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 "bignum_internal.h"
16 #include "rsa_alt_helpers.h"
17
18 /*
19 * Compute RSA prime factors from public and private exponents
20 *
21 * Summary of algorithm:
22 * Setting F := lcm(P-1,Q-1), the idea is as follows:
23 *
24 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2)
25 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the
26 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four
27 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1)
28 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime
29 * factors of N.
30 *
31 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same
32 * construction still applies since (-)^K is the identity on the set of
33 * roots of 1 in Z/NZ.
34 *
35 * The public and private key primitives (-)^E and (-)^D are mutually inverse
36 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e.
37 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L.
38 * Splitting L = 2^t * K with K odd, we have
39 *
40 * DE - 1 = FL = (F/2) * (2^(t+1)) * K,
41 *
42 * so (F / 2) * K is among the numbers
43 *
44 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord
45 *
46 * where ord is the order of 2 in (DE - 1).
47 * We can therefore iterate through these numbers apply the construction
48 * of (a) and (b) above to attempt to factor N.
49 *
50 */
mbedtls_rsa_deduce_primes(mbedtls_mpi const * N,mbedtls_mpi const * E,mbedtls_mpi const * D,mbedtls_mpi * P,mbedtls_mpi * Q)51 int mbedtls_rsa_deduce_primes(mbedtls_mpi const *N,
52 mbedtls_mpi const *E, mbedtls_mpi const *D,
53 mbedtls_mpi *P, mbedtls_mpi *Q)
54 {
55 int ret = 0;
56
57 uint16_t attempt; /* Number of current attempt */
58 uint16_t iter; /* Number of squares computed in the current attempt */
59
60 uint16_t order; /* Order of 2 in DE - 1 */
61
62 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */
63 mbedtls_mpi K; /* Temporary holding the current candidate */
64
65 const unsigned char primes[] = { 2,
66 3, 5, 7, 11, 13, 17, 19, 23,
67 29, 31, 37, 41, 43, 47, 53, 59,
68 61, 67, 71, 73, 79, 83, 89, 97,
69 101, 103, 107, 109, 113, 127, 131, 137,
70 139, 149, 151, 157, 163, 167, 173, 179,
71 181, 191, 193, 197, 199, 211, 223, 227,
72 229, 233, 239, 241, 251 };
73
74 const size_t num_primes = sizeof(primes) / sizeof(*primes);
75
76 if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) {
77 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
78 }
79
80 if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
81 mbedtls_mpi_cmp_int(D, 1) <= 0 ||
82 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
83 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
84 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
85 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
86 }
87
88 /*
89 * Initializations and temporary changes
90 */
91
92 mbedtls_mpi_init(&K);
93 mbedtls_mpi_init(&T);
94
95 /* T := DE - 1 */
96 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
97 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
98
99 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
100 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
101 goto cleanup;
102 }
103
104 /* After this operation, T holds the largest odd divisor of DE - 1. */
105 MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order));
106
107 /*
108 * Actual work
109 */
110
111 /* Skip trying 2 if N == 1 mod 8 */
112 attempt = 0;
113 if (N->p[0] % 8 == 1) {
114 attempt = 1;
115 }
116
117 for (; attempt < num_primes; ++attempt) {
118 MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt]));
119
120 /* Check if gcd(K,N) = 1 */
121 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N));
122 if (mbedtls_mpi_cmp_int(P, 1) != 0) {
123 continue;
124 }
125
126 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
127 * and check whether they have nontrivial GCD with N. */
128 MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N,
129 Q /* temporarily use Q for storing Montgomery
130 * multiplication helper values */));
131
132 for (iter = 1; iter <= order; ++iter) {
133 /* If we reach 1 prematurely, there's no point
134 * in continuing to square K */
135 if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
136 break;
137 }
138
139 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
140 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N));
141
142 if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
143 mbedtls_mpi_cmp_mpi(P, N) == -1) {
144 /*
145 * Have found a nontrivial divisor P of N.
146 * Set Q := N / P.
147 */
148
149 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
150 goto cleanup;
151 }
152
153 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
154 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
155 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N));
156 }
157
158 /*
159 * If we get here, then either we prematurely aborted the loop because
160 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
161 * be 1 if D,E,N were consistent.
162 * Check if that's the case and abort if not, to avoid very long,
163 * yet eventually failing, computations if N,D,E were not sane.
164 */
165 if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
166 break;
167 }
168 }
169
170 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
171
172 cleanup:
173
174 mbedtls_mpi_free(&K);
175 mbedtls_mpi_free(&T);
176 return ret;
177 }
178
179 /*
180 * Given P, Q and the public exponent E, deduce D.
181 * This is essentially a modular inversion.
182 */
mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const * P,mbedtls_mpi const * Q,mbedtls_mpi const * E,mbedtls_mpi * D)183 int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P,
184 mbedtls_mpi const *Q,
185 mbedtls_mpi const *E,
186 mbedtls_mpi *D)
187 {
188 int ret = 0;
189 mbedtls_mpi K, L;
190
191 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
192 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
193 }
194
195 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
196 mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
197 mbedtls_mpi_cmp_int(E, 0) == 0) {
198 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
199 }
200
201 if (mbedtls_mpi_get_bit(E, 0) != 1) {
202 return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
203 }
204
205 mbedtls_mpi_init(&K);
206 mbedtls_mpi_init(&L);
207
208 /* Temporarily put K := P-1 and L := Q-1 */
209 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
210 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
211
212 /* Temporarily put D := gcd(P-1, Q-1) */
213 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
214
215 /* K := LCM(P-1, Q-1) */
216 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
217 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D));
218
219 /* Compute modular inverse of E mod LCM(P-1, Q-1)
220 * This is FIPS 186-4 §B.3.1 criterion 3(b).
221 * This will return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if E is not coprime to
222 * (P-1)(Q-1), also validating FIPS 186-4 §B.3.1 criterion 2(a). */
223 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_even_in_range(D, E, &K));
224
225 cleanup:
226
227 mbedtls_mpi_free(&K);
228 mbedtls_mpi_free(&L);
229
230 return ret;
231 }
232
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)233 int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
234 const mbedtls_mpi *D, mbedtls_mpi *DP,
235 mbedtls_mpi *DQ, mbedtls_mpi *QP)
236 {
237 int ret = 0;
238 mbedtls_mpi K;
239 mbedtls_mpi_init(&K);
240
241 /* DP = D mod P-1 */
242 if (DP != NULL) {
243 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
244 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
245 }
246
247 /* DQ = D mod Q-1 */
248 if (DQ != NULL) {
249 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
250 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
251 }
252
253 /* QP = Q^{-1} mod P */
254 if (QP != NULL) {
255 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_odd(QP, Q, P));
256 }
257
258 cleanup:
259 mbedtls_mpi_free(&K);
260
261 return ret;
262 }
263
264 /*
265 * Check that core RSA parameters are sane.
266 */
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)267 int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P,
268 const mbedtls_mpi *Q, const mbedtls_mpi *D,
269 const mbedtls_mpi *E,
270 int (*f_rng)(void *, unsigned char *, size_t),
271 void *p_rng)
272 {
273 int ret = 0;
274 mbedtls_mpi K, L;
275
276 mbedtls_mpi_init(&K);
277 mbedtls_mpi_init(&L);
278
279 /*
280 * Step 1: If PRNG provided, check that P and Q are prime
281 */
282
283 #if defined(MBEDTLS_GENPRIME)
284 /*
285 * When generating keys, the strongest security we support aims for an error
286 * rate of at most 2^-100 and we are aiming for the same certainty here as
287 * well.
288 */
289 if (f_rng != NULL && P != NULL &&
290 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
291 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
292 goto cleanup;
293 }
294
295 if (f_rng != NULL && Q != NULL &&
296 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
297 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
298 goto cleanup;
299 }
300 #else
301 ((void) f_rng);
302 ((void) p_rng);
303 #endif /* MBEDTLS_GENPRIME */
304
305 /*
306 * Step 2: Check that 1 < N = P * Q
307 */
308
309 if (P != NULL && Q != NULL && N != NULL) {
310 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
311 if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
312 mbedtls_mpi_cmp_mpi(&K, N) != 0) {
313 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
314 goto cleanup;
315 }
316 }
317
318 /*
319 * Step 3: Check and 1 < D, E < N if present.
320 */
321
322 if (N != NULL && D != NULL && E != NULL) {
323 if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
324 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
325 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
326 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
327 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
328 goto cleanup;
329 }
330 }
331
332 /*
333 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
334 */
335
336 if (P != NULL && Q != NULL && D != NULL && E != NULL) {
337 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
338 mbedtls_mpi_cmp_int(Q, 1) <= 0) {
339 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
340 goto cleanup;
341 }
342
343 /* Compute DE-1 mod P-1 */
344 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
345 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
346 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
347 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
348 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
349 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
350 goto cleanup;
351 }
352
353 /* Compute DE-1 mod Q-1 */
354 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
355 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
356 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
357 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
358 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
359 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
360 goto cleanup;
361 }
362 }
363
364 cleanup:
365
366 mbedtls_mpi_free(&K);
367 mbedtls_mpi_free(&L);
368
369 /* Wrap MPI error codes by RSA check failure error code */
370 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
371 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
372 }
373
374 return ret;
375 }
376
377 /*
378 * Check that RSA CRT parameters are in accordance with core parameters.
379 */
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)380 int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
381 const mbedtls_mpi *D, const mbedtls_mpi *DP,
382 const mbedtls_mpi *DQ, const mbedtls_mpi *QP)
383 {
384 int ret = 0;
385
386 mbedtls_mpi K, L;
387 mbedtls_mpi_init(&K);
388 mbedtls_mpi_init(&L);
389
390 /* Check that DP - D == 0 mod P - 1 */
391 if (DP != NULL) {
392 if (P == NULL) {
393 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
394 goto cleanup;
395 }
396
397 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
398 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
399 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
400
401 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
402 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
403 goto cleanup;
404 }
405 }
406
407 /* Check that DQ - D == 0 mod Q - 1 */
408 if (DQ != NULL) {
409 if (Q == NULL) {
410 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
411 goto cleanup;
412 }
413
414 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
415 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
416 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
417
418 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
419 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
420 goto cleanup;
421 }
422 }
423
424 /* Check that QP * Q - 1 == 0 mod P */
425 if (QP != NULL) {
426 if (P == NULL || Q == NULL) {
427 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
428 goto cleanup;
429 }
430
431 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
432 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
433 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
434 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
435 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
436 goto cleanup;
437 }
438 }
439
440 cleanup:
441
442 /* Wrap MPI error codes by RSA check failure error code */
443 if (ret != 0 &&
444 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
445 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
446 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
447 }
448
449 mbedtls_mpi_free(&K);
450 mbedtls_mpi_free(&L);
451
452 return ret;
453 }
454
455 #endif /* MBEDTLS_RSA_C */
456