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 */
mbedtls_rsa_deduce_primes(mbedtls_mpi const * N,mbedtls_mpi const * E,mbedtls_mpi const * D,mbedtls_mpi * P,mbedtls_mpi * Q)62 int 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
86 const size_t num_primes = sizeof( primes ) / sizeof( *primes );
87
88 if( P == NULL || Q == NULL || P->p != NULL || Q->p != NULL )
89 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
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 {
97 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
98 }
99
100 /*
101 * Initializations and temporary changes
102 */
103
104 mbedtls_mpi_init( &K );
105 mbedtls_mpi_init( &T );
106
107 /* T := DE - 1 */
108 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, D, E ) );
109 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &T, &T, 1 ) );
110
111 if( ( order = (uint16_t) mbedtls_mpi_lsb( &T ) ) == 0 )
112 {
113 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
114 goto cleanup;
115 }
116
117 /* After this operation, T holds the largest odd divisor of DE - 1. */
118 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &T, order ) );
119
120 /*
121 * Actual work
122 */
123
124 /* Skip trying 2 if N == 1 mod 8 */
125 attempt = 0;
126 if( N->p[0] % 8 == 1 )
127 attempt = 1;
128
129 for( ; attempt < num_primes; ++attempt )
130 {
131 mbedtls_mpi_lset( &K, primes[attempt] );
132
133 /* Check if gcd(K,N) = 1 */
134 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
135 if( mbedtls_mpi_cmp_int( P, 1 ) != 0 )
136 continue;
137
138 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
139 * and check whether they have nontrivial GCD with N. */
140 MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &K, &K, &T, N,
141 Q /* temporarily use Q for storing Montgomery
142 * multiplication helper values */ ) );
143
144 for( iter = 1; iter <= order; ++iter )
145 {
146 /* If we reach 1 prematurely, there's no point
147 * in continuing to square K */
148 if( mbedtls_mpi_cmp_int( &K, 1 ) == 0 )
149 break;
150
151 MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &K, &K, 1 ) );
152 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
153
154 if( mbedtls_mpi_cmp_int( P, 1 ) == 1 &&
155 mbedtls_mpi_cmp_mpi( P, N ) == -1 )
156 {
157 /*
158 * Have found a nontrivial divisor P of N.
159 * Set Q := N / P.
160 */
161
162 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( Q, NULL, N, P ) );
163 goto cleanup;
164 }
165
166 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
167 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &K ) );
168 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, N ) );
169 }
170
171 /*
172 * If we get here, then either we prematurely aborted the loop because
173 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
174 * be 1 if D,E,N were consistent.
175 * Check if that's the case and abort if not, to avoid very long,
176 * yet eventually failing, computations if N,D,E were not sane.
177 */
178 if( mbedtls_mpi_cmp_int( &K, 1 ) != 0 )
179 {
180 break;
181 }
182 }
183
184 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
185
186 cleanup:
187
188 mbedtls_mpi_free( &K );
189 mbedtls_mpi_free( &T );
190 return( ret );
191 }
192
193 /*
194 * Given P, Q and the public exponent E, deduce D.
195 * This is essentially a modular inversion.
196 */
mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const * P,mbedtls_mpi const * Q,mbedtls_mpi const * E,mbedtls_mpi * D)197 int mbedtls_rsa_deduce_private_exponent( mbedtls_mpi const *P,
198 mbedtls_mpi const *Q,
199 mbedtls_mpi const *E,
200 mbedtls_mpi *D )
201 {
202 int ret = 0;
203 mbedtls_mpi K, L;
204
205 if( D == NULL || mbedtls_mpi_cmp_int( D, 0 ) != 0 )
206 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
207
208 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
209 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 ||
210 mbedtls_mpi_cmp_int( E, 0 ) == 0 )
211 {
212 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
213 }
214
215 mbedtls_mpi_init( &K );
216 mbedtls_mpi_init( &L );
217
218 /* Temporarily put K := P-1 and L := Q-1 */
219 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
220 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
221
222 /* Temporarily put D := gcd(P-1, Q-1) */
223 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( D, &K, &L ) );
224
225 /* K := LCM(P-1, Q-1) */
226 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &L ) );
227 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &K, NULL, &K, D ) );
228
229 /* Compute modular inverse of E in LCM(P-1, Q-1) */
230 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( D, E, &K ) );
231
232 cleanup:
233
234 mbedtls_mpi_free( &K );
235 mbedtls_mpi_free( &L );
236
237 return( ret );
238 }
239
240 /*
241 * Check that RSA CRT parameters are in accordance with core parameters.
242 */
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)243 int mbedtls_rsa_validate_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q,
244 const mbedtls_mpi *D, const mbedtls_mpi *DP,
245 const mbedtls_mpi *DQ, const mbedtls_mpi *QP )
246 {
247 int ret = 0;
248
249 mbedtls_mpi K, L;
250 mbedtls_mpi_init( &K );
251 mbedtls_mpi_init( &L );
252
253 /* Check that DP - D == 0 mod P - 1 */
254 if( DP != NULL )
255 {
256 if( P == NULL )
257 {
258 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
259 goto cleanup;
260 }
261
262 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
263 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DP, D ) );
264 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );
265
266 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
267 {
268 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
269 goto cleanup;
270 }
271 }
272
273 /* Check that DQ - D == 0 mod Q - 1 */
274 if( DQ != NULL )
275 {
276 if( Q == NULL )
277 {
278 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
279 goto cleanup;
280 }
281
282 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
283 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DQ, D ) );
284 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );
285
286 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
287 {
288 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
289 goto cleanup;
290 }
291 }
292
293 /* Check that QP * Q - 1 == 0 mod P */
294 if( QP != NULL )
295 {
296 if( P == NULL || Q == NULL )
297 {
298 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
299 goto cleanup;
300 }
301
302 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, QP, Q ) );
303 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
304 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, P ) );
305 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
306 {
307 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
308 goto cleanup;
309 }
310 }
311
312 cleanup:
313
314 /* Wrap MPI error codes by RSA check failure error code */
315 if( ret != 0 &&
316 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
317 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA )
318 {
319 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
320 }
321
322 mbedtls_mpi_free( &K );
323 mbedtls_mpi_free( &L );
324
325 return( ret );
326 }
327
328 /*
329 * Check that core RSA parameters are sane.
330 */
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)331 int mbedtls_rsa_validate_params( const mbedtls_mpi *N, const mbedtls_mpi *P,
332 const mbedtls_mpi *Q, const mbedtls_mpi *D,
333 const mbedtls_mpi *E,
334 int (*f_rng)(void *, unsigned char *, size_t),
335 void *p_rng )
336 {
337 int ret = 0;
338 mbedtls_mpi K, L;
339
340 mbedtls_mpi_init( &K );
341 mbedtls_mpi_init( &L );
342
343 /*
344 * Step 1: If PRNG provided, check that P and Q are prime
345 */
346
347 #if defined(MBEDTLS_GENPRIME)
348 /*
349 * When generating keys, the strongest security we support aims for an error
350 * rate of at most 2^-100 and we are aiming for the same certainty here as
351 * well.
352 */
353 if( f_rng != NULL && P != NULL &&
354 ( ret = mbedtls_mpi_is_prime_ext( P, 50, f_rng, p_rng ) ) != 0 )
355 {
356 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
357 goto cleanup;
358 }
359
360 if( f_rng != NULL && Q != NULL &&
361 ( ret = mbedtls_mpi_is_prime_ext( Q, 50, f_rng, p_rng ) ) != 0 )
362 {
363 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
364 goto cleanup;
365 }
366 #else
367 ((void) f_rng);
368 ((void) p_rng);
369 #endif /* MBEDTLS_GENPRIME */
370
371 /*
372 * Step 2: Check that 1 < N = P * Q
373 */
374
375 if( P != NULL && Q != NULL && N != NULL )
376 {
377 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, P, Q ) );
378 if( mbedtls_mpi_cmp_int( N, 1 ) <= 0 ||
379 mbedtls_mpi_cmp_mpi( &K, N ) != 0 )
380 {
381 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
382 goto cleanup;
383 }
384 }
385
386 /*
387 * Step 3: Check and 1 < D, E < N if present.
388 */
389
390 if( N != NULL && D != NULL && E != NULL )
391 {
392 if ( mbedtls_mpi_cmp_int( D, 1 ) <= 0 ||
393 mbedtls_mpi_cmp_int( E, 1 ) <= 0 ||
394 mbedtls_mpi_cmp_mpi( D, N ) >= 0 ||
395 mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
396 {
397 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
398 goto cleanup;
399 }
400 }
401
402 /*
403 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
404 */
405
406 if( P != NULL && Q != NULL && D != NULL && E != NULL )
407 {
408 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
409 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 )
410 {
411 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
412 goto cleanup;
413 }
414
415 /* Compute DE-1 mod P-1 */
416 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
417 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
418 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, P, 1 ) );
419 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
420 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
421 {
422 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
423 goto cleanup;
424 }
425
426 /* Compute DE-1 mod Q-1 */
427 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
428 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
429 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
430 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
431 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
432 {
433 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
434 goto cleanup;
435 }
436 }
437
438 cleanup:
439
440 mbedtls_mpi_free( &K );
441 mbedtls_mpi_free( &L );
442
443 /* Wrap MPI error codes by RSA check failure error code */
444 if( ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED )
445 {
446 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
447 }
448
449 return( ret );
450 }
451
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)452 int mbedtls_rsa_deduce_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q,
453 const mbedtls_mpi *D, mbedtls_mpi *DP,
454 mbedtls_mpi *DQ, mbedtls_mpi *QP )
455 {
456 int ret = 0;
457 mbedtls_mpi K;
458 mbedtls_mpi_init( &K );
459
460 /* DP = D mod P-1 */
461 if( DP != NULL )
462 {
463 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
464 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DP, D, &K ) );
465 }
466
467 /* DQ = D mod Q-1 */
468 if( DQ != NULL )
469 {
470 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
471 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DQ, D, &K ) );
472 }
473
474 /* QP = Q^{-1} mod P */
475 if( QP != NULL )
476 {
477 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( QP, Q, P ) );
478 }
479
480 cleanup:
481 mbedtls_mpi_free( &K );
482
483 return( ret );
484 }
485
486 #endif /* MBEDTLS_RSA_C */
487