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 */
50int 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
171cleanup:
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 */
182int 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
217cleanup:
218
219    mbedtls_mpi_free(&K);
220    mbedtls_mpi_free(&L);
221
222    return ret;
223}
224
225int 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
250cleanup:
251    mbedtls_mpi_free(&K);
252
253    return ret;
254}
255
256/*
257 * Check that core RSA parameters are sane.
258 */
259int 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
356cleanup:
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 */
372int 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
432cleanup:
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 */
448