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 75/* Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective 76 * coordinates. 77 * Uses algorithm Mdouble in appendix of 78 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 79 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 80 * modified to not require precomputation of c=b^{2^{m-1}}. 81 */ 82static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx) 83 { 84 BIGNUM *t1; 85 int ret = 0; 86 87 /* Since Mdouble is static we can guarantee that ctx != NULL. */ 88 BN_CTX_start(ctx); 89 t1 = BN_CTX_get(ctx); 90 if (t1 == NULL) goto err; 91 92 if (!group->meth->field_sqr(group, x, x, ctx)) goto err; 93 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err; 94 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err; 95 if (!group->meth->field_sqr(group, x, x, ctx)) goto err; 96 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err; 97 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err; 98 if (!BN_GF2m_add(x, x, t1)) goto err; 99 100 ret = 1; 101 102 err: 103 BN_CTX_end(ctx); 104 return ret; 105 } 106 107/* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery 108 * projective coordinates. 109 * Uses algorithm Madd in appendix of 110 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 111 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 112 */ 113static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1, 114 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx) 115 { 116 BIGNUM *t1, *t2; 117 int ret = 0; 118 119 /* Since Madd is static we can guarantee that ctx != NULL. */ 120 BN_CTX_start(ctx); 121 t1 = BN_CTX_get(ctx); 122 t2 = BN_CTX_get(ctx); 123 if (t2 == NULL) goto err; 124 125 if (!BN_copy(t1, x)) goto err; 126 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err; 127 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err; 128 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err; 129 if (!BN_GF2m_add(z1, z1, x1)) goto err; 130 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err; 131 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err; 132 if (!BN_GF2m_add(x1, x1, t2)) goto err; 133 134 ret = 1; 135 136 err: 137 BN_CTX_end(ctx); 138 return ret; 139 } 140 141/* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) 142 * using Montgomery point multiplication algorithm Mxy() in appendix of 143 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 144 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 145 * Returns: 146 * 0 on error 147 * 1 if return value should be the point at infinity 148 * 2 otherwise 149 */ 150static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1, 151 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx) 152 { 153 BIGNUM *t3, *t4, *t5; 154 int ret = 0; 155 156 if (BN_is_zero(z1)) 157 { 158 BN_zero(x2); 159 BN_zero(z2); 160 return 1; 161 } 162 163 if (BN_is_zero(z2)) 164 { 165 if (!BN_copy(x2, x)) return 0; 166 if (!BN_GF2m_add(z2, x, y)) return 0; 167 return 2; 168 } 169 170 /* Since Mxy is static we can guarantee that ctx != NULL. */ 171 BN_CTX_start(ctx); 172 t3 = BN_CTX_get(ctx); 173 t4 = BN_CTX_get(ctx); 174 t5 = BN_CTX_get(ctx); 175 if (t5 == NULL) goto err; 176 177 if (!BN_one(t5)) goto err; 178 179 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err; 180 181 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err; 182 if (!BN_GF2m_add(z1, z1, x1)) goto err; 183 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err; 184 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err; 185 if (!BN_GF2m_add(z2, z2, x2)) goto err; 186 187 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err; 188 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err; 189 if (!BN_GF2m_add(t4, t4, y)) goto err; 190 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err; 191 if (!BN_GF2m_add(t4, t4, z2)) goto err; 192 193 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err; 194 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err; 195 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err; 196 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err; 197 if (!BN_GF2m_add(z2, x2, x)) goto err; 198 199 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err; 200 if (!BN_GF2m_add(z2, z2, y)) goto err; 201 202 ret = 2; 203 204 err: 205 BN_CTX_end(ctx); 206 return ret; 207 } 208 209 210/* Computes scalar*point and stores the result in r. 211 * point can not equal r. 212 * Uses a modified algorithm 2P of 213 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 214 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 215 * 216 * To protect against side-channel attack the function uses constant time swap, 217 * avoiding conditional branches. 218 */ 219static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 220 const EC_POINT *point, BN_CTX *ctx) 221 { 222 BIGNUM *x1, *x2, *z1, *z2; 223 int ret = 0, i; 224 BN_ULONG mask,word; 225 226 if (r == point) 227 { 228 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT); 229 return 0; 230 } 231 232 /* if result should be point at infinity */ 233 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || 234 EC_POINT_is_at_infinity(group, point)) 235 { 236 return EC_POINT_set_to_infinity(group, r); 237 } 238 239 /* only support affine coordinates */ 240 if (!point->Z_is_one) return 0; 241 242 /* Since point_multiply is static we can guarantee that ctx != NULL. */ 243 BN_CTX_start(ctx); 244 x1 = BN_CTX_get(ctx); 245 z1 = BN_CTX_get(ctx); 246 if (z1 == NULL) goto err; 247 248 x2 = &r->X; 249 z2 = &r->Y; 250 251 bn_wexpand(x1, group->field.top); 252 bn_wexpand(z1, group->field.top); 253 bn_wexpand(x2, group->field.top); 254 bn_wexpand(z2, group->field.top); 255 256 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */ 257 if (!BN_one(z1)) goto err; /* z1 = 1 */ 258 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */ 259 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err; 260 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */ 261 262 /* find top most bit and go one past it */ 263 i = scalar->top - 1; 264 mask = BN_TBIT; 265 word = scalar->d[i]; 266 while (!(word & mask)) mask >>= 1; 267 mask >>= 1; 268 /* if top most bit was at word break, go to next word */ 269 if (!mask) 270 { 271 i--; 272 mask = BN_TBIT; 273 } 274 275 for (; i >= 0; i--) 276 { 277 word = scalar->d[i]; 278 while (mask) 279 { 280 BN_consttime_swap(word & mask, x1, x2, group->field.top); 281 BN_consttime_swap(word & mask, z1, z2, group->field.top); 282 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err; 283 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err; 284 BN_consttime_swap(word & mask, x1, x2, group->field.top); 285 BN_consttime_swap(word & mask, z1, z2, group->field.top); 286 mask >>= 1; 287 } 288 mask = BN_TBIT; 289 } 290 291 /* convert out of "projective" coordinates */ 292 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx); 293 if (i == 0) goto err; 294 else if (i == 1) 295 { 296 if (!EC_POINT_set_to_infinity(group, r)) goto err; 297 } 298 else 299 { 300 if (!BN_one(&r->Z)) goto err; 301 r->Z_is_one = 1; 302 } 303 304 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */ 305 BN_set_negative(&r->X, 0); 306 BN_set_negative(&r->Y, 0); 307 308 ret = 1; 309 310 err: 311 BN_CTX_end(ctx); 312 return ret; 313 } 314 315 316/* Computes the sum 317 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1] 318 * gracefully ignoring NULL scalar values. 319 */ 320int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 321 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) 322 { 323 BN_CTX *new_ctx = NULL; 324 int ret = 0; 325 size_t i; 326 EC_POINT *p=NULL; 327 EC_POINT *acc = NULL; 328 329 if (ctx == NULL) 330 { 331 ctx = new_ctx = BN_CTX_new(); 332 if (ctx == NULL) 333 return 0; 334 } 335 336 /* This implementation is more efficient than the wNAF implementation for 2 337 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points, 338 * or if we can perform a fast multiplication based on precomputation. 339 */ 340 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group))) 341 { 342 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx); 343 goto err; 344 } 345 346 if ((p = EC_POINT_new(group)) == NULL) goto err; 347 if ((acc = EC_POINT_new(group)) == NULL) goto err; 348 349 if (!EC_POINT_set_to_infinity(group, acc)) goto err; 350 351 if (scalar) 352 { 353 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err; 354 if (BN_is_negative(scalar)) 355 if (!group->meth->invert(group, p, ctx)) goto err; 356 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 357 } 358 359 for (i = 0; i < num; i++) 360 { 361 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err; 362 if (BN_is_negative(scalars[i])) 363 if (!group->meth->invert(group, p, ctx)) goto err; 364 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 365 } 366 367 if (!EC_POINT_copy(r, acc)) goto err; 368 369 ret = 1; 370 371 err: 372 if (p) EC_POINT_free(p); 373 if (acc) EC_POINT_free(acc); 374 if (new_ctx != NULL) 375 BN_CTX_free(new_ctx); 376 return ret; 377 } 378 379 380/* Precomputation for point multiplication: fall back to wNAF methods 381 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */ 382 383int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx) 384 { 385 return ec_wNAF_precompute_mult(group, ctx); 386 } 387 388int ec_GF2m_have_precompute_mult(const EC_GROUP *group) 389 { 390 return ec_wNAF_have_precompute_mult(group); 391 } 392