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 */
mbedtls_rsa_deduce_primes(mbedtls_mpi const * N,mbedtls_mpi const * E,mbedtls_mpi const * D,mbedtls_mpi * P,mbedtls_mpi * Q)50 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 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 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 const size_t num_primes = sizeof(primes) / sizeof(*primes);
74
75 if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) {
76 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
77 }
78
79 if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
80 mbedtls_mpi_cmp_int(D, 1) <= 0 ||
81 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
82 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
83 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
84 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
85 }
86
87 /*
88 * Initializations and temporary changes
89 */
90
91 mbedtls_mpi_init(&K);
92 mbedtls_mpi_init(&T);
93
94 /* T := DE - 1 */
95 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
96 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
97
98 if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
99 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
100 goto cleanup;
101 }
102
103 /* After this operation, T holds the largest odd divisor of DE - 1. */
104 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 attempt = 0;
112 if (N->p[0] % 8 == 1) {
113 attempt = 1;
114 }
115
116 for (; attempt < num_primes; ++attempt) {
117 MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&K, primes[attempt]));
118
119 /* Check if gcd(K,N) = 1 */
120 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
121 if (mbedtls_mpi_cmp_int(P, 1) != 0) {
122 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 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 for (iter = 1; iter <= order; ++iter) {
132 /* If we reach 1 prematurely, there's no point
133 * in continuing to square K */
134 if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
135 break;
136 }
137
138 MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
139 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
140
141 if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
142 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 MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
149 goto cleanup;
150 }
151
152 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
153 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
154 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 if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
165 break;
166 }
167 }
168
169 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
170
171 cleanup:
172
173 mbedtls_mpi_free(&K);
174 mbedtls_mpi_free(&T);
175 return ret;
176 }
177
178 /*
179 * Given P, Q and the public exponent E, deduce D.
180 * This is essentially a modular inversion.
181 */
mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const * P,mbedtls_mpi const * Q,mbedtls_mpi const * E,mbedtls_mpi * D)182 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 int ret = 0;
188 mbedtls_mpi K, L;
189
190 if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
191 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
192 }
193
194 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
195 mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
196 mbedtls_mpi_cmp_int(E, 0) == 0) {
197 return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
198 }
199
200 mbedtls_mpi_init(&K);
201 mbedtls_mpi_init(&L);
202
203 /* Temporarily put K := P-1 and L := Q-1 */
204 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
205 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
206
207 /* Temporarily put D := gcd(P-1, Q-1) */
208 MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
209
210 /* K := LCM(P-1, Q-1) */
211 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
212 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 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K));
216
217 cleanup:
218
219 mbedtls_mpi_free(&K);
220 mbedtls_mpi_free(&L);
221
222 return ret;
223 }
224
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)225 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 int ret = 0;
230 mbedtls_mpi K;
231 mbedtls_mpi_init(&K);
232
233 /* DP = D mod P-1 */
234 if (DP != NULL) {
235 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
236 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
237 }
238
239 /* DQ = D mod Q-1 */
240 if (DQ != NULL) {
241 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
242 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
243 }
244
245 /* QP = Q^{-1} mod P */
246 if (QP != NULL) {
247 MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
248 }
249
250 cleanup:
251 mbedtls_mpi_free(&K);
252
253 return ret;
254 }
255
256 /*
257 * Check that core RSA parameters are sane.
258 */
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)259 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 int ret = 0;
266 mbedtls_mpi K, L;
267
268 mbedtls_mpi_init(&K);
269 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 if (f_rng != NULL && P != NULL &&
282 (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
283 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
284 goto cleanup;
285 }
286
287 if (f_rng != NULL && Q != NULL &&
288 (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
289 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
290 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 if (P != NULL && Q != NULL && N != NULL) {
302 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
303 if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
304 mbedtls_mpi_cmp_mpi(&K, N) != 0) {
305 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
306 goto cleanup;
307 }
308 }
309
310 /*
311 * Step 3: Check and 1 < D, E < N if present.
312 */
313
314 if (N != NULL && D != NULL && E != NULL) {
315 if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
316 mbedtls_mpi_cmp_int(E, 1) <= 0 ||
317 mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
318 mbedtls_mpi_cmp_mpi(E, N) >= 0) {
319 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
320 goto cleanup;
321 }
322 }
323
324 /*
325 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
326 */
327
328 if (P != NULL && Q != NULL && D != NULL && E != NULL) {
329 if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
330 mbedtls_mpi_cmp_int(Q, 1) <= 0) {
331 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
332 goto cleanup;
333 }
334
335 /* Compute DE-1 mod P-1 */
336 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
337 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
338 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
339 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
340 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
341 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
342 goto cleanup;
343 }
344
345 /* Compute DE-1 mod Q-1 */
346 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
347 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
348 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
349 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
350 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
351 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
352 goto cleanup;
353 }
354 }
355
356 cleanup:
357
358 mbedtls_mpi_free(&K);
359 mbedtls_mpi_free(&L);
360
361 /* Wrap MPI error codes by RSA check failure error code */
362 if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
363 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
364 }
365
366 return ret;
367 }
368
369 /*
370 * Check that RSA CRT parameters are in accordance with core parameters.
371 */
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)372 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 int ret = 0;
377
378 mbedtls_mpi K, L;
379 mbedtls_mpi_init(&K);
380 mbedtls_mpi_init(&L);
381
382 /* Check that DP - D == 0 mod P - 1 */
383 if (DP != NULL) {
384 if (P == NULL) {
385 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
386 goto cleanup;
387 }
388
389 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
390 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
391 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
392
393 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
394 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
395 goto cleanup;
396 }
397 }
398
399 /* Check that DQ - D == 0 mod Q - 1 */
400 if (DQ != NULL) {
401 if (Q == NULL) {
402 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
403 goto cleanup;
404 }
405
406 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
407 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
408 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
409
410 if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
411 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
412 goto cleanup;
413 }
414 }
415
416 /* Check that QP * Q - 1 == 0 mod P */
417 if (QP != NULL) {
418 if (P == NULL || Q == NULL) {
419 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
420 goto cleanup;
421 }
422
423 MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
424 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
425 MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
426 if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
427 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
428 goto cleanup;
429 }
430 }
431
432 cleanup:
433
434 /* Wrap MPI error codes by RSA check failure error code */
435 if (ret != 0 &&
436 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
437 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
438 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
439 }
440
441 mbedtls_mpi_free(&K);
442 mbedtls_mpi_free(&L);
443
444 return ret;
445 }
446
447 #endif /* MBEDTLS_RSA_C */
448
