
|
f = lo128(c); q1[2] |= (f << 32) & 0xffffffffffffff; q1[3] = (f >> 24) & 0xffffffff; shr128(f, c, 56);
|
|
mul64x64_128(c, x[4], y[4]) add128_64(c, f)
|
|
f = lo128(c); q1[3] |= (f << 32) & 0xffffffffffffff; q1[4] = (f >> 24) & 0xffffffff; shr128(f, c, 56);
|
|
q1[4] |= (f << 32);
|
|
|
|
barrett_reduce256_modm(r, q1, r1);
|
|
}
|
|
|
|
static void
|
|
expand256_modm(bignum256modm out, const unsigned char *in, size_t len) {
|
|
unsigned char work[64] = {0};
|
|
bignum256modm_element_t x[16];
|
|
bignum256modm q1;
|
|
|
|
memcpy(work, in, len);
|
|
x[0] = U8TO64_LE(work + 0);
|
|
x[1] = U8TO64_LE(work + 8);
|
|
x[2] = U8TO64_LE(work + 16);
|
|
x[3] = U8TO64_LE(work + 24);
|
|
x[4] = U8TO64_LE(work + 32);
|
|
x[5] = U8TO64_LE(work + 40);
|
|
x[6] = U8TO64_LE(work + 48);
|
|
x[7] = U8TO64_LE(work + 56);
|
|
|
|
/* r1 = (x mod 256^(32+1)) = x mod (2^8)(31+1) = x & ((1 << 264) - 1) */
|
|
out[0] = ( x[0]) & 0xffffffffffffff;
|
|
out[1] = ((x[ 0] >> 56) | (x[ 1] << 8)) & 0xffffffffffffff;
|
|
out[2] = ((x[ 1] >> 48) | (x[ 2] << 16)) & 0xffffffffffffff;
|
|
out[3] = ((x[ 2] >> 40) | (x[ 3] << 24)) & 0xffffffffffffff;
|
|
out[4] = ((x[ 3] >> 32) | (x[ 4] << 32)) & 0x0000ffffffffff;
|
|
|
|
/* under 252 bits, no need to reduce */
|
|
if (len < 32)
|
|
return;
|
|
|
|
/* q1 = x >> 248 = 264 bits */
|
|
q1[0] = ((x[ 3] >> 56) | (x[ 4] << 8)) & 0xffffffffffffff;
|
|
q1[1] = ((x[ 4] >> 48) | (x[ 5] << 16)) & 0xffffffffffffff;
|
|
q1[2] = ((x[ 5] >> 40) | (x[ 6] << 24)) & 0xffffffffffffff;
|
|
q1[3] = ((x[ 6] >> 32) | (x[ 7] << 32)) & 0xffffffffffffff;
|
|
q1[4] = ((x[ 7] >> 24) );
|
|
|
|
barrett_reduce256_modm(out, q1, out);
|
|
}
|
|
|
|
static void
|
|
expand_raw256_modm(bignum256modm out, const unsigned char in[32]) {
|
|
bignum256modm_element_t x[4];
|
|
|
|
x[0] = U8TO64_LE(in + 0);
|
|
x[1] = U8TO64_LE(in + 8);
|
|
x[2] = U8TO64_LE(in + 16);
|
|
x[3] = U8TO64_LE(in + 24);
|
|
|
|
out[0] = ( x[0]) & 0xffffffffffffff;
|
|
out[1] = ((x[ 0] >> 56) | (x[ 1] << 8)) & 0xffffffffffffff;
|
|
out[2] = ((x[ 1] >> 48) | (x[ 2] << 16)) & 0xffffffffffffff;
|
|
out[3] = ((x[ 2] >> 40) | (x[ 3] << 24)) & 0xffffffffffffff;
|
|
out[4] = ((x[ 3] >> 32) ) & 0x000000ffffffff;
|
|
}
|
|
|
|
static void
|
|
contract256_modm(unsigned char out[32], const bignum256modm in) {
|
|
U64TO8_LE(out + 0, (in[0] ) | (in[1] << 56));
|
|
U64TO8_LE(out + 8, (in[1] >> 8) | (in[2] << 48));
|
|
U64TO8_LE(out + 16, (in[2] >> 16) | (in[3] << 40));
|
|
U64TO8_LE(out + 24, (in[3] >> 24) | (in[4] << 32));
|
|
}
|
|
|
|
static void
|
|
contract256_window4_modm(signed char r[64], const bignum256modm in) {
|
|
char carry;
|
|
signed char *quads = r;
|
|
bignum256modm_element_t i, j, v, m;
|
|
|
|
for (i = 0; i < 5; i++) {
|
|
v = in[i];
|
|
m = (i == 4) ? 8 : 14;
|
|
for (j = 0; j < m; j++) {
|
|
*quads++ = (v & 15);
|
|
v >>= 4;
|
|
}
|
|
}
|
|
|
|
/* making it signed */
|
|
carry = 0;
|
|
for(i = 0; i < 63; i++) {
|
|
r[i] += carry;
|
|
r[i+1] += (r[i] >> 4);
|
|
r[i] &= 15;
|
|
carry = (r[i] >> 3);
|
|
r[i] -= (carry << 4);
|
|
}
|
|
r[63] += carry;
|
|
}
|
|
|
|
static void
|
|
contract256_slidingwindow_modm(signed char r[256], const bignum256modm s, int windowsize) {
|
|
int i,j,k,b;
|
|
int m = (1 << (windowsize - 1)) - 1, soplen = 256;
|
|
signed char *bits = r;
|
|
bignum256modm_element_t v;
|
|
|
|
/* first put the binary expansion into r */
|
|
for (i = 0; i < 4; i++) {
|
|
v = s[i];
|
|
for (j = 0; j < 56; j++, v >>= 1)
|
|
*bits++ = (v & 1);
|
|
}
|
|
v = s[4];
|
|
for (j = 0; j < 32; j++, v >>= 1)
|
|
*bits++ = (v & 1);
|
|
|
|
/* Making it sliding window */
|
|
for (j = 0; j < soplen; j++) {
|
|
if (!r[j])
|
|
continue;
|
|
|
|
for (b = 1; (b < (soplen - j)) && (b <= 6); b++) {
|
|
if ((r[j] + (r[j + b] << b)) <= m) {
|
|
r[j] += r[j + b] << b;
|
|
r[j + b] = 0;
|
|
} else if ((r[j] - (r[j + b] << b)) >= -m) {
|
|
r[j] -= r[j + b] << b;
|
|
for (k = j + b; k < soplen; k++) {
|
|
if (!r[k]) {
|
|
r[k] = 1;
|
|
break;
|
|
}
|
|
r[k] = 0;
|
|
}
|
|
} else if (r[j + b]) {
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
helpers for batch verifcation, are allowed to be vartime
|
|
*/
|
|
|
|
/* out = a - b, a must be larger than b */
|
|
static void
|
|
sub256_modm_batch(bignum256modm out, const bignum256modm a, const bignum256modm b, size_t limbsize) {
|
|
size_t i = 0;
|
|
bignum256modm_element_t carry = 0;
|
|
switch (limbsize) {
|
|
case 4: out[i] = (a[i] - b[i]) ; carry = (out[i] >> 63); out[i] &= 0xffffffffffffff; i++;
|
|
case 3: out[i] = (a[i] - b[i]) - carry; carry = (out[i] >> 63); out[i] &= 0xffffffffffffff; i++;
|
|
case 2: out[i] = (a[i] - b[i]) - carry; carry = (out[i] >> 63); out[i] &= 0xffffffffffffff; i++;
|
|
case 1: out[i] = (a[i] - b[i]) - carry; carry = (out[i] >> 63); out[i] &= 0xffffffffffffff; i++;
|
|
case 0:
|
|
default: out[i] = (a[i] - b[i]) - carry;
|
|
}
|
|
}
|
|
|
|
|
|
/* is a < b */
|
|
static int
|
|
lt256_modm_batch(const bignum256modm a, const bignum256modm b, size_t limbsize) {
|
|
size_t i = 0;
|
|
bignum256modm_element_t t, carry = 0;
|
|
switch (limbsize) {
|
|
case 4: t = (a[i] - b[i]) ; carry = (t >> 63); i++;
|
|
case 3: t = (a[i] - b[i]) - carry; carry = (t >> 63); i++;
|
|
case 2: t = (a[i] - b[i]) - carry; carry = (t >> 63); i++;
|
|
case 1: t = (a[i] - b[i]) - carry; carry = (t >> 63); i++;
|
|
case 0: t = (a[i] - b[i]) - carry; carry = (t >> 63);
|
|
}
|
|
return (int)carry;
|
|
}
|
|
|
|
/* is a <= b */
|
|
static int
|
|
lte256_modm_batch(const bignum256modm a, const bignum256modm b, size_t limbsize) {
|
|
size_t i = 0;
|
|
bignum256modm_element_t t, carry = 0;
|
|
switch (limbsize) {
|
|
case 4: t = (b[i] - a[i]) ; carry = (t >> 63); i++;
|
|
case 3: t = (b[i] - a[i]) - carry; carry = (t >> 63); i++;
|
|
case 2: t = (b[i] - a[i]) - carry; carry = (t >> 63); i++;
|
|
case 1: t = (b[i] - a[i]) - carry; carry = (t >> 63); i++;
|
|
case 0: t = (b[i] - a[i]) - carry; carry = (t >> 63);
|
|
}
|
|
return (int)!carry;
|
|
}
|
|
|
|
/* is a == 0 */
|
|
static int
|
|
iszero256_modm_batch(const bignum256modm a) {
|
|
size_t i;
|
|
for (i = 0; i < 5; i++)
|
|
if (a[i])
|
|
return 0;
|
|
return 1;
|
|
}
|
|
|
|
/* is a == 1 */
|
|
static int
|
|
isone256_modm_batch(const bignum256modm a) {
|
|
size_t i;
|
|
for (i = 0; i < 5; i++)
|
|
if (a[i] != ((i) ? 0 : 1))
|
|
return 0;
|
|
return 1;
|
|
}
|
|
|
|
/* can a fit in to (at most) 128 bits */
|
|
static int
|
|
isatmost128bits256_modm_batch(const bignum256modm a) {
|
|
uint64_t mask =
|
|
((a[4] ) | /* 32 */
|
|
(a[3] ) | /* 88 */
|
|
(a[2] & 0xffffffffff0000));
|
|
|
|
return (mask == 0);
|
|
}
|