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1 /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
2  * All rights reserved.
3  *
4  * This package is an SSL implementation written
5  * by Eric Young (eay@cryptsoft.com).
6  * The implementation was written so as to conform with Netscapes SSL.
7  *
8  * This library is free for commercial and non-commercial use as long as
9  * the following conditions are aheared to.  The following conditions
10  * apply to all code found in this distribution, be it the RC4, RSA,
11  * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
12  * included with this distribution is covered by the same copyright terms
13  * except that the holder is Tim Hudson (tjh@cryptsoft.com).
14  *
15  * Copyright remains Eric Young's, and as such any Copyright notices in
16  * the code are not to be removed.
17  * If this package is used in a product, Eric Young should be given attribution
18  * as the author of the parts of the library used.
19  * This can be in the form of a textual message at program startup or
20  * in documentation (online or textual) provided with the package.
21  *
22  * Redistribution and use in source and binary forms, with or without
23  * modification, are permitted provided that the following conditions
24  * are met:
25  * 1. Redistributions of source code must retain the copyright
26  *    notice, this list of conditions and the following disclaimer.
27  * 2. Redistributions in binary form must reproduce the above copyright
28  *    notice, this list of conditions and the following disclaimer in the
29  *    documentation and/or other materials provided with the distribution.
30  * 3. All advertising materials mentioning features or use of this software
31  *    must display the following acknowledgement:
32  *    "This product includes cryptographic software written by
33  *     Eric Young (eay@cryptsoft.com)"
34  *    The word 'cryptographic' can be left out if the rouines from the library
35  *    being used are not cryptographic related :-).
36  * 4. If you include any Windows specific code (or a derivative thereof) from
37  *    the apps directory (application code) you must include an acknowledgement:
38  *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
39  *
40  * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
41  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
42  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
43  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
44  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
45  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
46  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
47  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
48  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
49  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
50  * SUCH DAMAGE.
51  *
52  * The licence and distribution terms for any publically available version or
53  * derivative of this code cannot be changed.  i.e. this code cannot simply be
54  * copied and put under another distribution licence
55  * [including the GNU Public Licence.]
56  */
57 /* ====================================================================
58  * Copyright (c) 1998-2001 The OpenSSL Project.  All rights reserved.
59  *
60  * Redistribution and use in source and binary forms, with or without
61  * modification, are permitted provided that the following conditions
62  * are met:
63  *
64  * 1. Redistributions of source code must retain the above copyright
65  *    notice, this list of conditions and the following disclaimer.
66  *
67  * 2. Redistributions in binary form must reproduce the above copyright
68  *    notice, this list of conditions and the following disclaimer in
69  *    the documentation and/or other materials provided with the
70  *    distribution.
71  *
72  * 3. All advertising materials mentioning features or use of this
73  *    software must display the following acknowledgment:
74  *    "This product includes software developed by the OpenSSL Project
75  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
76  *
77  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
78  *    endorse or promote products derived from this software without
79  *    prior written permission. For written permission, please contact
80  *    openssl-core@openssl.org.
81  *
82  * 5. Products derived from this software may not be called "OpenSSL"
83  *    nor may "OpenSSL" appear in their names without prior written
84  *    permission of the OpenSSL Project.
85  *
86  * 6. Redistributions of any form whatsoever must retain the following
87  *    acknowledgment:
88  *    "This product includes software developed by the OpenSSL Project
89  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
90  *
91  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
92  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
93  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
94  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
95  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
96  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
97  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
98  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
99  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
100  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
101  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
102  * OF THE POSSIBILITY OF SUCH DAMAGE.
103  * ====================================================================
104  *
105  * This product includes cryptographic software written by Eric Young
106  * (eay@cryptsoft.com).  This product includes software written by Tim
107  * Hudson (tjh@cryptsoft.com). */
108 
109 #include <openssl/bn.h>
110 
111 #include <openssl/err.h>
112 
113 #include "internal.h"
114 
115 
BN_mod_inverse_odd(BIGNUM * out,int * out_no_inverse,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)116 int BN_mod_inverse_odd(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
117                        const BIGNUM *n, BN_CTX *ctx) {
118   *out_no_inverse = 0;
119 
120   if (!BN_is_odd(n)) {
121     OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
122     return 0;
123   }
124 
125   if (BN_is_negative(a) || BN_cmp(a, n) >= 0) {
126     OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
127     return 0;
128   }
129 
130   BIGNUM *A, *B, *X, *Y;
131   int ret = 0;
132   int sign;
133 
134   BN_CTX_start(ctx);
135   A = BN_CTX_get(ctx);
136   B = BN_CTX_get(ctx);
137   X = BN_CTX_get(ctx);
138   Y = BN_CTX_get(ctx);
139   if (Y == NULL) {
140     goto err;
141   }
142 
143   BIGNUM *R = out;
144 
145   BN_zero(Y);
146   if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
147     goto err;
148   }
149   A->neg = 0;
150   sign = -1;
151   // From  B = a mod |n|,  A = |n|  it follows that
152   //
153   //      0 <= B < A,
154   //     -sign*X*a  ==  B   (mod |n|),
155   //      sign*Y*a  ==  A   (mod |n|).
156 
157   // Binary inversion algorithm; requires odd modulus. This is faster than the
158   // general algorithm if the modulus is sufficiently small (about 400 .. 500
159   // bits on 32-bit systems, but much more on 64-bit systems)
160   int shift;
161 
162   while (!BN_is_zero(B)) {
163     //      0 < B < |n|,
164     //      0 < A <= |n|,
165     // (1) -sign*X*a  ==  B   (mod |n|),
166     // (2)  sign*Y*a  ==  A   (mod |n|)
167 
168     // Now divide  B  by the maximum possible power of two in the integers,
169     // and divide  X  by the same value mod |n|.
170     // When we're done, (1) still holds.
171     shift = 0;
172     while (!BN_is_bit_set(B, shift)) {
173       // note that 0 < B
174       shift++;
175 
176       if (BN_is_odd(X)) {
177         if (!BN_uadd(X, X, n)) {
178           goto err;
179         }
180       }
181       // now X is even, so we can easily divide it by two
182       if (!BN_rshift1(X, X)) {
183         goto err;
184       }
185     }
186     if (shift > 0) {
187       if (!BN_rshift(B, B, shift)) {
188         goto err;
189       }
190     }
191 
192     // Same for A and Y. Afterwards, (2) still holds.
193     shift = 0;
194     while (!BN_is_bit_set(A, shift)) {
195       // note that 0 < A
196       shift++;
197 
198       if (BN_is_odd(Y)) {
199         if (!BN_uadd(Y, Y, n)) {
200           goto err;
201         }
202       }
203       // now Y is even
204       if (!BN_rshift1(Y, Y)) {
205         goto err;
206       }
207     }
208     if (shift > 0) {
209       if (!BN_rshift(A, A, shift)) {
210         goto err;
211       }
212     }
213 
214     // We still have (1) and (2).
215     // Both  A  and  B  are odd.
216     // The following computations ensure that
217     //
218     //     0 <= B < |n|,
219     //      0 < A < |n|,
220     // (1) -sign*X*a  ==  B   (mod |n|),
221     // (2)  sign*Y*a  ==  A   (mod |n|),
222     //
223     // and that either  A  or  B  is even in the next iteration.
224     if (BN_ucmp(B, A) >= 0) {
225       // -sign*(X + Y)*a == B - A  (mod |n|)
226       if (!BN_uadd(X, X, Y)) {
227         goto err;
228       }
229       // NB: we could use BN_mod_add_quick(X, X, Y, n), but that
230       // actually makes the algorithm slower
231       if (!BN_usub(B, B, A)) {
232         goto err;
233       }
234     } else {
235       //  sign*(X + Y)*a == A - B  (mod |n|)
236       if (!BN_uadd(Y, Y, X)) {
237         goto err;
238       }
239       // as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
240       if (!BN_usub(A, A, B)) {
241         goto err;
242       }
243     }
244   }
245 
246   if (!BN_is_one(A)) {
247     *out_no_inverse = 1;
248     OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
249     goto err;
250   }
251 
252   // The while loop (Euclid's algorithm) ends when
253   //      A == gcd(a,n);
254   // we have
255   //       sign*Y*a  ==  A  (mod |n|),
256   // where  Y  is non-negative.
257 
258   if (sign < 0) {
259     if (!BN_sub(Y, n, Y)) {
260       goto err;
261     }
262   }
263   // Now  Y*a  ==  A  (mod |n|).
264 
265   // Y*a == 1  (mod |n|)
266   if (!Y->neg && BN_ucmp(Y, n) < 0) {
267     if (!BN_copy(R, Y)) {
268       goto err;
269     }
270   } else {
271     if (!BN_nnmod(R, Y, n, ctx)) {
272       goto err;
273     }
274   }
275 
276   ret = 1;
277 
278 err:
279   BN_CTX_end(ctx);
280   return ret;
281 }
282 
BN_mod_inverse(BIGNUM * out,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)283 BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
284                        BN_CTX *ctx) {
285   BIGNUM *new_out = NULL;
286   if (out == NULL) {
287     new_out = BN_new();
288     if (new_out == NULL) {
289       OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
290       return NULL;
291     }
292     out = new_out;
293   }
294 
295   int ok = 0;
296   BIGNUM *a_reduced = NULL;
297   if (a->neg || BN_ucmp(a, n) >= 0) {
298     a_reduced = BN_dup(a);
299     if (a_reduced == NULL) {
300       goto err;
301     }
302     if (!BN_nnmod(a_reduced, a_reduced, n, ctx)) {
303       goto err;
304     }
305     a = a_reduced;
306   }
307 
308   int no_inverse;
309   if (!BN_is_odd(n)) {
310     if (!bn_mod_inverse_consttime(out, &no_inverse, a, n, ctx)) {
311       goto err;
312     }
313   } else if (!BN_mod_inverse_odd(out, &no_inverse, a, n, ctx)) {
314     goto err;
315   }
316 
317   ok = 1;
318 
319 err:
320   if (!ok) {
321     BN_free(new_out);
322     out = NULL;
323   }
324   BN_free(a_reduced);
325   return out;
326 }
327 
BN_mod_inverse_blinded(BIGNUM * out,int * out_no_inverse,const BIGNUM * a,const BN_MONT_CTX * mont,BN_CTX * ctx)328 int BN_mod_inverse_blinded(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
329                            const BN_MONT_CTX *mont, BN_CTX *ctx) {
330   *out_no_inverse = 0;
331 
332   if (BN_is_negative(a) || BN_cmp(a, &mont->N) >= 0) {
333     OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
334     return 0;
335   }
336 
337   int ret = 0;
338   BIGNUM blinding_factor;
339   BN_init(&blinding_factor);
340 
341   if (!BN_rand_range_ex(&blinding_factor, 1, &mont->N) ||
342       !BN_mod_mul_montgomery(out, &blinding_factor, a, mont, ctx) ||
343       !BN_mod_inverse_odd(out, out_no_inverse, out, &mont->N, ctx) ||
344       !BN_mod_mul_montgomery(out, &blinding_factor, out, mont, ctx)) {
345     OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
346     goto err;
347   }
348 
349   ret = 1;
350 
351 err:
352   BN_free(&blinding_factor);
353   return ret;
354 }
355 
bn_mod_inverse_prime(BIGNUM * out,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx,const BN_MONT_CTX * mont_p)356 int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
357                          BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
358   BN_CTX_start(ctx);
359   BIGNUM *p_minus_2 = BN_CTX_get(ctx);
360   int ok = p_minus_2 != NULL &&
361            BN_copy(p_minus_2, p) &&
362            BN_sub_word(p_minus_2, 2) &&
363            BN_mod_exp_mont(out, a, p_minus_2, p, ctx, mont_p);
364   BN_CTX_end(ctx);
365   return ok;
366 }
367 
bn_mod_inverse_secret_prime(BIGNUM * out,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx,const BN_MONT_CTX * mont_p)368 int bn_mod_inverse_secret_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
369                                 BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
370   BN_CTX_start(ctx);
371   BIGNUM *p_minus_2 = BN_CTX_get(ctx);
372   int ok = p_minus_2 != NULL &&
373            BN_copy(p_minus_2, p) &&
374            BN_sub_word(p_minus_2, 2) &&
375            BN_mod_exp_mont_consttime(out, a, p_minus_2, p, ctx, mont_p);
376   BN_CTX_end(ctx);
377   return ok;
378 }
379