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@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@cryptsoft.com). This product includes software written by Tim 52 * Hudson (tjh@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 #ifndef OPENSSL_HEADER_EC_INTERNAL_H 69 #define OPENSSL_HEADER_EC_INTERNAL_H 70 71 #include <openssl/base.h> 72 73 #include <openssl/bn.h> 74 #include <openssl/ec.h> 75 #include <openssl/ex_data.h> 76 #include <openssl/type_check.h> 77 78 #include "../bn/internal.h" 79 80 #if defined(__cplusplus) 81 extern "C" { 82 #endif 83 84 85 // EC internals. 86 87 88 // Cap the size of all field elements and scalars, including custom curves, to 89 // 66 bytes, large enough to fit secp521r1 and brainpoolP512r1, which appear to 90 // be the largest fields anyone plausibly uses. 91 #define EC_MAX_BYTES 66 92 #define EC_MAX_WORDS ((EC_MAX_BYTES + BN_BYTES - 1) / BN_BYTES) 93 94 OPENSSL_STATIC_ASSERT(EC_MAX_WORDS <= BN_SMALL_MAX_WORDS, 95 "bn_*_small functions not usable"); 96 97 98 // Scalars. 99 100 // An EC_SCALAR is an integer fully reduced modulo the order. Only the first 101 // |order->width| words are used. An |EC_SCALAR| is specific to an |EC_GROUP| 102 // and must not be mixed between groups. 103 typedef union { 104 // bytes is the representation of the scalar in little-endian order. 105 uint8_t bytes[EC_MAX_BYTES]; 106 BN_ULONG words[EC_MAX_WORDS]; 107 } EC_SCALAR; 108 109 // ec_bignum_to_scalar converts |in| to an |EC_SCALAR| and writes it to 110 // |*out|. It returns one on success and zero if |in| is out of range. 111 OPENSSL_EXPORT int ec_bignum_to_scalar(const EC_GROUP *group, EC_SCALAR *out, 112 const BIGNUM *in); 113 114 // ec_scalar_to_bytes serializes |in| as a big-endian bytestring to |out| and 115 // sets |*out_len| to the number of bytes written. The number of bytes written 116 // is |BN_num_bytes(&group->order)|, which is at most |EC_MAX_BYTES|. 117 OPENSSL_EXPORT void ec_scalar_to_bytes(const EC_GROUP *group, uint8_t *out, 118 size_t *out_len, const EC_SCALAR *in); 119 120 // ec_scalar_from_bytes deserializes |in| and stores the resulting scalar over 121 // group |group| to |out|. It returns one on success and zero if |in| is 122 // invalid. 123 int ec_scalar_from_bytes(const EC_GROUP *group, EC_SCALAR *out, 124 const uint8_t *in, size_t len); 125 126 // ec_scalar_reduce sets |out| to |words|, reduced modulo the group order. 127 // |words| must be less than order^2. |num| must be at most twice the width of 128 // group order. This function treats |words| as secret. 129 void ec_scalar_reduce(const EC_GROUP *group, EC_SCALAR *out, 130 const BN_ULONG *words, size_t num); 131 132 // ec_random_nonzero_scalar sets |out| to a uniformly selected random value from 133 // 1 to |group->order| - 1. It returns one on success and zero on error. 134 int ec_random_nonzero_scalar(const EC_GROUP *group, EC_SCALAR *out, 135 const uint8_t additional_data[32]); 136 137 // ec_scalar_equal_vartime returns one if |a| and |b| are equal and zero 138 // otherwise. Both values are treated as public. 139 int ec_scalar_equal_vartime(const EC_GROUP *group, const EC_SCALAR *a, 140 const EC_SCALAR *b); 141 142 // ec_scalar_is_zero returns one if |a| is zero and zero otherwise. 143 int ec_scalar_is_zero(const EC_GROUP *group, const EC_SCALAR *a); 144 145 // ec_scalar_add sets |r| to |a| + |b|. 146 void ec_scalar_add(const EC_GROUP *group, EC_SCALAR *r, const EC_SCALAR *a, 147 const EC_SCALAR *b); 148 149 // ec_scalar_sub sets |r| to |a| - |b|. 150 void ec_scalar_sub(const EC_GROUP *group, EC_SCALAR *r, const EC_SCALAR *a, 151 const EC_SCALAR *b); 152 153 // ec_scalar_neg sets |r| to -|a|. 154 void ec_scalar_neg(const EC_GROUP *group, EC_SCALAR *r, const EC_SCALAR *a); 155 156 // ec_scalar_to_montgomery sets |r| to |a| in Montgomery form. 157 void ec_scalar_to_montgomery(const EC_GROUP *group, EC_SCALAR *r, 158 const EC_SCALAR *a); 159 160 // ec_scalar_to_montgomery sets |r| to |a| converted from Montgomery form. 161 void ec_scalar_from_montgomery(const EC_GROUP *group, EC_SCALAR *r, 162 const EC_SCALAR *a); 163 164 // ec_scalar_mul_montgomery sets |r| to |a| * |b| where inputs and outputs are 165 // in Montgomery form. 166 void ec_scalar_mul_montgomery(const EC_GROUP *group, EC_SCALAR *r, 167 const EC_SCALAR *a, const EC_SCALAR *b); 168 169 // ec_scalar_inv0_montgomery sets |r| to |a|^-1 where inputs and outputs are in 170 // Montgomery form. If |a| is zero, |r| is set to zero. 171 void ec_scalar_inv0_montgomery(const EC_GROUP *group, EC_SCALAR *r, 172 const EC_SCALAR *a); 173 174 // ec_scalar_to_montgomery_inv_vartime sets |r| to |a|^-1 R. That is, it takes 175 // in |a| not in Montgomery form and computes the inverse in Montgomery form. It 176 // returns one on success and zero if |a| has no inverse. This function assumes 177 // |a| is public and may leak information about it via timing. 178 // 179 // Note this is not the same operation as |ec_scalar_inv0_montgomery|. 180 int ec_scalar_to_montgomery_inv_vartime(const EC_GROUP *group, EC_SCALAR *r, 181 const EC_SCALAR *a); 182 183 // ec_scalar_select, in constant time, sets |out| to |a| if |mask| is all ones 184 // and |b| if |mask| is all zeros. 185 void ec_scalar_select(const EC_GROUP *group, EC_SCALAR *out, BN_ULONG mask, 186 const EC_SCALAR *a, const EC_SCALAR *b); 187 188 189 // Field elements. 190 191 // An EC_FELEM represents a field element. Only the first |field->width| words 192 // are used. An |EC_FELEM| is specific to an |EC_GROUP| and must not be mixed 193 // between groups. Additionally, the representation (whether or not elements are 194 // represented in Montgomery-form) may vary between |EC_METHOD|s. 195 typedef union { 196 // bytes is the representation of the field element in little-endian order. 197 uint8_t bytes[EC_MAX_BYTES]; 198 BN_ULONG words[EC_MAX_WORDS]; 199 } EC_FELEM; 200 201 // ec_bignum_to_felem converts |in| to an |EC_FELEM|. It returns one on success 202 // and zero if |in| is out of range. 203 int ec_bignum_to_felem(const EC_GROUP *group, EC_FELEM *out, const BIGNUM *in); 204 205 // ec_felem_to_bignum converts |in| to a |BIGNUM|. It returns one on success and 206 // zero on allocation failure. 207 int ec_felem_to_bignum(const EC_GROUP *group, BIGNUM *out, const EC_FELEM *in); 208 209 // ec_felem_to_bytes serializes |in| as a big-endian bytestring to |out| and 210 // sets |*out_len| to the number of bytes written. The number of bytes written 211 // is |BN_num_bytes(&group->order)|, which is at most |EC_MAX_BYTES|. 212 void ec_felem_to_bytes(const EC_GROUP *group, uint8_t *out, size_t *out_len, 213 const EC_FELEM *in); 214 215 // ec_felem_from_bytes deserializes |in| and stores the resulting field element 216 // to |out|. It returns one on success and zero if |in| is invalid. 217 int ec_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out, const uint8_t *in, 218 size_t len); 219 220 // ec_felem_neg sets |out| to -|a|. 221 void ec_felem_neg(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a); 222 223 // ec_felem_add sets |out| to |a| + |b|. 224 void ec_felem_add(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a, 225 const EC_FELEM *b); 226 227 // ec_felem_add sets |out| to |a| - |b|. 228 void ec_felem_sub(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a, 229 const EC_FELEM *b); 230 231 // ec_felem_non_zero_mask returns all ones if |a| is non-zero and all zeros 232 // otherwise. 233 BN_ULONG ec_felem_non_zero_mask(const EC_GROUP *group, const EC_FELEM *a); 234 235 // ec_felem_select, in constant time, sets |out| to |a| if |mask| is all ones 236 // and |b| if |mask| is all zeros. 237 void ec_felem_select(const EC_GROUP *group, EC_FELEM *out, BN_ULONG mask, 238 const EC_FELEM *a, const EC_FELEM *b); 239 240 // ec_felem_equal returns one if |a| and |b| are equal and zero otherwise. 241 int ec_felem_equal(const EC_GROUP *group, const EC_FELEM *a, const EC_FELEM *b); 242 243 244 // Points. 245 // 246 // Points may represented in affine coordinates as |EC_AFFINE| or Jacobian 247 // coordinates as |EC_RAW_POINT|. Affine coordinates directly represent a 248 // point on the curve, but point addition over affine coordinates requires 249 // costly field inversions, so arithmetic is done in Jacobian coordinates. 250 // Converting from affine to Jacobian is cheap, while converting from Jacobian 251 // to affine costs a field inversion. (Jacobian coordinates amortize the field 252 // inversions needed in a sequence of point operations.) 253 // 254 // TODO(davidben): Rename |EC_RAW_POINT| to |EC_JACOBIAN|. 255 256 // An EC_RAW_POINT represents an elliptic curve point in Jacobian coordinates. 257 // Unlike |EC_POINT|, it is a plain struct which can be stack-allocated and 258 // needs no cleanup. It is specific to an |EC_GROUP| and must not be mixed 259 // between groups. 260 typedef struct { 261 // X, Y, and Z are Jacobian projective coordinates. They represent 262 // (X/Z^2, Y/Z^3) if Z != 0 and the point at infinity otherwise. 263 EC_FELEM X, Y, Z; 264 } EC_RAW_POINT; 265 266 // An EC_AFFINE represents an elliptic curve point in affine coordinates. 267 // coordinates. Note the point at infinity cannot be represented in affine 268 // coordinates. 269 typedef struct { 270 EC_FELEM X, Y; 271 } EC_AFFINE; 272 273 // ec_affine_to_jacobian converts |p| to Jacobian form and writes the result to 274 // |*out|. This operation is very cheap and only costs a few copies. 275 void ec_affine_to_jacobian(const EC_GROUP *group, EC_RAW_POINT *out, 276 const EC_AFFINE *p); 277 278 // ec_jacobian_to_affine converts |p| to affine form and writes the result to 279 // |*out|. It returns one on success and zero if |p| was the point at infinity. 280 // This operation performs a field inversion and should only be done once per 281 // point. 282 // 283 // If only extracting the x-coordinate, use |ec_get_x_coordinate_*| which is 284 // slightly faster. 285 int ec_jacobian_to_affine(const EC_GROUP *group, EC_AFFINE *out, 286 const EC_RAW_POINT *p); 287 288 // ec_jacobian_to_affine_batch converts |num| points in |in| from Jacobian 289 // coordinates to affine coordinates and writes the results to |out|. It returns 290 // one on success and zero if any of the input points were infinity. 291 // 292 // This function is not implemented for all curves. Add implementations as 293 // needed. 294 int ec_jacobian_to_affine_batch(const EC_GROUP *group, EC_AFFINE *out, 295 const EC_RAW_POINT *in, size_t num); 296 297 // ec_point_set_affine_coordinates sets |out|'s to a point with affine 298 // coordinates |x| and |y|. It returns one if the point is on the curve and 299 // zero otherwise. If the point is not on the curve, the value of |out| is 300 // undefined. 301 int ec_point_set_affine_coordinates(const EC_GROUP *group, EC_AFFINE *out, 302 const EC_FELEM *x, const EC_FELEM *y); 303 304 // ec_point_mul_scalar sets |r| to |p| * |scalar|. Both inputs are considered 305 // secret. 306 int ec_point_mul_scalar(const EC_GROUP *group, EC_RAW_POINT *r, 307 const EC_RAW_POINT *p, const EC_SCALAR *scalar); 308 309 // ec_point_mul_scalar_base sets |r| to generator * |scalar|. |scalar| is 310 // treated as secret. 311 int ec_point_mul_scalar_base(const EC_GROUP *group, EC_RAW_POINT *r, 312 const EC_SCALAR *scalar); 313 314 // ec_point_mul_scalar_batch sets |r| to |p0| * |scalar0| + |p1| * |scalar1| + 315 // |p2| * |scalar2|. |p2| may be NULL to skip that term. 316 // 317 // The inputs are treated as secret, however, this function leaks information 318 // about whether intermediate computations add a point to itself. Callers must 319 // ensure that discrete logs between |p0|, |p1|, and |p2| are uniformly 320 // distributed and independent of the scalars, which should be uniformly 321 // selected and not under the attackers control. This ensures the doubling case 322 // will occur with negligible probability. 323 // 324 // This function is not implemented for all curves. Add implementations as 325 // needed. 326 // 327 // TODO(davidben): This function does not use base point tables. For now, it is 328 // only used with the generic |EC_GFp_mont_method| implementation which has 329 // none. If generalizing to tuned curves, this may be useful. However, we still 330 // must double up to the least efficient input, so precomputed tables can only 331 // save table setup and allow a wider window size. 332 int ec_point_mul_scalar_batch(const EC_GROUP *group, EC_RAW_POINT *r, 333 const EC_RAW_POINT *p0, const EC_SCALAR *scalar0, 334 const EC_RAW_POINT *p1, const EC_SCALAR *scalar1, 335 const EC_RAW_POINT *p2, const EC_SCALAR *scalar2); 336 337 #define EC_MONT_PRECOMP_COMB_SIZE 5 338 339 // An |EC_PRECOMP| stores precomputed information about a point, to optimize 340 // repeated multiplications involving it. It is a union so different 341 // |EC_METHOD|s can store different information in it. 342 typedef union { 343 EC_AFFINE comb[(1 << EC_MONT_PRECOMP_COMB_SIZE) - 1]; 344 } EC_PRECOMP; 345 346 // ec_init_precomp precomputes multiples of |p| and writes the result to |out|. 347 // It returns one on success and zero on error. The resulting table may be used 348 // with |ec_point_mul_scalar_precomp|. This function will fail if |p| is the 349 // point at infinity. 350 // 351 // This function is not implemented for all curves. Add implementations as 352 // needed. 353 int ec_init_precomp(const EC_GROUP *group, EC_PRECOMP *out, 354 const EC_RAW_POINT *p); 355 356 // ec_point_mul_scalar_precomp sets |r| to |p0| * |scalar0| + |p1| * |scalar1| + 357 // |p2| * |scalar2|. |p1| or |p2| may be NULL to skip the corresponding term. 358 // The points are represented as |EC_PRECOMP| and must be initialized with 359 // |ec_init_precomp|. This function runs faster than |ec_point_mul_scalar_batch| 360 // but requires setup work per input point, so it is only appropriate for points 361 // which are used frequently. 362 // 363 // The inputs are treated as secret, however, this function leaks information 364 // about whether intermediate computations add a point to itself. Callers must 365 // ensure that discrete logs between |p0|, |p1|, and |p2| are uniformly 366 // distributed and independent of the scalars, which should be uniformly 367 // selected and not under the attackers control. This ensures the doubling case 368 // will occur with negligible probability. 369 // 370 // This function is not implemented for all curves. Add implementations as 371 // needed. 372 // 373 // TODO(davidben): This function does not use base point tables. For now, it is 374 // only used with the generic |EC_GFp_mont_method| implementation which has 375 // none. If generalizing to tuned curves, we should add a parameter for the base 376 // point and arrange for the generic implementation to have base point tables 377 // available. 378 int ec_point_mul_scalar_precomp(const EC_GROUP *group, EC_RAW_POINT *r, 379 const EC_PRECOMP *p0, const EC_SCALAR *scalar0, 380 const EC_PRECOMP *p1, const EC_SCALAR *scalar1, 381 const EC_PRECOMP *p2, const EC_SCALAR *scalar2); 382 383 // ec_point_mul_scalar_public sets |r| to 384 // generator * |g_scalar| + |p| * |p_scalar|. It assumes that the inputs are 385 // public so there is no concern about leaking their values through timing. 386 OPENSSL_EXPORT int ec_point_mul_scalar_public(const EC_GROUP *group, 387 EC_RAW_POINT *r, 388 const EC_SCALAR *g_scalar, 389 const EC_RAW_POINT *p, 390 const EC_SCALAR *p_scalar); 391 392 // ec_point_mul_scalar_public_batch sets |r| to the sum of generator * 393 // |g_scalar| and |points[i]| * |scalars[i]| where |points| and |scalars| have 394 // |num| elements. It assumes that the inputs are public so there is no concern 395 // about leaking their values through timing. |g_scalar| may be NULL to skip 396 // that term. 397 // 398 // This function is not implemented for all curves. Add implementations as 399 // needed. 400 int ec_point_mul_scalar_public_batch(const EC_GROUP *group, EC_RAW_POINT *r, 401 const EC_SCALAR *g_scalar, 402 const EC_RAW_POINT *points, 403 const EC_SCALAR *scalars, size_t num); 404 405 // ec_point_select, in constant time, sets |out| to |a| if |mask| is all ones 406 // and |b| if |mask| is all zeros. 407 void ec_point_select(const EC_GROUP *group, EC_RAW_POINT *out, BN_ULONG mask, 408 const EC_RAW_POINT *a, const EC_RAW_POINT *b); 409 410 // ec_affine_select behaves like |ec_point_select| but acts on affine points. 411 void ec_affine_select(const EC_GROUP *group, EC_AFFINE *out, BN_ULONG mask, 412 const EC_AFFINE *a, const EC_AFFINE *b); 413 414 // ec_precomp_select behaves like |ec_point_select| but acts on |EC_PRECOMP|. 415 void ec_precomp_select(const EC_GROUP *group, EC_PRECOMP *out, BN_ULONG mask, 416 const EC_PRECOMP *a, const EC_PRECOMP *b); 417 418 // ec_cmp_x_coordinate compares the x (affine) coordinate of |p|, mod the group 419 // order, with |r|. It returns one if the values match and zero if |p| is the 420 // point at infinity of the values do not match. 421 int ec_cmp_x_coordinate(const EC_GROUP *group, const EC_RAW_POINT *p, 422 const EC_SCALAR *r); 423 424 // ec_get_x_coordinate_as_scalar sets |*out| to |p|'s x-coordinate, modulo 425 // |group->order|. It returns one on success and zero if |p| is the point at 426 // infinity. 427 int ec_get_x_coordinate_as_scalar(const EC_GROUP *group, EC_SCALAR *out, 428 const EC_RAW_POINT *p); 429 430 // ec_get_x_coordinate_as_bytes writes |p|'s affine x-coordinate to |out|, which 431 // must have at must |max_out| bytes. It sets |*out_len| to the number of bytes 432 // written. The value is written big-endian and zero-padded to the size of the 433 // field. This function returns one on success and zero on failure. 434 int ec_get_x_coordinate_as_bytes(const EC_GROUP *group, uint8_t *out, 435 size_t *out_len, size_t max_out, 436 const EC_RAW_POINT *p); 437 438 // ec_point_to_bytes behaves like |EC_POINT_point2oct| but takes an 439 // |EC_AFFINE|. 440 size_t ec_point_to_bytes(const EC_GROUP *group, const EC_AFFINE *point, 441 point_conversion_form_t form, uint8_t *buf, 442 size_t len); 443 444 // ec_point_from_uncompressed parses |in| as a point in uncompressed form and 445 // sets the result to |out|. It returns one on success and zero if the input was 446 // invalid. 447 int ec_point_from_uncompressed(const EC_GROUP *group, EC_AFFINE *out, 448 const uint8_t *in, size_t len); 449 450 // ec_set_to_safe_point sets |out| to an arbitrary point on |group|, either the 451 // generator or the point at infinity. This is used to guard against callers of 452 // external APIs not checking the return value. 453 void ec_set_to_safe_point(const EC_GROUP *group, EC_RAW_POINT *out); 454 455 // ec_affine_jacobian_equal returns one if |a| and |b| represent the same point 456 // and zero otherwise. It treats both inputs as secret. 457 int ec_affine_jacobian_equal(const EC_GROUP *group, const EC_AFFINE *a, 458 const EC_RAW_POINT *b); 459 460 461 // Implementation details. 462 463 struct ec_method_st { 464 int (*group_init)(EC_GROUP *); 465 void (*group_finish)(EC_GROUP *); 466 int (*group_set_curve)(EC_GROUP *, const BIGNUM *p, const BIGNUM *a, 467 const BIGNUM *b, BN_CTX *); 468 469 // point_get_affine_coordinates sets |*x| and |*y| to the affine coordinates 470 // of |p|. Either |x| or |y| may be NULL to omit it. It returns one on success 471 // and zero if |p| is the point at infinity. 472 int (*point_get_affine_coordinates)(const EC_GROUP *, const EC_RAW_POINT *p, 473 EC_FELEM *x, EC_FELEM *y); 474 475 // jacobian_to_affine_batch implements |ec_jacobian_to_affine_batch|. 476 int (*jacobian_to_affine_batch)(const EC_GROUP *group, EC_AFFINE *out, 477 const EC_RAW_POINT *in, size_t num); 478 479 // add sets |r| to |a| + |b|. 480 void (*add)(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *a, 481 const EC_RAW_POINT *b); 482 // dbl sets |r| to |a| + |a|. 483 void (*dbl)(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *a); 484 485 // mul sets |r| to |scalar|*|p|. 486 void (*mul)(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *p, 487 const EC_SCALAR *scalar); 488 // mul_base sets |r| to |scalar|*generator. 489 void (*mul_base)(const EC_GROUP *group, EC_RAW_POINT *r, 490 const EC_SCALAR *scalar); 491 // mul_batch implements |ec_mul_scalar_batch|. 492 void (*mul_batch)(const EC_GROUP *group, EC_RAW_POINT *r, 493 const EC_RAW_POINT *p0, const EC_SCALAR *scalar0, 494 const EC_RAW_POINT *p1, const EC_SCALAR *scalar1, 495 const EC_RAW_POINT *p2, const EC_SCALAR *scalar2); 496 // mul_public sets |r| to |g_scalar|*generator + |p_scalar|*|p|. It assumes 497 // that the inputs are public so there is no concern about leaking their 498 // values through timing. 499 // 500 // This function may be omitted if |mul_public_batch| is provided. 501 void (*mul_public)(const EC_GROUP *group, EC_RAW_POINT *r, 502 const EC_SCALAR *g_scalar, const EC_RAW_POINT *p, 503 const EC_SCALAR *p_scalar); 504 // mul_public_batch implements |ec_point_mul_scalar_public_batch|. 505 int (*mul_public_batch)(const EC_GROUP *group, EC_RAW_POINT *r, 506 const EC_SCALAR *g_scalar, const EC_RAW_POINT *points, 507 const EC_SCALAR *scalars, size_t num); 508 509 // init_precomp implements |ec_init_precomp|. 510 int (*init_precomp)(const EC_GROUP *group, EC_PRECOMP *out, 511 const EC_RAW_POINT *p); 512 // mul_precomp implements |ec_point_mul_scalar_precomp|. 513 void (*mul_precomp)(const EC_GROUP *group, EC_RAW_POINT *r, 514 const EC_PRECOMP *p0, const EC_SCALAR *scalar0, 515 const EC_PRECOMP *p1, const EC_SCALAR *scalar1, 516 const EC_PRECOMP *p2, const EC_SCALAR *scalar2); 517 518 // felem_mul and felem_sqr implement multiplication and squaring, 519 // respectively, so that the generic |EC_POINT_add| and |EC_POINT_dbl| 520 // implementations can work both with |EC_GFp_mont_method| and the tuned 521 // operations. 522 // 523 // TODO(davidben): This constrains |EC_FELEM|'s internal representation, adds 524 // many indirect calls in the middle of the generic code, and a bunch of 525 // conversions. If p224-64.c were easily convertable to Montgomery form, we 526 // could say |EC_FELEM| is always in Montgomery form. If we routed the rest of 527 // simple.c to |EC_METHOD|, we could give |EC_POINT| an |EC_METHOD|-specific 528 // representation and say |EC_FELEM| is purely a |EC_GFp_mont_method| type. 529 void (*felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, 530 const EC_FELEM *b); 531 void (*felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a); 532 533 void (*felem_to_bytes)(const EC_GROUP *group, uint8_t *out, size_t *out_len, 534 const EC_FELEM *in); 535 int (*felem_from_bytes)(const EC_GROUP *group, EC_FELEM *out, 536 const uint8_t *in, size_t len); 537 538 // felem_reduce sets |out| to |words|, reduced modulo the field size, p. 539 // |words| must be less than p^2. |num| must be at most twice the width of p. 540 // This function treats |words| as secret. 541 // 542 // This function is only used in hash-to-curve and may be omitted in curves 543 // that do not support it. 544 void (*felem_reduce)(const EC_GROUP *group, EC_FELEM *out, 545 const BN_ULONG *words, size_t num); 546 547 // felem_exp sets |out| to |a|^|exp|. It treats |a| is secret but |exp| as 548 // public. 549 // 550 // This function is used in hash-to-curve and may be NULL in curves not used 551 // with hash-to-curve. 552 void (*felem_exp)(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a, 553 const BN_ULONG *exp, size_t num_exp); 554 555 // scalar_inv0_montgomery implements |ec_scalar_inv0_montgomery|. 556 void (*scalar_inv0_montgomery)(const EC_GROUP *group, EC_SCALAR *out, 557 const EC_SCALAR *in); 558 559 // scalar_to_montgomery_inv_vartime implements 560 // |ec_scalar_to_montgomery_inv_vartime|. 561 int (*scalar_to_montgomery_inv_vartime)(const EC_GROUP *group, EC_SCALAR *out, 562 const EC_SCALAR *in); 563 564 // cmp_x_coordinate compares the x (affine) coordinate of |p|, mod the group 565 // order, with |r|. It returns one if the values match and zero if |p| is the 566 // point at infinity of the values do not match. 567 int (*cmp_x_coordinate)(const EC_GROUP *group, const EC_RAW_POINT *p, 568 const EC_SCALAR *r); 569 } /* EC_METHOD */; 570 571 const EC_METHOD *EC_GFp_mont_method(void); 572 573 struct ec_group_st { 574 const EC_METHOD *meth; 575 576 // Unlike all other |EC_POINT|s, |generator| does not own |generator->group| 577 // to avoid a reference cycle. Additionally, Z is guaranteed to be one, so X 578 // and Y are suitable for use as an |EC_AFFINE|. 579 EC_POINT *generator; 580 BIGNUM order; 581 582 int curve_name; // optional NID for named curve 583 584 BN_MONT_CTX *order_mont; // data for ECDSA inverse 585 586 // The following members are handled by the method functions, 587 // even if they appear generic 588 589 BIGNUM field; // For curves over GF(p), this is the modulus. 590 591 EC_FELEM a, b; // Curve coefficients. 592 593 // a_is_minus3 is one if |a| is -3 mod |field| and zero otherwise. Point 594 // arithmetic is optimized for -3. 595 int a_is_minus3; 596 597 // field_greater_than_order is one if |field| is greate than |order| and zero 598 // otherwise. 599 int field_greater_than_order; 600 601 // field_minus_order, if |field_greater_than_order| is true, is |field| minus 602 // |order| represented as an |EC_FELEM|. Otherwise, it is zero. 603 // 604 // Note: unlike |EC_FELEM|s used as intermediate values internal to the 605 // |EC_METHOD|, this value is not encoded in Montgomery form. 606 EC_FELEM field_minus_order; 607 608 CRYPTO_refcount_t references; 609 610 BN_MONT_CTX *mont; // Montgomery structure. 611 612 EC_FELEM one; // The value one. 613 } /* EC_GROUP */; 614 615 struct ec_point_st { 616 // group is an owning reference to |group|, unless this is 617 // |group->generator|. 618 EC_GROUP *group; 619 // raw is the group-specific point data. Functions that take |EC_POINT| 620 // typically check consistency with |EC_GROUP| while functions that take 621 // |EC_RAW_POINT| do not. Thus accesses to this field should be externally 622 // checked for consistency. 623 EC_RAW_POINT raw; 624 } /* EC_POINT */; 625 626 EC_GROUP *ec_group_new(const EC_METHOD *meth); 627 628 void ec_GFp_mont_mul(const EC_GROUP *group, EC_RAW_POINT *r, 629 const EC_RAW_POINT *p, const EC_SCALAR *scalar); 630 void ec_GFp_mont_mul_base(const EC_GROUP *group, EC_RAW_POINT *r, 631 const EC_SCALAR *scalar); 632 void ec_GFp_mont_mul_batch(const EC_GROUP *group, EC_RAW_POINT *r, 633 const EC_RAW_POINT *p0, const EC_SCALAR *scalar0, 634 const EC_RAW_POINT *p1, const EC_SCALAR *scalar1, 635 const EC_RAW_POINT *p2, const EC_SCALAR *scalar2); 636 int ec_GFp_mont_init_precomp(const EC_GROUP *group, EC_PRECOMP *out, 637 const EC_RAW_POINT *p); 638 void ec_GFp_mont_mul_precomp(const EC_GROUP *group, EC_RAW_POINT *r, 639 const EC_PRECOMP *p0, const EC_SCALAR *scalar0, 640 const EC_PRECOMP *p1, const EC_SCALAR *scalar1, 641 const EC_PRECOMP *p2, const EC_SCALAR *scalar2); 642 643 // ec_compute_wNAF writes the modified width-(w+1) Non-Adjacent Form (wNAF) of 644 // |scalar| to |out|. |out| must have room for |bits| + 1 elements, each of 645 // which will be either zero or odd with an absolute value less than 2^w 646 // satisfying 647 // scalar = \sum_j out[j]*2^j 648 // where at most one of any w+1 consecutive digits is non-zero 649 // with the exception that the most significant digit may be only 650 // w-1 zeros away from that next non-zero digit. 651 void ec_compute_wNAF(const EC_GROUP *group, int8_t *out, 652 const EC_SCALAR *scalar, size_t bits, int w); 653 654 int ec_GFp_mont_mul_public_batch(const EC_GROUP *group, EC_RAW_POINT *r, 655 const EC_SCALAR *g_scalar, 656 const EC_RAW_POINT *points, 657 const EC_SCALAR *scalars, size_t num); 658 659 // method functions in simple.c 660 int ec_GFp_simple_group_init(EC_GROUP *); 661 void ec_GFp_simple_group_finish(EC_GROUP *); 662 int ec_GFp_simple_group_set_curve(EC_GROUP *, const BIGNUM *p, const BIGNUM *a, 663 const BIGNUM *b, BN_CTX *); 664 int ec_GFp_simple_group_get_curve(const EC_GROUP *, BIGNUM *p, BIGNUM *a, 665 BIGNUM *b); 666 void ec_GFp_simple_point_init(EC_RAW_POINT *); 667 void ec_GFp_simple_point_copy(EC_RAW_POINT *, const EC_RAW_POINT *); 668 void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *, EC_RAW_POINT *); 669 void ec_GFp_mont_add(const EC_GROUP *, EC_RAW_POINT *r, const EC_RAW_POINT *a, 670 const EC_RAW_POINT *b); 671 void ec_GFp_mont_dbl(const EC_GROUP *, EC_RAW_POINT *r, const EC_RAW_POINT *a); 672 void ec_GFp_simple_invert(const EC_GROUP *, EC_RAW_POINT *); 673 int ec_GFp_simple_is_at_infinity(const EC_GROUP *, const EC_RAW_POINT *); 674 int ec_GFp_simple_is_on_curve(const EC_GROUP *, const EC_RAW_POINT *); 675 int ec_GFp_simple_points_equal(const EC_GROUP *, const EC_RAW_POINT *a, 676 const EC_RAW_POINT *b); 677 void ec_simple_scalar_inv0_montgomery(const EC_GROUP *group, EC_SCALAR *r, 678 const EC_SCALAR *a); 679 680 int ec_simple_scalar_to_montgomery_inv_vartime(const EC_GROUP *group, 681 EC_SCALAR *r, 682 const EC_SCALAR *a); 683 684 int ec_GFp_simple_cmp_x_coordinate(const EC_GROUP *group, const EC_RAW_POINT *p, 685 const EC_SCALAR *r); 686 687 void ec_GFp_simple_felem_to_bytes(const EC_GROUP *group, uint8_t *out, 688 size_t *out_len, const EC_FELEM *in); 689 int ec_GFp_simple_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out, 690 const uint8_t *in, size_t len); 691 692 // method functions in montgomery.c 693 int ec_GFp_mont_group_init(EC_GROUP *); 694 int ec_GFp_mont_group_set_curve(EC_GROUP *, const BIGNUM *p, const BIGNUM *a, 695 const BIGNUM *b, BN_CTX *); 696 void ec_GFp_mont_group_finish(EC_GROUP *); 697 void ec_GFp_mont_felem_mul(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, 698 const EC_FELEM *b); 699 void ec_GFp_mont_felem_sqr(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a); 700 701 void ec_GFp_mont_felem_to_bytes(const EC_GROUP *group, uint8_t *out, 702 size_t *out_len, const EC_FELEM *in); 703 int ec_GFp_mont_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out, 704 const uint8_t *in, size_t len); 705 706 void ec_GFp_nistp_recode_scalar_bits(crypto_word_t *sign, crypto_word_t *digit, 707 crypto_word_t in); 708 709 const EC_METHOD *EC_GFp_nistp224_method(void); 710 const EC_METHOD *EC_GFp_nistp256_method(void); 711 712 // EC_GFp_nistz256_method is a GFp method using montgomery multiplication, with 713 // x86-64 optimized P256. See http://eprint.iacr.org/2013/816. 714 const EC_METHOD *EC_GFp_nistz256_method(void); 715 716 // An EC_WRAPPED_SCALAR is an |EC_SCALAR| with a parallel |BIGNUM| 717 // representation. It exists to support the |EC_KEY_get0_private_key| API. 718 typedef struct { 719 BIGNUM bignum; 720 EC_SCALAR scalar; 721 } EC_WRAPPED_SCALAR; 722 723 struct ec_key_st { 724 EC_GROUP *group; 725 726 // Ideally |pub_key| would be an |EC_AFFINE| so serializing it does not pay an 727 // inversion each time, but the |EC_KEY_get0_public_key| API implies public 728 // keys are stored in an |EC_POINT|-compatible form. 729 EC_POINT *pub_key; 730 EC_WRAPPED_SCALAR *priv_key; 731 732 // fixed_k may contain a specific value of 'k', to be used in ECDSA signing. 733 // This is only for the FIPS power-on tests. 734 BIGNUM *fixed_k; 735 736 unsigned int enc_flag; 737 point_conversion_form_t conv_form; 738 739 CRYPTO_refcount_t references; 740 741 ECDSA_METHOD *ecdsa_meth; 742 743 CRYPTO_EX_DATA ex_data; 744 } /* EC_KEY */; 745 746 struct built_in_curve { 747 int nid; 748 const uint8_t *oid; 749 uint8_t oid_len; 750 // comment is a human-readable string describing the curve. 751 const char *comment; 752 // param_len is the number of bytes needed to store a field element. 753 uint8_t param_len; 754 // params points to an array of 6*|param_len| bytes which hold the field 755 // elements of the following (in big-endian order): prime, a, b, generator x, 756 // generator y, order. 757 const uint8_t *params; 758 const EC_METHOD *method; 759 }; 760 761 #define OPENSSL_NUM_BUILT_IN_CURVES 4 762 763 struct built_in_curves { 764 struct built_in_curve curves[OPENSSL_NUM_BUILT_IN_CURVES]; 765 }; 766 767 // OPENSSL_built_in_curves returns a pointer to static information about 768 // standard curves. The array is terminated with an entry where |nid| is 769 // |NID_undef|. 770 const struct built_in_curves *OPENSSL_built_in_curves(void); 771 772 #if defined(__cplusplus) 773 } // extern C 774 #endif 775 776 #endif // OPENSSL_HEADER_EC_INTERNAL_H 777