1 /* Copyright (c) 2015, Google Inc. 2 * 3 * Permission to use, copy, modify, and/or distribute this software for any 4 * purpose with or without fee is hereby granted, provided that the above 5 * copyright notice and this permission notice appear in all copies. 6 * 7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY 10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION 12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN 13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ 14 15 /* A 64-bit implementation of the NIST P-224 elliptic curve point multiplication 16 * 17 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation 18 * and Adam Langley's public domain 64-bit C implementation of curve25519. */ 19 20 #include <openssl/base.h> 21 22 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ 23 !defined(OPENSSL_SMALL) 24 25 #include <openssl/bn.h> 26 #include <openssl/ec.h> 27 #include <openssl/err.h> 28 #include <openssl/mem.h> 29 #include <openssl/obj.h> 30 31 #include <string.h> 32 33 #include "internal.h" 34 35 36 typedef uint8_t u8; 37 typedef uint64_t u64; 38 typedef int64_t s64; 39 40 /* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 41 * using 64-bit coefficients called 'limbs', and sometimes (for multiplication 42 * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 43 * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-limb 44 * representation is an 'felem'; a 7-widelimb representation is a 'widefelem'. 45 * Even within felems, bits of adjacent limbs overlap, and we don't always 46 * reduce the representations: we ensure that inputs to each felem 47 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, and 48 * fit into a 128-bit word without overflow. The coefficients are then again 49 * partially reduced to obtain an felem satisfying a_i < 2^57. We only reduce 50 * to the unique minimal representation at the end of the computation. */ 51 52 typedef uint64_t limb; 53 typedef __uint128_t widelimb; 54 55 typedef limb felem[4]; 56 typedef widelimb widefelem[7]; 57 58 /* Field element represented as a byte arrary. 28*8 = 224 bits is also the 59 * group order size for the elliptic curve, and we also use this type for 60 * scalars for point multiplication. */ 61 typedef u8 felem_bytearray[28]; 62 63 static const felem_bytearray nistp224_curve_params[5] = { 64 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */ 65 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 66 0x00, 0x00, 0x00, 0x00, 0x00, 0x01}, 67 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */ 68 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 69 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE}, 70 {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */ 71 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA, 0x27, 0x0B, 72 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4}, 73 {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */ 74 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22, 0x34, 0x32, 75 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21}, 76 {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */ 77 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64, 0x44, 0xd5, 78 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}}; 79 80 /* Precomputed multiples of the standard generator 81 * Points are given in coordinates (X, Y, Z) where Z normally is 1 82 * (0 for the point at infinity). 83 * For each field element, slice a_0 is word 0, etc. 84 * 85 * The table has 2 * 16 elements, starting with the following: 86 * index | bits | point 87 * ------+---------+------------------------------ 88 * 0 | 0 0 0 0 | 0G 89 * 1 | 0 0 0 1 | 1G 90 * 2 | 0 0 1 0 | 2^56G 91 * 3 | 0 0 1 1 | (2^56 + 1)G 92 * 4 | 0 1 0 0 | 2^112G 93 * 5 | 0 1 0 1 | (2^112 + 1)G 94 * 6 | 0 1 1 0 | (2^112 + 2^56)G 95 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G 96 * 8 | 1 0 0 0 | 2^168G 97 * 9 | 1 0 0 1 | (2^168 + 1)G 98 * 10 | 1 0 1 0 | (2^168 + 2^56)G 99 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G 100 * 12 | 1 1 0 0 | (2^168 + 2^112)G 101 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G 102 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G 103 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G 104 * followed by a copy of this with each element multiplied by 2^28. 105 * 106 * The reason for this is so that we can clock bits into four different 107 * locations when doing simple scalar multiplies against the base point, 108 * and then another four locations using the second 16 elements. */ 109 static const felem g_pre_comp[2][16][3] = { 110 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 111 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, 112 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, 113 {1, 0, 0, 0}}, 114 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, 115 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, 116 {1, 0, 0, 0}}, 117 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, 118 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, 119 {1, 0, 0, 0}}, 120 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, 121 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, 122 {1, 0, 0, 0}}, 123 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, 124 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, 125 {1, 0, 0, 0}}, 126 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, 127 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, 128 {1, 0, 0, 0}}, 129 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, 130 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, 131 {1, 0, 0, 0}}, 132 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, 133 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, 134 {1, 0, 0, 0}}, 135 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, 136 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, 137 {1, 0, 0, 0}}, 138 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, 139 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, 140 {1, 0, 0, 0}}, 141 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, 142 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, 143 {1, 0, 0, 0}}, 144 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, 145 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, 146 {1, 0, 0, 0}}, 147 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, 148 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, 149 {1, 0, 0, 0}}, 150 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, 151 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, 152 {1, 0, 0, 0}}, 153 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, 154 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, 155 {1, 0, 0, 0}}}, 156 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 157 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, 158 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, 159 {1, 0, 0, 0}}, 160 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, 161 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, 162 {1, 0, 0, 0}}, 163 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, 164 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, 165 {1, 0, 0, 0}}, 166 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, 167 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, 168 {1, 0, 0, 0}}, 169 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, 170 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, 171 {1, 0, 0, 0}}, 172 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, 173 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, 174 {1, 0, 0, 0}}, 175 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, 176 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, 177 {1, 0, 0, 0}}, 178 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, 179 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, 180 {1, 0, 0, 0}}, 181 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, 182 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, 183 {1, 0, 0, 0}}, 184 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, 185 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, 186 {1, 0, 0, 0}}, 187 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, 188 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, 189 {1, 0, 0, 0}}, 190 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, 191 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, 192 {1, 0, 0, 0}}, 193 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, 194 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, 195 {1, 0, 0, 0}}, 196 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, 197 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, 198 {1, 0, 0, 0}}, 199 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, 200 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, 201 {1, 0, 0, 0}}}}; 202 203 /* Helper functions to convert field elements to/from internal representation */ 204 static void bin28_to_felem(felem out, const u8 in[28]) { 205 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff; 206 out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff; 207 out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff; 208 out[3] = (*((const uint64_t *)(in + 20))) >> 8; 209 } 210 211 static void felem_to_bin28(u8 out[28], const felem in) { 212 unsigned i; 213 for (i = 0; i < 7; ++i) { 214 out[i] = in[0] >> (8 * i); 215 out[i + 7] = in[1] >> (8 * i); 216 out[i + 14] = in[2] >> (8 * i); 217 out[i + 21] = in[3] >> (8 * i); 218 } 219 } 220 221 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */ 222 static void flip_endian(u8 *out, const u8 *in, unsigned len) { 223 unsigned i; 224 for (i = 0; i < len; ++i) { 225 out[i] = in[len - 1 - i]; 226 } 227 } 228 229 /* From OpenSSL BIGNUM to internal representation */ 230 static int BN_to_felem(felem out, const BIGNUM *bn) { 231 /* BN_bn2bin eats leading zeroes */ 232 felem_bytearray b_out; 233 memset(b_out, 0, sizeof(b_out)); 234 unsigned num_bytes = BN_num_bytes(bn); 235 if (num_bytes > sizeof(b_out) || 236 BN_is_negative(bn)) { 237 OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); 238 return 0; 239 } 240 241 felem_bytearray b_in; 242 num_bytes = BN_bn2bin(bn, b_in); 243 flip_endian(b_out, b_in, num_bytes); 244 bin28_to_felem(out, b_out); 245 return 1; 246 } 247 248 /* From internal representation to OpenSSL BIGNUM */ 249 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) { 250 felem_bytearray b_in, b_out; 251 felem_to_bin28(b_in, in); 252 flip_endian(b_out, b_in, sizeof(b_out)); 253 return BN_bin2bn(b_out, sizeof(b_out), out); 254 } 255 256 /* Field operations, using the internal representation of field elements. 257 * NB! These operations are specific to our point multiplication and cannot be 258 * expected to be correct in general - e.g., multiplication with a large scalar 259 * will cause an overflow. */ 260 261 static void felem_one(felem out) { 262 out[0] = 1; 263 out[1] = 0; 264 out[2] = 0; 265 out[3] = 0; 266 } 267 268 static void felem_assign(felem out, const felem in) { 269 out[0] = in[0]; 270 out[1] = in[1]; 271 out[2] = in[2]; 272 out[3] = in[3]; 273 } 274 275 /* Sum two field elements: out += in */ 276 static void felem_sum(felem out, const felem in) { 277 out[0] += in[0]; 278 out[1] += in[1]; 279 out[2] += in[2]; 280 out[3] += in[3]; 281 } 282 283 /* Get negative value: out = -in */ 284 /* Assumes in[i] < 2^57 */ 285 static void felem_neg(felem out, const felem in) { 286 static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); 287 static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); 288 static const limb two58m42m2 = 289 (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); 290 291 /* Set to 0 mod 2^224-2^96+1 to ensure out > in */ 292 out[0] = two58p2 - in[0]; 293 out[1] = two58m42m2 - in[1]; 294 out[2] = two58m2 - in[2]; 295 out[3] = two58m2 - in[3]; 296 } 297 298 /* Subtract field elements: out -= in */ 299 /* Assumes in[i] < 2^57 */ 300 static void felem_diff(felem out, const felem in) { 301 static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); 302 static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); 303 static const limb two58m42m2 = 304 (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); 305 306 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 307 out[0] += two58p2; 308 out[1] += two58m42m2; 309 out[2] += two58m2; 310 out[3] += two58m2; 311 312 out[0] -= in[0]; 313 out[1] -= in[1]; 314 out[2] -= in[2]; 315 out[3] -= in[3]; 316 } 317 318 /* Subtract in unreduced 128-bit mode: out -= in */ 319 /* Assumes in[i] < 2^119 */ 320 static void widefelem_diff(widefelem out, const widefelem in) { 321 static const widelimb two120 = ((widelimb)1) << 120; 322 static const widelimb two120m64 = 323 (((widelimb)1) << 120) - (((widelimb)1) << 64); 324 static const widelimb two120m104m64 = 325 (((widelimb)1) << 120) - (((widelimb)1) << 104) - (((widelimb)1) << 64); 326 327 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 328 out[0] += two120; 329 out[1] += two120m64; 330 out[2] += two120m64; 331 out[3] += two120; 332 out[4] += two120m104m64; 333 out[5] += two120m64; 334 out[6] += two120m64; 335 336 out[0] -= in[0]; 337 out[1] -= in[1]; 338 out[2] -= in[2]; 339 out[3] -= in[3]; 340 out[4] -= in[4]; 341 out[5] -= in[5]; 342 out[6] -= in[6]; 343 } 344 345 /* Subtract in mixed mode: out128 -= in64 */ 346 /* in[i] < 2^63 */ 347 static void felem_diff_128_64(widefelem out, const felem in) { 348 static const widelimb two64p8 = (((widelimb)1) << 64) + (((widelimb)1) << 8); 349 static const widelimb two64m8 = (((widelimb)1) << 64) - (((widelimb)1) << 8); 350 static const widelimb two64m48m8 = 351 (((widelimb)1) << 64) - (((widelimb)1) << 48) - (((widelimb)1) << 8); 352 353 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 354 out[0] += two64p8; 355 out[1] += two64m48m8; 356 out[2] += two64m8; 357 out[3] += two64m8; 358 359 out[0] -= in[0]; 360 out[1] -= in[1]; 361 out[2] -= in[2]; 362 out[3] -= in[3]; 363 } 364 365 /* Multiply a field element by a scalar: out = out * scalar 366 * The scalars we actually use are small, so results fit without overflow */ 367 static void felem_scalar(felem out, const limb scalar) { 368 out[0] *= scalar; 369 out[1] *= scalar; 370 out[2] *= scalar; 371 out[3] *= scalar; 372 } 373 374 /* Multiply an unreduced field element by a scalar: out = out * scalar 375 * The scalars we actually use are small, so results fit without overflow */ 376 static void widefelem_scalar(widefelem out, const widelimb scalar) { 377 out[0] *= scalar; 378 out[1] *= scalar; 379 out[2] *= scalar; 380 out[3] *= scalar; 381 out[4] *= scalar; 382 out[5] *= scalar; 383 out[6] *= scalar; 384 } 385 386 /* Square a field element: out = in^2 */ 387 static void felem_square(widefelem out, const felem in) { 388 limb tmp0, tmp1, tmp2; 389 tmp0 = 2 * in[0]; 390 tmp1 = 2 * in[1]; 391 tmp2 = 2 * in[2]; 392 out[0] = ((widelimb)in[0]) * in[0]; 393 out[1] = ((widelimb)in[0]) * tmp1; 394 out[2] = ((widelimb)in[0]) * tmp2 + ((widelimb)in[1]) * in[1]; 395 out[3] = ((widelimb)in[3]) * tmp0 + ((widelimb)in[1]) * tmp2; 396 out[4] = ((widelimb)in[3]) * tmp1 + ((widelimb)in[2]) * in[2]; 397 out[5] = ((widelimb)in[3]) * tmp2; 398 out[6] = ((widelimb)in[3]) * in[3]; 399 } 400 401 /* Multiply two field elements: out = in1 * in2 */ 402 static void felem_mul(widefelem out, const felem in1, const felem in2) { 403 out[0] = ((widelimb)in1[0]) * in2[0]; 404 out[1] = ((widelimb)in1[0]) * in2[1] + ((widelimb)in1[1]) * in2[0]; 405 out[2] = ((widelimb)in1[0]) * in2[2] + ((widelimb)in1[1]) * in2[1] + 406 ((widelimb)in1[2]) * in2[0]; 407 out[3] = ((widelimb)in1[0]) * in2[3] + ((widelimb)in1[1]) * in2[2] + 408 ((widelimb)in1[2]) * in2[1] + ((widelimb)in1[3]) * in2[0]; 409 out[4] = ((widelimb)in1[1]) * in2[3] + ((widelimb)in1[2]) * in2[2] + 410 ((widelimb)in1[3]) * in2[1]; 411 out[5] = ((widelimb)in1[2]) * in2[3] + ((widelimb)in1[3]) * in2[2]; 412 out[6] = ((widelimb)in1[3]) * in2[3]; 413 } 414 415 /* Reduce seven 128-bit coefficients to four 64-bit coefficients. 416 * Requires in[i] < 2^126, 417 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */ 418 static void felem_reduce(felem out, const widefelem in) { 419 static const widelimb two127p15 = 420 (((widelimb)1) << 127) + (((widelimb)1) << 15); 421 static const widelimb two127m71 = 422 (((widelimb)1) << 127) - (((widelimb)1) << 71); 423 static const widelimb two127m71m55 = 424 (((widelimb)1) << 127) - (((widelimb)1) << 71) - (((widelimb)1) << 55); 425 widelimb output[5]; 426 427 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */ 428 output[0] = in[0] + two127p15; 429 output[1] = in[1] + two127m71m55; 430 output[2] = in[2] + two127m71; 431 output[3] = in[3]; 432 output[4] = in[4]; 433 434 /* Eliminate in[4], in[5], in[6] */ 435 output[4] += in[6] >> 16; 436 output[3] += (in[6] & 0xffff) << 40; 437 output[2] -= in[6]; 438 439 output[3] += in[5] >> 16; 440 output[2] += (in[5] & 0xffff) << 40; 441 output[1] -= in[5]; 442 443 output[2] += output[4] >> 16; 444 output[1] += (output[4] & 0xffff) << 40; 445 output[0] -= output[4]; 446 447 /* Carry 2 -> 3 -> 4 */ 448 output[3] += output[2] >> 56; 449 output[2] &= 0x00ffffffffffffff; 450 451 output[4] = output[3] >> 56; 452 output[3] &= 0x00ffffffffffffff; 453 454 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */ 455 456 /* Eliminate output[4] */ 457 output[2] += output[4] >> 16; 458 /* output[2] < 2^56 + 2^56 = 2^57 */ 459 output[1] += (output[4] & 0xffff) << 40; 460 output[0] -= output[4]; 461 462 /* Carry 0 -> 1 -> 2 -> 3 */ 463 output[1] += output[0] >> 56; 464 out[0] = output[0] & 0x00ffffffffffffff; 465 466 output[2] += output[1] >> 56; 467 /* output[2] < 2^57 + 2^72 */ 468 out[1] = output[1] & 0x00ffffffffffffff; 469 output[3] += output[2] >> 56; 470 /* output[3] <= 2^56 + 2^16 */ 471 out[2] = output[2] & 0x00ffffffffffffff; 472 473 /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, 474 * out[3] <= 2^56 + 2^16 (due to final carry), 475 * so out < 2*p */ 476 out[3] = output[3]; 477 } 478 479 static void felem_square_reduce(felem out, const felem in) { 480 widefelem tmp; 481 felem_square(tmp, in); 482 felem_reduce(out, tmp); 483 } 484 485 static void felem_mul_reduce(felem out, const felem in1, const felem in2) { 486 widefelem tmp; 487 felem_mul(tmp, in1, in2); 488 felem_reduce(out, tmp); 489 } 490 491 /* Reduce to unique minimal representation. 492 * Requires 0 <= in < 2*p (always call felem_reduce first) */ 493 static void felem_contract(felem out, const felem in) { 494 static const int64_t two56 = ((limb)1) << 56; 495 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */ 496 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */ 497 int64_t tmp[4], a; 498 tmp[0] = in[0]; 499 tmp[1] = in[1]; 500 tmp[2] = in[2]; 501 tmp[3] = in[3]; 502 /* Case 1: a = 1 iff in >= 2^224 */ 503 a = (in[3] >> 56); 504 tmp[0] -= a; 505 tmp[1] += a << 40; 506 tmp[3] &= 0x00ffffffffffffff; 507 /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and 508 * the lower part is non-zero */ 509 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | 510 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); 511 a &= 0x00ffffffffffffff; 512 /* turn a into an all-one mask (if a = 0) or an all-zero mask */ 513 a = (a - 1) >> 63; 514 /* subtract 2^224 - 2^96 + 1 if a is all-one */ 515 tmp[3] &= a ^ 0xffffffffffffffff; 516 tmp[2] &= a ^ 0xffffffffffffffff; 517 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; 518 tmp[0] -= 1 & a; 519 520 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must 521 * be non-zero, so we only need one step */ 522 a = tmp[0] >> 63; 523 tmp[0] += two56 & a; 524 tmp[1] -= 1 & a; 525 526 /* carry 1 -> 2 -> 3 */ 527 tmp[2] += tmp[1] >> 56; 528 tmp[1] &= 0x00ffffffffffffff; 529 530 tmp[3] += tmp[2] >> 56; 531 tmp[2] &= 0x00ffffffffffffff; 532 533 /* Now 0 <= out < p */ 534 out[0] = tmp[0]; 535 out[1] = tmp[1]; 536 out[2] = tmp[2]; 537 out[3] = tmp[3]; 538 } 539 540 /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field 541 * elements are reduced to in < 2^225, so we only need to check three cases: 0, 542 * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */ 543 static limb felem_is_zero(const felem in) { 544 limb zero = in[0] | in[1] | in[2] | in[3]; 545 zero = (((int64_t)(zero)-1) >> 63) & 1; 546 547 limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | 548 (in[2] ^ 0x00ffffffffffffff) | 549 (in[3] ^ 0x00ffffffffffffff); 550 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; 551 limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | 552 (in[2] ^ 0x00ffffffffffffff) | 553 (in[3] ^ 0x01ffffffffffffff); 554 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; 555 return (zero | two224m96p1 | two225m97p2); 556 } 557 558 static limb felem_is_zero_int(const felem in) { 559 return (int)(felem_is_zero(in) & ((limb)1)); 560 } 561 562 /* Invert a field element */ 563 /* Computation chain copied from djb's code */ 564 static void felem_inv(felem out, const felem in) { 565 felem ftmp, ftmp2, ftmp3, ftmp4; 566 widefelem tmp; 567 unsigned i; 568 569 felem_square(tmp, in); 570 felem_reduce(ftmp, tmp); /* 2 */ 571 felem_mul(tmp, in, ftmp); 572 felem_reduce(ftmp, tmp); /* 2^2 - 1 */ 573 felem_square(tmp, ftmp); 574 felem_reduce(ftmp, tmp); /* 2^3 - 2 */ 575 felem_mul(tmp, in, ftmp); 576 felem_reduce(ftmp, tmp); /* 2^3 - 1 */ 577 felem_square(tmp, ftmp); 578 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */ 579 felem_square(tmp, ftmp2); 580 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */ 581 felem_square(tmp, ftmp2); 582 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */ 583 felem_mul(tmp, ftmp2, ftmp); 584 felem_reduce(ftmp, tmp); /* 2^6 - 1 */ 585 felem_square(tmp, ftmp); 586 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */ 587 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */ 588 felem_square(tmp, ftmp2); 589 felem_reduce(ftmp2, tmp); 590 } 591 felem_mul(tmp, ftmp2, ftmp); 592 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */ 593 felem_square(tmp, ftmp2); 594 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */ 595 for (i = 0; i < 11; ++i) {/* 2^24 - 2^12 */ 596 felem_square(tmp, ftmp3); 597 felem_reduce(ftmp3, tmp); 598 } 599 felem_mul(tmp, ftmp3, ftmp2); 600 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */ 601 felem_square(tmp, ftmp2); 602 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */ 603 for (i = 0; i < 23; ++i) {/* 2^48 - 2^24 */ 604 felem_square(tmp, ftmp3); 605 felem_reduce(ftmp3, tmp); 606 } 607 felem_mul(tmp, ftmp3, ftmp2); 608 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */ 609 felem_square(tmp, ftmp3); 610 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */ 611 for (i = 0; i < 47; ++i) {/* 2^96 - 2^48 */ 612 felem_square(tmp, ftmp4); 613 felem_reduce(ftmp4, tmp); 614 } 615 felem_mul(tmp, ftmp3, ftmp4); 616 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */ 617 felem_square(tmp, ftmp3); 618 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */ 619 for (i = 0; i < 23; ++i) {/* 2^120 - 2^24 */ 620 felem_square(tmp, ftmp4); 621 felem_reduce(ftmp4, tmp); 622 } 623 felem_mul(tmp, ftmp2, ftmp4); 624 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */ 625 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */ 626 felem_square(tmp, ftmp2); 627 felem_reduce(ftmp2, tmp); 628 } 629 felem_mul(tmp, ftmp2, ftmp); 630 felem_reduce(ftmp, tmp); /* 2^126 - 1 */ 631 felem_square(tmp, ftmp); 632 felem_reduce(ftmp, tmp); /* 2^127 - 2 */ 633 felem_mul(tmp, ftmp, in); 634 felem_reduce(ftmp, tmp); /* 2^127 - 1 */ 635 for (i = 0; i < 97; ++i) {/* 2^224 - 2^97 */ 636 felem_square(tmp, ftmp); 637 felem_reduce(ftmp, tmp); 638 } 639 felem_mul(tmp, ftmp, ftmp3); 640 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */ 641 } 642 643 /* Copy in constant time: 644 * if icopy == 1, copy in to out, 645 * if icopy == 0, copy out to itself. */ 646 static void copy_conditional(felem out, const felem in, limb icopy) { 647 unsigned i; 648 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ 649 const limb copy = -icopy; 650 for (i = 0; i < 4; ++i) { 651 const limb tmp = copy & (in[i] ^ out[i]); 652 out[i] ^= tmp; 653 } 654 } 655 656 /* ELLIPTIC CURVE POINT OPERATIONS 657 * 658 * Points are represented in Jacobian projective coordinates: 659 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), 660 * or to the point at infinity if Z == 0. */ 661 662 /* Double an elliptic curve point: 663 * (X', Y', Z') = 2 * (X, Y, Z), where 664 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 665 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 666 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z 667 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, 668 * while x_out == y_in is not (maybe this works, but it's not tested). */ 669 static void point_double(felem x_out, felem y_out, felem z_out, 670 const felem x_in, const felem y_in, const felem z_in) { 671 widefelem tmp, tmp2; 672 felem delta, gamma, beta, alpha, ftmp, ftmp2; 673 674 felem_assign(ftmp, x_in); 675 felem_assign(ftmp2, x_in); 676 677 /* delta = z^2 */ 678 felem_square(tmp, z_in); 679 felem_reduce(delta, tmp); 680 681 /* gamma = y^2 */ 682 felem_square(tmp, y_in); 683 felem_reduce(gamma, tmp); 684 685 /* beta = x*gamma */ 686 felem_mul(tmp, x_in, gamma); 687 felem_reduce(beta, tmp); 688 689 /* alpha = 3*(x-delta)*(x+delta) */ 690 felem_diff(ftmp, delta); 691 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ 692 felem_sum(ftmp2, delta); 693 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ 694 felem_scalar(ftmp2, 3); 695 /* ftmp2[i] < 3 * 2^58 < 2^60 */ 696 felem_mul(tmp, ftmp, ftmp2); 697 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */ 698 felem_reduce(alpha, tmp); 699 700 /* x' = alpha^2 - 8*beta */ 701 felem_square(tmp, alpha); 702 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 703 felem_assign(ftmp, beta); 704 felem_scalar(ftmp, 8); 705 /* ftmp[i] < 8 * 2^57 = 2^60 */ 706 felem_diff_128_64(tmp, ftmp); 707 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 708 felem_reduce(x_out, tmp); 709 710 /* z' = (y + z)^2 - gamma - delta */ 711 felem_sum(delta, gamma); 712 /* delta[i] < 2^57 + 2^57 = 2^58 */ 713 felem_assign(ftmp, y_in); 714 felem_sum(ftmp, z_in); 715 /* ftmp[i] < 2^57 + 2^57 = 2^58 */ 716 felem_square(tmp, ftmp); 717 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */ 718 felem_diff_128_64(tmp, delta); 719 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */ 720 felem_reduce(z_out, tmp); 721 722 /* y' = alpha*(4*beta - x') - 8*gamma^2 */ 723 felem_scalar(beta, 4); 724 /* beta[i] < 4 * 2^57 = 2^59 */ 725 felem_diff(beta, x_out); 726 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */ 727 felem_mul(tmp, alpha, beta); 728 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */ 729 felem_square(tmp2, gamma); 730 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */ 731 widefelem_scalar(tmp2, 8); 732 /* tmp2[i] < 8 * 2^116 = 2^119 */ 733 widefelem_diff(tmp, tmp2); 734 /* tmp[i] < 2^119 + 2^120 < 2^121 */ 735 felem_reduce(y_out, tmp); 736 } 737 738 /* Add two elliptic curve points: 739 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where 740 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - 741 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 742 * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * 743 * X_1)^2 - X_3) - 744 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 745 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) 746 * 747 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */ 748 749 /* This function is not entirely constant-time: it includes a branch for 750 * checking whether the two input points are equal, (while not equal to the 751 * point at infinity). This case never happens during single point 752 * multiplication, so there is no timing leak for ECDH or ECDSA signing. */ 753 static void point_add(felem x3, felem y3, felem z3, const felem x1, 754 const felem y1, const felem z1, const int mixed, 755 const felem x2, const felem y2, const felem z2) { 756 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; 757 widefelem tmp, tmp2; 758 limb z1_is_zero, z2_is_zero, x_equal, y_equal; 759 760 if (!mixed) { 761 /* ftmp2 = z2^2 */ 762 felem_square(tmp, z2); 763 felem_reduce(ftmp2, tmp); 764 765 /* ftmp4 = z2^3 */ 766 felem_mul(tmp, ftmp2, z2); 767 felem_reduce(ftmp4, tmp); 768 769 /* ftmp4 = z2^3*y1 */ 770 felem_mul(tmp2, ftmp4, y1); 771 felem_reduce(ftmp4, tmp2); 772 773 /* ftmp2 = z2^2*x1 */ 774 felem_mul(tmp2, ftmp2, x1); 775 felem_reduce(ftmp2, tmp2); 776 } else { 777 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */ 778 779 /* ftmp4 = z2^3*y1 */ 780 felem_assign(ftmp4, y1); 781 782 /* ftmp2 = z2^2*x1 */ 783 felem_assign(ftmp2, x1); 784 } 785 786 /* ftmp = z1^2 */ 787 felem_square(tmp, z1); 788 felem_reduce(ftmp, tmp); 789 790 /* ftmp3 = z1^3 */ 791 felem_mul(tmp, ftmp, z1); 792 felem_reduce(ftmp3, tmp); 793 794 /* tmp = z1^3*y2 */ 795 felem_mul(tmp, ftmp3, y2); 796 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 797 798 /* ftmp3 = z1^3*y2 - z2^3*y1 */ 799 felem_diff_128_64(tmp, ftmp4); 800 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 801 felem_reduce(ftmp3, tmp); 802 803 /* tmp = z1^2*x2 */ 804 felem_mul(tmp, ftmp, x2); 805 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 806 807 /* ftmp = z1^2*x2 - z2^2*x1 */ 808 felem_diff_128_64(tmp, ftmp2); 809 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 810 felem_reduce(ftmp, tmp); 811 812 /* the formulae are incorrect if the points are equal 813 * so we check for this and do doubling if this happens */ 814 x_equal = felem_is_zero(ftmp); 815 y_equal = felem_is_zero(ftmp3); 816 z1_is_zero = felem_is_zero(z1); 817 z2_is_zero = felem_is_zero(z2); 818 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */ 819 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 820 point_double(x3, y3, z3, x1, y1, z1); 821 return; 822 } 823 824 /* ftmp5 = z1*z2 */ 825 if (!mixed) { 826 felem_mul(tmp, z1, z2); 827 felem_reduce(ftmp5, tmp); 828 } else { 829 /* special case z2 = 0 is handled later */ 830 felem_assign(ftmp5, z1); 831 } 832 833 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */ 834 felem_mul(tmp, ftmp, ftmp5); 835 felem_reduce(z_out, tmp); 836 837 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ 838 felem_assign(ftmp5, ftmp); 839 felem_square(tmp, ftmp); 840 felem_reduce(ftmp, tmp); 841 842 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */ 843 felem_mul(tmp, ftmp, ftmp5); 844 felem_reduce(ftmp5, tmp); 845 846 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 847 felem_mul(tmp, ftmp2, ftmp); 848 felem_reduce(ftmp2, tmp); 849 850 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 851 felem_mul(tmp, ftmp4, ftmp5); 852 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 853 854 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */ 855 felem_square(tmp2, ftmp3); 856 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */ 857 858 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */ 859 felem_diff_128_64(tmp2, ftmp5); 860 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */ 861 862 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 863 felem_assign(ftmp5, ftmp2); 864 felem_scalar(ftmp5, 2); 865 /* ftmp5[i] < 2 * 2^57 = 2^58 */ 866 867 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - 868 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 869 felem_diff_128_64(tmp2, ftmp5); 870 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */ 871 felem_reduce(x_out, tmp2); 872 873 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */ 874 felem_diff(ftmp2, x_out); 875 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */ 876 877 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */ 878 felem_mul(tmp2, ftmp3, ftmp2); 879 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ 880 881 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - 882 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 883 widefelem_diff(tmp2, tmp); 884 /* tmp2[i] < 2^118 + 2^120 < 2^121 */ 885 felem_reduce(y_out, tmp2); 886 887 /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is 888 * the point at infinity, so we need to check for this separately */ 889 890 /* if point 1 is at infinity, copy point 2 to output, and vice versa */ 891 copy_conditional(x_out, x2, z1_is_zero); 892 copy_conditional(x_out, x1, z2_is_zero); 893 copy_conditional(y_out, y2, z1_is_zero); 894 copy_conditional(y_out, y1, z2_is_zero); 895 copy_conditional(z_out, z2, z1_is_zero); 896 copy_conditional(z_out, z1, z2_is_zero); 897 felem_assign(x3, x_out); 898 felem_assign(y3, y_out); 899 felem_assign(z3, z_out); 900 } 901 902 /* select_point selects the |idx|th point from a precomputation table and 903 * copies it to out. */ 904 static void select_point(const u64 idx, unsigned int size, 905 const felem pre_comp[/*size*/][3], felem out[3]) { 906 unsigned i, j; 907 limb *outlimbs = &out[0][0]; 908 memset(outlimbs, 0, 3 * sizeof(felem)); 909 910 for (i = 0; i < size; i++) { 911 const limb *inlimbs = &pre_comp[i][0][0]; 912 u64 mask = i ^ idx; 913 mask |= mask >> 4; 914 mask |= mask >> 2; 915 mask |= mask >> 1; 916 mask &= 1; 917 mask--; 918 for (j = 0; j < 4 * 3; j++) { 919 outlimbs[j] |= inlimbs[j] & mask; 920 } 921 } 922 } 923 924 /* get_bit returns the |i|th bit in |in| */ 925 static char get_bit(const felem_bytearray in, unsigned i) { 926 if (i >= 224) { 927 return 0; 928 } 929 return (in[i >> 3] >> (i & 7)) & 1; 930 } 931 932 /* Interleaved point multiplication using precomputed point multiples: 933 * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], 934 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple 935 * of the generator, using certain (large) precomputed multiples in g_pre_comp. 936 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */ 937 static void batch_mul(felem x_out, felem y_out, felem z_out, 938 const felem_bytearray scalars[], 939 const unsigned num_points, const u8 *g_scalar, 940 const int mixed, const felem pre_comp[][17][3]) { 941 int i, skip; 942 unsigned num; 943 unsigned gen_mul = (g_scalar != NULL); 944 felem nq[3], tmp[4]; 945 u64 bits; 946 u8 sign, digit; 947 948 /* set nq to the point at infinity */ 949 memset(nq, 0, 3 * sizeof(felem)); 950 951 /* Loop over all scalars msb-to-lsb, interleaving additions 952 * of multiples of the generator (two in each of the last 28 rounds) 953 * and additions of other points multiples (every 5th round). */ 954 skip = 1; /* save two point operations in the first round */ 955 for (i = (num_points ? 220 : 27); i >= 0; --i) { 956 /* double */ 957 if (!skip) { 958 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 959 } 960 961 /* add multiples of the generator */ 962 if (gen_mul && (i <= 27)) { 963 /* first, look 28 bits upwards */ 964 bits = get_bit(g_scalar, i + 196) << 3; 965 bits |= get_bit(g_scalar, i + 140) << 2; 966 bits |= get_bit(g_scalar, i + 84) << 1; 967 bits |= get_bit(g_scalar, i + 28); 968 /* select the point to add, in constant time */ 969 select_point(bits, 16, g_pre_comp[1], tmp); 970 971 if (!skip) { 972 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 973 tmp[0], tmp[1], tmp[2]); 974 } else { 975 memcpy(nq, tmp, 3 * sizeof(felem)); 976 skip = 0; 977 } 978 979 /* second, look at the current position */ 980 bits = get_bit(g_scalar, i + 168) << 3; 981 bits |= get_bit(g_scalar, i + 112) << 2; 982 bits |= get_bit(g_scalar, i + 56) << 1; 983 bits |= get_bit(g_scalar, i); 984 /* select the point to add, in constant time */ 985 select_point(bits, 16, g_pre_comp[0], tmp); 986 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], 987 tmp[1], tmp[2]); 988 } 989 990 /* do other additions every 5 doublings */ 991 if (num_points && (i % 5 == 0)) { 992 /* loop over all scalars */ 993 for (num = 0; num < num_points; ++num) { 994 bits = get_bit(scalars[num], i + 4) << 5; 995 bits |= get_bit(scalars[num], i + 3) << 4; 996 bits |= get_bit(scalars[num], i + 2) << 3; 997 bits |= get_bit(scalars[num], i + 1) << 2; 998 bits |= get_bit(scalars[num], i) << 1; 999 bits |= get_bit(scalars[num], i - 1); 1000 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 1001 1002 /* select the point to add or subtract */ 1003 select_point(digit, 17, pre_comp[num], tmp); 1004 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */ 1005 copy_conditional(tmp[1], tmp[3], sign); 1006 1007 if (!skip) { 1008 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], mixed, tmp[0], 1009 tmp[1], tmp[2]); 1010 } else { 1011 memcpy(nq, tmp, 3 * sizeof(felem)); 1012 skip = 0; 1013 } 1014 } 1015 } 1016 } 1017 felem_assign(x_out, nq[0]); 1018 felem_assign(y_out, nq[1]); 1019 felem_assign(z_out, nq[2]); 1020 } 1021 1022 int ec_GFp_nistp224_group_init(EC_GROUP *group) { 1023 int ret; 1024 ret = ec_GFp_simple_group_init(group); 1025 group->a_is_minus3 = 1; 1026 return ret; 1027 } 1028 1029 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p, 1030 const BIGNUM *a, const BIGNUM *b, 1031 BN_CTX *ctx) { 1032 int ret = 0; 1033 BN_CTX *new_ctx = NULL; 1034 BIGNUM *curve_p, *curve_a, *curve_b; 1035 1036 if (ctx == NULL) { 1037 ctx = BN_CTX_new(); 1038 new_ctx = ctx; 1039 if (ctx == NULL) { 1040 return 0; 1041 } 1042 } 1043 BN_CTX_start(ctx); 1044 if (((curve_p = BN_CTX_get(ctx)) == NULL) || 1045 ((curve_a = BN_CTX_get(ctx)) == NULL) || 1046 ((curve_b = BN_CTX_get(ctx)) == NULL)) { 1047 goto err; 1048 } 1049 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p); 1050 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a); 1051 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b); 1052 if (BN_cmp(curve_p, p) || 1053 BN_cmp(curve_a, a) || 1054 BN_cmp(curve_b, b)) { 1055 OPENSSL_PUT_ERROR(EC, EC_R_WRONG_CURVE_PARAMETERS); 1056 goto err; 1057 } 1058 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); 1059 1060 err: 1061 BN_CTX_end(ctx); 1062 BN_CTX_free(new_ctx); 1063 return ret; 1064 } 1065 1066 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns 1067 * (X', Y') = (X/Z^2, Y/Z^3) */ 1068 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, 1069 const EC_POINT *point, 1070 BIGNUM *x, BIGNUM *y, 1071 BN_CTX *ctx) { 1072 felem z1, z2, x_in, y_in, x_out, y_out; 1073 widefelem tmp; 1074 1075 if (EC_POINT_is_at_infinity(group, point)) { 1076 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); 1077 return 0; 1078 } 1079 1080 if (!BN_to_felem(x_in, &point->X) || 1081 !BN_to_felem(y_in, &point->Y) || 1082 !BN_to_felem(z1, &point->Z)) { 1083 return 0; 1084 } 1085 1086 felem_inv(z2, z1); 1087 felem_square(tmp, z2); 1088 felem_reduce(z1, tmp); 1089 felem_mul(tmp, x_in, z1); 1090 felem_reduce(x_in, tmp); 1091 felem_contract(x_out, x_in); 1092 if (x != NULL && !felem_to_BN(x, x_out)) { 1093 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1094 return 0; 1095 } 1096 1097 felem_mul(tmp, z1, z2); 1098 felem_reduce(z1, tmp); 1099 felem_mul(tmp, y_in, z1); 1100 felem_reduce(y_in, tmp); 1101 felem_contract(y_out, y_in); 1102 if (y != NULL && !felem_to_BN(y, y_out)) { 1103 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1104 return 0; 1105 } 1106 1107 return 1; 1108 } 1109 1110 static void make_points_affine(size_t num, felem points[/*num*/][3], 1111 felem tmp_felems[/*num+1*/]) { 1112 /* Runs in constant time, unless an input is the point at infinity 1113 * (which normally shouldn't happen). */ 1114 ec_GFp_nistp_points_make_affine_internal( 1115 num, points, sizeof(felem), tmp_felems, (void (*)(void *))felem_one, 1116 (int (*)(const void *))felem_is_zero_int, 1117 (void (*)(void *, const void *))felem_assign, 1118 (void (*)(void *, const void *))felem_square_reduce, 1119 (void (*)(void *, const void *, const void *))felem_mul_reduce, 1120 (void (*)(void *, const void *))felem_inv, 1121 (void (*)(void *, const void *))felem_contract); 1122 } 1123 1124 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, 1125 const BIGNUM *g_scalar, const EC_POINT *p_, 1126 const BIGNUM *p_scalar_, BN_CTX *ctx) { 1127 /* TODO: This function used to take |points| and |scalars| as arrays of 1128 * |num| elements. The code below should be simplified to work in terms of 1129 * |p_| and |p_scalar_|. */ 1130 size_t num = p_ != NULL ? 1 : 0; 1131 const EC_POINT **points = p_ != NULL ? &p_ : NULL; 1132 BIGNUM const *const *scalars = p_ != NULL ? &p_scalar_ : NULL; 1133 1134 int ret = 0; 1135 int j; 1136 unsigned i; 1137 int mixed = 0; 1138 BN_CTX *new_ctx = NULL; 1139 BIGNUM *x, *y, *z, *tmp_scalar; 1140 felem_bytearray g_secret; 1141 felem_bytearray *secrets = NULL; 1142 felem(*pre_comp)[17][3] = NULL; 1143 felem *tmp_felems = NULL; 1144 felem_bytearray tmp; 1145 unsigned num_bytes; 1146 size_t num_points = num; 1147 felem x_in, y_in, z_in, x_out, y_out, z_out; 1148 const EC_POINT *p = NULL; 1149 const BIGNUM *p_scalar = NULL; 1150 1151 if (ctx == NULL) { 1152 ctx = BN_CTX_new(); 1153 new_ctx = ctx; 1154 if (ctx == NULL) { 1155 return 0; 1156 } 1157 } 1158 1159 BN_CTX_start(ctx); 1160 if ((x = BN_CTX_get(ctx)) == NULL || 1161 (y = BN_CTX_get(ctx)) == NULL || 1162 (z = BN_CTX_get(ctx)) == NULL || 1163 (tmp_scalar = BN_CTX_get(ctx)) == NULL) { 1164 goto err; 1165 } 1166 1167 if (num_points > 0) { 1168 if (num_points >= 3) { 1169 /* unless we precompute multiples for just one or two points, 1170 * converting those into affine form is time well spent */ 1171 mixed = 1; 1172 } 1173 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); 1174 pre_comp = OPENSSL_malloc(num_points * sizeof(felem[17][3])); 1175 if (mixed) { 1176 tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem)); 1177 } 1178 if (secrets == NULL || 1179 pre_comp == NULL || 1180 (mixed && tmp_felems == NULL)) { 1181 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 1182 goto err; 1183 } 1184 1185 /* we treat NULL scalars as 0, and NULL points as points at infinity, 1186 * i.e., they contribute nothing to the linear combination */ 1187 memset(secrets, 0, num_points * sizeof(felem_bytearray)); 1188 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem)); 1189 for (i = 0; i < num_points; ++i) { 1190 if (i == num) { 1191 /* the generator */ 1192 p = EC_GROUP_get0_generator(group); 1193 p_scalar = g_scalar; 1194 } else { 1195 /* the i^th point */ 1196 p = points[i]; 1197 p_scalar = scalars[i]; 1198 } 1199 1200 if (p_scalar != NULL && p != NULL) { 1201 /* reduce g_scalar to 0 <= g_scalar < 2^224 */ 1202 if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) { 1203 /* this is an unusual input, and we don't guarantee 1204 * constant-timeness */ 1205 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { 1206 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1207 goto err; 1208 } 1209 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1210 } else { 1211 num_bytes = BN_bn2bin(p_scalar, tmp); 1212 } 1213 1214 flip_endian(secrets[i], tmp, num_bytes); 1215 /* precompute multiples */ 1216 if (!BN_to_felem(x_out, &p->X) || 1217 !BN_to_felem(y_out, &p->Y) || 1218 !BN_to_felem(z_out, &p->Z)) { 1219 goto err; 1220 } 1221 1222 felem_assign(pre_comp[i][1][0], x_out); 1223 felem_assign(pre_comp[i][1][1], y_out); 1224 felem_assign(pre_comp[i][1][2], z_out); 1225 1226 for (j = 2; j <= 16; ++j) { 1227 if (j & 1) { 1228 point_add(pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2], 1229 pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2], 1230 0, pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], 1231 pre_comp[i][j - 1][2]); 1232 } else { 1233 point_double(pre_comp[i][j][0], pre_comp[i][j][1], 1234 pre_comp[i][j][2], pre_comp[i][j / 2][0], 1235 pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); 1236 } 1237 } 1238 } 1239 } 1240 1241 if (mixed) { 1242 make_points_affine(num_points * 17, pre_comp[0], tmp_felems); 1243 } 1244 } 1245 1246 if (g_scalar != NULL) { 1247 memset(g_secret, 0, sizeof(g_secret)); 1248 /* reduce g_scalar to 0 <= g_scalar < 2^224 */ 1249 if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) { 1250 /* this is an unusual input, and we don't guarantee constant-timeness */ 1251 if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) { 1252 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1253 goto err; 1254 } 1255 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1256 } else { 1257 num_bytes = BN_bn2bin(g_scalar, tmp); 1258 } 1259 1260 flip_endian(g_secret, tmp, num_bytes); 1261 } 1262 batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, 1263 num_points, g_scalar != NULL ? g_secret : NULL, mixed, 1264 (const felem(*)[17][3])pre_comp); 1265 1266 /* reduce the output to its unique minimal representation */ 1267 felem_contract(x_in, x_out); 1268 felem_contract(y_in, y_out); 1269 felem_contract(z_in, z_out); 1270 if (!felem_to_BN(x, x_in) || 1271 !felem_to_BN(y, y_in) || 1272 !felem_to_BN(z, z_in)) { 1273 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1274 goto err; 1275 } 1276 ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); 1277 1278 err: 1279 BN_CTX_end(ctx); 1280 BN_CTX_free(new_ctx); 1281 OPENSSL_free(secrets); 1282 OPENSSL_free(pre_comp); 1283 OPENSSL_free(tmp_felems); 1284 return ret; 1285 } 1286 1287 const EC_METHOD *EC_GFp_nistp224_method(void) { 1288 static const EC_METHOD ret = {ec_GFp_nistp224_group_init, 1289 ec_GFp_simple_group_finish, 1290 ec_GFp_simple_group_clear_finish, 1291 ec_GFp_simple_group_copy, 1292 ec_GFp_nistp224_group_set_curve, 1293 ec_GFp_nistp224_point_get_affine_coordinates, 1294 ec_GFp_nistp224_points_mul, 1295 0 /* check_pub_key_order */, 1296 ec_GFp_simple_field_mul, 1297 ec_GFp_simple_field_sqr, 1298 0 /* field_encode */, 1299 0 /* field_decode */, 1300 0 /* field_set_to_one */}; 1301 1302 return &ret; 1303 } 1304 1305 #endif /* 64_BIT && !WINDOWS && !SMALL */ 1306
