/* * Copyright (c) 2018 Thomas Pornin * * 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 #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; 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); /* * 2^256 = 2^224 - 2^192 - 2^96 + 1 in the field. */ 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 */ d[3] += (uint64_t)(w >> 64) + (t << 32) - t; #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]); (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; /* * 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 */ 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 carry, then we need to add p. */ t = cc; cc = _addcarry_u64(0, d[0], -t, &d[0]); cc = _addcarry_u64(cc, d[1], (-t) >> 32, &d[1]); cc = _addcarry_u64(cc, d[2], 0, &d[2]); (void)_addcarry_u64(cc, d[3], t - (t << 32), &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; memcpy(t, a, sizeof t); for (i = 0; i < 30; i ++) { f256_montysquare(t, t); f256_montymul(t, t, a); } 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; } } 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). */ memcpy(P->x, x, sizeof x); memcpy(P->y, y, sizeof y); 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). */ memcpy(t1, P1->x, sizeof t1); 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). */ memcpy(t1, P1->x, sizeof t1); 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; 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. */ 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) { memcpy(zt, jac[i].z, sizeof zt); memcpy(jac[i].z, jac[i + 1].z, sizeof zt); memcpy(jac[i + 1].z, zt, sizeof zt); f256_montymul(z[i >> 1], jac[i].z, jac[i + 1].z); } if ((num & 1) != 0) { memcpy(z[num >> 1], jac[num - 1].z, sizeof zt); 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) { 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