1 /* crypto/bn/bn_exp.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) 1998-2000 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).
113 #include "cryptlib.h"
116 #define TABLE_SIZE 32
118 /* this one works - simple but works */
119 int BN_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
125 if ((r == a) || (r == p))
126 rr = BN_CTX_get(ctx);
129 if ((v = BN_CTX_get(ctx)) == NULL) goto err;
131 if (BN_copy(v,a) == NULL) goto err;
135 { if (BN_copy(rr,a) == NULL) goto err; }
136 else { if (!BN_one(rr)) goto err; }
138 for (i=1; i<bits; i++)
140 if (!BN_sqr(v,v,ctx)) goto err;
141 if (BN_is_bit_set(p,i))
143 if (!BN_mul(rr,rr,v,ctx)) goto err;
148 if (r != rr) BN_copy(r,rr);
155 int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
164 /* For even modulus m = 2^k*m_odd, it might make sense to compute
165 * a^p mod m_odd and a^p mod 2^k separately (with Montgomery
166 * exponentiation for the odd part), using appropriate exponent
167 * reductions, and combine the results using the CRT.
169 * For now, we use Montgomery only if the modulus is odd; otherwise,
170 * exponentiation using the reciprocal-based quick remaindering
173 * (Timing obtained with expspeed.c [computations a^p mod m
174 * where a, p, m are of the same length: 256, 512, 1024, 2048,
175 * 4096, 8192 bits], compared to the running time of the
176 * standard algorithm:
178 * BN_mod_exp_mont 33 .. 40 % [AMD K6-2, Linux, debug configuration]
179 * 55 .. 77 % [UltraSparc processor, but
180 * debug-solaris-sparcv8-gcc conf.]
182 * BN_mod_exp_recp 50 .. 70 % [AMD K6-2, Linux, debug configuration]
183 * 62 .. 118 % [UltraSparc, debug-solaris-sparcv8-gcc]
185 * On the Sparc, BN_mod_exp_recp was faster than BN_mod_exp_mont
186 * at 2048 and more bits, but at 512 and 1024 bits, it was
187 * slower even than the standard algorithm!
189 * "Real" timings [linux-elf, solaris-sparcv9-gcc configurations]
190 * should be obtained when the new Montgomery reduction code
191 * has been integrated into OpenSSL.)
195 #define MONT_EXP_WORD
199 /* I have finally been able to take out this pre-condition of
200 * the top bit being set. It was caused by an error in BN_div
201 * with negatives. There was also another problem when for a^b%m
202 * a >= m. eay 07-May-97 */
203 /* if ((m->d[m->top-1]&BN_TBIT) && BN_is_odd(m)) */
207 # ifdef MONT_EXP_WORD
208 if (a->top == 1 && !a->neg)
210 BN_ULONG A = a->d[0];
211 ret=BN_mod_exp_mont_word(r,A,p,m,ctx,NULL);
215 ret=BN_mod_exp_mont(r,a,p,m,ctx,NULL);
220 { ret=BN_mod_exp_recp(r,a,p,m,ctx); }
222 { ret=BN_mod_exp_simple(r,a,p,m,ctx); }
230 int BN_mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
231 const BIGNUM *m, BN_CTX *ctx)
233 int i,j,bits,ret=0,wstart,wend,window,wvalue;
236 BIGNUM val[TABLE_SIZE];
248 if ((aa = BN_CTX_get(ctx)) == NULL) goto err;
250 BN_RECP_CTX_init(&recp);
253 /* ignore sign of 'm' */
254 if (!BN_copy(aa, m)) goto err;
256 if (BN_RECP_CTX_set(&recp,aa,ctx) <= 0) goto err;
260 if (BN_RECP_CTX_set(&recp,m,ctx) <= 0) goto err;
266 if (!BN_nnmod(&(val[0]),a,m,ctx)) goto err; /* 1 */
267 if (BN_is_zero(&(val[0])))
273 window = BN_window_bits_for_exponent_size(bits);
276 if (!BN_mod_mul_reciprocal(aa,&(val[0]),&(val[0]),&recp,ctx))
282 if (!BN_mod_mul_reciprocal(&(val[i]),&(val[i-1]),aa,&recp,ctx))
288 start=1; /* This is used to avoid multiplication etc
289 * when there is only the value '1' in the
291 wvalue=0; /* The 'value' of the window */
292 wstart=bits-1; /* The top bit of the window */
293 wend=0; /* The bottom bit of the window */
295 if (!BN_one(r)) goto err;
299 if (BN_is_bit_set(p,wstart) == 0)
302 if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx))
304 if (wstart == 0) break;
308 /* We now have wstart on a 'set' bit, we now need to work out
309 * how bit a window to do. To do this we need to scan
310 * forward until the last set bit before the end of the
315 for (i=1; i<window; i++)
317 if (wstart-i < 0) break;
318 if (BN_is_bit_set(p,wstart-i))
326 /* wend is the size of the current window */
328 /* add the 'bytes above' */
332 if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx))
336 /* wvalue will be an odd number < 2^window */
337 if (!BN_mod_mul_reciprocal(r,r,&(val[wvalue>>1]),&recp,ctx))
340 /* move the 'window' down further */
344 if (wstart < 0) break;
350 BN_clear_free(&(val[i]));
351 BN_RECP_CTX_free(&recp);
357 int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
358 const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont)
360 int i,j,bits,ret=0,wstart,wend,window,wvalue;
364 BIGNUM val[TABLE_SIZE];
365 BN_MONT_CTX *mont=NULL;
373 BNerr(BN_F_BN_MOD_EXP_MONT,BN_R_CALLED_WITH_EVEN_MODULUS);
386 if (d == NULL || r == NULL) goto err;
388 /* If this is not done, things will break in the montgomery
395 if ((mont=BN_MONT_CTX_new()) == NULL) goto err;
396 if (!BN_MONT_CTX_set(mont,m,ctx)) goto err;
401 if (a->neg || BN_ucmp(a,m) >= 0)
403 if (!BN_nnmod(&(val[0]),a,m,ctx))
414 if (!BN_to_montgomery(&(val[0]),aa,mont,ctx)) goto err; /* 1 */
416 window = BN_window_bits_for_exponent_size(bits);
419 if (!BN_mod_mul_montgomery(d,&(val[0]),&(val[0]),mont,ctx)) goto err; /* 2 */
424 if (!BN_mod_mul_montgomery(&(val[i]),&(val[i-1]),d,mont,ctx))
430 start=1; /* This is used to avoid multiplication etc
431 * when there is only the value '1' in the
433 wvalue=0; /* The 'value' of the window */
434 wstart=bits-1; /* The top bit of the window */
435 wend=0; /* The bottom bit of the window */
437 if (!BN_to_montgomery(r,BN_value_one(),mont,ctx)) goto err;
440 if (BN_is_bit_set(p,wstart) == 0)
444 if (!BN_mod_mul_montgomery(r,r,r,mont,ctx))
447 if (wstart == 0) break;
451 /* We now have wstart on a 'set' bit, we now need to work out
452 * how bit a window to do. To do this we need to scan
453 * forward until the last set bit before the end of the
458 for (i=1; i<window; i++)
460 if (wstart-i < 0) break;
461 if (BN_is_bit_set(p,wstart-i))
469 /* wend is the size of the current window */
471 /* add the 'bytes above' */
475 if (!BN_mod_mul_montgomery(r,r,r,mont,ctx))
479 /* wvalue will be an odd number < 2^window */
480 if (!BN_mod_mul_montgomery(r,r,&(val[wvalue>>1]),mont,ctx))
483 /* move the 'window' down further */
487 if (wstart < 0) break;
489 if (!BN_from_montgomery(rr,r,mont,ctx)) goto err;
492 if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont);
495 BN_clear_free(&(val[i]));
500 int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
501 const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont)
503 BN_MONT_CTX *mont = NULL;
509 #define BN_MOD_MUL_WORD(r, w, m) \
510 (BN_mul_word(r, (w)) && \
511 (/* BN_ucmp(r, (m)) < 0 ? 1 :*/ \
512 (BN_mod(t, r, m, ctx) && (swap_tmp = r, r = t, t = swap_tmp, 1))))
513 /* BN_MOD_MUL_WORD is only used with 'w' large,
514 * so the BN_ucmp test is probably more overhead
515 * than always using BN_mod (which uses BN_copy if
516 * a similar test returns true). */
517 /* We can use BN_mod and do not need BN_nnmod because our
518 * accumulator is never negative (the result of BN_mod does
519 * not depend on the sign of the modulus).
521 #define BN_TO_MONTGOMERY_WORD(r, w, mont) \
522 (BN_set_word(r, (w)) && BN_to_montgomery(r, r, (mont), ctx))
527 if (m->top == 0 || !(m->d[0] & 1))
529 BNerr(BN_F_BN_MOD_EXP_MONT_WORD,BN_R_CALLED_WITH_EVEN_MODULUS);
533 a %= m->d[0]; /* make sure that 'a' is reduced */
535 bits = BN_num_bits(p);
551 if (d == NULL || r == NULL || t == NULL) goto err;
557 if ((mont = BN_MONT_CTX_new()) == NULL) goto err;
558 if (!BN_MONT_CTX_set(mont, m, ctx)) goto err;
561 r_is_one = 1; /* except for Montgomery factor */
565 /* The result is accumulated in the product r*w. */
566 w = a; /* bit 'bits-1' of 'p' is always set */
567 for (b = bits-2; b >= 0; b--)
569 /* First, square r*w. */
571 if ((next_w/w) != w) /* overflow */
575 if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
580 if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
587 if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) goto err;
590 /* Second, multiply r*w by 'a' if exponent bit is set. */
591 if (BN_is_bit_set(p, b))
594 if ((next_w/a) != w) /* overflow */
598 if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
603 if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
611 /* Finally, set r:=r*w. */
616 if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
621 if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
625 if (r_is_one) /* can happen only if a == 1*/
627 if (!BN_one(rr)) goto err;
631 if (!BN_from_montgomery(rr, r, mont, ctx)) goto err;
635 if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont);
642 /* The old fallback, simple version :-) */
643 int BN_mod_exp_simple(BIGNUM *r,
644 const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
647 int i,j,bits,ret=0,wstart,wend,window,wvalue,ts=0;
650 BIGNUM val[TABLE_SIZE];
661 if ((d = BN_CTX_get(ctx)) == NULL) goto err;
665 if (!BN_nnmod(&(val[0]),a,m,ctx)) goto err; /* 1 */
666 if (BN_is_zero(&(val[0])))
672 window = BN_window_bits_for_exponent_size(bits);
675 if (!BN_mod_mul(d,&(val[0]),&(val[0]),m,ctx))
681 if (!BN_mod_mul(&(val[i]),&(val[i-1]),d,m,ctx))
687 start=1; /* This is used to avoid multiplication etc
688 * when there is only the value '1' in the
690 wvalue=0; /* The 'value' of the window */
691 wstart=bits-1; /* The top bit of the window */
692 wend=0; /* The bottom bit of the window */
694 if (!BN_one(r)) goto err;
698 if (BN_is_bit_set(p,wstart) == 0)
701 if (!BN_mod_mul(r,r,r,m,ctx))
703 if (wstart == 0) break;
707 /* We now have wstart on a 'set' bit, we now need to work out
708 * how bit a window to do. To do this we need to scan
709 * forward until the last set bit before the end of the
714 for (i=1; i<window; i++)
716 if (wstart-i < 0) break;
717 if (BN_is_bit_set(p,wstart-i))
725 /* wend is the size of the current window */
727 /* add the 'bytes above' */
731 if (!BN_mod_mul(r,r,r,m,ctx))
735 /* wvalue will be an odd number < 2^window */
736 if (!BN_mod_mul(r,r,&(val[wvalue>>1]),m,ctx))
739 /* move the 'window' down further */
743 if (wstart < 0) break;
749 BN_clear_free(&(val[i]));