1 /* 2 * Copyright 2011-2019 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the OpenSSL license (the "License"). You may not use 5 * this file except in compliance with the License. You can obtain a copy 6 * in the file LICENSE in the source distribution or at 7 * https://www.openssl.org/source/license.html 8 */ 9 10 /* Copyright 2011 Google Inc. 11 * 12 * Licensed under the Apache License, Version 2.0 (the "License"); 13 * 14 * you may not use this file except in compliance with the License. 15 * You may obtain a copy of the License at 16 * 17 * http://www.apache.org/licenses/LICENSE-2.0 18 * 19 * Unless required by applicable law or agreed to in writing, software 20 * distributed under the License is distributed on an "AS IS" BASIS, 21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 22 * See the License for the specific language governing permissions and 23 * limitations under the License. 24 */ 25 26 /* 27 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication 28 * 29 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. 30 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519 31 * work which got its smarts from Daniel J. Bernstein's work on the same. 32 */ 33 34 #include <openssl/e_os2.h> 35 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128 36 NON_EMPTY_TRANSLATION_UNIT 37 #else 38 39 # include <string.h> 40 # include <openssl/err.h> 41 # include "ec_lcl.h" 42 43 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16 44 /* even with gcc, the typedef won't work for 32-bit platforms */ 45 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit 46 * platforms */ 47 # else 48 # error "Your compiler doesn't appear to support 128-bit integer types" 49 # endif 50 51 typedef uint8_t u8; 52 typedef uint64_t u64; 53 54 /* 55 * The underlying field. P521 operates over GF(2^521-1). We can serialise an 56 * element of this field into 66 bytes where the most significant byte 57 * contains only a single bit. We call this an felem_bytearray. 58 */ 59 60 typedef u8 felem_bytearray[66]; 61 62 /* 63 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5. 64 * These values are big-endian. 65 */ 66 static const felem_bytearray nistp521_curve_params[5] = { 67 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */ 68 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 69 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 70 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 71 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 72 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 73 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 74 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 75 0xff, 0xff}, 76 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */ 77 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 78 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 79 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 80 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 81 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 82 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 83 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 84 0xff, 0xfc}, 85 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */ 86 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 87 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 88 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 89 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 90 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 91 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 92 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 93 0x3f, 0x00}, 94 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */ 95 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 96 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f, 97 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 98 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 99 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 100 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 101 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 102 0xbd, 0x66}, 103 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */ 104 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 105 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 106 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 107 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 108 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad, 109 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 110 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 111 0x66, 0x50} 112 }; 113 114 /*- 115 * The representation of field elements. 116 * ------------------------------------ 117 * 118 * We represent field elements with nine values. These values are either 64 or 119 * 128 bits and the field element represented is: 120 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p) 121 * Each of the nine values is called a 'limb'. Since the limbs are spaced only 122 * 58 bits apart, but are greater than 58 bits in length, the most significant 123 * bits of each limb overlap with the least significant bits of the next. 124 * 125 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a 126 * 'largefelem' */ 127 128 # define NLIMBS 9 129 130 typedef uint64_t limb; 131 typedef limb felem[NLIMBS]; 132 typedef uint128_t largefelem[NLIMBS]; 133 134 static const limb bottom57bits = 0x1ffffffffffffff; 135 static const limb bottom58bits = 0x3ffffffffffffff; 136 137 /* 138 * bin66_to_felem takes a little-endian byte array and converts it into felem 139 * form. This assumes that the CPU is little-endian. 140 */ 141 static void bin66_to_felem(felem out, const u8 in[66]) 142 { 143 out[0] = (*((limb *) & in[0])) & bottom58bits; 144 out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits; 145 out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits; 146 out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits; 147 out[4] = (*((limb *) & in[29])) & bottom58bits; 148 out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits; 149 out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits; 150 out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits; 151 out[8] = (*((limb *) & in[58])) & bottom57bits; 152 } 153 154 /* 155 * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte 156 * array. This assumes that the CPU is little-endian. 157 */ 158 static void felem_to_bin66(u8 out[66], const felem in) 159 { 160 memset(out, 0, 66); 161 (*((limb *) & out[0])) = in[0]; 162 (*((limb *) & out[7])) |= in[1] << 2; 163 (*((limb *) & out[14])) |= in[2] << 4; 164 (*((limb *) & out[21])) |= in[3] << 6; 165 (*((limb *) & out[29])) = in[4]; 166 (*((limb *) & out[36])) |= in[5] << 2; 167 (*((limb *) & out[43])) |= in[6] << 4; 168 (*((limb *) & out[50])) |= in[7] << 6; 169 (*((limb *) & out[58])) = in[8]; 170 } 171 172 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */ 173 static int BN_to_felem(felem out, const BIGNUM *bn) 174 { 175 felem_bytearray b_out; 176 int num_bytes; 177 178 if (BN_is_negative(bn)) { 179 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); 180 return 0; 181 } 182 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out)); 183 if (num_bytes < 0) { 184 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); 185 return 0; 186 } 187 bin66_to_felem(out, b_out); 188 return 1; 189 } 190 191 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */ 192 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) 193 { 194 felem_bytearray b_out; 195 felem_to_bin66(b_out, in); 196 return BN_lebin2bn(b_out, sizeof(b_out), out); 197 } 198 199 /*- 200 * Field operations 201 * ---------------- 202 */ 203 204 static void felem_one(felem out) 205 { 206 out[0] = 1; 207 out[1] = 0; 208 out[2] = 0; 209 out[3] = 0; 210 out[4] = 0; 211 out[5] = 0; 212 out[6] = 0; 213 out[7] = 0; 214 out[8] = 0; 215 } 216 217 static void felem_assign(felem out, const felem in) 218 { 219 out[0] = in[0]; 220 out[1] = in[1]; 221 out[2] = in[2]; 222 out[3] = in[3]; 223 out[4] = in[4]; 224 out[5] = in[5]; 225 out[6] = in[6]; 226 out[7] = in[7]; 227 out[8] = in[8]; 228 } 229 230 /* felem_sum64 sets out = out + in. */ 231 static void felem_sum64(felem out, const felem in) 232 { 233 out[0] += in[0]; 234 out[1] += in[1]; 235 out[2] += in[2]; 236 out[3] += in[3]; 237 out[4] += in[4]; 238 out[5] += in[5]; 239 out[6] += in[6]; 240 out[7] += in[7]; 241 out[8] += in[8]; 242 } 243 244 /* felem_scalar sets out = in * scalar */ 245 static void felem_scalar(felem out, const felem in, limb scalar) 246 { 247 out[0] = in[0] * scalar; 248 out[1] = in[1] * scalar; 249 out[2] = in[2] * scalar; 250 out[3] = in[3] * scalar; 251 out[4] = in[4] * scalar; 252 out[5] = in[5] * scalar; 253 out[6] = in[6] * scalar; 254 out[7] = in[7] * scalar; 255 out[8] = in[8] * scalar; 256 } 257 258 /* felem_scalar64 sets out = out * scalar */ 259 static void felem_scalar64(felem out, limb scalar) 260 { 261 out[0] *= scalar; 262 out[1] *= scalar; 263 out[2] *= scalar; 264 out[3] *= scalar; 265 out[4] *= scalar; 266 out[5] *= scalar; 267 out[6] *= scalar; 268 out[7] *= scalar; 269 out[8] *= scalar; 270 } 271 272 /* felem_scalar128 sets out = out * scalar */ 273 static void felem_scalar128(largefelem out, limb scalar) 274 { 275 out[0] *= scalar; 276 out[1] *= scalar; 277 out[2] *= scalar; 278 out[3] *= scalar; 279 out[4] *= scalar; 280 out[5] *= scalar; 281 out[6] *= scalar; 282 out[7] *= scalar; 283 out[8] *= scalar; 284 } 285 286 /*- 287 * felem_neg sets |out| to |-in| 288 * On entry: 289 * in[i] < 2^59 + 2^14 290 * On exit: 291 * out[i] < 2^62 292 */ 293 static void felem_neg(felem out, const felem in) 294 { 295 /* In order to prevent underflow, we subtract from 0 mod p. */ 296 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); 297 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); 298 299 out[0] = two62m3 - in[0]; 300 out[1] = two62m2 - in[1]; 301 out[2] = two62m2 - in[2]; 302 out[3] = two62m2 - in[3]; 303 out[4] = two62m2 - in[4]; 304 out[5] = two62m2 - in[5]; 305 out[6] = two62m2 - in[6]; 306 out[7] = two62m2 - in[7]; 307 out[8] = two62m2 - in[8]; 308 } 309 310 /*- 311 * felem_diff64 subtracts |in| from |out| 312 * On entry: 313 * in[i] < 2^59 + 2^14 314 * On exit: 315 * out[i] < out[i] + 2^62 316 */ 317 static void felem_diff64(felem out, const felem in) 318 { 319 /* 320 * In order to prevent underflow, we add 0 mod p before subtracting. 321 */ 322 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); 323 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); 324 325 out[0] += two62m3 - in[0]; 326 out[1] += two62m2 - in[1]; 327 out[2] += two62m2 - in[2]; 328 out[3] += two62m2 - in[3]; 329 out[4] += two62m2 - in[4]; 330 out[5] += two62m2 - in[5]; 331 out[6] += two62m2 - in[6]; 332 out[7] += two62m2 - in[7]; 333 out[8] += two62m2 - in[8]; 334 } 335 336 /*- 337 * felem_diff_128_64 subtracts |in| from |out| 338 * On entry: 339 * in[i] < 2^62 + 2^17 340 * On exit: 341 * out[i] < out[i] + 2^63 342 */ 343 static void felem_diff_128_64(largefelem out, const felem in) 344 { 345 /* 346 * In order to prevent underflow, we add 64p mod p (which is equivalent 347 * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521 348 * digit number with all bits set to 1. See "The representation of field 349 * elements" comment above for a description of how limbs are used to 350 * represent a number. 64p is represented with 8 limbs containing a number 351 * with 58 bits set and one limb with a number with 57 bits set. 352 */ 353 static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6); 354 static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5); 355 356 out[0] += two63m6 - in[0]; 357 out[1] += two63m5 - in[1]; 358 out[2] += two63m5 - in[2]; 359 out[3] += two63m5 - in[3]; 360 out[4] += two63m5 - in[4]; 361 out[5] += two63m5 - in[5]; 362 out[6] += two63m5 - in[6]; 363 out[7] += two63m5 - in[7]; 364 out[8] += two63m5 - in[8]; 365 } 366 367 /*- 368 * felem_diff_128_64 subtracts |in| from |out| 369 * On entry: 370 * in[i] < 2^126 371 * On exit: 372 * out[i] < out[i] + 2^127 - 2^69 373 */ 374 static void felem_diff128(largefelem out, const largefelem in) 375 { 376 /* 377 * In order to prevent underflow, we add 0 mod p before subtracting. 378 */ 379 static const uint128_t two127m70 = 380 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70); 381 static const uint128_t two127m69 = 382 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69); 383 384 out[0] += (two127m70 - in[0]); 385 out[1] += (two127m69 - in[1]); 386 out[2] += (two127m69 - in[2]); 387 out[3] += (two127m69 - in[3]); 388 out[4] += (two127m69 - in[4]); 389 out[5] += (two127m69 - in[5]); 390 out[6] += (two127m69 - in[6]); 391 out[7] += (two127m69 - in[7]); 392 out[8] += (two127m69 - in[8]); 393 } 394 395 /*- 396 * felem_square sets |out| = |in|^2 397 * On entry: 398 * in[i] < 2^62 399 * On exit: 400 * out[i] < 17 * max(in[i]) * max(in[i]) 401 */ 402 static void felem_square(largefelem out, const felem in) 403 { 404 felem inx2, inx4; 405 felem_scalar(inx2, in, 2); 406 felem_scalar(inx4, in, 4); 407 408 /*- 409 * We have many cases were we want to do 410 * in[x] * in[y] + 411 * in[y] * in[x] 412 * This is obviously just 413 * 2 * in[x] * in[y] 414 * However, rather than do the doubling on the 128 bit result, we 415 * double one of the inputs to the multiplication by reading from 416 * |inx2| 417 */ 418 419 out[0] = ((uint128_t) in[0]) * in[0]; 420 out[1] = ((uint128_t) in[0]) * inx2[1]; 421 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1]; 422 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2]; 423 out[4] = ((uint128_t) in[0]) * inx2[4] + 424 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2]; 425 out[5] = ((uint128_t) in[0]) * inx2[5] + 426 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3]; 427 out[6] = ((uint128_t) in[0]) * inx2[6] + 428 ((uint128_t) in[1]) * inx2[5] + 429 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3]; 430 out[7] = ((uint128_t) in[0]) * inx2[7] + 431 ((uint128_t) in[1]) * inx2[6] + 432 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4]; 433 out[8] = ((uint128_t) in[0]) * inx2[8] + 434 ((uint128_t) in[1]) * inx2[7] + 435 ((uint128_t) in[2]) * inx2[6] + 436 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4]; 437 438 /* 439 * The remaining limbs fall above 2^521, with the first falling at 2^522. 440 * They correspond to locations one bit up from the limbs produced above 441 * so we would have to multiply by two to align them. Again, rather than 442 * operate on the 128-bit result, we double one of the inputs to the 443 * multiplication. If we want to double for both this reason, and the 444 * reason above, then we end up multiplying by four. 445 */ 446 447 /* 9 */ 448 out[0] += ((uint128_t) in[1]) * inx4[8] + 449 ((uint128_t) in[2]) * inx4[7] + 450 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5]; 451 452 /* 10 */ 453 out[1] += ((uint128_t) in[2]) * inx4[8] + 454 ((uint128_t) in[3]) * inx4[7] + 455 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5]; 456 457 /* 11 */ 458 out[2] += ((uint128_t) in[3]) * inx4[8] + 459 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6]; 460 461 /* 12 */ 462 out[3] += ((uint128_t) in[4]) * inx4[8] + 463 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6]; 464 465 /* 13 */ 466 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7]; 467 468 /* 14 */ 469 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7]; 470 471 /* 15 */ 472 out[6] += ((uint128_t) in[7]) * inx4[8]; 473 474 /* 16 */ 475 out[7] += ((uint128_t) in[8]) * inx2[8]; 476 } 477 478 /*- 479 * felem_mul sets |out| = |in1| * |in2| 480 * On entry: 481 * in1[i] < 2^64 482 * in2[i] < 2^63 483 * On exit: 484 * out[i] < 17 * max(in1[i]) * max(in2[i]) 485 */ 486 static void felem_mul(largefelem out, const felem in1, const felem in2) 487 { 488 felem in2x2; 489 felem_scalar(in2x2, in2, 2); 490 491 out[0] = ((uint128_t) in1[0]) * in2[0]; 492 493 out[1] = ((uint128_t) in1[0]) * in2[1] + 494 ((uint128_t) in1[1]) * in2[0]; 495 496 out[2] = ((uint128_t) in1[0]) * in2[2] + 497 ((uint128_t) in1[1]) * in2[1] + 498 ((uint128_t) in1[2]) * in2[0]; 499 500 out[3] = ((uint128_t) in1[0]) * in2[3] + 501 ((uint128_t) in1[1]) * in2[2] + 502 ((uint128_t) in1[2]) * in2[1] + 503 ((uint128_t) in1[3]) * in2[0]; 504 505 out[4] = ((uint128_t) in1[0]) * in2[4] + 506 ((uint128_t) in1[1]) * in2[3] + 507 ((uint128_t) in1[2]) * in2[2] + 508 ((uint128_t) in1[3]) * in2[1] + 509 ((uint128_t) in1[4]) * in2[0]; 510 511 out[5] = ((uint128_t) in1[0]) * in2[5] + 512 ((uint128_t) in1[1]) * in2[4] + 513 ((uint128_t) in1[2]) * in2[3] + 514 ((uint128_t) in1[3]) * in2[2] + 515 ((uint128_t) in1[4]) * in2[1] + 516 ((uint128_t) in1[5]) * in2[0]; 517 518 out[6] = ((uint128_t) in1[0]) * in2[6] + 519 ((uint128_t) in1[1]) * in2[5] + 520 ((uint128_t) in1[2]) * in2[4] + 521 ((uint128_t) in1[3]) * in2[3] + 522 ((uint128_t) in1[4]) * in2[2] + 523 ((uint128_t) in1[5]) * in2[1] + 524 ((uint128_t) in1[6]) * in2[0]; 525 526 out[7] = ((uint128_t) in1[0]) * in2[7] + 527 ((uint128_t) in1[1]) * in2[6] + 528 ((uint128_t) in1[2]) * in2[5] + 529 ((uint128_t) in1[3]) * in2[4] + 530 ((uint128_t) in1[4]) * in2[3] + 531 ((uint128_t) in1[5]) * in2[2] + 532 ((uint128_t) in1[6]) * in2[1] + 533 ((uint128_t) in1[7]) * in2[0]; 534 535 out[8] = ((uint128_t) in1[0]) * in2[8] + 536 ((uint128_t) in1[1]) * in2[7] + 537 ((uint128_t) in1[2]) * in2[6] + 538 ((uint128_t) in1[3]) * in2[5] + 539 ((uint128_t) in1[4]) * in2[4] + 540 ((uint128_t) in1[5]) * in2[3] + 541 ((uint128_t) in1[6]) * in2[2] + 542 ((uint128_t) in1[7]) * in2[1] + 543 ((uint128_t) in1[8]) * in2[0]; 544 545 /* See comment in felem_square about the use of in2x2 here */ 546 547 out[0] += ((uint128_t) in1[1]) * in2x2[8] + 548 ((uint128_t) in1[2]) * in2x2[7] + 549 ((uint128_t) in1[3]) * in2x2[6] + 550 ((uint128_t) in1[4]) * in2x2[5] + 551 ((uint128_t) in1[5]) * in2x2[4] + 552 ((uint128_t) in1[6]) * in2x2[3] + 553 ((uint128_t) in1[7]) * in2x2[2] + 554 ((uint128_t) in1[8]) * in2x2[1]; 555 556 out[1] += ((uint128_t) in1[2]) * in2x2[8] + 557 ((uint128_t) in1[3]) * in2x2[7] + 558 ((uint128_t) in1[4]) * in2x2[6] + 559 ((uint128_t) in1[5]) * in2x2[5] + 560 ((uint128_t) in1[6]) * in2x2[4] + 561 ((uint128_t) in1[7]) * in2x2[3] + 562 ((uint128_t) in1[8]) * in2x2[2]; 563 564 out[2] += ((uint128_t) in1[3]) * in2x2[8] + 565 ((uint128_t) in1[4]) * in2x2[7] + 566 ((uint128_t) in1[5]) * in2x2[6] + 567 ((uint128_t) in1[6]) * in2x2[5] + 568 ((uint128_t) in1[7]) * in2x2[4] + 569 ((uint128_t) in1[8]) * in2x2[3]; 570 571 out[3] += ((uint128_t) in1[4]) * in2x2[8] + 572 ((uint128_t) in1[5]) * in2x2[7] + 573 ((uint128_t) in1[6]) * in2x2[6] + 574 ((uint128_t) in1[7]) * in2x2[5] + 575 ((uint128_t) in1[8]) * in2x2[4]; 576 577 out[4] += ((uint128_t) in1[5]) * in2x2[8] + 578 ((uint128_t) in1[6]) * in2x2[7] + 579 ((uint128_t) in1[7]) * in2x2[6] + 580 ((uint128_t) in1[8]) * in2x2[5]; 581 582 out[5] += ((uint128_t) in1[6]) * in2x2[8] + 583 ((uint128_t) in1[7]) * in2x2[7] + 584 ((uint128_t) in1[8]) * in2x2[6]; 585 586 out[6] += ((uint128_t) in1[7]) * in2x2[8] + 587 ((uint128_t) in1[8]) * in2x2[7]; 588 589 out[7] += ((uint128_t) in1[8]) * in2x2[8]; 590 } 591 592 static const limb bottom52bits = 0xfffffffffffff; 593 594 /*- 595 * felem_reduce converts a largefelem to an felem. 596 * On entry: 597 * in[i] < 2^128 598 * On exit: 599 * out[i] < 2^59 + 2^14 600 */ 601 static void felem_reduce(felem out, const largefelem in) 602 { 603 u64 overflow1, overflow2; 604 605 out[0] = ((limb) in[0]) & bottom58bits; 606 out[1] = ((limb) in[1]) & bottom58bits; 607 out[2] = ((limb) in[2]) & bottom58bits; 608 out[3] = ((limb) in[3]) & bottom58bits; 609 out[4] = ((limb) in[4]) & bottom58bits; 610 out[5] = ((limb) in[5]) & bottom58bits; 611 out[6] = ((limb) in[6]) & bottom58bits; 612 out[7] = ((limb) in[7]) & bottom58bits; 613 out[8] = ((limb) in[8]) & bottom58bits; 614 615 /* out[i] < 2^58 */ 616 617 out[1] += ((limb) in[0]) >> 58; 618 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6; 619 /*- 620 * out[1] < 2^58 + 2^6 + 2^58 621 * = 2^59 + 2^6 622 */ 623 out[2] += ((limb) (in[0] >> 64)) >> 52; 624 625 out[2] += ((limb) in[1]) >> 58; 626 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6; 627 out[3] += ((limb) (in[1] >> 64)) >> 52; 628 629 out[3] += ((limb) in[2]) >> 58; 630 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6; 631 out[4] += ((limb) (in[2] >> 64)) >> 52; 632 633 out[4] += ((limb) in[3]) >> 58; 634 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6; 635 out[5] += ((limb) (in[3] >> 64)) >> 52; 636 637 out[5] += ((limb) in[4]) >> 58; 638 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6; 639 out[6] += ((limb) (in[4] >> 64)) >> 52; 640 641 out[6] += ((limb) in[5]) >> 58; 642 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6; 643 out[7] += ((limb) (in[5] >> 64)) >> 52; 644 645 out[7] += ((limb) in[6]) >> 58; 646 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6; 647 out[8] += ((limb) (in[6] >> 64)) >> 52; 648 649 out[8] += ((limb) in[7]) >> 58; 650 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6; 651 /*- 652 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12 653 * < 2^59 + 2^13 654 */ 655 overflow1 = ((limb) (in[7] >> 64)) >> 52; 656 657 overflow1 += ((limb) in[8]) >> 58; 658 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6; 659 overflow2 = ((limb) (in[8] >> 64)) >> 52; 660 661 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */ 662 overflow2 <<= 1; /* overflow2 < 2^13 */ 663 664 out[0] += overflow1; /* out[0] < 2^60 */ 665 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */ 666 667 out[1] += out[0] >> 58; 668 out[0] &= bottom58bits; 669 /*- 670 * out[0] < 2^58 671 * out[1] < 2^59 + 2^6 + 2^13 + 2^2 672 * < 2^59 + 2^14 673 */ 674 } 675 676 static void felem_square_reduce(felem out, const felem in) 677 { 678 largefelem tmp; 679 felem_square(tmp, in); 680 felem_reduce(out, tmp); 681 } 682 683 static void felem_mul_reduce(felem out, const felem in1, const felem in2) 684 { 685 largefelem tmp; 686 felem_mul(tmp, in1, in2); 687 felem_reduce(out, tmp); 688 } 689 690 /*- 691 * felem_inv calculates |out| = |in|^{-1} 692 * 693 * Based on Fermat's Little Theorem: 694 * a^p = a (mod p) 695 * a^{p-1} = 1 (mod p) 696 * a^{p-2} = a^{-1} (mod p) 697 */ 698 static void felem_inv(felem out, const felem in) 699 { 700 felem ftmp, ftmp2, ftmp3, ftmp4; 701 largefelem tmp; 702 unsigned i; 703 704 felem_square(tmp, in); 705 felem_reduce(ftmp, tmp); /* 2^1 */ 706 felem_mul(tmp, in, ftmp); 707 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ 708 felem_assign(ftmp2, ftmp); 709 felem_square(tmp, ftmp); 710 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ 711 felem_mul(tmp, in, ftmp); 712 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */ 713 felem_square(tmp, ftmp); 714 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */ 715 716 felem_square(tmp, ftmp2); 717 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */ 718 felem_square(tmp, ftmp3); 719 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */ 720 felem_mul(tmp, ftmp3, ftmp2); 721 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */ 722 723 felem_assign(ftmp2, ftmp3); 724 felem_square(tmp, ftmp3); 725 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */ 726 felem_square(tmp, ftmp3); 727 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */ 728 felem_square(tmp, ftmp3); 729 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */ 730 felem_square(tmp, ftmp3); 731 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */ 732 felem_assign(ftmp4, ftmp3); 733 felem_mul(tmp, ftmp3, ftmp); 734 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */ 735 felem_square(tmp, ftmp4); 736 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */ 737 felem_mul(tmp, ftmp3, ftmp2); 738 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */ 739 felem_assign(ftmp2, ftmp3); 740 741 for (i = 0; i < 8; i++) { 742 felem_square(tmp, ftmp3); 743 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */ 744 } 745 felem_mul(tmp, ftmp3, ftmp2); 746 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */ 747 felem_assign(ftmp2, ftmp3); 748 749 for (i = 0; i < 16; i++) { 750 felem_square(tmp, ftmp3); 751 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */ 752 } 753 felem_mul(tmp, ftmp3, ftmp2); 754 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */ 755 felem_assign(ftmp2, ftmp3); 756 757 for (i = 0; i < 32; i++) { 758 felem_square(tmp, ftmp3); 759 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */ 760 } 761 felem_mul(tmp, ftmp3, ftmp2); 762 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */ 763 felem_assign(ftmp2, ftmp3); 764 765 for (i = 0; i < 64; i++) { 766 felem_square(tmp, ftmp3); 767 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */ 768 } 769 felem_mul(tmp, ftmp3, ftmp2); 770 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */ 771 felem_assign(ftmp2, ftmp3); 772 773 for (i = 0; i < 128; i++) { 774 felem_square(tmp, ftmp3); 775 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */ 776 } 777 felem_mul(tmp, ftmp3, ftmp2); 778 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */ 779 felem_assign(ftmp2, ftmp3); 780 781 for (i = 0; i < 256; i++) { 782 felem_square(tmp, ftmp3); 783 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */ 784 } 785 felem_mul(tmp, ftmp3, ftmp2); 786 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */ 787 788 for (i = 0; i < 9; i++) { 789 felem_square(tmp, ftmp3); 790 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */ 791 } 792 felem_mul(tmp, ftmp3, ftmp4); 793 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */ 794 felem_mul(tmp, ftmp3, in); 795 felem_reduce(out, tmp); /* 2^512 - 3 */ 796 } 797 798 /* This is 2^521-1, expressed as an felem */ 799 static const felem kPrime = { 800 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 801 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 802 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff 803 }; 804 805 /*- 806 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 807 * otherwise. 808 * On entry: 809 * in[i] < 2^59 + 2^14 810 */ 811 static limb felem_is_zero(const felem in) 812 { 813 felem ftmp; 814 limb is_zero, is_p; 815 felem_assign(ftmp, in); 816 817 ftmp[0] += ftmp[8] >> 57; 818 ftmp[8] &= bottom57bits; 819 /* ftmp[8] < 2^57 */ 820 ftmp[1] += ftmp[0] >> 58; 821 ftmp[0] &= bottom58bits; 822 ftmp[2] += ftmp[1] >> 58; 823 ftmp[1] &= bottom58bits; 824 ftmp[3] += ftmp[2] >> 58; 825 ftmp[2] &= bottom58bits; 826 ftmp[4] += ftmp[3] >> 58; 827 ftmp[3] &= bottom58bits; 828 ftmp[5] += ftmp[4] >> 58; 829 ftmp[4] &= bottom58bits; 830 ftmp[6] += ftmp[5] >> 58; 831 ftmp[5] &= bottom58bits; 832 ftmp[7] += ftmp[6] >> 58; 833 ftmp[6] &= bottom58bits; 834 ftmp[8] += ftmp[7] >> 58; 835 ftmp[7] &= bottom58bits; 836 /* ftmp[8] < 2^57 + 4 */ 837 838 /* 839 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater 840 * than our bound for ftmp[8]. Therefore we only have to check if the 841 * zero is zero or 2^521-1. 842 */ 843 844 is_zero = 0; 845 is_zero |= ftmp[0]; 846 is_zero |= ftmp[1]; 847 is_zero |= ftmp[2]; 848 is_zero |= ftmp[3]; 849 is_zero |= ftmp[4]; 850 is_zero |= ftmp[5]; 851 is_zero |= ftmp[6]; 852 is_zero |= ftmp[7]; 853 is_zero |= ftmp[8]; 854 855 is_zero--; 856 /* 857 * We know that ftmp[i] < 2^63, therefore the only way that the top bit 858 * can be set is if is_zero was 0 before the decrement. 859 */ 860 is_zero = 0 - (is_zero >> 63); 861 862 is_p = ftmp[0] ^ kPrime[0]; 863 is_p |= ftmp[1] ^ kPrime[1]; 864 is_p |= ftmp[2] ^ kPrime[2]; 865 is_p |= ftmp[3] ^ kPrime[3]; 866 is_p |= ftmp[4] ^ kPrime[4]; 867 is_p |= ftmp[5] ^ kPrime[5]; 868 is_p |= ftmp[6] ^ kPrime[6]; 869 is_p |= ftmp[7] ^ kPrime[7]; 870 is_p |= ftmp[8] ^ kPrime[8]; 871 872 is_p--; 873 is_p = 0 - (is_p >> 63); 874 875 is_zero |= is_p; 876 return is_zero; 877 } 878 879 static int felem_is_zero_int(const void *in) 880 { 881 return (int)(felem_is_zero(in) & ((limb) 1)); 882 } 883 884 /*- 885 * felem_contract converts |in| to its unique, minimal representation. 886 * On entry: 887 * in[i] < 2^59 + 2^14 888 */ 889 static void felem_contract(felem out, const felem in) 890 { 891 limb is_p, is_greater, sign; 892 static const limb two58 = ((limb) 1) << 58; 893 894 felem_assign(out, in); 895 896 out[0] += out[8] >> 57; 897 out[8] &= bottom57bits; 898 /* out[8] < 2^57 */ 899 out[1] += out[0] >> 58; 900 out[0] &= bottom58bits; 901 out[2] += out[1] >> 58; 902 out[1] &= bottom58bits; 903 out[3] += out[2] >> 58; 904 out[2] &= bottom58bits; 905 out[4] += out[3] >> 58; 906 out[3] &= bottom58bits; 907 out[5] += out[4] >> 58; 908 out[4] &= bottom58bits; 909 out[6] += out[5] >> 58; 910 out[5] &= bottom58bits; 911 out[7] += out[6] >> 58; 912 out[6] &= bottom58bits; 913 out[8] += out[7] >> 58; 914 out[7] &= bottom58bits; 915 /* out[8] < 2^57 + 4 */ 916 917 /* 918 * If the value is greater than 2^521-1 then we have to subtract 2^521-1 919 * out. See the comments in felem_is_zero regarding why we don't test for 920 * other multiples of the prime. 921 */ 922 923 /* 924 * First, if |out| is equal to 2^521-1, we subtract it out to get zero. 925 */ 926 927 is_p = out[0] ^ kPrime[0]; 928 is_p |= out[1] ^ kPrime[1]; 929 is_p |= out[2] ^ kPrime[2]; 930 is_p |= out[3] ^ kPrime[3]; 931 is_p |= out[4] ^ kPrime[4]; 932 is_p |= out[5] ^ kPrime[5]; 933 is_p |= out[6] ^ kPrime[6]; 934 is_p |= out[7] ^ kPrime[7]; 935 is_p |= out[8] ^ kPrime[8]; 936 937 is_p--; 938 is_p &= is_p << 32; 939 is_p &= is_p << 16; 940 is_p &= is_p << 8; 941 is_p &= is_p << 4; 942 is_p &= is_p << 2; 943 is_p &= is_p << 1; 944 is_p = 0 - (is_p >> 63); 945 is_p = ~is_p; 946 947 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */ 948 949 out[0] &= is_p; 950 out[1] &= is_p; 951 out[2] &= is_p; 952 out[3] &= is_p; 953 out[4] &= is_p; 954 out[5] &= is_p; 955 out[6] &= is_p; 956 out[7] &= is_p; 957 out[8] &= is_p; 958 959 /* 960 * In order to test that |out| >= 2^521-1 we need only test if out[8] >> 961 * 57 is greater than zero as (2^521-1) + x >= 2^522 962 */ 963 is_greater = out[8] >> 57; 964 is_greater |= is_greater << 32; 965 is_greater |= is_greater << 16; 966 is_greater |= is_greater << 8; 967 is_greater |= is_greater << 4; 968 is_greater |= is_greater << 2; 969 is_greater |= is_greater << 1; 970 is_greater = 0 - (is_greater >> 63); 971 972 out[0] -= kPrime[0] & is_greater; 973 out[1] -= kPrime[1] & is_greater; 974 out[2] -= kPrime[2] & is_greater; 975 out[3] -= kPrime[3] & is_greater; 976 out[4] -= kPrime[4] & is_greater; 977 out[5] -= kPrime[5] & is_greater; 978 out[6] -= kPrime[6] & is_greater; 979 out[7] -= kPrime[7] & is_greater; 980 out[8] -= kPrime[8] & is_greater; 981 982 /* Eliminate negative coefficients */ 983 sign = -(out[0] >> 63); 984 out[0] += (two58 & sign); 985 out[1] -= (1 & sign); 986 sign = -(out[1] >> 63); 987 out[1] += (two58 & sign); 988 out[2] -= (1 & sign); 989 sign = -(out[2] >> 63); 990 out[2] += (two58 & sign); 991 out[3] -= (1 & sign); 992 sign = -(out[3] >> 63); 993 out[3] += (two58 & sign); 994 out[4] -= (1 & sign); 995 sign = -(out[4] >> 63); 996 out[4] += (two58 & sign); 997 out[5] -= (1 & sign); 998 sign = -(out[0] >> 63); 999 out[5] += (two58 & sign); 1000 out[6] -= (1 & sign); 1001 sign = -(out[6] >> 63); 1002 out[6] += (two58 & sign); 1003 out[7] -= (1 & sign); 1004 sign = -(out[7] >> 63); 1005 out[7] += (two58 & sign); 1006 out[8] -= (1 & sign); 1007 sign = -(out[5] >> 63); 1008 out[5] += (two58 & sign); 1009 out[6] -= (1 & sign); 1010 sign = -(out[6] >> 63); 1011 out[6] += (two58 & sign); 1012 out[7] -= (1 & sign); 1013 sign = -(out[7] >> 63); 1014 out[7] += (two58 & sign); 1015 out[8] -= (1 & sign); 1016 } 1017 1018 /*- 1019 * Group operations 1020 * ---------------- 1021 * 1022 * Building on top of the field operations we have the operations on the 1023 * elliptic curve group itself. Points on the curve are represented in Jacobian 1024 * coordinates */ 1025 1026 /*- 1027 * point_double calculates 2*(x_in, y_in, z_in) 1028 * 1029 * The method is taken from: 1030 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b 1031 * 1032 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. 1033 * while x_out == y_in is not (maybe this works, but it's not tested). */ 1034 static void 1035 point_double(felem x_out, felem y_out, felem z_out, 1036 const felem x_in, const felem y_in, const felem z_in) 1037 { 1038 largefelem tmp, tmp2; 1039 felem delta, gamma, beta, alpha, ftmp, ftmp2; 1040 1041 felem_assign(ftmp, x_in); 1042 felem_assign(ftmp2, x_in); 1043 1044 /* delta = z^2 */ 1045 felem_square(tmp, z_in); 1046 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */ 1047 1048 /* gamma = y^2 */ 1049 felem_square(tmp, y_in); 1050 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */ 1051 1052 /* beta = x*gamma */ 1053 felem_mul(tmp, x_in, gamma); 1054 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */ 1055 1056 /* alpha = 3*(x-delta)*(x+delta) */ 1057 felem_diff64(ftmp, delta); 1058 /* ftmp[i] < 2^61 */ 1059 felem_sum64(ftmp2, delta); 1060 /* ftmp2[i] < 2^60 + 2^15 */ 1061 felem_scalar64(ftmp2, 3); 1062 /* ftmp2[i] < 3*2^60 + 3*2^15 */ 1063 felem_mul(tmp, ftmp, ftmp2); 1064 /*- 1065 * tmp[i] < 17(3*2^121 + 3*2^76) 1066 * = 61*2^121 + 61*2^76 1067 * < 64*2^121 + 64*2^76 1068 * = 2^127 + 2^82 1069 * < 2^128 1070 */ 1071 felem_reduce(alpha, tmp); 1072 1073 /* x' = alpha^2 - 8*beta */ 1074 felem_square(tmp, alpha); 1075 /* 1076 * tmp[i] < 17*2^120 < 2^125 1077 */ 1078 felem_assign(ftmp, beta); 1079 felem_scalar64(ftmp, 8); 1080 /* ftmp[i] < 2^62 + 2^17 */ 1081 felem_diff_128_64(tmp, ftmp); 1082 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */ 1083 felem_reduce(x_out, tmp); 1084 1085 /* z' = (y + z)^2 - gamma - delta */ 1086 felem_sum64(delta, gamma); 1087 /* delta[i] < 2^60 + 2^15 */ 1088 felem_assign(ftmp, y_in); 1089 felem_sum64(ftmp, z_in); 1090 /* ftmp[i] < 2^60 + 2^15 */ 1091 felem_square(tmp, ftmp); 1092 /* 1093 * tmp[i] < 17(2^122) < 2^127 1094 */ 1095 felem_diff_128_64(tmp, delta); 1096 /* tmp[i] < 2^127 + 2^63 */ 1097 felem_reduce(z_out, tmp); 1098 1099 /* y' = alpha*(4*beta - x') - 8*gamma^2 */ 1100 felem_scalar64(beta, 4); 1101 /* beta[i] < 2^61 + 2^16 */ 1102 felem_diff64(beta, x_out); 1103 /* beta[i] < 2^61 + 2^60 + 2^16 */ 1104 felem_mul(tmp, alpha, beta); 1105 /*- 1106 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16)) 1107 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30) 1108 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30) 1109 * < 2^128 1110 */ 1111 felem_square(tmp2, gamma); 1112 /*- 1113 * tmp2[i] < 17*(2^59 + 2^14)^2 1114 * = 17*(2^118 + 2^74 + 2^28) 1115 */ 1116 felem_scalar128(tmp2, 8); 1117 /*- 1118 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28) 1119 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31 1120 * < 2^126 1121 */ 1122 felem_diff128(tmp, tmp2); 1123 /*- 1124 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30) 1125 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 + 1126 * 2^74 + 2^69 + 2^34 + 2^30 1127 * < 2^128 1128 */ 1129 felem_reduce(y_out, tmp); 1130 } 1131 1132 /* copy_conditional copies in to out iff mask is all ones. */ 1133 static void copy_conditional(felem out, const felem in, limb mask) 1134 { 1135 unsigned i; 1136 for (i = 0; i < NLIMBS; ++i) { 1137 const limb tmp = mask & (in[i] ^ out[i]); 1138 out[i] ^= tmp; 1139 } 1140 } 1141 1142 /*- 1143 * point_add calculates (x1, y1, z1) + (x2, y2, z2) 1144 * 1145 * The method is taken from 1146 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, 1147 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). 1148 * 1149 * This function includes a branch for checking whether the two input points 1150 * are equal (while not equal to the point at infinity). See comment below 1151 * on constant-time. 1152 */ 1153 static void point_add(felem x3, felem y3, felem z3, 1154 const felem x1, const felem y1, const felem z1, 1155 const int mixed, const felem x2, const felem y2, 1156 const felem z2) 1157 { 1158 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; 1159 largefelem tmp, tmp2; 1160 limb x_equal, y_equal, z1_is_zero, z2_is_zero; 1161 1162 z1_is_zero = felem_is_zero(z1); 1163 z2_is_zero = felem_is_zero(z2); 1164 1165 /* ftmp = z1z1 = z1**2 */ 1166 felem_square(tmp, z1); 1167 felem_reduce(ftmp, tmp); 1168 1169 if (!mixed) { 1170 /* ftmp2 = z2z2 = z2**2 */ 1171 felem_square(tmp, z2); 1172 felem_reduce(ftmp2, tmp); 1173 1174 /* u1 = ftmp3 = x1*z2z2 */ 1175 felem_mul(tmp, x1, ftmp2); 1176 felem_reduce(ftmp3, tmp); 1177 1178 /* ftmp5 = z1 + z2 */ 1179 felem_assign(ftmp5, z1); 1180 felem_sum64(ftmp5, z2); 1181 /* ftmp5[i] < 2^61 */ 1182 1183 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */ 1184 felem_square(tmp, ftmp5); 1185 /* tmp[i] < 17*2^122 */ 1186 felem_diff_128_64(tmp, ftmp); 1187 /* tmp[i] < 17*2^122 + 2^63 */ 1188 felem_diff_128_64(tmp, ftmp2); 1189 /* tmp[i] < 17*2^122 + 2^64 */ 1190 felem_reduce(ftmp5, tmp); 1191 1192 /* ftmp2 = z2 * z2z2 */ 1193 felem_mul(tmp, ftmp2, z2); 1194 felem_reduce(ftmp2, tmp); 1195 1196 /* s1 = ftmp6 = y1 * z2**3 */ 1197 felem_mul(tmp, y1, ftmp2); 1198 felem_reduce(ftmp6, tmp); 1199 } else { 1200 /* 1201 * We'll assume z2 = 1 (special case z2 = 0 is handled later) 1202 */ 1203 1204 /* u1 = ftmp3 = x1*z2z2 */ 1205 felem_assign(ftmp3, x1); 1206 1207 /* ftmp5 = 2*z1z2 */ 1208 felem_scalar(ftmp5, z1, 2); 1209 1210 /* s1 = ftmp6 = y1 * z2**3 */ 1211 felem_assign(ftmp6, y1); 1212 } 1213 1214 /* u2 = x2*z1z1 */ 1215 felem_mul(tmp, x2, ftmp); 1216 /* tmp[i] < 17*2^120 */ 1217 1218 /* h = ftmp4 = u2 - u1 */ 1219 felem_diff_128_64(tmp, ftmp3); 1220 /* tmp[i] < 17*2^120 + 2^63 */ 1221 felem_reduce(ftmp4, tmp); 1222 1223 x_equal = felem_is_zero(ftmp4); 1224 1225 /* z_out = ftmp5 * h */ 1226 felem_mul(tmp, ftmp5, ftmp4); 1227 felem_reduce(z_out, tmp); 1228 1229 /* ftmp = z1 * z1z1 */ 1230 felem_mul(tmp, ftmp, z1); 1231 felem_reduce(ftmp, tmp); 1232 1233 /* s2 = tmp = y2 * z1**3 */ 1234 felem_mul(tmp, y2, ftmp); 1235 /* tmp[i] < 17*2^120 */ 1236 1237 /* r = ftmp5 = (s2 - s1)*2 */ 1238 felem_diff_128_64(tmp, ftmp6); 1239 /* tmp[i] < 17*2^120 + 2^63 */ 1240 felem_reduce(ftmp5, tmp); 1241 y_equal = felem_is_zero(ftmp5); 1242 felem_scalar64(ftmp5, 2); 1243 /* ftmp5[i] < 2^61 */ 1244 1245 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 1246 /* 1247 * This is obviously not constant-time but it will almost-never happen 1248 * for ECDH / ECDSA. The case where it can happen is during scalar-mult 1249 * where the intermediate value gets very close to the group order. 1250 * Since |ec_GFp_nistp_recode_scalar_bits| produces signed digits for 1251 * the scalar, it's possible for the intermediate value to be a small 1252 * negative multiple of the base point, and for the final signed digit 1253 * to be the same value. We believe that this only occurs for the scalar 1254 * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 1255 * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb 1256 * 71e913863f7, in that case the penultimate intermediate is -9G and 1257 * the final digit is also -9G. Since this only happens for a single 1258 * scalar, the timing leak is irrelevant. (Any attacker who wanted to 1259 * check whether a secret scalar was that exact value, can already do 1260 * so.) 1261 */ 1262 point_double(x3, y3, z3, x1, y1, z1); 1263 return; 1264 } 1265 1266 /* I = ftmp = (2h)**2 */ 1267 felem_assign(ftmp, ftmp4); 1268 felem_scalar64(ftmp, 2); 1269 /* ftmp[i] < 2^61 */ 1270 felem_square(tmp, ftmp); 1271 /* tmp[i] < 17*2^122 */ 1272 felem_reduce(ftmp, tmp); 1273 1274 /* J = ftmp2 = h * I */ 1275 felem_mul(tmp, ftmp4, ftmp); 1276 felem_reduce(ftmp2, tmp); 1277 1278 /* V = ftmp4 = U1 * I */ 1279 felem_mul(tmp, ftmp3, ftmp); 1280 felem_reduce(ftmp4, tmp); 1281 1282 /* x_out = r**2 - J - 2V */ 1283 felem_square(tmp, ftmp5); 1284 /* tmp[i] < 17*2^122 */ 1285 felem_diff_128_64(tmp, ftmp2); 1286 /* tmp[i] < 17*2^122 + 2^63 */ 1287 felem_assign(ftmp3, ftmp4); 1288 felem_scalar64(ftmp4, 2); 1289 /* ftmp4[i] < 2^61 */ 1290 felem_diff_128_64(tmp, ftmp4); 1291 /* tmp[i] < 17*2^122 + 2^64 */ 1292 felem_reduce(x_out, tmp); 1293 1294 /* y_out = r(V-x_out) - 2 * s1 * J */ 1295 felem_diff64(ftmp3, x_out); 1296 /* 1297 * ftmp3[i] < 2^60 + 2^60 = 2^61 1298 */ 1299 felem_mul(tmp, ftmp5, ftmp3); 1300 /* tmp[i] < 17*2^122 */ 1301 felem_mul(tmp2, ftmp6, ftmp2); 1302 /* tmp2[i] < 17*2^120 */ 1303 felem_scalar128(tmp2, 2); 1304 /* tmp2[i] < 17*2^121 */ 1305 felem_diff128(tmp, tmp2); 1306 /*- 1307 * tmp[i] < 2^127 - 2^69 + 17*2^122 1308 * = 2^126 - 2^122 - 2^6 - 2^2 - 1 1309 * < 2^127 1310 */ 1311 felem_reduce(y_out, tmp); 1312 1313 copy_conditional(x_out, x2, z1_is_zero); 1314 copy_conditional(x_out, x1, z2_is_zero); 1315 copy_conditional(y_out, y2, z1_is_zero); 1316 copy_conditional(y_out, y1, z2_is_zero); 1317 copy_conditional(z_out, z2, z1_is_zero); 1318 copy_conditional(z_out, z1, z2_is_zero); 1319 felem_assign(x3, x_out); 1320 felem_assign(y3, y_out); 1321 felem_assign(z3, z_out); 1322 } 1323 1324 /*- 1325 * Base point pre computation 1326 * -------------------------- 1327 * 1328 * Two different sorts of precomputed tables are used in the following code. 1329 * Each contain various points on the curve, where each point is three field 1330 * elements (x, y, z). 1331 * 1332 * For the base point table, z is usually 1 (0 for the point at infinity). 1333 * This table has 16 elements: 1334 * index | bits | point 1335 * ------+---------+------------------------------ 1336 * 0 | 0 0 0 0 | 0G 1337 * 1 | 0 0 0 1 | 1G 1338 * 2 | 0 0 1 0 | 2^130G 1339 * 3 | 0 0 1 1 | (2^130 + 1)G 1340 * 4 | 0 1 0 0 | 2^260G 1341 * 5 | 0 1 0 1 | (2^260 + 1)G 1342 * 6 | 0 1 1 0 | (2^260 + 2^130)G 1343 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G 1344 * 8 | 1 0 0 0 | 2^390G 1345 * 9 | 1 0 0 1 | (2^390 + 1)G 1346 * 10 | 1 0 1 0 | (2^390 + 2^130)G 1347 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G 1348 * 12 | 1 1 0 0 | (2^390 + 2^260)G 1349 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G 1350 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G 1351 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G 1352 * 1353 * The reason for this is so that we can clock bits into four different 1354 * locations when doing simple scalar multiplies against the base point. 1355 * 1356 * Tables for other points have table[i] = iG for i in 0 .. 16. */ 1357 1358 /* gmul is the table of precomputed base points */ 1359 static const felem gmul[16][3] = { 1360 {{0, 0, 0, 0, 0, 0, 0, 0, 0}, 1361 {0, 0, 0, 0, 0, 0, 0, 0, 0}, 1362 {0, 0, 0, 0, 0, 0, 0, 0, 0}}, 1363 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334, 1364 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8, 1365 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404}, 1366 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353, 1367 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45, 1368 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b}, 1369 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1370 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad, 1371 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e, 1372 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5}, 1373 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58, 1374 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c, 1375 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7}, 1376 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1377 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873, 1378 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c, 1379 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9}, 1380 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52, 1381 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e, 1382 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe}, 1383 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1384 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2, 1385 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561, 1386 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065}, 1387 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a, 1388 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e, 1389 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524}, 1390 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1391 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6, 1392 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51, 1393 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe}, 1394 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d, 1395 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c, 1396 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7}, 1397 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1398 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27, 1399 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f, 1400 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256}, 1401 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa, 1402 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2, 1403 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd}, 1404 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1405 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890, 1406 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74, 1407 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23}, 1408 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516, 1409 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1, 1410 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e}, 1411 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1412 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce, 1413 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7, 1414 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5}, 1415 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318, 1416 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83, 1417 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242}, 1418 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1419 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae, 1420 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef, 1421 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203}, 1422 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447, 1423 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283, 1424 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f}, 1425 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1426 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5, 1427 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c, 1428 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a}, 1429 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df, 1430 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645, 1431 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a}, 1432 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1433 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292, 1434 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422, 1435 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b}, 1436 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30, 1437 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb, 1438 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f}, 1439 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1440 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767, 1441 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3, 1442 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf}, 1443 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2, 1444 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692, 1445 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d}, 1446 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1447 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3, 1448 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade, 1449 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684}, 1450 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8, 1451 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a, 1452 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81}, 1453 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1454 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608, 1455 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610, 1456 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d}, 1457 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006, 1458 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86, 1459 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42}, 1460 {1, 0, 0, 0, 0, 0, 0, 0, 0}}, 1461 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c, 1462 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9, 1463 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f}, 1464 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7, 1465 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c, 1466 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055}, 1467 {1, 0, 0, 0, 0, 0, 0, 0, 0}} 1468 }; 1469 1470 /* 1471 * select_point selects the |idx|th point from a precomputation table and 1472 * copies it to out. 1473 */ 1474 /* pre_comp below is of the size provided in |size| */ 1475 static void select_point(const limb idx, unsigned int size, 1476 const felem pre_comp[][3], felem out[3]) 1477 { 1478 unsigned i, j; 1479 limb *outlimbs = &out[0][0]; 1480 1481 memset(out, 0, sizeof(*out) * 3); 1482 1483 for (i = 0; i < size; i++) { 1484 const limb *inlimbs = &pre_comp[i][0][0]; 1485 limb mask = i ^ idx; 1486 mask |= mask >> 4; 1487 mask |= mask >> 2; 1488 mask |= mask >> 1; 1489 mask &= 1; 1490 mask--; 1491 for (j = 0; j < NLIMBS * 3; j++) 1492 outlimbs[j] |= inlimbs[j] & mask; 1493 } 1494 } 1495 1496 /* get_bit returns the |i|th bit in |in| */ 1497 static char get_bit(const felem_bytearray in, int i) 1498 { 1499 if (i < 0) 1500 return 0; 1501 return (in[i >> 3] >> (i & 7)) & 1; 1502 } 1503 1504 /* 1505 * Interleaved point multiplication using precomputed point multiples: The 1506 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars 1507 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the 1508 * generator, using certain (large) precomputed multiples in g_pre_comp. 1509 * Output point (X, Y, Z) is stored in x_out, y_out, z_out 1510 */ 1511 static void batch_mul(felem x_out, felem y_out, felem z_out, 1512 const felem_bytearray scalars[], 1513 const unsigned num_points, const u8 *g_scalar, 1514 const int mixed, const felem pre_comp[][17][3], 1515 const felem g_pre_comp[16][3]) 1516 { 1517 int i, skip; 1518 unsigned num, gen_mul = (g_scalar != NULL); 1519 felem nq[3], tmp[4]; 1520 limb bits; 1521 u8 sign, digit; 1522 1523 /* set nq to the point at infinity */ 1524 memset(nq, 0, sizeof(nq)); 1525 1526 /* 1527 * Loop over all scalars msb-to-lsb, interleaving additions of multiples 1528 * of the generator (last quarter of rounds) and additions of other 1529 * points multiples (every 5th round). 1530 */ 1531 skip = 1; /* save two point operations in the first 1532 * round */ 1533 for (i = (num_points ? 520 : 130); i >= 0; --i) { 1534 /* double */ 1535 if (!skip) 1536 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 1537 1538 /* add multiples of the generator */ 1539 if (gen_mul && (i <= 130)) { 1540 bits = get_bit(g_scalar, i + 390) << 3; 1541 if (i < 130) { 1542 bits |= get_bit(g_scalar, i + 260) << 2; 1543 bits |= get_bit(g_scalar, i + 130) << 1; 1544 bits |= get_bit(g_scalar, i); 1545 } 1546 /* select the point to add, in constant time */ 1547 select_point(bits, 16, g_pre_comp, tmp); 1548 if (!skip) { 1549 /* The 1 argument below is for "mixed" */ 1550 point_add(nq[0], nq[1], nq[2], 1551 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]); 1552 } else { 1553 memcpy(nq, tmp, 3 * sizeof(felem)); 1554 skip = 0; 1555 } 1556 } 1557 1558 /* do other additions every 5 doublings */ 1559 if (num_points && (i % 5 == 0)) { 1560 /* loop over all scalars */ 1561 for (num = 0; num < num_points; ++num) { 1562 bits = get_bit(scalars[num], i + 4) << 5; 1563 bits |= get_bit(scalars[num], i + 3) << 4; 1564 bits |= get_bit(scalars[num], i + 2) << 3; 1565 bits |= get_bit(scalars[num], i + 1) << 2; 1566 bits |= get_bit(scalars[num], i) << 1; 1567 bits |= get_bit(scalars[num], i - 1); 1568 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 1569 1570 /* 1571 * select the point to add or subtract, in constant time 1572 */ 1573 select_point(digit, 17, pre_comp[num], tmp); 1574 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative 1575 * point */ 1576 copy_conditional(tmp[1], tmp[3], (-(limb) sign)); 1577 1578 if (!skip) { 1579 point_add(nq[0], nq[1], nq[2], 1580 nq[0], nq[1], nq[2], 1581 mixed, tmp[0], tmp[1], tmp[2]); 1582 } else { 1583 memcpy(nq, tmp, 3 * sizeof(felem)); 1584 skip = 0; 1585 } 1586 } 1587 } 1588 } 1589 felem_assign(x_out, nq[0]); 1590 felem_assign(y_out, nq[1]); 1591 felem_assign(z_out, nq[2]); 1592 } 1593 1594 /* Precomputation for the group generator. */ 1595 struct nistp521_pre_comp_st { 1596 felem g_pre_comp[16][3]; 1597 CRYPTO_REF_COUNT references; 1598 CRYPTO_RWLOCK *lock; 1599 }; 1600 1601 const EC_METHOD *EC_GFp_nistp521_method(void) 1602 { 1603 static const EC_METHOD ret = { 1604 EC_FLAGS_DEFAULT_OCT, 1605 NID_X9_62_prime_field, 1606 ec_GFp_nistp521_group_init, 1607 ec_GFp_simple_group_finish, 1608 ec_GFp_simple_group_clear_finish, 1609 ec_GFp_nist_group_copy, 1610 ec_GFp_nistp521_group_set_curve, 1611 ec_GFp_simple_group_get_curve, 1612 ec_GFp_simple_group_get_degree, 1613 ec_group_simple_order_bits, 1614 ec_GFp_simple_group_check_discriminant, 1615 ec_GFp_simple_point_init, 1616 ec_GFp_simple_point_finish, 1617 ec_GFp_simple_point_clear_finish, 1618 ec_GFp_simple_point_copy, 1619 ec_GFp_simple_point_set_to_infinity, 1620 ec_GFp_simple_set_Jprojective_coordinates_GFp, 1621 ec_GFp_simple_get_Jprojective_coordinates_GFp, 1622 ec_GFp_simple_point_set_affine_coordinates, 1623 ec_GFp_nistp521_point_get_affine_coordinates, 1624 0 /* point_set_compressed_coordinates */ , 1625 0 /* point2oct */ , 1626 0 /* oct2point */ , 1627 ec_GFp_simple_add, 1628 ec_GFp_simple_dbl, 1629 ec_GFp_simple_invert, 1630 ec_GFp_simple_is_at_infinity, 1631 ec_GFp_simple_is_on_curve, 1632 ec_GFp_simple_cmp, 1633 ec_GFp_simple_make_affine, 1634 ec_GFp_simple_points_make_affine, 1635 ec_GFp_nistp521_points_mul, 1636 ec_GFp_nistp521_precompute_mult, 1637 ec_GFp_nistp521_have_precompute_mult, 1638 ec_GFp_nist_field_mul, 1639 ec_GFp_nist_field_sqr, 1640 0 /* field_div */ , 1641 ec_GFp_simple_field_inv, 1642 0 /* field_encode */ , 1643 0 /* field_decode */ , 1644 0, /* field_set_to_one */ 1645 ec_key_simple_priv2oct, 1646 ec_key_simple_oct2priv, 1647 0, /* set private */ 1648 ec_key_simple_generate_key, 1649 ec_key_simple_check_key, 1650 ec_key_simple_generate_public_key, 1651 0, /* keycopy */ 1652 0, /* keyfinish */ 1653 ecdh_simple_compute_key, 1654 0, /* field_inverse_mod_ord */ 1655 0, /* blind_coordinates */ 1656 0, /* ladder_pre */ 1657 0, /* ladder_step */ 1658 0 /* ladder_post */ 1659 }; 1660 1661 return &ret; 1662 } 1663 1664 /******************************************************************************/ 1665 /* 1666 * FUNCTIONS TO MANAGE PRECOMPUTATION 1667 */ 1668 1669 static NISTP521_PRE_COMP *nistp521_pre_comp_new(void) 1670 { 1671 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret)); 1672 1673 if (ret == NULL) { 1674 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); 1675 return ret; 1676 } 1677 1678 ret->references = 1; 1679 1680 ret->lock = CRYPTO_THREAD_lock_new(); 1681 if (ret->lock == NULL) { 1682 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); 1683 OPENSSL_free(ret); 1684 return NULL; 1685 } 1686 return ret; 1687 } 1688 1689 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p) 1690 { 1691 int i; 1692 if (p != NULL) 1693 CRYPTO_UP_REF(&p->references, &i, p->lock); 1694 return p; 1695 } 1696 1697 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p) 1698 { 1699 int i; 1700 1701 if (p == NULL) 1702 return; 1703 1704 CRYPTO_DOWN_REF(&p->references, &i, p->lock); 1705 REF_PRINT_COUNT("EC_nistp521", x); 1706 if (i > 0) 1707 return; 1708 REF_ASSERT_ISNT(i < 0); 1709 1710 CRYPTO_THREAD_lock_free(p->lock); 1711 OPENSSL_free(p); 1712 } 1713 1714 /******************************************************************************/ 1715 /* 1716 * OPENSSL EC_METHOD FUNCTIONS 1717 */ 1718 1719 int ec_GFp_nistp521_group_init(EC_GROUP *group) 1720 { 1721 int ret; 1722 ret = ec_GFp_simple_group_init(group); 1723 group->a_is_minus3 = 1; 1724 return ret; 1725 } 1726 1727 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p, 1728 const BIGNUM *a, const BIGNUM *b, 1729 BN_CTX *ctx) 1730 { 1731 int ret = 0; 1732 BN_CTX *new_ctx = NULL; 1733 BIGNUM *curve_p, *curve_a, *curve_b; 1734 1735 if (ctx == NULL) 1736 if ((ctx = new_ctx = BN_CTX_new()) == NULL) 1737 return 0; 1738 BN_CTX_start(ctx); 1739 curve_p = BN_CTX_get(ctx); 1740 curve_a = BN_CTX_get(ctx); 1741 curve_b = BN_CTX_get(ctx); 1742 if (curve_b == NULL) 1743 goto err; 1744 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p); 1745 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a); 1746 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b); 1747 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) { 1748 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE, 1749 EC_R_WRONG_CURVE_PARAMETERS); 1750 goto err; 1751 } 1752 group->field_mod_func = BN_nist_mod_521; 1753 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); 1754 err: 1755 BN_CTX_end(ctx); 1756 BN_CTX_free(new_ctx); 1757 return ret; 1758 } 1759 1760 /* 1761 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = 1762 * (X/Z^2, Y/Z^3) 1763 */ 1764 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group, 1765 const EC_POINT *point, 1766 BIGNUM *x, BIGNUM *y, 1767 BN_CTX *ctx) 1768 { 1769 felem z1, z2, x_in, y_in, x_out, y_out; 1770 largefelem tmp; 1771 1772 if (EC_POINT_is_at_infinity(group, point)) { 1773 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, 1774 EC_R_POINT_AT_INFINITY); 1775 return 0; 1776 } 1777 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) || 1778 (!BN_to_felem(z1, point->Z))) 1779 return 0; 1780 felem_inv(z2, z1); 1781 felem_square(tmp, z2); 1782 felem_reduce(z1, tmp); 1783 felem_mul(tmp, x_in, z1); 1784 felem_reduce(x_in, tmp); 1785 felem_contract(x_out, x_in); 1786 if (x != NULL) { 1787 if (!felem_to_BN(x, x_out)) { 1788 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, 1789 ERR_R_BN_LIB); 1790 return 0; 1791 } 1792 } 1793 felem_mul(tmp, z1, z2); 1794 felem_reduce(z1, tmp); 1795 felem_mul(tmp, y_in, z1); 1796 felem_reduce(y_in, tmp); 1797 felem_contract(y_out, y_in); 1798 if (y != NULL) { 1799 if (!felem_to_BN(y, y_out)) { 1800 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, 1801 ERR_R_BN_LIB); 1802 return 0; 1803 } 1804 } 1805 return 1; 1806 } 1807 1808 /* points below is of size |num|, and tmp_felems is of size |num+1/ */ 1809 static void make_points_affine(size_t num, felem points[][3], 1810 felem tmp_felems[]) 1811 { 1812 /* 1813 * Runs in constant time, unless an input is the point at infinity (which 1814 * normally shouldn't happen). 1815 */ 1816 ec_GFp_nistp_points_make_affine_internal(num, 1817 points, 1818 sizeof(felem), 1819 tmp_felems, 1820 (void (*)(void *))felem_one, 1821 felem_is_zero_int, 1822 (void (*)(void *, const void *)) 1823 felem_assign, 1824 (void (*)(void *, const void *)) 1825 felem_square_reduce, (void (*) 1826 (void *, 1827 const void 1828 *, 1829 const void 1830 *)) 1831 felem_mul_reduce, 1832 (void (*)(void *, const void *)) 1833 felem_inv, 1834 (void (*)(void *, const void *)) 1835 felem_contract); 1836 } 1837 1838 /* 1839 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL 1840 * values Result is stored in r (r can equal one of the inputs). 1841 */ 1842 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r, 1843 const BIGNUM *scalar, size_t num, 1844 const EC_POINT *points[], 1845 const BIGNUM *scalars[], BN_CTX *ctx) 1846 { 1847 int ret = 0; 1848 int j; 1849 int mixed = 0; 1850 BIGNUM *x, *y, *z, *tmp_scalar; 1851 felem_bytearray g_secret; 1852 felem_bytearray *secrets = NULL; 1853 felem (*pre_comp)[17][3] = NULL; 1854 felem *tmp_felems = NULL; 1855 unsigned i; 1856 int num_bytes; 1857 int have_pre_comp = 0; 1858 size_t num_points = num; 1859 felem x_in, y_in, z_in, x_out, y_out, z_out; 1860 NISTP521_PRE_COMP *pre = NULL; 1861 felem(*g_pre_comp)[3] = NULL; 1862 EC_POINT *generator = NULL; 1863 const EC_POINT *p = NULL; 1864 const BIGNUM *p_scalar = NULL; 1865 1866 BN_CTX_start(ctx); 1867 x = BN_CTX_get(ctx); 1868 y = BN_CTX_get(ctx); 1869 z = BN_CTX_get(ctx); 1870 tmp_scalar = BN_CTX_get(ctx); 1871 if (tmp_scalar == NULL) 1872 goto err; 1873 1874 if (scalar != NULL) { 1875 pre = group->pre_comp.nistp521; 1876 if (pre) 1877 /* we have precomputation, try to use it */ 1878 g_pre_comp = &pre->g_pre_comp[0]; 1879 else 1880 /* try to use the standard precomputation */ 1881 g_pre_comp = (felem(*)[3]) gmul; 1882 generator = EC_POINT_new(group); 1883 if (generator == NULL) 1884 goto err; 1885 /* get the generator from precomputation */ 1886 if (!felem_to_BN(x, g_pre_comp[1][0]) || 1887 !felem_to_BN(y, g_pre_comp[1][1]) || 1888 !felem_to_BN(z, g_pre_comp[1][2])) { 1889 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); 1890 goto err; 1891 } 1892 if (!EC_POINT_set_Jprojective_coordinates_GFp(group, 1893 generator, x, y, z, 1894 ctx)) 1895 goto err; 1896 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) 1897 /* precomputation matches generator */ 1898 have_pre_comp = 1; 1899 else 1900 /* 1901 * we don't have valid precomputation: treat the generator as a 1902 * random point 1903 */ 1904 num_points++; 1905 } 1906 1907 if (num_points > 0) { 1908 if (num_points >= 2) { 1909 /* 1910 * unless we precompute multiples for just one point, converting 1911 * those into affine form is time well spent 1912 */ 1913 mixed = 1; 1914 } 1915 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points); 1916 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points); 1917 if (mixed) 1918 tmp_felems = 1919 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1)); 1920 if ((secrets == NULL) || (pre_comp == NULL) 1921 || (mixed && (tmp_felems == NULL))) { 1922 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE); 1923 goto err; 1924 } 1925 1926 /* 1927 * we treat NULL scalars as 0, and NULL points as points at infinity, 1928 * i.e., they contribute nothing to the linear combination 1929 */ 1930 for (i = 0; i < num_points; ++i) { 1931 if (i == num) { 1932 /* 1933 * we didn't have a valid precomputation, so we pick the 1934 * generator 1935 */ 1936 p = EC_GROUP_get0_generator(group); 1937 p_scalar = scalar; 1938 } else { 1939 /* the i^th point */ 1940 p = points[i]; 1941 p_scalar = scalars[i]; 1942 } 1943 if ((p_scalar != NULL) && (p != NULL)) { 1944 /* reduce scalar to 0 <= scalar < 2^521 */ 1945 if ((BN_num_bits(p_scalar) > 521) 1946 || (BN_is_negative(p_scalar))) { 1947 /* 1948 * this is an unusual input, and we don't guarantee 1949 * constant-timeness 1950 */ 1951 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) { 1952 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); 1953 goto err; 1954 } 1955 num_bytes = BN_bn2lebinpad(tmp_scalar, 1956 secrets[i], sizeof(secrets[i])); 1957 } else { 1958 num_bytes = BN_bn2lebinpad(p_scalar, 1959 secrets[i], sizeof(secrets[i])); 1960 } 1961 if (num_bytes < 0) { 1962 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); 1963 goto err; 1964 } 1965 /* precompute multiples */ 1966 if ((!BN_to_felem(x_out, p->X)) || 1967 (!BN_to_felem(y_out, p->Y)) || 1968 (!BN_to_felem(z_out, p->Z))) 1969 goto err; 1970 memcpy(pre_comp[i][1][0], x_out, sizeof(felem)); 1971 memcpy(pre_comp[i][1][1], y_out, sizeof(felem)); 1972 memcpy(pre_comp[i][1][2], z_out, sizeof(felem)); 1973 for (j = 2; j <= 16; ++j) { 1974 if (j & 1) { 1975 point_add(pre_comp[i][j][0], pre_comp[i][j][1], 1976 pre_comp[i][j][2], pre_comp[i][1][0], 1977 pre_comp[i][1][1], pre_comp[i][1][2], 0, 1978 pre_comp[i][j - 1][0], 1979 pre_comp[i][j - 1][1], 1980 pre_comp[i][j - 1][2]); 1981 } else { 1982 point_double(pre_comp[i][j][0], pre_comp[i][j][1], 1983 pre_comp[i][j][2], pre_comp[i][j / 2][0], 1984 pre_comp[i][j / 2][1], 1985 pre_comp[i][j / 2][2]); 1986 } 1987 } 1988 } 1989 } 1990 if (mixed) 1991 make_points_affine(num_points * 17, pre_comp[0], tmp_felems); 1992 } 1993 1994 /* the scalar for the generator */ 1995 if ((scalar != NULL) && (have_pre_comp)) { 1996 memset(g_secret, 0, sizeof(g_secret)); 1997 /* reduce scalar to 0 <= scalar < 2^521 */ 1998 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) { 1999 /* 2000 * this is an unusual input, and we don't guarantee 2001 * constant-timeness 2002 */ 2003 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) { 2004 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); 2005 goto err; 2006 } 2007 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret)); 2008 } else { 2009 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret)); 2010 } 2011 /* do the multiplication with generator precomputation */ 2012 batch_mul(x_out, y_out, z_out, 2013 (const felem_bytearray(*))secrets, num_points, 2014 g_secret, 2015 mixed, (const felem(*)[17][3])pre_comp, 2016 (const felem(*)[3])g_pre_comp); 2017 } else { 2018 /* do the multiplication without generator precomputation */ 2019 batch_mul(x_out, y_out, z_out, 2020 (const felem_bytearray(*))secrets, num_points, 2021 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL); 2022 } 2023 /* reduce the output to its unique minimal representation */ 2024 felem_contract(x_in, x_out); 2025 felem_contract(y_in, y_out); 2026 felem_contract(z_in, z_out); 2027 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) || 2028 (!felem_to_BN(z, z_in))) { 2029 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); 2030 goto err; 2031 } 2032 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); 2033 2034 err: 2035 BN_CTX_end(ctx); 2036 EC_POINT_free(generator); 2037 OPENSSL_free(secrets); 2038 OPENSSL_free(pre_comp); 2039 OPENSSL_free(tmp_felems); 2040 return ret; 2041 } 2042 2043 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx) 2044 { 2045 int ret = 0; 2046 NISTP521_PRE_COMP *pre = NULL; 2047 int i, j; 2048 BN_CTX *new_ctx = NULL; 2049 BIGNUM *x, *y; 2050 EC_POINT *generator = NULL; 2051 felem tmp_felems[16]; 2052 2053 /* throw away old precomputation */ 2054 EC_pre_comp_free(group); 2055 if (ctx == NULL) 2056 if ((ctx = new_ctx = BN_CTX_new()) == NULL) 2057 return 0; 2058 BN_CTX_start(ctx); 2059 x = BN_CTX_get(ctx); 2060 y = BN_CTX_get(ctx); 2061 if (y == NULL) 2062 goto err; 2063 /* get the generator */ 2064 if (group->generator == NULL) 2065 goto err; 2066 generator = EC_POINT_new(group); 2067 if (generator == NULL) 2068 goto err; 2069 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x); 2070 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y); 2071 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx)) 2072 goto err; 2073 if ((pre = nistp521_pre_comp_new()) == NULL) 2074 goto err; 2075 /* 2076 * if the generator is the standard one, use built-in precomputation 2077 */ 2078 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { 2079 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp)); 2080 goto done; 2081 } 2082 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) || 2083 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) || 2084 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z))) 2085 goto err; 2086 /* compute 2^130*G, 2^260*G, 2^390*G */ 2087 for (i = 1; i <= 4; i <<= 1) { 2088 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1], 2089 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0], 2090 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]); 2091 for (j = 0; j < 129; ++j) { 2092 point_double(pre->g_pre_comp[2 * i][0], 2093 pre->g_pre_comp[2 * i][1], 2094 pre->g_pre_comp[2 * i][2], 2095 pre->g_pre_comp[2 * i][0], 2096 pre->g_pre_comp[2 * i][1], 2097 pre->g_pre_comp[2 * i][2]); 2098 } 2099 } 2100 /* g_pre_comp[0] is the point at infinity */ 2101 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0])); 2102 /* the remaining multiples */ 2103 /* 2^130*G + 2^260*G */ 2104 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1], 2105 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0], 2106 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2], 2107 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], 2108 pre->g_pre_comp[2][2]); 2109 /* 2^130*G + 2^390*G */ 2110 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1], 2111 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0], 2112 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], 2113 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], 2114 pre->g_pre_comp[2][2]); 2115 /* 2^260*G + 2^390*G */ 2116 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1], 2117 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0], 2118 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], 2119 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1], 2120 pre->g_pre_comp[4][2]); 2121 /* 2^130*G + 2^260*G + 2^390*G */ 2122 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1], 2123 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0], 2124 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2], 2125 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], 2126 pre->g_pre_comp[2][2]); 2127 for (i = 1; i < 8; ++i) { 2128 /* odd multiples: add G */ 2129 point_add(pre->g_pre_comp[2 * i + 1][0], 2130 pre->g_pre_comp[2 * i + 1][1], 2131 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0], 2132 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0, 2133 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1], 2134 pre->g_pre_comp[1][2]); 2135 } 2136 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems); 2137 2138 done: 2139 SETPRECOMP(group, nistp521, pre); 2140 ret = 1; 2141 pre = NULL; 2142 err: 2143 BN_CTX_end(ctx); 2144 EC_POINT_free(generator); 2145 BN_CTX_free(new_ctx); 2146 EC_nistp521_pre_comp_free(pre); 2147 return ret; 2148 } 2149 2150 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group) 2151 { 2152 return HAVEPRECOMP(group, nistp521); 2153 } 2154 2155 #endif 2156