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 "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 : */
50 2 : int 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 2 : 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 2 : 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 2 : const size_t num_primes = sizeof(primes) / sizeof(*primes);
74 :
75 2 : if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) {
76 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
77 : }
78 :
79 4 : if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
80 4 : mbedtls_mpi_cmp_int(D, 1) <= 0 ||
81 4 : mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
82 4 : mbedtls_mpi_cmp_int(E, 1) <= 0 ||
83 2 : mbedtls_mpi_cmp_mpi(E, N) >= 0) {
84 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
85 : }
86 :
87 : /*
88 : * Initializations and temporary changes
89 : */
90 :
91 2 : mbedtls_mpi_init(&K);
92 2 : mbedtls_mpi_init(&T);
93 :
94 : /* T := DE - 1 */
95 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
96 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
97 :
98 2 : if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
99 0 : ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
100 0 : goto cleanup;
101 : }
102 :
103 : /* After this operation, T holds the largest odd divisor of DE - 1. */
104 2 : 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 2 : attempt = 0;
112 2 : if (N->p[0] % 8 == 1) {
113 0 : attempt = 1;
114 : }
115 :
116 2 : for (; attempt < num_primes; ++attempt) {
117 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt]));
118 :
119 : /* Check if gcd(K,N) = 1 */
120 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
121 2 : if (mbedtls_mpi_cmp_int(P, 1) != 0) {
122 0 : 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 2 : 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 2 : for (iter = 1; iter <= order; ++iter) {
132 : /* If we reach 1 prematurely, there's no point
133 : * in continuing to square K */
134 2 : if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
135 0 : break;
136 : }
137 :
138 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
139 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
140 :
141 4 : if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
142 2 : 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 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
149 2 : goto cleanup;
150 : }
151 :
152 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
153 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
154 0 : 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 0 : if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
165 0 : break;
166 : }
167 : }
168 :
169 0 : ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
170 :
171 2 : cleanup:
172 :
173 2 : mbedtls_mpi_free(&K);
174 2 : mbedtls_mpi_free(&T);
175 2 : return ret;
176 : }
177 :
178 : /*
179 : * Given P, Q and the public exponent E, deduce D.
180 : * This is essentially a modular inversion.
181 : */
182 0 : int 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 0 : int ret = 0;
188 : mbedtls_mpi K, L;
189 :
190 0 : if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
191 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
192 : }
193 :
194 0 : if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
195 0 : mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
196 0 : mbedtls_mpi_cmp_int(E, 0) == 0) {
197 0 : return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
198 : }
199 :
200 0 : mbedtls_mpi_init(&K);
201 0 : mbedtls_mpi_init(&L);
202 :
203 : /* Temporarily put K := P-1 and L := Q-1 */
204 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
205 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
206 :
207 : /* Temporarily put D := gcd(P-1, Q-1) */
208 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
209 :
210 : /* K := LCM(P-1, Q-1) */
211 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
212 0 : 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 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K));
216 :
217 0 : cleanup:
218 :
219 0 : mbedtls_mpi_free(&K);
220 0 : mbedtls_mpi_free(&L);
221 :
222 0 : return ret;
223 : }
224 :
225 2 : int 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 2 : int ret = 0;
230 : mbedtls_mpi K;
231 2 : mbedtls_mpi_init(&K);
232 :
233 : /* DP = D mod P-1 */
234 2 : if (DP != NULL) {
235 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
236 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
237 : }
238 :
239 : /* DQ = D mod Q-1 */
240 2 : if (DQ != NULL) {
241 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
242 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
243 : }
244 :
245 : /* QP = Q^{-1} mod P */
246 2 : if (QP != NULL) {
247 2 : MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
248 : }
249 :
250 2 : cleanup:
251 2 : mbedtls_mpi_free(&K);
252 :
253 2 : return ret;
254 : }
255 :
256 : /*
257 : * Check that core RSA parameters are sane.
258 : */
259 0 : int 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 0 : int ret = 0;
266 : mbedtls_mpi K, L;
267 :
268 0 : mbedtls_mpi_init(&K);
269 0 : 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 0 : if (f_rng != NULL && P != NULL &&
282 0 : (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
283 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
284 0 : goto cleanup;
285 : }
286 :
287 0 : if (f_rng != NULL && Q != NULL &&
288 0 : (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
289 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
290 0 : 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 0 : if (P != NULL && Q != NULL && N != NULL) {
302 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
303 0 : if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
304 0 : mbedtls_mpi_cmp_mpi(&K, N) != 0) {
305 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
306 0 : goto cleanup;
307 : }
308 : }
309 :
310 : /*
311 : * Step 3: Check and 1 < D, E < N if present.
312 : */
313 :
314 0 : if (N != NULL && D != NULL && E != NULL) {
315 0 : if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
316 0 : mbedtls_mpi_cmp_int(E, 1) <= 0 ||
317 0 : mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
318 0 : mbedtls_mpi_cmp_mpi(E, N) >= 0) {
319 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
320 0 : goto cleanup;
321 : }
322 : }
323 :
324 : /*
325 : * Step 4: Check that D, E are inverse modulo P-1 and Q-1
326 : */
327 :
328 0 : if (P != NULL && Q != NULL && D != NULL && E != NULL) {
329 0 : if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
330 0 : mbedtls_mpi_cmp_int(Q, 1) <= 0) {
331 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
332 0 : goto cleanup;
333 : }
334 :
335 : /* Compute DE-1 mod P-1 */
336 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
337 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
338 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
339 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
340 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
341 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
342 0 : goto cleanup;
343 : }
344 :
345 : /* Compute DE-1 mod Q-1 */
346 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
347 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
348 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
349 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
350 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
351 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
352 0 : goto cleanup;
353 : }
354 : }
355 :
356 0 : cleanup:
357 :
358 0 : mbedtls_mpi_free(&K);
359 0 : mbedtls_mpi_free(&L);
360 :
361 : /* Wrap MPI error codes by RSA check failure error code */
362 0 : if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
363 0 : ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
364 : }
365 :
366 0 : return ret;
367 : }
368 :
369 : /*
370 : * Check that RSA CRT parameters are in accordance with core parameters.
371 : */
372 0 : int 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 0 : int ret = 0;
377 :
378 : mbedtls_mpi K, L;
379 0 : mbedtls_mpi_init(&K);
380 0 : mbedtls_mpi_init(&L);
381 :
382 : /* Check that DP - D == 0 mod P - 1 */
383 0 : if (DP != NULL) {
384 0 : if (P == NULL) {
385 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
386 0 : goto cleanup;
387 : }
388 :
389 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
390 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
391 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
392 :
393 0 : if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
394 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
395 0 : goto cleanup;
396 : }
397 : }
398 :
399 : /* Check that DQ - D == 0 mod Q - 1 */
400 0 : if (DQ != NULL) {
401 0 : if (Q == NULL) {
402 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
403 0 : goto cleanup;
404 : }
405 :
406 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
407 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
408 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
409 :
410 0 : if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
411 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
412 0 : goto cleanup;
413 : }
414 : }
415 :
416 : /* Check that QP * Q - 1 == 0 mod P */
417 0 : if (QP != NULL) {
418 0 : if (P == NULL || Q == NULL) {
419 0 : ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
420 0 : goto cleanup;
421 : }
422 :
423 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
424 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
425 0 : MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
426 0 : if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
427 0 : ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
428 0 : goto cleanup;
429 : }
430 : }
431 :
432 0 : cleanup:
433 :
434 : /* Wrap MPI error codes by RSA check failure error code */
435 0 : if (ret != 0 &&
436 0 : ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
437 : ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
438 0 : ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
439 : }
440 :
441 0 : mbedtls_mpi_free(&K);
442 0 : mbedtls_mpi_free(&L);
443 :
444 0 : return ret;
445 : }
446 :
447 : #endif /* MBEDTLS_RSA_C */
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