1 /*
2 * Helper functions for the RSA module
3 *
4 * Copyright (C) 2006-2017, ARM Limited, All Rights Reserved
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 * This file is part of mbed TLS (https://tls.mbed.org)
20 *
21 */
22
23 #if !defined(MBEDTLS_CONFIG_FILE)
24 #include "mbedtls/config.h"
25 #else
26 #include MBEDTLS_CONFIG_FILE
27 #endif
28
29 #if defined(MBEDTLS_RSA_C)
30
31 #include "mbedtls/rsa.h"
32 #include "mbedtls/bignum.h"
33 #include "mbedtls/rsa_internal.h"
34
35 /*
36 * Compute RSA prime factors from public and private exponents
37 *
38 * Summary of algorithm:
39 * Setting F := lcm(P-1,Q-1), the idea is as follows:
40 *
41 * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2)
42 * is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the
43 * square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four
44 * possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1)
45 * or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime
46 * factors of N.
47 *
48 * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same
49 * construction still applies since (-)^K is the identity on the set of
50 * roots of 1 in Z/NZ.
51 *
52 * The public and private key primitives (-)^E and (-)^D are mutually inverse
53 * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e.
54 * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L.
55 * Splitting L = 2^t * K with K odd, we have
56 *
57 * DE - 1 = FL = (F/2) * (2^(t+1)) * K,
58 *
59 * so (F / 2) * K is among the numbers
60 *
61 * (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord
62 *
63 * where ord is the order of 2 in (DE - 1).
64 * We can therefore iterate through these numbers apply the construction
65 * of (a) and (b) above to attempt to factor N.
66 *
67 */
mbedtls_rsa_deduce_primes(mbedtls_mpi const * N,mbedtls_mpi const * E,mbedtls_mpi const * D,mbedtls_mpi * P,mbedtls_mpi * Q)68 int mbedtls_rsa_deduce_primes( mbedtls_mpi const *N,
69 mbedtls_mpi const *E, mbedtls_mpi const *D,
70 mbedtls_mpi *P, mbedtls_mpi *Q )
71 {
72 int ret = 0;
73
74 uint16_t attempt; /* Number of current attempt */
75 uint16_t iter; /* Number of squares computed in the current attempt */
76
77 uint16_t order; /* Order of 2 in DE - 1 */
78
79 mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */
80 mbedtls_mpi K; /* Temporary holding the current candidate */
81
82 const unsigned char primes[] = { 2,
83 3, 5, 7, 11, 13, 17, 19, 23,
84 29, 31, 37, 41, 43, 47, 53, 59,
85 61, 67, 71, 73, 79, 83, 89, 97,
86 101, 103, 107, 109, 113, 127, 131, 137,
87 139, 149, 151, 157, 163, 167, 173, 179,
88 181, 191, 193, 197, 199, 211, 223, 227,
89 229, 233, 239, 241, 251
90 };
91
92 const size_t num_primes = sizeof( primes ) / sizeof( *primes );
93
94 if( P == NULL || Q == NULL || P->p != NULL || Q->p != NULL )
95 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
96
97 if( mbedtls_mpi_cmp_int( N, 0 ) <= 0 ||
98 mbedtls_mpi_cmp_int( D, 1 ) <= 0 ||
99 mbedtls_mpi_cmp_mpi( D, N ) >= 0 ||
100 mbedtls_mpi_cmp_int( E, 1 ) <= 0 ||
101 mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
102 {
103 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
104 }
105
106 /*
107 * Initializations and temporary changes
108 */
109
110 mbedtls_mpi_init( &K );
111 mbedtls_mpi_init( &T );
112
113 /* T := DE - 1 */
114 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, D, E ) );
115 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &T, &T, 1 ) );
116
117 if( ( order = (uint16_t) mbedtls_mpi_lsb( &T ) ) == 0 )
118 {
119 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
120 goto cleanup;
121 }
122
123 /* After this operation, T holds the largest odd divisor of DE - 1. */
124 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &T, order ) );
125
126 /*
127 * Actual work
128 */
129
130 /* Skip trying 2 if N == 1 mod 8 */
131 attempt = 0;
132 if( N->p[0] % 8 == 1 )
133 attempt = 1;
134
135 for( ; attempt < num_primes; ++attempt )
136 {
137 mbedtls_mpi_lset( &K, primes[attempt] );
138
139 /* Check if gcd(K,N) = 1 */
140 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
141 if( mbedtls_mpi_cmp_int( P, 1 ) != 0 )
142 continue;
143
144 /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
145 * and check whether they have nontrivial GCD with N. */
146 MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &K, &K, &T, N,
147 Q /* temporarily use Q for storing Montgomery
148 * multiplication helper values */ ) );
149
150 for( iter = 1; iter <= order; ++iter )
151 {
152 /* If we reach 1 prematurely, there's no point
153 * in continuing to square K */
154 if( mbedtls_mpi_cmp_int( &K, 1 ) == 0 )
155 break;
156
157 MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &K, &K, 1 ) );
158 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
159
160 if( mbedtls_mpi_cmp_int( P, 1 ) == 1 &&
161 mbedtls_mpi_cmp_mpi( P, N ) == -1 )
162 {
163 /*
164 * Have found a nontrivial divisor P of N.
165 * Set Q := N / P.
166 */
167
168 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( Q, NULL, N, P ) );
169 goto cleanup;
170 }
171
172 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
173 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &K ) );
174 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, N ) );
175 }
176
177 /*
178 * If we get here, then either we prematurely aborted the loop because
179 * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
180 * be 1 if D,E,N were consistent.
181 * Check if that's the case and abort if not, to avoid very long,
182 * yet eventually failing, computations if N,D,E were not sane.
183 */
184 if( mbedtls_mpi_cmp_int( &K, 1 ) != 0 )
185 {
186 break;
187 }
188 }
189
190 ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
191
192 cleanup:
193
194 mbedtls_mpi_free( &K );
195 mbedtls_mpi_free( &T );
196 return( ret );
197 }
198
199 /*
200 * Given P, Q and the public exponent E, deduce D.
201 * This is essentially a modular inversion.
202 */
mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const * P,mbedtls_mpi const * Q,mbedtls_mpi const * E,mbedtls_mpi * D)203 int mbedtls_rsa_deduce_private_exponent( mbedtls_mpi const *P,
204 mbedtls_mpi const *Q,
205 mbedtls_mpi const *E,
206 mbedtls_mpi *D )
207 {
208 int ret = 0;
209 mbedtls_mpi K, L;
210
211 if( D == NULL || mbedtls_mpi_cmp_int( D, 0 ) != 0 )
212 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
213
214 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
215 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 ||
216 mbedtls_mpi_cmp_int( E, 0 ) == 0 )
217 {
218 return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
219 }
220
221 mbedtls_mpi_init( &K );
222 mbedtls_mpi_init( &L );
223
224 /* Temporarily put K := P-1 and L := Q-1 */
225 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
226 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
227
228 /* Temporarily put D := gcd(P-1, Q-1) */
229 MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( D, &K, &L ) );
230
231 /* K := LCM(P-1, Q-1) */
232 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &L ) );
233 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &K, NULL, &K, D ) );
234
235 /* Compute modular inverse of E in LCM(P-1, Q-1) */
236 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( D, E, &K ) );
237
238 cleanup:
239
240 mbedtls_mpi_free( &K );
241 mbedtls_mpi_free( &L );
242
243 return( ret );
244 }
245
246 /*
247 * Check that RSA CRT parameters are in accordance with core parameters.
248 */
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)249 int mbedtls_rsa_validate_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q,
250 const mbedtls_mpi *D, const mbedtls_mpi *DP,
251 const mbedtls_mpi *DQ, const mbedtls_mpi *QP )
252 {
253 int ret = 0;
254
255 mbedtls_mpi K, L;
256 mbedtls_mpi_init( &K );
257 mbedtls_mpi_init( &L );
258
259 /* Check that DP - D == 0 mod P - 1 */
260 if( DP != NULL )
261 {
262 if( P == NULL )
263 {
264 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
265 goto cleanup;
266 }
267
268 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
269 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DP, D ) );
270 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );
271
272 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
273 {
274 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
275 goto cleanup;
276 }
277 }
278
279 /* Check that DQ - D == 0 mod Q - 1 */
280 if( DQ != NULL )
281 {
282 if( Q == NULL )
283 {
284 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
285 goto cleanup;
286 }
287
288 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
289 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DQ, D ) );
290 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );
291
292 if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
293 {
294 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
295 goto cleanup;
296 }
297 }
298
299 /* Check that QP * Q - 1 == 0 mod P */
300 if( QP != NULL )
301 {
302 if( P == NULL || Q == NULL )
303 {
304 ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
305 goto cleanup;
306 }
307
308 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, QP, Q ) );
309 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
310 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, P ) );
311 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
312 {
313 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
314 goto cleanup;
315 }
316 }
317
318 cleanup:
319
320 /* Wrap MPI error codes by RSA check failure error code */
321 if( ret != 0 &&
322 ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
323 ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA )
324 {
325 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
326 }
327
328 mbedtls_mpi_free( &K );
329 mbedtls_mpi_free( &L );
330
331 return( ret );
332 }
333
334 /*
335 * Check that core RSA parameters are sane.
336 */
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)337 int mbedtls_rsa_validate_params( const mbedtls_mpi *N, const mbedtls_mpi *P,
338 const mbedtls_mpi *Q, const mbedtls_mpi *D,
339 const mbedtls_mpi *E,
340 int (*f_rng)(void *, unsigned char *, size_t),
341 void *p_rng )
342 {
343 int ret = 0;
344 mbedtls_mpi K, L;
345
346 mbedtls_mpi_init( &K );
347 mbedtls_mpi_init( &L );
348
349 /*
350 * Step 1: If PRNG provided, check that P and Q are prime
351 */
352
353 #if defined(MBEDTLS_GENPRIME)
354 /*
355 * When generating keys, the strongest security we support aims for an error
356 * rate of at most 2^-100 and we are aiming for the same certainty here as
357 * well.
358 */
359 if( f_rng != NULL && P != NULL &&
360 ( ret = mbedtls_mpi_is_prime_ext( P, 50, f_rng, p_rng ) ) != 0 )
361 {
362 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
363 goto cleanup;
364 }
365
366 if( f_rng != NULL && Q != NULL &&
367 ( ret = mbedtls_mpi_is_prime_ext( Q, 50, f_rng, p_rng ) ) != 0 )
368 {
369 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
370 goto cleanup;
371 }
372 #else
373 ((void) f_rng);
374 ((void) p_rng);
375 #endif /* MBEDTLS_GENPRIME */
376
377 /*
378 * Step 2: Check that 1 < N = P * Q
379 */
380
381 if( P != NULL && Q != NULL && N != NULL )
382 {
383 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, P, Q ) );
384 if( mbedtls_mpi_cmp_int( N, 1 ) <= 0 ||
385 mbedtls_mpi_cmp_mpi( &K, N ) != 0 )
386 {
387 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
388 goto cleanup;
389 }
390 }
391
392 /*
393 * Step 3: Check and 1 < D, E < N if present.
394 */
395
396 if( N != NULL && D != NULL && E != NULL )
397 {
398 if ( mbedtls_mpi_cmp_int( D, 1 ) <= 0 ||
399 mbedtls_mpi_cmp_int( E, 1 ) <= 0 ||
400 mbedtls_mpi_cmp_mpi( D, N ) >= 0 ||
401 mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
402 {
403 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
404 goto cleanup;
405 }
406 }
407
408 /*
409 * Step 4: Check that D, E are inverse modulo P-1 and Q-1
410 */
411
412 if( P != NULL && Q != NULL && D != NULL && E != NULL )
413 {
414 if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
415 mbedtls_mpi_cmp_int( Q, 1 ) <= 0 )
416 {
417 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
418 goto cleanup;
419 }
420
421 /* Compute DE-1 mod P-1 */
422 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
423 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
424 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, P, 1 ) );
425 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
426 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
427 {
428 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
429 goto cleanup;
430 }
431
432 /* Compute DE-1 mod Q-1 */
433 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
434 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
435 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
436 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
437 if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
438 {
439 ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
440 goto cleanup;
441 }
442 }
443
444 cleanup:
445
446 mbedtls_mpi_free( &K );
447 mbedtls_mpi_free( &L );
448
449 /* Wrap MPI error codes by RSA check failure error code */
450 if( ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED )
451 {
452 ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
453 }
454
455 return( ret );
456 }
457
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)458 int mbedtls_rsa_deduce_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q,
459 const mbedtls_mpi *D, mbedtls_mpi *DP,
460 mbedtls_mpi *DQ, mbedtls_mpi *QP )
461 {
462 int ret = 0;
463 mbedtls_mpi K;
464 mbedtls_mpi_init( &K );
465
466 /* DP = D mod P-1 */
467 if( DP != NULL )
468 {
469 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
470 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DP, D, &K ) );
471 }
472
473 /* DQ = D mod Q-1 */
474 if( DQ != NULL )
475 {
476 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
477 MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DQ, D, &K ) );
478 }
479
480 /* QP = Q^{-1} mod P */
481 if( QP != NULL )
482 {
483 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( QP, Q, P ) );
484 }
485
486 cleanup:
487 mbedtls_mpi_free( &K );
488
489 return( ret );
490 }
491
492 #endif /* MBEDTLS_RSA_C */
493