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 "rsa_alt_helpers.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 */
mbedtls_rsa_deduce_primes(mbedtls_mpi const * N,mbedtls_mpi const * E,mbedtls_mpi const * D,mbedtls_mpi * P,mbedtls_mpi * Q)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 {
66 int ret = 0;
67
68 uint16_t attempt; /* Number of current attempt */
69 uint16_t iter; /* Number of squares computed in the current attempt */
70
71 uint16_t order; /* Order of 2 in DE - 1 */
72
73 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */
74 mbedtls_mpi K; /* Temporary holding the current candidate */
75
76 const unsigned char primes[] = { 2,
77 3, 5, 7, 11, 13, 17, 19, 23,
78 29, 31, 37, 41, 43, 47, 53, 59,
79 61, 67, 71, 73, 79, 83, 89, 97,
80 101, 103, 107, 109, 113, 127, 131, 137,
81 139, 149, 151, 157, 163, 167, 173, 179,
82 181, 191, 193, 197, 199, 211, 223, 227,
83 229, 233, 239, 241, 251 };
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
91 if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
92 mbedtls_mpi_cmp_int(D, 1) <= 0 ||
93 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
94 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
95 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
96 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
97 }
98
99 /*
100 * Initializations and temporary changes
101 */
102
103 mbedtls_mpi_init(&K);
104 mbedtls_mpi_init(&T);
105
106 /* T := DE - 1 */
107 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
108 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
109
110 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
111 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
112 goto cleanup;
113 }
114
115 /* After this operation, T holds the largest odd divisor of DE - 1. */
116 MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order));
117
118 /*
119 * Actual work
120 */
121
122 /* Skip trying 2 if N == 1 mod 8 */
123 attempt = 0;
124 if (N->p[0] % 8 == 1) {
125 attempt = 1;
126 }
127
128 for (; attempt < num_primes; ++attempt) {
129 mbedtls_mpi_lset(&K, primes[attempt]);
130
131 /* Check if gcd(K,N) = 1 */
132 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
133 if (mbedtls_mpi_cmp_int(P, 1) != 0) {
134 continue;
135 }
136
137 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
138 * and check whether they have nontrivial GCD with N. */
139 MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N,
140 Q /* temporarily use Q for storing Montgomery
141 * multiplication helper values */));
142
143 for (iter = 1; iter <= order; ++iter) {
144 /* If we reach 1 prematurely, there's no point
145 * in continuing to square K */
146 if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
147 break;
148 }
149
150 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
151 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
152
153 if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
154 mbedtls_mpi_cmp_mpi(P, N) == -1) {
155 /*
156 * Have found a nontrivial divisor P of N.
157 * Set Q := N / P.
158 */
159
160 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
161 goto cleanup;
162 }
163
164 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
165 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
166 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N));
167 }
168
169 /*
170 * If we get here, then either we prematurely aborted the loop because
171 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
172 * be 1 if D,E,N were consistent.
173 * Check if that's the case and abort if not, to avoid very long,
174 * yet eventually failing, computations if N,D,E were not sane.
175 */
176 if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
177 break;
178 }
179 }
180
181 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
182
183 cleanup:
184
185 mbedtls_mpi_free(&K);
186 mbedtls_mpi_free(&T);
187 return ret;
188 }
189
190 /*
191 * Given P, Q and the public exponent E, deduce D.
192 * This is essentially a modular inversion.
193 */
mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const * P,mbedtls_mpi const * Q,mbedtls_mpi const * E,mbedtls_mpi * D)194 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P,
195 mbedtls_mpi const *Q,
196 mbedtls_mpi const *E,
197 mbedtls_mpi *D)
198 {
199 int ret = 0;
200 mbedtls_mpi K, L;
201
202 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
203 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
204 }
205
206 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
207 mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
208 mbedtls_mpi_cmp_int(E, 0) == 0) {
209 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
210 }
211
212 mbedtls_mpi_init(&K);
213 mbedtls_mpi_init(&L);
214
215 /* Temporarily put K := P-1 and L := Q-1 */
216 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
217 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
218
219 /* Temporarily put D := gcd(P-1, Q-1) */
220 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
221
222 /* K := LCM(P-1, Q-1) */
223 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
224 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D));
225
226 /* Compute modular inverse of E in LCM(P-1, Q-1) */
227 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K));
228
229 cleanup:
230
231 mbedtls_mpi_free(&K);
232 mbedtls_mpi_free(&L);
233
234 return ret;
235 }
236
mbedtls_rsa_deduce_crt(const mbedtls_mpi * P,const mbedtls_mpi * Q,const mbedtls_mpi * D,mbedtls_mpi * DP,mbedtls_mpi * DQ,mbedtls_mpi * QP)237 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
238 const mbedtls_mpi *D, mbedtls_mpi *DP,
239 mbedtls_mpi *DQ, mbedtls_mpi *QP)
240 {
241 int ret = 0;
242 mbedtls_mpi K;
243 mbedtls_mpi_init(&K);
244
245 /* DP = D mod P-1 */
246 if (DP != NULL) {
247 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
248 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
249 }
250
251 /* DQ = D mod Q-1 */
252 if (DQ != NULL) {
253 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
254 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
255 }
256
257 /* QP = Q^{-1} mod P */
258 if (QP != NULL) {
259 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
260 }
261
262 cleanup:
263 mbedtls_mpi_free(&K);
264
265 return ret;
266 }
267
268 /*
269 * Check that core RSA parameters are sane.
270 */
mbedtls_rsa_validate_params(const mbedtls_mpi * N,const mbedtls_mpi * P,const mbedtls_mpi * Q,const mbedtls_mpi * D,const mbedtls_mpi * E,int (* f_rng)(void *,unsigned char *,size_t),void * p_rng)271 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P,
272 const mbedtls_mpi *Q, const mbedtls_mpi *D,
273 const mbedtls_mpi *E,
274 int (*f_rng)(void *, unsigned char *, size_t),
275 void *p_rng)
276 {
277 int ret = 0;
278 mbedtls_mpi K, L;
279
280 mbedtls_mpi_init(&K);
281 mbedtls_mpi_init(&L);
282
283 /*
284 * Step 1: If PRNG provided, check that P and Q are prime
285 */
286
287 #if defined(MBEDTLS_GENPRIME)
288 /*
289 * When generating keys, the strongest security we support aims for an error
290 * rate of at most 2^-100 and we are aiming for the same certainty here as
291 * well.
292 */
293 if (f_rng != NULL && P != NULL &&
294 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
295 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
296 goto cleanup;
297 }
298
299 if (f_rng != NULL && Q != NULL &&
300 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
301 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
302 goto cleanup;
303 }
304 #else
305 ((void) f_rng);
306 ((void) p_rng);
307 #endif /* MBEDTLS_GENPRIME */
308
309 /*
310 * Step 2: Check that 1 < N = P * Q
311 */
312
313 if (P != NULL && Q != NULL && N != NULL) {
314 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
315 if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
316 mbedtls_mpi_cmp_mpi(&K, N) != 0) {
317 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
318 goto cleanup;
319 }
320 }
321
322 /*
323 * Step 3: Check and 1 < D, E < N if present.
324 */
325
326 if (N != NULL && D != NULL && E != NULL) {
327 if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
328 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
329 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
330 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
331 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
332 goto cleanup;
333 }
334 }
335
336 /*
337 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
338 */
339
340 if (P != NULL && Q != NULL && D != NULL && E != NULL) {
341 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
342 mbedtls_mpi_cmp_int(Q, 1) <= 0) {
343 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
344 goto cleanup;
345 }
346
347 /* Compute DE-1 mod P-1 */
348 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
349 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
350 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
351 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
352 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
353 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
354 goto cleanup;
355 }
356
357 /* Compute DE-1 mod Q-1 */
358 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
359 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
360 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
361 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
362 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
363 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
364 goto cleanup;
365 }
366 }
367
368 cleanup:
369
370 mbedtls_mpi_free(&K);
371 mbedtls_mpi_free(&L);
372
373 /* Wrap MPI error codes by RSA check failure error code */
374 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
375 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
376 }
377
378 return ret;
379 }
380
381 /*
382 * Check that RSA CRT parameters are in accordance with core parameters.
383 */
mbedtls_rsa_validate_crt(const mbedtls_mpi * P,const mbedtls_mpi * Q,const mbedtls_mpi * D,const mbedtls_mpi * DP,const mbedtls_mpi * DQ,const mbedtls_mpi * QP)384 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
385 const mbedtls_mpi *D, const mbedtls_mpi *DP,
386 const mbedtls_mpi *DQ, const mbedtls_mpi *QP)
387 {
388 int ret = 0;
389
390 mbedtls_mpi K, L;
391 mbedtls_mpi_init(&K);
392 mbedtls_mpi_init(&L);
393
394 /* Check that DP - D == 0 mod P - 1 */
395 if (DP != NULL) {
396 if (P == NULL) {
397 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
398 goto cleanup;
399 }
400
401 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
402 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
403 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
404
405 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
406 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
407 goto cleanup;
408 }
409 }
410
411 /* Check that DQ - D == 0 mod Q - 1 */
412 if (DQ != NULL) {
413 if (Q == NULL) {
414 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
415 goto cleanup;
416 }
417
418 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
419 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
420 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
421
422 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
423 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
424 goto cleanup;
425 }
426 }
427
428 /* Check that QP * Q - 1 == 0 mod P */
429 if (QP != NULL) {
430 if (P == NULL || Q == NULL) {
431 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
432 goto cleanup;
433 }
434
435 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
436 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
437 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
438 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
439 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
440 goto cleanup;
441 }
442 }
443
444 cleanup:
445
446 /* Wrap MPI error codes by RSA check failure error code */
447 if (ret != 0 &&
448 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
449 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
450 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
451 }
452
453 mbedtls_mpi_free(&K);
454 mbedtls_mpi_free(&L);
455
456 return ret;
457 }
458
459 #endif /* MBEDTLS_RSA_C */
460