1 /* 2 * Helper functions for the RSA module 3 * 4 * Copyright The Mbed TLS Contributors 5 * SPDX-License-Identifier: Apache-2.0 6 * 7 * Licensed under the Apache License, Version 2.0 (the "License"); you may 8 * not use this file except in compliance with the License. 9 * You may obtain a copy of the License at 10 * 11 * http://www.apache.org/licenses/LICENSE-2.0 12 * 13 * Unless required by applicable law or agreed to in writing, software 14 * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT 15 * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 16 * See the License for the specific language governing permissions and 17 * limitations under the License. 18 * 19 */ 20 21 #include "common.h" 22 23 #if defined(MBEDTLS_RSA_C) 24 25 #include "mbedtls/rsa.h" 26 #include "mbedtls/bignum.h" 27 #include "rsa_alt_helpers.h" 28 29 /* 30 * Compute RSA prime factors from public and private exponents 31 * 32 * Summary of algorithm: 33 * Setting F := lcm(P-1,Q-1), the idea is as follows: 34 * 35 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2) 36 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the 37 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four 38 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1) 39 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime 40 * factors of N. 41 * 42 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same 43 * construction still applies since (-)^K is the identity on the set of 44 * roots of 1 in Z/NZ. 45 * 46 * The public and private key primitives (-)^E and (-)^D are mutually inverse 47 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e. 48 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L. 49 * Splitting L = 2^t * K with K odd, we have 50 * 51 * DE - 1 = FL = (F/2) * (2^(t+1)) * K, 52 * 53 * so (F / 2) * K is among the numbers 54 * 55 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord 56 * 57 * where ord is the order of 2 in (DE - 1). 58 * We can therefore iterate through these numbers apply the construction 59 * of (a) and (b) above to attempt to factor N. 60 * 61 */ 62 int mbedtls_rsa_deduce_primes(mbedtls_mpi const *N, 63 mbedtls_mpi const *E, mbedtls_mpi const *D, 64 mbedtls_mpi *P, mbedtls_mpi *Q) 65 { 66 int ret = 0; 67 68 uint16_t attempt; /* Number of current attempt */ 69 uint16_t iter; /* Number of squares computed in the current attempt */ 70 71 uint16_t order; /* Order of 2 in DE - 1 */ 72 73 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */ 74 mbedtls_mpi K; /* Temporary holding the current candidate */ 75 76 const unsigned char primes[] = { 2, 77 3, 5, 7, 11, 13, 17, 19, 23, 78 29, 31, 37, 41, 43, 47, 53, 59, 79 61, 67, 71, 73, 79, 83, 89, 97, 80 101, 103, 107, 109, 113, 127, 131, 137, 81 139, 149, 151, 157, 163, 167, 173, 179, 82 181, 191, 193, 197, 199, 211, 223, 227, 83 229, 233, 239, 241, 251 }; 84 85 const size_t num_primes = sizeof(primes) / sizeof(*primes); 86 87 if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) { 88 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 89 } 90 91 if (mbedtls_mpi_cmp_int(N, 0) <= 0 || 92 mbedtls_mpi_cmp_int(D, 1) <= 0 || 93 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 94 mbedtls_mpi_cmp_int(E, 1) <= 0 || 95 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 96 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 97 } 98 99 /* 100 * Initializations and temporary changes 101 */ 102 103 mbedtls_mpi_init(&K); 104 mbedtls_mpi_init(&T); 105 106 /* T := DE - 1 */ 107 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E)); 108 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1)); 109 110 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) { 111 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 112 goto cleanup; 113 } 114 115 /* After this operation, T holds the largest odd divisor of DE - 1. */ 116 MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order)); 117 118 /* 119 * Actual work 120 */ 121 122 /* Skip trying 2 if N == 1 mod 8 */ 123 attempt = 0; 124 if (N->p[0] % 8 == 1) { 125 attempt = 1; 126 } 127 128 for (; attempt < num_primes; ++attempt) { 129 mbedtls_mpi_lset(&K, primes[attempt]); 130 131 /* Check if gcd(K,N) = 1 */ 132 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N)); 133 if (mbedtls_mpi_cmp_int(P, 1) != 0) { 134 continue; 135 } 136 137 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ... 138 * and check whether they have nontrivial GCD with N. */ 139 MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N, 140 Q /* temporarily use Q for storing Montgomery 141 * multiplication helper values */)); 142 143 for (iter = 1; iter <= order; ++iter) { 144 /* If we reach 1 prematurely, there's no point 145 * in continuing to square K */ 146 if (mbedtls_mpi_cmp_int(&K, 1) == 0) { 147 break; 148 } 149 150 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1)); 151 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N)); 152 153 if (mbedtls_mpi_cmp_int(P, 1) == 1 && 154 mbedtls_mpi_cmp_mpi(P, N) == -1) { 155 /* 156 * Have found a nontrivial divisor P of N. 157 * Set Q := N / P. 158 */ 159 160 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P)); 161 goto cleanup; 162 } 163 164 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 165 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K)); 166 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N)); 167 } 168 169 /* 170 * If we get here, then either we prematurely aborted the loop because 171 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must 172 * be 1 if D,E,N were consistent. 173 * Check if that's the case and abort if not, to avoid very long, 174 * yet eventually failing, computations if N,D,E were not sane. 175 */ 176 if (mbedtls_mpi_cmp_int(&K, 1) != 0) { 177 break; 178 } 179 } 180 181 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 182 183 cleanup: 184 185 mbedtls_mpi_free(&K); 186 mbedtls_mpi_free(&T); 187 return ret; 188 } 189 190 /* 191 * Given P, Q and the public exponent E, deduce D. 192 * This is essentially a modular inversion. 193 */ 194 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P, 195 mbedtls_mpi const *Q, 196 mbedtls_mpi const *E, 197 mbedtls_mpi *D) 198 { 199 int ret = 0; 200 mbedtls_mpi K, L; 201 202 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) { 203 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 204 } 205 206 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 207 mbedtls_mpi_cmp_int(Q, 1) <= 0 || 208 mbedtls_mpi_cmp_int(E, 0) == 0) { 209 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 210 } 211 212 mbedtls_mpi_init(&K); 213 mbedtls_mpi_init(&L); 214 215 /* Temporarily put K := P-1 and L := Q-1 */ 216 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 217 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 218 219 /* Temporarily put D := gcd(P-1, Q-1) */ 220 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L)); 221 222 /* K := LCM(P-1, Q-1) */ 223 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L)); 224 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D)); 225 226 /* Compute modular inverse of E in LCM(P-1, Q-1) */ 227 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K)); 228 229 cleanup: 230 231 mbedtls_mpi_free(&K); 232 mbedtls_mpi_free(&L); 233 234 return ret; 235 } 236 237 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 238 const mbedtls_mpi *D, mbedtls_mpi *DP, 239 mbedtls_mpi *DQ, mbedtls_mpi *QP) 240 { 241 int ret = 0; 242 mbedtls_mpi K; 243 mbedtls_mpi_init(&K); 244 245 /* DP = D mod P-1 */ 246 if (DP != NULL) { 247 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 248 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K)); 249 } 250 251 /* DQ = D mod Q-1 */ 252 if (DQ != NULL) { 253 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 254 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K)); 255 } 256 257 /* QP = Q^{-1} mod P */ 258 if (QP != NULL) { 259 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P)); 260 } 261 262 cleanup: 263 mbedtls_mpi_free(&K); 264 265 return ret; 266 } 267 268 /* 269 * Check that core RSA parameters are sane. 270 */ 271 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P, 272 const mbedtls_mpi *Q, const mbedtls_mpi *D, 273 const mbedtls_mpi *E, 274 int (*f_rng)(void *, unsigned char *, size_t), 275 void *p_rng) 276 { 277 int ret = 0; 278 mbedtls_mpi K, L; 279 280 mbedtls_mpi_init(&K); 281 mbedtls_mpi_init(&L); 282 283 /* 284 * Step 1: If PRNG provided, check that P and Q are prime 285 */ 286 287 #if defined(MBEDTLS_GENPRIME) 288 /* 289 * When generating keys, the strongest security we support aims for an error 290 * rate of at most 2^-100 and we are aiming for the same certainty here as 291 * well. 292 */ 293 if (f_rng != NULL && P != NULL && 294 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) { 295 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 296 goto cleanup; 297 } 298 299 if (f_rng != NULL && Q != NULL && 300 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) { 301 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 302 goto cleanup; 303 } 304 #else 305 ((void) f_rng); 306 ((void) p_rng); 307 #endif /* MBEDTLS_GENPRIME */ 308 309 /* 310 * Step 2: Check that 1 < N = P * Q 311 */ 312 313 if (P != NULL && Q != NULL && N != NULL) { 314 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q)); 315 if (mbedtls_mpi_cmp_int(N, 1) <= 0 || 316 mbedtls_mpi_cmp_mpi(&K, N) != 0) { 317 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 318 goto cleanup; 319 } 320 } 321 322 /* 323 * Step 3: Check and 1 < D, E < N if present. 324 */ 325 326 if (N != NULL && D != NULL && E != NULL) { 327 if (mbedtls_mpi_cmp_int(D, 1) <= 0 || 328 mbedtls_mpi_cmp_int(E, 1) <= 0 || 329 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 330 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 331 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 332 goto cleanup; 333 } 334 } 335 336 /* 337 * Step 4: Check that D, E are inverse modulo P-1 and Q-1 338 */ 339 340 if (P != NULL && Q != NULL && D != NULL && E != NULL) { 341 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 342 mbedtls_mpi_cmp_int(Q, 1) <= 0) { 343 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 344 goto cleanup; 345 } 346 347 /* Compute DE-1 mod P-1 */ 348 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 349 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 350 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1)); 351 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 352 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 353 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 354 goto cleanup; 355 } 356 357 /* Compute DE-1 mod Q-1 */ 358 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 359 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 360 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 361 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 362 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 363 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 364 goto cleanup; 365 } 366 } 367 368 cleanup: 369 370 mbedtls_mpi_free(&K); 371 mbedtls_mpi_free(&L); 372 373 /* Wrap MPI error codes by RSA check failure error code */ 374 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) { 375 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 376 } 377 378 return ret; 379 } 380 381 /* 382 * Check that RSA CRT parameters are in accordance with core parameters. 383 */ 384 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 385 const mbedtls_mpi *D, const mbedtls_mpi *DP, 386 const mbedtls_mpi *DQ, const mbedtls_mpi *QP) 387 { 388 int ret = 0; 389 390 mbedtls_mpi K, L; 391 mbedtls_mpi_init(&K); 392 mbedtls_mpi_init(&L); 393 394 /* Check that DP - D == 0 mod P - 1 */ 395 if (DP != NULL) { 396 if (P == NULL) { 397 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 398 goto cleanup; 399 } 400 401 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 402 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D)); 403 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 404 405 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 406 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 407 goto cleanup; 408 } 409 } 410 411 /* Check that DQ - D == 0 mod Q - 1 */ 412 if (DQ != NULL) { 413 if (Q == NULL) { 414 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 415 goto cleanup; 416 } 417 418 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 419 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D)); 420 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 421 422 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 423 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 424 goto cleanup; 425 } 426 } 427 428 /* Check that QP * Q - 1 == 0 mod P */ 429 if (QP != NULL) { 430 if (P == NULL || Q == NULL) { 431 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 432 goto cleanup; 433 } 434 435 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q)); 436 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 437 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P)); 438 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 439 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 440 goto cleanup; 441 } 442 } 443 444 cleanup: 445 446 /* Wrap MPI error codes by RSA check failure error code */ 447 if (ret != 0 && 448 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED && 449 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) { 450 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 451 } 452 453 mbedtls_mpi_free(&K); 454 mbedtls_mpi_free(&L); 455 456 return ret; 457 } 458 459 #endif /* MBEDTLS_RSA_C */ 460