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 30 #include <string.h> 31 32 #include "internal.h" 33 #include "../delocate.h" 34 #include "../../internal.h" 35 36 37 /* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 38 * using 64-bit coefficients called 'limbs', and sometimes (for multiplication 39 * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 40 * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-p224_limb 41 * representation is an 'p224_felem'; a 7-p224_widelimb representation is a 42 * 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we 43 * don't always reduce the representations: we ensure that inputs to each 44 * p224_felem multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 45 * 4*2^60*2^60, and fit into a 128-bit word without overflow. The coefficients 46 * are then again partially reduced to obtain an p224_felem satisfying a_i < 47 * 2^57. We only reduce to the unique minimal representation at the end of the 48 * computation. */ 49 50 typedef uint64_t p224_limb; 51 typedef uint128_t p224_widelimb; 52 53 typedef p224_limb p224_felem[4]; 54 typedef p224_widelimb p224_widefelem[7]; 55 56 /* Field element represented as a byte arrary. 28*8 = 224 bits is also the 57 * group order size for the elliptic curve, and we also use this type for 58 * scalars for point multiplication. */ 59 typedef uint8_t p224_felem_bytearray[28]; 60 61 /* Precomputed multiples of the standard generator 62 * Points are given in coordinates (X, Y, Z) where Z normally is 1 63 * (0 for the point at infinity). 64 * For each field element, slice a_0 is word 0, etc. 65 * 66 * The table has 2 * 16 elements, starting with the following: 67 * index | bits | point 68 * ------+---------+------------------------------ 69 * 0 | 0 0 0 0 | 0G 70 * 1 | 0 0 0 1 | 1G 71 * 2 | 0 0 1 0 | 2^56G 72 * 3 | 0 0 1 1 | (2^56 + 1)G 73 * 4 | 0 1 0 0 | 2^112G 74 * 5 | 0 1 0 1 | (2^112 + 1)G 75 * 6 | 0 1 1 0 | (2^112 + 2^56)G 76 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G 77 * 8 | 1 0 0 0 | 2^168G 78 * 9 | 1 0 0 1 | (2^168 + 1)G 79 * 10 | 1 0 1 0 | (2^168 + 2^56)G 80 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G 81 * 12 | 1 1 0 0 | (2^168 + 2^112)G 82 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G 83 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G 84 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G 85 * followed by a copy of this with each element multiplied by 2^28. 86 * 87 * The reason for this is so that we can clock bits into four different 88 * locations when doing simple scalar multiplies against the base point, 89 * and then another four locations using the second 16 elements. */ 90 static const p224_felem g_p224_pre_comp[2][16][3] = { 91 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 92 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, 93 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, 94 {1, 0, 0, 0}}, 95 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, 96 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, 97 {1, 0, 0, 0}}, 98 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, 99 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, 100 {1, 0, 0, 0}}, 101 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, 102 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, 103 {1, 0, 0, 0}}, 104 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, 105 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, 106 {1, 0, 0, 0}}, 107 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, 108 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, 109 {1, 0, 0, 0}}, 110 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, 111 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, 112 {1, 0, 0, 0}}, 113 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, 114 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, 115 {1, 0, 0, 0}}, 116 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, 117 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, 118 {1, 0, 0, 0}}, 119 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, 120 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, 121 {1, 0, 0, 0}}, 122 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, 123 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, 124 {1, 0, 0, 0}}, 125 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, 126 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, 127 {1, 0, 0, 0}}, 128 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, 129 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, 130 {1, 0, 0, 0}}, 131 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, 132 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, 133 {1, 0, 0, 0}}, 134 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, 135 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, 136 {1, 0, 0, 0}}}, 137 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 138 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, 139 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, 140 {1, 0, 0, 0}}, 141 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, 142 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, 143 {1, 0, 0, 0}}, 144 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, 145 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, 146 {1, 0, 0, 0}}, 147 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, 148 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, 149 {1, 0, 0, 0}}, 150 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, 151 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, 152 {1, 0, 0, 0}}, 153 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, 154 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, 155 {1, 0, 0, 0}}, 156 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, 157 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, 158 {1, 0, 0, 0}}, 159 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, 160 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, 161 {1, 0, 0, 0}}, 162 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, 163 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, 164 {1, 0, 0, 0}}, 165 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, 166 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, 167 {1, 0, 0, 0}}, 168 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, 169 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, 170 {1, 0, 0, 0}}, 171 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, 172 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, 173 {1, 0, 0, 0}}, 174 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, 175 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, 176 {1, 0, 0, 0}}, 177 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, 178 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, 179 {1, 0, 0, 0}}, 180 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, 181 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, 182 {1, 0, 0, 0}}}}; 183 184 static uint64_t p224_load_u64(const uint8_t in[8]) { 185 uint64_t ret; 186 OPENSSL_memcpy(&ret, in, sizeof(ret)); 187 return ret; 188 } 189 190 /* Helper functions to convert field elements to/from internal representation */ 191 static void p224_bin28_to_felem(p224_felem out, const uint8_t in[28]) { 192 out[0] = p224_load_u64(in) & 0x00ffffffffffffff; 193 out[1] = p224_load_u64(in + 7) & 0x00ffffffffffffff; 194 out[2] = p224_load_u64(in + 14) & 0x00ffffffffffffff; 195 out[3] = p224_load_u64(in + 20) >> 8; 196 } 197 198 static void p224_felem_to_bin28(uint8_t out[28], const p224_felem in) { 199 for (size_t i = 0; i < 7; ++i) { 200 out[i] = in[0] >> (8 * i); 201 out[i + 7] = in[1] >> (8 * i); 202 out[i + 14] = in[2] >> (8 * i); 203 out[i + 21] = in[3] >> (8 * i); 204 } 205 } 206 207 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */ 208 static void p224_flip_endian(uint8_t *out, const uint8_t *in, size_t len) { 209 for (size_t i = 0; i < len; ++i) { 210 out[i] = in[len - 1 - i]; 211 } 212 } 213 214 /* From OpenSSL BIGNUM to internal representation */ 215 static int p224_BN_to_felem(p224_felem out, const BIGNUM *bn) { 216 /* BN_bn2bin eats leading zeroes */ 217 p224_felem_bytearray b_out; 218 OPENSSL_memset(b_out, 0, sizeof(b_out)); 219 size_t num_bytes = BN_num_bytes(bn); 220 if (num_bytes > sizeof(b_out) || 221 BN_is_negative(bn)) { 222 OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); 223 return 0; 224 } 225 226 p224_felem_bytearray b_in; 227 num_bytes = BN_bn2bin(bn, b_in); 228 p224_flip_endian(b_out, b_in, num_bytes); 229 p224_bin28_to_felem(out, b_out); 230 return 1; 231 } 232 233 /* From internal representation to OpenSSL BIGNUM */ 234 static BIGNUM *p224_felem_to_BN(BIGNUM *out, const p224_felem in) { 235 p224_felem_bytearray b_in, b_out; 236 p224_felem_to_bin28(b_in, in); 237 p224_flip_endian(b_out, b_in, sizeof(b_out)); 238 return BN_bin2bn(b_out, sizeof(b_out), out); 239 } 240 241 /* Field operations, using the internal representation of field elements. 242 * NB! These operations are specific to our point multiplication and cannot be 243 * expected to be correct in general - e.g., multiplication with a large scalar 244 * will cause an overflow. */ 245 246 static void p224_felem_assign(p224_felem out, const p224_felem in) { 247 out[0] = in[0]; 248 out[1] = in[1]; 249 out[2] = in[2]; 250 out[3] = in[3]; 251 } 252 253 /* Sum two field elements: out += in */ 254 static void p224_felem_sum(p224_felem out, const p224_felem in) { 255 out[0] += in[0]; 256 out[1] += in[1]; 257 out[2] += in[2]; 258 out[3] += in[3]; 259 } 260 261 /* Get negative value: out = -in */ 262 /* Assumes in[i] < 2^57 */ 263 static void p224_felem_neg(p224_felem out, const p224_felem in) { 264 static const p224_limb two58p2 = 265 (((p224_limb)1) << 58) + (((p224_limb)1) << 2); 266 static const p224_limb two58m2 = 267 (((p224_limb)1) << 58) - (((p224_limb)1) << 2); 268 static const p224_limb two58m42m2 = 269 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2); 270 271 /* Set to 0 mod 2^224-2^96+1 to ensure out > in */ 272 out[0] = two58p2 - in[0]; 273 out[1] = two58m42m2 - in[1]; 274 out[2] = two58m2 - in[2]; 275 out[3] = two58m2 - in[3]; 276 } 277 278 /* Subtract field elements: out -= in */ 279 /* Assumes in[i] < 2^57 */ 280 static void p224_felem_diff(p224_felem out, const p224_felem in) { 281 static const p224_limb two58p2 = 282 (((p224_limb)1) << 58) + (((p224_limb)1) << 2); 283 static const p224_limb two58m2 = 284 (((p224_limb)1) << 58) - (((p224_limb)1) << 2); 285 static const p224_limb two58m42m2 = 286 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2); 287 288 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 289 out[0] += two58p2; 290 out[1] += two58m42m2; 291 out[2] += two58m2; 292 out[3] += two58m2; 293 294 out[0] -= in[0]; 295 out[1] -= in[1]; 296 out[2] -= in[2]; 297 out[3] -= in[3]; 298 } 299 300 /* Subtract in unreduced 128-bit mode: out -= in */ 301 /* Assumes in[i] < 2^119 */ 302 static void p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) { 303 static const p224_widelimb two120 = ((p224_widelimb)1) << 120; 304 static const p224_widelimb two120m64 = 305 (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64); 306 static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) - 307 (((p224_widelimb)1) << 104) - 308 (((p224_widelimb)1) << 64); 309 310 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 311 out[0] += two120; 312 out[1] += two120m64; 313 out[2] += two120m64; 314 out[3] += two120; 315 out[4] += two120m104m64; 316 out[5] += two120m64; 317 out[6] += two120m64; 318 319 out[0] -= in[0]; 320 out[1] -= in[1]; 321 out[2] -= in[2]; 322 out[3] -= in[3]; 323 out[4] -= in[4]; 324 out[5] -= in[5]; 325 out[6] -= in[6]; 326 } 327 328 /* Subtract in mixed mode: out128 -= in64 */ 329 /* in[i] < 2^63 */ 330 static void p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) { 331 static const p224_widelimb two64p8 = 332 (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8); 333 static const p224_widelimb two64m8 = 334 (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8); 335 static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) - 336 (((p224_widelimb)1) << 48) - 337 (((p224_widelimb)1) << 8); 338 339 /* Add 0 mod 2^224-2^96+1 to ensure out > in */ 340 out[0] += two64p8; 341 out[1] += two64m48m8; 342 out[2] += two64m8; 343 out[3] += two64m8; 344 345 out[0] -= in[0]; 346 out[1] -= in[1]; 347 out[2] -= in[2]; 348 out[3] -= in[3]; 349 } 350 351 /* Multiply a field element by a scalar: out = out * scalar 352 * The scalars we actually use are small, so results fit without overflow */ 353 static void p224_felem_scalar(p224_felem out, const p224_limb scalar) { 354 out[0] *= scalar; 355 out[1] *= scalar; 356 out[2] *= scalar; 357 out[3] *= scalar; 358 } 359 360 /* Multiply an unreduced field element by a scalar: out = out * scalar 361 * The scalars we actually use are small, so results fit without overflow */ 362 static void p224_widefelem_scalar(p224_widefelem out, 363 const p224_widelimb scalar) { 364 out[0] *= scalar; 365 out[1] *= scalar; 366 out[2] *= scalar; 367 out[3] *= scalar; 368 out[4] *= scalar; 369 out[5] *= scalar; 370 out[6] *= scalar; 371 } 372 373 /* Square a field element: out = in^2 */ 374 static void p224_felem_square(p224_widefelem out, const p224_felem in) { 375 p224_limb tmp0, tmp1, tmp2; 376 tmp0 = 2 * in[0]; 377 tmp1 = 2 * in[1]; 378 tmp2 = 2 * in[2]; 379 out[0] = ((p224_widelimb)in[0]) * in[0]; 380 out[1] = ((p224_widelimb)in[0]) * tmp1; 381 out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1]; 382 out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2; 383 out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2]; 384 out[5] = ((p224_widelimb)in[3]) * tmp2; 385 out[6] = ((p224_widelimb)in[3]) * in[3]; 386 } 387 388 /* Multiply two field elements: out = in1 * in2 */ 389 static void p224_felem_mul(p224_widefelem out, const p224_felem in1, 390 const p224_felem in2) { 391 out[0] = ((p224_widelimb)in1[0]) * in2[0]; 392 out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0]; 393 out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] + 394 ((p224_widelimb)in1[2]) * in2[0]; 395 out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] + 396 ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0]; 397 out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] + 398 ((p224_widelimb)in1[3]) * in2[1]; 399 out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2]; 400 out[6] = ((p224_widelimb)in1[3]) * in2[3]; 401 } 402 403 /* Reduce seven 128-bit coefficients to four 64-bit coefficients. 404 * Requires in[i] < 2^126, 405 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */ 406 static void p224_felem_reduce(p224_felem out, const p224_widefelem in) { 407 static const p224_widelimb two127p15 = 408 (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15); 409 static const p224_widelimb two127m71 = 410 (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71); 411 static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) - 412 (((p224_widelimb)1) << 71) - 413 (((p224_widelimb)1) << 55); 414 p224_widelimb output[5]; 415 416 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */ 417 output[0] = in[0] + two127p15; 418 output[1] = in[1] + two127m71m55; 419 output[2] = in[2] + two127m71; 420 output[3] = in[3]; 421 output[4] = in[4]; 422 423 /* Eliminate in[4], in[5], in[6] */ 424 output[4] += in[6] >> 16; 425 output[3] += (in[6] & 0xffff) << 40; 426 output[2] -= in[6]; 427 428 output[3] += in[5] >> 16; 429 output[2] += (in[5] & 0xffff) << 40; 430 output[1] -= in[5]; 431 432 output[2] += output[4] >> 16; 433 output[1] += (output[4] & 0xffff) << 40; 434 output[0] -= output[4]; 435 436 /* Carry 2 -> 3 -> 4 */ 437 output[3] += output[2] >> 56; 438 output[2] &= 0x00ffffffffffffff; 439 440 output[4] = output[3] >> 56; 441 output[3] &= 0x00ffffffffffffff; 442 443 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */ 444 445 /* Eliminate output[4] */ 446 output[2] += output[4] >> 16; 447 /* output[2] < 2^56 + 2^56 = 2^57 */ 448 output[1] += (output[4] & 0xffff) << 40; 449 output[0] -= output[4]; 450 451 /* Carry 0 -> 1 -> 2 -> 3 */ 452 output[1] += output[0] >> 56; 453 out[0] = output[0] & 0x00ffffffffffffff; 454 455 output[2] += output[1] >> 56; 456 /* output[2] < 2^57 + 2^72 */ 457 out[1] = output[1] & 0x00ffffffffffffff; 458 output[3] += output[2] >> 56; 459 /* output[3] <= 2^56 + 2^16 */ 460 out[2] = output[2] & 0x00ffffffffffffff; 461 462 /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, 463 * out[3] <= 2^56 + 2^16 (due to final carry), 464 * so out < 2*p */ 465 out[3] = output[3]; 466 } 467 468 /* Reduce to unique minimal representation. 469 * Requires 0 <= in < 2*p (always call p224_felem_reduce first) */ 470 static void p224_felem_contract(p224_felem out, const p224_felem in) { 471 static const int64_t two56 = ((p224_limb)1) << 56; 472 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */ 473 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */ 474 int64_t tmp[4], a; 475 tmp[0] = in[0]; 476 tmp[1] = in[1]; 477 tmp[2] = in[2]; 478 tmp[3] = in[3]; 479 /* Case 1: a = 1 iff in >= 2^224 */ 480 a = (in[3] >> 56); 481 tmp[0] -= a; 482 tmp[1] += a << 40; 483 tmp[3] &= 0x00ffffffffffffff; 484 /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and 485 * the lower part is non-zero */ 486 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | 487 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); 488 a &= 0x00ffffffffffffff; 489 /* turn a into an all-one mask (if a = 0) or an all-zero mask */ 490 a = (a - 1) >> 63; 491 /* subtract 2^224 - 2^96 + 1 if a is all-one */ 492 tmp[3] &= a ^ 0xffffffffffffffff; 493 tmp[2] &= a ^ 0xffffffffffffffff; 494 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; 495 tmp[0] -= 1 & a; 496 497 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must 498 * be non-zero, so we only need one step */ 499 a = tmp[0] >> 63; 500 tmp[0] += two56 & a; 501 tmp[1] -= 1 & a; 502 503 /* carry 1 -> 2 -> 3 */ 504 tmp[2] += tmp[1] >> 56; 505 tmp[1] &= 0x00ffffffffffffff; 506 507 tmp[3] += tmp[2] >> 56; 508 tmp[2] &= 0x00ffffffffffffff; 509 510 /* Now 0 <= out < p */ 511 out[0] = tmp[0]; 512 out[1] = tmp[1]; 513 out[2] = tmp[2]; 514 out[3] = tmp[3]; 515 } 516 517 /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field 518 * elements are reduced to in < 2^225, so we only need to check three cases: 0, 519 * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */ 520 static p224_limb p224_felem_is_zero(const p224_felem in) { 521 p224_limb zero = in[0] | in[1] | in[2] | in[3]; 522 zero = (((int64_t)(zero)-1) >> 63) & 1; 523 524 p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | 525 (in[2] ^ 0x00ffffffffffffff) | 526 (in[3] ^ 0x00ffffffffffffff); 527 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; 528 p224_limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | 529 (in[2] ^ 0x00ffffffffffffff) | 530 (in[3] ^ 0x01ffffffffffffff); 531 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; 532 return (zero | two224m96p1 | two225m97p2); 533 } 534 535 /* Invert a field element */ 536 /* Computation chain copied from djb's code */ 537 static void p224_felem_inv(p224_felem out, const p224_felem in) { 538 p224_felem ftmp, ftmp2, ftmp3, ftmp4; 539 p224_widefelem tmp; 540 541 p224_felem_square(tmp, in); 542 p224_felem_reduce(ftmp, tmp); /* 2 */ 543 p224_felem_mul(tmp, in, ftmp); 544 p224_felem_reduce(ftmp, tmp); /* 2^2 - 1 */ 545 p224_felem_square(tmp, ftmp); 546 p224_felem_reduce(ftmp, tmp); /* 2^3 - 2 */ 547 p224_felem_mul(tmp, in, ftmp); 548 p224_felem_reduce(ftmp, tmp); /* 2^3 - 1 */ 549 p224_felem_square(tmp, ftmp); 550 p224_felem_reduce(ftmp2, tmp); /* 2^4 - 2 */ 551 p224_felem_square(tmp, ftmp2); 552 p224_felem_reduce(ftmp2, tmp); /* 2^5 - 4 */ 553 p224_felem_square(tmp, ftmp2); 554 p224_felem_reduce(ftmp2, tmp); /* 2^6 - 8 */ 555 p224_felem_mul(tmp, ftmp2, ftmp); 556 p224_felem_reduce(ftmp, tmp); /* 2^6 - 1 */ 557 p224_felem_square(tmp, ftmp); 558 p224_felem_reduce(ftmp2, tmp); /* 2^7 - 2 */ 559 for (size_t i = 0; i < 5; ++i) { /* 2^12 - 2^6 */ 560 p224_felem_square(tmp, ftmp2); 561 p224_felem_reduce(ftmp2, tmp); 562 } 563 p224_felem_mul(tmp, ftmp2, ftmp); 564 p224_felem_reduce(ftmp2, tmp); /* 2^12 - 1 */ 565 p224_felem_square(tmp, ftmp2); 566 p224_felem_reduce(ftmp3, tmp); /* 2^13 - 2 */ 567 for (size_t i = 0; i < 11; ++i) {/* 2^24 - 2^12 */ 568 p224_felem_square(tmp, ftmp3); 569 p224_felem_reduce(ftmp3, tmp); 570 } 571 p224_felem_mul(tmp, ftmp3, ftmp2); 572 p224_felem_reduce(ftmp2, tmp); /* 2^24 - 1 */ 573 p224_felem_square(tmp, ftmp2); 574 p224_felem_reduce(ftmp3, tmp); /* 2^25 - 2 */ 575 for (size_t i = 0; i < 23; ++i) {/* 2^48 - 2^24 */ 576 p224_felem_square(tmp, ftmp3); 577 p224_felem_reduce(ftmp3, tmp); 578 } 579 p224_felem_mul(tmp, ftmp3, ftmp2); 580 p224_felem_reduce(ftmp3, tmp); /* 2^48 - 1 */ 581 p224_felem_square(tmp, ftmp3); 582 p224_felem_reduce(ftmp4, tmp); /* 2^49 - 2 */ 583 for (size_t i = 0; i < 47; ++i) {/* 2^96 - 2^48 */ 584 p224_felem_square(tmp, ftmp4); 585 p224_felem_reduce(ftmp4, tmp); 586 } 587 p224_felem_mul(tmp, ftmp3, ftmp4); 588 p224_felem_reduce(ftmp3, tmp); /* 2^96 - 1 */ 589 p224_felem_square(tmp, ftmp3); 590 p224_felem_reduce(ftmp4, tmp); /* 2^97 - 2 */ 591 for (size_t i = 0; i < 23; ++i) {/* 2^120 - 2^24 */ 592 p224_felem_square(tmp, ftmp4); 593 p224_felem_reduce(ftmp4, tmp); 594 } 595 p224_felem_mul(tmp, ftmp2, ftmp4); 596 p224_felem_reduce(ftmp2, tmp); /* 2^120 - 1 */ 597 for (size_t i = 0; i < 6; ++i) { /* 2^126 - 2^6 */ 598 p224_felem_square(tmp, ftmp2); 599 p224_felem_reduce(ftmp2, tmp); 600 } 601 p224_felem_mul(tmp, ftmp2, ftmp); 602 p224_felem_reduce(ftmp, tmp); /* 2^126 - 1 */ 603 p224_felem_square(tmp, ftmp); 604 p224_felem_reduce(ftmp, tmp); /* 2^127 - 2 */ 605 p224_felem_mul(tmp, ftmp, in); 606 p224_felem_reduce(ftmp, tmp); /* 2^127 - 1 */ 607 for (size_t i = 0; i < 97; ++i) {/* 2^224 - 2^97 */ 608 p224_felem_square(tmp, ftmp); 609 p224_felem_reduce(ftmp, tmp); 610 } 611 p224_felem_mul(tmp, ftmp, ftmp3); 612 p224_felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */ 613 } 614 615 /* Copy in constant time: 616 * if icopy == 1, copy in to out, 617 * if icopy == 0, copy out to itself. */ 618 static void p224_copy_conditional(p224_felem out, const p224_felem in, 619 p224_limb icopy) { 620 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ 621 const p224_limb copy = -icopy; 622 for (size_t i = 0; i < 4; ++i) { 623 const p224_limb tmp = copy & (in[i] ^ out[i]); 624 out[i] ^= tmp; 625 } 626 } 627 628 /* ELLIPTIC CURVE POINT OPERATIONS 629 * 630 * Points are represented in Jacobian projective coordinates: 631 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), 632 * or to the point at infinity if Z == 0. */ 633 634 /* Double an elliptic curve point: 635 * (X', Y', Z') = 2 * (X, Y, Z), where 636 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 637 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 638 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z 639 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, 640 * while x_out == y_in is not (maybe this works, but it's not tested). */ 641 static void p224_point_double(p224_felem x_out, p224_felem y_out, 642 p224_felem z_out, const p224_felem x_in, 643 const p224_felem y_in, const p224_felem z_in) { 644 p224_widefelem tmp, tmp2; 645 p224_felem delta, gamma, beta, alpha, ftmp, ftmp2; 646 647 p224_felem_assign(ftmp, x_in); 648 p224_felem_assign(ftmp2, x_in); 649 650 /* delta = z^2 */ 651 p224_felem_square(tmp, z_in); 652 p224_felem_reduce(delta, tmp); 653 654 /* gamma = y^2 */ 655 p224_felem_square(tmp, y_in); 656 p224_felem_reduce(gamma, tmp); 657 658 /* beta = x*gamma */ 659 p224_felem_mul(tmp, x_in, gamma); 660 p224_felem_reduce(beta, tmp); 661 662 /* alpha = 3*(x-delta)*(x+delta) */ 663 p224_felem_diff(ftmp, delta); 664 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ 665 p224_felem_sum(ftmp2, delta); 666 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ 667 p224_felem_scalar(ftmp2, 3); 668 /* ftmp2[i] < 3 * 2^58 < 2^60 */ 669 p224_felem_mul(tmp, ftmp, ftmp2); 670 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */ 671 p224_felem_reduce(alpha, tmp); 672 673 /* x' = alpha^2 - 8*beta */ 674 p224_felem_square(tmp, alpha); 675 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 676 p224_felem_assign(ftmp, beta); 677 p224_felem_scalar(ftmp, 8); 678 /* ftmp[i] < 8 * 2^57 = 2^60 */ 679 p224_felem_diff_128_64(tmp, ftmp); 680 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 681 p224_felem_reduce(x_out, tmp); 682 683 /* z' = (y + z)^2 - gamma - delta */ 684 p224_felem_sum(delta, gamma); 685 /* delta[i] < 2^57 + 2^57 = 2^58 */ 686 p224_felem_assign(ftmp, y_in); 687 p224_felem_sum(ftmp, z_in); 688 /* ftmp[i] < 2^57 + 2^57 = 2^58 */ 689 p224_felem_square(tmp, ftmp); 690 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */ 691 p224_felem_diff_128_64(tmp, delta); 692 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */ 693 p224_felem_reduce(z_out, tmp); 694 695 /* y' = alpha*(4*beta - x') - 8*gamma^2 */ 696 p224_felem_scalar(beta, 4); 697 /* beta[i] < 4 * 2^57 = 2^59 */ 698 p224_felem_diff(beta, x_out); 699 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */ 700 p224_felem_mul(tmp, alpha, beta); 701 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */ 702 p224_felem_square(tmp2, gamma); 703 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */ 704 p224_widefelem_scalar(tmp2, 8); 705 /* tmp2[i] < 8 * 2^116 = 2^119 */ 706 p224_widefelem_diff(tmp, tmp2); 707 /* tmp[i] < 2^119 + 2^120 < 2^121 */ 708 p224_felem_reduce(y_out, tmp); 709 } 710 711 /* Add two elliptic curve points: 712 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where 713 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - 714 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 715 * 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 * 716 * X_1)^2 - X_3) - 717 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 718 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) 719 * 720 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */ 721 722 /* This function is not entirely constant-time: it includes a branch for 723 * checking whether the two input points are equal, (while not equal to the 724 * point at infinity). This case never happens during single point 725 * multiplication, so there is no timing leak for ECDH or ECDSA signing. */ 726 static void p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3, 727 const p224_felem x1, const p224_felem y1, 728 const p224_felem z1, const int mixed, 729 const p224_felem x2, const p224_felem y2, 730 const p224_felem z2) { 731 p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; 732 p224_widefelem tmp, tmp2; 733 p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal; 734 735 if (!mixed) { 736 /* ftmp2 = z2^2 */ 737 p224_felem_square(tmp, z2); 738 p224_felem_reduce(ftmp2, tmp); 739 740 /* ftmp4 = z2^3 */ 741 p224_felem_mul(tmp, ftmp2, z2); 742 p224_felem_reduce(ftmp4, tmp); 743 744 /* ftmp4 = z2^3*y1 */ 745 p224_felem_mul(tmp2, ftmp4, y1); 746 p224_felem_reduce(ftmp4, tmp2); 747 748 /* ftmp2 = z2^2*x1 */ 749 p224_felem_mul(tmp2, ftmp2, x1); 750 p224_felem_reduce(ftmp2, tmp2); 751 } else { 752 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */ 753 754 /* ftmp4 = z2^3*y1 */ 755 p224_felem_assign(ftmp4, y1); 756 757 /* ftmp2 = z2^2*x1 */ 758 p224_felem_assign(ftmp2, x1); 759 } 760 761 /* ftmp = z1^2 */ 762 p224_felem_square(tmp, z1); 763 p224_felem_reduce(ftmp, tmp); 764 765 /* ftmp3 = z1^3 */ 766 p224_felem_mul(tmp, ftmp, z1); 767 p224_felem_reduce(ftmp3, tmp); 768 769 /* tmp = z1^3*y2 */ 770 p224_felem_mul(tmp, ftmp3, y2); 771 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 772 773 /* ftmp3 = z1^3*y2 - z2^3*y1 */ 774 p224_felem_diff_128_64(tmp, ftmp4); 775 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 776 p224_felem_reduce(ftmp3, tmp); 777 778 /* tmp = z1^2*x2 */ 779 p224_felem_mul(tmp, ftmp, x2); 780 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 781 782 /* ftmp = z1^2*x2 - z2^2*x1 */ 783 p224_felem_diff_128_64(tmp, ftmp2); 784 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ 785 p224_felem_reduce(ftmp, tmp); 786 787 /* the formulae are incorrect if the points are equal 788 * so we check for this and do doubling if this happens */ 789 x_equal = p224_felem_is_zero(ftmp); 790 y_equal = p224_felem_is_zero(ftmp3); 791 z1_is_zero = p224_felem_is_zero(z1); 792 z2_is_zero = p224_felem_is_zero(z2); 793 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */ 794 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 795 p224_point_double(x3, y3, z3, x1, y1, z1); 796 return; 797 } 798 799 /* ftmp5 = z1*z2 */ 800 if (!mixed) { 801 p224_felem_mul(tmp, z1, z2); 802 p224_felem_reduce(ftmp5, tmp); 803 } else { 804 /* special case z2 = 0 is handled later */ 805 p224_felem_assign(ftmp5, z1); 806 } 807 808 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */ 809 p224_felem_mul(tmp, ftmp, ftmp5); 810 p224_felem_reduce(z_out, tmp); 811 812 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ 813 p224_felem_assign(ftmp5, ftmp); 814 p224_felem_square(tmp, ftmp); 815 p224_felem_reduce(ftmp, tmp); 816 817 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */ 818 p224_felem_mul(tmp, ftmp, ftmp5); 819 p224_felem_reduce(ftmp5, tmp); 820 821 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 822 p224_felem_mul(tmp, ftmp2, ftmp); 823 p224_felem_reduce(ftmp2, tmp); 824 825 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 826 p224_felem_mul(tmp, ftmp4, ftmp5); 827 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ 828 829 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */ 830 p224_felem_square(tmp2, ftmp3); 831 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */ 832 833 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */ 834 p224_felem_diff_128_64(tmp2, ftmp5); 835 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */ 836 837 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 838 p224_felem_assign(ftmp5, ftmp2); 839 p224_felem_scalar(ftmp5, 2); 840 /* ftmp5[i] < 2 * 2^57 = 2^58 */ 841 842 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - 843 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 844 p224_felem_diff_128_64(tmp2, ftmp5); 845 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */ 846 p224_felem_reduce(x_out, tmp2); 847 848 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */ 849 p224_felem_diff(ftmp2, x_out); 850 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */ 851 852 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */ 853 p224_felem_mul(tmp2, ftmp3, ftmp2); 854 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ 855 856 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - 857 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 858 p224_widefelem_diff(tmp2, tmp); 859 /* tmp2[i] < 2^118 + 2^120 < 2^121 */ 860 p224_felem_reduce(y_out, tmp2); 861 862 /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is 863 * the point at infinity, so we need to check for this separately */ 864 865 /* if point 1 is at infinity, copy point 2 to output, and vice versa */ 866 p224_copy_conditional(x_out, x2, z1_is_zero); 867 p224_copy_conditional(x_out, x1, z2_is_zero); 868 p224_copy_conditional(y_out, y2, z1_is_zero); 869 p224_copy_conditional(y_out, y1, z2_is_zero); 870 p224_copy_conditional(z_out, z2, z1_is_zero); 871 p224_copy_conditional(z_out, z1, z2_is_zero); 872 p224_felem_assign(x3, x_out); 873 p224_felem_assign(y3, y_out); 874 p224_felem_assign(z3, z_out); 875 } 876 877 /* p224_select_point selects the |idx|th point from a precomputation table and 878 * copies it to out. */ 879 static void p224_select_point(const uint64_t idx, size_t size, 880 const p224_felem pre_comp[/*size*/][3], 881 p224_felem out[3]) { 882 p224_limb *outlimbs = &out[0][0]; 883 OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem)); 884 885 for (size_t i = 0; i < size; i++) { 886 const p224_limb *inlimbs = &pre_comp[i][0][0]; 887 uint64_t mask = i ^ idx; 888 mask |= mask >> 4; 889 mask |= mask >> 2; 890 mask |= mask >> 1; 891 mask &= 1; 892 mask--; 893 for (size_t j = 0; j < 4 * 3; j++) { 894 outlimbs[j] |= inlimbs[j] & mask; 895 } 896 } 897 } 898 899 /* p224_get_bit returns the |i|th bit in |in| */ 900 static char p224_get_bit(const p224_felem_bytearray in, size_t i) { 901 if (i >= 224) { 902 return 0; 903 } 904 return (in[i >> 3] >> (i & 7)) & 1; 905 } 906 907 /* Interleaved point multiplication using precomputed point multiples: 908 * The small point multiples 0*P, 1*P, ..., 16*P are in p_pre_comp, the scalars 909 * in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple 910 * of the generator, using certain (large) precomputed multiples in 911 * g_p224_pre_comp. Output point (X, Y, Z) is stored in x_out, y_out, z_out */ 912 static void p224_batch_mul(p224_felem x_out, p224_felem y_out, p224_felem z_out, 913 const uint8_t *p_scalar, const uint8_t *g_scalar, 914 const p224_felem p_pre_comp[17][3]) { 915 p224_felem nq[3], tmp[4]; 916 uint64_t bits; 917 uint8_t sign, digit; 918 919 /* set nq to the point at infinity */ 920 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem)); 921 922 /* Loop over both scalars msb-to-lsb, interleaving additions of multiples of 923 * the generator (two in each of the last 28 rounds) and additions of p (every 924 * 5th round). */ 925 int skip = 1; /* save two point operations in the first round */ 926 size_t i = p_scalar != NULL ? 220 : 27; 927 for (;;) { 928 /* double */ 929 if (!skip) { 930 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 931 } 932 933 /* add multiples of the generator */ 934 if (g_scalar != NULL && i <= 27) { 935 /* first, look 28 bits upwards */ 936 bits = p224_get_bit(g_scalar, i + 196) << 3; 937 bits |= p224_get_bit(g_scalar, i + 140) << 2; 938 bits |= p224_get_bit(g_scalar, i + 84) << 1; 939 bits |= p224_get_bit(g_scalar, i + 28); 940 /* select the point to add, in constant time */ 941 p224_select_point(bits, 16, g_p224_pre_comp[1], tmp); 942 943 if (!skip) { 944 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 945 tmp[0], tmp[1], tmp[2]); 946 } else { 947 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 948 skip = 0; 949 } 950 951 /* second, look at the current position */ 952 bits = p224_get_bit(g_scalar, i + 168) << 3; 953 bits |= p224_get_bit(g_scalar, i + 112) << 2; 954 bits |= p224_get_bit(g_scalar, i + 56) << 1; 955 bits |= p224_get_bit(g_scalar, i); 956 /* select the point to add, in constant time */ 957 p224_select_point(bits, 16, g_p224_pre_comp[0], tmp); 958 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 959 tmp[0], tmp[1], tmp[2]); 960 } 961 962 /* do other additions every 5 doublings */ 963 if (p_scalar != NULL && i % 5 == 0) { 964 bits = p224_get_bit(p_scalar, i + 4) << 5; 965 bits |= p224_get_bit(p_scalar, i + 3) << 4; 966 bits |= p224_get_bit(p_scalar, i + 2) << 3; 967 bits |= p224_get_bit(p_scalar, i + 1) << 2; 968 bits |= p224_get_bit(p_scalar, i) << 1; 969 bits |= p224_get_bit(p_scalar, i - 1); 970 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 971 972 /* select the point to add or subtract */ 973 p224_select_point(digit, 17, p_pre_comp, tmp); 974 p224_felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */ 975 p224_copy_conditional(tmp[1], tmp[3], sign); 976 977 if (!skip) { 978 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, 979 tmp[0], tmp[1], tmp[2]); 980 } else { 981 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 982 skip = 0; 983 } 984 } 985 986 if (i == 0) { 987 break; 988 } 989 --i; 990 } 991 p224_felem_assign(x_out, nq[0]); 992 p224_felem_assign(y_out, nq[1]); 993 p224_felem_assign(z_out, nq[2]); 994 } 995 996 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns 997 * (X', Y') = (X/Z^2, Y/Z^3) */ 998 static int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, 999 const EC_POINT *point, 1000 BIGNUM *x, BIGNUM *y, 1001 BN_CTX *ctx) { 1002 p224_felem z1, z2, x_in, y_in, x_out, y_out; 1003 p224_widefelem tmp; 1004 1005 if (EC_POINT_is_at_infinity(group, point)) { 1006 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); 1007 return 0; 1008 } 1009 1010 if (!p224_BN_to_felem(x_in, &point->X) || 1011 !p224_BN_to_felem(y_in, &point->Y) || 1012 !p224_BN_to_felem(z1, &point->Z)) { 1013 return 0; 1014 } 1015 1016 p224_felem_inv(z2, z1); 1017 p224_felem_square(tmp, z2); 1018 p224_felem_reduce(z1, tmp); 1019 p224_felem_mul(tmp, x_in, z1); 1020 p224_felem_reduce(x_in, tmp); 1021 p224_felem_contract(x_out, x_in); 1022 if (x != NULL && !p224_felem_to_BN(x, x_out)) { 1023 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1024 return 0; 1025 } 1026 1027 p224_felem_mul(tmp, z1, z2); 1028 p224_felem_reduce(z1, tmp); 1029 p224_felem_mul(tmp, y_in, z1); 1030 p224_felem_reduce(y_in, tmp); 1031 p224_felem_contract(y_out, y_in); 1032 if (y != NULL && !p224_felem_to_BN(y, y_out)) { 1033 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1034 return 0; 1035 } 1036 1037 return 1; 1038 } 1039 1040 static int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, 1041 const BIGNUM *g_scalar, const EC_POINT *p, 1042 const BIGNUM *p_scalar, BN_CTX *ctx) { 1043 int ret = 0; 1044 BN_CTX *new_ctx = NULL; 1045 BIGNUM *x, *y, *z, *tmp_scalar; 1046 p224_felem_bytearray g_secret, p_secret; 1047 p224_felem p_pre_comp[17][3]; 1048 p224_felem_bytearray tmp; 1049 p224_felem x_in, y_in, z_in, x_out, y_out, z_out; 1050 1051 if (ctx == NULL) { 1052 ctx = BN_CTX_new(); 1053 new_ctx = ctx; 1054 if (ctx == NULL) { 1055 return 0; 1056 } 1057 } 1058 1059 BN_CTX_start(ctx); 1060 if ((x = BN_CTX_get(ctx)) == NULL || 1061 (y = BN_CTX_get(ctx)) == NULL || 1062 (z = BN_CTX_get(ctx)) == NULL || 1063 (tmp_scalar = BN_CTX_get(ctx)) == NULL) { 1064 goto err; 1065 } 1066 1067 if (p != NULL && p_scalar != NULL) { 1068 /* We treat NULL scalars as 0, and NULL points as points at infinity, i.e., 1069 * they contribute nothing to the linear combination. */ 1070 OPENSSL_memset(&p_secret, 0, sizeof(p_secret)); 1071 OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp)); 1072 size_t num_bytes; 1073 /* reduce g_scalar to 0 <= g_scalar < 2^224 */ 1074 if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) { 1075 /* this is an unusual input, and we don't guarantee 1076 * constant-timeness */ 1077 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { 1078 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1079 goto err; 1080 } 1081 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1082 } else { 1083 num_bytes = BN_bn2bin(p_scalar, tmp); 1084 } 1085 1086 p224_flip_endian(p_secret, tmp, num_bytes); 1087 /* precompute multiples */ 1088 if (!p224_BN_to_felem(x_out, &p->X) || 1089 !p224_BN_to_felem(y_out, &p->Y) || 1090 !p224_BN_to_felem(z_out, &p->Z)) { 1091 goto err; 1092 } 1093 1094 p224_felem_assign(p_pre_comp[1][0], x_out); 1095 p224_felem_assign(p_pre_comp[1][1], y_out); 1096 p224_felem_assign(p_pre_comp[1][2], z_out); 1097 1098 for (size_t j = 2; j <= 16; ++j) { 1099 if (j & 1) { 1100 p224_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2], 1101 p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2], 1102 0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], 1103 p_pre_comp[j - 1][2]); 1104 } else { 1105 p224_point_double(p_pre_comp[j][0], p_pre_comp[j][1], 1106 p_pre_comp[j][2], p_pre_comp[j / 2][0], 1107 p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]); 1108 } 1109 } 1110 } 1111 1112 if (g_scalar != NULL) { 1113 OPENSSL_memset(g_secret, 0, sizeof(g_secret)); 1114 size_t num_bytes; 1115 /* reduce g_scalar to 0 <= g_scalar < 2^224 */ 1116 if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) { 1117 /* this is an unusual input, and we don't guarantee constant-timeness */ 1118 if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) { 1119 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1120 goto err; 1121 } 1122 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1123 } else { 1124 num_bytes = BN_bn2bin(g_scalar, tmp); 1125 } 1126 1127 p224_flip_endian(g_secret, tmp, num_bytes); 1128 } 1129 p224_batch_mul( 1130 x_out, y_out, z_out, (p != NULL && p_scalar != NULL) ? p_secret : NULL, 1131 g_scalar != NULL ? g_secret : NULL, (const p224_felem(*)[3])p_pre_comp); 1132 1133 /* reduce the output to its unique minimal representation */ 1134 p224_felem_contract(x_in, x_out); 1135 p224_felem_contract(y_in, y_out); 1136 p224_felem_contract(z_in, z_out); 1137 if (!p224_felem_to_BN(x, x_in) || 1138 !p224_felem_to_BN(y, y_in) || 1139 !p224_felem_to_BN(z, z_in)) { 1140 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); 1141 goto err; 1142 } 1143 ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); 1144 1145 err: 1146 BN_CTX_end(ctx); 1147 BN_CTX_free(new_ctx); 1148 return ret; 1149 } 1150 1151 DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) { 1152 out->group_init = ec_GFp_simple_group_init; 1153 out->group_finish = ec_GFp_simple_group_finish; 1154 out->group_copy = ec_GFp_simple_group_copy; 1155 out->group_set_curve = ec_GFp_simple_group_set_curve; 1156 out->point_get_affine_coordinates = 1157 ec_GFp_nistp224_point_get_affine_coordinates; 1158 out->mul = ec_GFp_nistp224_points_mul; 1159 out->field_mul = ec_GFp_simple_field_mul; 1160 out->field_sqr = ec_GFp_simple_field_sqr; 1161 out->field_encode = NULL; 1162 out->field_decode = NULL; 1163 }; 1164 1165 #endif /* 64_BIT && !WINDOWS && !SMALL */ 1166
