1 /* 2 * Copyright 2001-2019 The OpenSSL Project Authors. All Rights Reserved. 3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved 4 * 5 * Licensed under the OpenSSL license (the "License"). You may not use 6 * this file except in compliance with the License. You can obtain a copy 7 * in the file LICENSE in the source distribution or at 8 * https://www.openssl.org/source/license.html 9 */ 10 11 #include <openssl/err.h> 12 #include <openssl/symhacks.h> 13 14 #include "ec_lcl.h" 15 16 const EC_METHOD *EC_GFp_simple_method(void) 17 { 18 static const EC_METHOD ret = { 19 EC_FLAGS_DEFAULT_OCT, 20 NID_X9_62_prime_field, 21 ec_GFp_simple_group_init, 22 ec_GFp_simple_group_finish, 23 ec_GFp_simple_group_clear_finish, 24 ec_GFp_simple_group_copy, 25 ec_GFp_simple_group_set_curve, 26 ec_GFp_simple_group_get_curve, 27 ec_GFp_simple_group_get_degree, 28 ec_group_simple_order_bits, 29 ec_GFp_simple_group_check_discriminant, 30 ec_GFp_simple_point_init, 31 ec_GFp_simple_point_finish, 32 ec_GFp_simple_point_clear_finish, 33 ec_GFp_simple_point_copy, 34 ec_GFp_simple_point_set_to_infinity, 35 ec_GFp_simple_set_Jprojective_coordinates_GFp, 36 ec_GFp_simple_get_Jprojective_coordinates_GFp, 37 ec_GFp_simple_point_set_affine_coordinates, 38 ec_GFp_simple_point_get_affine_coordinates, 39 0, 0, 0, 40 ec_GFp_simple_add, 41 ec_GFp_simple_dbl, 42 ec_GFp_simple_invert, 43 ec_GFp_simple_is_at_infinity, 44 ec_GFp_simple_is_on_curve, 45 ec_GFp_simple_cmp, 46 ec_GFp_simple_make_affine, 47 ec_GFp_simple_points_make_affine, 48 0 /* mul */ , 49 0 /* precompute_mult */ , 50 0 /* have_precompute_mult */ , 51 ec_GFp_simple_field_mul, 52 ec_GFp_simple_field_sqr, 53 0 /* field_div */ , 54 ec_GFp_simple_field_inv, 55 0 /* field_encode */ , 56 0 /* field_decode */ , 57 0, /* field_set_to_one */ 58 ec_key_simple_priv2oct, 59 ec_key_simple_oct2priv, 60 0, /* set private */ 61 ec_key_simple_generate_key, 62 ec_key_simple_check_key, 63 ec_key_simple_generate_public_key, 64 0, /* keycopy */ 65 0, /* keyfinish */ 66 ecdh_simple_compute_key, 67 0, /* field_inverse_mod_ord */ 68 ec_GFp_simple_blind_coordinates, 69 ec_GFp_simple_ladder_pre, 70 ec_GFp_simple_ladder_step, 71 ec_GFp_simple_ladder_post 72 }; 73 74 return &ret; 75 } 76 77 /* 78 * Most method functions in this file are designed to work with 79 * non-trivial representations of field elements if necessary 80 * (see ecp_mont.c): while standard modular addition and subtraction 81 * are used, the field_mul and field_sqr methods will be used for 82 * multiplication, and field_encode and field_decode (if defined) 83 * will be used for converting between representations. 84 * 85 * Functions ec_GFp_simple_points_make_affine() and 86 * ec_GFp_simple_point_get_affine_coordinates() specifically assume 87 * that if a non-trivial representation is used, it is a Montgomery 88 * representation (i.e. 'encoding' means multiplying by some factor R). 89 */ 90 91 int ec_GFp_simple_group_init(EC_GROUP *group) 92 { 93 group->field = BN_new(); 94 group->a = BN_new(); 95 group->b = BN_new(); 96 if (group->field == NULL || group->a == NULL || group->b == NULL) { 97 BN_free(group->field); 98 BN_free(group->a); 99 BN_free(group->b); 100 return 0; 101 } 102 group->a_is_minus3 = 0; 103 return 1; 104 } 105 106 void ec_GFp_simple_group_finish(EC_GROUP *group) 107 { 108 BN_free(group->field); 109 BN_free(group->a); 110 BN_free(group->b); 111 } 112 113 void ec_GFp_simple_group_clear_finish(EC_GROUP *group) 114 { 115 BN_clear_free(group->field); 116 BN_clear_free(group->a); 117 BN_clear_free(group->b); 118 } 119 120 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src) 121 { 122 if (!BN_copy(dest->field, src->field)) 123 return 0; 124 if (!BN_copy(dest->a, src->a)) 125 return 0; 126 if (!BN_copy(dest->b, src->b)) 127 return 0; 128 129 dest->a_is_minus3 = src->a_is_minus3; 130 131 return 1; 132 } 133 134 int ec_GFp_simple_group_set_curve(EC_GROUP *group, 135 const BIGNUM *p, const BIGNUM *a, 136 const BIGNUM *b, BN_CTX *ctx) 137 { 138 int ret = 0; 139 BN_CTX *new_ctx = NULL; 140 BIGNUM *tmp_a; 141 142 /* p must be a prime > 3 */ 143 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) { 144 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD); 145 return 0; 146 } 147 148 if (ctx == NULL) { 149 ctx = new_ctx = BN_CTX_new(); 150 if (ctx == NULL) 151 return 0; 152 } 153 154 BN_CTX_start(ctx); 155 tmp_a = BN_CTX_get(ctx); 156 if (tmp_a == NULL) 157 goto err; 158 159 /* group->field */ 160 if (!BN_copy(group->field, p)) 161 goto err; 162 BN_set_negative(group->field, 0); 163 164 /* group->a */ 165 if (!BN_nnmod(tmp_a, a, p, ctx)) 166 goto err; 167 if (group->meth->field_encode) { 168 if (!group->meth->field_encode(group, group->a, tmp_a, ctx)) 169 goto err; 170 } else if (!BN_copy(group->a, tmp_a)) 171 goto err; 172 173 /* group->b */ 174 if (!BN_nnmod(group->b, b, p, ctx)) 175 goto err; 176 if (group->meth->field_encode) 177 if (!group->meth->field_encode(group, group->b, group->b, ctx)) 178 goto err; 179 180 /* group->a_is_minus3 */ 181 if (!BN_add_word(tmp_a, 3)) 182 goto err; 183 group->a_is_minus3 = (0 == BN_cmp(tmp_a, group->field)); 184 185 ret = 1; 186 187 err: 188 BN_CTX_end(ctx); 189 BN_CTX_free(new_ctx); 190 return ret; 191 } 192 193 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, 194 BIGNUM *b, BN_CTX *ctx) 195 { 196 int ret = 0; 197 BN_CTX *new_ctx = NULL; 198 199 if (p != NULL) { 200 if (!BN_copy(p, group->field)) 201 return 0; 202 } 203 204 if (a != NULL || b != NULL) { 205 if (group->meth->field_decode) { 206 if (ctx == NULL) { 207 ctx = new_ctx = BN_CTX_new(); 208 if (ctx == NULL) 209 return 0; 210 } 211 if (a != NULL) { 212 if (!group->meth->field_decode(group, a, group->a, ctx)) 213 goto err; 214 } 215 if (b != NULL) { 216 if (!group->meth->field_decode(group, b, group->b, ctx)) 217 goto err; 218 } 219 } else { 220 if (a != NULL) { 221 if (!BN_copy(a, group->a)) 222 goto err; 223 } 224 if (b != NULL) { 225 if (!BN_copy(b, group->b)) 226 goto err; 227 } 228 } 229 } 230 231 ret = 1; 232 233 err: 234 BN_CTX_free(new_ctx); 235 return ret; 236 } 237 238 int ec_GFp_simple_group_get_degree(const EC_GROUP *group) 239 { 240 return BN_num_bits(group->field); 241 } 242 243 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx) 244 { 245 int ret = 0; 246 BIGNUM *a, *b, *order, *tmp_1, *tmp_2; 247 const BIGNUM *p = group->field; 248 BN_CTX *new_ctx = NULL; 249 250 if (ctx == NULL) { 251 ctx = new_ctx = BN_CTX_new(); 252 if (ctx == NULL) { 253 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT, 254 ERR_R_MALLOC_FAILURE); 255 goto err; 256 } 257 } 258 BN_CTX_start(ctx); 259 a = BN_CTX_get(ctx); 260 b = BN_CTX_get(ctx); 261 tmp_1 = BN_CTX_get(ctx); 262 tmp_2 = BN_CTX_get(ctx); 263 order = BN_CTX_get(ctx); 264 if (order == NULL) 265 goto err; 266 267 if (group->meth->field_decode) { 268 if (!group->meth->field_decode(group, a, group->a, ctx)) 269 goto err; 270 if (!group->meth->field_decode(group, b, group->b, ctx)) 271 goto err; 272 } else { 273 if (!BN_copy(a, group->a)) 274 goto err; 275 if (!BN_copy(b, group->b)) 276 goto err; 277 } 278 279 /*- 280 * check the discriminant: 281 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p) 282 * 0 =< a, b < p 283 */ 284 if (BN_is_zero(a)) { 285 if (BN_is_zero(b)) 286 goto err; 287 } else if (!BN_is_zero(b)) { 288 if (!BN_mod_sqr(tmp_1, a, p, ctx)) 289 goto err; 290 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx)) 291 goto err; 292 if (!BN_lshift(tmp_1, tmp_2, 2)) 293 goto err; 294 /* tmp_1 = 4*a^3 */ 295 296 if (!BN_mod_sqr(tmp_2, b, p, ctx)) 297 goto err; 298 if (!BN_mul_word(tmp_2, 27)) 299 goto err; 300 /* tmp_2 = 27*b^2 */ 301 302 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx)) 303 goto err; 304 if (BN_is_zero(a)) 305 goto err; 306 } 307 ret = 1; 308 309 err: 310 if (ctx != NULL) 311 BN_CTX_end(ctx); 312 BN_CTX_free(new_ctx); 313 return ret; 314 } 315 316 int ec_GFp_simple_point_init(EC_POINT *point) 317 { 318 point->X = BN_new(); 319 point->Y = BN_new(); 320 point->Z = BN_new(); 321 point->Z_is_one = 0; 322 323 if (point->X == NULL || point->Y == NULL || point->Z == NULL) { 324 BN_free(point->X); 325 BN_free(point->Y); 326 BN_free(point->Z); 327 return 0; 328 } 329 return 1; 330 } 331 332 void ec_GFp_simple_point_finish(EC_POINT *point) 333 { 334 BN_free(point->X); 335 BN_free(point->Y); 336 BN_free(point->Z); 337 } 338 339 void ec_GFp_simple_point_clear_finish(EC_POINT *point) 340 { 341 BN_clear_free(point->X); 342 BN_clear_free(point->Y); 343 BN_clear_free(point->Z); 344 point->Z_is_one = 0; 345 } 346 347 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src) 348 { 349 if (!BN_copy(dest->X, src->X)) 350 return 0; 351 if (!BN_copy(dest->Y, src->Y)) 352 return 0; 353 if (!BN_copy(dest->Z, src->Z)) 354 return 0; 355 dest->Z_is_one = src->Z_is_one; 356 dest->curve_name = src->curve_name; 357 358 return 1; 359 } 360 361 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, 362 EC_POINT *point) 363 { 364 point->Z_is_one = 0; 365 BN_zero(point->Z); 366 return 1; 367 } 368 369 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group, 370 EC_POINT *point, 371 const BIGNUM *x, 372 const BIGNUM *y, 373 const BIGNUM *z, 374 BN_CTX *ctx) 375 { 376 BN_CTX *new_ctx = NULL; 377 int ret = 0; 378 379 if (ctx == NULL) { 380 ctx = new_ctx = BN_CTX_new(); 381 if (ctx == NULL) 382 return 0; 383 } 384 385 if (x != NULL) { 386 if (!BN_nnmod(point->X, x, group->field, ctx)) 387 goto err; 388 if (group->meth->field_encode) { 389 if (!group->meth->field_encode(group, point->X, point->X, ctx)) 390 goto err; 391 } 392 } 393 394 if (y != NULL) { 395 if (!BN_nnmod(point->Y, y, group->field, ctx)) 396 goto err; 397 if (group->meth->field_encode) { 398 if (!group->meth->field_encode(group, point->Y, point->Y, ctx)) 399 goto err; 400 } 401 } 402 403 if (z != NULL) { 404 int Z_is_one; 405 406 if (!BN_nnmod(point->Z, z, group->field, ctx)) 407 goto err; 408 Z_is_one = BN_is_one(point->Z); 409 if (group->meth->field_encode) { 410 if (Z_is_one && (group->meth->field_set_to_one != 0)) { 411 if (!group->meth->field_set_to_one(group, point->Z, ctx)) 412 goto err; 413 } else { 414 if (!group-> 415 meth->field_encode(group, point->Z, point->Z, ctx)) 416 goto err; 417 } 418 } 419 point->Z_is_one = Z_is_one; 420 } 421 422 ret = 1; 423 424 err: 425 BN_CTX_free(new_ctx); 426 return ret; 427 } 428 429 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group, 430 const EC_POINT *point, 431 BIGNUM *x, BIGNUM *y, 432 BIGNUM *z, BN_CTX *ctx) 433 { 434 BN_CTX *new_ctx = NULL; 435 int ret = 0; 436 437 if (group->meth->field_decode != 0) { 438 if (ctx == NULL) { 439 ctx = new_ctx = BN_CTX_new(); 440 if (ctx == NULL) 441 return 0; 442 } 443 444 if (x != NULL) { 445 if (!group->meth->field_decode(group, x, point->X, ctx)) 446 goto err; 447 } 448 if (y != NULL) { 449 if (!group->meth->field_decode(group, y, point->Y, ctx)) 450 goto err; 451 } 452 if (z != NULL) { 453 if (!group->meth->field_decode(group, z, point->Z, ctx)) 454 goto err; 455 } 456 } else { 457 if (x != NULL) { 458 if (!BN_copy(x, point->X)) 459 goto err; 460 } 461 if (y != NULL) { 462 if (!BN_copy(y, point->Y)) 463 goto err; 464 } 465 if (z != NULL) { 466 if (!BN_copy(z, point->Z)) 467 goto err; 468 } 469 } 470 471 ret = 1; 472 473 err: 474 BN_CTX_free(new_ctx); 475 return ret; 476 } 477 478 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, 479 EC_POINT *point, 480 const BIGNUM *x, 481 const BIGNUM *y, BN_CTX *ctx) 482 { 483 if (x == NULL || y == NULL) { 484 /* 485 * unlike for projective coordinates, we do not tolerate this 486 */ 487 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES, 488 ERR_R_PASSED_NULL_PARAMETER); 489 return 0; 490 } 491 492 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, 493 BN_value_one(), ctx); 494 } 495 496 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group, 497 const EC_POINT *point, 498 BIGNUM *x, BIGNUM *y, 499 BN_CTX *ctx) 500 { 501 BN_CTX *new_ctx = NULL; 502 BIGNUM *Z, *Z_1, *Z_2, *Z_3; 503 const BIGNUM *Z_; 504 int ret = 0; 505 506 if (EC_POINT_is_at_infinity(group, point)) { 507 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, 508 EC_R_POINT_AT_INFINITY); 509 return 0; 510 } 511 512 if (ctx == NULL) { 513 ctx = new_ctx = BN_CTX_new(); 514 if (ctx == NULL) 515 return 0; 516 } 517 518 BN_CTX_start(ctx); 519 Z = BN_CTX_get(ctx); 520 Z_1 = BN_CTX_get(ctx); 521 Z_2 = BN_CTX_get(ctx); 522 Z_3 = BN_CTX_get(ctx); 523 if (Z_3 == NULL) 524 goto err; 525 526 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */ 527 528 if (group->meth->field_decode) { 529 if (!group->meth->field_decode(group, Z, point->Z, ctx)) 530 goto err; 531 Z_ = Z; 532 } else { 533 Z_ = point->Z; 534 } 535 536 if (BN_is_one(Z_)) { 537 if (group->meth->field_decode) { 538 if (x != NULL) { 539 if (!group->meth->field_decode(group, x, point->X, ctx)) 540 goto err; 541 } 542 if (y != NULL) { 543 if (!group->meth->field_decode(group, y, point->Y, ctx)) 544 goto err; 545 } 546 } else { 547 if (x != NULL) { 548 if (!BN_copy(x, point->X)) 549 goto err; 550 } 551 if (y != NULL) { 552 if (!BN_copy(y, point->Y)) 553 goto err; 554 } 555 } 556 } else { 557 if (!group->meth->field_inv(group, Z_1, Z_, ctx)) { 558 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, 559 ERR_R_BN_LIB); 560 goto err; 561 } 562 563 if (group->meth->field_encode == 0) { 564 /* field_sqr works on standard representation */ 565 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx)) 566 goto err; 567 } else { 568 if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx)) 569 goto err; 570 } 571 572 if (x != NULL) { 573 /* 574 * in the Montgomery case, field_mul will cancel out Montgomery 575 * factor in X: 576 */ 577 if (!group->meth->field_mul(group, x, point->X, Z_2, ctx)) 578 goto err; 579 } 580 581 if (y != NULL) { 582 if (group->meth->field_encode == 0) { 583 /* 584 * field_mul works on standard representation 585 */ 586 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx)) 587 goto err; 588 } else { 589 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx)) 590 goto err; 591 } 592 593 /* 594 * in the Montgomery case, field_mul will cancel out Montgomery 595 * factor in Y: 596 */ 597 if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx)) 598 goto err; 599 } 600 } 601 602 ret = 1; 603 604 err: 605 BN_CTX_end(ctx); 606 BN_CTX_free(new_ctx); 607 return ret; 608 } 609 610 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, 611 const EC_POINT *b, BN_CTX *ctx) 612 { 613 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, 614 const BIGNUM *, BN_CTX *); 615 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *); 616 const BIGNUM *p; 617 BN_CTX *new_ctx = NULL; 618 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6; 619 int ret = 0; 620 621 if (a == b) 622 return EC_POINT_dbl(group, r, a, ctx); 623 if (EC_POINT_is_at_infinity(group, a)) 624 return EC_POINT_copy(r, b); 625 if (EC_POINT_is_at_infinity(group, b)) 626 return EC_POINT_copy(r, a); 627 628 field_mul = group->meth->field_mul; 629 field_sqr = group->meth->field_sqr; 630 p = group->field; 631 632 if (ctx == NULL) { 633 ctx = new_ctx = BN_CTX_new(); 634 if (ctx == NULL) 635 return 0; 636 } 637 638 BN_CTX_start(ctx); 639 n0 = BN_CTX_get(ctx); 640 n1 = BN_CTX_get(ctx); 641 n2 = BN_CTX_get(ctx); 642 n3 = BN_CTX_get(ctx); 643 n4 = BN_CTX_get(ctx); 644 n5 = BN_CTX_get(ctx); 645 n6 = BN_CTX_get(ctx); 646 if (n6 == NULL) 647 goto end; 648 649 /* 650 * Note that in this function we must not read components of 'a' or 'b' 651 * once we have written the corresponding components of 'r'. ('r' might 652 * be one of 'a' or 'b'.) 653 */ 654 655 /* n1, n2 */ 656 if (b->Z_is_one) { 657 if (!BN_copy(n1, a->X)) 658 goto end; 659 if (!BN_copy(n2, a->Y)) 660 goto end; 661 /* n1 = X_a */ 662 /* n2 = Y_a */ 663 } else { 664 if (!field_sqr(group, n0, b->Z, ctx)) 665 goto end; 666 if (!field_mul(group, n1, a->X, n0, ctx)) 667 goto end; 668 /* n1 = X_a * Z_b^2 */ 669 670 if (!field_mul(group, n0, n0, b->Z, ctx)) 671 goto end; 672 if (!field_mul(group, n2, a->Y, n0, ctx)) 673 goto end; 674 /* n2 = Y_a * Z_b^3 */ 675 } 676 677 /* n3, n4 */ 678 if (a->Z_is_one) { 679 if (!BN_copy(n3, b->X)) 680 goto end; 681 if (!BN_copy(n4, b->Y)) 682 goto end; 683 /* n3 = X_b */ 684 /* n4 = Y_b */ 685 } else { 686 if (!field_sqr(group, n0, a->Z, ctx)) 687 goto end; 688 if (!field_mul(group, n3, b->X, n0, ctx)) 689 goto end; 690 /* n3 = X_b * Z_a^2 */ 691 692 if (!field_mul(group, n0, n0, a->Z, ctx)) 693 goto end; 694 if (!field_mul(group, n4, b->Y, n0, ctx)) 695 goto end; 696 /* n4 = Y_b * Z_a^3 */ 697 } 698 699 /* n5, n6 */ 700 if (!BN_mod_sub_quick(n5, n1, n3, p)) 701 goto end; 702 if (!BN_mod_sub_quick(n6, n2, n4, p)) 703 goto end; 704 /* n5 = n1 - n3 */ 705 /* n6 = n2 - n4 */ 706 707 if (BN_is_zero(n5)) { 708 if (BN_is_zero(n6)) { 709 /* a is the same point as b */ 710 BN_CTX_end(ctx); 711 ret = EC_POINT_dbl(group, r, a, ctx); 712 ctx = NULL; 713 goto end; 714 } else { 715 /* a is the inverse of b */ 716 BN_zero(r->Z); 717 r->Z_is_one = 0; 718 ret = 1; 719 goto end; 720 } 721 } 722 723 /* 'n7', 'n8' */ 724 if (!BN_mod_add_quick(n1, n1, n3, p)) 725 goto end; 726 if (!BN_mod_add_quick(n2, n2, n4, p)) 727 goto end; 728 /* 'n7' = n1 + n3 */ 729 /* 'n8' = n2 + n4 */ 730 731 /* Z_r */ 732 if (a->Z_is_one && b->Z_is_one) { 733 if (!BN_copy(r->Z, n5)) 734 goto end; 735 } else { 736 if (a->Z_is_one) { 737 if (!BN_copy(n0, b->Z)) 738 goto end; 739 } else if (b->Z_is_one) { 740 if (!BN_copy(n0, a->Z)) 741 goto end; 742 } else { 743 if (!field_mul(group, n0, a->Z, b->Z, ctx)) 744 goto end; 745 } 746 if (!field_mul(group, r->Z, n0, n5, ctx)) 747 goto end; 748 } 749 r->Z_is_one = 0; 750 /* Z_r = Z_a * Z_b * n5 */ 751 752 /* X_r */ 753 if (!field_sqr(group, n0, n6, ctx)) 754 goto end; 755 if (!field_sqr(group, n4, n5, ctx)) 756 goto end; 757 if (!field_mul(group, n3, n1, n4, ctx)) 758 goto end; 759 if (!BN_mod_sub_quick(r->X, n0, n3, p)) 760 goto end; 761 /* X_r = n6^2 - n5^2 * 'n7' */ 762 763 /* 'n9' */ 764 if (!BN_mod_lshift1_quick(n0, r->X, p)) 765 goto end; 766 if (!BN_mod_sub_quick(n0, n3, n0, p)) 767 goto end; 768 /* n9 = n5^2 * 'n7' - 2 * X_r */ 769 770 /* Y_r */ 771 if (!field_mul(group, n0, n0, n6, ctx)) 772 goto end; 773 if (!field_mul(group, n5, n4, n5, ctx)) 774 goto end; /* now n5 is n5^3 */ 775 if (!field_mul(group, n1, n2, n5, ctx)) 776 goto end; 777 if (!BN_mod_sub_quick(n0, n0, n1, p)) 778 goto end; 779 if (BN_is_odd(n0)) 780 if (!BN_add(n0, n0, p)) 781 goto end; 782 /* now 0 <= n0 < 2*p, and n0 is even */ 783 if (!BN_rshift1(r->Y, n0)) 784 goto end; 785 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */ 786 787 ret = 1; 788 789 end: 790 if (ctx) /* otherwise we already called BN_CTX_end */ 791 BN_CTX_end(ctx); 792 BN_CTX_free(new_ctx); 793 return ret; 794 } 795 796 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, 797 BN_CTX *ctx) 798 { 799 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, 800 const BIGNUM *, BN_CTX *); 801 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *); 802 const BIGNUM *p; 803 BN_CTX *new_ctx = NULL; 804 BIGNUM *n0, *n1, *n2, *n3; 805 int ret = 0; 806 807 if (EC_POINT_is_at_infinity(group, a)) { 808 BN_zero(r->Z); 809 r->Z_is_one = 0; 810 return 1; 811 } 812 813 field_mul = group->meth->field_mul; 814 field_sqr = group->meth->field_sqr; 815 p = group->field; 816 817 if (ctx == NULL) { 818 ctx = new_ctx = BN_CTX_new(); 819 if (ctx == NULL) 820 return 0; 821 } 822 823 BN_CTX_start(ctx); 824 n0 = BN_CTX_get(ctx); 825 n1 = BN_CTX_get(ctx); 826 n2 = BN_CTX_get(ctx); 827 n3 = BN_CTX_get(ctx); 828 if (n3 == NULL) 829 goto err; 830 831 /* 832 * Note that in this function we must not read components of 'a' once we 833 * have written the corresponding components of 'r'. ('r' might the same 834 * as 'a'.) 835 */ 836 837 /* n1 */ 838 if (a->Z_is_one) { 839 if (!field_sqr(group, n0, a->X, ctx)) 840 goto err; 841 if (!BN_mod_lshift1_quick(n1, n0, p)) 842 goto err; 843 if (!BN_mod_add_quick(n0, n0, n1, p)) 844 goto err; 845 if (!BN_mod_add_quick(n1, n0, group->a, p)) 846 goto err; 847 /* n1 = 3 * X_a^2 + a_curve */ 848 } else if (group->a_is_minus3) { 849 if (!field_sqr(group, n1, a->Z, ctx)) 850 goto err; 851 if (!BN_mod_add_quick(n0, a->X, n1, p)) 852 goto err; 853 if (!BN_mod_sub_quick(n2, a->X, n1, p)) 854 goto err; 855 if (!field_mul(group, n1, n0, n2, ctx)) 856 goto err; 857 if (!BN_mod_lshift1_quick(n0, n1, p)) 858 goto err; 859 if (!BN_mod_add_quick(n1, n0, n1, p)) 860 goto err; 861 /*- 862 * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2) 863 * = 3 * X_a^2 - 3 * Z_a^4 864 */ 865 } else { 866 if (!field_sqr(group, n0, a->X, ctx)) 867 goto err; 868 if (!BN_mod_lshift1_quick(n1, n0, p)) 869 goto err; 870 if (!BN_mod_add_quick(n0, n0, n1, p)) 871 goto err; 872 if (!field_sqr(group, n1, a->Z, ctx)) 873 goto err; 874 if (!field_sqr(group, n1, n1, ctx)) 875 goto err; 876 if (!field_mul(group, n1, n1, group->a, ctx)) 877 goto err; 878 if (!BN_mod_add_quick(n1, n1, n0, p)) 879 goto err; 880 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */ 881 } 882 883 /* Z_r */ 884 if (a->Z_is_one) { 885 if (!BN_copy(n0, a->Y)) 886 goto err; 887 } else { 888 if (!field_mul(group, n0, a->Y, a->Z, ctx)) 889 goto err; 890 } 891 if (!BN_mod_lshift1_quick(r->Z, n0, p)) 892 goto err; 893 r->Z_is_one = 0; 894 /* Z_r = 2 * Y_a * Z_a */ 895 896 /* n2 */ 897 if (!field_sqr(group, n3, a->Y, ctx)) 898 goto err; 899 if (!field_mul(group, n2, a->X, n3, ctx)) 900 goto err; 901 if (!BN_mod_lshift_quick(n2, n2, 2, p)) 902 goto err; 903 /* n2 = 4 * X_a * Y_a^2 */ 904 905 /* X_r */ 906 if (!BN_mod_lshift1_quick(n0, n2, p)) 907 goto err; 908 if (!field_sqr(group, r->X, n1, ctx)) 909 goto err; 910 if (!BN_mod_sub_quick(r->X, r->X, n0, p)) 911 goto err; 912 /* X_r = n1^2 - 2 * n2 */ 913 914 /* n3 */ 915 if (!field_sqr(group, n0, n3, ctx)) 916 goto err; 917 if (!BN_mod_lshift_quick(n3, n0, 3, p)) 918 goto err; 919 /* n3 = 8 * Y_a^4 */ 920 921 /* Y_r */ 922 if (!BN_mod_sub_quick(n0, n2, r->X, p)) 923 goto err; 924 if (!field_mul(group, n0, n1, n0, ctx)) 925 goto err; 926 if (!BN_mod_sub_quick(r->Y, n0, n3, p)) 927 goto err; 928 /* Y_r = n1 * (n2 - X_r) - n3 */ 929 930 ret = 1; 931 932 err: 933 BN_CTX_end(ctx); 934 BN_CTX_free(new_ctx); 935 return ret; 936 } 937 938 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx) 939 { 940 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y)) 941 /* point is its own inverse */ 942 return 1; 943 944 return BN_usub(point->Y, group->field, point->Y); 945 } 946 947 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point) 948 { 949 return BN_is_zero(point->Z); 950 } 951 952 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, 953 BN_CTX *ctx) 954 { 955 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, 956 const BIGNUM *, BN_CTX *); 957 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *); 958 const BIGNUM *p; 959 BN_CTX *new_ctx = NULL; 960 BIGNUM *rh, *tmp, *Z4, *Z6; 961 int ret = -1; 962 963 if (EC_POINT_is_at_infinity(group, point)) 964 return 1; 965 966 field_mul = group->meth->field_mul; 967 field_sqr = group->meth->field_sqr; 968 p = group->field; 969 970 if (ctx == NULL) { 971 ctx = new_ctx = BN_CTX_new(); 972 if (ctx == NULL) 973 return -1; 974 } 975 976 BN_CTX_start(ctx); 977 rh = BN_CTX_get(ctx); 978 tmp = BN_CTX_get(ctx); 979 Z4 = BN_CTX_get(ctx); 980 Z6 = BN_CTX_get(ctx); 981 if (Z6 == NULL) 982 goto err; 983 984 /*- 985 * We have a curve defined by a Weierstrass equation 986 * y^2 = x^3 + a*x + b. 987 * The point to consider is given in Jacobian projective coordinates 988 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3). 989 * Substituting this and multiplying by Z^6 transforms the above equation into 990 * Y^2 = X^3 + a*X*Z^4 + b*Z^6. 991 * To test this, we add up the right-hand side in 'rh'. 992 */ 993 994 /* rh := X^2 */ 995 if (!field_sqr(group, rh, point->X, ctx)) 996 goto err; 997 998 if (!point->Z_is_one) { 999 if (!field_sqr(group, tmp, point->Z, ctx)) 1000 goto err; 1001 if (!field_sqr(group, Z4, tmp, ctx)) 1002 goto err; 1003 if (!field_mul(group, Z6, Z4, tmp, ctx)) 1004 goto err; 1005 1006 /* rh := (rh + a*Z^4)*X */ 1007 if (group->a_is_minus3) { 1008 if (!BN_mod_lshift1_quick(tmp, Z4, p)) 1009 goto err; 1010 if (!BN_mod_add_quick(tmp, tmp, Z4, p)) 1011 goto err; 1012 if (!BN_mod_sub_quick(rh, rh, tmp, p)) 1013 goto err; 1014 if (!field_mul(group, rh, rh, point->X, ctx)) 1015 goto err; 1016 } else { 1017 if (!field_mul(group, tmp, Z4, group->a, ctx)) 1018 goto err; 1019 if (!BN_mod_add_quick(rh, rh, tmp, p)) 1020 goto err; 1021 if (!field_mul(group, rh, rh, point->X, ctx)) 1022 goto err; 1023 } 1024 1025 /* rh := rh + b*Z^6 */ 1026 if (!field_mul(group, tmp, group->b, Z6, ctx)) 1027 goto err; 1028 if (!BN_mod_add_quick(rh, rh, tmp, p)) 1029 goto err; 1030 } else { 1031 /* point->Z_is_one */ 1032 1033 /* rh := (rh + a)*X */ 1034 if (!BN_mod_add_quick(rh, rh, group->a, p)) 1035 goto err; 1036 if (!field_mul(group, rh, rh, point->X, ctx)) 1037 goto err; 1038 /* rh := rh + b */ 1039 if (!BN_mod_add_quick(rh, rh, group->b, p)) 1040 goto err; 1041 } 1042 1043 /* 'lh' := Y^2 */ 1044 if (!field_sqr(group, tmp, point->Y, ctx)) 1045 goto err; 1046 1047 ret = (0 == BN_ucmp(tmp, rh)); 1048 1049 err: 1050 BN_CTX_end(ctx); 1051 BN_CTX_free(new_ctx); 1052 return ret; 1053 } 1054 1055 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a, 1056 const EC_POINT *b, BN_CTX *ctx) 1057 { 1058 /*- 1059 * return values: 1060 * -1 error 1061 * 0 equal (in affine coordinates) 1062 * 1 not equal 1063 */ 1064 1065 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, 1066 const BIGNUM *, BN_CTX *); 1067 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *); 1068 BN_CTX *new_ctx = NULL; 1069 BIGNUM *tmp1, *tmp2, *Za23, *Zb23; 1070 const BIGNUM *tmp1_, *tmp2_; 1071 int ret = -1; 1072 1073 if (EC_POINT_is_at_infinity(group, a)) { 1074 return EC_POINT_is_at_infinity(group, b) ? 0 : 1; 1075 } 1076 1077 if (EC_POINT_is_at_infinity(group, b)) 1078 return 1; 1079 1080 if (a->Z_is_one && b->Z_is_one) { 1081 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1; 1082 } 1083 1084 field_mul = group->meth->field_mul; 1085 field_sqr = group->meth->field_sqr; 1086 1087 if (ctx == NULL) { 1088 ctx = new_ctx = BN_CTX_new(); 1089 if (ctx == NULL) 1090 return -1; 1091 } 1092 1093 BN_CTX_start(ctx); 1094 tmp1 = BN_CTX_get(ctx); 1095 tmp2 = BN_CTX_get(ctx); 1096 Za23 = BN_CTX_get(ctx); 1097 Zb23 = BN_CTX_get(ctx); 1098 if (Zb23 == NULL) 1099 goto end; 1100 1101 /*- 1102 * We have to decide whether 1103 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3), 1104 * or equivalently, whether 1105 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3). 1106 */ 1107 1108 if (!b->Z_is_one) { 1109 if (!field_sqr(group, Zb23, b->Z, ctx)) 1110 goto end; 1111 if (!field_mul(group, tmp1, a->X, Zb23, ctx)) 1112 goto end; 1113 tmp1_ = tmp1; 1114 } else 1115 tmp1_ = a->X; 1116 if (!a->Z_is_one) { 1117 if (!field_sqr(group, Za23, a->Z, ctx)) 1118 goto end; 1119 if (!field_mul(group, tmp2, b->X, Za23, ctx)) 1120 goto end; 1121 tmp2_ = tmp2; 1122 } else 1123 tmp2_ = b->X; 1124 1125 /* compare X_a*Z_b^2 with X_b*Z_a^2 */ 1126 if (BN_cmp(tmp1_, tmp2_) != 0) { 1127 ret = 1; /* points differ */ 1128 goto end; 1129 } 1130 1131 if (!b->Z_is_one) { 1132 if (!field_mul(group, Zb23, Zb23, b->Z, ctx)) 1133 goto end; 1134 if (!field_mul(group, tmp1, a->Y, Zb23, ctx)) 1135 goto end; 1136 /* tmp1_ = tmp1 */ 1137 } else 1138 tmp1_ = a->Y; 1139 if (!a->Z_is_one) { 1140 if (!field_mul(group, Za23, Za23, a->Z, ctx)) 1141 goto end; 1142 if (!field_mul(group, tmp2, b->Y, Za23, ctx)) 1143 goto end; 1144 /* tmp2_ = tmp2 */ 1145 } else 1146 tmp2_ = b->Y; 1147 1148 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */ 1149 if (BN_cmp(tmp1_, tmp2_) != 0) { 1150 ret = 1; /* points differ */ 1151 goto end; 1152 } 1153 1154 /* points are equal */ 1155 ret = 0; 1156 1157 end: 1158 BN_CTX_end(ctx); 1159 BN_CTX_free(new_ctx); 1160 return ret; 1161 } 1162 1163 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point, 1164 BN_CTX *ctx) 1165 { 1166 BN_CTX *new_ctx = NULL; 1167 BIGNUM *x, *y; 1168 int ret = 0; 1169 1170 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point)) 1171 return 1; 1172 1173 if (ctx == NULL) { 1174 ctx = new_ctx = BN_CTX_new(); 1175 if (ctx == NULL) 1176 return 0; 1177 } 1178 1179 BN_CTX_start(ctx); 1180 x = BN_CTX_get(ctx); 1181 y = BN_CTX_get(ctx); 1182 if (y == NULL) 1183 goto err; 1184 1185 if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx)) 1186 goto err; 1187 if (!EC_POINT_set_affine_coordinates(group, point, x, y, ctx)) 1188 goto err; 1189 if (!point->Z_is_one) { 1190 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR); 1191 goto err; 1192 } 1193 1194 ret = 1; 1195 1196 err: 1197 BN_CTX_end(ctx); 1198 BN_CTX_free(new_ctx); 1199 return ret; 1200 } 1201 1202 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num, 1203 EC_POINT *points[], BN_CTX *ctx) 1204 { 1205 BN_CTX *new_ctx = NULL; 1206 BIGNUM *tmp, *tmp_Z; 1207 BIGNUM **prod_Z = NULL; 1208 size_t i; 1209 int ret = 0; 1210 1211 if (num == 0) 1212 return 1; 1213 1214 if (ctx == NULL) { 1215 ctx = new_ctx = BN_CTX_new(); 1216 if (ctx == NULL) 1217 return 0; 1218 } 1219 1220 BN_CTX_start(ctx); 1221 tmp = BN_CTX_get(ctx); 1222 tmp_Z = BN_CTX_get(ctx); 1223 if (tmp_Z == NULL) 1224 goto err; 1225 1226 prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0])); 1227 if (prod_Z == NULL) 1228 goto err; 1229 for (i = 0; i < num; i++) { 1230 prod_Z[i] = BN_new(); 1231 if (prod_Z[i] == NULL) 1232 goto err; 1233 } 1234 1235 /* 1236 * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z, 1237 * skipping any zero-valued inputs (pretend that they're 1). 1238 */ 1239 1240 if (!BN_is_zero(points[0]->Z)) { 1241 if (!BN_copy(prod_Z[0], points[0]->Z)) 1242 goto err; 1243 } else { 1244 if (group->meth->field_set_to_one != 0) { 1245 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx)) 1246 goto err; 1247 } else { 1248 if (!BN_one(prod_Z[0])) 1249 goto err; 1250 } 1251 } 1252 1253 for (i = 1; i < num; i++) { 1254 if (!BN_is_zero(points[i]->Z)) { 1255 if (!group-> 1256 meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z, 1257 ctx)) 1258 goto err; 1259 } else { 1260 if (!BN_copy(prod_Z[i], prod_Z[i - 1])) 1261 goto err; 1262 } 1263 } 1264 1265 /* 1266 * Now use a single explicit inversion to replace every non-zero 1267 * points[i]->Z by its inverse. 1268 */ 1269 1270 if (!group->meth->field_inv(group, tmp, prod_Z[num - 1], ctx)) { 1271 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB); 1272 goto err; 1273 } 1274 if (group->meth->field_encode != 0) { 1275 /* 1276 * In the Montgomery case, we just turned R*H (representing H) into 1277 * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to 1278 * multiply by the Montgomery factor twice. 1279 */ 1280 if (!group->meth->field_encode(group, tmp, tmp, ctx)) 1281 goto err; 1282 if (!group->meth->field_encode(group, tmp, tmp, ctx)) 1283 goto err; 1284 } 1285 1286 for (i = num - 1; i > 0; --i) { 1287 /* 1288 * Loop invariant: tmp is the product of the inverses of points[0]->Z 1289 * .. points[i]->Z (zero-valued inputs skipped). 1290 */ 1291 if (!BN_is_zero(points[i]->Z)) { 1292 /* 1293 * Set tmp_Z to the inverse of points[i]->Z (as product of Z 1294 * inverses 0 .. i, Z values 0 .. i - 1). 1295 */ 1296 if (!group-> 1297 meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) 1298 goto err; 1299 /* 1300 * Update tmp to satisfy the loop invariant for i - 1. 1301 */ 1302 if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx)) 1303 goto err; 1304 /* Replace points[i]->Z by its inverse. */ 1305 if (!BN_copy(points[i]->Z, tmp_Z)) 1306 goto err; 1307 } 1308 } 1309 1310 if (!BN_is_zero(points[0]->Z)) { 1311 /* Replace points[0]->Z by its inverse. */ 1312 if (!BN_copy(points[0]->Z, tmp)) 1313 goto err; 1314 } 1315 1316 /* Finally, fix up the X and Y coordinates for all points. */ 1317 1318 for (i = 0; i < num; i++) { 1319 EC_POINT *p = points[i]; 1320 1321 if (!BN_is_zero(p->Z)) { 1322 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */ 1323 1324 if (!group->meth->field_sqr(group, tmp, p->Z, ctx)) 1325 goto err; 1326 if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx)) 1327 goto err; 1328 1329 if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx)) 1330 goto err; 1331 if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx)) 1332 goto err; 1333 1334 if (group->meth->field_set_to_one != 0) { 1335 if (!group->meth->field_set_to_one(group, p->Z, ctx)) 1336 goto err; 1337 } else { 1338 if (!BN_one(p->Z)) 1339 goto err; 1340 } 1341 p->Z_is_one = 1; 1342 } 1343 } 1344 1345 ret = 1; 1346 1347 err: 1348 BN_CTX_end(ctx); 1349 BN_CTX_free(new_ctx); 1350 if (prod_Z != NULL) { 1351 for (i = 0; i < num; i++) { 1352 if (prod_Z[i] == NULL) 1353 break; 1354 BN_clear_free(prod_Z[i]); 1355 } 1356 OPENSSL_free(prod_Z); 1357 } 1358 return ret; 1359 } 1360 1361 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, 1362 const BIGNUM *b, BN_CTX *ctx) 1363 { 1364 return BN_mod_mul(r, a, b, group->field, ctx); 1365 } 1366 1367 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, 1368 BN_CTX *ctx) 1369 { 1370 return BN_mod_sqr(r, a, group->field, ctx); 1371 } 1372 1373 /*- 1374 * Computes the multiplicative inverse of a in GF(p), storing the result in r. 1375 * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error. 1376 * Since we don't have a Mont structure here, SCA hardening is with blinding. 1377 */ 1378 int ec_GFp_simple_field_inv(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, 1379 BN_CTX *ctx) 1380 { 1381 BIGNUM *e = NULL; 1382 BN_CTX *new_ctx = NULL; 1383 int ret = 0; 1384 1385 if (ctx == NULL && (ctx = new_ctx = BN_CTX_secure_new()) == NULL) 1386 return 0; 1387 1388 BN_CTX_start(ctx); 1389 if ((e = BN_CTX_get(ctx)) == NULL) 1390 goto err; 1391 1392 do { 1393 if (!BN_priv_rand_range(e, group->field)) 1394 goto err; 1395 } while (BN_is_zero(e)); 1396 1397 /* r := a * e */ 1398 if (!group->meth->field_mul(group, r, a, e, ctx)) 1399 goto err; 1400 /* r := 1/(a * e) */ 1401 if (!BN_mod_inverse(r, r, group->field, ctx)) { 1402 ECerr(EC_F_EC_GFP_SIMPLE_FIELD_INV, EC_R_CANNOT_INVERT); 1403 goto err; 1404 } 1405 /* r := e/(a * e) = 1/a */ 1406 if (!group->meth->field_mul(group, r, r, e, ctx)) 1407 goto err; 1408 1409 ret = 1; 1410 1411 err: 1412 BN_CTX_end(ctx); 1413 BN_CTX_free(new_ctx); 1414 return ret; 1415 } 1416 1417 /*- 1418 * Apply randomization of EC point projective coordinates: 1419 * 1420 * (X, Y ,Z ) = (lambda^2*X, lambda^3*Y, lambda*Z) 1421 * lambda = [1,group->field) 1422 * 1423 */ 1424 int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p, 1425 BN_CTX *ctx) 1426 { 1427 int ret = 0; 1428 BIGNUM *lambda = NULL; 1429 BIGNUM *temp = NULL; 1430 1431 BN_CTX_start(ctx); 1432 lambda = BN_CTX_get(ctx); 1433 temp = BN_CTX_get(ctx); 1434 if (temp == NULL) { 1435 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_MALLOC_FAILURE); 1436 goto err; 1437 } 1438 1439 /* make sure lambda is not zero */ 1440 do { 1441 if (!BN_priv_rand_range(lambda, group->field)) { 1442 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_BN_LIB); 1443 goto err; 1444 } 1445 } while (BN_is_zero(lambda)); 1446 1447 /* if field_encode defined convert between representations */ 1448 if (group->meth->field_encode != NULL 1449 && !group->meth->field_encode(group, lambda, lambda, ctx)) 1450 goto err; 1451 if (!group->meth->field_mul(group, p->Z, p->Z, lambda, ctx)) 1452 goto err; 1453 if (!group->meth->field_sqr(group, temp, lambda, ctx)) 1454 goto err; 1455 if (!group->meth->field_mul(group, p->X, p->X, temp, ctx)) 1456 goto err; 1457 if (!group->meth->field_mul(group, temp, temp, lambda, ctx)) 1458 goto err; 1459 if (!group->meth->field_mul(group, p->Y, p->Y, temp, ctx)) 1460 goto err; 1461 p->Z_is_one = 0; 1462 1463 ret = 1; 1464 1465 err: 1466 BN_CTX_end(ctx); 1467 return ret; 1468 } 1469 1470 /*- 1471 * Set s := p, r := 2p. 1472 * 1473 * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve 1474 * multiplication resistant against side channel attacks" appendix, as described 1475 * at 1476 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2 1477 * 1478 * The input point p will be in randomized Jacobian projective coords: 1479 * x = X/Z**2, y=Y/Z**3 1480 * 1481 * The output points p, s, and r are converted to standard (homogeneous) 1482 * projective coords: 1483 * x = X/Z, y=Y/Z 1484 */ 1485 int ec_GFp_simple_ladder_pre(const EC_GROUP *group, 1486 EC_POINT *r, EC_POINT *s, 1487 EC_POINT *p, BN_CTX *ctx) 1488 { 1489 BIGNUM *t1, *t2, *t3, *t4, *t5, *t6 = NULL; 1490 1491 t1 = r->Z; 1492 t2 = r->Y; 1493 t3 = s->X; 1494 t4 = r->X; 1495 t5 = s->Y; 1496 t6 = s->Z; 1497 1498 /* convert p: (X,Y,Z) -> (XZ,Y,Z**3) */ 1499 if (!group->meth->field_mul(group, p->X, p->X, p->Z, ctx) 1500 || !group->meth->field_sqr(group, t1, p->Z, ctx) 1501 || !group->meth->field_mul(group, p->Z, p->Z, t1, ctx) 1502 /* r := 2p */ 1503 || !group->meth->field_sqr(group, t2, p->X, ctx) 1504 || !group->meth->field_sqr(group, t3, p->Z, ctx) 1505 || !group->meth->field_mul(group, t4, t3, group->a, ctx) 1506 || !BN_mod_sub_quick(t5, t2, t4, group->field) 1507 || !BN_mod_add_quick(t2, t2, t4, group->field) 1508 || !group->meth->field_sqr(group, t5, t5, ctx) 1509 || !group->meth->field_mul(group, t6, t3, group->b, ctx) 1510 || !group->meth->field_mul(group, t1, p->X, p->Z, ctx) 1511 || !group->meth->field_mul(group, t4, t1, t6, ctx) 1512 || !BN_mod_lshift_quick(t4, t4, 3, group->field) 1513 /* r->X coord output */ 1514 || !BN_mod_sub_quick(r->X, t5, t4, group->field) 1515 || !group->meth->field_mul(group, t1, t1, t2, ctx) 1516 || !group->meth->field_mul(group, t2, t3, t6, ctx) 1517 || !BN_mod_add_quick(t1, t1, t2, group->field) 1518 /* r->Z coord output */ 1519 || !BN_mod_lshift_quick(r->Z, t1, 2, group->field) 1520 || !EC_POINT_copy(s, p)) 1521 return 0; 1522 1523 r->Z_is_one = 0; 1524 s->Z_is_one = 0; 1525 p->Z_is_one = 0; 1526 1527 return 1; 1528 } 1529 1530 /*- 1531 * Differential addition-and-doubling using Eq. (9) and (10) from Izu-Takagi 1532 * "A fast parallel elliptic curve multiplication resistant against side channel 1533 * attacks", as described at 1534 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-4 1535 */ 1536 int ec_GFp_simple_ladder_step(const EC_GROUP *group, 1537 EC_POINT *r, EC_POINT *s, 1538 EC_POINT *p, BN_CTX *ctx) 1539 { 1540 int ret = 0; 1541 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6, *t7 = NULL; 1542 1543 BN_CTX_start(ctx); 1544 t0 = BN_CTX_get(ctx); 1545 t1 = BN_CTX_get(ctx); 1546 t2 = BN_CTX_get(ctx); 1547 t3 = BN_CTX_get(ctx); 1548 t4 = BN_CTX_get(ctx); 1549 t5 = BN_CTX_get(ctx); 1550 t6 = BN_CTX_get(ctx); 1551 t7 = BN_CTX_get(ctx); 1552 1553 if (t7 == NULL 1554 || !group->meth->field_mul(group, t0, r->X, s->X, ctx) 1555 || !group->meth->field_mul(group, t1, r->Z, s->Z, ctx) 1556 || !group->meth->field_mul(group, t2, r->X, s->Z, ctx) 1557 || !group->meth->field_mul(group, t3, r->Z, s->X, ctx) 1558 || !group->meth->field_mul(group, t4, group->a, t1, ctx) 1559 || !BN_mod_add_quick(t0, t0, t4, group->field) 1560 || !BN_mod_add_quick(t4, t3, t2, group->field) 1561 || !group->meth->field_mul(group, t0, t4, t0, ctx) 1562 || !group->meth->field_sqr(group, t1, t1, ctx) 1563 || !BN_mod_lshift_quick(t7, group->b, 2, group->field) 1564 || !group->meth->field_mul(group, t1, t7, t1, ctx) 1565 || !BN_mod_lshift1_quick(t0, t0, group->field) 1566 || !BN_mod_add_quick(t0, t1, t0, group->field) 1567 || !BN_mod_sub_quick(t1, t2, t3, group->field) 1568 || !group->meth->field_sqr(group, t1, t1, ctx) 1569 || !group->meth->field_mul(group, t3, t1, p->X, ctx) 1570 || !group->meth->field_mul(group, t0, p->Z, t0, ctx) 1571 /* s->X coord output */ 1572 || !BN_mod_sub_quick(s->X, t0, t3, group->field) 1573 /* s->Z coord output */ 1574 || !group->meth->field_mul(group, s->Z, p->Z, t1, ctx) 1575 || !group->meth->field_sqr(group, t3, r->X, ctx) 1576 || !group->meth->field_sqr(group, t2, r->Z, ctx) 1577 || !group->meth->field_mul(group, t4, t2, group->a, ctx) 1578 || !BN_mod_add_quick(t5, r->X, r->Z, group->field) 1579 || !group->meth->field_sqr(group, t5, t5, ctx) 1580 || !BN_mod_sub_quick(t5, t5, t3, group->field) 1581 || !BN_mod_sub_quick(t5, t5, t2, group->field) 1582 || !BN_mod_sub_quick(t6, t3, t4, group->field) 1583 || !group->meth->field_sqr(group, t6, t6, ctx) 1584 || !group->meth->field_mul(group, t0, t2, t5, ctx) 1585 || !group->meth->field_mul(group, t0, t7, t0, ctx) 1586 /* r->X coord output */ 1587 || !BN_mod_sub_quick(r->X, t6, t0, group->field) 1588 || !BN_mod_add_quick(t6, t3, t4, group->field) 1589 || !group->meth->field_sqr(group, t3, t2, ctx) 1590 || !group->meth->field_mul(group, t7, t3, t7, ctx) 1591 || !group->meth->field_mul(group, t5, t5, t6, ctx) 1592 || !BN_mod_lshift1_quick(t5, t5, group->field) 1593 /* r->Z coord output */ 1594 || !BN_mod_add_quick(r->Z, t7, t5, group->field)) 1595 goto err; 1596 1597 ret = 1; 1598 1599 err: 1600 BN_CTX_end(ctx); 1601 return ret; 1602 } 1603 1604 /*- 1605 * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass 1606 * Elliptic Curves and Side-Channel Attacks", modified to work in projective 1607 * coordinates and return r in Jacobian projective coordinates. 1608 * 1609 * X4 = two*Y1*X2*Z3*Z2*Z1; 1610 * Y4 = two*b*Z3*SQR(Z2*Z1) + Z3*(a*Z2*Z1+X1*X2)*(X1*Z2+X2*Z1) - X3*SQR(X1*Z2-X2*Z1); 1611 * Z4 = two*Y1*Z3*SQR(Z2)*Z1; 1612 * 1613 * Z4 != 0 because: 1614 * - Z1==0 implies p is at infinity, which would have caused an early exit in 1615 * the caller; 1616 * - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch); 1617 * - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch); 1618 * - Y1==0 implies p has order 2, so either r or s are infinity and handled by 1619 * one of the BN_is_zero(...) branches. 1620 */ 1621 int ec_GFp_simple_ladder_post(const EC_GROUP *group, 1622 EC_POINT *r, EC_POINT *s, 1623 EC_POINT *p, BN_CTX *ctx) 1624 { 1625 int ret = 0; 1626 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL; 1627 1628 if (BN_is_zero(r->Z)) 1629 return EC_POINT_set_to_infinity(group, r); 1630 1631 if (BN_is_zero(s->Z)) { 1632 /* (X,Y,Z) -> (XZ,YZ**2,Z) */ 1633 if (!group->meth->field_mul(group, r->X, p->X, p->Z, ctx) 1634 || !group->meth->field_sqr(group, r->Z, p->Z, ctx) 1635 || !group->meth->field_mul(group, r->Y, p->Y, r->Z, ctx) 1636 || !BN_copy(r->Z, p->Z) 1637 || !EC_POINT_invert(group, r, ctx)) 1638 return 0; 1639 return 1; 1640 } 1641 1642 BN_CTX_start(ctx); 1643 t0 = BN_CTX_get(ctx); 1644 t1 = BN_CTX_get(ctx); 1645 t2 = BN_CTX_get(ctx); 1646 t3 = BN_CTX_get(ctx); 1647 t4 = BN_CTX_get(ctx); 1648 t5 = BN_CTX_get(ctx); 1649 t6 = BN_CTX_get(ctx); 1650 1651 if (t6 == NULL 1652 || !BN_mod_lshift1_quick(t0, p->Y, group->field) 1653 || !group->meth->field_mul(group, t1, r->X, p->Z, ctx) 1654 || !group->meth->field_mul(group, t2, r->Z, s->Z, ctx) 1655 || !group->meth->field_mul(group, t2, t1, t2, ctx) 1656 || !group->meth->field_mul(group, t3, t2, t0, ctx) 1657 || !group->meth->field_mul(group, t2, r->Z, p->Z, ctx) 1658 || !group->meth->field_sqr(group, t4, t2, ctx) 1659 || !BN_mod_lshift1_quick(t5, group->b, group->field) 1660 || !group->meth->field_mul(group, t4, t4, t5, ctx) 1661 || !group->meth->field_mul(group, t6, t2, group->a, ctx) 1662 || !group->meth->field_mul(group, t5, r->X, p->X, ctx) 1663 || !BN_mod_add_quick(t5, t6, t5, group->field) 1664 || !group->meth->field_mul(group, t6, r->Z, p->X, ctx) 1665 || !BN_mod_add_quick(t2, t6, t1, group->field) 1666 || !group->meth->field_mul(group, t5, t5, t2, ctx) 1667 || !BN_mod_sub_quick(t6, t6, t1, group->field) 1668 || !group->meth->field_sqr(group, t6, t6, ctx) 1669 || !group->meth->field_mul(group, t6, t6, s->X, ctx) 1670 || !BN_mod_add_quick(t4, t5, t4, group->field) 1671 || !group->meth->field_mul(group, t4, t4, s->Z, ctx) 1672 || !BN_mod_sub_quick(t4, t4, t6, group->field) 1673 || !group->meth->field_sqr(group, t5, r->Z, ctx) 1674 || !group->meth->field_mul(group, r->Z, p->Z, s->Z, ctx) 1675 || !group->meth->field_mul(group, r->Z, t5, r->Z, ctx) 1676 || !group->meth->field_mul(group, r->Z, r->Z, t0, ctx) 1677 /* t3 := X, t4 := Y */ 1678 /* (X,Y,Z) -> (XZ,YZ**2,Z) */ 1679 || !group->meth->field_mul(group, r->X, t3, r->Z, ctx) 1680 || !group->meth->field_sqr(group, t3, r->Z, ctx) 1681 || !group->meth->field_mul(group, r->Y, t4, t3, ctx)) 1682 goto err; 1683 1684 ret = 1; 1685 1686 err: 1687 BN_CTX_end(ctx); 1688 return ret; 1689 } 1690