Line data Source code
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 : */
51 2 : 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 2 : 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 2 : 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 2 : const size_t num_primes = sizeof(primes) / sizeof(*primes);
75 :
76 2 : if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) {
77 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
78 : }
79 :
80 4 : if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
81 4 : mbedtls_mpi_cmp_int(D, 1) <= 0 ||
82 4 : mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
83 4 : mbedtls_mpi_cmp_int(E, 1) <= 0 ||
84 2 : mbedtls_mpi_cmp_mpi(E, N) >= 0) {
85 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
86 : }
87 :
88 : /*
89 : * Initializations and temporary changes
90 : */
91 :
92 2 : mbedtls_mpi_init(&K);
93 2 : mbedtls_mpi_init(&T);
94 :
95 : /* T := DE - 1 */
96 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
97 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
98 :
99 2 : if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
100 0 : ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
101 0 : goto cleanup;
102 : }
103 :
104 : /* After this operation, T holds the largest odd divisor of DE - 1. */
105 2 : 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 2 : attempt = 0;
113 2 : if (N->p[0] % 8 == 1) {
114 0 : attempt = 1;
115 : }
116 :
117 2 : for (; attempt < num_primes; ++attempt) {
118 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt]));
119 :
120 : /* Check if gcd(K,N) = 1 */
121 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N));
122 2 : if (mbedtls_mpi_cmp_int(P, 1) != 0) {
123 0 : 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 2 : 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 2 : for (iter = 1; iter <= order; ++iter) {
133 : /* If we reach 1 prematurely, there's no point
134 : * in continuing to square K */
135 2 : if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
136 0 : break;
137 : }
138 :
139 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
140 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd_modinv_odd(P, NULL, &K, N));
141 :
142 4 : if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
143 2 : 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 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
150 2 : goto cleanup;
151 : }
152 :
153 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
154 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
155 0 : 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 0 : if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
166 0 : break;
167 : }
168 : }
169 :
170 0 : ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
171 :
172 2 : cleanup:
173 :
174 2 : mbedtls_mpi_free(&K);
175 2 : mbedtls_mpi_free(&T);
176 2 : return ret;
177 : }
178 :
179 : /*
180 : * Given P, Q and the public exponent E, deduce D.
181 : * This is essentially a modular inversion.
182 : */
183 0 : 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 0 : int ret = 0;
189 : mbedtls_mpi K, L;
190 :
191 0 : if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
192 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
193 : }
194 :
195 0 : if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
196 0 : mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
197 0 : mbedtls_mpi_cmp_int(E, 0) == 0) {
198 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
199 : }
200 :
201 0 : if (mbedtls_mpi_get_bit(E, 0) != 1) {
202 0 : return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
203 : }
204 :
205 0 : mbedtls_mpi_init(&K);
206 0 : mbedtls_mpi_init(&L);
207 :
208 : /* Temporarily put K := P-1 and L := Q-1 */
209 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
210 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
211 :
212 : /* Temporarily put D := gcd(P-1, Q-1) */
213 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
214 :
215 : /* K := LCM(P-1, Q-1) */
216 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
217 0 : 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 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_even_in_range(D, E, &K));
224 :
225 0 : cleanup:
226 :
227 0 : mbedtls_mpi_free(&K);
228 0 : mbedtls_mpi_free(&L);
229 :
230 0 : return ret;
231 : }
232 :
233 2 : 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 2 : int ret = 0;
238 : mbedtls_mpi K;
239 2 : mbedtls_mpi_init(&K);
240 :
241 : /* DP = D mod P-1 */
242 2 : if (DP != NULL) {
243 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
244 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
245 : }
246 :
247 : /* DQ = D mod Q-1 */
248 2 : if (DQ != NULL) {
249 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
250 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
251 : }
252 :
253 : /* QP = Q^{-1} mod P */
254 2 : if (QP != NULL) {
255 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod_odd(QP, Q, P));
256 : }
257 :
258 2 : cleanup:
259 2 : mbedtls_mpi_free(&K);
260 :
261 2 : return ret;
262 : }
263 :
264 : /*
265 : * Check that core RSA parameters are sane.
266 : */
267 0 : 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 0 : int ret = 0;
274 : mbedtls_mpi K, L;
275 :
276 0 : mbedtls_mpi_init(&K);
277 0 : 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 0 : if (f_rng != NULL && P != NULL &&
290 0 : (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
291 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
292 0 : goto cleanup;
293 : }
294 :
295 0 : if (f_rng != NULL && Q != NULL &&
296 0 : (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
297 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
298 0 : 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 0 : if (P != NULL && Q != NULL && N != NULL) {
310 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
311 0 : if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
312 0 : mbedtls_mpi_cmp_mpi(&K, N) != 0) {
313 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
314 0 : goto cleanup;
315 : }
316 : }
317 :
318 : /*
319 : * Step 3: Check and 1 < D, E < N if present.
320 : */
321 :
322 0 : if (N != NULL && D != NULL && E != NULL) {
323 0 : if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
324 0 : mbedtls_mpi_cmp_int(E, 1) <= 0 ||
325 0 : mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
326 0 : mbedtls_mpi_cmp_mpi(E, N) >= 0) {
327 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
328 0 : goto cleanup;
329 : }
330 : }
331 :
332 : /*
333 : * Step 4: Check that D, E are inverse modulo P-1 and Q-1
334 : */
335 :
336 0 : if (P != NULL && Q != NULL && D != NULL && E != NULL) {
337 0 : if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
338 0 : mbedtls_mpi_cmp_int(Q, 1) <= 0) {
339 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
340 0 : goto cleanup;
341 : }
342 :
343 : /* Compute DE-1 mod P-1 */
344 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
345 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
346 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
347 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
348 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
349 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
350 0 : goto cleanup;
351 : }
352 :
353 : /* Compute DE-1 mod Q-1 */
354 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
355 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
356 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
357 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
358 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
359 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
360 0 : goto cleanup;
361 : }
362 : }
363 :
364 0 : cleanup:
365 :
366 0 : mbedtls_mpi_free(&K);
367 0 : mbedtls_mpi_free(&L);
368 :
369 : /* Wrap MPI error codes by RSA check failure error code */
370 0 : if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
371 0 : ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
372 : }
373 :
374 0 : return ret;
375 : }
376 :
377 : /*
378 : * Check that RSA CRT parameters are in accordance with core parameters.
379 : */
380 0 : 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 0 : int ret = 0;
385 :
386 : mbedtls_mpi K, L;
387 0 : mbedtls_mpi_init(&K);
388 0 : mbedtls_mpi_init(&L);
389 :
390 : /* Check that DP - D == 0 mod P - 1 */
391 0 : if (DP != NULL) {
392 0 : if (P == NULL) {
393 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
394 0 : goto cleanup;
395 : }
396 :
397 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
398 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
399 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
400 :
401 0 : if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
402 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
403 0 : goto cleanup;
404 : }
405 : }
406 :
407 : /* Check that DQ - D == 0 mod Q - 1 */
408 0 : if (DQ != NULL) {
409 0 : if (Q == NULL) {
410 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
411 0 : goto cleanup;
412 : }
413 :
414 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
415 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
416 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
417 :
418 0 : if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
419 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
420 0 : goto cleanup;
421 : }
422 : }
423 :
424 : /* Check that QP * Q - 1 == 0 mod P */
425 0 : if (QP != NULL) {
426 0 : if (P == NULL || Q == NULL) {
427 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
428 0 : goto cleanup;
429 : }
430 :
431 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
432 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
433 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
434 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
435 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
436 0 : goto cleanup;
437 : }
438 : }
439 :
440 0 : cleanup:
441 :
442 : /* Wrap MPI error codes by RSA check failure error code */
443 0 : if (ret != 0 &&
444 0 : ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
445 : ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
446 0 : ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
447 : }
448 :
449 0 : mbedtls_mpi_free(&K);
450 0 : mbedtls_mpi_free(&L);
451 :
452 0 : return ret;
453 : }
454 :
455 : #endif /* MBEDTLS_RSA_C */
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