1 /* Originally written by Bodo Moeller for the OpenSSL project. 2 * ==================================================================== 3 * Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in 14 * the documentation and/or other materials provided with the 15 * distribution. 16 * 17 * 3. All advertising materials mentioning features or use of this 18 * software must display the following acknowledgment: 19 * "This product includes software developed by the OpenSSL Project 20 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 21 * 22 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 23 * endorse or promote products derived from this software without 24 * prior written permission. For written permission, please contact 25 * openssl-core (at) openssl.org. 26 * 27 * 5. Products derived from this software may not be called "OpenSSL" 28 * nor may "OpenSSL" appear in their names without prior written 29 * permission of the OpenSSL Project. 30 * 31 * 6. Redistributions of any form whatsoever must retain the following 32 * acknowledgment: 33 * "This product includes software developed by the OpenSSL Project 34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 35 * 36 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 37 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 38 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 39 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 40 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 41 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 42 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 43 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 44 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 45 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 46 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 47 * OF THE POSSIBILITY OF SUCH DAMAGE. 48 * ==================================================================== 49 * 50 * This product includes cryptographic software written by Eric Young 51 * (eay (at) cryptsoft.com). This product includes software written by Tim 52 * Hudson (tjh (at) cryptsoft.com). 53 * 54 */ 55 /* ==================================================================== 56 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. 57 * 58 * Portions of the attached software ("Contribution") are developed by 59 * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project. 60 * 61 * The Contribution is licensed pursuant to the OpenSSL open source 62 * license provided above. 63 * 64 * The elliptic curve binary polynomial software is originally written by 65 * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems 66 * Laboratories. */ 67 68 #include <openssl/ec.h> 69 70 #include <string.h> 71 72 #include <openssl/bn.h> 73 #include <openssl/err.h> 74 #include <openssl/mem.h> 75 76 #include "internal.h" 77 #include "../../internal.h" 78 79 80 // Most method functions in this file are designed to work with non-trivial 81 // representations of field elements if necessary (see ecp_mont.c): while 82 // standard modular addition and subtraction are used, the field_mul and 83 // field_sqr methods will be used for multiplication, and field_encode and 84 // field_decode (if defined) will be used for converting between 85 // representations. 86 // 87 // Functions here specifically assume that if a non-trivial representation is 88 // used, it is a Montgomery representation (i.e. 'encoding' means multiplying 89 // by some factor R). 90 91 int ec_GFp_simple_group_init(EC_GROUP *group) { 92 BN_init(&group->field); 93 group->a_is_minus3 = 0; 94 return 1; 95 } 96 97 void ec_GFp_simple_group_finish(EC_GROUP *group) { 98 BN_free(&group->field); 99 } 100 101 int ec_GFp_simple_group_set_curve(EC_GROUP *group, const BIGNUM *p, 102 const BIGNUM *a, const BIGNUM *b, 103 BN_CTX *ctx) { 104 int ret = 0; 105 BN_CTX *new_ctx = NULL; 106 107 // p must be a prime > 3 108 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) { 109 OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD); 110 return 0; 111 } 112 113 if (ctx == NULL) { 114 ctx = new_ctx = BN_CTX_new(); 115 if (ctx == NULL) { 116 return 0; 117 } 118 } 119 120 BN_CTX_start(ctx); 121 BIGNUM *tmp = BN_CTX_get(ctx); 122 if (tmp == NULL) { 123 goto err; 124 } 125 126 // group->field 127 if (!BN_copy(&group->field, p)) { 128 goto err; 129 } 130 BN_set_negative(&group->field, 0); 131 // Store the field in minimal form, so it can be used with |BN_ULONG| arrays. 132 bn_set_minimal_width(&group->field); 133 134 // group->a 135 if (!BN_nnmod(tmp, a, &group->field, ctx) || 136 !ec_bignum_to_felem(group, &group->a, tmp)) { 137 goto err; 138 } 139 140 // group->a_is_minus3 141 if (!BN_add_word(tmp, 3)) { 142 goto err; 143 } 144 group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field)); 145 146 // group->b 147 if (!BN_nnmod(tmp, b, &group->field, ctx) || 148 !ec_bignum_to_felem(group, &group->b, tmp)) { 149 goto err; 150 } 151 152 if (!ec_bignum_to_felem(group, &group->one, BN_value_one())) { 153 goto err; 154 } 155 156 ret = 1; 157 158 err: 159 BN_CTX_end(ctx); 160 BN_CTX_free(new_ctx); 161 return ret; 162 } 163 164 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, 165 BIGNUM *b) { 166 if ((p != NULL && !BN_copy(p, &group->field)) || 167 (a != NULL && !ec_felem_to_bignum(group, a, &group->a)) || 168 (b != NULL && !ec_felem_to_bignum(group, b, &group->b))) { 169 return 0; 170 } 171 return 1; 172 } 173 174 void ec_GFp_simple_point_init(EC_RAW_POINT *point) { 175 OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM)); 176 OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM)); 177 OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM)); 178 } 179 180 void ec_GFp_simple_point_copy(EC_RAW_POINT *dest, const EC_RAW_POINT *src) { 181 OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM)); 182 OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM)); 183 OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM)); 184 } 185 186 void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, 187 EC_RAW_POINT *point) { 188 // Although it is strictly only necessary to zero Z, we zero the entire point 189 // in case |point| was stack-allocated and yet to be initialized. 190 ec_GFp_simple_point_init(point); 191 } 192 193 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, 194 EC_RAW_POINT *point, 195 const BIGNUM *x, 196 const BIGNUM *y) { 197 if (x == NULL || y == NULL) { 198 OPENSSL_PUT_ERROR(EC, ERR_R_PASSED_NULL_PARAMETER); 199 return 0; 200 } 201 202 if (!ec_bignum_to_felem(group, &point->X, x) || 203 !ec_bignum_to_felem(group, &point->Y, y)) { 204 return 0; 205 } 206 OPENSSL_memcpy(&point->Z, &group->one, sizeof(EC_FELEM)); 207 208 return 1; 209 } 210 211 void ec_GFp_simple_invert(const EC_GROUP *group, EC_RAW_POINT *point) { 212 ec_felem_neg(group, &point->Y, &point->Y); 213 } 214 215 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, 216 const EC_RAW_POINT *point) { 217 return ec_felem_non_zero_mask(group, &point->Z) == 0; 218 } 219 220 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, 221 const EC_RAW_POINT *point) { 222 if (ec_GFp_simple_is_at_infinity(group, point)) { 223 return 1; 224 } 225 226 // We have a curve defined by a Weierstrass equation 227 // y^2 = x^3 + a*x + b. 228 // The point to consider is given in Jacobian projective coordinates 229 // where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3). 230 // Substituting this and multiplying by Z^6 transforms the above equation 231 // into 232 // Y^2 = X^3 + a*X*Z^4 + b*Z^6. 233 // To test this, we add up the right-hand side in 'rh'. 234 235 void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, 236 const EC_FELEM *b) = group->meth->felem_mul; 237 void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) = 238 group->meth->felem_sqr; 239 240 // rh := X^2 241 EC_FELEM rh; 242 felem_sqr(group, &rh, &point->X); 243 244 EC_FELEM tmp, Z4, Z6; 245 if (!ec_felem_equal(group, &point->Z, &group->one)) { 246 felem_sqr(group, &tmp, &point->Z); 247 felem_sqr(group, &Z4, &tmp); 248 felem_mul(group, &Z6, &Z4, &tmp); 249 250 // rh := (rh + a*Z^4)*X 251 if (group->a_is_minus3) { 252 ec_felem_add(group, &tmp, &Z4, &Z4); 253 ec_felem_add(group, &tmp, &tmp, &Z4); 254 ec_felem_sub(group, &rh, &rh, &tmp); 255 felem_mul(group, &rh, &rh, &point->X); 256 } else { 257 felem_mul(group, &tmp, &Z4, &group->a); 258 ec_felem_add(group, &rh, &rh, &tmp); 259 felem_mul(group, &rh, &rh, &point->X); 260 } 261 262 // rh := rh + b*Z^6 263 felem_mul(group, &tmp, &group->b, &Z6); 264 ec_felem_add(group, &rh, &rh, &tmp); 265 } else { 266 // rh := (rh + a)*X 267 ec_felem_add(group, &rh, &rh, &group->a); 268 felem_mul(group, &rh, &rh, &point->X); 269 // rh := rh + b 270 ec_felem_add(group, &rh, &rh, &group->b); 271 } 272 273 // 'lh' := Y^2 274 felem_sqr(group, &tmp, &point->Y); 275 return ec_felem_equal(group, &tmp, &rh); 276 } 277 278 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_RAW_POINT *a, 279 const EC_RAW_POINT *b) { 280 // Note this function returns zero if |a| and |b| are equal and 1 if they are 281 // not equal. 282 if (ec_GFp_simple_is_at_infinity(group, a)) { 283 return ec_GFp_simple_is_at_infinity(group, b) ? 0 : 1; 284 } 285 286 if (ec_GFp_simple_is_at_infinity(group, b)) { 287 return 1; 288 } 289 290 int a_Z_is_one = ec_felem_equal(group, &a->Z, &group->one); 291 int b_Z_is_one = ec_felem_equal(group, &b->Z, &group->one); 292 293 if (a_Z_is_one && b_Z_is_one) { 294 return !ec_felem_equal(group, &a->X, &b->X) || 295 !ec_felem_equal(group, &a->Y, &b->Y); 296 } 297 298 void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, 299 const EC_FELEM *b) = group->meth->felem_mul; 300 void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) = 301 group->meth->felem_sqr; 302 303 // We have to decide whether 304 // (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3), 305 // or equivalently, whether 306 // (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3). 307 308 EC_FELEM tmp1, tmp2, Za23, Zb23; 309 const EC_FELEM *tmp1_, *tmp2_; 310 if (!b_Z_is_one) { 311 felem_sqr(group, &Zb23, &b->Z); 312 felem_mul(group, &tmp1, &a->X, &Zb23); 313 tmp1_ = &tmp1; 314 } else { 315 tmp1_ = &a->X; 316 } 317 if (!a_Z_is_one) { 318 felem_sqr(group, &Za23, &a->Z); 319 felem_mul(group, &tmp2, &b->X, &Za23); 320 tmp2_ = &tmp2; 321 } else { 322 tmp2_ = &b->X; 323 } 324 325 // Compare X_a*Z_b^2 with X_b*Z_a^2. 326 if (!ec_felem_equal(group, tmp1_, tmp2_)) { 327 return 1; // The points differ. 328 } 329 330 if (!b_Z_is_one) { 331 felem_mul(group, &Zb23, &Zb23, &b->Z); 332 felem_mul(group, &tmp1, &a->Y, &Zb23); 333 // tmp1_ = &tmp1 334 } else { 335 tmp1_ = &a->Y; 336 } 337 if (!a_Z_is_one) { 338 felem_mul(group, &Za23, &Za23, &a->Z); 339 felem_mul(group, &tmp2, &b->Y, &Za23); 340 // tmp2_ = &tmp2 341 } else { 342 tmp2_ = &b->Y; 343 } 344 345 // Compare Y_a*Z_b^3 with Y_b*Z_a^3. 346 if (!ec_felem_equal(group, tmp1_, tmp2_)) { 347 return 1; // The points differ. 348 } 349 350 // The points are equal. 351 return 0; 352 } 353 354 int ec_GFp_simple_mont_inv_mod_ord_vartime(const EC_GROUP *group, 355 EC_SCALAR *out, 356 const EC_SCALAR *in) { 357 // This implementation (in fact) runs in constant time, 358 // even though for this interface it is not mandatory. 359 360 // out = in^-1 in the Montgomery domain. This is 361 // |ec_scalar_to_montgomery| followed by |ec_scalar_inv_montgomery|, but 362 // |ec_scalar_inv_montgomery| followed by |ec_scalar_from_montgomery| is 363 // equivalent and slightly more efficient. 364 ec_scalar_inv_montgomery(group, out, in); 365 ec_scalar_from_montgomery(group, out, out); 366 return 1; 367 } 368 369 int ec_GFp_simple_cmp_x_coordinate(const EC_GROUP *group, const EC_RAW_POINT *p, 370 const EC_SCALAR *r) { 371 if (ec_GFp_simple_is_at_infinity(group, p)) { 372 // |ec_get_x_coordinate_as_scalar| will check this internally, but this way 373 // we do not push to the error queue. 374 return 0; 375 } 376 377 EC_SCALAR x; 378 return ec_get_x_coordinate_as_scalar(group, &x, p) && 379 ec_scalar_equal_vartime(group, &x, r); 380 } 381
