1/* crypto/ec/ec2_mult.c */ 2/* ==================================================================== 3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. 4 * 5 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included 6 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed 7 * to the OpenSSL project. 8 * 9 * The ECC Code is licensed pursuant to the OpenSSL open source 10 * license provided below. 11 * 12 * The software is originally written by Sheueling Chang Shantz and 13 * Douglas Stebila of Sun Microsystems Laboratories. 14 * 15 */ 16/* ==================================================================== 17 * Copyright (c) 1998-2003 The OpenSSL Project. All rights reserved. 18 * 19 * Redistribution and use in source and binary forms, with or without 20 * modification, are permitted provided that the following conditions 21 * are met: 22 * 23 * 1. Redistributions of source code must retain the above copyright 24 * notice, this list of conditions and the following disclaimer. 25 * 26 * 2. Redistributions in binary form must reproduce the above copyright 27 * notice, this list of conditions and the following disclaimer in 28 * the documentation and/or other materials provided with the 29 * distribution. 30 * 31 * 3. All advertising materials mentioning features or use of this 32 * software must display the following acknowledgment: 33 * "This product includes software developed by the OpenSSL Project 34 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 35 * 36 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 37 * endorse or promote products derived from this software without 38 * prior written permission. For written permission, please contact 39 * openssl-core@openssl.org. 40 * 41 * 5. Products derived from this software may not be called "OpenSSL" 42 * nor may "OpenSSL" appear in their names without prior written 43 * permission of the OpenSSL Project. 44 * 45 * 6. Redistributions of any form whatsoever must retain the following 46 * acknowledgment: 47 * "This product includes software developed by the OpenSSL Project 48 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 49 * 50 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 51 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 52 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 53 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 54 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 55 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 56 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 57 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 58 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 59 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 60 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 61 * OF THE POSSIBILITY OF SUCH DAMAGE. 62 * ==================================================================== 63 * 64 * This product includes cryptographic software written by Eric Young 65 * (eay@cryptsoft.com). This product includes software written by Tim 66 * Hudson (tjh@cryptsoft.com). 67 * 68 */ 69 70#include <openssl/err.h> 71 72#include "ec_lcl.h" 73 74#ifndef OPENSSL_NO_EC2M 75 76/*- 77 * Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective 78 * coordinates. 79 * Uses algorithm Mdouble in appendix of 80 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 81 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 82 * modified to not require precomputation of c=b^{2^{m-1}}. 83 */ 84static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, 85 BN_CTX *ctx) 86{ 87 BIGNUM *t1; 88 int ret = 0; 89 90 /* Since Mdouble is static we can guarantee that ctx != NULL. */ 91 BN_CTX_start(ctx); 92 t1 = BN_CTX_get(ctx); 93 if (t1 == NULL) 94 goto err; 95 96 if (!group->meth->field_sqr(group, x, x, ctx)) 97 goto err; 98 if (!group->meth->field_sqr(group, t1, z, ctx)) 99 goto err; 100 if (!group->meth->field_mul(group, z, x, t1, ctx)) 101 goto err; 102 if (!group->meth->field_sqr(group, x, x, ctx)) 103 goto err; 104 if (!group->meth->field_sqr(group, t1, t1, ctx)) 105 goto err; 106 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) 107 goto err; 108 if (!BN_GF2m_add(x, x, t1)) 109 goto err; 110 111 ret = 1; 112 113 err: 114 BN_CTX_end(ctx); 115 return ret; 116} 117 118/*- 119 * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery 120 * projective coordinates. 121 * Uses algorithm Madd in appendix of 122 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 123 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 124 */ 125static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, 126 BIGNUM *z1, const BIGNUM *x2, const BIGNUM *z2, 127 BN_CTX *ctx) 128{ 129 BIGNUM *t1, *t2; 130 int ret = 0; 131 132 /* Since Madd is static we can guarantee that ctx != NULL. */ 133 BN_CTX_start(ctx); 134 t1 = BN_CTX_get(ctx); 135 t2 = BN_CTX_get(ctx); 136 if (t2 == NULL) 137 goto err; 138 139 if (!BN_copy(t1, x)) 140 goto err; 141 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) 142 goto err; 143 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) 144 goto err; 145 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) 146 goto err; 147 if (!BN_GF2m_add(z1, z1, x1)) 148 goto err; 149 if (!group->meth->field_sqr(group, z1, z1, ctx)) 150 goto err; 151 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) 152 goto err; 153 if (!BN_GF2m_add(x1, x1, t2)) 154 goto err; 155 156 ret = 1; 157 158 err: 159 BN_CTX_end(ctx); 160 return ret; 161} 162 163/*- 164 * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) 165 * using Montgomery point multiplication algorithm Mxy() in appendix of 166 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 167 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 168 * Returns: 169 * 0 on error 170 * 1 if return value should be the point at infinity 171 * 2 otherwise 172 */ 173static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, 174 BIGNUM *x1, BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, 175 BN_CTX *ctx) 176{ 177 BIGNUM *t3, *t4, *t5; 178 int ret = 0; 179 180 if (BN_is_zero(z1)) { 181 BN_zero(x2); 182 BN_zero(z2); 183 return 1; 184 } 185 186 if (BN_is_zero(z2)) { 187 if (!BN_copy(x2, x)) 188 return 0; 189 if (!BN_GF2m_add(z2, x, y)) 190 return 0; 191 return 2; 192 } 193 194 /* Since Mxy is static we can guarantee that ctx != NULL. */ 195 BN_CTX_start(ctx); 196 t3 = BN_CTX_get(ctx); 197 t4 = BN_CTX_get(ctx); 198 t5 = BN_CTX_get(ctx); 199 if (t5 == NULL) 200 goto err; 201 202 if (!BN_one(t5)) 203 goto err; 204 205 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) 206 goto err; 207 208 if (!group->meth->field_mul(group, z1, z1, x, ctx)) 209 goto err; 210 if (!BN_GF2m_add(z1, z1, x1)) 211 goto err; 212 if (!group->meth->field_mul(group, z2, z2, x, ctx)) 213 goto err; 214 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) 215 goto err; 216 if (!BN_GF2m_add(z2, z2, x2)) 217 goto err; 218 219 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) 220 goto err; 221 if (!group->meth->field_sqr(group, t4, x, ctx)) 222 goto err; 223 if (!BN_GF2m_add(t4, t4, y)) 224 goto err; 225 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) 226 goto err; 227 if (!BN_GF2m_add(t4, t4, z2)) 228 goto err; 229 230 if (!group->meth->field_mul(group, t3, t3, x, ctx)) 231 goto err; 232 if (!group->meth->field_div(group, t3, t5, t3, ctx)) 233 goto err; 234 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) 235 goto err; 236 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) 237 goto err; 238 if (!BN_GF2m_add(z2, x2, x)) 239 goto err; 240 241 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) 242 goto err; 243 if (!BN_GF2m_add(z2, z2, y)) 244 goto err; 245 246 ret = 2; 247 248 err: 249 BN_CTX_end(ctx); 250 return ret; 251} 252 253/*- 254 * Computes scalar*point and stores the result in r. 255 * point can not equal r. 256 * Uses a modified algorithm 2P of 257 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 258 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 259 * 260 * To protect against side-channel attack the function uses constant time swap, 261 * avoiding conditional branches. 262 */ 263static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, 264 EC_POINT *r, 265 const BIGNUM *scalar, 266 const EC_POINT *point, 267 BN_CTX *ctx) 268{ 269 BIGNUM *x1, *x2, *z1, *z2; 270 int ret = 0, i; 271 BN_ULONG mask, word; 272 273 if (r == point) { 274 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT); 275 return 0; 276 } 277 278 /* if result should be point at infinity */ 279 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || 280 EC_POINT_is_at_infinity(group, point)) { 281 return EC_POINT_set_to_infinity(group, r); 282 } 283 284 /* only support affine coordinates */ 285 if (!point->Z_is_one) 286 return 0; 287 288 /* 289 * Since point_multiply is static we can guarantee that ctx != NULL. 290 */ 291 BN_CTX_start(ctx); 292 x1 = BN_CTX_get(ctx); 293 z1 = BN_CTX_get(ctx); 294 if (z1 == NULL) 295 goto err; 296 297 x2 = &r->X; 298 z2 = &r->Y; 299 300 bn_wexpand(x1, group->field.top); 301 bn_wexpand(z1, group->field.top); 302 bn_wexpand(x2, group->field.top); 303 bn_wexpand(z2, group->field.top); 304 305 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) 306 goto err; /* x1 = x */ 307 if (!BN_one(z1)) 308 goto err; /* z1 = 1 */ 309 if (!group->meth->field_sqr(group, z2, x1, ctx)) 310 goto err; /* z2 = x1^2 = x^2 */ 311 if (!group->meth->field_sqr(group, x2, z2, ctx)) 312 goto err; 313 if (!BN_GF2m_add(x2, x2, &group->b)) 314 goto err; /* x2 = x^4 + b */ 315 316 /* find top most bit and go one past it */ 317 i = scalar->top - 1; 318 mask = BN_TBIT; 319 word = scalar->d[i]; 320 while (!(word & mask)) 321 mask >>= 1; 322 mask >>= 1; 323 /* if top most bit was at word break, go to next word */ 324 if (!mask) { 325 i--; 326 mask = BN_TBIT; 327 } 328 329 for (; i >= 0; i--) { 330 word = scalar->d[i]; 331 while (mask) { 332 BN_consttime_swap(word & mask, x1, x2, group->field.top); 333 BN_consttime_swap(word & mask, z1, z2, group->field.top); 334 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) 335 goto err; 336 if (!gf2m_Mdouble(group, x1, z1, ctx)) 337 goto err; 338 BN_consttime_swap(word & mask, x1, x2, group->field.top); 339 BN_consttime_swap(word & mask, z1, z2, group->field.top); 340 mask >>= 1; 341 } 342 mask = BN_TBIT; 343 } 344 345 /* convert out of "projective" coordinates */ 346 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx); 347 if (i == 0) 348 goto err; 349 else if (i == 1) { 350 if (!EC_POINT_set_to_infinity(group, r)) 351 goto err; 352 } else { 353 if (!BN_one(&r->Z)) 354 goto err; 355 r->Z_is_one = 1; 356 } 357 358 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */ 359 BN_set_negative(&r->X, 0); 360 BN_set_negative(&r->Y, 0); 361 362 ret = 1; 363 364 err: 365 BN_CTX_end(ctx); 366 return ret; 367} 368 369/*- 370 * Computes the sum 371 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1] 372 * gracefully ignoring NULL scalar values. 373 */ 374int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, 375 const BIGNUM *scalar, size_t num, 376 const EC_POINT *points[], const BIGNUM *scalars[], 377 BN_CTX *ctx) 378{ 379 BN_CTX *new_ctx = NULL; 380 int ret = 0; 381 size_t i; 382 EC_POINT *p = NULL; 383 EC_POINT *acc = NULL; 384 385 if (ctx == NULL) { 386 ctx = new_ctx = BN_CTX_new(); 387 if (ctx == NULL) 388 return 0; 389 } 390 391 /* 392 * This implementation is more efficient than the wNAF implementation for 393 * 2 or fewer points. Use the ec_wNAF_mul implementation for 3 or more 394 * points, or if we can perform a fast multiplication based on 395 * precomputation. 396 */ 397 if ((scalar && (num > 1)) || (num > 2) 398 || (num == 0 && EC_GROUP_have_precompute_mult(group))) { 399 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx); 400 goto err; 401 } 402 403 if ((p = EC_POINT_new(group)) == NULL) 404 goto err; 405 if ((acc = EC_POINT_new(group)) == NULL) 406 goto err; 407 408 if (!EC_POINT_set_to_infinity(group, acc)) 409 goto err; 410 411 if (scalar) { 412 if (!ec_GF2m_montgomery_point_multiply 413 (group, p, scalar, group->generator, ctx)) 414 goto err; 415 if (BN_is_negative(scalar)) 416 if (!group->meth->invert(group, p, ctx)) 417 goto err; 418 if (!group->meth->add(group, acc, acc, p, ctx)) 419 goto err; 420 } 421 422 for (i = 0; i < num; i++) { 423 if (!ec_GF2m_montgomery_point_multiply 424 (group, p, scalars[i], points[i], ctx)) 425 goto err; 426 if (BN_is_negative(scalars[i])) 427 if (!group->meth->invert(group, p, ctx)) 428 goto err; 429 if (!group->meth->add(group, acc, acc, p, ctx)) 430 goto err; 431 } 432 433 if (!EC_POINT_copy(r, acc)) 434 goto err; 435 436 ret = 1; 437 438 err: 439 if (p) 440 EC_POINT_free(p); 441 if (acc) 442 EC_POINT_free(acc); 443 if (new_ctx != NULL) 444 BN_CTX_free(new_ctx); 445 return ret; 446} 447 448/* 449 * Precomputation for point multiplication: fall back to wNAF methods because 450 * ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate 451 */ 452 453int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx) 454{ 455 return ec_wNAF_precompute_mult(group, ctx); 456} 457 458int ec_GF2m_have_precompute_mult(const EC_GROUP *group) 459{ 460 return ec_wNAF_have_precompute_mult(group); 461} 462 463#endif 464