common/mbedtls/rsa_internal.c

   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 "mbedtls/rsa_internal.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     int ret = 0;
  66
  67     uint16_t attempt;  /* Number of current attempt  */
  68     uint16_t iter;     /* Number of squares computed in the current attempt */
  69
  70     uint16_t order;    /* Order of 2 in DE - 1 */
  71
  72     mbedtls_mpi T;  /* Holds largest odd divisor of DE - 1     */
  73     mbedtls_mpi K;  /* Temporary holding the current candidate */
  74
  75     const unsigned char primes[] = { 2,
  76                                      3,    5,    7,   11,   13,   17,   19,   23,
  77                                      29,   31,   37,   41,   43,   47,   53,   59,
  78                                      61,   67,   71,   73,   79,   83,   89,   97,
  79                                      101,  103,  107,  109,  113,  127,  131,  137,
  80                                      139,  149,  151,  157,  163,  167,  173,  179,
  81                                      181,  191,  193,  197,  199,  211,  223,  227,
  82                                      229,  233,  239,  241,  251
  83                                    };
  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     if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
  91             mbedtls_mpi_cmp_int(D, 1) <= 0 ||
  92             mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
  93             mbedtls_mpi_cmp_int(E, 1) <= 0 ||
  94             mbedtls_mpi_cmp_mpi(E, N) >= 0) {
  95         return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA);
  96     }
  97
  98     /*
  99      * Initializations and temporary changes
 100      */
 101
 102     mbedtls_mpi_init(&K);
 103     mbedtls_mpi_init(&T);
 104
 105     /* T := DE - 1 */
 106     MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D,  E));
 107     MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
 108
 109     if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
 110         ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
 111         goto cleanup;
 112     }
 113
 114     /* After this operation, T holds the largest odd divisor of DE - 1. */
 115     MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order));
 116
 117     /*
 118      * Actual work
 119      */
 120
 121     /* Skip trying 2 if N == 1 mod 8 */
 122     attempt = 0;
 123     if (N->p[0] % 8 == 1)
 124         attempt = 1;
 125
 126     for (; attempt < num_primes; ++attempt) {
 127         mbedtls_mpi_lset(&K, primes[attempt]);
 128
 129         /* Check if gcd(K,N) = 1 */
 130         MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
 131         if (mbedtls_mpi_cmp_int(P, 1) != 0)
 132             continue;
 133
 134         /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
 135          * and check whether they have nontrivial GCD with N. */
 136         MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N,
 137                                             Q /* temporarily use Q for storing Montgomery
 138                                 * multiplication helper values */));
 139
 140         for (iter = 1; iter <= order; ++iter) {
 141             /* If we reach 1 prematurely, there's no point
 142              * in continuing to square K */
 143             if (mbedtls_mpi_cmp_int(&K, 1) == 0)
 144                 break;
 145
 146             MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
 147             MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
 148
 149             if (mbedtls_mpi_cmp_int(P, 1) ==  1 &&
 150                     mbedtls_mpi_cmp_mpi(P, N) == -1) {
 151                 /*
 152                  * Have found a nontrivial divisor P of N.
 153                  * Set Q := N / P.
 154                  */
 155
 156                 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
 157                 goto cleanup;
 158             }
 159
 160             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
 161             MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
 162             MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N));
 163         }
 164
 165         /*
 166          * If we get here, then either we prematurely aborted the loop because
 167          * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
 168          * be 1 if D,E,N were consistent.
 169          * Check if that's the case and abort if not, to avoid very long,
 170          * yet eventually failing, computations if N,D,E were not sane.
 171          */
 172         if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
 173             break;
 174         }
 175     }
 176
 177     ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
 178
 179 cleanup:
 180
 181     mbedtls_mpi_free(&K);
 182     mbedtls_mpi_free(&T);
 183     return (ret);
 184 }
 185
 186 /*
 187  * Given P, Q and the public exponent E, deduce D.
 188  * This is essentially a modular inversion.
 189  */
 190 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P,
 191                                         mbedtls_mpi const *Q,
 192                                         mbedtls_mpi const *E,
 193                                         mbedtls_mpi *D) {
 194     int ret = 0;
 195     mbedtls_mpi K, L;
 196
 197     if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0)
 198         return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA);
 199
 200     if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
 201             mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
 202             mbedtls_mpi_cmp_int(E, 0) == 0) {
 203         return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA);
 204     }
 205
 206     mbedtls_mpi_init(&K);
 207     mbedtls_mpi_init(&L);
 208
 209     /* Temporarily put K := P-1 and L := Q-1 */
 210     MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
 211     MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
 212
 213     /* Temporarily put D := gcd(P-1, Q-1) */
 214     MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
 215
 216     /* K := LCM(P-1, Q-1) */
 217     MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
 218     MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D));
 219
 220     /* Compute modular inverse of E in LCM(P-1, Q-1) */
 221     MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K));
 222
 223 cleanup:
 224
 225     mbedtls_mpi_free(&K);
 226     mbedtls_mpi_free(&L);
 227
 228     return (ret);
 229 }
 230
 231 /*
 232  * Check that RSA CRT parameters are in accordance with core parameters.
 233  */
 234 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P,  const mbedtls_mpi *Q,
 235                              const mbedtls_mpi *D,  const mbedtls_mpi *DP,
 236                              const mbedtls_mpi *DQ, const mbedtls_mpi *QP) {
 237     int ret = 0;
 238
 239     mbedtls_mpi K, L;
 240     mbedtls_mpi_init(&K);
 241     mbedtls_mpi_init(&L);
 242
 243     /* Check that DP - D == 0 mod P - 1 */
 244     if (DP != NULL) {
 245         if (P == NULL) {
 246             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
 247             goto cleanup;
 248         }
 249
 250         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
 251         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
 252         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
 253
 254         if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
 255             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 256             goto cleanup;
 257         }
 258     }
 259
 260     /* Check that DQ - D == 0 mod Q - 1 */
 261     if (DQ != NULL) {
 262         if (Q == NULL) {
 263             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
 264             goto cleanup;
 265         }
 266
 267         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
 268         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
 269         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
 270
 271         if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
 272             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 273             goto cleanup;
 274         }
 275     }
 276
 277     /* Check that QP * Q - 1 == 0 mod P */
 278     if (QP != NULL) {
 279         if (P == NULL || Q == NULL) {
 280             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
 281             goto cleanup;
 282         }
 283
 284         MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
 285         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
 286         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
 287         if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
 288             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 289             goto cleanup;
 290         }
 291     }
 292
 293 cleanup:
 294
 295     /* Wrap MPI error codes by RSA check failure error code */
 296     if (ret != 0 &&
 297             ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
 298             ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
 299         ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 300     }
 301
 302     mbedtls_mpi_free(&K);
 303     mbedtls_mpi_free(&L);
 304
 305     return (ret);
 306 }
 307
 308 /*
 309  * Check that core RSA parameters are sane.
 310  */
 311 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P,
 312                                 const mbedtls_mpi *Q, const mbedtls_mpi *D,
 313                                 const mbedtls_mpi *E,
 314                                 int (*f_rng)(void *, unsigned char *, size_t),
 315                                 void *p_rng) {
 316     int ret = 0;
 317     mbedtls_mpi K, L;
 318
 319     mbedtls_mpi_init(&K);
 320     mbedtls_mpi_init(&L);
 321
 322     /*
 323      * Step 1: If PRNG provided, check that P and Q are prime
 324      */
 325
 326 #if defined(MBEDTLS_GENPRIME)
 327     /*
 328      * When generating keys, the strongest security we support aims for an error
 329      * rate of at most 2^-100 and we are aiming for the same certainty here as
 330      * well.
 331      */
 332     if (f_rng != NULL && P != NULL &&
 333             (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
 334         ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 335         goto cleanup;
 336     }
 337
 338     if (f_rng != NULL && Q != NULL &&
 339             (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
 340         ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 341         goto cleanup;
 342     }
 343 #else
 344     ((void) f_rng);
 345     ((void) p_rng);
 346 #endif /* MBEDTLS_GENPRIME */
 347
 348     /*
 349      * Step 2: Check that 1 < N = P * Q
 350      */
 351
 352     if (P != NULL && Q != NULL && N != NULL) {
 353         MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
 354         if (mbedtls_mpi_cmp_int(N, 1)  <= 0 ||
 355                 mbedtls_mpi_cmp_mpi(&K, N) != 0) {
 356             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 357             goto cleanup;
 358         }
 359     }
 360
 361     /*
 362      * Step 3: Check and 1 < D, E < N if present.
 363      */
 364
 365     if (N != NULL && D != NULL && E != NULL) {
 366         if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
 367                 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
 368                 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
 369                 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
 370             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 371             goto cleanup;
 372         }
 373     }
 374
 375     /*
 376      * Step 4: Check that D, E are inverse modulo P-1 and Q-1
 377      */
 378
 379     if (P != NULL && Q != NULL && D != NULL && E != NULL) {
 380         if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
 381                 mbedtls_mpi_cmp_int(Q, 1) <= 0) {
 382             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 383             goto cleanup;
 384         }
 385
 386         /* Compute DE-1 mod P-1 */
 387         MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
 388         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
 389         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
 390         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
 391         if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
 392             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 393             goto cleanup;
 394         }
 395
 396         /* Compute DE-1 mod Q-1 */
 397         MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
 398         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
 399         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
 400         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
 401         if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
 402             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 403             goto cleanup;
 404         }
 405     }
 406
 407 cleanup:
 408
 409     mbedtls_mpi_free(&K);
 410     mbedtls_mpi_free(&L);
 411
 412     /* Wrap MPI error codes by RSA check failure error code */
 413     if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
 414         ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
 415     }
 416
 417     return (ret);
 418 }
 419
 420 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
 421                            const mbedtls_mpi *D, mbedtls_mpi *DP,
 422                            mbedtls_mpi *DQ, mbedtls_mpi *QP) {
 423     int ret = 0;
 424     mbedtls_mpi K;
 425     mbedtls_mpi_init(&K);
 426
 427     /* DP = D mod P-1 */
 428     if (DP != NULL) {
 429         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
 430         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
 431     }
 432
 433     /* DQ = D mod Q-1 */
 434     if (DQ != NULL) {
 435         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
 436         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
 437     }
 438
 439     /* QP = Q^{-1} mod P */
 440     if (QP != NULL) {
 441         MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
 442     }
 443
 444 cleanup:
 445     mbedtls_mpi_free(&K);
 446
 447     return (ret);
 448 }
 449
 450 #endif /* MBEDTLS_RSA_C */