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, BN_CTX *ctx) 85 { 86 BIGNUM *t1; 87 int ret = 0; 88 89 /* Since Mdouble is static we can guarantee that ctx != NULL. */ 90 BN_CTX_start(ctx); 91 t1 = BN_CTX_get(ctx); 92 if (t1 == NULL) goto err; 93 94 if (!group->meth->field_sqr(group, x, x, ctx)) goto err; 95 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err; 96 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err; 97 if (!group->meth->field_sqr(group, x, x, ctx)) goto err; 98 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err; 99 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err; 100 if (!BN_GF2m_add(x, x, t1)) goto err; 101 102 ret = 1; 103 104 err: 105 BN_CTX_end(ctx); 106 return ret; 107 } 108 109/* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery 110 * projective coordinates. 111 * Uses algorithm Madd in appendix of 112 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 113 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 114 */ 115static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1, 116 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx) 117 { 118 BIGNUM *t1, *t2; 119 int ret = 0; 120 121 /* Since Madd is static we can guarantee that ctx != NULL. */ 122 BN_CTX_start(ctx); 123 t1 = BN_CTX_get(ctx); 124 t2 = BN_CTX_get(ctx); 125 if (t2 == NULL) goto err; 126 127 if (!BN_copy(t1, x)) goto err; 128 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err; 129 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err; 130 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err; 131 if (!BN_GF2m_add(z1, z1, x1)) goto err; 132 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err; 133 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err; 134 if (!BN_GF2m_add(x1, x1, t2)) goto err; 135 136 ret = 1; 137 138 err: 139 BN_CTX_end(ctx); 140 return ret; 141 } 142 143/* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) 144 * using Montgomery point multiplication algorithm Mxy() in appendix of 145 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 146 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 147 * Returns: 148 * 0 on error 149 * 1 if return value should be the point at infinity 150 * 2 otherwise 151 */ 152static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1, 153 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx) 154 { 155 BIGNUM *t3, *t4, *t5; 156 int ret = 0; 157 158 if (BN_is_zero(z1)) 159 { 160 BN_zero(x2); 161 BN_zero(z2); 162 return 1; 163 } 164 165 if (BN_is_zero(z2)) 166 { 167 if (!BN_copy(x2, x)) return 0; 168 if (!BN_GF2m_add(z2, x, y)) return 0; 169 return 2; 170 } 171 172 /* Since Mxy is static we can guarantee that ctx != NULL. */ 173 BN_CTX_start(ctx); 174 t3 = BN_CTX_get(ctx); 175 t4 = BN_CTX_get(ctx); 176 t5 = BN_CTX_get(ctx); 177 if (t5 == NULL) goto err; 178 179 if (!BN_one(t5)) goto err; 180 181 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err; 182 183 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err; 184 if (!BN_GF2m_add(z1, z1, x1)) goto err; 185 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err; 186 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err; 187 if (!BN_GF2m_add(z2, z2, x2)) goto err; 188 189 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err; 190 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err; 191 if (!BN_GF2m_add(t4, t4, y)) goto err; 192 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err; 193 if (!BN_GF2m_add(t4, t4, z2)) goto err; 194 195 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err; 196 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err; 197 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err; 198 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err; 199 if (!BN_GF2m_add(z2, x2, x)) goto err; 200 201 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err; 202 if (!BN_GF2m_add(z2, z2, y)) goto err; 203 204 ret = 2; 205 206 err: 207 BN_CTX_end(ctx); 208 return ret; 209 } 210 211 212/* Computes scalar*point and stores the result in r. 213 * point can not equal r. 214 * Uses a modified algorithm 2P of 215 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 216 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 217 * 218 * To protect against side-channel attack the function uses constant time swap, 219 * avoiding conditional branches. 220 */ 221static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 222 const EC_POINT *point, BN_CTX *ctx) 223 { 224 BIGNUM *x1, *x2, *z1, *z2; 225 int ret = 0, i; 226 BN_ULONG mask,word; 227 228 if (r == point) 229 { 230 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT); 231 return 0; 232 } 233 234 /* if result should be point at infinity */ 235 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || 236 EC_POINT_is_at_infinity(group, point)) 237 { 238 return EC_POINT_set_to_infinity(group, r); 239 } 240 241 /* only support affine coordinates */ 242 if (!point->Z_is_one) return 0; 243 244 /* Since point_multiply is static we can guarantee that ctx != NULL. */ 245 BN_CTX_start(ctx); 246 x1 = BN_CTX_get(ctx); 247 z1 = BN_CTX_get(ctx); 248 if (z1 == NULL) goto err; 249 250 x2 = &r->X; 251 z2 = &r->Y; 252 253 bn_wexpand(x1, group->field.top); 254 bn_wexpand(z1, group->field.top); 255 bn_wexpand(x2, group->field.top); 256 bn_wexpand(z2, group->field.top); 257 258 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */ 259 if (!BN_one(z1)) goto err; /* z1 = 1 */ 260 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */ 261 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err; 262 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */ 263 264 /* find top most bit and go one past it */ 265 i = scalar->top - 1; 266 mask = BN_TBIT; 267 word = scalar->d[i]; 268 while (!(word & mask)) mask >>= 1; 269 mask >>= 1; 270 /* if top most bit was at word break, go to next word */ 271 if (!mask) 272 { 273 i--; 274 mask = BN_TBIT; 275 } 276 277 for (; i >= 0; i--) 278 { 279 word = scalar->d[i]; 280 while (mask) 281 { 282 BN_consttime_swap(word & mask, x1, x2, group->field.top); 283 BN_consttime_swap(word & mask, z1, z2, group->field.top); 284 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err; 285 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err; 286 BN_consttime_swap(word & mask, x1, x2, group->field.top); 287 BN_consttime_swap(word & mask, z1, z2, group->field.top); 288 mask >>= 1; 289 } 290 mask = BN_TBIT; 291 } 292 293 /* convert out of "projective" coordinates */ 294 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx); 295 if (i == 0) goto err; 296 else if (i == 1) 297 { 298 if (!EC_POINT_set_to_infinity(group, r)) goto err; 299 } 300 else 301 { 302 if (!BN_one(&r->Z)) goto err; 303 r->Z_is_one = 1; 304 } 305 306 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */ 307 BN_set_negative(&r->X, 0); 308 BN_set_negative(&r->Y, 0); 309 310 ret = 1; 311 312 err: 313 BN_CTX_end(ctx); 314 return ret; 315 } 316 317 318/* Computes the sum 319 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1] 320 * gracefully ignoring NULL scalar values. 321 */ 322int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 323 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) 324 { 325 BN_CTX *new_ctx = NULL; 326 int ret = 0; 327 size_t i; 328 EC_POINT *p=NULL; 329 EC_POINT *acc = NULL; 330 331 if (ctx == NULL) 332 { 333 ctx = new_ctx = BN_CTX_new(); 334 if (ctx == NULL) 335 return 0; 336 } 337 338 /* This implementation is more efficient than the wNAF implementation for 2 339 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points, 340 * or if we can perform a fast multiplication based on precomputation. 341 */ 342 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group))) 343 { 344 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx); 345 goto err; 346 } 347 348 if ((p = EC_POINT_new(group)) == NULL) goto err; 349 if ((acc = EC_POINT_new(group)) == NULL) goto err; 350 351 if (!EC_POINT_set_to_infinity(group, acc)) goto err; 352 353 if (scalar) 354 { 355 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err; 356 if (BN_is_negative(scalar)) 357 if (!group->meth->invert(group, p, ctx)) goto err; 358 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 359 } 360 361 for (i = 0; i < num; i++) 362 { 363 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err; 364 if (BN_is_negative(scalars[i])) 365 if (!group->meth->invert(group, p, ctx)) goto err; 366 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 367 } 368 369 if (!EC_POINT_copy(r, acc)) goto err; 370 371 ret = 1; 372 373 err: 374 if (p) EC_POINT_free(p); 375 if (acc) EC_POINT_free(acc); 376 if (new_ctx != NULL) 377 BN_CTX_free(new_ctx); 378 return ret; 379 } 380 381 382/* Precomputation for point multiplication: fall back to wNAF methods 383 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */ 384 385int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx) 386 { 387 return ec_wNAF_precompute_mult(group, ctx); 388 } 389 390int ec_GF2m_have_precompute_mult(const EC_GROUP *group) 391 { 392 return ec_wNAF_have_precompute_mult(group); 393 } 394 395#endif 396