/* 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); }