1 /* crypto/bn/bn_gcd.c */
2 /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
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.
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).
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.
23 * Redistribution and use in source and binary forms, with or without
24 * modification, are permitted provided that the following conditions
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)"
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
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.]
58 /* ====================================================================
59 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
61 * Redistribution and use in source and binary forms, with or without
62 * modification, are permitted provided that the following conditions
65 * 1. Redistributions of source code must retain the above copyright
66 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
87 * 6. Redistributions of any form whatsoever must retain the following
89 * "This product includes software developed by the OpenSSL Project
90 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
112 #include "cryptlib.h"
115 static BIGNUM
*euclid(BIGNUM
*a
, BIGNUM
*b
);
117 int BN_gcd(BIGNUM
*r
, const BIGNUM
*in_a
, const BIGNUM
*in_b
, BN_CTX
*ctx
)
128 if (a
== NULL
|| b
== NULL
) goto err
;
130 if (BN_copy(a
,in_a
) == NULL
) goto err
;
131 if (BN_copy(b
,in_b
) == NULL
) goto err
;
135 if (BN_cmp(a
,b
) < 0) { t
=a
; a
=b
; b
=t
; }
137 if (t
== NULL
) goto err
;
139 if (BN_copy(r
,t
) == NULL
) goto err
;
146 static BIGNUM
*euclid(BIGNUM
*a
, BIGNUM
*b
)
155 while (!BN_is_zero(b
))
163 if (!BN_sub(a
,a
,b
)) goto err
;
164 if (!BN_rshift1(a
,a
)) goto err
;
168 else /* a odd - b even */
170 if (!BN_rshift1(b
,b
)) goto err
;
179 if (!BN_rshift1(a
,a
)) goto err
;
183 else /* a even - b even */
185 if (!BN_rshift1(a
,a
)) goto err
;
186 if (!BN_rshift1(b
,b
)) goto err
;
195 if (!BN_lshift(a
,a
,shifts
)) goto err
;
203 /* solves ax == 1 (mod n) */
204 BIGNUM
*BN_mod_inverse(BIGNUM
*in
,
205 const BIGNUM
*a
, const BIGNUM
*n
, BN_CTX
*ctx
)
207 BIGNUM
*A
,*B
,*X
,*Y
,*M
,*D
,*T
,*R
=NULL
;
222 if (T
== NULL
) goto err
;
228 if (R
== NULL
) goto err
;
232 if (BN_copy(B
,a
) == NULL
) goto err
;
233 if (BN_copy(A
,n
) == NULL
) goto err
;
235 if (B
->neg
|| (BN_ucmp(B
, A
) >= 0))
237 if (!BN_nnmod(B
, B
, A
, ctx
)) goto err
;
240 /* From B = a mod |n|, A = |n| it follows that
243 * -sign*X*a == B (mod |n|),
244 * sign*Y*a == A (mod |n|).
247 if (BN_is_odd(n
) && (BN_num_bits(n
) <= (BN_BITS
<= 32 ? 450 : 2048)))
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) */
255 while (!BN_is_zero(B
))
260 * (1) -sign*X*a == B (mod |n|),
261 * (2) sign*Y*a == A (mod |n|)
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. */
268 while (!BN_is_bit_set(B
, shift
)) /* note that 0 < B */
274 if (!BN_uadd(X
, X
, n
)) goto err
;
276 /* now X is even, so we can easily divide it by two */
277 if (!BN_rshift1(X
, X
)) goto err
;
281 if (!BN_rshift(B
, B
, shift
)) goto err
;
285 /* Same for A and Y. Afterwards, (2) still holds. */
287 while (!BN_is_bit_set(A
, shift
)) /* note that 0 < A */
293 if (!BN_uadd(Y
, Y
, n
)) goto err
;
296 if (!BN_rshift1(Y
, Y
)) goto err
;
300 if (!BN_rshift(A
, A
, shift
)) goto err
;
304 /* We still have (1) and (2).
305 * Both A and B are odd.
306 * The following computations ensure that
310 * (1) -sign*X*a == B (mod |n|),
311 * (2) sign*Y*a == A (mod |n|),
313 * and that either A or B is even in the next iteration.
315 if (BN_ucmp(B
, A
) >= 0)
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
;
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
;
334 /* general inversion algorithm */
336 while (!BN_is_zero(B
))
342 * (*) -sign*X*a == B (mod |n|),
343 * sign*Y*a == A (mod |n|)
346 /* (D, M) := (A/B, A%B) ... */
347 if (BN_num_bits(A
) == BN_num_bits(B
))
349 if (!BN_one(D
)) goto err
;
350 if (!BN_sub(M
,A
,B
)) goto err
;
352 else if (BN_num_bits(A
) == BN_num_bits(B
) + 1)
354 /* A/B is 1, 2, or 3 */
355 if (!BN_lshift1(T
,B
)) goto err
;
356 if (BN_ucmp(A
,T
) < 0)
358 /* A < 2*B, so D=1 */
359 if (!BN_one(D
)) goto err
;
360 if (!BN_sub(M
,A
,B
)) goto err
;
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)
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 */
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
;
384 if (!BN_div(D
,M
,A
,B
,ctx
)) goto err
;
390 * (**) sign*Y*a == D*B + M (mod |n|).
393 tmp
=A
; /* keep the BIGNUM object, the value does not matter */
395 /* (A, B) := (B, A mod B) ... */
398 /* ... so we have 0 <= B < A again */
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|),
404 * sign*Y*a - D*A == B (mod |n|).
405 * Similarly, (*) translates into
406 * -sign*X*a == A (mod |n|).
409 * sign*Y*a + D*sign*X*a == B (mod |n|),
411 * sign*(Y + D*X)*a == B (mod |n|).
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.
419 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
422 if (!BN_add(tmp
,X
,Y
)) goto err
;
428 if (!BN_lshift1(tmp
,X
)) goto err
;
430 else if (BN_is_word(D
,4))
432 if (!BN_lshift(tmp
,X
,2)) goto err
;
434 else if (D
->top
== 1)
436 if (!BN_copy(tmp
,X
)) goto err
;
437 if (!BN_mul_word(tmp
,D
->d
[0])) goto err
;
441 if (!BN_mul(tmp
,D
,X
,ctx
)) goto err
;
443 if (!BN_add(tmp
,tmp
,Y
)) goto err
;
446 M
=Y
; /* keep the BIGNUM object, the value does not matter */
454 * The while loop (Euclid's algorithm) ends when
457 * sign*Y*a == A (mod |n|),
458 * where Y is non-negative.
463 if (!BN_sub(Y
,n
,Y
)) goto err
;
465 /* Now Y*a == A (mod |n|). */
470 /* Y*a == 1 (mod |n|) */
471 if (!Y
->neg
&& BN_ucmp(Y
,n
) < 0)
473 if (!BN_copy(R
,Y
)) goto err
;
477 if (!BN_nnmod(R
,Y
,n
,ctx
)) goto err
;
482 BNerr(BN_F_BN_MOD_INVERSE
,BN_R_NO_INVERSE
);
487 if ((ret
== NULL
) && (in
== NULL
)) BN_free(R
);