/* * 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 five 64-bit integers, in basis 2^52. * Limbs may occasionally exceed 2^52. * * A _partially reduced_ value is such that the following hold: * - top limb is less than 2^48 + 2^30 * - the other limbs fit on 53 bits each * In particular, such a value is less than twice the modulus p. */ #define BIT(n) ((uint64_t)1 << (n)) #define MASK48 (BIT(48) - BIT(0)) #define MASK52 (BIT(52) - BIT(0)) /* R = 2^260 mod p */ static const uint64_t F256_R[] = { 0x0000000000010, 0xF000000000000, 0xFFFFFFFFFFFFF, 0xFFEFFFFFFFFFF, 0x00000000FFFFF }; /* 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[] = { 0xDF6229C4BDDFD, 0xCA8843090D89C, 0x212ED6ACF005C, 0x83415A220ABF7, 0x0C30061DD4874 }; /* * Addition in the field. Carry propagation is not performed. * On input, limbs may be up to 63 bits each; on output, they will * be up to one bit more than on input. */ static inline void f256_add(uint64_t *d, const uint64_t *a, const uint64_t *b) { d[0] = a[0] + b[0]; d[1] = a[1] + b[1]; d[2] = a[2] + b[2]; d[3] = a[3] + b[3]; d[4] = a[4] + b[4]; } /* * Partially reduce the provided value. * Input: limbs can go up to 61 bits each. * Output: partially reduced. */ static inline void f256_partial_reduce(uint64_t *a) { uint64_t w, cc, s; /* * Propagate carries. */ w = a[0]; a[0] = w & MASK52; cc = w >> 52; w = a[1] + cc; a[1] = w & MASK52; cc = w >> 52; w = a[2] + cc; a[2] = w & MASK52; cc = w >> 52; w = a[3] + cc; a[3] = w & MASK52; cc = w >> 52; a[4] += cc; s = a[4] >> 48; /* s < 2^14 */ a[0] += s; /* a[0] < 2^52 + 2^14 */ w = a[1] - (s << 44); a[1] = w & MASK52; /* a[1] < 2^52 */ cc = -(w >> 52) & 0xFFF; /* cc < 16 */ w = a[2] - cc; a[2] = w & MASK52; /* a[2] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = a[3] - cc - (s << 36); a[3] = w & MASK52; /* a[3] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = a[4] & MASK48; a[4] = w + (s << 16) - cc; /* a[4] < 2^48 + 2^30 */ } /* * Subtraction in the field. * Input: limbs must fit on 60 bits each; in particular, the complete * integer will be less than 2^268 + 2^217. * Output: partially reduced. */ static inline void f256_sub(uint64_t *d, const uint64_t *a, const uint64_t *b) { uint64_t t[5], w, s, cc; /* * We compute d = 2^13*p + a - b; this ensures a positive * intermediate value. * * Each individual addition/subtraction may yield a positive or * negative result; thus, we need to handle a signed carry, thus * with sign extension. We prefer not to use signed types (int64_t) * because conversion from unsigned to signed is cumbersome (a * direct cast with the top bit set is undefined behavior; instead, * we have to use pointer aliasing, using the guaranteed properties * of exact-width types, but this requires the compiler to optimize * away the writes and reads from RAM), and right-shifting a * signed negative value is implementation-defined. Therefore, * we use a custom sign extension. */ w = a[0] - b[0] - BIT(13); t[0] = w & MASK52; cc = w >> 52; cc |= -(cc & BIT(11)); w = a[1] - b[1] + cc; t[1] = w & MASK52; cc = w >> 52; cc |= -(cc & BIT(11)); w = a[2] - b[2] + cc; t[2] = (w & MASK52) + BIT(5); cc = w >> 52; cc |= -(cc & BIT(11)); w = a[3] - b[3] + cc; t[3] = (w & MASK52) + BIT(49); cc = w >> 52; cc |= -(cc & BIT(11)); t[4] = (BIT(61) - BIT(29)) + a[4] - b[4] + cc; /* * Perform partial reduction. Rule is: * 2^256 = 2^224 - 2^192 - 2^96 + 1 mod p * * At that point: * 0 <= t[0] <= 2^52 - 1 * 0 <= t[1] <= 2^52 - 1 * 2^5 <= t[2] <= 2^52 + 2^5 - 1 * 2^49 <= t[3] <= 2^52 + 2^49 - 1 * 2^59 < t[4] <= 2^61 + 2^60 - 2^29 * * Thus, the value 's' (t[4] / 2^48) will be necessarily * greater than 2048, and less than 12288. */ s = t[4] >> 48; d[0] = t[0] + s; /* d[0] <= 2^52 + 12287 */ w = t[1] - (s << 44); d[1] = w & MASK52; /* d[1] <= 2^52 - 1 */ cc = -(w >> 52) & 0xFFF; /* cc <= 48 */ w = t[2] - cc; cc = w >> 63; /* cc = 0 or 1 */ d[2] = w + (cc << 52); /* d[2] <= 2^52 + 31 */ w = t[3] - cc - (s << 36); cc = w >> 63; /* cc = 0 or 1 */ d[3] = w + (cc << 52); /* t[3] <= 2^52 + 2^49 - 1 */ d[4] = (t[4] & MASK48) + (s << 16) - cc; /* d[4] < 2^48 + 2^30 */ /* * If s = 0, then none of the limbs is modified, and there cannot * be an overflow; if s != 0, then (s << 16) > cc, and there is * no overflow either. */ } /* * Montgomery multiplication in the field. * Input: limbs must fit on 56 bits each. * Output: partially reduced. */ static void f256_montymul(uint64_t *d, const uint64_t *a, const uint64_t *b) { #if BR_INT128 int i; uint64_t t[5]; t[0] = 0; t[1] = 0; t[2] = 0; t[3] = 0; t[4] = 0; for (i = 0; i < 5; i ++) { uint64_t x, f, cc, w, s; unsigned __int128 z; /* * Since limbs of a[] and b[] fit on 56 bits each, * each individual product fits on 112 bits. Also, * the factor f fits on 52 bits, so f<<48 fits on * 112 bits too. This guarantees that carries (cc) * will fit on 62 bits, thus no overflow. * * The operations below compute: * t <- (t + x*b + f*p) / 2^64 */ x = a[i]; z = (unsigned __int128)b[0] * (unsigned __int128)x + (unsigned __int128)t[0]; f = (uint64_t)z & MASK52; cc = (uint64_t)(z >> 52); z = (unsigned __int128)b[1] * (unsigned __int128)x + (unsigned __int128)t[1] + cc + ((unsigned __int128)f << 44); t[0] = (uint64_t)z & MASK52; cc = (uint64_t)(z >> 52); z = (unsigned __int128)b[2] * (unsigned __int128)x + (unsigned __int128)t[2] + cc; t[1] = (uint64_t)z & MASK52; cc = (uint64_t)(z >> 52); z = (unsigned __int128)b[3] * (unsigned __int128)x + (unsigned __int128)t[3] + cc + ((unsigned __int128)f << 36); t[2] = (uint64_t)z & MASK52; cc = (uint64_t)(z >> 52); z = (unsigned __int128)b[4] * (unsigned __int128)x + (unsigned __int128)t[4] + cc + ((unsigned __int128)f << 48) - ((unsigned __int128)f << 16); t[3] = (uint64_t)z & MASK52; t[4] = (uint64_t)(z >> 52); /* * t[4] may be up to 62 bits here; we need to do a * partial reduction. Note that limbs t[0] to t[3] * fit on 52 bits each. */ s = t[4] >> 48; /* s < 2^14 */ t[0] += s; /* t[0] < 2^52 + 2^14 */ w = t[1] - (s << 44); t[1] = w & MASK52; /* t[1] < 2^52 */ cc = -(w >> 52) & 0xFFF; /* cc < 16 */ w = t[2] - cc; t[2] = w & MASK52; /* t[2] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = t[3] - cc - (s << 36); t[3] = w & MASK52; /* t[3] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = t[4] & MASK48; t[4] = w + (s << 16) - cc; /* t[4] < 2^48 + 2^30 */ /* * The final t[4] cannot overflow because cc is 0 or 1, * and cc can be 1 only if s != 0. */ } d[0] = t[0]; d[1] = t[1]; d[2] = t[2]; d[3] = t[3]; d[4] = t[4]; #elif BR_UMUL128 int i; uint64_t t[5]; t[0] = 0; t[1] = 0; t[2] = 0; t[3] = 0; t[4] = 0; for (i = 0; i < 5; i ++) { uint64_t x, f, cc, w, s, zh, zl; unsigned char k; /* * Since limbs of a[] and b[] fit on 56 bits each, * each individual product fits on 112 bits. Also, * the factor f fits on 52 bits, so f<<48 fits on * 112 bits too. This guarantees that carries (cc) * will fit on 62 bits, thus no overflow. * * The operations below compute: * t <- (t + x*b + f*p) / 2^64 */ x = a[i]; zl = _umul128(b[0], x, &zh); k = _addcarry_u64(0, t[0], zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); f = zl & MASK52; cc = (zl >> 52) | (zh << 12); zl = _umul128(b[1], x, &zh); k = _addcarry_u64(0, t[1], zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, cc, zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, f << 44, zl, &zl); (void)_addcarry_u64(k, f >> 20, zh, &zh); t[0] = zl & MASK52; cc = (zl >> 52) | (zh << 12); zl = _umul128(b[2], x, &zh); k = _addcarry_u64(0, t[2], zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, cc, zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); t[1] = zl & MASK52; cc = (zl >> 52) | (zh << 12); zl = _umul128(b[3], x, &zh); k = _addcarry_u64(0, t[3], zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, cc, zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, f << 36, zl, &zl); (void)_addcarry_u64(k, f >> 28, zh, &zh); t[2] = zl & MASK52; cc = (zl >> 52) | (zh << 12); zl = _umul128(b[4], x, &zh); k = _addcarry_u64(0, t[4], zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, cc, zl, &zl); (void)_addcarry_u64(k, 0, zh, &zh); k = _addcarry_u64(0, f << 48, zl, &zl); (void)_addcarry_u64(k, f >> 16, zh, &zh); k = _subborrow_u64(0, zl, f << 16, &zl); (void)_subborrow_u64(k, zh, f >> 48, &zh); t[3] = zl & MASK52; t[4] = (zl >> 52) | (zh << 12); /* * t[4] may be up to 62 bits here; we need to do a * partial reduction. Note that limbs t[0] to t[3] * fit on 52 bits each. */ s = t[4] >> 48; /* s < 2^14 */ t[0] += s; /* t[0] < 2^52 + 2^14 */ w = t[1] - (s << 44); t[1] = w & MASK52; /* t[1] < 2^52 */ cc = -(w >> 52) & 0xFFF; /* cc < 16 */ w = t[2] - cc; t[2] = w & MASK52; /* t[2] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = t[3] - cc - (s << 36); t[3] = w & MASK52; /* t[3] < 2^52 */ cc = w >> 63; /* cc = 0 or 1 */ w = t[4] & MASK48; t[4] = w + (s << 16) - cc; /* t[4] < 2^48 + 2^30 */ /* * The final t[4] cannot overflow because cc is 0 or 1, * and cc can be 1 only if s != 0. */ } d[0] = t[0]; d[1] = t[1]; d[2] = t[2]; d[3] = t[3]; d[4] = t[4]; #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 = 2^520 mod p. * If R = 2^260 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[] = { 0x0000000000300, 0xFFFFFFFF00000, 0xFFFFEFFFFFFFB, 0xFDFFFFFFFFFFF, 0x0000004FFFFFF }; 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 2^260 modulo p. */ static const uint64_t one[] = { 1, 0, 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[5], t[5]; 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 should be partially reduced. * On output, limbs a[0] to a[3] fit on 52 bits each, limb a[4] fits * on 48 bits, and the integer is less than p. */ static inline void f256_final_reduce(uint64_t *a) { uint64_t r[5], t[5], w, cc; int i; /* * Propagate carries to ensure that limbs 0 to 3 fit on 52 bits. */ cc = 0; for (i = 0; i < 5; i ++) { w = a[i] + cc; r[i] = w & MASK52; cc = w >> 52; } /* * We compute t = r + (2^256 - p) = r + 2^224 - 2^192 - 2^96 + 1. * If t < 2^256, then r < p, and we return r. Otherwise, we * want to return r - p = t - 2^256. */ /* * Add 2^224 + 1, and propagate carries to ensure that limbs * t[0] to t[3] fit in 52 bits each. */ w = r[0] + 1; t[0] = w & MASK52; cc = w >> 52; w = r[1] + cc; t[1] = w & MASK52; cc = w >> 52; w = r[2] + cc; t[2] = w & MASK52; cc = w >> 52; w = r[3] + cc; t[3] = w & MASK52; cc = w >> 52; t[4] = r[4] + cc + BIT(16); /* * Subtract 2^192 + 2^96. Since we just added 2^224 + 1, the * result cannot be negative. */ w = t[1] - BIT(44); t[1] = w & MASK52; cc = w >> 63; w = t[2] - cc; t[2] = w & MASK52; cc = w >> 63; w = t[3] - BIT(36) - cc; t[3] = w & MASK52; cc = w >> 63; t[4] -= cc; /* * If the top limb t[4] fits on 48 bits, then r[] is already * in the proper range. Otherwise, t[] is the value to return * (truncated to 256 bits). */ cc = -(t[4] >> 48); t[4] &= MASK48; for (i = 0; i < 5; i ++) { a[i] = r[i] ^ (cc & (r[i] ^ t[i])); } } /* * 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[5]; uint64_t y[5]; } p256_affine; typedef struct { uint64_t x[5]; uint64_t y[5]; uint64_t z[5]; } p256_jacobian; /* * Decode a field element (unsigned big endian notation). */ static void f256_decode(uint64_t *a, const unsigned char *buf) { uint64_t w0, w1, w2, w3; w3 = br_dec64be(buf + 0); w2 = br_dec64be(buf + 8); w1 = br_dec64be(buf + 16); w0 = br_dec64be(buf + 24); a[0] = w0 & MASK52; a[1] = ((w0 >> 52) | (w1 << 12)) & MASK52; a[2] = ((w1 >> 40) | (w2 << 24)) & MASK52; a[3] = ((w2 >> 28) | (w3 << 36)) & MASK52; a[4] = w3 >> 16; } /* * Encode a field element (unsigned big endian notation). The field * element MUST be fully reduced. */ static void f256_encode(unsigned char *buf, const uint64_t *a) { uint64_t w0, w1, w2, w3; w0 = a[0] | (a[1] << 52); w1 = (a[1] >> 12) | (a[2] << 40); w2 = (a[2] >> 24) | (a[3] << 28); w3 = (a[3] >> 36) | (a[4] << 16); br_enc64be(buf + 0, w3); br_enc64be(buf + 8, w2); br_enc64be(buf + 16, w1); br_enc64be(buf + 24, w0); } /* * 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[5], y[5], t[5], x3[5], 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. */ f256_decode(x, buf + 1); f256_decode(y, buf + 33); 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] | t[4]; 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[5], t2[5], 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; f256_encode(buf + 1, t1); f256_encode(buf + 33, t2); /* Return success if and only if P->z != 0. */ z = P->z[0] | P->z[1] | P->z[2] | P->z[3] | P->z[4]; 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. */ 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[5], t2[5], t3[5], t4[5]; /* * 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); f256_partial_reduce(P->z); /* * 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[5], t2[5], t3[5], t4[5], t5[5], t6[5], t7[5], 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] | t4[4]; 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[5], t2[5], t3[5], t4[5], t5[5], t6[5], t7[5], 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] | t4[4]; 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[5], t2[5], t3[5], t4[5], t5[5], t6[5], t7[5], 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] | P1->z[4]; 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] | t2[4] | t4[0] | t4[1] | t4[2] | t4[3] | t4[4]; 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); f256_partial_reduce(t1); /* * 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 < 5; 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 < 5; 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.x[4] |= m & W[n].x[4]; 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]; T.y[4] |= m & W[n].y[4]; } 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 < 5; j ++) { Q.x[j] ^= m & (Q.x[j] ^ T.x[j]); Q.y[j] ^= m & (Q.y[j] ^ T.y[j]); Q.z[j] ^= m & (Q.z[j] ^ 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 invert 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][5]; 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[] = { { { 0x30D418A9143C1, 0xC4FEDB60179E7, 0x62251075BA95F, 0x5C669FB732B77, 0x08905F76B5375 }, { 0x5357CE95560A8, 0x43A19E45CDDF2, 0x21F3258B4AB8E, 0xD8552E88688DD, 0x0571FF18A5885 } }, { { 0x46D410DDD64DF, 0x0B433827D8500, 0x1490D9AA6AE3C, 0xA3A832205038D, 0x06BB32E52DCF3 }, { 0x48D361BEE1A57, 0xB7B236FF82F36, 0x042DBE152CD7C, 0xA3AA9A8FB0E92, 0x08C577517A5B8 } }, { { 0x3F904EEBC1272, 0x9E87D81FBFFAC, 0xCBBC98B027F84, 0x47E46AD77DD87, 0x06936A3FD6FF7 }, { 0x5C1FC983A7EBD, 0xC3861FE1AB04C, 0x2EE98E583E47A, 0xC06A88208311A, 0x05F06A2AB587C } }, { { 0xB50D46918DCC5, 0xD7623C17374B0, 0x100AF24650A6E, 0x76ABCDAACACE8, 0x077362F591B01 }, { 0xF24CE4CBABA68, 0x17AD6F4472D96, 0xDDD22E1762847, 0x862EB6C36DEE5, 0x04B14C39CC5AB } }, { { 0x8AAEC45C61F5C, 0x9D4B9537DBE1B, 0x76C20C90EC649, 0x3C7D41CB5AAD0, 0x0907960649052 }, { 0x9B4AE7BA4F107, 0xF75EB882BEB30, 0x7A1F6873C568E, 0x915C540A9877E, 0x03A076BB9DD1E } }, { { 0x47373E77664A1, 0xF246CEE3E4039, 0x17A3AD55AE744, 0x673C50A961A5B, 0x03074B5964213 }, { 0x6220D377E44BA, 0x30DFF14B593D3, 0x639F11299C2B5, 0x75F5424D44CEF, 0x04C9916DEA07F } }, { { 0x354EA0173B4F1, 0x3C23C00F70746, 0x23BB082BD2021, 0xE03E43EAAB50C, 0x03BA5119D3123 }, { 0xD0303F5B9D4DE, 0x17DA67BDD2847, 0xC941956742F2F, 0x8670F933BDC77, 0x0AEDD9164E240 } }, { { 0x4CD19499A78FB, 0x4BF9B345527F1, 0x2CFC6B462AB5C, 0x30CDF90F02AF0, 0x0763891F62652 }, { 0xA3A9532D49775, 0xD7F9EBA15F59D, 0x60BBF021E3327, 0xF75C23C7B84BE, 0x06EC12F2C706D } }, { { 0x6E8F264E20E8E, 0xC79A7A84175C9, 0xC8EB00ABE6BFE, 0x16A4CC09C0444, 0x005B3081D0C4E }, { 0x777AA45F33140, 0xDCE5D45E31EB7, 0xB12F1A56AF7BE, 0xF9B2B6E019A88, 0x086659CDFD835 } }, { { 0xDBD19DC21EC8C, 0x94FCF81392C18, 0x250B4998F9868, 0x28EB37D2CD648, 0x0C61C947E4B34 }, { 0x407880DD9E767, 0x0C83FBE080C2B, 0x9BE5D2C43A899, 0xAB4EF7D2D6577, 0x08719A555B3B4 } }, { { 0x260A6245E4043, 0x53E7FDFE0EA7D, 0xAC1AB59DE4079, 0x072EFF3A4158D, 0x0E7090F1949C9 }, { 0x85612B944E886, 0xE857F61C81A76, 0xAD643D250F939, 0x88DAC0DAA891E, 0x089300244125B } }, { { 0x1AA7D26977684, 0x58A345A3304B7, 0x37385EABDEDEF, 0x155E409D29DEE, 0x0EE1DF780B83E }, { 0x12D91CBB5B437, 0x65A8956370CAC, 0xDE6D66170ED2F, 0xAC9B8228CFA8A, 0x0FF57C95C3238 } }, { { 0x25634B2ED7097, 0x9156FD30DCCC4, 0x9E98110E35676, 0x7594CBCD43F55, 0x038477ACC395B }, { 0x2B90C00EE17FF, 0xF842ED2E33575, 0x1F5BC16874838, 0x7968CD06422BD, 0x0BC0876AB9E7B } }, { { 0xA35BB0CF664AF, 0x68F9707E3A242, 0x832660126E48F, 0x72D2717BF54C6, 0x0AAE7333ED12C }, { 0x2DB7995D586B1, 0xE732237C227B5, 0x65E7DBBE29569, 0xBBBD8E4193E2A, 0x052706DC3EAA1 } }, { { 0xD8B7BC60055BE, 0xD76E27E4B72BC, 0x81937003CC23E, 0xA090E337424E4, 0x02AA0E43EAD3D }, { 0x524F6383C45D2, 0x422A41B2540B8, 0x8A4797D766355, 0xDF444EFA6DE77, 0x0042170A9079A } }, }; /* * 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] | P.z[4]; 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_m62 = { (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_m62_get(void) { return &br_ec_p256_m62; } #else /* see bearssl_ec.h */ const br_ec_impl * br_ec_p256_m62_get(void) { return 0; } #endif