1 /* crypto/bn/bn_mul.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.]
60 # undef NDEBUG /* avoid conflicting definitions */
65 #include "internal/cryptlib.h"
68 #if defined(OPENSSL_NO_ASM) || !defined(OPENSSL_BN_ASM_PART_WORDS)
70 * Here follows specialised variants of bn_add_words() and bn_sub_words().
71 * They have the property performing operations on arrays of different sizes.
72 * The sizes of those arrays is expressed through cl, which is the common
73 * length ( basicall, min(len(a),len(b)) ), and dl, which is the delta
74 * between the two lengths, calculated as len(a)-len(b). All lengths are the
75 * number of BN_ULONGs... For the operations that require a result array as
76 * parameter, it must have the length cl+abs(dl). These functions should
77 * probably end up in bn_asm.c as soon as there are assembler counterparts
78 * for the systems that use assembler files.
81 BN_ULONG bn_sub_part_words(BN_ULONG *r,
82 const BN_ULONG *a, const BN_ULONG *b,
88 c = bn_sub_words(r, a, b, cl);
100 r[0] = (0 - t - c) & BN_MASK2;
107 r[1] = (0 - t - c) & BN_MASK2;
114 r[2] = (0 - t - c) & BN_MASK2;
121 r[3] = (0 - t - c) & BN_MASK2;
134 r[0] = (t - c) & BN_MASK2;
141 r[1] = (t - c) & BN_MASK2;
148 r[2] = (t - c) & BN_MASK2;
155 r[3] = (t - c) & BN_MASK2;
167 switch (save_dl - dl) {
209 BN_ULONG bn_add_part_words(BN_ULONG *r,
210 const BN_ULONG *a, const BN_ULONG *b,
216 c = bn_add_words(r, a, b, cl);
228 l = (c + b[0]) & BN_MASK2;
234 l = (c + b[1]) & BN_MASK2;
240 l = (c + b[2]) & BN_MASK2;
246 l = (c + b[3]) & BN_MASK2;
258 switch (dl - save_dl) {
298 t = (a[0] + c) & BN_MASK2;
304 t = (a[1] + c) & BN_MASK2;
310 t = (a[2] + c) & BN_MASK2;
316 t = (a[3] + c) & BN_MASK2;
328 switch (save_dl - dl) {
371 * Karatsuba recursive multiplication algorithm (cf. Knuth, The Art of
372 * Computer Programming, Vol. 2)
376 * r is 2*n2 words in size,
377 * a and b are both n2 words in size.
378 * n2 must be a power of 2.
379 * We multiply and return the result.
380 * t must be 2*n2 words in size
383 * a[0]*b[0]+a[1]*b[1]+(a[0]-a[1])*(b[1]-b[0])
386 /* dnX may not be positive, but n2/2+dnX has to be */
387 void bn_mul_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
388 int dna, int dnb, BN_ULONG *t)
390 int n = n2 / 2, c1, c2;
391 int tna = n + dna, tnb = n + dnb;
392 unsigned int neg, zero;
398 bn_mul_comba4(r, a, b);
403 * Only call bn_mul_comba 8 if n2 == 8 and the two arrays are complete
406 if (n2 == 8 && dna == 0 && dnb == 0) {
407 bn_mul_comba8(r, a, b);
410 # endif /* BN_MUL_COMBA */
411 /* Else do normal multiply */
412 if (n2 < BN_MUL_RECURSIVE_SIZE_NORMAL) {
413 bn_mul_normal(r, a, n2 + dna, b, n2 + dnb);
415 memset(&r[2 * n2 + dna + dnb], 0,
416 sizeof(BN_ULONG) * -(dna + dnb));
419 /* r=(a[0]-a[1])*(b[1]-b[0]) */
420 c1 = bn_cmp_part_words(a, &(a[n]), tna, n - tna);
421 c2 = bn_cmp_part_words(&(b[n]), b, tnb, tnb - n);
423 switch (c1 * 3 + c2) {
425 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
426 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
432 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
433 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n); /* + */
442 bn_sub_part_words(t, a, &(a[n]), tna, n - tna); /* + */
443 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
450 bn_sub_part_words(t, a, &(a[n]), tna, n - tna);
451 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n);
456 if (n == 4 && dna == 0 && dnb == 0) { /* XXX: bn_mul_comba4 could take
457 * extra args to do this well */
459 bn_mul_comba4(&(t[n2]), t, &(t[n]));
461 memset(&t[n2], 0, sizeof(*t) * 8);
463 bn_mul_comba4(r, a, b);
464 bn_mul_comba4(&(r[n2]), &(a[n]), &(b[n]));
465 } else if (n == 8 && dna == 0 && dnb == 0) { /* XXX: bn_mul_comba8 could
466 * take extra args to do
469 bn_mul_comba8(&(t[n2]), t, &(t[n]));
471 memset(&t[n2], 0, sizeof(*t) * 16);
473 bn_mul_comba8(r, a, b);
474 bn_mul_comba8(&(r[n2]), &(a[n]), &(b[n]));
476 # endif /* BN_MUL_COMBA */
480 bn_mul_recursive(&(t[n2]), t, &(t[n]), n, 0, 0, p);
482 memset(&t[n2], 0, sizeof(*t) * n2);
483 bn_mul_recursive(r, a, b, n, 0, 0, p);
484 bn_mul_recursive(&(r[n2]), &(a[n]), &(b[n]), n, dna, dnb, p);
488 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
489 * r[10] holds (a[0]*b[0])
490 * r[32] holds (b[1]*b[1])
493 c1 = (int)(bn_add_words(t, r, &(r[n2]), n2));
495 if (neg) { /* if t[32] is negative */
496 c1 -= (int)(bn_sub_words(&(t[n2]), t, &(t[n2]), n2));
498 /* Might have a carry */
499 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), t, n2));
503 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
504 * r[10] holds (a[0]*b[0])
505 * r[32] holds (b[1]*b[1])
506 * c1 holds the carry bits
508 c1 += (int)(bn_add_words(&(r[n]), &(r[n]), &(t[n2]), n2));
512 ln = (lo + c1) & BN_MASK2;
516 * The overflow will stop before we over write words we should not
519 if (ln < (BN_ULONG)c1) {
523 ln = (lo + 1) & BN_MASK2;
531 * n+tn is the word length t needs to be n*4 is size, as does r
533 /* tnX may not be negative but less than n */
534 void bn_mul_part_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n,
535 int tna, int tnb, BN_ULONG *t)
537 int i, j, n2 = n * 2;
542 bn_mul_normal(r, a, n + tna, b, n + tnb);
546 /* r=(a[0]-a[1])*(b[1]-b[0]) */
547 c1 = bn_cmp_part_words(a, &(a[n]), tna, n - tna);
548 c2 = bn_cmp_part_words(&(b[n]), b, tnb, tnb - n);
550 switch (c1 * 3 + c2) {
552 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
553 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
558 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
559 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n); /* + */
567 bn_sub_part_words(t, a, &(a[n]), tna, n - tna); /* + */
568 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
574 bn_sub_part_words(t, a, &(a[n]), tna, n - tna);
575 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n);
579 * The zero case isn't yet implemented here. The speedup would probably
584 bn_mul_comba4(&(t[n2]), t, &(t[n]));
585 bn_mul_comba4(r, a, b);
586 bn_mul_normal(&(r[n2]), &(a[n]), tn, &(b[n]), tn);
587 memset(&r[n2 + tn * 2], 0, sizeof(*r) * (n2 - tn * 2));
591 bn_mul_comba8(&(t[n2]), t, &(t[n]));
592 bn_mul_comba8(r, a, b);
593 bn_mul_normal(&(r[n2]), &(a[n]), tna, &(b[n]), tnb);
594 memset(&r[n2 + tna + tnb], 0, sizeof(*r) * (n2 - tna - tnb));
597 bn_mul_recursive(&(t[n2]), t, &(t[n]), n, 0, 0, p);
598 bn_mul_recursive(r, a, b, n, 0, 0, p);
601 * If there is only a bottom half to the number, just do it
608 bn_mul_recursive(&(r[n2]), &(a[n]), &(b[n]),
609 i, tna - i, tnb - i, p);
610 memset(&r[n2 + i * 2], 0, sizeof(*r) * (n2 - i * 2));
611 } else if (j > 0) { /* eg, n == 16, i == 8 and tn == 11 */
612 bn_mul_part_recursive(&(r[n2]), &(a[n]), &(b[n]),
613 i, tna - i, tnb - i, p);
614 memset(&(r[n2 + tna + tnb]), 0,
615 sizeof(BN_ULONG) * (n2 - tna - tnb));
616 } else { /* (j < 0) eg, n == 16, i == 8 and tn == 5 */
618 memset(&r[n2], 0, sizeof(*r) * n2);
619 if (tna < BN_MUL_RECURSIVE_SIZE_NORMAL
620 && tnb < BN_MUL_RECURSIVE_SIZE_NORMAL) {
621 bn_mul_normal(&(r[n2]), &(a[n]), tna, &(b[n]), tnb);
626 * these simplified conditions work exclusively because
627 * difference between tna and tnb is 1 or 0
629 if (i < tna || i < tnb) {
630 bn_mul_part_recursive(&(r[n2]),
632 i, tna - i, tnb - i, p);
634 } else if (i == tna || i == tnb) {
635 bn_mul_recursive(&(r[n2]),
637 i, tna - i, tnb - i, p);
646 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
647 * r[10] holds (a[0]*b[0])
648 * r[32] holds (b[1]*b[1])
651 c1 = (int)(bn_add_words(t, r, &(r[n2]), n2));
653 if (neg) { /* if t[32] is negative */
654 c1 -= (int)(bn_sub_words(&(t[n2]), t, &(t[n2]), n2));
656 /* Might have a carry */
657 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), t, n2));
661 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
662 * r[10] holds (a[0]*b[0])
663 * r[32] holds (b[1]*b[1])
664 * c1 holds the carry bits
666 c1 += (int)(bn_add_words(&(r[n]), &(r[n]), &(t[n2]), n2));
670 ln = (lo + c1) & BN_MASK2;
674 * The overflow will stop before we over write words we should not
677 if (ln < (BN_ULONG)c1) {
681 ln = (lo + 1) & BN_MASK2;
689 * a and b must be the same size, which is n2.
690 * r needs to be n2 words and t needs to be n2*2
692 void bn_mul_low_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
697 bn_mul_recursive(r, a, b, n, 0, 0, &(t[0]));
698 if (n >= BN_MUL_LOW_RECURSIVE_SIZE_NORMAL) {
699 bn_mul_low_recursive(&(t[0]), &(a[0]), &(b[n]), n, &(t[n2]));
700 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
701 bn_mul_low_recursive(&(t[0]), &(a[n]), &(b[0]), n, &(t[n2]));
702 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
704 bn_mul_low_normal(&(t[0]), &(a[0]), &(b[n]), n);
705 bn_mul_low_normal(&(t[n]), &(a[n]), &(b[0]), n);
706 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
707 bn_add_words(&(r[n]), &(r[n]), &(t[n]), n);
712 * a and b must be the same size, which is n2.
713 * r needs to be n2 words and t needs to be n2*2
714 * l is the low words of the output.
717 void bn_mul_high(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, BN_ULONG *l, int n2,
723 BN_ULONG ll, lc, *lp, *mp;
727 /* Calculate (al-ah)*(bh-bl) */
729 c1 = bn_cmp_words(&(a[0]), &(a[n]), n);
730 c2 = bn_cmp_words(&(b[n]), &(b[0]), n);
731 switch (c1 * 3 + c2) {
733 bn_sub_words(&(r[0]), &(a[n]), &(a[0]), n);
734 bn_sub_words(&(r[n]), &(b[0]), &(b[n]), n);
740 bn_sub_words(&(r[0]), &(a[n]), &(a[0]), n);
741 bn_sub_words(&(r[n]), &(b[n]), &(b[0]), n);
750 bn_sub_words(&(r[0]), &(a[0]), &(a[n]), n);
751 bn_sub_words(&(r[n]), &(b[0]), &(b[n]), n);
758 bn_sub_words(&(r[0]), &(a[0]), &(a[n]), n);
759 bn_sub_words(&(r[n]), &(b[n]), &(b[0]), n);
764 /* t[10] = (a[0]-a[1])*(b[1]-b[0]) */
765 /* r[10] = (a[1]*b[1]) */
768 bn_mul_comba8(&(t[0]), &(r[0]), &(r[n]));
769 bn_mul_comba8(r, &(a[n]), &(b[n]));
773 bn_mul_recursive(&(t[0]), &(r[0]), &(r[n]), n, 0, 0, &(t[n2]));
774 bn_mul_recursive(r, &(a[n]), &(b[n]), n, 0, 0, &(t[n2]));
779 * s1 == low(ah*bh)+low((al-ah)*(bh-bl))+low(al*bl)+high(al*bl)
780 * We know s0 and s1 so the only unknown is high(al*bl)
781 * high(al*bl) == s1 - low(ah*bh+s0+(al-ah)*(bh-bl))
782 * high(al*bl) == s1 - (r[0]+l[0]+t[0])
786 c1 = (int)(bn_add_words(lp, &(r[0]), &(l[0]), n));
793 neg = (int)(bn_sub_words(&(t[n2]), lp, &(t[0]), n));
795 bn_add_words(&(t[n2]), lp, &(t[0]), n);
800 bn_sub_words(&(t[n2 + n]), &(l[n]), &(t[n2]), n);
804 for (i = 0; i < n; i++)
805 lp[i] = ((~mp[i]) + 1) & BN_MASK2;
811 * t[10] = (a[0]-a[1])*(b[1]-b[0]) neg is the sign
812 * r[10] = (a[1]*b[1])
816 * R[21] = al*bl + ah*bh + (a[0]-a[1])*(b[1]-b[0])
820 * R[1]=t[3]+l[0]+r[0](+-)t[0] (have carry/borrow)
821 * R[2]=r[0]+t[3]+r[1](+-)t[1] (have carry/borrow)
822 * R[3]=r[1]+(carry/borrow)
826 c1 = (int)(bn_add_words(lp, &(t[n2 + n]), &(l[0]), n));
831 c1 += (int)(bn_add_words(&(t[n2]), lp, &(r[0]), n));
833 c1 -= (int)(bn_sub_words(&(t[n2]), &(t[n2]), &(t[0]), n));
835 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), &(t[0]), n));
837 c2 = (int)(bn_add_words(&(r[0]), &(r[0]), &(t[n2 + n]), n));
838 c2 += (int)(bn_add_words(&(r[0]), &(r[0]), &(r[n]), n));
840 c2 -= (int)(bn_sub_words(&(r[0]), &(r[0]), &(t[n]), n));
842 c2 += (int)(bn_add_words(&(r[0]), &(r[0]), &(t[n]), n));
844 if (c1 != 0) { /* Add starting at r[0], could be +ve or -ve */
849 ll = (r[i] + lc) & BN_MASK2;
857 r[i++] = (ll - lc) & BN_MASK2;
862 if (c2 != 0) { /* Add starting at r[1] */
867 ll = (r[i] + lc) & BN_MASK2;
875 r[i++] = (ll - lc) & BN_MASK2;
881 #endif /* BN_RECURSION */
883 int BN_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
888 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
903 if ((al == 0) || (bl == 0)) {
910 if ((r == a) || (r == b)) {
911 if ((rr = BN_CTX_get(ctx)) == NULL)
915 rr->neg = a->neg ^ b->neg;
917 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
924 if (bn_wexpand(rr, 8) == NULL)
927 bn_mul_comba4(rr->d, a->d, b->d);
932 if (bn_wexpand(rr, 16) == NULL)
935 bn_mul_comba8(rr->d, a->d, b->d);
939 #endif /* BN_MUL_COMBA */
941 if ((al >= BN_MULL_SIZE_NORMAL) && (bl >= BN_MULL_SIZE_NORMAL)) {
942 if (i >= -1 && i <= 1) {
944 * Find out the power of two lower or equal to the longest of the
948 j = BN_num_bits_word((BN_ULONG)al);
951 j = BN_num_bits_word((BN_ULONG)bl);
954 assert(j <= al || j <= bl);
959 if (al > j || bl > j) {
960 if (bn_wexpand(t, k * 4) == NULL)
962 if (bn_wexpand(rr, k * 4) == NULL)
964 bn_mul_part_recursive(rr->d, a->d, b->d,
965 j, al - j, bl - j, t->d);
966 } else { /* al <= j || bl <= j */
968 if (bn_wexpand(t, k * 2) == NULL)
970 if (bn_wexpand(rr, k * 2) == NULL)
972 bn_mul_recursive(rr->d, a->d, b->d, j, al - j, bl - j, t->d);
978 if (i == 1 && !BN_get_flags(b, BN_FLG_STATIC_DATA)) {
979 BIGNUM *tmp_bn = (BIGNUM *)b;
980 if (bn_wexpand(tmp_bn, al) == NULL)
985 } else if (i == -1 && !BN_get_flags(a, BN_FLG_STATIC_DATA)) {
986 BIGNUM *tmp_bn = (BIGNUM *)a;
987 if (bn_wexpand(tmp_bn, bl) == NULL)
994 /* symmetric and > 4 */
996 j = BN_num_bits_word((BN_ULONG)al);
1000 if (al == j) { /* exact multiple */
1001 if (bn_wexpand(t, k * 2) == NULL)
1003 if (bn_wexpand(rr, k * 2) == NULL)
1005 bn_mul_recursive(rr->d, a->d, b->d, al, t->d);
1007 if (bn_wexpand(t, k * 4) == NULL)
1009 if (bn_wexpand(rr, k * 4) == NULL)
1011 bn_mul_part_recursive(rr->d, a->d, b->d, al - j, j, t->d);
1018 #endif /* BN_RECURSION */
1019 if (bn_wexpand(rr, top) == NULL)
1022 bn_mul_normal(rr->d, a->d, al, b->d, bl);
1024 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
1037 void bn_mul_normal(BN_ULONG *r, BN_ULONG *a, int na, BN_ULONG *b, int nb)
1055 (void)bn_mul_words(r, a, na, 0);
1058 rr[0] = bn_mul_words(r, a, na, b[0]);
1063 rr[1] = bn_mul_add_words(&(r[1]), a, na, b[1]);
1066 rr[2] = bn_mul_add_words(&(r[2]), a, na, b[2]);
1069 rr[3] = bn_mul_add_words(&(r[3]), a, na, b[3]);
1072 rr[4] = bn_mul_add_words(&(r[4]), a, na, b[4]);
1079 void bn_mul_low_normal(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n)
1081 bn_mul_words(r, a, n, b[0]);
1086 bn_mul_add_words(&(r[1]), a, n, b[1]);
1089 bn_mul_add_words(&(r[2]), a, n, b[2]);
1092 bn_mul_add_words(&(r[3]), a, n, b[3]);
1095 bn_mul_add_words(&(r[4]), a, n, b[4]);