rsa_alt_helpers.c (13626B)
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 "rsa_alt_helpers.h" 16 17 /* 18 * Compute RSA prime factors from public and private exponents 19 * 20 * Summary of algorithm: 21 * Setting F := lcm(P-1,Q-1), the idea is as follows: 22 * 23 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2) 24 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the 25 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four 26 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1) 27 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime 28 * factors of N. 29 * 30 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same 31 * construction still applies since (-)^K is the identity on the set of 32 * roots of 1 in Z/NZ. 33 * 34 * The public and private key primitives (-)^E and (-)^D are mutually inverse 35 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e. 36 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L. 37 * Splitting L = 2^t * K with K odd, we have 38 * 39 * DE - 1 = FL = (F/2) * (2^(t+1)) * K, 40 * 41 * so (F / 2) * K is among the numbers 42 * 43 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord 44 * 45 * where ord is the order of 2 in (DE - 1). 46 * We can therefore iterate through these numbers apply the construction 47 * of (a) and (b) above to attempt to factor N. 48 * 49 */ 50 int mbedtls_rsa_deduce_primes(mbedtls_mpi const *N, 51 mbedtls_mpi const *E, mbedtls_mpi const *D, 52 mbedtls_mpi *P, mbedtls_mpi *Q) 53 { 54 int ret = 0; 55 56 uint16_t attempt; /* Number of current attempt */ 57 uint16_t iter; /* Number of squares computed in the current attempt */ 58 59 uint16_t order; /* Order of 2 in DE - 1 */ 60 61 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */ 62 mbedtls_mpi K; /* Temporary holding the current candidate */ 63 64 const unsigned char primes[] = { 2, 65 3, 5, 7, 11, 13, 17, 19, 23, 66 29, 31, 37, 41, 43, 47, 53, 59, 67 61, 67, 71, 73, 79, 83, 89, 97, 68 101, 103, 107, 109, 113, 127, 131, 137, 69 139, 149, 151, 157, 163, 167, 173, 179, 70 181, 191, 193, 197, 199, 211, 223, 227, 71 229, 233, 239, 241, 251 }; 72 73 const size_t num_primes = sizeof(primes) / sizeof(*primes); 74 75 if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) { 76 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 77 } 78 79 if (mbedtls_mpi_cmp_int(N, 0) <= 0 || 80 mbedtls_mpi_cmp_int(D, 1) <= 0 || 81 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 82 mbedtls_mpi_cmp_int(E, 1) <= 0 || 83 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 84 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 85 } 86 87 /* 88 * Initializations and temporary changes 89 */ 90 91 mbedtls_mpi_init(&K); 92 mbedtls_mpi_init(&T); 93 94 /* T := DE - 1 */ 95 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E)); 96 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1)); 97 98 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) { 99 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 100 goto cleanup; 101 } 102 103 /* After this operation, T holds the largest odd divisor of DE - 1. */ 104 MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order)); 105 106 /* 107 * Actual work 108 */ 109 110 /* Skip trying 2 if N == 1 mod 8 */ 111 attempt = 0; 112 if (N->p[0] % 8 == 1) { 113 attempt = 1; 114 } 115 116 for (; attempt < num_primes; ++attempt) { 117 MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt])); 118 119 /* Check if gcd(K,N) = 1 */ 120 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N)); 121 if (mbedtls_mpi_cmp_int(P, 1) != 0) { 122 continue; 123 } 124 125 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ... 126 * and check whether they have nontrivial GCD with N. */ 127 MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N, 128 Q /* temporarily use Q for storing Montgomery 129 * multiplication helper values */)); 130 131 for (iter = 1; iter <= order; ++iter) { 132 /* If we reach 1 prematurely, there's no point 133 * in continuing to square K */ 134 if (mbedtls_mpi_cmp_int(&K, 1) == 0) { 135 break; 136 } 137 138 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1)); 139 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N)); 140 141 if (mbedtls_mpi_cmp_int(P, 1) == 1 && 142 mbedtls_mpi_cmp_mpi(P, N) == -1) { 143 /* 144 * Have found a nontrivial divisor P of N. 145 * Set Q := N / P. 146 */ 147 148 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P)); 149 goto cleanup; 150 } 151 152 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 153 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K)); 154 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N)); 155 } 156 157 /* 158 * If we get here, then either we prematurely aborted the loop because 159 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must 160 * be 1 if D,E,N were consistent. 161 * Check if that's the case and abort if not, to avoid very long, 162 * yet eventually failing, computations if N,D,E were not sane. 163 */ 164 if (mbedtls_mpi_cmp_int(&K, 1) != 0) { 165 break; 166 } 167 } 168 169 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 170 171 cleanup: 172 173 mbedtls_mpi_free(&K); 174 mbedtls_mpi_free(&T); 175 return ret; 176 } 177 178 /* 179 * Given P, Q and the public exponent E, deduce D. 180 * This is essentially a modular inversion. 181 */ 182 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P, 183 mbedtls_mpi const *Q, 184 mbedtls_mpi const *E, 185 mbedtls_mpi *D) 186 { 187 int ret = 0; 188 mbedtls_mpi K, L; 189 190 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) { 191 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 192 } 193 194 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 195 mbedtls_mpi_cmp_int(Q, 1) <= 0 || 196 mbedtls_mpi_cmp_int(E, 0) == 0) { 197 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 198 } 199 200 mbedtls_mpi_init(&K); 201 mbedtls_mpi_init(&L); 202 203 /* Temporarily put K := P-1 and L := Q-1 */ 204 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 205 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 206 207 /* Temporarily put D := gcd(P-1, Q-1) */ 208 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L)); 209 210 /* K := LCM(P-1, Q-1) */ 211 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L)); 212 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D)); 213 214 /* Compute modular inverse of E in LCM(P-1, Q-1) */ 215 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K)); 216 217 cleanup: 218 219 mbedtls_mpi_free(&K); 220 mbedtls_mpi_free(&L); 221 222 return ret; 223 } 224 225 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 226 const mbedtls_mpi *D, mbedtls_mpi *DP, 227 mbedtls_mpi *DQ, mbedtls_mpi *QP) 228 { 229 int ret = 0; 230 mbedtls_mpi K; 231 mbedtls_mpi_init(&K); 232 233 /* DP = D mod P-1 */ 234 if (DP != NULL) { 235 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 236 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K)); 237 } 238 239 /* DQ = D mod Q-1 */ 240 if (DQ != NULL) { 241 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 242 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K)); 243 } 244 245 /* QP = Q^{-1} mod P */ 246 if (QP != NULL) { 247 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P)); 248 } 249 250 cleanup: 251 mbedtls_mpi_free(&K); 252 253 return ret; 254 } 255 256 /* 257 * Check that core RSA parameters are sane. 258 */ 259 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P, 260 const mbedtls_mpi *Q, const mbedtls_mpi *D, 261 const mbedtls_mpi *E, 262 int (*f_rng)(void *, unsigned char *, size_t), 263 void *p_rng) 264 { 265 int ret = 0; 266 mbedtls_mpi K, L; 267 268 mbedtls_mpi_init(&K); 269 mbedtls_mpi_init(&L); 270 271 /* 272 * Step 1: If PRNG provided, check that P and Q are prime 273 */ 274 275 #if defined(MBEDTLS_GENPRIME) 276 /* 277 * When generating keys, the strongest security we support aims for an error 278 * rate of at most 2^-100 and we are aiming for the same certainty here as 279 * well. 280 */ 281 if (f_rng != NULL && P != NULL && 282 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) { 283 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 284 goto cleanup; 285 } 286 287 if (f_rng != NULL && Q != NULL && 288 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) { 289 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 290 goto cleanup; 291 } 292 #else 293 ((void) f_rng); 294 ((void) p_rng); 295 #endif /* MBEDTLS_GENPRIME */ 296 297 /* 298 * Step 2: Check that 1 < N = P * Q 299 */ 300 301 if (P != NULL && Q != NULL && N != NULL) { 302 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q)); 303 if (mbedtls_mpi_cmp_int(N, 1) <= 0 || 304 mbedtls_mpi_cmp_mpi(&K, N) != 0) { 305 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 306 goto cleanup; 307 } 308 } 309 310 /* 311 * Step 3: Check and 1 < D, E < N if present. 312 */ 313 314 if (N != NULL && D != NULL && E != NULL) { 315 if (mbedtls_mpi_cmp_int(D, 1) <= 0 || 316 mbedtls_mpi_cmp_int(E, 1) <= 0 || 317 mbedtls_mpi_cmp_mpi(D, N) >= 0 || 318 mbedtls_mpi_cmp_mpi(E, N) >= 0) { 319 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 320 goto cleanup; 321 } 322 } 323 324 /* 325 * Step 4: Check that D, E are inverse modulo P-1 and Q-1 326 */ 327 328 if (P != NULL && Q != NULL && D != NULL && E != NULL) { 329 if (mbedtls_mpi_cmp_int(P, 1) <= 0 || 330 mbedtls_mpi_cmp_int(Q, 1) <= 0) { 331 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 332 goto cleanup; 333 } 334 335 /* Compute DE-1 mod P-1 */ 336 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 337 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 338 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1)); 339 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 340 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 341 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 342 goto cleanup; 343 } 344 345 /* Compute DE-1 mod Q-1 */ 346 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E)); 347 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 348 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1)); 349 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L)); 350 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 351 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 352 goto cleanup; 353 } 354 } 355 356 cleanup: 357 358 mbedtls_mpi_free(&K); 359 mbedtls_mpi_free(&L); 360 361 /* Wrap MPI error codes by RSA check failure error code */ 362 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) { 363 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 364 } 365 366 return ret; 367 } 368 369 /* 370 * Check that RSA CRT parameters are in accordance with core parameters. 371 */ 372 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q, 373 const mbedtls_mpi *D, const mbedtls_mpi *DP, 374 const mbedtls_mpi *DQ, const mbedtls_mpi *QP) 375 { 376 int ret = 0; 377 378 mbedtls_mpi K, L; 379 mbedtls_mpi_init(&K); 380 mbedtls_mpi_init(&L); 381 382 /* Check that DP - D == 0 mod P - 1 */ 383 if (DP != NULL) { 384 if (P == NULL) { 385 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 386 goto cleanup; 387 } 388 389 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1)); 390 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D)); 391 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 392 393 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 394 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 395 goto cleanup; 396 } 397 } 398 399 /* Check that DQ - D == 0 mod Q - 1 */ 400 if (DQ != NULL) { 401 if (Q == NULL) { 402 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 403 goto cleanup; 404 } 405 406 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1)); 407 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D)); 408 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K)); 409 410 if (mbedtls_mpi_cmp_int(&L, 0) != 0) { 411 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 412 goto cleanup; 413 } 414 } 415 416 /* Check that QP * Q - 1 == 0 mod P */ 417 if (QP != NULL) { 418 if (P == NULL || Q == NULL) { 419 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 420 goto cleanup; 421 } 422 423 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q)); 424 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1)); 425 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P)); 426 if (mbedtls_mpi_cmp_int(&K, 0) != 0) { 427 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 428 goto cleanup; 429 } 430 } 431 432 cleanup: 433 434 /* Wrap MPI error codes by RSA check failure error code */ 435 if (ret != 0 && 436 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED && 437 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) { 438 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 439 } 440 441 mbedtls_mpi_free(&K); 442 mbedtls_mpi_free(&L); 443 444 return ret; 445 } 446 447 #endif /* MBEDTLS_RSA_C */