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1/* crypto/bn/bn_gcd.c */
2/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
3 * All rights reserved.
4 *
5 * This package is an SSL implementation written
6 * by Eric Young (eay@cryptsoft.com).
7 * The implementation was written so as to conform with Netscapes SSL.
8 *
9 * This library is free for commercial and non-commercial use as long as
10 * the following conditions are aheared to. The following conditions
11 * apply to all code found in this distribution, be it the RC4, RSA,
12 * lhash, DES, etc., code; not just the SSL code. The SSL documentation
13 * included with this distribution is covered by the same copyright terms
14 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
15 *
16 * Copyright remains Eric Young's, and as such any Copyright notices in
17 * the code are not to be removed.
18 * If this package is used in a product, Eric Young should be given attribution
19 * as the author of the parts of the library used.
20 * This can be in the form of a textual message at program startup or
21 * in documentation (online or textual) provided with the package.
22 *
23 * Redistribution and use in source and binary forms, with or without
24 * modification, are permitted provided that the following conditions
25 * are met:
26 * 1. Redistributions of source code must retain the copyright
27 * notice, this list of conditions and the following disclaimer.
28 * 2. Redistributions in binary form must reproduce the above copyright
29 * notice, this list of conditions and the following disclaimer in the
30 * documentation and/or other materials provided with the distribution.
31 * 3. All advertising materials mentioning features or use of this software
32 * must display the following acknowledgement:
33 * "This product includes cryptographic software written by
34 * Eric Young (eay@cryptsoft.com)"
35 * The word 'cryptographic' can be left out if the rouines from the library
36 * being used are not cryptographic related :-).
37 * 4. If you include any Windows specific code (or a derivative thereof) from
38 * the apps directory (application code) you must include an acknowledgement:
39 * "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
40 *
41 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
42 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
43 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
44 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
45 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
46 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
47 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
48 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
49 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
50 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
51 * SUCH DAMAGE.
52 *
53 * The licence and distribution terms for any publically available version or
54 * derivative of this code cannot be changed. i.e. this code cannot simply be
55 * copied and put under another distribution licence
56 * [including the GNU Public Licence.]
57 */
58/* ====================================================================
59 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
60 *
61 * Redistribution and use in source and binary forms, with or without
62 * modification, are permitted provided that the following conditions
63 * are met:
64 *
65 * 1. Redistributions of source code must retain the above copyright
66 * notice, this list of conditions and the following disclaimer.
67 *
68 * 2. Redistributions in binary form must reproduce the above copyright
69 * notice, this list of conditions and the following disclaimer in
70 * the documentation and/or other materials provided with the
71 * distribution.
72 *
73 * 3. All advertising materials mentioning features or use of this
74 * software must display the following acknowledgment:
75 * "This product includes software developed by the OpenSSL Project
76 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
77 *
78 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
79 * endorse or promote products derived from this software without
80 * prior written permission. For written permission, please contact
81 * openssl-core@openssl.org.
82 *
83 * 5. Products derived from this software may not be called "OpenSSL"
84 * nor may "OpenSSL" appear in their names without prior written
85 * permission of the OpenSSL Project.
86 *
87 * 6. Redistributions of any form whatsoever must retain the following
88 * acknowledgment:
89 * "This product includes software developed by the OpenSSL Project
90 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
91 *
92 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
93 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
94 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
95 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
96 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
97 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
98 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
99 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
100 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
101 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
102 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
103 * OF THE POSSIBILITY OF SUCH DAMAGE.
104 * ====================================================================
105 *
106 * This product includes cryptographic software written by Eric Young
107 * (eay@cryptsoft.com). This product includes software written by Tim
108 * Hudson (tjh@cryptsoft.com).
109 *
110 */
111
112#include "cryptlib.h"
113#include "bn_lcl.h"
114
115static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
116
117int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
118 {
119 BIGNUM *a,*b,*t;
120 int ret=0;
121
122 bn_check_top(in_a);
123 bn_check_top(in_b);
124
125 BN_CTX_start(ctx);
126 a = BN_CTX_get(ctx);
127 b = BN_CTX_get(ctx);
128 if (a == NULL || b == NULL) goto err;
129
130 if (BN_copy(a,in_a) == NULL) goto err;
131 if (BN_copy(b,in_b) == NULL) goto err;
132 a->neg = 0;
133 b->neg = 0;
134
135 if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; }
136 t=euclid(a,b);
137 if (t == NULL) goto err;
138
139 if (BN_copy(r,t) == NULL) goto err;
140 ret=1;
141err:
142 BN_CTX_end(ctx);
143 return(ret);
144 }
145
146static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
147 {
148 BIGNUM *t;
149 int shifts=0;
150
151 bn_check_top(a);
152 bn_check_top(b);
153
154 /* 0 <= b <= a */
155 while (!BN_is_zero(b))
156 {
157 /* 0 < b <= a */
158
159 if (BN_is_odd(a))
160 {
161 if (BN_is_odd(b))
162 {
163 if (!BN_sub(a,a,b)) goto err;
164 if (!BN_rshift1(a,a)) goto err;
165 if (BN_cmp(a,b) < 0)
166 { t=a; a=b; b=t; }
167 }
168 else /* a odd - b even */
169 {
170 if (!BN_rshift1(b,b)) goto err;
171 if (BN_cmp(a,b) < 0)
172 { t=a; a=b; b=t; }
173 }
174 }
175 else /* a is even */
176 {
177 if (BN_is_odd(b))
178 {
179 if (!BN_rshift1(a,a)) goto err;
180 if (BN_cmp(a,b) < 0)
181 { t=a; a=b; b=t; }
182 }
183 else /* a even - b even */
184 {
185 if (!BN_rshift1(a,a)) goto err;
186 if (!BN_rshift1(b,b)) goto err;
187 shifts++;
188 }
189 }
190 /* 0 <= b <= a */
191 }
192
193 if (shifts)
194 {
195 if (!BN_lshift(a,a,shifts)) goto err;
196 }
197 return(a);
198err:
199 return(NULL);
200 }
201
202
203/* solves ax == 1 (mod n) */
204BIGNUM *BN_mod_inverse(BIGNUM *in,
205 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
206 {
207 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
208 BIGNUM *ret=NULL;
209 int sign;
210
211 bn_check_top(a);
212 bn_check_top(n);
213
214 BN_CTX_start(ctx);
215 A = BN_CTX_get(ctx);
216 B = BN_CTX_get(ctx);
217 X = BN_CTX_get(ctx);
218 D = BN_CTX_get(ctx);
219 M = BN_CTX_get(ctx);
220 Y = BN_CTX_get(ctx);
221 T = BN_CTX_get(ctx);
222 if (T == NULL) goto err;
223
224 if (in == NULL)
225 R=BN_new();
226 else
227 R=in;
228 if (R == NULL) goto err;
229
230 BN_one(X);
231 BN_zero(Y);
232 if (BN_copy(B,a) == NULL) goto err;
233 if (BN_copy(A,n) == NULL) goto err;
234 A->neg = 0;
235 if (B->neg || (BN_ucmp(B, A) >= 0))
236 {
237 if (!BN_nnmod(B, B, A, ctx)) goto err;
238 }
239 sign = -1;
240 /* From B = a mod |n|, A = |n| it follows that
241 *
242 * 0 <= B < A,
243 * -sign*X*a == B (mod |n|),
244 * sign*Y*a == A (mod |n|).
245 */
246
247 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
248 {
249 /* Binary inversion algorithm; requires odd modulus.
250 * This is faster than the general algorithm if the modulus
251 * is sufficiently small (about 400 .. 500 bits on 32-bit
252 * sytems, but much more on 64-bit systems) */
253 int shift;
254
255 while (!BN_is_zero(B))
256 {
257 /*
258 * 0 < B < |n|,
259 * 0 < A <= |n|,
260 * (1) -sign*X*a == B (mod |n|),
261 * (2) sign*Y*a == A (mod |n|)
262 */
263
264 /* Now divide B by the maximum possible power of two in the integers,
265 * and divide X by the same value mod |n|.
266 * When we're done, (1) still holds. */
267 shift = 0;
268 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
269 {
270 shift++;
271
272 if (BN_is_odd(X))
273 {
274 if (!BN_uadd(X, X, n)) goto err;
275 }
276 /* now X is even, so we can easily divide it by two */
277 if (!BN_rshift1(X, X)) goto err;
278 }
279 if (shift > 0)
280 {
281 if (!BN_rshift(B, B, shift)) goto err;
282 }
283
284
285 /* Same for A and Y. Afterwards, (2) still holds. */
286 shift = 0;
287 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
288 {
289 shift++;
290
291 if (BN_is_odd(Y))
292 {
293 if (!BN_uadd(Y, Y, n)) goto err;
294 }
295 /* now Y is even */
296 if (!BN_rshift1(Y, Y)) goto err;
297 }
298 if (shift > 0)
299 {
300 if (!BN_rshift(A, A, shift)) goto err;
301 }
302
303
304 /* We still have (1) and (2).
305 * Both A and B are odd.
306 * The following computations ensure that
307 *
308 * 0 <= B < |n|,
309 * 0 < A < |n|,
310 * (1) -sign*X*a == B (mod |n|),
311 * (2) sign*Y*a == A (mod |n|),
312 *
313 * and that either A or B is even in the next iteration.
314 */
315 if (BN_ucmp(B, A) >= 0)
316 {
317 /* -sign*(X + Y)*a == B - A (mod |n|) */
318 if (!BN_uadd(X, X, Y)) goto err;
319 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
320 * actually makes the algorithm slower */
321 if (!BN_usub(B, B, A)) goto err;
322 }
323 else
324 {
325 /* sign*(X + Y)*a == A - B (mod |n|) */
326 if (!BN_uadd(Y, Y, X)) goto err;
327 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
328 if (!BN_usub(A, A, B)) goto err;
329 }
330 }
331 }
332 else
333 {
334 /* general inversion algorithm */
335
336 while (!BN_is_zero(B))
337 {
338 BIGNUM *tmp;
339
340 /*
341 * 0 < B < A,
342 * (*) -sign*X*a == B (mod |n|),
343 * sign*Y*a == A (mod |n|)
344 */
345
346 /* (D, M) := (A/B, A%B) ... */
347 if (BN_num_bits(A) == BN_num_bits(B))
348 {
349 if (!BN_one(D)) goto err;
350 if (!BN_sub(M,A,B)) goto err;
351 }
352 else if (BN_num_bits(A) == BN_num_bits(B) + 1)
353 {
354 /* A/B is 1, 2, or 3 */
355 if (!BN_lshift1(T,B)) goto err;
356 if (BN_ucmp(A,T) < 0)
357 {
358 /* A < 2*B, so D=1 */
359 if (!BN_one(D)) goto err;
360 if (!BN_sub(M,A,B)) goto err;
361 }
362 else
363 {
364 /* A >= 2*B, so D=2 or D=3 */
365 if (!BN_sub(M,A,T)) goto err;
366 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
367 if (BN_ucmp(A,D) < 0)
368 {
369 /* A < 3*B, so D=2 */
370 if (!BN_set_word(D,2)) goto err;
371 /* M (= A - 2*B) already has the correct value */
372 }
373 else
374 {
375 /* only D=3 remains */
376 if (!BN_set_word(D,3)) goto err;
377 /* currently M = A - 2*B, but we need M = A - 3*B */
378 if (!BN_sub(M,M,B)) goto err;
379 }
380 }
381 }
382 else
383 {
384 if (!BN_div(D,M,A,B,ctx)) goto err;
385 }
386
387 /* Now
388 * A = D*B + M;
389 * thus we have
390 * (**) sign*Y*a == D*B + M (mod |n|).
391 */
392
393 tmp=A; /* keep the BIGNUM object, the value does not matter */
394
395 /* (A, B) := (B, A mod B) ... */
396 A=B;
397 B=M;
398 /* ... so we have 0 <= B < A again */
399
400 /* Since the former M is now B and the former B is now A,
401 * (**) translates into
402 * sign*Y*a == D*A + B (mod |n|),
403 * i.e.
404 * sign*Y*a - D*A == B (mod |n|).
405 * Similarly, (*) translates into
406 * -sign*X*a == A (mod |n|).
407 *
408 * Thus,
409 * sign*Y*a + D*sign*X*a == B (mod |n|),
410 * i.e.
411 * sign*(Y + D*X)*a == B (mod |n|).
412 *
413 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
414 * -sign*X*a == B (mod |n|),
415 * sign*Y*a == A (mod |n|).
416 * Note that X and Y stay non-negative all the time.
417 */
418
419 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
420 if (BN_is_one(D))
421 {
422 if (!BN_add(tmp,X,Y)) goto err;
423 }
424 else
425 {
426 if (BN_is_word(D,2))
427 {
428 if (!BN_lshift1(tmp,X)) goto err;
429 }
430 else if (BN_is_word(D,4))
431 {
432 if (!BN_lshift(tmp,X,2)) goto err;
433 }
434 else if (D->top == 1)
435 {
436 if (!BN_copy(tmp,X)) goto err;
437 if (!BN_mul_word(tmp,D->d[0])) goto err;
438 }
439 else
440 {
441 if (!BN_mul(tmp,D,X,ctx)) goto err;
442 }
443 if (!BN_add(tmp,tmp,Y)) goto err;
444 }
445
446 M=Y; /* keep the BIGNUM object, the value does not matter */
447 Y=X;
448 X=tmp;
449 sign = -sign;
450 }
451 }
452
453 /*
454 * The while loop (Euclid's algorithm) ends when
455 * A == gcd(a,n);
456 * we have
457 * sign*Y*a == A (mod |n|),
458 * where Y is non-negative.
459 */
460
461 if (sign < 0)
462 {
463 if (!BN_sub(Y,n,Y)) goto err;
464 }
465 /* Now Y*a == A (mod |n|). */
466
467
468 if (BN_is_one(A))
469 {
470 /* Y*a == 1 (mod |n|) */
471 if (!Y->neg && BN_ucmp(Y,n) < 0)
472 {
473 if (!BN_copy(R,Y)) goto err;
474 }
475 else
476 {
477 if (!BN_nnmod(R,Y,n,ctx)) goto err;
478 }
479 }
480 else
481 {
482 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
483 goto err;
484 }
485 ret=R;
486err:
487 if ((ret == NULL) && (in == NULL)) BN_free(R);
488 BN_CTX_end(ctx);
489 return(ret);
490 }
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