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 */
62int 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
183cleanup:
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 */
194int 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
229cleanup:
230
231 mbedtls_mpi_free(&K);
232 mbedtls_mpi_free(&L);
233
234 return ret;
235}
236
237/*
238 * Check that RSA CRT parameters are in accordance with core parameters.
239 */
240int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
241 const mbedtls_mpi *D, const mbedtls_mpi *DP,
242 const mbedtls_mpi *DQ, const mbedtls_mpi *QP)
243{
244 int ret = 0;
245
246 mbedtls_mpi K, L;
247 mbedtls_mpi_init(&K);
248 mbedtls_mpi_init(&L);
249
250 /* Check that DP - D == 0 mod P - 1 */
251 if (DP != NULL) {
252 if (P == NULL) {
253 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
254 goto cleanup;
255 }
256
257 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
258 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
259 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
260
261 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
262 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
263 goto cleanup;
264 }
265 }
266
267 /* Check that DQ - D == 0 mod Q - 1 */
268 if (DQ != NULL) {
269 if (Q == NULL) {
270 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
271 goto cleanup;
272 }
273
274 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
275 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
276 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
277
278 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
279 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
280 goto cleanup;
281 }
282 }
283
284 /* Check that QP * Q - 1 == 0 mod P */
285 if (QP != NULL) {
286 if (P == NULL || Q == NULL) {
287 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
288 goto cleanup;
289 }
290
291 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
292 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
293 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
294 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
295 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
296 goto cleanup;
297 }
298 }
299
300cleanup:
301
302 /* Wrap MPI error codes by RSA check failure error code */
303 if (ret != 0 &&
304 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
305 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
306 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
307 }
308
309 mbedtls_mpi_free(&K);
310 mbedtls_mpi_free(&L);
311
312 return ret;
313}
314
315/*
316 * Check that core RSA parameters are sane.
317 */
318int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P,
319 const mbedtls_mpi *Q, const mbedtls_mpi *D,
320 const mbedtls_mpi *E,
321 int (*f_rng)(void *, unsigned char *, size_t),
322 void *p_rng)
323{
324 int ret = 0;
325 mbedtls_mpi K, L;
326
327 mbedtls_mpi_init(&K);
328 mbedtls_mpi_init(&L);
329
330 /*
331 * Step 1: If PRNG provided, check that P and Q are prime
332 */
333
334#if defined(MBEDTLS_GENPRIME)
335 /*
336 * When generating keys, the strongest security we support aims for an error
337 * rate of at most 2^-100 and we are aiming for the same certainty here as
338 * well.
339 */
340 if (f_rng != NULL && P != NULL &&
341 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
342 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
343 goto cleanup;
344 }
345
346 if (f_rng != NULL && Q != NULL &&
347 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
348 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
349 goto cleanup;
350 }
351#else
352 ((void) f_rng);
353 ((void) p_rng);
354#endif /* MBEDTLS_GENPRIME */
355
356 /*
357 * Step 2: Check that 1 < N = P * Q
358 */
359
360 if (P != NULL && Q != NULL && N != NULL) {
361 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
362 if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
363 mbedtls_mpi_cmp_mpi(&K, N) != 0) {
364 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
365 goto cleanup;
366 }
367 }
368
369 /*
370 * Step 3: Check and 1 < D, E < N if present.
371 */
372
373 if (N != NULL && D != NULL && E != NULL) {
374 if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
375 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
376 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
377 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
378 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
379 goto cleanup;
380 }
381 }
382
383 /*
384 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
385 */
386
387 if (P != NULL && Q != NULL && D != NULL && E != NULL) {
388 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
389 mbedtls_mpi_cmp_int(Q, 1) <= 0) {
390 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
391 goto cleanup;
392 }
393
394 /* Compute DE-1 mod P-1 */
395 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
396 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
397 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
398 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
399 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
400 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
401 goto cleanup;
402 }
403
404 /* Compute DE-1 mod Q-1 */
405 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
406 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
407 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
408 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
409 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
410 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
411 goto cleanup;
412 }
413 }
414
415cleanup:
416
417 mbedtls_mpi_free(&K);
418 mbedtls_mpi_free(&L);
419
420 /* Wrap MPI error codes by RSA check failure error code */
421 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
422 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
423 }
424
425 return ret;
426}
427
428int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
429 const mbedtls_mpi *D, mbedtls_mpi *DP,
430 mbedtls_mpi *DQ, mbedtls_mpi *QP)
431{
432 int ret = 0;
433 mbedtls_mpi K;
434 mbedtls_mpi_init(&K);
435
436 /* DP = D mod P-1 */
437 if (DP != NULL) {
438 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
439 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
440 }
441
442 /* DQ = D mod Q-1 */
443 if (DQ != NULL) {
444 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
445 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
446 }
447
448 /* QP = Q^{-1} mod P */
449 if (QP != NULL) {
450 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
451 }
452
453cleanup:
454 mbedtls_mpi_free(&K);
455
456 return ret;
457}
458
459#endif /* MBEDTLS_RSA_C */
460