1 /* 2 * Helper functions for the RSA module 3 * 4 * Copyright The Mbed TLS Contributors 5 * SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later 6 * 7 */ 8 9 #include "common.h" 10 11 #if defined(MBEDTLS_RSA_C) 12 13 #include "mbedtls/rsa.h" 14 #include "mbedtls/bignum.h" 15 #include "bignum_internal.h" 16 #include "rsa_alt_helpers.h" 17 18 /* 19 * Compute RSA prime factors from public and private exponents 20 * 21 * Summary of algorithm: 22 * Setting F := lcm(P-1,Q-1), the idea is as follows: 23 * 24 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2) 25 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the 26 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four 27 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1) 28 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime 29 * factors of N. 30 * 31 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same 32 * construction still applies since (-)^K is the identity on the set of 33 * roots of 1 in Z/NZ. 34 * 35 * The public and private key primitives (-)^E and (-)^D are mutually inverse 36 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e. 37 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L. 38 * Splitting L = 2^t * K with K odd, we have 39 * 40 * DE - 1 = FL = (F/2) * (2^(t+1)) * K, 41 * 42 * so (F / 2) * K is among the numbers 43 * 44 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord 45 * 46 * where ord is the order of 2 in (DE - 1). 47 * We can therefore iterate through these numbers apply the construction 48 * of (a) and (b) above to attempt to factor N. 49 * 50 */ 51 int mbedtls_rsa_deduce_primes(mbedtls_mpi const *N, 52 mbedtls_mpi const *E, mbedtls_mpi const *D, 53 mbedtls_mpi *P, mbedtls_mpi *Q) 54 { 55 int ret = 0; 56 57 uint16_t attempt; /* Number of current attempt */ 58 uint16_t iter; /* Number of squares computed in the current attempt */ 59 60 uint16_t order; /* Order of 2 in DE - 1 */ 61 62 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */ 63 mbedtls_mpi K; /* Temporary holding the current candidate */ 64 65 const unsigned char primes[] = { 2, 66 3, 5, 7, 11, 13, 17, 19, 23, 67 29, 31, 37, 41, 43, 47, 53, 59, 68 61, 67, 71, 73, 79, 83, 89, 97, 69 101, 103, 107, 109, 113, 127, 131, 137, 70 139, 149, 151, 157, 163, 167, 173, 179, 71 181, 191, 193, 197, 199, 211, 223, 227, 72 229, 233, 239, 241, 251 }; 73 74 const size_t num_primes = sizeof(primes) / sizeof(*primes); 75 76 if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) { 77 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 78 } 79 80 if (mbedtls_mpi_cmp_int(N, 0) <= 0 || 81 mbedtls_mpi_cmp_int(D, 1) <= 0 || 82 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 83 mbedtls_mpi_cmp_int(E, 1) <= 0 || 84 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 85 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 86 } 87 88 /* 89 * Initializations and temporary changes 90 */ 91 92 mbedtls_mpi_init(&K); 93 mbedtls_mpi_init(&T); 94 95 /* T := DE - 1 */ 96 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E)); 97 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1)); 98 99 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) { 100 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 101 goto cleanup; 102 } 103 104 /* After this operation, T holds the largest odd divisor of DE - 1. */ 105 MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order)); 106 107 /* 108 * Actual work 109 */ 110 111 /* Skip trying 2 if N == 1 mod 8 */ 112 attempt = 0; 113 if (N->p[0] % 8 == 1) { 114 attempt = 1; 115 } 116 117 for (; attempt < num_primes; ++attempt) { 118 MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt])); 119 120 /* Check if gcd(K,N) = 1 */ 121 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N)); 122 if (mbedtls_mpi_cmp_int(P, 1) != 0) { 123 continue; 124 } 125 126 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ... 127 * and check whether they have nontrivial GCD with N. */ 128 MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N, 129 Q /* temporarily use Q for storing Montgomery 130 * multiplication helper values */)); 131 132 for (iter = 1; iter <= order; ++iter) { 133 /* If we reach 1 prematurely, there's no point 134 * in continuing to square K */ 135 if (mbedtls_mpi_cmp_int(&K, 1) == 0) { 136 break; 137 } 138 139 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1)); 140 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N)); 141 142 if (mbedtls_mpi_cmp_int(P, 1) == 1 && 143 mbedtls_mpi_cmp_mpi(P, N) == -1) { 144 /* 145 * Have found a nontrivial divisor P of N. 146 * Set Q := N / P. 147 */ 148 149 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P)); 150 goto cleanup; 151 } 152 153 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 154 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K)); 155 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N)); 156 } 157 158 /* 159 * If we get here, then either we prematurely aborted the loop because 160 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must 161 * be 1 if D,E,N were consistent. 162 * Check if that's the case and abort if not, to avoid very long, 163 * yet eventually failing, computations if N,D,E were not sane. 164 */ 165 if (mbedtls_mpi_cmp_int(&K, 1) != 0) { 166 break; 167 } 168 } 169 170 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 171 172 cleanup: 173 174 mbedtls_mpi_free(&K); 175 mbedtls_mpi_free(&T); 176 return ret; 177 } 178 179 /* 180 * Given P, Q and the public exponent E, deduce D. 181 * This is essentially a modular inversion. 182 */ 183 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P, 184 mbedtls_mpi const *Q, 185 mbedtls_mpi const *E, 186 mbedtls_mpi *D) 187 { 188 int ret = 0; 189 mbedtls_mpi K, L; 190 191 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) { 192 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 193 } 194 195 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 196 mbedtls_mpi_cmp_int(Q, 1) <= 0 || 197 mbedtls_mpi_cmp_int(E, 0) == 0) { 198 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 199 } 200 201 if (mbedtls_mpi_get_bit(E, 0) != 1) { 202 return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; 203 } 204 205 mbedtls_mpi_init(&K); 206 mbedtls_mpi_init(&L); 207 208 /* Temporarily put K := P-1 and L := Q-1 */ 209 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 210 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 211 212 /* Temporarily put D := gcd(P-1, Q-1) */ 213 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L)); 214 215 /* K := LCM(P-1, Q-1) */ 216 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L)); 217 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D)); 218 219 /* Compute modular inverse of E mod LCM(P-1, Q-1) 220 * This is FIPS 186-4 §B.3.1 criterion 3(b). 221 * This will return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if E is not coprime to 222 * (P-1)(Q-1), also validating FIPS 186-4 §B.3.1 criterion 2(a). */ 223 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_even_in_range(D, E, &K)); 224 225 cleanup: 226 227 mbedtls_mpi_free(&K); 228 mbedtls_mpi_free(&L); 229 230 return ret; 231 } 232 233 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 234 const mbedtls_mpi *D, mbedtls_mpi *DP, 235 mbedtls_mpi *DQ, mbedtls_mpi *QP) 236 { 237 int ret = 0; 238 mbedtls_mpi K; 239 mbedtls_mpi_init(&K); 240 241 /* DP = D mod P-1 */ 242 if (DP != NULL) { 243 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 244 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K)); 245 } 246 247 /* DQ = D mod Q-1 */ 248 if (DQ != NULL) { 249 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 250 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K)); 251 } 252 253 /* QP = Q^{-1} mod P */ 254 if (QP != NULL) { 255 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_odd(QP, Q, P)); 256 } 257 258 cleanup: 259 mbedtls_mpi_free(&K); 260 261 return ret; 262 } 263 264 /* 265 * Check that core RSA parameters are sane. 266 */ 267 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P, 268 const mbedtls_mpi *Q, const mbedtls_mpi *D, 269 const mbedtls_mpi *E, 270 int (*f_rng)(void *, unsigned char *, size_t), 271 void *p_rng) 272 { 273 int ret = 0; 274 mbedtls_mpi K, L; 275 276 mbedtls_mpi_init(&K); 277 mbedtls_mpi_init(&L); 278 279 /* 280 * Step 1: If PRNG provided, check that P and Q are prime 281 */ 282 283 #if defined(MBEDTLS_GENPRIME) 284 /* 285 * When generating keys, the strongest security we support aims for an error 286 * rate of at most 2^-100 and we are aiming for the same certainty here as 287 * well. 288 */ 289 if (f_rng != NULL && P != NULL && 290 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) { 291 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 292 goto cleanup; 293 } 294 295 if (f_rng != NULL && Q != NULL && 296 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) { 297 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 298 goto cleanup; 299 } 300 #else 301 ((void) f_rng); 302 ((void) p_rng); 303 #endif /* MBEDTLS_GENPRIME */ 304 305 /* 306 * Step 2: Check that 1 < N = P * Q 307 */ 308 309 if (P != NULL && Q != NULL && N != NULL) { 310 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q)); 311 if (mbedtls_mpi_cmp_int(N, 1) <= 0 || 312 mbedtls_mpi_cmp_mpi(&K, N) != 0) { 313 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 314 goto cleanup; 315 } 316 } 317 318 /* 319 * Step 3: Check and 1 < D, E < N if present. 320 */ 321 322 if (N != NULL && D != NULL && E != NULL) { 323 if (mbedtls_mpi_cmp_int(D, 1) <= 0 || 324 mbedtls_mpi_cmp_int(E, 1) <= 0 || 325 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 326 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 327 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 328 goto cleanup; 329 } 330 } 331 332 /* 333 * Step 4: Check that D, E are inverse modulo P-1 and Q-1 334 */ 335 336 if (P != NULL && Q != NULL && D != NULL && E != NULL) { 337 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 338 mbedtls_mpi_cmp_int(Q, 1) <= 0) { 339 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 340 goto cleanup; 341 } 342 343 /* Compute DE-1 mod P-1 */ 344 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 345 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 346 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1)); 347 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 348 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 349 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 350 goto cleanup; 351 } 352 353 /* Compute DE-1 mod Q-1 */ 354 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 355 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 356 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 357 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 358 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 359 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 360 goto cleanup; 361 } 362 } 363 364 cleanup: 365 366 mbedtls_mpi_free(&K); 367 mbedtls_mpi_free(&L); 368 369 /* Wrap MPI error codes by RSA check failure error code */ 370 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) { 371 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 372 } 373 374 return ret; 375 } 376 377 /* 378 * Check that RSA CRT parameters are in accordance with core parameters. 379 */ 380 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 381 const mbedtls_mpi *D, const mbedtls_mpi *DP, 382 const mbedtls_mpi *DQ, const mbedtls_mpi *QP) 383 { 384 int ret = 0; 385 386 mbedtls_mpi K, L; 387 mbedtls_mpi_init(&K); 388 mbedtls_mpi_init(&L); 389 390 /* Check that DP - D == 0 mod P - 1 */ 391 if (DP != NULL) { 392 if (P == NULL) { 393 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 394 goto cleanup; 395 } 396 397 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 398 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D)); 399 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 400 401 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 402 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 403 goto cleanup; 404 } 405 } 406 407 /* Check that DQ - D == 0 mod Q - 1 */ 408 if (DQ != NULL) { 409 if (Q == NULL) { 410 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 411 goto cleanup; 412 } 413 414 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 415 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D)); 416 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 417 418 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 419 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 420 goto cleanup; 421 } 422 } 423 424 /* Check that QP * Q - 1 == 0 mod P */ 425 if (QP != NULL) { 426 if (P == NULL || Q == NULL) { 427 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 428 goto cleanup; 429 } 430 431 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q)); 432 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 433 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P)); 434 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 435 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 436 goto cleanup; 437 } 438 } 439 440 cleanup: 441 442 /* Wrap MPI error codes by RSA check failure error code */ 443 if (ret != 0 && 444 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED && 445 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) { 446 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 447 } 448 449 mbedtls_mpi_free(&K); 450 mbedtls_mpi_free(&L); 451 452 return ret; 453 } 454 455 #endif /* MBEDTLS_RSA_C */ 456