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 (at) 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 (at) cryptsoft.com). This product includes software written by Tim 66 * Hudson (tjh (at) 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 */ 82 static 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 */ 113 static 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 */ 150 static 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 /* Computes scalar*point and stores the result in r. 210 * point can not equal r. 211 * Uses algorithm 2P of 212 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over 213 * GF(2^m) without precomputation" (CHES '99, LNCS 1717). 214 */ 215 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 216 const EC_POINT *point, BN_CTX *ctx) 217 { 218 BIGNUM *x1, *x2, *z1, *z2; 219 int ret = 0, i; 220 BN_ULONG mask,word; 221 222 if (r == point) 223 { 224 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT); 225 return 0; 226 } 227 228 /* if result should be point at infinity */ 229 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || 230 EC_POINT_is_at_infinity(group, point)) 231 { 232 return EC_POINT_set_to_infinity(group, r); 233 } 234 235 /* only support affine coordinates */ 236 if (!point->Z_is_one) return 0; 237 238 /* Since point_multiply is static we can guarantee that ctx != NULL. */ 239 BN_CTX_start(ctx); 240 x1 = BN_CTX_get(ctx); 241 z1 = BN_CTX_get(ctx); 242 if (z1 == NULL) goto err; 243 244 x2 = &r->X; 245 z2 = &r->Y; 246 247 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */ 248 if (!BN_one(z1)) goto err; /* z1 = 1 */ 249 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */ 250 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err; 251 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */ 252 253 /* find top most bit and go one past it */ 254 i = scalar->top - 1; 255 mask = BN_TBIT; 256 word = scalar->d[i]; 257 while (!(word & mask)) mask >>= 1; 258 mask >>= 1; 259 /* if top most bit was at word break, go to next word */ 260 if (!mask) 261 { 262 i--; 263 mask = BN_TBIT; 264 } 265 266 for (; i >= 0; i--) 267 { 268 word = scalar->d[i]; 269 while (mask) 270 { 271 if (word & mask) 272 { 273 if (!gf2m_Madd(group, &point->X, x1, z1, x2, z2, ctx)) goto err; 274 if (!gf2m_Mdouble(group, x2, z2, ctx)) goto err; 275 } 276 else 277 { 278 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err; 279 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err; 280 } 281 mask >>= 1; 282 } 283 mask = BN_TBIT; 284 } 285 286 /* convert out of "projective" coordinates */ 287 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx); 288 if (i == 0) goto err; 289 else if (i == 1) 290 { 291 if (!EC_POINT_set_to_infinity(group, r)) goto err; 292 } 293 else 294 { 295 if (!BN_one(&r->Z)) goto err; 296 r->Z_is_one = 1; 297 } 298 299 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */ 300 BN_set_negative(&r->X, 0); 301 BN_set_negative(&r->Y, 0); 302 303 ret = 1; 304 305 err: 306 BN_CTX_end(ctx); 307 return ret; 308 } 309 310 311 /* Computes the sum 312 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1] 313 * gracefully ignoring NULL scalar values. 314 */ 315 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 316 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) 317 { 318 BN_CTX *new_ctx = NULL; 319 int ret = 0; 320 size_t i; 321 EC_POINT *p=NULL; 322 EC_POINT *acc = NULL; 323 324 if (ctx == NULL) 325 { 326 ctx = new_ctx = BN_CTX_new(); 327 if (ctx == NULL) 328 return 0; 329 } 330 331 /* This implementation is more efficient than the wNAF implementation for 2 332 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points, 333 * or if we can perform a fast multiplication based on precomputation. 334 */ 335 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group))) 336 { 337 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx); 338 goto err; 339 } 340 341 if ((p = EC_POINT_new(group)) == NULL) goto err; 342 if ((acc = EC_POINT_new(group)) == NULL) goto err; 343 344 if (!EC_POINT_set_to_infinity(group, acc)) goto err; 345 346 if (scalar) 347 { 348 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err; 349 if (BN_is_negative(scalar)) 350 if (!group->meth->invert(group, p, ctx)) goto err; 351 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 352 } 353 354 for (i = 0; i < num; i++) 355 { 356 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err; 357 if (BN_is_negative(scalars[i])) 358 if (!group->meth->invert(group, p, ctx)) goto err; 359 if (!group->meth->add(group, acc, acc, p, ctx)) goto err; 360 } 361 362 if (!EC_POINT_copy(r, acc)) goto err; 363 364 ret = 1; 365 366 err: 367 if (p) EC_POINT_free(p); 368 if (acc) EC_POINT_free(acc); 369 if (new_ctx != NULL) 370 BN_CTX_free(new_ctx); 371 return ret; 372 } 373 374 375 /* Precomputation for point multiplication: fall back to wNAF methods 376 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */ 377 378 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx) 379 { 380 return ec_wNAF_precompute_mult(group, ctx); 381 } 382 383 int ec_GF2m_have_precompute_mult(const EC_GROUP *group) 384 { 385 return ec_wNAF_have_precompute_mult(group); 386 } 387
