1 /* 2 * Helper functions for the RSA module 3 * 4 * Copyright (C) 2006-2017, ARM Limited, All Rights Reserved 5 * SPDX-License-Identifier: GPL-2.0 6 * 7 * This program is free software; you can redistribute it and/or modify 8 * it under the terms of the GNU General Public License as published by 9 * the Free Software Foundation; either version 2 of the License, or 10 * (at your option) any later version. 11 * 12 * This program is distributed in the hope that it will be useful, 13 * but WITHOUT ANY WARRANTY; without even the implied warranty of 14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 * GNU General Public License for more details. 16 * 17 * You should have received a copy of the GNU General Public License along 18 * with this program; if not, write to the Free Software Foundation, Inc., 19 * 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * This file is part of mbed TLS (https://tls.mbed.org) 22 * 23 */ 24 25 #if !defined(MBEDTLS_CONFIG_FILE) 26 #include "mbedtls/config.h" 27 #else 28 #include MBEDTLS_CONFIG_FILE 29 #endif 30 31 #if defined(MBEDTLS_RSA_C) 32 33 #include "mbedtls/rsa.h" 34 #include "mbedtls/bignum.h" 35 #include "mbedtls/rsa_internal.h" 36 37 /* 38 * Compute RSA prime factors from public and private exponents 39 * 40 * Summary of algorithm: 41 * Setting F := lcm(P-1,Q-1), the idea is as follows: 42 * 43 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2) 44 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the 45 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four 46 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1) 47 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime 48 * factors of N. 49 * 50 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same 51 * construction still applies since (-)^K is the identity on the set of 52 * roots of 1 in Z/NZ. 53 * 54 * The public and private key primitives (-)^E and (-)^D are mutually inverse 55 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e. 56 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L. 57 * Splitting L = 2^t * K with K odd, we have 58 * 59 * DE - 1 = FL = (F/2) * (2^(t+1)) * K, 60 * 61 * so (F / 2) * K is among the numbers 62 * 63 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord 64 * 65 * where ord is the order of 2 in (DE - 1). 66 * We can therefore iterate through these numbers apply the construction 67 * of (a) and (b) above to attempt to factor N. 68 * 69 */ 70 int mbedtls_rsa_deduce_primes( mbedtls_mpi const *N, 71 mbedtls_mpi const *E, mbedtls_mpi const *D, 72 mbedtls_mpi *P, mbedtls_mpi *Q ) 73 { 74 int ret = 0; 75 76 uint16_t attempt; /* Number of current attempt */ 77 uint16_t iter; /* Number of squares computed in the current attempt */ 78 79 uint16_t order; /* Order of 2 in DE - 1 */ 80 81 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */ 82 mbedtls_mpi K; /* Temporary holding the current candidate */ 83 84 const unsigned char primes[] = { 2, 85 3, 5, 7, 11, 13, 17, 19, 23, 86 29, 31, 37, 41, 43, 47, 53, 59, 87 61, 67, 71, 73, 79, 83, 89, 97, 88 101, 103, 107, 109, 113, 127, 131, 137, 89 139, 149, 151, 157, 163, 167, 173, 179, 90 181, 191, 193, 197, 199, 211, 223, 227, 91 229, 233, 239, 241, 251 92 }; 93 94 const size_t num_primes = sizeof( primes ) / sizeof( *primes ); 95 96 if( P == NULL || Q == NULL || P->p != NULL || Q->p != NULL ) 97 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); 98 99 if( mbedtls_mpi_cmp_int( N, 0 ) <= 0 || 100 mbedtls_mpi_cmp_int( D, 1 ) <= 0 || 101 mbedtls_mpi_cmp_mpi( D, N ) >= 0 || 102 mbedtls_mpi_cmp_int( E, 1 ) <= 0 || 103 mbedtls_mpi_cmp_mpi( E, N ) >= 0 ) 104 { 105 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); 106 } 107 108 /* 109 * Initializations and temporary changes 110 */ 111 112 mbedtls_mpi_init( &K ); 113 mbedtls_mpi_init( &T ); 114 115 /* T := DE - 1 */ 116 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, D, E ) ); 117 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &T, &T, 1 ) ); 118 119 if( ( order = (uint16_t) mbedtls_mpi_lsb( &T ) ) == 0 ) 120 { 121 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 122 goto cleanup; 123 } 124 125 /* After this operation, T holds the largest odd divisor of DE - 1. */ 126 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &T, order ) ); 127 128 /* 129 * Actual work 130 */ 131 132 /* Skip trying 2 if N == 1 mod 8 */ 133 attempt = 0; 134 if( N->p[0] % 8 == 1 ) 135 attempt = 1; 136 137 for( ; attempt < num_primes; ++attempt ) 138 { 139 mbedtls_mpi_lset( &K, primes[attempt] ); 140 141 /* Check if gcd(K,N) = 1 */ 142 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) ); 143 if( mbedtls_mpi_cmp_int( P, 1 ) != 0 ) 144 continue; 145 146 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ... 147 * and check whether they have nontrivial GCD with N. */ 148 MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &K, &K, &T, N, 149 Q /* temporarily use Q for storing Montgomery 150 * multiplication helper values */ ) ); 151 152 for( iter = 1; iter <= order; ++iter ) 153 { 154 /* If we reach 1 prematurely, there's no point 155 * in continuing to square K */ 156 if( mbedtls_mpi_cmp_int( &K, 1 ) == 0 ) 157 break; 158 159 MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &K, &K, 1 ) ); 160 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) ); 161 162 if( mbedtls_mpi_cmp_int( P, 1 ) == 1 && 163 mbedtls_mpi_cmp_mpi( P, N ) == -1 ) 164 { 165 /* 166 * Have found a nontrivial divisor P of N. 167 * Set Q := N / P. 168 */ 169 170 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( Q, NULL, N, P ) ); 171 goto cleanup; 172 } 173 174 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) ); 175 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &K ) ); 176 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, N ) ); 177 } 178 179 /* 180 * If we get here, then either we prematurely aborted the loop because 181 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must 182 * be 1 if D,E,N were consistent. 183 * Check if that's the case and abort if not, to avoid very long, 184 * yet eventually failing, computations if N,D,E were not sane. 185 */ 186 if( mbedtls_mpi_cmp_int( &K, 1 ) != 0 ) 187 { 188 break; 189 } 190 } 191 192 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; 193 194 cleanup: 195 196 mbedtls_mpi_free( &K ); 197 mbedtls_mpi_free( &T ); 198 return( ret ); 199 } 200 201 /* 202 * Given P, Q and the public exponent E, deduce D. 203 * This is essentially a modular inversion. 204 */ 205 int mbedtls_rsa_deduce_private_exponent( mbedtls_mpi const *P, 206 mbedtls_mpi const *Q, 207 mbedtls_mpi const *E, 208 mbedtls_mpi *D ) 209 { 210 int ret = 0; 211 mbedtls_mpi K, L; 212 213 if( D == NULL || mbedtls_mpi_cmp_int( D, 0 ) != 0 ) 214 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); 215 216 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 || 217 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 || 218 mbedtls_mpi_cmp_int( E, 0 ) == 0 ) 219 { 220 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); 221 } 222 223 mbedtls_mpi_init( &K ); 224 mbedtls_mpi_init( &L ); 225 226 /* Temporarily put K := P-1 and L := Q-1 */ 227 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) ); 228 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) ); 229 230 /* Temporarily put D := gcd(P-1, Q-1) */ 231 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( D, &K, &L ) ); 232 233 /* K := LCM(P-1, Q-1) */ 234 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &L ) ); 235 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &K, NULL, &K, D ) ); 236 237 /* Compute modular inverse of E in LCM(P-1, Q-1) */ 238 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( D, E, &K ) ); 239 240 cleanup: 241 242 mbedtls_mpi_free( &K ); 243 mbedtls_mpi_free( &L ); 244 245 return( ret ); 246 } 247 248 /* 249 * Check that RSA CRT parameters are in accordance with core parameters. 250 */ 251 int mbedtls_rsa_validate_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q, 252 const mbedtls_mpi *D, const mbedtls_mpi *DP, 253 const mbedtls_mpi *DQ, const mbedtls_mpi *QP ) 254 { 255 int ret = 0; 256 257 mbedtls_mpi K, L; 258 mbedtls_mpi_init( &K ); 259 mbedtls_mpi_init( &L ); 260 261 /* Check that DP - D == 0 mod P - 1 */ 262 if( DP != NULL ) 263 { 264 if( P == NULL ) 265 { 266 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 267 goto cleanup; 268 } 269 270 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) ); 271 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DP, D ) ); 272 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) ); 273 274 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 ) 275 { 276 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 277 goto cleanup; 278 } 279 } 280 281 /* Check that DQ - D == 0 mod Q - 1 */ 282 if( DQ != NULL ) 283 { 284 if( Q == NULL ) 285 { 286 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 287 goto cleanup; 288 } 289 290 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) ); 291 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DQ, D ) ); 292 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) ); 293 294 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 ) 295 { 296 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 297 goto cleanup; 298 } 299 } 300 301 /* Check that QP * Q - 1 == 0 mod P */ 302 if( QP != NULL ) 303 { 304 if( P == NULL || Q == NULL ) 305 { 306 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; 307 goto cleanup; 308 } 309 310 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, QP, Q ) ); 311 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) ); 312 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, P ) ); 313 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 ) 314 { 315 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 316 goto cleanup; 317 } 318 } 319 320 cleanup: 321 322 /* Wrap MPI error codes by RSA check failure error code */ 323 if( ret != 0 && 324 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED && 325 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA ) 326 { 327 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 328 } 329 330 mbedtls_mpi_free( &K ); 331 mbedtls_mpi_free( &L ); 332 333 return( ret ); 334 } 335 336 /* 337 * Check that core RSA parameters are sane. 338 */ 339 int mbedtls_rsa_validate_params( const mbedtls_mpi *N, const mbedtls_mpi *P, 340 const mbedtls_mpi *Q, const mbedtls_mpi *D, 341 const mbedtls_mpi *E, 342 int (*f_rng)(void *, unsigned char *, size_t), 343 void *p_rng ) 344 { 345 int ret = 0; 346 mbedtls_mpi K, L; 347 348 mbedtls_mpi_init( &K ); 349 mbedtls_mpi_init( &L ); 350 351 /* 352 * Step 1: If PRNG provided, check that P and Q are prime 353 */ 354 355 #if defined(MBEDTLS_GENPRIME) 356 if( f_rng != NULL && P != NULL && 357 ( ret = mbedtls_mpi_is_prime( P, f_rng, p_rng ) ) != 0 ) 358 { 359 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 360 goto cleanup; 361 } 362 363 if( f_rng != NULL && Q != NULL && 364 ( ret = mbedtls_mpi_is_prime( Q, f_rng, p_rng ) ) != 0 ) 365 { 366 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 367 goto cleanup; 368 } 369 #else 370 ((void) f_rng); 371 ((void) p_rng); 372 #endif /* MBEDTLS_GENPRIME */ 373 374 /* 375 * Step 2: Check that 1 < N = P * Q 376 */ 377 378 if( P != NULL && Q != NULL && N != NULL ) 379 { 380 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, P, Q ) ); 381 if( mbedtls_mpi_cmp_int( N, 1 ) <= 0 || 382 mbedtls_mpi_cmp_mpi( &K, N ) != 0 ) 383 { 384 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 385 goto cleanup; 386 } 387 } 388 389 /* 390 * Step 3: Check and 1 < D, E < N if present. 391 */ 392 393 if( N != NULL && D != NULL && E != NULL ) 394 { 395 if ( mbedtls_mpi_cmp_int( D, 1 ) <= 0 || 396 mbedtls_mpi_cmp_int( E, 1 ) <= 0 || 397 mbedtls_mpi_cmp_mpi( D, N ) >= 0 || 398 mbedtls_mpi_cmp_mpi( E, N ) >= 0 ) 399 { 400 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 401 goto cleanup; 402 } 403 } 404 405 /* 406 * Step 4: Check that D, E are inverse modulo P-1 and Q-1 407 */ 408 409 if( P != NULL && Q != NULL && D != NULL && E != NULL ) 410 { 411 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 || 412 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 ) 413 { 414 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 415 goto cleanup; 416 } 417 418 /* Compute DE-1 mod P-1 */ 419 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) ); 420 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) ); 421 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, P, 1 ) ); 422 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) ); 423 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 ) 424 { 425 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 426 goto cleanup; 427 } 428 429 /* Compute DE-1 mod Q-1 */ 430 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) ); 431 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) ); 432 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) ); 433 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) ); 434 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 ) 435 { 436 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 437 goto cleanup; 438 } 439 } 440 441 cleanup: 442 443 mbedtls_mpi_free( &K ); 444 mbedtls_mpi_free( &L ); 445 446 /* Wrap MPI error codes by RSA check failure error code */ 447 if( ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED ) 448 { 449 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED; 450 } 451 452 return( ret ); 453 } 454 455 int mbedtls_rsa_deduce_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q, 456 const mbedtls_mpi *D, mbedtls_mpi *DP, 457 mbedtls_mpi *DQ, mbedtls_mpi *QP ) 458 { 459 int ret = 0; 460 mbedtls_mpi K; 461 mbedtls_mpi_init( &K ); 462 463 /* DP = D mod P-1 */ 464 if( DP != NULL ) 465 { 466 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) ); 467 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DP, D, &K ) ); 468 } 469 470 /* DQ = D mod Q-1 */ 471 if( DQ != NULL ) 472 { 473 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) ); 474 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DQ, D, &K ) ); 475 } 476 477 /* QP = Q^{-1} mod P */ 478 if( QP != NULL ) 479 { 480 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( QP, Q, P ) ); 481 } 482 483 cleanup: 484 mbedtls_mpi_free( &K ); 485 486 return( ret ); 487 } 488 489 #endif /* MBEDTLS_RSA_C */ 490