/* fe_448.c * * Copyright (C) 2006-2020 wolfSSL Inc. * * This file is part of wolfSSL. * * wolfSSL is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * wolfSSL is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1335, USA */ /* Based On Daniel J Bernstein's curve25519 Public Domain ref10 work. * Small implementation based on Daniel Beer's curve25519 public domain work. * Reworked for curve448 by Sean Parkinson. */ #ifdef HAVE_CONFIG_H #include #endif #include #if defined(HAVE_CURVE448) || defined(HAVE_ED448) #include #ifdef NO_INLINE #include #else #define WOLFSSL_MISC_INCLUDED #include #endif #if defined(CURVE448_SMALL) || defined(ED448_SMALL) /* Initialize the field element operations. */ void fe448_init(void) { } /* Normalize the field element. * Ensure result is in range: 0..2^448-2^224-2 * * a [in] Field element in range 0..2^448-1. */ void fe448_norm(uint8_t* a) { int i; int16_t c = 0; int16_t o = 0; for (i = 0; i < 56; i++) { c += a[i]; if ((i == 0) || (i == 28)) c += 1; c >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) o += c; o += a[i]; a[i] = (uint8_t)o; o >>= 8; } } /* Copy one field element into another: d = a. * * d [in] Destination field element. * a [in] Source field element. */ void fe448_copy(uint8_t* d, const uint8_t* a) { int i; for (i = 0; i < 56; i++) { d[i] = a[i]; } } /* Conditionally swap the elements. * Constant time implementation. * * a [in] First field element. * b [in] Second field element. * c [in] Swap when 1. Valid values: 0, 1. */ static void fe448_cswap(uint8_t* a, uint8_t* b, int c) { int i; uint8_t mask = -(uint8_t)c; uint8_t t[56]; for (i = 0; i < 56; i++) t[i] = (a[i] ^ b[i]) & mask; for (i = 0; i < 56; i++) a[i] ^= t[i]; for (i = 0; i < 56; i++) b[i] ^= t[i]; } /* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold sum. * a [in] Field element to add. * b [in] Field element to add. */ void fe448_add(uint8_t* r, const uint8_t* a, const uint8_t* b) { int i; int16_t c = 0; int16_t o = 0; for (i = 0; i < 56; i++) { c += a[i]; c += b[i]; r[i] = (uint8_t)c; c >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) o += c; o += r[i]; r[i] = (uint8_t)o; o >>= 8; } } /* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold difference. * a [in] Field element to subtract from. * b [in] Field element to subtract. */ void fe448_sub(uint8_t* r, const uint8_t* a, const uint8_t* b) { int i; int16_t c = 0; int16_t o = 0; for (i = 0; i < 56; i++) { if (i == 28) c += 0x1fc; else c += 0x1fe; c += a[i]; c -= b[i]; r[i] = (uint8_t)c; c >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) o += c; o += r[i]; r[i] = (uint8_t)o; o >>= 8; } } /* Mulitply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. */ void fe448_mul39081(uint8_t* r, const uint8_t* a) { int i; int32_t c = 0; int32_t o = 0; for (i = 0; i < 56; i++) { c += a[i] * (int32_t)39081; r[i] = (uint8_t)c; c >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) o += c; o += r[i]; r[i] = (uint8_t)o; o >>= 8; } } /* Mulitply two field elements. r = (a * b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. * b [in] Field element to multiply. */ void fe448_mul(uint8_t* r, const uint8_t* a, const uint8_t* b) { int i, k; int32_t c = 0; int16_t o = 0, cc = 0; uint8_t t[112]; for (k = 0; k < 56; k++) { i = 0; for (; i <= k; i++) { c += (int32_t)a[i] * b[k - i]; } t[k] = (uint8_t)c; c >>= 8; } for (; k < 111; k++) { i = k - 55; for (; i < 56; i++) { c += (int32_t)a[i] * b[k - i]; } t[k] = (uint8_t)c; c >>= 8; } t[k] = (uint8_t)c; for (i = 0; i < 28; i++) { o += t[i]; o += t[i + 56]; o += t[i + 84]; r[i] = (uint8_t)o; o >>= 8; } for (i = 28; i < 56; i++) { o += t[i]; o += t[i + 56]; o += t[i + 28]; o += t[i + 56]; r[i] = (uint8_t)o; o >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) cc += o; cc += r[i]; r[i] = (uint8_t)cc; cc >>= 8; } } /* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to square. */ void fe448_sqr(uint8_t* r, const uint8_t* a) { int i, k; int32_t c = 0; int32_t p; int16_t o = 0, cc = 0; uint8_t t[112]; for (k = 0; k < 56; k++) { i = 0; for (; i <= k; i++) { if (k - i < i) break; p = (int32_t)a[i] * a[k - i]; if (k - i != i) p *= 2; c += p; } t[k] = (uint8_t)c; c >>= 8; } for (; k < 111; k++) { i = k - 55; for (; i < 56; i++) { if (k - i < i) break; p = (int32_t)a[i] * a[k - i]; if (k - i != i) p *= 2; c += p; } t[k] = (uint8_t)c; c >>= 8; } t[k] = (uint8_t)c; for (i = 0; i < 28; i++) { o += t[i]; o += t[i + 56]; o += t[i + 84]; r[i] = (uint8_t)o; o >>= 8; } for (i = 28; i < 56; i++) { o += t[i]; o += t[i + 56]; o += t[i + 28]; o += t[i + 56]; r[i] = (uint8_t)o; o >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) cc += o; cc += r[i]; r[i] = (uint8_t)cc; cc >>= 8; } fe448_norm(r); } /* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1 * Constant time implementation - using Fermat's little theorem: * a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a * For curve448: p - 2 = 2^448 - 2^224 - 3 * * r [in] Field element to hold result. * a [in] Field element to invert. */ void fe448_invert(uint8_t* r, const uint8_t* a) { int i; uint8_t t[56]; fe448_sqr(t, a); fe448_mul(t, t, a); for (i = 0; i < 221; i++) { fe448_sqr(t, t); fe448_mul(t, t, a); } fe448_sqr(t, t); for (i = 0; i < 222; i++) { fe448_sqr(t, t); fe448_mul(t, t, a); } fe448_sqr(t, t); fe448_sqr(t, t); fe448_mul(r, t, a); } /* Scalar multiply the point by a number. r = n.a * Uses Montogmery ladder and only requires the x-ordinate. * * r [in] Field element to hold result. * n [in] Scalar as an array of bytes. * a [in] Point to multiply - x-ordinate only. */ int curve448(byte* r, const byte* n, const byte* a) { uint8_t x1[56]; uint8_t x2[56] = {1}; uint8_t z2[56] = {0}; uint8_t x3[56]; uint8_t z3[56] = {1}; uint8_t t0[56]; uint8_t t1[56]; int i; unsigned int swap; unsigned int b; fe448_copy(x1, a); fe448_copy(x3, a); swap = 0; for (i = 447; i >= 0; --i) { b = (n[i >> 3] >> (i & 7)) & 1; swap ^= b; fe448_cswap(x2, x3, swap); fe448_cswap(z2, z3, swap); swap = b; /* Montgomery Ladder - double and add */ fe448_add(t0, x2, z2); fe448_add(t1, x3, z3); fe448_sub(x2, x2, z2); fe448_sub(x3, x3, z3); fe448_mul(t1, t1, x2); fe448_mul(z3, x3, t0); fe448_sqr(t0, t0); fe448_sqr(x2, x2); fe448_add(x3, z3, t1); fe448_sqr(x3, x3); fe448_sub(z3, z3, t1); fe448_sqr(z3, z3); fe448_mul(z3, z3, x1); fe448_sub(t1, t0, x2); fe448_mul(x2, t0, x2); fe448_mul39081(z2, t1); fe448_add(z2, t0, z2); fe448_mul(z2, z2, t1); } fe448_cswap(x2, x3, swap); fe448_cswap(z2, z3, swap); fe448_invert(z2, z2); fe448_mul(r, x2, z2); fe448_norm(r); return 0; } #ifdef HAVE_ED448 /* Check whether field element is not 0. * Field element must have been normalized before call. * * a [in] Field element. * returns 0 when zero, and any other value otherwise. */ int fe448_isnonzero(const uint8_t* a) { int i; uint8_t c = 0; for (i = 0; i < 56; i++) c |= a[i]; return c; } /* Negates the field element. r = -a mod (2^448 - 2^224 - 1) * Add 0x200 to each element and subtract 2 from next. * Top element overflow handled by subtracting 2 from index 0 and 28. * * r [in] Field element to hold result. * a [in] Field element. */ void fe448_neg(uint8_t* r, const uint8_t* a) { int i; int16_t c = 0; int16_t o = 0; for (i = 0; i < 56; i++) { if (i == 28) c += 0x1fc; else c += 0x1fe; c -= a[i]; r[i] = (uint8_t)c; c >>= 8; } for (i = 0; i < 56; i++) { if ((i == 0) || (i == 28)) o += c; o += r[i]; r[i] = (uint8_t)o; o >>= 8; } } /* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1 * Used for calcualting y-ordinate from x-ordinate for Ed448. * * r [in] Field element to hold result. * a [in] Field element to exponentiate. */ void fe448_pow_2_446_222_1(uint8_t* r, const uint8_t* a) { int i; uint8_t t[56]; fe448_sqr(t, a); fe448_mul(t, t, a); for (i = 0; i < 221; i++) { fe448_sqr(t, t); fe448_mul(t, t, a); } fe448_sqr(t, t); for (i = 0; i < 221; i++) { fe448_sqr(t, t); fe448_mul(t, t, a); } fe448_sqr(t, t); fe448_mul(r, t, a); } /* Constant time, conditional move of b into a. * a is not changed if the condition is 0. * * a A field element. * b A field element. * c If 1 then copy and if 0 then don't copy. */ void fe448_cmov(uint8_t* a, const uint8_t* b, int c) { int i; uint8_t m = -(uint8_t)c; uint8_t t[56]; for (i = 0; i < 56; i++) t[i] = m & (a[i] ^ b[i]); for (i = 0; i < 56; i++) a[i] ^= t[i]; } #endif /* HAVE_ED448 */ #elif defined(CURVED448_128BIT) /* Initialize the field element operations. */ void fe448_init(void) { } /* Convert the field element from a byte array to an array of 56-bits. * * r [in] Array to encode into. * b [in] Byte array. */ void fe448_from_bytes(int64_t* r, const unsigned char* b) { r[ 0] = ((int64_t) (b[ 0]) << 0) | ((int64_t) (b[ 1]) << 8) | ((int64_t) (b[ 2]) << 16) | ((int64_t) (b[ 3]) << 24) | ((int64_t) (b[ 4]) << 32) | ((int64_t) (b[ 5]) << 40) | ((int64_t) (b[ 6]) << 48); r[ 1] = ((int64_t) (b[ 7]) << 0) | ((int64_t) (b[ 8]) << 8) | ((int64_t) (b[ 9]) << 16) | ((int64_t) (b[10]) << 24) | ((int64_t) (b[11]) << 32) | ((int64_t) (b[12]) << 40) | ((int64_t) (b[13]) << 48); r[ 2] = ((int64_t) (b[14]) << 0) | ((int64_t) (b[15]) << 8) | ((int64_t) (b[16]) << 16) | ((int64_t) (b[17]) << 24) | ((int64_t) (b[18]) << 32) | ((int64_t) (b[19]) << 40) | ((int64_t) (b[20]) << 48); r[ 3] = ((int64_t) (b[21]) << 0) | ((int64_t) (b[22]) << 8) | ((int64_t) (b[23]) << 16) | ((int64_t) (b[24]) << 24) | ((int64_t) (b[25]) << 32) | ((int64_t) (b[26]) << 40) | ((int64_t) (b[27]) << 48); r[ 4] = ((int64_t) (b[28]) << 0) | ((int64_t) (b[29]) << 8) | ((int64_t) (b[30]) << 16) | ((int64_t) (b[31]) << 24) | ((int64_t) (b[32]) << 32) | ((int64_t) (b[33]) << 40) | ((int64_t) (b[34]) << 48); r[ 5] = ((int64_t) (b[35]) << 0) | ((int64_t) (b[36]) << 8) | ((int64_t) (b[37]) << 16) | ((int64_t) (b[38]) << 24) | ((int64_t) (b[39]) << 32) | ((int64_t) (b[40]) << 40) | ((int64_t) (b[41]) << 48); r[ 6] = ((int64_t) (b[42]) << 0) | ((int64_t) (b[43]) << 8) | ((int64_t) (b[44]) << 16) | ((int64_t) (b[45]) << 24) | ((int64_t) (b[46]) << 32) | ((int64_t) (b[47]) << 40) | ((int64_t) (b[48]) << 48); r[ 7] = ((int64_t) (b[49]) << 0) | ((int64_t) (b[50]) << 8) | ((int64_t) (b[51]) << 16) | ((int64_t) (b[52]) << 24) | ((int64_t) (b[53]) << 32) | ((int64_t) (b[54]) << 40) | ((int64_t) (b[55]) << 48); } /* Convert the field element to a byte array from an array of 56-bits. * * b [in] Byte array. * a [in] Array to encode into. */ void fe448_to_bytes(unsigned char* b, const int64_t* a) { int128_t t; /* Mod */ int64_t in0 = a[0]; int64_t in1 = a[1]; int64_t in2 = a[2]; int64_t in3 = a[3]; int64_t in4 = a[4]; int64_t in5 = a[5]; int64_t in6 = a[6]; int64_t in7 = a[7]; int64_t o = in7 >> 56; in7 -= o << 56; in0 += o; in4 += o; o = (in0 + 1) >> 56; o = (o + in1) >> 56; o = (o + in2) >> 56; o = (o + in3) >> 56; o = (o + in4 + 1) >> 56; o = (o + in5) >> 56; o = (o + in6) >> 56; o = (o + in7) >> 56; in0 += o; in4 += o; in7 -= o << 56; o = (in0 >> 56); in1 += o; t = o << 56; in0 -= t; o = (in1 >> 56); in2 += o; t = o << 56; in1 -= t; o = (in2 >> 56); in3 += o; t = o << 56; in2 -= t; o = (in3 >> 56); in4 += o; t = o << 56; in3 -= t; o = (in4 >> 56); in5 += o; t = o << 56; in4 -= t; o = (in5 >> 56); in6 += o; t = o << 56; in5 -= t; o = (in6 >> 56); in7 += o; t = o << 56; in6 -= t; o = (in7 >> 56); in0 += o; in4 += o; t = o << 56; in7 -= t; /* Output as bytes */ b[ 0] = (in0 >> 0); b[ 1] = (in0 >> 8); b[ 2] = (in0 >> 16); b[ 3] = (in0 >> 24); b[ 4] = (in0 >> 32); b[ 5] = (in0 >> 40); b[ 6] = (in0 >> 48); b[ 7] = (in1 >> 0); b[ 8] = (in1 >> 8); b[ 9] = (in1 >> 16); b[10] = (in1 >> 24); b[11] = (in1 >> 32); b[12] = (in1 >> 40); b[13] = (in1 >> 48); b[14] = (in2 >> 0); b[15] = (in2 >> 8); b[16] = (in2 >> 16); b[17] = (in2 >> 24); b[18] = (in2 >> 32); b[19] = (in2 >> 40); b[20] = (in2 >> 48); b[21] = (in3 >> 0); b[22] = (in3 >> 8); b[23] = (in3 >> 16); b[24] = (in3 >> 24); b[25] = (in3 >> 32); b[26] = (in3 >> 40); b[27] = (in3 >> 48); b[28] = (in4 >> 0); b[29] = (in4 >> 8); b[30] = (in4 >> 16); b[31] = (in4 >> 24); b[32] = (in4 >> 32); b[33] = (in4 >> 40); b[34] = (in4 >> 48); b[35] = (in5 >> 0); b[36] = (in5 >> 8); b[37] = (in5 >> 16); b[38] = (in5 >> 24); b[39] = (in5 >> 32); b[40] = (in5 >> 40); b[41] = (in5 >> 48); b[42] = (in6 >> 0); b[43] = (in6 >> 8); b[44] = (in6 >> 16); b[45] = (in6 >> 24); b[46] = (in6 >> 32); b[47] = (in6 >> 40); b[48] = (in6 >> 48); b[49] = (in7 >> 0); b[50] = (in7 >> 8); b[51] = (in7 >> 16); b[52] = (in7 >> 24); b[53] = (in7 >> 32); b[54] = (in7 >> 40); b[55] = (in7 >> 48); } /* Set the field element to 0. * * a [in] Field element. */ void fe448_1(int64_t* a) { a[0] = 1; a[1] = 0; a[2] = 0; a[3] = 0; a[4] = 0; a[5] = 0; a[6] = 0; a[7] = 0; } /* Set the field element to 0. * * a [in] Field element. */ void fe448_0(int64_t* a) { a[0] = 0; a[1] = 0; a[2] = 0; a[3] = 0; a[4] = 0; a[5] = 0; a[6] = 0; a[7] = 0; } /* Copy one field element into another: d = a. * * d [in] Destination field element. * a [in] Source field element. */ void fe448_copy(int64_t* d, const int64_t* a) { d[0] = a[0]; d[1] = a[1]; d[2] = a[2]; d[3] = a[3]; d[4] = a[4]; d[5] = a[5]; d[6] = a[6]; d[7] = a[7]; } /* Conditionally swap the elements. * Constant time implementation. * * a [in] First field element. * b [in] Second field element. * c [in] Swap when 1. Valid values: 0, 1. */ static void fe448_cswap(int64_t* a, int64_t* b, int c) { int64_t mask = -(int64_t)c; int64_t t0 = (a[0] ^ b[0]) & mask; int64_t t1 = (a[1] ^ b[1]) & mask; int64_t t2 = (a[2] ^ b[2]) & mask; int64_t t3 = (a[3] ^ b[3]) & mask; int64_t t4 = (a[4] ^ b[4]) & mask; int64_t t5 = (a[5] ^ b[5]) & mask; int64_t t6 = (a[6] ^ b[6]) & mask; int64_t t7 = (a[7] ^ b[7]) & mask; a[0] ^= t0; a[1] ^= t1; a[2] ^= t2; a[3] ^= t3; a[4] ^= t4; a[5] ^= t5; a[6] ^= t6; a[7] ^= t7; b[0] ^= t0; b[1] ^= t1; b[2] ^= t2; b[3] ^= t3; b[4] ^= t4; b[5] ^= t5; b[6] ^= t6; b[7] ^= t7; } /* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold sum. * a [in] Field element to add. * b [in] Field element to add. */ void fe448_add(int64_t* r, const int64_t* a, const int64_t* b) { r[0] = a[0] + b[0]; r[1] = a[1] + b[1]; r[2] = a[2] + b[2]; r[3] = a[3] + b[3]; r[4] = a[4] + b[4]; r[5] = a[5] + b[5]; r[6] = a[6] + b[6]; r[7] = a[7] + b[7]; } /* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold difference. * a [in] Field element to subtract from. * b [in] Field element to subtract. */ void fe448_sub(int64_t* r, const int64_t* a, const int64_t* b) { r[0] = a[0] - b[0]; r[1] = a[1] - b[1]; r[2] = a[2] - b[2]; r[3] = a[3] - b[3]; r[4] = a[4] - b[4]; r[5] = a[5] - b[5]; r[6] = a[6] - b[6]; r[7] = a[7] - b[7]; } /* Mulitply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. */ void fe448_mul39081(int64_t* r, const int64_t* a) { int128_t t; int64_t o; int128_t t0 = a[0] * (int128_t)39081; int128_t t1 = a[1] * (int128_t)39081; int128_t t2 = a[2] * (int128_t)39081; int128_t t3 = a[3] * (int128_t)39081; int128_t t4 = a[4] * (int128_t)39081; int128_t t5 = a[5] * (int128_t)39081; int128_t t6 = a[6] * (int128_t)39081; int128_t t7 = a[7] * (int128_t)39081; o = (int64_t)(t0 >> 56); t1 += o; t = (int128_t)o << 56; t0 -= t; o = (int64_t)(t1 >> 56); t2 += o; t = (int128_t)o << 56; t1 -= t; o = (int64_t)(t2 >> 56); t3 += o; t = (int128_t)o << 56; t2 -= t; o = (int64_t)(t3 >> 56); t4 += o; t = (int128_t)o << 56; t3 -= t; o = (int64_t)(t4 >> 56); t5 += o; t = (int128_t)o << 56; t4 -= t; o = (int64_t)(t5 >> 56); t6 += o; t = (int128_t)o << 56; t5 -= t; o = (int64_t)(t6 >> 56); t7 += o; t = (int128_t)o << 56; t6 -= t; o = (int64_t)(t7 >> 56); t0 += o; t4 += o; t = (int128_t)o << 56; t7 -= t; /* Store */ r[0] = (int64_t)t0; r[1] = (int64_t)t1; r[2] = (int64_t)t2; r[3] = (int64_t)t3; r[4] = (int64_t)t4; r[5] = (int64_t)t5; r[6] = (int64_t)t6; r[7] = (int64_t)t7; } /* Mulitply two field elements. r = (a * b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. * b [in] Field element to multiply. */ void fe448_mul(int64_t* r, const int64_t* a, const int64_t* b) { int128_t t; int64_t o; int128_t t0 = (int128_t)a[ 0] * b[ 0]; int128_t t1 = (int128_t)a[ 0] * b[ 1]; int128_t t101 = (int128_t)a[ 1] * b[ 0]; int128_t t2 = (int128_t)a[ 0] * b[ 2]; int128_t t102 = (int128_t)a[ 1] * b[ 1]; int128_t t202 = (int128_t)a[ 2] * b[ 0]; int128_t t3 = (int128_t)a[ 0] * b[ 3]; int128_t t103 = (int128_t)a[ 1] * b[ 2]; int128_t t203 = (int128_t)a[ 2] * b[ 1]; int128_t t303 = (int128_t)a[ 3] * b[ 0]; int128_t t4 = (int128_t)a[ 0] * b[ 4]; int128_t t104 = (int128_t)a[ 1] * b[ 3]; int128_t t204 = (int128_t)a[ 2] * b[ 2]; int128_t t304 = (int128_t)a[ 3] * b[ 1]; int128_t t404 = (int128_t)a[ 4] * b[ 0]; int128_t t5 = (int128_t)a[ 0] * b[ 5]; int128_t t105 = (int128_t)a[ 1] * b[ 4]; int128_t t205 = (int128_t)a[ 2] * b[ 3]; int128_t t305 = (int128_t)a[ 3] * b[ 2]; int128_t t405 = (int128_t)a[ 4] * b[ 1]; int128_t t505 = (int128_t)a[ 5] * b[ 0]; int128_t t6 = (int128_t)a[ 0] * b[ 6]; int128_t t106 = (int128_t)a[ 1] * b[ 5]; int128_t t206 = (int128_t)a[ 2] * b[ 4]; int128_t t306 = (int128_t)a[ 3] * b[ 3]; int128_t t406 = (int128_t)a[ 4] * b[ 2]; int128_t t506 = (int128_t)a[ 5] * b[ 1]; int128_t t606 = (int128_t)a[ 6] * b[ 0]; int128_t t7 = (int128_t)a[ 0] * b[ 7]; int128_t t107 = (int128_t)a[ 1] * b[ 6]; int128_t t207 = (int128_t)a[ 2] * b[ 5]; int128_t t307 = (int128_t)a[ 3] * b[ 4]; int128_t t407 = (int128_t)a[ 4] * b[ 3]; int128_t t507 = (int128_t)a[ 5] * b[ 2]; int128_t t607 = (int128_t)a[ 6] * b[ 1]; int128_t t707 = (int128_t)a[ 7] * b[ 0]; int128_t t8 = (int128_t)a[ 1] * b[ 7]; int128_t t108 = (int128_t)a[ 2] * b[ 6]; int128_t t208 = (int128_t)a[ 3] * b[ 5]; int128_t t308 = (int128_t)a[ 4] * b[ 4]; int128_t t408 = (int128_t)a[ 5] * b[ 3]; int128_t t508 = (int128_t)a[ 6] * b[ 2]; int128_t t608 = (int128_t)a[ 7] * b[ 1]; int128_t t9 = (int128_t)a[ 2] * b[ 7]; int128_t t109 = (int128_t)a[ 3] * b[ 6]; int128_t t209 = (int128_t)a[ 4] * b[ 5]; int128_t t309 = (int128_t)a[ 5] * b[ 4]; int128_t t409 = (int128_t)a[ 6] * b[ 3]; int128_t t509 = (int128_t)a[ 7] * b[ 2]; int128_t t10 = (int128_t)a[ 3] * b[ 7]; int128_t t110 = (int128_t)a[ 4] * b[ 6]; int128_t t210 = (int128_t)a[ 5] * b[ 5]; int128_t t310 = (int128_t)a[ 6] * b[ 4]; int128_t t410 = (int128_t)a[ 7] * b[ 3]; int128_t t11 = (int128_t)a[ 4] * b[ 7]; int128_t t111 = (int128_t)a[ 5] * b[ 6]; int128_t t211 = (int128_t)a[ 6] * b[ 5]; int128_t t311 = (int128_t)a[ 7] * b[ 4]; int128_t t12 = (int128_t)a[ 5] * b[ 7]; int128_t t112 = (int128_t)a[ 6] * b[ 6]; int128_t t212 = (int128_t)a[ 7] * b[ 5]; int128_t t13 = (int128_t)a[ 6] * b[ 7]; int128_t t113 = (int128_t)a[ 7] * b[ 6]; int128_t t14 = (int128_t)a[ 7] * b[ 7]; t1 += t101; t2 += t102; t2 += t202; t3 += t103; t3 += t203; t3 += t303; t4 += t104; t4 += t204; t4 += t304; t4 += t404; t5 += t105; t5 += t205; t5 += t305; t5 += t405; t5 += t505; t6 += t106; t6 += t206; t6 += t306; t6 += t406; t6 += t506; t6 += t606; t7 += t107; t7 += t207; t7 += t307; t7 += t407; t7 += t507; t7 += t607; t7 += t707; t8 += t108; t8 += t208; t8 += t308; t8 += t408; t8 += t508; t8 += t608; t9 += t109; t9 += t209; t9 += t309; t9 += t409; t9 += t509; t10 += t110; t10 += t210; t10 += t310; t10 += t410; t11 += t111; t11 += t211; t11 += t311; t12 += t112; t12 += t212; t13 += t113; /* Reduce */ t0 += t8 + t12; t1 += t9 + t13; t2 += t10 + t14; t3 += t11; t4 += t12 + t8 + t12; t5 += t13 + t9 + t13; t6 += t14 + t10 + t14; t7 += t11; o = t7 >> 56; t0 += o; t4 += o; t = (int128_t)o << 56; t7 -= t; o = (int64_t)(t0 >> 56); t1 += o; t = (int128_t)o << 56; t0 -= t; o = (int64_t)(t1 >> 56); t2 += o; t = (int128_t)o << 56; t1 -= t; o = (int64_t)(t2 >> 56); t3 += o; t = (int128_t)o << 56; t2 -= t; o = (int64_t)(t3 >> 56); t4 += o; t = (int128_t)o << 56; t3 -= t; o = (int64_t)(t4 >> 56); t5 += o; t = (int128_t)o << 56; t4 -= t; o = (int64_t)(t5 >> 56); t6 += o; t = (int128_t)o << 56; t5 -= t; o = (int64_t)(t6 >> 56); t7 += o; t = (int128_t)o << 56; t6 -= t; o = (int64_t)(t7 >> 56); t0 += o; t4 += o; t = (int128_t)o << 56; t7 -= t; /* Store */ r[0] = (int64_t)t0; r[1] = (int64_t)t1; r[2] = (int64_t)t2; r[3] = (int64_t)t3; r[4] = (int64_t)t4; r[5] = (int64_t)t5; r[6] = (int64_t)t6; r[7] = (int64_t)t7; } /* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to square. */ void fe448_sqr(int64_t* r, const int64_t* a) { int128_t t; int64_t o; int128_t t0 = (int128_t)a[ 0] * a[ 0]; int128_t t1 = 2 * (int128_t)a[ 0] * a[ 1]; int128_t t2 = 2 * (int128_t)a[ 0] * a[ 2]; int128_t t102 = (int128_t)a[ 1] * a[ 1]; int128_t t3 = 2 * (int128_t)a[ 0] * a[ 3]; int128_t t103 = 2 * (int128_t)a[ 1] * a[ 2]; int128_t t4 = 2 * (int128_t)a[ 0] * a[ 4]; int128_t t104 = 2 * (int128_t)a[ 1] * a[ 3]; int128_t t204 = (int128_t)a[ 2] * a[ 2]; int128_t t5 = 2 * (int128_t)a[ 0] * a[ 5]; int128_t t105 = 2 * (int128_t)a[ 1] * a[ 4]; int128_t t205 = 2 * (int128_t)a[ 2] * a[ 3]; int128_t t6 = 2 * (int128_t)a[ 0] * a[ 6]; int128_t t106 = 2 * (int128_t)a[ 1] * a[ 5]; int128_t t206 = 2 * (int128_t)a[ 2] * a[ 4]; int128_t t306 = (int128_t)a[ 3] * a[ 3]; int128_t t7 = 2 * (int128_t)a[ 0] * a[ 7]; int128_t t107 = 2 * (int128_t)a[ 1] * a[ 6]; int128_t t207 = 2 * (int128_t)a[ 2] * a[ 5]; int128_t t307 = 2 * (int128_t)a[ 3] * a[ 4]; int128_t t8 = 2 * (int128_t)a[ 1] * a[ 7]; int128_t t108 = 2 * (int128_t)a[ 2] * a[ 6]; int128_t t208 = 2 * (int128_t)a[ 3] * a[ 5]; int128_t t308 = (int128_t)a[ 4] * a[ 4]; int128_t t9 = 2 * (int128_t)a[ 2] * a[ 7]; int128_t t109 = 2 * (int128_t)a[ 3] * a[ 6]; int128_t t209 = 2 * (int128_t)a[ 4] * a[ 5]; int128_t t10 = 2 * (int128_t)a[ 3] * a[ 7]; int128_t t110 = 2 * (int128_t)a[ 4] * a[ 6]; int128_t t210 = (int128_t)a[ 5] * a[ 5]; int128_t t11 = 2 * (int128_t)a[ 4] * a[ 7]; int128_t t111 = 2 * (int128_t)a[ 5] * a[ 6]; int128_t t12 = 2 * (int128_t)a[ 5] * a[ 7]; int128_t t112 = (int128_t)a[ 6] * a[ 6]; int128_t t13 = 2 * (int128_t)a[ 6] * a[ 7]; int128_t t14 = (int128_t)a[ 7] * a[ 7]; t2 += t102; t3 += t103; t4 += t104; t4 += t204; t5 += t105; t5 += t205; t6 += t106; t6 += t206; t6 += t306; t7 += t107; t7 += t207; t7 += t307; t8 += t108; t8 += t208; t8 += t308; t9 += t109; t9 += t209; t10 += t110; t10 += t210; t11 += t111; t12 += t112; /* Reduce */ t0 += t8 + t12; t1 += t9 + t13; t2 += t10 + t14; t3 += t11; t4 += t12 + t8 + t12; t5 += t13 + t9 + t13; t6 += t14 + t10 + t14; t7 += t11; o = t7 >> 56; t0 += o; t4 += o; t = (int128_t)o << 56; t7 -= t; o = (int64_t)(t0 >> 56); t1 += o; t = (int128_t)o << 56; t0 -= t; o = (int64_t)(t1 >> 56); t2 += o; t = (int128_t)o << 56; t1 -= t; o = (int64_t)(t2 >> 56); t3 += o; t = (int128_t)o << 56; t2 -= t; o = (int64_t)(t3 >> 56); t4 += o; t = (int128_t)o << 56; t3 -= t; o = (int64_t)(t4 >> 56); t5 += o; t = (int128_t)o << 56; t4 -= t; o = (int64_t)(t5 >> 56); t6 += o; t = (int128_t)o << 56; t5 -= t; o = (int64_t)(t6 >> 56); t7 += o; t = (int128_t)o << 56; t6 -= t; o = (int64_t)(t7 >> 56); t0 += o; t4 += o; t = (int128_t)o << 56; t7 -= t; /* Store */ r[0] = (int64_t)t0; r[1] = (int64_t)t1; r[2] = (int64_t)t2; r[3] = (int64_t)t3; r[4] = (int64_t)t4; r[5] = (int64_t)t5; r[6] = (int64_t)t6; r[7] = (int64_t)t7; } /* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1 * Constant time implementation - using Fermat's little theorem: * a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a * For curve448: p - 2 = 2^448 - 2^224 - 3 * * r [in] Field element to hold result. * a [in] Field element to invert. */ void fe448_invert(int64_t* r, const int64_t* a) { int64_t t1[8]; int64_t t2[8]; int64_t t3[8]; int64_t t4[8]; int i; fe448_sqr(t1, a); /* t1 = 2 */ fe448_mul(t1, t1, a); /* t1 = 3 */ fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2); /* t2 = c */ fe448_mul(t3, t2, a); /* t3 = d */ fe448_mul(t1, t2, t1); /* t1 = f */ fe448_sqr(t2, t1); /* t2 = 1e */ fe448_mul(t4, t2, a); /* t4 = 1f */ fe448_sqr(t2, t4); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 3e0 */ fe448_mul(t1, t2, t4); /* t1 = 3ff */ fe448_sqr(t2, t1); for (i = 1; i < 10; ++i) fe448_sqr(t2, t2); /* t2 = ffc00 */ fe448_mul(t1, t2, t1); /* t1 = fffff */ fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 1ffffe0 */ fe448_mul(t1, t2, t4); /* t1 = 1ffffff */ fe448_sqr(t2, t1); for (i = 1; i < 25; ++i) fe448_sqr(t2, t2); /* t2 = 3fffffe000000 */ fe448_mul(t1, t2, t1); /* t1 = 3ffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 7fffffffffffe0 */ fe448_mul(t1, t2, t4); /* t1 = 7fffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 55; ++i) fe448_sqr(t2, t2); /* t2 = 3fffffffffffff80000000000000 */ fe448_mul(t1, t2, t1); /* t1 = 3fffffffffffffffffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 110; ++i) fe448_sqr(t2, t2); /* t2 = fffffffffffffffffffffffffffc000000000000000000000000000 */ fe448_mul(t1, t2, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2); /* t2 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff0 */ fe448_mul(t3, t3, t2); /* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffd */ fe448_mul(t1, t3, a); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */ fe448_sqr(t1, t1); for (i = 1; i < 224; ++i) fe448_sqr(t1, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe00000000000000000000000000000000000000000000000000000000 */ fe448_mul(r, t3, t1); /* r = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffffffffffffffffffffffffffffffffffffffffffffffffffffd */ } /* Scalar multiply the point by a number. r = n.a * Uses Montogmery ladder and only requires the x-ordinate. * * r [in] Field element to hold result. * n [in] Scalar as an array of bytes. * a [in] Point to multiply - x-ordinate only. */ int curve448(byte* r, const byte* n, const byte* a) { int64_t x1[8]; int64_t x2[8]; int64_t z2[8]; int64_t x3[8]; int64_t z3[8]; int64_t t0[8]; int64_t t1[8]; int i; unsigned int swap; unsigned int b; fe448_from_bytes(x1, a); fe448_1(x2); fe448_0(z2); fe448_copy(x3, x1); fe448_1(z3); swap = 0; for (i = 447; i >= 0; --i) { b = (n[i >> 3] >> (i & 7)) & 1; swap ^= b; fe448_cswap(x2, x3, swap); fe448_cswap(z2, z3, swap); swap = b; /* Montgomery Ladder - double and add */ fe448_add(t0, x2, z2); fe448_reduce(t0); fe448_add(t1, x3, z3); fe448_reduce(t1); fe448_sub(x2, x2, z2); fe448_sub(x3, x3, z3); fe448_mul(t1, t1, x2); fe448_mul(z3, x3, t0); fe448_sqr(t0, t0); fe448_sqr(x2, x2); fe448_add(x3, z3, t1); fe448_reduce(x3); fe448_sqr(x3, x3); fe448_sub(z3, z3, t1); fe448_sqr(z3, z3); fe448_mul(z3, z3, x1); fe448_sub(t1, t0, x2); fe448_mul(x2, t0, x2); fe448_mul39081(z2, t1); fe448_add(z2, t0, z2); fe448_mul(z2, z2, t1); } /* Last two bits are 0 - no final swap check required. */ fe448_invert(z2, z2); fe448_mul(x2, x2, z2); fe448_to_bytes(r, x2); return 0; } #ifdef HAVE_ED448 /* Check whether field element is not 0. * Must convert to a normalized form before checking. * * a [in] Field element. * returns 0 when zero, and any other value otherwise. */ int fe448_isnonzero(const int64_t* a) { uint8_t b[56]; int i; uint8_t c = 0; fe448_to_bytes(b, a); for (i = 0; i < 56; i++) c |= b[i]; return c; } /* Check whether field element is negative. * Must convert to a normalized form before checking. * * a [in] Field element. * returns 1 when negative, and 0 otherwise. */ int fe448_isnegative(const int64_t* a) { uint8_t b[56]; fe448_to_bytes(b, a); return b[0] & 1; } /* Negates the field element. r = -a * * r [in] Field element to hold result. * a [in] Field element. */ void fe448_neg(int64_t* r, const int64_t* a) { r[0] = -a[0]; r[1] = -a[1]; r[2] = -a[2]; r[3] = -a[3]; r[4] = -a[4]; r[5] = -a[5]; r[6] = -a[6]; r[7] = -a[7]; } /* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1 * Used for calcualting y-ordinate from x-ordinate for Ed448. * * r [in] Field element to hold result. * a [in] Field element to exponentiate. */ void fe448_pow_2_446_222_1(int64_t* r, const int64_t* a) { int64_t t1[8]; int64_t t2[8]; int64_t t3[8]; int64_t t4[8]; int64_t t5[8]; int i; fe448_sqr(t3, a); /* t3 = 2 */ fe448_mul(t1, t3, a); /* t1 = 3 */ fe448_sqr(t5, t1); /* t5 = 6 */ fe448_mul(t5, t5, a); /* t5 = 7 */ fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2); /* t2 = c */ fe448_mul(t3, t2, t3); /* t3 = e */ fe448_mul(t1, t2, t1); /* t1 = f */ fe448_sqr(t2, t1); for (i = 1; i < 3; ++i) fe448_sqr(t2, t2); /* t2 = 78 */ fe448_mul(t5, t2, t5); /* t5 = 7f */ fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2); /* t2 = f0 */ fe448_mul(t1, t2, t1); /* t1 = ff */ fe448_mul(t3, t3, t2); /* t3 = fe */ fe448_sqr(t2, t1); for (i = 1; i < 7; ++i) fe448_sqr(t2, t2); /* t2 = 7f80 */ fe448_mul(t5, t2, t5); /* t5 = 7fff */ fe448_sqr(t2, t1); for (i = 1; i < 8; ++i) fe448_sqr(t2, t2); /* t2 = ff00 */ fe448_mul(t1, t2, t1); /* t1 = ffff */ fe448_mul(t3, t3, t2); /* t3 = fffe */ fe448_sqr(t2, t5); for (i = 1; i < 15; ++i) fe448_sqr(t2, t2); /* t2 = 3fff8000 */ fe448_mul(t5, t2, t5); /* t5 = 3fffffff */ fe448_sqr(t2, t1); for (i = 1; i < 16; ++i) fe448_sqr(t2, t2); /* t2 = ffff0000 */ fe448_mul(t1, t2, t1); /* t1 = ffffffff */ fe448_mul(t3, t3, t2); /* t3 = fffffffe */ fe448_sqr(t2, t1); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2); /* t2 = ffffffff00000000 */ fe448_mul(t2, t2, t1); /* t2 = ffffffffffffffff */ fe448_sqr(t1, t2); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1); /* t1 = ffffffffffffffff0000000000000000 */ fe448_mul(t1, t1, t2); /* t1 = ffffffffffffffffffffffffffffffff */ fe448_sqr(t1, t1); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1); /* t1 = ffffffffffffffffffffffffffffffff0000000000000000 */ fe448_mul(t4, t1, t2); /* t4 = ffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t2, t4); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2); /* t2 = ffffffffffffffffffffffffffffffffffffffffffffffff00000000 */ fe448_mul(t3, t3, t2); /* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */ fe448_sqr(t1, t3); for (i = 1; i < 192; ++i) fe448_sqr(t1, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe000000000000000000000000000000000000000000000000 */ fe448_mul(t1, t1, t4); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t1, t1); for (i = 1; i < 30; ++i) fe448_sqr(t1, t1); /* t1 = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffc0000000 */ fe448_mul(r, t5, t1); /* r = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffffffffff */ } /* Constant time, conditional move of b into a. * a is not changed if the condition is 0. * * a A field element. * b A field element. * c If 1 then copy and if 0 then don't copy. */ void fe448_cmov(int64_t* a, const int64_t* b, int c) { int64_t m = -(int64_t)c; int64_t t0 = m & (a[0] ^ b[0]); int64_t t1 = m & (a[1] ^ b[1]); int64_t t2 = m & (a[2] ^ b[2]); int64_t t3 = m & (a[3] ^ b[3]); int64_t t4 = m & (a[4] ^ b[4]); int64_t t5 = m & (a[5] ^ b[5]); int64_t t6 = m & (a[6] ^ b[6]); int64_t t7 = m & (a[7] ^ b[7]); a[0] ^= t0; a[1] ^= t1; a[2] ^= t2; a[3] ^= t3; a[4] ^= t4; a[5] ^= t5; a[6] ^= t6; a[7] ^= t7; } #endif /* HAVE_ED448 */ #else /* Initialize the field element operations. */ void fe448_init(void) { } /* Convert the field element from a byte array to an array of 28-bits. * * r [in] Array to encode into. * b [in] Byte array. */ void fe448_from_bytes(int32_t* r, const unsigned char* b) { r[ 0] = (((int32_t)((b[ 0] ) >> 0)) << 0) | (((int32_t)((b[ 1] ) >> 0)) << 8) | (((int32_t)((b[ 2] ) >> 0)) << 16) | ((((int32_t)((b[ 3] & 0xf )) >> 0)) << 24); r[ 1] = (((int32_t)((b[ 3] ) >> 4)) << 0) | (((int32_t)((b[ 4] ) >> 0)) << 4) | (((int32_t)((b[ 5] ) >> 0)) << 12) | (((int32_t)((b[ 6] ) >> 0)) << 20); r[ 2] = (((int32_t)((b[ 7] ) >> 0)) << 0) | (((int32_t)((b[ 8] ) >> 0)) << 8) | (((int32_t)((b[ 9] ) >> 0)) << 16) | ((((int32_t)((b[10] & 0xf )) >> 0)) << 24); r[ 3] = (((int32_t)((b[10] ) >> 4)) << 0) | (((int32_t)((b[11] ) >> 0)) << 4) | (((int32_t)((b[12] ) >> 0)) << 12) | (((int32_t)((b[13] ) >> 0)) << 20); r[ 4] = (((int32_t)((b[14] ) >> 0)) << 0) | (((int32_t)((b[15] ) >> 0)) << 8) | (((int32_t)((b[16] ) >> 0)) << 16) | ((((int32_t)((b[17] & 0xf )) >> 0)) << 24); r[ 5] = (((int32_t)((b[17] ) >> 4)) << 0) | (((int32_t)((b[18] ) >> 0)) << 4) | (((int32_t)((b[19] ) >> 0)) << 12) | (((int32_t)((b[20] ) >> 0)) << 20); r[ 6] = (((int32_t)((b[21] ) >> 0)) << 0) | (((int32_t)((b[22] ) >> 0)) << 8) | (((int32_t)((b[23] ) >> 0)) << 16) | ((((int32_t)((b[24] & 0xf )) >> 0)) << 24); r[ 7] = (((int32_t)((b[24] ) >> 4)) << 0) | (((int32_t)((b[25] ) >> 0)) << 4) | (((int32_t)((b[26] ) >> 0)) << 12) | (((int32_t)((b[27] ) >> 0)) << 20); r[ 8] = (((int32_t)((b[28] ) >> 0)) << 0) | (((int32_t)((b[29] ) >> 0)) << 8) | (((int32_t)((b[30] ) >> 0)) << 16) | ((((int32_t)((b[31] & 0xf )) >> 0)) << 24); r[ 9] = (((int32_t)((b[31] ) >> 4)) << 0) | (((int32_t)((b[32] ) >> 0)) << 4) | (((int32_t)((b[33] ) >> 0)) << 12) | (((int32_t)((b[34] ) >> 0)) << 20); r[10] = (((int32_t)((b[35] ) >> 0)) << 0) | (((int32_t)((b[36] ) >> 0)) << 8) | (((int32_t)((b[37] ) >> 0)) << 16) | ((((int32_t)((b[38] & 0xf )) >> 0)) << 24); r[11] = (((int32_t)((b[38] ) >> 4)) << 0) | (((int32_t)((b[39] ) >> 0)) << 4) | (((int32_t)((b[40] ) >> 0)) << 12) | (((int32_t)((b[41] ) >> 0)) << 20); r[12] = (((int32_t)((b[42] ) >> 0)) << 0) | (((int32_t)((b[43] ) >> 0)) << 8) | (((int32_t)((b[44] ) >> 0)) << 16) | ((((int32_t)((b[45] & 0xf )) >> 0)) << 24); r[13] = (((int32_t)((b[45] ) >> 4)) << 0) | (((int32_t)((b[46] ) >> 0)) << 4) | (((int32_t)((b[47] ) >> 0)) << 12) | (((int32_t)((b[48] ) >> 0)) << 20); r[14] = (((int32_t)((b[49] ) >> 0)) << 0) | (((int32_t)((b[50] ) >> 0)) << 8) | (((int32_t)((b[51] ) >> 0)) << 16) | ((((int32_t)((b[52] & 0xf )) >> 0)) << 24); r[15] = (((int32_t)((b[52] ) >> 4)) << 0) | (((int32_t)((b[53] ) >> 0)) << 4) | (((int32_t)((b[54] ) >> 0)) << 12) | (((int32_t)((b[55] ) >> 0)) << 20); } /* Convert the field element to a byte array from an array of 28-bits. * * b [in] Byte array. * a [in] Array to encode into. */ void fe448_to_bytes(unsigned char* b, const int32_t* a) { int64_t t; /* Mod */ int32_t in0 = a[0]; int32_t in1 = a[1]; int32_t in2 = a[2]; int32_t in3 = a[3]; int32_t in4 = a[4]; int32_t in5 = a[5]; int32_t in6 = a[6]; int32_t in7 = a[7]; int32_t in8 = a[8]; int32_t in9 = a[9]; int32_t in10 = a[10]; int32_t in11 = a[11]; int32_t in12 = a[12]; int32_t in13 = a[13]; int32_t in14 = a[14]; int32_t in15 = a[15]; int32_t o = in15 >> 28; in15 -= o << 28; in0 += o; in8 += o; o = (in0 + 1) >> 28; o = (o + in1) >> 28; o = (o + in2) >> 28; o = (o + in3) >> 28; o = (o + in4) >> 28; o = (o + in5) >> 28; o = (o + in6) >> 28; o = (o + in7) >> 28; o = (o + in8 + 1) >> 28; o = (o + in9) >> 28; o = (o + in10) >> 28; o = (o + in11) >> 28; o = (o + in12) >> 28; o = (o + in13) >> 28; o = (o + in14) >> 28; o = (o + in15) >> 28; in0 += o; in8 += o; in15 -= o << 28; o = (in0 >> 28); in1 += o; t = o << 28; in0 -= t; o = (in1 >> 28); in2 += o; t = o << 28; in1 -= t; o = (in2 >> 28); in3 += o; t = o << 28; in2 -= t; o = (in3 >> 28); in4 += o; t = o << 28; in3 -= t; o = (in4 >> 28); in5 += o; t = o << 28; in4 -= t; o = (in5 >> 28); in6 += o; t = o << 28; in5 -= t; o = (in6 >> 28); in7 += o; t = o << 28; in6 -= t; o = (in7 >> 28); in8 += o; t = o << 28; in7 -= t; o = (in8 >> 28); in9 += o; t = o << 28; in8 -= t; o = (in9 >> 28); in10 += o; t = o << 28; in9 -= t; o = (in10 >> 28); in11 += o; t = o << 28; in10 -= t; o = (in11 >> 28); in12 += o; t = o << 28; in11 -= t; o = (in12 >> 28); in13 += o; t = o << 28; in12 -= t; o = (in13 >> 28); in14 += o; t = o << 28; in13 -= t; o = (in14 >> 28); in15 += o; t = o << 28; in14 -= t; o = (in15 >> 28); in0 += o; in8 += o; t = o << 28; in15 -= t; /* Output as bytes */ b[ 0] = (in0 >> 0); b[ 1] = (in0 >> 8); b[ 2] = (in0 >> 16); b[ 3] = (in0 >> 24) + ((in1 >> 0) << 4); b[ 4] = (in1 >> 4); b[ 5] = (in1 >> 12); b[ 6] = (in1 >> 20); b[ 7] = (in2 >> 0); b[ 8] = (in2 >> 8); b[ 9] = (in2 >> 16); b[10] = (in2 >> 24) + ((in3 >> 0) << 4); b[11] = (in3 >> 4); b[12] = (in3 >> 12); b[13] = (in3 >> 20); b[14] = (in4 >> 0); b[15] = (in4 >> 8); b[16] = (in4 >> 16); b[17] = (in4 >> 24) + ((in5 >> 0) << 4); b[18] = (in5 >> 4); b[19] = (in5 >> 12); b[20] = (in5 >> 20); b[21] = (in6 >> 0); b[22] = (in6 >> 8); b[23] = (in6 >> 16); b[24] = (in6 >> 24) + ((in7 >> 0) << 4); b[25] = (in7 >> 4); b[26] = (in7 >> 12); b[27] = (in7 >> 20); b[28] = (in8 >> 0); b[29] = (in8 >> 8); b[30] = (in8 >> 16); b[31] = (in8 >> 24) + ((in9 >> 0) << 4); b[32] = (in9 >> 4); b[33] = (in9 >> 12); b[34] = (in9 >> 20); b[35] = (in10 >> 0); b[36] = (in10 >> 8); b[37] = (in10 >> 16); b[38] = (in10 >> 24) + ((in11 >> 0) << 4); b[39] = (in11 >> 4); b[40] = (in11 >> 12); b[41] = (in11 >> 20); b[42] = (in12 >> 0); b[43] = (in12 >> 8); b[44] = (in12 >> 16); b[45] = (in12 >> 24) + ((in13 >> 0) << 4); b[46] = (in13 >> 4); b[47] = (in13 >> 12); b[48] = (in13 >> 20); b[49] = (in14 >> 0); b[50] = (in14 >> 8); b[51] = (in14 >> 16); b[52] = (in14 >> 24) + ((in15 >> 0) << 4); b[53] = (in15 >> 4); b[54] = (in15 >> 12); b[55] = (in15 >> 20); } /* Set the field element to 0. * * a [in] Field element. */ void fe448_1(int32_t* a) { a[0] = 1; a[1] = 0; a[2] = 0; a[3] = 0; a[4] = 0; a[5] = 0; a[6] = 0; a[7] = 0; a[8] = 0; a[9] = 0; a[10] = 0; a[11] = 0; a[12] = 0; a[13] = 0; a[14] = 0; a[15] = 0; } /* Set the field element to 0. * * a [in] Field element. */ void fe448_0(int32_t* a) { a[0] = 0; a[1] = 0; a[2] = 0; a[3] = 0; a[4] = 0; a[5] = 0; a[6] = 0; a[7] = 0; a[8] = 0; a[9] = 0; a[10] = 0; a[11] = 0; a[12] = 0; a[13] = 0; a[14] = 0; a[15] = 0; } /* Copy one field element into another: d = a. * * d [in] Destination field element. * a [in] Source field element. */ void fe448_copy(int32_t* d, const int32_t* a) { d[0] = a[0]; d[1] = a[1]; d[2] = a[2]; d[3] = a[3]; d[4] = a[4]; d[5] = a[5]; d[6] = a[6]; d[7] = a[7]; d[8] = a[8]; d[9] = a[9]; d[10] = a[10]; d[11] = a[11]; d[12] = a[12]; d[13] = a[13]; d[14] = a[14]; d[15] = a[15]; } /* Conditionally swap the elements. * Constant time implementation. * * a [in] First field element. * b [in] Second field element. * c [in] Swap when 1. Valid values: 0, 1. */ static void fe448_cswap(int32_t* a, int32_t* b, int c) { int32_t mask = -(int32_t)c; int32_t t0 = (a[0] ^ b[0]) & mask; int32_t t1 = (a[1] ^ b[1]) & mask; int32_t t2 = (a[2] ^ b[2]) & mask; int32_t t3 = (a[3] ^ b[3]) & mask; int32_t t4 = (a[4] ^ b[4]) & mask; int32_t t5 = (a[5] ^ b[5]) & mask; int32_t t6 = (a[6] ^ b[6]) & mask; int32_t t7 = (a[7] ^ b[7]) & mask; int32_t t8 = (a[8] ^ b[8]) & mask; int32_t t9 = (a[9] ^ b[9]) & mask; int32_t t10 = (a[10] ^ b[10]) & mask; int32_t t11 = (a[11] ^ b[11]) & mask; int32_t t12 = (a[12] ^ b[12]) & mask; int32_t t13 = (a[13] ^ b[13]) & mask; int32_t t14 = (a[14] ^ b[14]) & mask; int32_t t15 = (a[15] ^ b[15]) & mask; a[0] ^= t0; a[1] ^= t1; a[2] ^= t2; a[3] ^= t3; a[4] ^= t4; a[5] ^= t5; a[6] ^= t6; a[7] ^= t7; a[8] ^= t8; a[9] ^= t9; a[10] ^= t10; a[11] ^= t11; a[12] ^= t12; a[13] ^= t13; a[14] ^= t14; a[15] ^= t15; b[0] ^= t0; b[1] ^= t1; b[2] ^= t2; b[3] ^= t3; b[4] ^= t4; b[5] ^= t5; b[6] ^= t6; b[7] ^= t7; b[8] ^= t8; b[9] ^= t9; b[10] ^= t10; b[11] ^= t11; b[12] ^= t12; b[13] ^= t13; b[14] ^= t14; b[15] ^= t15; } /* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold sum. * a [in] Field element to add. * b [in] Field element to add. */ void fe448_add(int32_t* r, const int32_t* a, const int32_t* b) { r[0] = a[0] + b[0]; r[1] = a[1] + b[1]; r[2] = a[2] + b[2]; r[3] = a[3] + b[3]; r[4] = a[4] + b[4]; r[5] = a[5] + b[5]; r[6] = a[6] + b[6]; r[7] = a[7] + b[7]; r[8] = a[8] + b[8]; r[9] = a[9] + b[9]; r[10] = a[10] + b[10]; r[11] = a[11] + b[11]; r[12] = a[12] + b[12]; r[13] = a[13] + b[13]; r[14] = a[14] + b[14]; r[15] = a[15] + b[15]; } /* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold difference. * a [in] Field element to subtract from. * b [in] Field element to subtract. */ void fe448_sub(int32_t* r, const int32_t* a, const int32_t* b) { r[0] = a[0] - b[0]; r[1] = a[1] - b[1]; r[2] = a[2] - b[2]; r[3] = a[3] - b[3]; r[4] = a[4] - b[4]; r[5] = a[5] - b[5]; r[6] = a[6] - b[6]; r[7] = a[7] - b[7]; r[8] = a[8] - b[8]; r[9] = a[9] - b[9]; r[10] = a[10] - b[10]; r[11] = a[11] - b[11]; r[12] = a[12] - b[12]; r[13] = a[13] - b[13]; r[14] = a[14] - b[14]; r[15] = a[15] - b[15]; } void fe448_reduce(int32_t* a) { int64_t o; o = a[0 ] >> 28; a[1 ] += o; a[0 ] -= o << 28; o = a[1 ] >> 28; a[2 ] += o; a[1 ] -= o << 28; o = a[2 ] >> 28; a[3 ] += o; a[2 ] -= o << 28; o = a[3 ] >> 28; a[4 ] += o; a[3 ] -= o << 28; o = a[4 ] >> 28; a[5 ] += o; a[4 ] -= o << 28; o = a[5 ] >> 28; a[6 ] += o; a[5 ] -= o << 28; o = a[6 ] >> 28; a[7 ] += o; a[6 ] -= o << 28; o = a[7 ] >> 28; a[8 ] += o; a[7 ] -= o << 28; o = a[8 ] >> 28; a[9 ] += o; a[8 ] -= o << 28; o = a[9 ] >> 28; a[10] += o; a[9 ] -= o << 28; o = a[10] >> 28; a[11] += o; a[10] -= o << 28; o = a[11] >> 28; a[12] += o; a[11] -= o << 28; o = a[12] >> 28; a[13] += o; a[12] -= o << 28; o = a[13] >> 28; a[14] += o; a[13] -= o << 28; o = a[14] >> 28; a[15] += o; a[14] -= o << 28; o = a[15] >> 28; a[0] += o; a[8] += o; a[15] -= o << 28; } /* Mulitply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. */ void fe448_mul39081(int32_t* r, const int32_t* a) { int64_t t; int32_t o; int64_t t0 = a[0] * (int64_t)39081; int64_t t1 = a[1] * (int64_t)39081; int64_t t2 = a[2] * (int64_t)39081; int64_t t3 = a[3] * (int64_t)39081; int64_t t4 = a[4] * (int64_t)39081; int64_t t5 = a[5] * (int64_t)39081; int64_t t6 = a[6] * (int64_t)39081; int64_t t7 = a[7] * (int64_t)39081; int64_t t8 = a[8] * (int64_t)39081; int64_t t9 = a[9] * (int64_t)39081; int64_t t10 = a[10] * (int64_t)39081; int64_t t11 = a[11] * (int64_t)39081; int64_t t12 = a[12] * (int64_t)39081; int64_t t13 = a[13] * (int64_t)39081; int64_t t14 = a[14] * (int64_t)39081; int64_t t15 = a[15] * (int64_t)39081; o = (int32_t)(t0 >> 28); t1 += o; t = (int64_t)o << 28; t0 -= t; o = (int32_t)(t1 >> 28); t2 += o; t = (int64_t)o << 28; t1 -= t; o = (int32_t)(t2 >> 28); t3 += o; t = (int64_t)o << 28; t2 -= t; o = (int32_t)(t3 >> 28); t4 += o; t = (int64_t)o << 28; t3 -= t; o = (int32_t)(t4 >> 28); t5 += o; t = (int64_t)o << 28; t4 -= t; o = (int32_t)(t5 >> 28); t6 += o; t = (int64_t)o << 28; t5 -= t; o = (int32_t)(t6 >> 28); t7 += o; t = (int64_t)o << 28; t6 -= t; o = (int32_t)(t7 >> 28); t8 += o; t = (int64_t)o << 28; t7 -= t; o = (int32_t)(t8 >> 28); t9 += o; t = (int64_t)o << 28; t8 -= t; o = (int32_t)(t9 >> 28); t10 += o; t = (int64_t)o << 28; t9 -= t; o = (int32_t)(t10 >> 28); t11 += o; t = (int64_t)o << 28; t10 -= t; o = (int32_t)(t11 >> 28); t12 += o; t = (int64_t)o << 28; t11 -= t; o = (int32_t)(t12 >> 28); t13 += o; t = (int64_t)o << 28; t12 -= t; o = (int32_t)(t13 >> 28); t14 += o; t = (int64_t)o << 28; t13 -= t; o = (int32_t)(t14 >> 28); t15 += o; t = (int64_t)o << 28; t14 -= t; o = (int32_t)(t15 >> 28); t0 += o; t8 += o; t = (int64_t)o << 28; t15 -= t; /* Store */ r[0] = (int32_t)t0; r[1] = (int32_t)t1; r[2] = (int32_t)t2; r[3] = (int32_t)t3; r[4] = (int32_t)t4; r[5] = (int32_t)t5; r[6] = (int32_t)t6; r[7] = (int32_t)t7; r[8] = (int32_t)t8; r[9] = (int32_t)t9; r[10] = (int32_t)t10; r[11] = (int32_t)t11; r[12] = (int32_t)t12; r[13] = (int32_t)t13; r[14] = (int32_t)t14; r[15] = (int32_t)t15; } /* Mulitply two field elements. r = a * b * * r [in] Field element to hold result. * a [in] Field element to multiply. * b [in] Field element to multiply. */ static WC_INLINE void fe448_mul_8(int32_t* r, const int32_t* a, const int32_t* b) { int64_t t; int64_t t0 = (int64_t)a[ 0] * b[ 0]; int64_t t1 = (int64_t)a[ 0] * b[ 1]; int64_t t101 = (int64_t)a[ 1] * b[ 0]; int64_t t2 = (int64_t)a[ 0] * b[ 2]; int64_t t102 = (int64_t)a[ 1] * b[ 1]; int64_t t202 = (int64_t)a[ 2] * b[ 0]; int64_t t3 = (int64_t)a[ 0] * b[ 3]; int64_t t103 = (int64_t)a[ 1] * b[ 2]; int64_t t203 = (int64_t)a[ 2] * b[ 1]; int64_t t303 = (int64_t)a[ 3] * b[ 0]; int64_t t4 = (int64_t)a[ 0] * b[ 4]; int64_t t104 = (int64_t)a[ 1] * b[ 3]; int64_t t204 = (int64_t)a[ 2] * b[ 2]; int64_t t304 = (int64_t)a[ 3] * b[ 1]; int64_t t404 = (int64_t)a[ 4] * b[ 0]; int64_t t5 = (int64_t)a[ 0] * b[ 5]; int64_t t105 = (int64_t)a[ 1] * b[ 4]; int64_t t205 = (int64_t)a[ 2] * b[ 3]; int64_t t305 = (int64_t)a[ 3] * b[ 2]; int64_t t405 = (int64_t)a[ 4] * b[ 1]; int64_t t505 = (int64_t)a[ 5] * b[ 0]; int64_t t6 = (int64_t)a[ 0] * b[ 6]; int64_t t106 = (int64_t)a[ 1] * b[ 5]; int64_t t206 = (int64_t)a[ 2] * b[ 4]; int64_t t306 = (int64_t)a[ 3] * b[ 3]; int64_t t406 = (int64_t)a[ 4] * b[ 2]; int64_t t506 = (int64_t)a[ 5] * b[ 1]; int64_t t606 = (int64_t)a[ 6] * b[ 0]; int64_t t7 = (int64_t)a[ 0] * b[ 7]; int64_t t107 = (int64_t)a[ 1] * b[ 6]; int64_t t207 = (int64_t)a[ 2] * b[ 5]; int64_t t307 = (int64_t)a[ 3] * b[ 4]; int64_t t407 = (int64_t)a[ 4] * b[ 3]; int64_t t507 = (int64_t)a[ 5] * b[ 2]; int64_t t607 = (int64_t)a[ 6] * b[ 1]; int64_t t707 = (int64_t)a[ 7] * b[ 0]; int64_t t8 = (int64_t)a[ 1] * b[ 7]; int64_t t108 = (int64_t)a[ 2] * b[ 6]; int64_t t208 = (int64_t)a[ 3] * b[ 5]; int64_t t308 = (int64_t)a[ 4] * b[ 4]; int64_t t408 = (int64_t)a[ 5] * b[ 3]; int64_t t508 = (int64_t)a[ 6] * b[ 2]; int64_t t608 = (int64_t)a[ 7] * b[ 1]; int64_t t9 = (int64_t)a[ 2] * b[ 7]; int64_t t109 = (int64_t)a[ 3] * b[ 6]; int64_t t209 = (int64_t)a[ 4] * b[ 5]; int64_t t309 = (int64_t)a[ 5] * b[ 4]; int64_t t409 = (int64_t)a[ 6] * b[ 3]; int64_t t509 = (int64_t)a[ 7] * b[ 2]; int64_t t10 = (int64_t)a[ 3] * b[ 7]; int64_t t110 = (int64_t)a[ 4] * b[ 6]; int64_t t210 = (int64_t)a[ 5] * b[ 5]; int64_t t310 = (int64_t)a[ 6] * b[ 4]; int64_t t410 = (int64_t)a[ 7] * b[ 3]; int64_t t11 = (int64_t)a[ 4] * b[ 7]; int64_t t111 = (int64_t)a[ 5] * b[ 6]; int64_t t211 = (int64_t)a[ 6] * b[ 5]; int64_t t311 = (int64_t)a[ 7] * b[ 4]; int64_t t12 = (int64_t)a[ 5] * b[ 7]; int64_t t112 = (int64_t)a[ 6] * b[ 6]; int64_t t212 = (int64_t)a[ 7] * b[ 5]; int64_t t13 = (int64_t)a[ 6] * b[ 7]; int64_t t113 = (int64_t)a[ 7] * b[ 6]; int64_t t14 = (int64_t)a[ 7] * b[ 7]; t1 += t101; t2 += t102; t2 += t202; t3 += t103; t3 += t203; t3 += t303; t4 += t104; t4 += t204; t4 += t304; t4 += t404; t5 += t105; t5 += t205; t5 += t305; t5 += t405; t5 += t505; t6 += t106; t6 += t206; t6 += t306; t6 += t406; t6 += t506; t6 += t606; t7 += t107; t7 += t207; t7 += t307; t7 += t407; t7 += t507; t7 += t607; t7 += t707; t8 += t108; t8 += t208; t8 += t308; t8 += t408; t8 += t508; t8 += t608; t9 += t109; t9 += t209; t9 += t309; t9 += t409; t9 += t509; t10 += t110; t10 += t210; t10 += t310; t10 += t410; t11 += t111; t11 += t211; t11 += t311; t12 += t112; t12 += t212; t13 += t113; int64_t o = t14 >> 28; int64_t t15 = o; t14 -= o << 28; o = (t0 >> 28); t1 += o; t = o << 28; t0 -= t; o = (t1 >> 28); t2 += o; t = o << 28; t1 -= t; o = (t2 >> 28); t3 += o; t = o << 28; t2 -= t; o = (t3 >> 28); t4 += o; t = o << 28; t3 -= t; o = (t4 >> 28); t5 += o; t = o << 28; t4 -= t; o = (t5 >> 28); t6 += o; t = o << 28; t5 -= t; o = (t6 >> 28); t7 += o; t = o << 28; t6 -= t; o = (t7 >> 28); t8 += o; t = o << 28; t7 -= t; o = (t8 >> 28); t9 += o; t = o << 28; t8 -= t; o = (t9 >> 28); t10 += o; t = o << 28; t9 -= t; o = (t10 >> 28); t11 += o; t = o << 28; t10 -= t; o = (t11 >> 28); t12 += o; t = o << 28; t11 -= t; o = (t12 >> 28); t13 += o; t = o << 28; t12 -= t; o = (t13 >> 28); t14 += o; t = o << 28; t13 -= t; o = (t14 >> 28); t15 += o; t = o << 28; t14 -= t; o = (t15 >> 28); t0 += o; t8 += o; t = o << 28; t15 -= t; /* Store */ r[0] = (int32_t)t0; r[1] = (int32_t)t1; r[2] = (int32_t)t2; r[3] = (int32_t)t3; r[4] = (int32_t)t4; r[5] = (int32_t)t5; r[6] = (int32_t)t6; r[7] = (int32_t)t7; r[8] = (int32_t)t8; r[9] = (int32_t)t9; r[10] = (int32_t)t10; r[11] = (int32_t)t11; r[12] = (int32_t)t12; r[13] = (int32_t)t13; r[14] = (int32_t)t14; r[15] = (int32_t)t15; } /* Mulitply two field elements. r = (a * b) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to multiply. * b [in] Field element to multiply. */ void fe448_mul(int32_t* r, const int32_t* a, const int32_t* b) { int32_t r0[16]; int32_t r1[16]; int32_t* a1 = r1; int32_t b1[8]; int32_t r2[16]; a1[0] = a[0] + a[8]; a1[1] = a[1] + a[9]; a1[2] = a[2] + a[10]; a1[3] = a[3] + a[11]; a1[4] = a[4] + a[12]; a1[5] = a[5] + a[13]; a1[6] = a[6] + a[14]; a1[7] = a[7] + a[15]; b1[0] = b[0] + b[8]; b1[1] = b[1] + b[9]; b1[2] = b[2] + b[10]; b1[3] = b[3] + b[11]; b1[4] = b[4] + b[12]; b1[5] = b[5] + b[13]; b1[6] = b[6] + b[14]; b1[7] = b[7] + b[15]; fe448_mul_8(r2, a + 8, b + 8); fe448_mul_8(r0, a, b); fe448_mul_8(r1, a1, b1); r[ 0] = r0[ 0] + r2[ 0] + r1[ 8] - r0[ 8]; r[ 1] = r0[ 1] + r2[ 1] + r1[ 9] - r0[ 9]; r[ 2] = r0[ 2] + r2[ 2] + r1[10] - r0[10]; r[ 3] = r0[ 3] + r2[ 3] + r1[11] - r0[11]; r[ 4] = r0[ 4] + r2[ 4] + r1[12] - r0[12]; r[ 5] = r0[ 5] + r2[ 5] + r1[13] - r0[13]; r[ 6] = r0[ 6] + r2[ 6] + r1[14] - r0[14]; r[ 7] = r0[ 7] + r2[ 7] + r1[15] - r0[15]; r[ 8] = r2[ 8] + r1[ 0] - r0[ 0] + r1[ 8]; r[ 9] = r2[ 9] + r1[ 1] - r0[ 1] + r1[ 9]; r[10] = r2[10] + r1[ 2] - r0[ 2] + r1[10]; r[11] = r2[11] + r1[ 3] - r0[ 3] + r1[11]; r[12] = r2[12] + r1[ 4] - r0[ 4] + r1[12]; r[13] = r2[13] + r1[ 5] - r0[ 5] + r1[13]; r[14] = r2[14] + r1[ 6] - r0[ 6] + r1[14]; r[15] = r2[15] + r1[ 7] - r0[ 7] + r1[15]; } /* Square a field element. r = a * a * * r [in] Field element to hold result. * a [in] Field element to square. */ static WC_INLINE void fe448_sqr_8(int32_t* r, const int32_t* a) { int64_t t; int64_t t0 = (int64_t)a[ 0] * a[ 0]; int64_t t1 = 2 * (int64_t)a[ 0] * a[ 1]; int64_t t2 = 2 * (int64_t)a[ 0] * a[ 2]; int64_t t102 = (int64_t)a[ 1] * a[ 1]; int64_t t3 = 2 * (int64_t)a[ 0] * a[ 3]; int64_t t103 = 2 * (int64_t)a[ 1] * a[ 2]; int64_t t4 = 2 * (int64_t)a[ 0] * a[ 4]; int64_t t104 = 2 * (int64_t)a[ 1] * a[ 3]; int64_t t204 = (int64_t)a[ 2] * a[ 2]; int64_t t5 = 2 * (int64_t)a[ 0] * a[ 5]; int64_t t105 = 2 * (int64_t)a[ 1] * a[ 4]; int64_t t205 = 2 * (int64_t)a[ 2] * a[ 3]; int64_t t6 = 2 * (int64_t)a[ 0] * a[ 6]; int64_t t106 = 2 * (int64_t)a[ 1] * a[ 5]; int64_t t206 = 2 * (int64_t)a[ 2] * a[ 4]; int64_t t306 = (int64_t)a[ 3] * a[ 3]; int64_t t7 = 2 * (int64_t)a[ 0] * a[ 7]; int64_t t107 = 2 * (int64_t)a[ 1] * a[ 6]; int64_t t207 = 2 * (int64_t)a[ 2] * a[ 5]; int64_t t307 = 2 * (int64_t)a[ 3] * a[ 4]; int64_t t8 = 2 * (int64_t)a[ 1] * a[ 7]; int64_t t108 = 2 * (int64_t)a[ 2] * a[ 6]; int64_t t208 = 2 * (int64_t)a[ 3] * a[ 5]; int64_t t308 = (int64_t)a[ 4] * a[ 4]; int64_t t9 = 2 * (int64_t)a[ 2] * a[ 7]; int64_t t109 = 2 * (int64_t)a[ 3] * a[ 6]; int64_t t209 = 2 * (int64_t)a[ 4] * a[ 5]; int64_t t10 = 2 * (int64_t)a[ 3] * a[ 7]; int64_t t110 = 2 * (int64_t)a[ 4] * a[ 6]; int64_t t210 = (int64_t)a[ 5] * a[ 5]; int64_t t11 = 2 * (int64_t)a[ 4] * a[ 7]; int64_t t111 = 2 * (int64_t)a[ 5] * a[ 6]; int64_t t12 = 2 * (int64_t)a[ 5] * a[ 7]; int64_t t112 = (int64_t)a[ 6] * a[ 6]; int64_t t13 = 2 * (int64_t)a[ 6] * a[ 7]; int64_t t14 = (int64_t)a[ 7] * a[ 7]; t2 += t102; t3 += t103; t4 += t104; t4 += t204; t5 += t105; t5 += t205; t6 += t106; t6 += t206; t6 += t306; t7 += t107; t7 += t207; t7 += t307; t8 += t108; t8 += t208; t8 += t308; t9 += t109; t9 += t209; t10 += t110; t10 += t210; t11 += t111; t12 += t112; int64_t o = t14 >> 28; int64_t t15 = o; t14 -= o << 28; o = (t0 >> 28); t1 += o; t = o << 28; t0 -= t; o = (t1 >> 28); t2 += o; t = o << 28; t1 -= t; o = (t2 >> 28); t3 += o; t = o << 28; t2 -= t; o = (t3 >> 28); t4 += o; t = o << 28; t3 -= t; o = (t4 >> 28); t5 += o; t = o << 28; t4 -= t; o = (t5 >> 28); t6 += o; t = o << 28; t5 -= t; o = (t6 >> 28); t7 += o; t = o << 28; t6 -= t; o = (t7 >> 28); t8 += o; t = o << 28; t7 -= t; o = (t8 >> 28); t9 += o; t = o << 28; t8 -= t; o = (t9 >> 28); t10 += o; t = o << 28; t9 -= t; o = (t10 >> 28); t11 += o; t = o << 28; t10 -= t; o = (t11 >> 28); t12 += o; t = o << 28; t11 -= t; o = (t12 >> 28); t13 += o; t = o << 28; t12 -= t; o = (t13 >> 28); t14 += o; t = o << 28; t13 -= t; o = (t14 >> 28); t15 += o; t = o << 28; t14 -= t; o = (t15 >> 28); t0 += o; t8 += o; t = o << 28; t15 -= t; /* Store */ r[0] = (int32_t)t0; r[1] = (int32_t)t1; r[2] = (int32_t)t2; r[3] = (int32_t)t3; r[4] = (int32_t)t4; r[5] = (int32_t)t5; r[6] = (int32_t)t6; r[7] = (int32_t)t7; r[8] = (int32_t)t8; r[9] = (int32_t)t9; r[10] = (int32_t)t10; r[11] = (int32_t)t11; r[12] = (int32_t)t12; r[13] = (int32_t)t13; r[14] = (int32_t)t14; r[15] = (int32_t)t15; } /* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1) * * r [in] Field element to hold result. * a [in] Field element to square. */ void fe448_sqr(int32_t* r, const int32_t* a) { int32_t r0[16]; int32_t r1[16]; int32_t* a1 = r1; int32_t r2[16]; a1[0] = a[0] + a[8]; a1[1] = a[1] + a[9]; a1[2] = a[2] + a[10]; a1[3] = a[3] + a[11]; a1[4] = a[4] + a[12]; a1[5] = a[5] + a[13]; a1[6] = a[6] + a[14]; a1[7] = a[7] + a[15]; fe448_sqr_8(r2, a + 8); fe448_sqr_8(r0, a); fe448_sqr_8(r1, a1); r[ 0] = r0[ 0] + r2[ 0] + r1[ 8] - r0[ 8]; r[ 1] = r0[ 1] + r2[ 1] + r1[ 9] - r0[ 9]; r[ 2] = r0[ 2] + r2[ 2] + r1[10] - r0[10]; r[ 3] = r0[ 3] + r2[ 3] + r1[11] - r0[11]; r[ 4] = r0[ 4] + r2[ 4] + r1[12] - r0[12]; r[ 5] = r0[ 5] + r2[ 5] + r1[13] - r0[13]; r[ 6] = r0[ 6] + r2[ 6] + r1[14] - r0[14]; r[ 7] = r0[ 7] + r2[ 7] + r1[15] - r0[15]; r[ 8] = r2[ 8] + r1[ 0] - r0[ 0] + r1[ 8]; r[ 9] = r2[ 9] + r1[ 1] - r0[ 1] + r1[ 9]; r[10] = r2[10] + r1[ 2] - r0[ 2] + r1[10]; r[11] = r2[11] + r1[ 3] - r0[ 3] + r1[11]; r[12] = r2[12] + r1[ 4] - r0[ 4] + r1[12]; r[13] = r2[13] + r1[ 5] - r0[ 5] + r1[13]; r[14] = r2[14] + r1[ 6] - r0[ 6] + r1[14]; r[15] = r2[15] + r1[ 7] - r0[ 7] + r1[15]; } /* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1 * Constant time implementation - using Fermat's little theorem: * a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a * For curve448: p - 2 = 2^448 - 2^224 - 3 * * r [in] Field element to hold result. * a [in] Field element to invert. */ void fe448_invert(int32_t* r, const int32_t* a) { int32_t t1[16]; int32_t t2[16]; int32_t t3[16]; int32_t t4[16]; int i; fe448_sqr(t1, a); /* t1 = 2 */ fe448_mul(t1, t1, a); /* t1 = 3 */ fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2); /* t2 = c */ fe448_mul(t3, t2, a); /* t3 = d */ fe448_mul(t1, t2, t1); /* t1 = f */ fe448_sqr(t2, t1); /* t2 = 1e */ fe448_mul(t4, t2, a); /* t4 = 1f */ fe448_sqr(t2, t4); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 3e0 */ fe448_mul(t1, t2, t4); /* t1 = 3ff */ fe448_sqr(t2, t1); for (i = 1; i < 10; ++i) fe448_sqr(t2, t2); /* t2 = ffc00 */ fe448_mul(t1, t2, t1); /* t1 = fffff */ fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 1ffffe0 */ fe448_mul(t1, t2, t4); /* t1 = 1ffffff */ fe448_sqr(t2, t1); for (i = 1; i < 25; ++i) fe448_sqr(t2, t2); /* t2 = 3fffffe000000 */ fe448_mul(t1, t2, t1); /* t1 = 3ffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2); /* t2 = 7fffffffffffe0 */ fe448_mul(t1, t2, t4); /* t1 = 7fffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 55; ++i) fe448_sqr(t2, t2); /* t2 = 3fffffffffffff80000000000000 */ fe448_mul(t1, t2, t1); /* t1 = 3fffffffffffffffffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 110; ++i) fe448_sqr(t2, t2); /* t2 = fffffffffffffffffffffffffffc000000000000000000000000000 */ fe448_mul(t1, t2, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2); /* t2 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff0 */ fe448_mul(t3, t3, t2); /* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffd */ fe448_mul(t1, t3, a); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */ fe448_sqr(t1, t1); for (i = 1; i < 224; ++i) fe448_sqr(t1, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe00000000000000000000000000000000000000000000000000000000 */ fe448_mul(r, t3, t1); /* r = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffffffffffffffffffffffffffffffffffffffffffffffffffffd */ } /* Scalar multiply the point by a number. r = n.a * Uses Montogmery ladder and only requires the x-ordinate. * * r [in] Field element to hold result. * n [in] Scalar as an array of bytes. * a [in] Point to multiply - x-ordinate only. */ int curve448(byte* r, const byte* n, const byte* a) { int32_t x1[16]; int32_t x2[16]; int32_t z2[16]; int32_t x3[16]; int32_t z3[16]; int32_t t0[16]; int32_t t1[16]; int i; unsigned int swap; unsigned int b; fe448_from_bytes(x1, a); fe448_1(x2); fe448_0(z2); fe448_copy(x3, x1); fe448_1(z3); swap = 0; for (i = 447; i >= 0; --i) { b = (n[i >> 3] >> (i & 7)) & 1; swap ^= b; fe448_cswap(x2, x3, swap); fe448_cswap(z2, z3, swap); swap = b; /* Montgomery Ladder - double and add */ fe448_add(t0, x2, z2); fe448_reduce(t0); fe448_add(t1, x3, z3); fe448_reduce(t1); fe448_sub(x2, x2, z2); fe448_sub(x3, x3, z3); fe448_mul(t1, t1, x2); fe448_mul(z3, x3, t0); fe448_sqr(t0, t0); fe448_sqr(x2, x2); fe448_add(x3, z3, t1); fe448_reduce(x3); fe448_sqr(x3, x3); fe448_sub(z3, z3, t1); fe448_sqr(z3, z3); fe448_mul(z3, z3, x1); fe448_sub(t1, t0, x2); fe448_mul(x2, t0, x2); fe448_mul39081(z2, t1); fe448_add(z2, t0, z2); fe448_mul(z2, z2, t1); } /* Last two bits are 0 - no final swap check required. */ fe448_invert(z2, z2); fe448_mul(x2, x2, z2); fe448_to_bytes(r, x2); return 0; } #ifdef HAVE_ED448 /* Check whether field element is not 0. * Must convert to a normalized form before checking. * * a [in] Field element. * returns 0 when zero, and any other value otherwise. */ int fe448_isnonzero(const int32_t* a) { uint8_t b[56]; int i; uint8_t c = 0; fe448_to_bytes(b, a); for (i = 0; i < 56; i++) c |= b[i]; return c; } /* Check whether field element is negative. * Must convert to a normalized form before checking. * * a [in] Field element. * returns 1 when negative, and 0 otherwise. */ int fe448_isnegative(const int32_t* a) { uint8_t b[56]; fe448_to_bytes(b, a); return b[0] & 1; } /* Negates the field element. r = -a * * r [in] Field element to hold result. * a [in] Field element. */ void fe448_neg(int32_t* r, const int32_t* a) { r[0] = -a[0]; r[1] = -a[1]; r[2] = -a[2]; r[3] = -a[3]; r[4] = -a[4]; r[5] = -a[5]; r[6] = -a[6]; r[7] = -a[7]; r[8] = -a[8]; r[9] = -a[9]; r[10] = -a[10]; r[11] = -a[11]; r[12] = -a[12]; r[13] = -a[13]; r[14] = -a[14]; r[15] = -a[15]; } /* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1 * Used for calcualting y-ordinate from x-ordinate for Ed448. * * r [in] Field element to hold result. * a [in] Field element to exponentiate. */ void fe448_pow_2_446_222_1(int32_t* r, const int32_t* a) { int32_t t1[16]; int32_t t2[16]; int32_t t3[16]; int32_t t4[16]; int32_t t5[16]; int i; fe448_sqr(t3, a); /* t3 = 2 */ fe448_mul(t1, t3, a); /* t1 = 3 */ fe448_sqr(t5, t1); /* t5 = 6 */ fe448_mul(t5, t5, a); /* t5 = 7 */ fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2); /* t2 = c */ fe448_mul(t3, t2, t3); /* t3 = e */ fe448_mul(t1, t2, t1); /* t1 = f */ fe448_sqr(t2, t1); for (i = 1; i < 3; ++i) fe448_sqr(t2, t2); /* t2 = 78 */ fe448_mul(t5, t2, t5); /* t5 = 7f */ fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2); /* t2 = f0 */ fe448_mul(t1, t2, t1); /* t1 = ff */ fe448_mul(t3, t3, t2); /* t3 = fe */ fe448_sqr(t2, t1); for (i = 1; i < 7; ++i) fe448_sqr(t2, t2); /* t2 = 7f80 */ fe448_mul(t5, t2, t5); /* t5 = 7fff */ fe448_sqr(t2, t1); for (i = 1; i < 8; ++i) fe448_sqr(t2, t2); /* t2 = ff00 */ fe448_mul(t1, t2, t1); /* t1 = ffff */ fe448_mul(t3, t3, t2); /* t3 = fffe */ fe448_sqr(t2, t5); for (i = 1; i < 15; ++i) fe448_sqr(t2, t2); /* t2 = 3fff8000 */ fe448_mul(t5, t2, t5); /* t5 = 3fffffff */ fe448_sqr(t2, t1); for (i = 1; i < 16; ++i) fe448_sqr(t2, t2); /* t2 = ffff0000 */ fe448_mul(t1, t2, t1); /* t1 = ffffffff */ fe448_mul(t3, t3, t2); /* t3 = fffffffe */ fe448_sqr(t2, t1); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2); /* t2 = ffffffff00000000 */ fe448_mul(t2, t2, t1); /* t2 = ffffffffffffffff */ fe448_sqr(t1, t2); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1); /* t1 = ffffffffffffffff0000000000000000 */ fe448_mul(t1, t1, t2); /* t1 = ffffffffffffffffffffffffffffffff */ fe448_sqr(t1, t1); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1); /* t1 = ffffffffffffffffffffffffffffffff0000000000000000 */ fe448_mul(t4, t1, t2); /* t4 = ffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t2, t4); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2); /* t2 = ffffffffffffffffffffffffffffffffffffffffffffffff00000000 */ fe448_mul(t3, t3, t2); /* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */ fe448_sqr(t1, t3); for (i = 1; i < 192; ++i) fe448_sqr(t1, t1); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe000000000000000000000000000000000000000000000000 */ fe448_mul(t1, t1, t4); /* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffff */ fe448_sqr(t1, t1); for (i = 1; i < 30; ++i) fe448_sqr(t1, t1); /* t1 = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffc0000000 */ fe448_mul(r, t5, t1); /* r = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffffffffff */ } /* Constant time, conditional move of b into a. * a is not changed if the condition is 0. * * a A field element. * b A field element. * c If 1 then copy and if 0 then don't copy. */ void fe448_cmov(int32_t* a, const int32_t* b, int c) { int32_t m = -(int32_t)c; int32_t t0 = m & (a[0] ^ b[0]); int32_t t1 = m & (a[1] ^ b[1]); int32_t t2 = m & (a[2] ^ b[2]); int32_t t3 = m & (a[3] ^ b[3]); int32_t t4 = m & (a[4] ^ b[4]); int32_t t5 = m & (a[5] ^ b[5]); int32_t t6 = m & (a[6] ^ b[6]); int32_t t7 = m & (a[7] ^ b[7]); int32_t t8 = m & (a[8] ^ b[8]); int32_t t9 = m & (a[9] ^ b[9]); int32_t t10 = m & (a[10] ^ b[10]); int32_t t11 = m & (a[11] ^ b[11]); int32_t t12 = m & (a[12] ^ b[12]); int32_t t13 = m & (a[13] ^ b[13]); int32_t t14 = m & (a[14] ^ b[14]); int32_t t15 = m & (a[15] ^ b[15]); a[0] ^= t0; a[1] ^= t1; a[2] ^= t2; a[3] ^= t3; a[4] ^= t4; a[5] ^= t5; a[6] ^= t6; a[7] ^= t7; a[8] ^= t8; a[9] ^= t9; a[10] ^= t10; a[11] ^= t11; a[12] ^= t12; a[13] ^= t13; a[14] ^= t14; a[15] ^= t15; } #endif /* HAVE_ED448 */ #endif #endif /* HAVE_CURVE448 || HAVE_ED448 */