1 /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
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.
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).
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.
22 * Redistribution and use in source and binary forms, with or without
23 * modification, are permitted provided that the following conditions
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)"
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
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.]
59 #include "internal/cryptlib.h"
62 #if defined(OPENSSL_NO_ASM) || !defined(OPENSSL_BN_ASM_PART_WORDS)
64 * Here follows specialised variants of bn_add_words() and bn_sub_words().
65 * They have the property performing operations on arrays of different sizes.
66 * The sizes of those arrays is expressed through cl, which is the common
67 * length ( basically, min(len(a),len(b)) ), and dl, which is the delta
68 * between the two lengths, calculated as len(a)-len(b). All lengths are the
69 * number of BN_ULONGs... For the operations that require a result array as
70 * parameter, it must have the length cl+abs(dl). These functions should
71 * probably end up in bn_asm.c as soon as there are assembler counterparts
72 * for the systems that use assembler files.
75 BN_ULONG bn_sub_part_words(BN_ULONG *r,
76 const BN_ULONG *a, const BN_ULONG *b,
82 c = bn_sub_words(r, a, b, cl);
94 r[0] = (0 - t - c) & BN_MASK2;
101 r[1] = (0 - t - c) & BN_MASK2;
108 r[2] = (0 - t - c) & BN_MASK2;
115 r[3] = (0 - t - c) & BN_MASK2;
128 r[0] = (t - c) & BN_MASK2;
135 r[1] = (t - c) & BN_MASK2;
142 r[2] = (t - c) & BN_MASK2;
149 r[3] = (t - c) & BN_MASK2;
161 switch (save_dl - dl) {
203 BN_ULONG bn_add_part_words(BN_ULONG *r,
204 const BN_ULONG *a, const BN_ULONG *b,
210 c = bn_add_words(r, a, b, cl);
222 l = (c + b[0]) & BN_MASK2;
228 l = (c + b[1]) & BN_MASK2;
234 l = (c + b[2]) & BN_MASK2;
240 l = (c + b[3]) & BN_MASK2;
252 switch (dl - save_dl) {
292 t = (a[0] + c) & BN_MASK2;
298 t = (a[1] + c) & BN_MASK2;
304 t = (a[2] + c) & BN_MASK2;
310 t = (a[3] + c) & BN_MASK2;
322 switch (save_dl - dl) {
365 * Karatsuba recursive multiplication algorithm (cf. Knuth, The Art of
366 * Computer Programming, Vol. 2)
370 * r is 2*n2 words in size,
371 * a and b are both n2 words in size.
372 * n2 must be a power of 2.
373 * We multiply and return the result.
374 * t must be 2*n2 words in size
377 * a[0]*b[0]+a[1]*b[1]+(a[0]-a[1])*(b[1]-b[0])
380 /* dnX may not be positive, but n2/2+dnX has to be */
381 void bn_mul_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
382 int dna, int dnb, BN_ULONG *t)
384 int n = n2 / 2, c1, c2;
385 int tna = n + dna, tnb = n + dnb;
386 unsigned int neg, zero;
392 bn_mul_comba4(r, a, b);
397 * Only call bn_mul_comba 8 if n2 == 8 and the two arrays are complete
400 if (n2 == 8 && dna == 0 && dnb == 0) {
401 bn_mul_comba8(r, a, b);
404 # endif /* BN_MUL_COMBA */
405 /* Else do normal multiply */
406 if (n2 < BN_MUL_RECURSIVE_SIZE_NORMAL) {
407 bn_mul_normal(r, a, n2 + dna, b, n2 + dnb);
409 memset(&r[2 * n2 + dna + dnb], 0,
410 sizeof(BN_ULONG) * -(dna + dnb));
413 /* r=(a[0]-a[1])*(b[1]-b[0]) */
414 c1 = bn_cmp_part_words(a, &(a[n]), tna, n - tna);
415 c2 = bn_cmp_part_words(&(b[n]), b, tnb, tnb - n);
417 switch (c1 * 3 + c2) {
419 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
420 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
426 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
427 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n); /* + */
436 bn_sub_part_words(t, a, &(a[n]), tna, n - tna); /* + */
437 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
444 bn_sub_part_words(t, a, &(a[n]), tna, n - tna);
445 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n);
450 if (n == 4 && dna == 0 && dnb == 0) { /* XXX: bn_mul_comba4 could take
451 * extra args to do this well */
453 bn_mul_comba4(&(t[n2]), t, &(t[n]));
455 memset(&t[n2], 0, sizeof(*t) * 8);
457 bn_mul_comba4(r, a, b);
458 bn_mul_comba4(&(r[n2]), &(a[n]), &(b[n]));
459 } else if (n == 8 && dna == 0 && dnb == 0) { /* XXX: bn_mul_comba8 could
460 * take extra args to do
463 bn_mul_comba8(&(t[n2]), t, &(t[n]));
465 memset(&t[n2], 0, sizeof(*t) * 16);
467 bn_mul_comba8(r, a, b);
468 bn_mul_comba8(&(r[n2]), &(a[n]), &(b[n]));
470 # endif /* BN_MUL_COMBA */
474 bn_mul_recursive(&(t[n2]), t, &(t[n]), n, 0, 0, p);
476 memset(&t[n2], 0, sizeof(*t) * n2);
477 bn_mul_recursive(r, a, b, n, 0, 0, p);
478 bn_mul_recursive(&(r[n2]), &(a[n]), &(b[n]), n, dna, dnb, p);
482 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
483 * r[10] holds (a[0]*b[0])
484 * r[32] holds (b[1]*b[1])
487 c1 = (int)(bn_add_words(t, r, &(r[n2]), n2));
489 if (neg) { /* if t[32] is negative */
490 c1 -= (int)(bn_sub_words(&(t[n2]), t, &(t[n2]), n2));
492 /* Might have a carry */
493 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), t, n2));
497 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
498 * r[10] holds (a[0]*b[0])
499 * r[32] holds (b[1]*b[1])
500 * c1 holds the carry bits
502 c1 += (int)(bn_add_words(&(r[n]), &(r[n]), &(t[n2]), n2));
506 ln = (lo + c1) & BN_MASK2;
510 * The overflow will stop before we over write words we should not
513 if (ln < (BN_ULONG)c1) {
517 ln = (lo + 1) & BN_MASK2;
525 * n+tn is the word length t needs to be n*4 is size, as does r
527 /* tnX may not be negative but less than n */
528 void bn_mul_part_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n,
529 int tna, int tnb, BN_ULONG *t)
531 int i, j, n2 = n * 2;
536 bn_mul_normal(r, a, n + tna, b, n + tnb);
540 /* r=(a[0]-a[1])*(b[1]-b[0]) */
541 c1 = bn_cmp_part_words(a, &(a[n]), tna, n - tna);
542 c2 = bn_cmp_part_words(&(b[n]), b, tnb, tnb - n);
544 switch (c1 * 3 + c2) {
546 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
547 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
552 bn_sub_part_words(t, &(a[n]), a, tna, tna - n); /* - */
553 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n); /* + */
561 bn_sub_part_words(t, a, &(a[n]), tna, n - tna); /* + */
562 bn_sub_part_words(&(t[n]), b, &(b[n]), tnb, n - tnb); /* - */
568 bn_sub_part_words(t, a, &(a[n]), tna, n - tna);
569 bn_sub_part_words(&(t[n]), &(b[n]), b, tnb, tnb - n);
573 * The zero case isn't yet implemented here. The speedup would probably
578 bn_mul_comba4(&(t[n2]), t, &(t[n]));
579 bn_mul_comba4(r, a, b);
580 bn_mul_normal(&(r[n2]), &(a[n]), tn, &(b[n]), tn);
581 memset(&r[n2 + tn * 2], 0, sizeof(*r) * (n2 - tn * 2));
585 bn_mul_comba8(&(t[n2]), t, &(t[n]));
586 bn_mul_comba8(r, a, b);
587 bn_mul_normal(&(r[n2]), &(a[n]), tna, &(b[n]), tnb);
588 memset(&r[n2 + tna + tnb], 0, sizeof(*r) * (n2 - tna - tnb));
591 bn_mul_recursive(&(t[n2]), t, &(t[n]), n, 0, 0, p);
592 bn_mul_recursive(r, a, b, n, 0, 0, p);
595 * If there is only a bottom half to the number, just do it
602 bn_mul_recursive(&(r[n2]), &(a[n]), &(b[n]),
603 i, tna - i, tnb - i, p);
604 memset(&r[n2 + i * 2], 0, sizeof(*r) * (n2 - i * 2));
605 } else if (j > 0) { /* eg, n == 16, i == 8 and tn == 11 */
606 bn_mul_part_recursive(&(r[n2]), &(a[n]), &(b[n]),
607 i, tna - i, tnb - i, p);
608 memset(&(r[n2 + tna + tnb]), 0,
609 sizeof(BN_ULONG) * (n2 - tna - tnb));
610 } else { /* (j < 0) eg, n == 16, i == 8 and tn == 5 */
612 memset(&r[n2], 0, sizeof(*r) * n2);
613 if (tna < BN_MUL_RECURSIVE_SIZE_NORMAL
614 && tnb < BN_MUL_RECURSIVE_SIZE_NORMAL) {
615 bn_mul_normal(&(r[n2]), &(a[n]), tna, &(b[n]), tnb);
620 * these simplified conditions work exclusively because
621 * difference between tna and tnb is 1 or 0
623 if (i < tna || i < tnb) {
624 bn_mul_part_recursive(&(r[n2]),
626 i, tna - i, tnb - i, p);
628 } else if (i == tna || i == tnb) {
629 bn_mul_recursive(&(r[n2]),
631 i, tna - i, tnb - i, p);
640 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
641 * r[10] holds (a[0]*b[0])
642 * r[32] holds (b[1]*b[1])
645 c1 = (int)(bn_add_words(t, r, &(r[n2]), n2));
647 if (neg) { /* if t[32] is negative */
648 c1 -= (int)(bn_sub_words(&(t[n2]), t, &(t[n2]), n2));
650 /* Might have a carry */
651 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), t, n2));
655 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
656 * r[10] holds (a[0]*b[0])
657 * r[32] holds (b[1]*b[1])
658 * c1 holds the carry bits
660 c1 += (int)(bn_add_words(&(r[n]), &(r[n]), &(t[n2]), n2));
664 ln = (lo + c1) & BN_MASK2;
668 * The overflow will stop before we over write words we should not
671 if (ln < (BN_ULONG)c1) {
675 ln = (lo + 1) & BN_MASK2;
683 * a and b must be the same size, which is n2.
684 * r needs to be n2 words and t needs to be n2*2
686 void bn_mul_low_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
691 bn_mul_recursive(r, a, b, n, 0, 0, &(t[0]));
692 if (n >= BN_MUL_LOW_RECURSIVE_SIZE_NORMAL) {
693 bn_mul_low_recursive(&(t[0]), &(a[0]), &(b[n]), n, &(t[n2]));
694 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
695 bn_mul_low_recursive(&(t[0]), &(a[n]), &(b[0]), n, &(t[n2]));
696 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
698 bn_mul_low_normal(&(t[0]), &(a[0]), &(b[n]), n);
699 bn_mul_low_normal(&(t[n]), &(a[n]), &(b[0]), n);
700 bn_add_words(&(r[n]), &(r[n]), &(t[0]), n);
701 bn_add_words(&(r[n]), &(r[n]), &(t[n]), n);
706 * a and b must be the same size, which is n2.
707 * r needs to be n2 words and t needs to be n2*2
708 * l is the low words of the output.
711 void bn_mul_high(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, BN_ULONG *l, int n2,
717 BN_ULONG ll, lc, *lp, *mp;
721 /* Calculate (al-ah)*(bh-bl) */
723 c1 = bn_cmp_words(&(a[0]), &(a[n]), n);
724 c2 = bn_cmp_words(&(b[n]), &(b[0]), n);
725 switch (c1 * 3 + c2) {
727 bn_sub_words(&(r[0]), &(a[n]), &(a[0]), n);
728 bn_sub_words(&(r[n]), &(b[0]), &(b[n]), n);
734 bn_sub_words(&(r[0]), &(a[n]), &(a[0]), n);
735 bn_sub_words(&(r[n]), &(b[n]), &(b[0]), n);
744 bn_sub_words(&(r[0]), &(a[0]), &(a[n]), n);
745 bn_sub_words(&(r[n]), &(b[0]), &(b[n]), n);
752 bn_sub_words(&(r[0]), &(a[0]), &(a[n]), n);
753 bn_sub_words(&(r[n]), &(b[n]), &(b[0]), n);
758 /* t[10] = (a[0]-a[1])*(b[1]-b[0]) */
759 /* r[10] = (a[1]*b[1]) */
762 bn_mul_comba8(&(t[0]), &(r[0]), &(r[n]));
763 bn_mul_comba8(r, &(a[n]), &(b[n]));
767 bn_mul_recursive(&(t[0]), &(r[0]), &(r[n]), n, 0, 0, &(t[n2]));
768 bn_mul_recursive(r, &(a[n]), &(b[n]), n, 0, 0, &(t[n2]));
773 * s1 == low(ah*bh)+low((al-ah)*(bh-bl))+low(al*bl)+high(al*bl)
774 * We know s0 and s1 so the only unknown is high(al*bl)
775 * high(al*bl) == s1 - low(ah*bh+s0+(al-ah)*(bh-bl))
776 * high(al*bl) == s1 - (r[0]+l[0]+t[0])
780 c1 = (int)(bn_add_words(lp, &(r[0]), &(l[0]), n));
787 neg = (int)(bn_sub_words(&(t[n2]), lp, &(t[0]), n));
789 bn_add_words(&(t[n2]), lp, &(t[0]), n);
794 bn_sub_words(&(t[n2 + n]), &(l[n]), &(t[n2]), n);
798 for (i = 0; i < n; i++)
799 lp[i] = ((~mp[i]) + 1) & BN_MASK2;
805 * t[10] = (a[0]-a[1])*(b[1]-b[0]) neg is the sign
806 * r[10] = (a[1]*b[1])
810 * R[21] = al*bl + ah*bh + (a[0]-a[1])*(b[1]-b[0])
814 * R[1]=t[3]+l[0]+r[0](+-)t[0] (have carry/borrow)
815 * R[2]=r[0]+t[3]+r[1](+-)t[1] (have carry/borrow)
816 * R[3]=r[1]+(carry/borrow)
820 c1 = (int)(bn_add_words(lp, &(t[n2 + n]), &(l[0]), n));
825 c1 += (int)(bn_add_words(&(t[n2]), lp, &(r[0]), n));
827 c1 -= (int)(bn_sub_words(&(t[n2]), &(t[n2]), &(t[0]), n));
829 c1 += (int)(bn_add_words(&(t[n2]), &(t[n2]), &(t[0]), n));
831 c2 = (int)(bn_add_words(&(r[0]), &(r[0]), &(t[n2 + n]), n));
832 c2 += (int)(bn_add_words(&(r[0]), &(r[0]), &(r[n]), n));
834 c2 -= (int)(bn_sub_words(&(r[0]), &(r[0]), &(t[n]), n));
836 c2 += (int)(bn_add_words(&(r[0]), &(r[0]), &(t[n]), n));
838 if (c1 != 0) { /* Add starting at r[0], could be +ve or -ve */
843 ll = (r[i] + lc) & BN_MASK2;
851 r[i++] = (ll - lc) & BN_MASK2;
856 if (c2 != 0) { /* Add starting at r[1] */
861 ll = (r[i] + lc) & BN_MASK2;
869 r[i++] = (ll - lc) & BN_MASK2;
875 #endif /* BN_RECURSION */
877 int BN_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
882 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
897 if ((al == 0) || (bl == 0)) {
904 if ((r == a) || (r == b)) {
905 if ((rr = BN_CTX_get(ctx)) == NULL)
909 rr->neg = a->neg ^ b->neg;
911 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
918 if (bn_wexpand(rr, 8) == NULL)
921 bn_mul_comba4(rr->d, a->d, b->d);
926 if (bn_wexpand(rr, 16) == NULL)
929 bn_mul_comba8(rr->d, a->d, b->d);
933 #endif /* BN_MUL_COMBA */
935 if ((al >= BN_MULL_SIZE_NORMAL) && (bl >= BN_MULL_SIZE_NORMAL)) {
936 if (i >= -1 && i <= 1) {
938 * Find out the power of two lower or equal to the longest of the
942 j = BN_num_bits_word((BN_ULONG)al);
945 j = BN_num_bits_word((BN_ULONG)bl);
948 assert(j <= al || j <= bl);
953 if (al > j || bl > j) {
954 if (bn_wexpand(t, k * 4) == NULL)
956 if (bn_wexpand(rr, k * 4) == NULL)
958 bn_mul_part_recursive(rr->d, a->d, b->d,
959 j, al - j, bl - j, t->d);
960 } else { /* al <= j || bl <= j */
962 if (bn_wexpand(t, k * 2) == NULL)
964 if (bn_wexpand(rr, k * 2) == NULL)
966 bn_mul_recursive(rr->d, a->d, b->d, j, al - j, bl - j, t->d);
972 if (i == 1 && !BN_get_flags(b, BN_FLG_STATIC_DATA)) {
973 BIGNUM *tmp_bn = (BIGNUM *)b;
974 if (bn_wexpand(tmp_bn, al) == NULL)
979 } else if (i == -1 && !BN_get_flags(a, BN_FLG_STATIC_DATA)) {
980 BIGNUM *tmp_bn = (BIGNUM *)a;
981 if (bn_wexpand(tmp_bn, bl) == NULL)
988 /* symmetric and > 4 */
990 j = BN_num_bits_word((BN_ULONG)al);
994 if (al == j) { /* exact multiple */
995 if (bn_wexpand(t, k * 2) == NULL)
997 if (bn_wexpand(rr, k * 2) == NULL)
999 bn_mul_recursive(rr->d, a->d, b->d, al, t->d);
1001 if (bn_wexpand(t, k * 4) == NULL)
1003 if (bn_wexpand(rr, k * 4) == NULL)
1005 bn_mul_part_recursive(rr->d, a->d, b->d, al - j, j, t->d);
1012 #endif /* BN_RECURSION */
1013 if (bn_wexpand(rr, top) == NULL)
1016 bn_mul_normal(rr->d, a->d, al, b->d, bl);
1018 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
1031 void bn_mul_normal(BN_ULONG *r, BN_ULONG *a, int na, BN_ULONG *b, int nb)
1049 (void)bn_mul_words(r, a, na, 0);
1052 rr[0] = bn_mul_words(r, a, na, b[0]);
1057 rr[1] = bn_mul_add_words(&(r[1]), a, na, b[1]);
1060 rr[2] = bn_mul_add_words(&(r[2]), a, na, b[2]);
1063 rr[3] = bn_mul_add_words(&(r[3]), a, na, b[3]);
1066 rr[4] = bn_mul_add_words(&(r[4]), a, na, b[4]);
1073 void bn_mul_low_normal(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n)
1075 bn_mul_words(r, a, n, b[0]);
1080 bn_mul_add_words(&(r[1]), a, n, b[1]);
1083 bn_mul_add_words(&(r[2]), a, n, b[2]);
1086 bn_mul_add_words(&(r[3]), a, n, b[3]);
1089 bn_mul_add_words(&(r[4]), a, n, b[4]);