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 #include <openssl/bn.h> 23 #include <openssl/ec.h> 24 #include <openssl/err.h> 25 #include <openssl/mem.h> 26 27 #include <string.h> 28 29 #include "internal.h" 30 #include "../delocate.h" 31 #include "../../internal.h" 32 33 34 #if defined(BORINGSSL_HAS_UINT128) && !defined(OPENSSL_SMALL) 35 36 // Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 37 // using 64-bit coefficients called 'limbs', and sometimes (for multiplication 38 // results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 39 // 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-p224_limb 40 // representation is an 'p224_felem'; a 7-p224_widelimb representation is a 41 // 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we 42 // don't always reduce the representations: we ensure that inputs to each 43 // p224_felem multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 44 // 4*2^60*2^60, and fit into a 128-bit word without overflow. The coefficients 45 // are then again partially reduced to obtain an p224_felem satisfying a_i < 46 // 2^57. We only reduce to the unique minimal representation at the end of the 47 // computation. 48 49 typedef uint64_t p224_limb; 50 typedef uint128_t p224_widelimb; 51 52 typedef p224_limb p224_felem[4]; 53 typedef p224_widelimb p224_widefelem[7]; 54 55 // Field element represented as a byte arrary. 28*8 = 224 bits is also the 56 // group order size for the elliptic curve, and we also use this type for 57 // scalars for point multiplication. 58 typedef uint8_t p224_felem_bytearray[28]; 59 60 // Precomputed multiples of the standard generator 61 // Points are given in coordinates (X, Y, Z) where Z normally is 1 62 // (0 for the point at infinity). 63 // For each field element, slice a_0 is word 0, etc. 64 // 65 // The table has 2 * 16 elements, starting with the following: 66 // index | bits | point 67 // ------+---------+------------------------------ 68 // 0 | 0 0 0 0 | 0G 69 // 1 | 0 0 0 1 | 1G 70 // 2 | 0 0 1 0 | 2^56G 71 // 3 | 0 0 1 1 | (2^56 + 1)G 72 // 4 | 0 1 0 0 | 2^112G 73 // 5 | 0 1 0 1 | (2^112 + 1)G 74 // 6 | 0 1 1 0 | (2^112 + 2^56)G 75 // 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G 76 // 8 | 1 0 0 0 | 2^168G 77 // 9 | 1 0 0 1 | (2^168 + 1)G 78 // 10 | 1 0 1 0 | (2^168 + 2^56)G 79 // 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G 80 // 12 | 1 1 0 0 | (2^168 + 2^112)G 81 // 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G 82 // 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G 83 // 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G 84 // followed by a copy of this with each element multiplied by 2^28. 85 // 86 // The reason for this is so that we can clock bits into four different 87 // locations when doing simple scalar multiplies against the base point, 88 // and then another four locations using the second 16 elements. 89 static const p224_felem g_p224_pre_comp[2][16][3] = { 90 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 91 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, 92 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, 93 {1, 0, 0, 0}}, 94 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, 95 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, 96 {1, 0, 0, 0}}, 97 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, 98 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, 99 {1, 0, 0, 0}}, 100 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, 101 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, 102 {1, 0, 0, 0}}, 103 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, 104 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, 105 {1, 0, 0, 0}}, 106 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, 107 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, 108 {1, 0, 0, 0}}, 109 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, 110 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, 111 {1, 0, 0, 0}}, 112 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, 113 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, 114 {1, 0, 0, 0}}, 115 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, 116 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, 117 {1, 0, 0, 0}}, 118 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, 119 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, 120 {1, 0, 0, 0}}, 121 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, 122 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, 123 {1, 0, 0, 0}}, 124 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, 125 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, 126 {1, 0, 0, 0}}, 127 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, 128 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, 129 {1, 0, 0, 0}}, 130 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, 131 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, 132 {1, 0, 0, 0}}, 133 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, 134 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, 135 {1, 0, 0, 0}}}, 136 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 137 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, 138 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, 139 {1, 0, 0, 0}}, 140 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, 141 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, 142 {1, 0, 0, 0}}, 143 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, 144 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, 145 {1, 0, 0, 0}}, 146 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, 147 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, 148 {1, 0, 0, 0}}, 149 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, 150 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, 151 {1, 0, 0, 0}}, 152 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, 153 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, 154 {1, 0, 0, 0}}, 155 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, 156 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, 157 {1, 0, 0, 0}}, 158 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, 159 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, 160 {1, 0, 0, 0}}, 161 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, 162 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, 163 {1, 0, 0, 0}}, 164 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, 165 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, 166 {1, 0, 0, 0}}, 167 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, 168 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, 169 {1, 0, 0, 0}}, 170 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, 171 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, 172 {1, 0, 0, 0}}, 173 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, 174 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, 175 {1, 0, 0, 0}}, 176 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, 177 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, 178 {1, 0, 0, 0}}, 179 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, 180 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, 181 {1, 0, 0, 0}}}}; 182 183 static uint64_t p224_load_u64(const uint8_t in[8]) { 184 uint64_t ret; 185 OPENSSL_memcpy(&ret, in, sizeof(ret)); 186 return ret; 187 } 188 189 // Helper functions to convert field elements to/from internal representation 190 static void p224_bin28_to_felem(p224_felem out, const uint8_t in[28]) { 191 out[0] = p224_load_u64(in) & 0x00ffffffffffffff; 192 out[1] = p224_load_u64(in + 7) & 0x00ffffffffffffff; 193 out[2] = p224_load_u64(in + 14) & 0x00ffffffffffffff; 194 out[3] = p224_load_u64(in + 20) >> 8; 195 } 196 197 static void p224_felem_to_bin28(uint8_t out[28], const p224_felem in) { 198 for (size_t i = 0; i < 7; ++i) { 199 out[i] = in[0] >> (8 * i); 200 out[i + 7] = in[1] >> (8 * i); 201 out[i + 14] = in[2] >> (8 * i); 202 out[i + 21] = in[3] >> (8 * i); 203 } 204 } 205 206 // To preserve endianness when using BN_bn2bin and BN_bin2bn 207 static void p224_flip_endian(uint8_t *out, const uint8_t *in, size_t len) { 208 for (size_t i = 0; i < len; ++i) { 209 out[i] = in[len - 1 - i]; 210 } 211 } 212 213 // From OpenSSL BIGNUM to internal representation 214 static int p224_BN_to_felem(p224_felem out, const BIGNUM *bn) { 215 // BN_bn2bin eats leading zeroes 216 p224_felem_bytearray b_out; 217 OPENSSL_memset(b_out, 0, sizeof(b_out)); 218 size_t num_bytes = BN_num_bytes(bn); 219 if (num_bytes > sizeof(b_out) || 220 BN_is_negative(bn)) { 221 OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); 222 return 0; 223 } 224 225 p224_felem_bytearray b_in; 226 num_bytes = BN_bn2bin(bn, b_in); 227 p224_flip_endian(b_out, b_in, num_bytes); 228 p224_bin28_to_felem(out, b_out); 229 return 1; 230 } 231 232 // From internal representation to OpenSSL BIGNUM 233 static BIGNUM *p224_felem_to_BN(BIGNUM *out, const p224_felem in) { 234 p224_felem_bytearray b_in, b_out; 235 p224_felem_to_bin28(b_in, in); 236 p224_flip_endian(b_out, b_in, sizeof(b_out)); 237 return BN_bin2bn(b_out, sizeof(b_out), out); 238 } 239 240 // Field operations, using the internal representation of field elements. 241 // NB! These operations are specific to our point multiplication and cannot be 242 // expected to be correct in general - e.g., multiplication with a large scalar 243 // will cause an overflow. 244 245 static void p224_felem_assign(p224_felem out, const p224_felem in) { 246 out[0] = in[0]; 247 out[1] = in[1]; 248 out[2] = in[2]; 249 out[3] = in[3]; 250 } 251 252 // Sum two field elements: out += in 253 static void p224_felem_sum(p224_felem out, const p224_felem in) { 254 out[0] += in[0]; 255 out[1] += in[1]; 256 out[2] += in[2]; 257 out[3] += in[3]; 258 } 259 260 // Get negative value: out = -in 261 // Assumes in[i] < 2^57 262 static void p224_felem_neg(p224_felem out, const p224_felem in) { 263 static const p224_limb two58p2 = 264 (((p224_limb)1) << 58) + (((p224_limb)1) << 2); 265 static const p224_limb two58m2 = 266 (((p224_limb)1) << 58) - (((p224_limb)1) << 2); 267 static const p224_limb two58m42m2 = 268 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2); 269 270 // Set to 0 mod 2^224-2^96+1 to ensure out > in 271 out[0] = two58p2 - in[0]; 272 out[1] = two58m42m2 - in[1]; 273 out[2] = two58m2 - in[2]; 274 out[3] = two58m2 - in[3]; 275 } 276 277 // Subtract field elements: out -= in 278 // Assumes in[i] < 2^57 279 static void p224_felem_diff(p224_felem out, const p224_felem in) { 280 static const p224_limb two58p2 = 281 (((p224_limb)1) << 58) + (((p224_limb)1) << 2); 282 static const p224_limb two58m2 = 283 (((p224_limb)1) << 58) - (((p224_limb)1) << 2); 284 static const p224_limb two58m42m2 = 285 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2); 286 287 // Add 0 mod 2^224-2^96+1 to ensure out > in 288 out[0] += two58p2; 289 out[1] += two58m42m2; 290 out[2] += two58m2; 291 out[3] += two58m2; 292 293 out[0] -= in[0]; 294 out[1] -= in[1]; 295 out[2] -= in[2]; 296 out[3] -= in[3]; 297 } 298 299 // Subtract in unreduced 128-bit mode: out -= in 300 // Assumes in[i] < 2^119 301 static void p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) { 302 static const p224_widelimb two120 = ((p224_widelimb)1) << 120; 303 static const p224_widelimb two120m64 = 304 (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64); 305 static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) - 306 (((p224_widelimb)1) << 104) - 307 (((p224_widelimb)1) << 64); 308 309 // Add 0 mod 2^224-2^96+1 to ensure out > in 310 out[0] += two120; 311 out[1] += two120m64; 312 out[2] += two120m64; 313 out[3] += two120; 314 out[4] += two120m104m64; 315 out[5] += two120m64; 316 out[6] += two120m64; 317 318 out[0] -= in[0]; 319 out[1] -= in[1]; 320 out[2] -= in[2]; 321 out[3] -= in[3]; 322 out[4] -= in[4]; 323 out[5] -= in[5]; 324 out[6] -= in[6]; 325 } 326 327 // Subtract in mixed mode: out128 -= in64 328 // in[i] < 2^63 329 static void p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) { 330 static const p224_widelimb two64p8 = 331 (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8); 332 static const p224_widelimb two64m8 = 333 (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8); 334 static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) - 335 (((p224_widelimb)1) << 48) - 336 (((p224_widelimb)1) << 8); 337 338 // Add 0 mod 2^224-2^96+1 to ensure out > in 339 out[0] += two64p8; 340 out[1] += two64m48m8; 341 out[2] += two64m8; 342 out[3] += two64m8; 343 344 out[0] -= in[0]; 345 out[1] -= in[1]; 346 out[2] -= in[2]; 347 out[3] -= in[3]; 348 } 349 350 // Multiply a field element by a scalar: out = out * scalar 351 // The scalars we actually use are small, so results fit without overflow 352 static void p224_felem_scalar(p224_felem out, const p224_limb scalar) { 353 out[0] *= scalar; 354 out[1] *= scalar; 355 out[2] *= scalar; 356 out[3] *= scalar; 357 } 358 359 // Multiply an unreduced field element by a scalar: out = out * scalar 360 // The scalars we actually use are small, so results fit without overflow 361 static void p224_widefelem_scalar(p224_widefelem out, 362 const p224_widelimb scalar) { 363 out[0] *= scalar; 364 out[1] *= scalar; 365 out[2] *= scalar; 366 out[3] *= scalar; 367 out[4] *= scalar; 368 out[5] *= scalar; 369 out[6] *= scalar; 370 } 371 372 // Square a field element: out = in^2 373 static void p224_felem_square(p224_widefelem out, const p224_felem in) { 374 p224_limb tmp0, tmp1, tmp2; 375 tmp0 = 2 * in[0]; 376 tmp1 = 2 * in[1]; 377 tmp2 = 2 * in[2]; 378 out[0] = ((p224_widelimb)in[0]) * in[0]; 379 out[1] = ((p224_widelimb)in[0]) * tmp1; 380 out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1]; 381 out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2; 382 out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2]; 383 out[5] = ((p224_widelimb)in[3]) * tmp2; 384 out[6] = ((p224_widelimb)in[3]) * in[3]; 385 } 386 387 // Multiply two field elements: out = in1 * in2 388 static void p224_felem_mul(p224_widefelem out, const p224_felem in1, 389 const p224_felem in2) { 390 out[0] = ((p224_widelimb)in1[0]) * in2[0]; 391 out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0]; 392 out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] + 393 ((p224_widelimb)in1[2]) * in2[0]; 394 out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] + 395 ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0]; 396 out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] + 397 ((p224_widelimb)in1[3]) * in2[1]; 398 out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2]; 399 out[6] = ((p224_widelimb)in1[3]) * in2[3]; 400 } 401 402 // Reduce seven 128-bit coefficients to four 64-bit coefficients. 403 // Requires in[i] < 2^126, 404 // ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 405 static void p224_felem_reduce(p224_felem out, const p224_widefelem in) { 406 static const p224_widelimb two127p15 = 407 (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15); 408 static const p224_widelimb two127m71 = 409 (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71); 410 static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) - 411 (((p224_widelimb)1) << 71) - 412 (((p224_widelimb)1) << 55); 413 p224_widelimb output[5]; 414 415 // Add 0 mod 2^224-2^96+1 to ensure all differences are positive 416 output[0] = in[0] + two127p15; 417 output[1] = in[1] + two127m71m55; 418 output[2] = in[2] + two127m71; 419 output[3] = in[3]; 420 output[4] = in[4]; 421 422 // Eliminate in[4], in[5], in[6] 423 output[4] += in[6] >> 16; 424 output[3] += (in[6] & 0xffff) << 40; 425 output[2] -= in[6]; 426 427 output[3] += in[5] >> 16; 428 output[2] += (in[5] & 0xffff) << 40; 429 output[1] -= in[5]; 430 431 output[2] += output[4] >> 16; 432 output[1] += (output[4] & 0xffff) << 40; 433 output[0] -= output[4]; 434 435 // Carry 2 -> 3 -> 4 436 output[3] += output[2] >> 56; 437 output[2] &= 0x00ffffffffffffff; 438 439 output[4] = output[3] >> 56; 440 output[3] &= 0x00ffffffffffffff; 441 442 // Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 443 444 // Eliminate output[4] 445 output[2] += output[4] >> 16; 446 // output[2] < 2^56 + 2^56 = 2^57 447 output[1] += (output[4] & 0xffff) << 40; 448 output[0] -= output[4]; 449 450 // Carry 0 -> 1 -> 2 -> 3 451 output[1] += output[0] >> 56; 452 out[0] = output[0] & 0x00ffffffffffffff; 453 454 output[2] += output[1] >> 56; 455 // output[2] < 2^57 + 2^72 456 out[1] = output[1] & 0x00ffffffffffffff; 457 output[3] += output[2] >> 56; 458 // output[3] <= 2^56 + 2^16 459 out[2] = output[2] & 0x00ffffffffffffff; 460 461 // out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, 462 // out[3] <= 2^56 + 2^16 (due to final carry), 463 // so out < 2*p 464 out[3] = output[3]; 465 } 466 467 // Reduce to unique minimal representation. 468 // Requires 0 <= in < 2*p (always call p224_felem_reduce first) 469 static void p224_felem_contract(p224_felem out, const p224_felem in) { 470 static const int64_t two56 = ((p224_limb)1) << 56; 471 // 0 <= in < 2*p, p = 2^224 - 2^96 + 1 472 // if in > p , reduce in = in - 2^224 + 2^96 - 1 473 int64_t tmp[4], a; 474 tmp[0] = in[0]; 475 tmp[1] = in[1]; 476 tmp[2] = in[2]; 477 tmp[3] = in[3]; 478 // Case 1: a = 1 iff in >= 2^224 479 a = (in[3] >> 56); 480 tmp[0] -= a; 481 tmp[1] += a << 40; 482 tmp[3] &= 0x00ffffffffffffff; 483 // Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and 484 // the lower part is non-zero 485 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | 486 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); 487 a &= 0x00ffffffffffffff; 488 // turn a into an all-one mask (if a = 0) or an all-zero mask 489 a = (a - 1) >> 63; 490 // subtract 2^224 - 2^96 + 1 if a is all-one 491 tmp[3] &= a ^ 0xffffffffffffffff; 492 tmp[2] &= a ^ 0xffffffffffffffff; 493 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; 494 tmp[0] -= 1 & a; 495 496 // eliminate negative coefficients: if tmp[0] is negative, tmp[1] must 497 // be non-zero, so we only need one step 498 a = tmp[0] >> 63; 499 tmp[0] += two56 & a; 500 tmp[1] -= 1 & a; 501 502 // carry 1 -> 2 -> 3 503 tmp[2] += tmp[1] >> 56; 504 tmp[1] &= 0x00ffffffffffffff; 505 506 tmp[3] += tmp[2] >> 56; 507 tmp[2] &= 0x00ffffffffffffff; 508 509 // Now 0 <= out < p 510 out[0] = tmp[0]; 511 out[1] = tmp[1]; 512 out[2] = tmp[2]; 513 out[3] = tmp[3]; 514 } 515 516 // Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field 517 // elements are reduced to in < 2^225, so we only need to check three cases: 0, 518 // 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 519 static p224_limb p224_felem_is_zero(const p224_felem in) { 520 p224_limb zero = in[0] | in[1] | in[2] | in[3]; 521 zero = (((int64_t)(zero)-1) >> 63) & 1; 522 523 p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | 524 (in[2] ^ 0x00ffffffffffffff) | 525 (in[3] ^ 0x00ffffffffffffff); 526 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; 527 p224_limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | 528 (in[2] ^ 0x00ffffffffffffff) | 529 (in[3] ^ 0x01ffffffffffffff); 530 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; 531 return (zero | two224m96p1 | two225m97p2); 532 } 533 534 // Invert a field element 535 // Computation chain copied from djb's code 536 static void p224_felem_inv(p224_felem out, const p224_felem in) { 537 p224_felem ftmp, ftmp2, ftmp3, ftmp4; 538 p224_widefelem tmp; 539 540 p224_felem_square(tmp, in); 541 p224_felem_reduce(ftmp, tmp); // 2 542 p224_felem_mul(tmp, in, ftmp); 543 p224_felem_reduce(ftmp, tmp); // 2^2 - 1 544 p224_felem_square(tmp, ftmp); 545 p224_felem_reduce(ftmp, tmp); // 2^3 - 2 546 p224_felem_mul(tmp, in, ftmp); 547 p224_felem_reduce(ftmp, tmp); // 2^3 - 1 548 p224_felem_square(tmp, ftmp); 549 p224_felem_reduce(ftmp2, tmp); // 2^4 - 2 550 p224_felem_square(tmp, ftmp2); 551 p224_felem_reduce(ftmp2, tmp); // 2^5 - 4 552 p224_felem_square(tmp, ftmp2); 553 p224_felem_reduce(ftmp2, tmp); // 2^6 - 8 554 p224_felem_mul(tmp, ftmp2, ftmp); 555 p224_felem_reduce(ftmp, tmp); // 2^6 - 1 556 p224_felem_square(tmp, ftmp); 557 p224_felem_reduce(ftmp2, tmp); // 2^7 - 2 558 for (size_t i = 0; i < 5; ++i) { // 2^12 - 2^6 559 p224_felem_square(tmp, ftmp2); 560 p224_felem_reduce(ftmp2, tmp); 561 } 562 p224_felem_mul(tmp, ftmp2, ftmp); 563 p224_felem_reduce(ftmp2, tmp); // 2^12 - 1 564 p224_felem_square(tmp, ftmp2); 565 p224_felem_reduce(ftmp3, tmp); // 2^13 - 2 566 for (size_t i = 0; i < 11; ++i) { // 2^24 - 2^12 567 p224_felem_square(tmp, ftmp3); 568 p224_felem_reduce(ftmp3, tmp); 569 } 570 p224_felem_mul(tmp, ftmp3, ftmp2); 571 p224_felem_reduce(ftmp2, tmp); // 2^24 - 1 572 p224_felem_square(tmp, ftmp2); 573 p224_felem_reduce(ftmp3, tmp); // 2^25 - 2 574 for (size_t i = 0; i < 23; ++i) { // 2^48 - 2^24 575 p224_felem_square(tmp, ftmp3); 576 p224_felem_reduce(ftmp3, tmp); 577 } 578 p224_felem_mul(tmp, ftmp3, ftmp2); 579 p224_felem_reduce(ftmp3, tmp); // 2^48 - 1 580 p224_felem_square(tmp, ftmp3); 581 p224_felem_reduce(ftmp4, tmp); // 2^49 - 2 582 for (size_t i = 0; i < 47; ++i) { // 2^96 - 2^48 583 p224_felem_square(tmp, ftmp4); 584 p224_felem_reduce(ftmp4, tmp); 585 } 586 p224_felem_mul(tmp, ftmp3, ftmp4); 587 p224_felem_reduce(ftmp3, tmp); // 2^96 - 1 588 p224_felem_square(tmp, ftmp3); 589 p224_felem_reduce(ftmp4, tmp); // 2^97 - 2 590 for (size_t i = 0; i < 23; ++i) { // 2^120 - 2^24 591 p224_felem_square(tmp, ftmp4); 592 p224_felem_reduce(ftmp4, tmp); 593 } 594 p224_felem_mul(tmp, ftmp2, ftmp4); 595 p224_felem_reduce(ftmp2, tmp); // 2^120 - 1 596 for (size_t i = 0; i < 6; ++i) { // 2^126 - 2^6 597 p224_felem_square(tmp, ftmp2); 598 p224_felem_reduce(ftmp2, tmp); 599 } 600 p224_felem_mul(tmp, ftmp2, ftmp); 601 p224_felem_reduce(ftmp, tmp); // 2^126 - 1 602 p224_felem_square(tmp, ftmp); 603 p224_felem_reduce(ftmp, tmp); // 2^127 - 2 604 p224_felem_mul(tmp, ftmp, in); 605 p224_felem_reduce(ftmp, tmp); // 2^127 - 1 606 for (size_t i = 0; i < 97; ++i) { // 2^224 - 2^97 607 p224_felem_square(tmp, ftmp); 608 p224_felem_reduce(ftmp, tmp); 609 } 610 p224_felem_mul(tmp, ftmp, ftmp3); 611 p224_felem_reduce(out, tmp); // 2^224 - 2^96 - 1 612 } 613 614 // Copy in constant time: 615 // if icopy == 1, copy in to out, 616 // if icopy == 0, copy out to itself. 617 static void p224_copy_conditional(p224_felem out, const p224_felem in, 618 p224_limb icopy) { 619 // icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one 620 const p224_limb copy = -icopy; 621 for (size_t i = 0; i < 4; ++i) { 622 const p224_limb tmp = copy & (in[i] ^ out[i]); 623 out[i] ^= tmp; 624 } 625 } 626 627 // ELLIPTIC CURVE POINT OPERATIONS 628 // 629 // Points are represented in Jacobian projective coordinates: 630 // (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), 631 // or to the point at infinity if Z == 0. 632 633 // Double an elliptic curve point: 634 // (X', Y', Z') = 2 * (X, Y, Z), where 635 // X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 636 // Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 637 // Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z 638 // Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, 639 // while x_out == y_in is not (maybe this works, but it's not tested). 640 static void p224_point_double(p224_felem x_out, p224_felem y_out, 641 p224_felem z_out, const p224_felem x_in, 642 const p224_felem y_in, const p224_felem z_in) { 643 p224_widefelem tmp, tmp2; 644 p224_felem delta, gamma, beta, alpha, ftmp, ftmp2; 645 646 p224_felem_assign(ftmp, x_in); 647 p224_felem_assign(ftmp2, x_in); 648 649 // delta = z^2 650 p224_felem_square(tmp, z_in); 651 p224_felem_reduce(delta, tmp); 652 653 // gamma = y^2 654 p224_felem_square(tmp, y_in); 655 p224_felem_reduce(gamma, tmp); 656 657 // beta = x*gamma 658 p224_felem_mul(tmp, x_in, gamma); 659 p224_felem_reduce(beta, tmp); 660 661 // alpha = 3*(x-delta)*(x+delta) 662 p224_felem_diff(ftmp, delta); 663 // ftmp[i] < 2^57 + 2^58 + 2 < 2^59 664 p224_felem_sum(ftmp2, delta); 665 // ftmp2[i] < 2^57 + 2^57 = 2^58 666 p224_felem_scalar(ftmp2, 3); 667 // ftmp2[i] < 3 * 2^58 < 2^60 668 p224_felem_mul(tmp, ftmp, ftmp2); 669 // tmp[i] < 2^60 * 2^59 * 4 = 2^121 670 p224_felem_reduce(alpha, tmp); 671 672 // x' = alpha^2 - 8*beta 673 p224_felem_square(tmp, alpha); 674 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 675 p224_felem_assign(ftmp, beta); 676 p224_felem_scalar(ftmp, 8); 677 // ftmp[i] < 8 * 2^57 = 2^60 678 p224_felem_diff_128_64(tmp, ftmp); 679 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 680 p224_felem_reduce(x_out, tmp); 681 682 // z' = (y + z)^2 - gamma - delta 683 p224_felem_sum(delta, gamma); 684 // delta[i] < 2^57 + 2^57 = 2^58 685 p224_felem_assign(ftmp, y_in); 686 p224_felem_sum(ftmp, z_in); 687 // ftmp[i] < 2^57 + 2^57 = 2^58 688 p224_felem_square(tmp, ftmp); 689 // tmp[i] < 4 * 2^58 * 2^58 = 2^118 690 p224_felem_diff_128_64(tmp, delta); 691 // tmp[i] < 2^118 + 2^64 + 8 < 2^119 692 p224_felem_reduce(z_out, tmp); 693 694 // y' = alpha*(4*beta - x') - 8*gamma^2 695 p224_felem_scalar(beta, 4); 696 // beta[i] < 4 * 2^57 = 2^59 697 p224_felem_diff(beta, x_out); 698 // beta[i] < 2^59 + 2^58 + 2 < 2^60 699 p224_felem_mul(tmp, alpha, beta); 700 // tmp[i] < 4 * 2^57 * 2^60 = 2^119 701 p224_felem_square(tmp2, gamma); 702 // tmp2[i] < 4 * 2^57 * 2^57 = 2^116 703 p224_widefelem_scalar(tmp2, 8); 704 // tmp2[i] < 8 * 2^116 = 2^119 705 p224_widefelem_diff(tmp, tmp2); 706 // tmp[i] < 2^119 + 2^120 < 2^121 707 p224_felem_reduce(y_out, tmp); 708 } 709 710 // Add two elliptic curve points: 711 // (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where 712 // X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - 713 // 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 714 // 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 * 715 // X_1)^2 - X_3) - 716 // Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 717 // Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) 718 // 719 // This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. 720 721 // This function is not entirely constant-time: it includes a branch for 722 // checking whether the two input points are equal, (while not equal to the 723 // point at infinity). This case never happens during single point 724 // multiplication, so there is no timing leak for ECDH or ECDSA signing. 725 static void p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3, 726 const p224_felem x1, const p224_felem y1, 727 const p224_felem z1, const int mixed, 728 const p224_felem x2, const p224_felem y2, 729 const p224_felem z2) { 730 p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; 731 p224_widefelem tmp, tmp2; 732 p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal; 733 734 if (!mixed) { 735 // ftmp2 = z2^2 736 p224_felem_square(tmp, z2); 737 p224_felem_reduce(ftmp2, tmp); 738 739 // ftmp4 = z2^3 740 p224_felem_mul(tmp, ftmp2, z2); 741 p224_felem_reduce(ftmp4, tmp); 742 743 // ftmp4 = z2^3*y1 744 p224_felem_mul(tmp2, ftmp4, y1); 745 p224_felem_reduce(ftmp4, tmp2); 746 747 // ftmp2 = z2^2*x1 748 p224_felem_mul(tmp2, ftmp2, x1); 749 p224_felem_reduce(ftmp2, tmp2); 750 } else { 751 // We'll assume z2 = 1 (special case z2 = 0 is handled later) 752 753 // ftmp4 = z2^3*y1 754 p224_felem_assign(ftmp4, y1); 755 756 // ftmp2 = z2^2*x1 757 p224_felem_assign(ftmp2, x1); 758 } 759 760 // ftmp = z1^2 761 p224_felem_square(tmp, z1); 762 p224_felem_reduce(ftmp, tmp); 763 764 // ftmp3 = z1^3 765 p224_felem_mul(tmp, ftmp, z1); 766 p224_felem_reduce(ftmp3, tmp); 767 768 // tmp = z1^3*y2 769 p224_felem_mul(tmp, ftmp3, y2); 770 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 771 772 // ftmp3 = z1^3*y2 - z2^3*y1 773 p224_felem_diff_128_64(tmp, ftmp4); 774 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 775 p224_felem_reduce(ftmp3, tmp); 776 777 // tmp = z1^2*x2 778 p224_felem_mul(tmp, ftmp, x2); 779 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 780 781 // ftmp = z1^2*x2 - z2^2*x1 782 p224_felem_diff_128_64(tmp, ftmp2); 783 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 784 p224_felem_reduce(ftmp, tmp); 785 786 // the formulae are incorrect if the points are equal 787 // so we check for this and do doubling if this happens 788 x_equal = p224_felem_is_zero(ftmp); 789 y_equal = p224_felem_is_zero(ftmp3); 790 z1_is_zero = p224_felem_is_zero(z1); 791 z2_is_zero = p224_felem_is_zero(z2); 792 // In affine coordinates, (X_1, Y_1) == (X_2, Y_2) 793 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 794 p224_point_double(x3, y3, z3, x1, y1, z1); 795 return; 796 } 797 798 // ftmp5 = z1*z2 799 if (!mixed) { 800 p224_felem_mul(tmp, z1, z2); 801 p224_felem_reduce(ftmp5, tmp); 802 } else { 803 // special case z2 = 0 is handled later 804 p224_felem_assign(ftmp5, z1); 805 } 806 807 // z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) 808 p224_felem_mul(tmp, ftmp, ftmp5); 809 p224_felem_reduce(z_out, tmp); 810 811 // ftmp = (z1^2*x2 - z2^2*x1)^2 812 p224_felem_assign(ftmp5, ftmp); 813 p224_felem_square(tmp, ftmp); 814 p224_felem_reduce(ftmp, tmp); 815 816 // ftmp5 = (z1^2*x2 - z2^2*x1)^3 817 p224_felem_mul(tmp, ftmp, ftmp5); 818 p224_felem_reduce(ftmp5, tmp); 819 820 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 821 p224_felem_mul(tmp, ftmp2, ftmp); 822 p224_felem_reduce(ftmp2, tmp); 823 824 // tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 825 p224_felem_mul(tmp, ftmp4, ftmp5); 826 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 827 828 // tmp2 = (z1^3*y2 - z2^3*y1)^2 829 p224_felem_square(tmp2, ftmp3); 830 // tmp2[i] < 4 * 2^57 * 2^57 < 2^116 831 832 // tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 833 p224_felem_diff_128_64(tmp2, ftmp5); 834 // tmp2[i] < 2^116 + 2^64 + 8 < 2^117 835 836 // ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 837 p224_felem_assign(ftmp5, ftmp2); 838 p224_felem_scalar(ftmp5, 2); 839 // ftmp5[i] < 2 * 2^57 = 2^58 840 841 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - 842 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 843 p224_felem_diff_128_64(tmp2, ftmp5); 844 // tmp2[i] < 2^117 + 2^64 + 8 < 2^118 845 p224_felem_reduce(x_out, tmp2); 846 847 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out 848 p224_felem_diff(ftmp2, x_out); 849 // ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 850 851 // tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) 852 p224_felem_mul(tmp2, ftmp3, ftmp2); 853 // tmp2[i] < 4 * 2^57 * 2^59 = 2^118 854 855 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - 856 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 857 p224_widefelem_diff(tmp2, tmp); 858 // tmp2[i] < 2^118 + 2^120 < 2^121 859 p224_felem_reduce(y_out, tmp2); 860 861 // the result (x_out, y_out, z_out) is incorrect if one of the inputs is 862 // the point at infinity, so we need to check for this separately 863 864 // if point 1 is at infinity, copy point 2 to output, and vice versa 865 p224_copy_conditional(x_out, x2, z1_is_zero); 866 p224_copy_conditional(x_out, x1, z2_is_zero); 867 p224_copy_conditional(y_out, y2, z1_is_zero); 868 p224_copy_conditional(y_out, y1, z2_is_zero); 869 p224_copy_conditional(z_out, z2, z1_is_zero); 870 p224_copy_conditional(z_out, z1, z2_is_zero); 871 p224_felem_assign(x3, x_out); 872 p224_felem_assign(y3, y_out); 873 p224_felem_assign(z3, z_out); 874 } 875 876 // p224_select_point selects the |idx|th point from a precomputation table and 877 // copies it to out. 878 static void p224_select_point(const uint64_t idx, size_t size, 879 const p224_felem pre_comp[/*size*/][3], 880 p224_felem out[3]) { 881 p224_limb *outlimbs = &out[0][0]; 882 OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem)); 883 884 for (size_t i = 0; i < size; i++) { 885 const p224_limb *inlimbs = &pre_comp[i][0][0]; 886 uint64_t mask = i ^ idx; 887 mask |= mask >> 4; 888 mask |= mask >> 2; 889 mask |= mask >> 1; 890 mask &= 1; 891 mask--; 892 for (size_t j = 0; j < 4 * 3; j++) { 893 outlimbs[j] |= inlimbs[j] & mask; 894 } 895 } 896 } 897 898 // p224_get_bit returns the |i|th bit in |in| 899 static char p224_get_bit(const p224_felem_bytearray in, size_t i) { 900 if (i >= 224) { 901 return 0; 902 } 903 return (in[i >> 3] >> (i & 7)) & 1; 904 } 905 906 // Interleaved point multiplication using precomputed point multiples: 907 // The small point multiples 0*P, 1*P, ..., 16*P are in p_pre_comp, the scalars 908 // in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple 909 // of the generator, using certain (large) precomputed multiples in 910 // g_p224_pre_comp. Output point (X, Y, Z) is stored in x_out, y_out, z_out 911 static void p224_batch_mul(p224_felem x_out, p224_felem y_out, p224_felem z_out, 912 const uint8_t *p_scalar, const uint8_t *g_scalar, 913 const p224_felem p_pre_comp[17][3]) { 914 p224_felem nq[3], tmp[4]; 915 uint64_t bits; 916 uint8_t sign, digit; 917 918 // set nq to the point at infinity 919 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem)); 920 921 // Loop over both scalars msb-to-lsb, interleaving additions of multiples of 922 // the generator (two in each of the last 28 rounds) and additions of p (every 923 // 5th round). 924 int skip = 1; // save two point operations in the first round 925 size_t i = p_scalar != NULL ? 220 : 27; 926 for (;;) { 927 // double 928 if (!skip) { 929 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 930 } 931 932 // add multiples of the generator 933 if (g_scalar != NULL && i <= 27) { 934 // first, look 28 bits upwards 935 bits = p224_get_bit(g_scalar, i + 196) << 3; 936 bits |= p224_get_bit(g_scalar, i + 140) << 2; 937 bits |= p224_get_bit(g_scalar, i + 84) << 1; 938 bits |= p224_get_bit(g_scalar, i + 28); 939 // select the point to add, in constant time 940 p224_select_point(bits, 16, g_p224_pre_comp[1], tmp); 941 942 if (!skip) { 943 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 944 tmp[0], tmp[1], tmp[2]); 945 } else { 946 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 947 skip = 0; 948 } 949 950 // second, look at the current position 951 bits = p224_get_bit(g_scalar, i + 168) << 3; 952 bits |= p224_get_bit(g_scalar, i + 112) << 2; 953 bits |= p224_get_bit(g_scalar, i + 56) << 1; 954 bits |= p224_get_bit(g_scalar, i); 955 // select the point to add, in constant time 956 p224_select_point(bits, 16, g_p224_pre_comp[0], tmp); 957 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 958 tmp[0], tmp[1], tmp[2]); 959 } 960 961 // do other additions every 5 doublings 962 if (p_scalar != NULL && i % 5 == 0) { 963 bits = p224_get_bit(p_scalar, i + 4) << 5; 964 bits |= p224_get_bit(p_scalar, i + 3) << 4; 965 bits |= p224_get_bit(p_scalar, i + 2) << 3; 966 bits |= p224_get_bit(p_scalar, i + 1) << 2; 967 bits |= p224_get_bit(p_scalar, i) << 1; 968 bits |= p224_get_bit(p_scalar, i - 1); 969 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 970 971 // select the point to add or subtract 972 p224_select_point(digit, 17, p_pre_comp, tmp); 973 p224_felem_neg(tmp[3], tmp[1]); // (X, -Y, Z) is the negative point 974 p224_copy_conditional(tmp[1], tmp[3], sign); 975 976 if (!skip) { 977 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, 978 tmp[0], tmp[1], tmp[2]); 979 } else { 980 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 981 skip = 0; 982 } 983 } 984 985 if (i == 0) { 986 break; 987 } 988 --i; 989 } 990 p224_felem_assign(x_out, nq[0]); 991 p224_felem_assign(y_out, nq[1]); 992 p224_felem_assign(z_out, nq[2]); 993 } 994 995 // Takes the Jacobian coordinates (X, Y, Z) of a point and returns 996 // (X', Y') = (X/Z^2, Y/Z^3) 997 static int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, 998 const EC_POINT *point, 999 BIGNUM *x, BIGNUM *y, 1000 BN_CTX *ctx) { 1001 p224_felem z1, z2, x_in, y_in, x_out, y_out; 1002 p224_widefelem tmp; 1003 1004 if (EC_POINT_is_at_infinity(group, point)) { 1005 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); 1006 return 0; 1007 } 1008 1009 if (!p224_BN_to_felem(x_in, &point->X) || 1010 !p224_BN_to_felem(y_in, &point->Y) || 1011 !p224_BN_to_felem(z1, &point->Z)) { 1012 return 0; 1013 } 1014 1015 p224_felem_inv(z2, z1); 1016 p224_felem_square(tmp, z2); 1017 p224_felem_reduce(z1, tmp); 1018 1019 if (x != NULL) { 1020 p224_felem_mul(tmp, x_in, z1); 1021 p224_felem_reduce(x_in, tmp); 1022 p224_felem_contract(x_out, x_in); 1023 if (!p224_felem_to_BN(x, x_out)) { 1024 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1025 return 0; 1026 } 1027 } 1028 1029 if (y != NULL) { 1030 p224_felem_mul(tmp, z1, z2); 1031 p224_felem_reduce(z1, tmp); 1032 p224_felem_mul(tmp, y_in, z1); 1033 p224_felem_reduce(y_in, tmp); 1034 p224_felem_contract(y_out, y_in); 1035 if (!p224_felem_to_BN(y, y_out)) { 1036 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1037 return 0; 1038 } 1039 } 1040 1041 return 1; 1042 } 1043 1044 static int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, 1045 const EC_SCALAR *g_scalar, 1046 const EC_POINT *p, 1047 const EC_SCALAR *p_scalar, BN_CTX *ctx) { 1048 p224_felem p_pre_comp[17][3]; 1049 p224_felem x_in, y_in, z_in, x_out, y_out, z_out; 1050 1051 if (p != NULL && p_scalar != NULL) { 1052 // We treat NULL scalars as 0, and NULL points as points at infinity, i.e., 1053 // they contribute nothing to the linear combination. 1054 OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp)); 1055 // precompute multiples 1056 if (!p224_BN_to_felem(x_out, &p->X) || 1057 !p224_BN_to_felem(y_out, &p->Y) || 1058 !p224_BN_to_felem(z_out, &p->Z)) { 1059 return 0; 1060 } 1061 1062 p224_felem_assign(p_pre_comp[1][0], x_out); 1063 p224_felem_assign(p_pre_comp[1][1], y_out); 1064 p224_felem_assign(p_pre_comp[1][2], z_out); 1065 1066 for (size_t j = 2; j <= 16; ++j) { 1067 if (j & 1) { 1068 p224_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2], 1069 p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2], 1070 0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], 1071 p_pre_comp[j - 1][2]); 1072 } else { 1073 p224_point_double(p_pre_comp[j][0], p_pre_comp[j][1], 1074 p_pre_comp[j][2], p_pre_comp[j / 2][0], 1075 p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]); 1076 } 1077 } 1078 } 1079 1080 p224_batch_mul(x_out, y_out, z_out, 1081 (p != NULL && p_scalar != NULL) ? p_scalar->bytes : NULL, 1082 g_scalar != NULL ? g_scalar->bytes : NULL, 1083 (const p224_felem(*)[3])p_pre_comp); 1084 1085 // reduce the output to its unique minimal representation 1086 p224_felem_contract(x_in, x_out); 1087 p224_felem_contract(y_in, y_out); 1088 p224_felem_contract(z_in, z_out); 1089 if (!p224_felem_to_BN(&r->X, x_in) || 1090 !p224_felem_to_BN(&r->Y, y_in) || 1091 !p224_felem_to_BN(&r->Z, z_in)) { 1092 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1093 return 0; 1094 } 1095 return 1; 1096 } 1097 1098 DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) { 1099 out->group_init = ec_GFp_simple_group_init; 1100 out->group_finish = ec_GFp_simple_group_finish; 1101 out->group_set_curve = ec_GFp_simple_group_set_curve; 1102 out->point_get_affine_coordinates = 1103 ec_GFp_nistp224_point_get_affine_coordinates; 1104 out->mul = ec_GFp_nistp224_points_mul; 1105 out->mul_public = ec_GFp_nistp224_points_mul; 1106 out->field_mul = ec_GFp_simple_field_mul; 1107 out->field_sqr = ec_GFp_simple_field_sqr; 1108 out->field_encode = NULL; 1109 out->field_decode = NULL; 1110 }; 1111 1112 #endif // BORINGSSL_HAS_UINT128 && !SMALL 1113
