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.]
64 /* Here follows specialised variants of bn_cmp_words(), bn_add_words() and
65 bn_sub_words(). They all have the property performing operations on
66 arrays of different sizes. The sizes of those arrays is expressed through
67 cl, which is the common length ( basicall, min(len(a),len(b)) ), and dl,
68 which is the delta between the two lengths, calculated as len(a)-len(b).
69 All lengths are the number of BN_ULONGs... For the operations that require
70 a result array as parameter, it must have the length cl+abs(dl).
71 These functions should probably end up in bn_asm.c as soon as there are
72 assembler counterparts for the systems that use assembler files. */
74 int bn_cmp_part_words(const BN_ULONG *a, const BN_ULONG *b,
77 if (dl < 0) /* a < b */
79 if (dl > 0) /* a > b */
82 return bn_cmp_words(a,b,cl);
85 BN_ULONG bn_sub_part_words(BN_ULONG *r,
86 const BN_ULONG *a, const BN_ULONG *b,
92 c = bn_sub_words(r, a, b, cl);
104 fprintf(stderr, " bn_sub_part_words %d + %d (dl < 0, c = %d)\n", cl, dl, c);
109 r[0] = (0-t-c)&BN_MASK2;
111 if (++dl >= 0) break;
114 r[1] = (0-t-c)&BN_MASK2;
116 if (++dl >= 0) break;
119 r[2] = (0-t-c)&BN_MASK2;
121 if (++dl >= 0) break;
124 r[3] = (0-t-c)&BN_MASK2;
126 if (++dl >= 0) break;
136 fprintf(stderr, " bn_sub_part_words %d + %d (dl > 0, c = %d)\n", cl, dl, c);
141 r[0] = (t-c)&BN_MASK2;
143 if (--dl <= 0) break;
146 r[1] = (t-c)&BN_MASK2;
148 if (--dl <= 0) break;
151 r[2] = (t-c)&BN_MASK2;
153 if (--dl <= 0) break;
156 r[3] = (t-c)&BN_MASK2;
158 if (--dl <= 0) break;
167 fprintf(stderr, " bn_sub_part_words %d + %d (dl > 0, c == 0)\n", cl, dl);
171 switch (save_dl - dl)
175 if (--dl <= 0) break;
178 if (--dl <= 0) break;
181 if (--dl <= 0) break;
190 fprintf(stderr, " bn_sub_part_words %d + %d (dl > 0, copy)\n", cl, dl);
195 if (--dl <= 0) break;
197 if (--dl <= 0) break;
199 if (--dl <= 0) break;
201 if (--dl <= 0) break;
211 BN_ULONG bn_add_part_words(BN_ULONG *r,
212 const BN_ULONG *a, const BN_ULONG *b,
218 c = bn_sub_words(r, a, b, cl);
231 fprintf(stderr, " bn_add_part_words %d + %d (dl < 0, c = %d)\n", cl, dl, c);
238 if (++dl >= 0) break;
243 if (++dl >= 0) break;
248 if (++dl >= 0) break;
253 if (++dl >= 0) break;
262 fprintf(stderr, " bn_add_part_words %d + %d (dl < 0, c == 0)\n", cl, dl);
266 switch (dl - save_dl)
270 if (++dl >= 0) break;
273 if (++dl >= 0) break;
276 if (++dl >= 0) break;
285 fprintf(stderr, " bn_add_part_words %d + %d (dl < 0, copy)\n", cl, dl);
290 if (++dl >= 0) break;
292 if (++dl >= 0) break;
294 if (++dl >= 0) break;
296 if (++dl >= 0) break;
307 fprintf(stderr, " bn_add_part_words %d + %d (dl > 0)\n", cl, dl);
314 if (--dl <= 0) break;
319 if (--dl <= 0) break;
324 if (--dl <= 0) break;
329 if (--dl <= 0) break;
336 fprintf(stderr, " bn_add_part_words %d + %d (dl > 0, c == 0)\n", cl, dl);
342 switch (save_dl - dl)
346 if (--dl <= 0) break;
349 if (--dl <= 0) break;
352 if (--dl <= 0) break;
361 fprintf(stderr, " bn_add_part_words %d + %d (dl > 0, copy)\n", cl, dl);
366 if (--dl <= 0) break;
368 if (--dl <= 0) break;
370 if (--dl <= 0) break;
372 if (--dl <= 0) break;
383 /* Karatsuba recursive multiplication algorithm
384 * (cf. Knuth, The Art of Computer Programming, Vol. 2) */
386 /* r is 2*n2 words in size,
387 * a and b are both n2 words in size.
388 * n2 must be a power of 2.
389 * We multiply and return the result.
390 * t must be 2*n2 words in size
393 * a[0]*b[0]+a[1]*b[1]+(a[0]-a[1])*(b[1]-b[0])
396 void bn_mul_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
400 unsigned int neg,zero;
404 fprintf(stderr," bn_mul_recursive %d * %d\n",n2,n2);
410 bn_mul_comba4(r,a,b);
416 bn_mul_comba8(r,a,b);
419 # endif /* BN_MUL_COMBA */
420 if (n2 < BN_MUL_RECURSIVE_SIZE_NORMAL)
422 /* This should not happen */
423 bn_mul_normal(r,a,n2,b,n2);
426 /* r=(a[0]-a[1])*(b[1]-b[0]) */
427 c1=bn_cmp_words(a,&(a[n]),n);
428 c2=bn_cmp_words(&(b[n]),b,n);
433 bn_sub_words(t, &(a[n]),a, n); /* - */
434 bn_sub_words(&(t[n]),b, &(b[n]),n); /* - */
440 bn_sub_words(t, &(a[n]),a, n); /* - */
441 bn_sub_words(&(t[n]),&(b[n]),b, n); /* + */
450 bn_sub_words(t, a, &(a[n]),n); /* + */
451 bn_sub_words(&(t[n]),b, &(b[n]),n); /* - */
458 bn_sub_words(t, a, &(a[n]),n);
459 bn_sub_words(&(t[n]),&(b[n]),b, n);
467 bn_mul_comba4(&(t[n2]),t,&(t[n]));
469 memset(&(t[n2]),0,8*sizeof(BN_ULONG));
471 bn_mul_comba4(r,a,b);
472 bn_mul_comba4(&(r[n2]),&(a[n]),&(b[n]));
477 bn_mul_comba8(&(t[n2]),t,&(t[n]));
479 memset(&(t[n2]),0,16*sizeof(BN_ULONG));
481 bn_mul_comba8(r,a,b);
482 bn_mul_comba8(&(r[n2]),&(a[n]),&(b[n]));
485 # endif /* BN_MUL_COMBA */
489 bn_mul_recursive(&(t[n2]),t,&(t[n]),n,p);
491 memset(&(t[n2]),0,n2*sizeof(BN_ULONG));
492 bn_mul_recursive(r,a,b,n,p);
493 bn_mul_recursive(&(r[n2]),&(a[n]),&(b[n]),n,p);
496 /* t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
497 * r[10] holds (a[0]*b[0])
498 * r[32] holds (b[1]*b[1])
501 c1=(int)(bn_add_words(t,r,&(r[n2]),n2));
503 if (neg) /* if t[32] is negative */
505 c1-=(int)(bn_sub_words(&(t[n2]),t,&(t[n2]),n2));
509 /* Might have a carry */
510 c1+=(int)(bn_add_words(&(t[n2]),&(t[n2]),t,n2));
513 /* t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
514 * r[10] holds (a[0]*b[0])
515 * r[32] holds (b[1]*b[1])
516 * c1 holds the carry bits
518 c1+=(int)(bn_add_words(&(r[n]),&(r[n]),&(t[n2]),n2));
526 /* The overflow will stop before we over write
527 * words we should not overwrite */
528 if (ln < (BN_ULONG)c1)
540 /* n+tn is the word length
541 * t needs to be n*4 is size, as does r */
542 void bn_mul_part_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int tn,
546 unsigned int c1,c2,neg,zero;
550 fprintf(stderr," bn_mul_part_recursive (%d+%d) * (%d+%d)\n",
556 bn_mul_normal(r,a,i,b,i);
560 /* r=(a[0]-a[1])*(b[1]-b[0]) */
561 c1=bn_cmp_part_words(a,&(a[n]),tn,n-tn);
562 c2=bn_cmp_part_words(&(b[n]),b,tn,tn-n);
567 bn_sub_part_words(t, &(a[n]),a, tn,tn-n); /* - */
568 bn_sub_part_words(&(t[n]),b, &(b[n]),tn,n-tn); /* - */
574 bn_sub_part_words(t, &(a[n]),a, tn,tn-n); /* - */
575 bn_sub_part_words(&(t[n]),&(b[n]),b, tn,tn-n); /* + */
584 bn_sub_part_words(t, a, &(a[n]),tn,n-tn); /* + */
585 bn_sub_part_words(&(t[n]),b, &(b[n]),tn,n-tn); /* - */
592 bn_sub_part_words(t, a, &(a[n]),tn,n-tn);
593 bn_sub_part_words(&(t[n]),&(b[n]),b, tn,tn-n);
596 /* The zero case isn't yet implemented here. The speedup
597 would probably be negligible. */
601 bn_mul_comba4(&(t[n2]),t,&(t[n]));
602 bn_mul_comba4(r,a,b);
603 bn_mul_normal(&(r[n2]),&(a[n]),tn,&(b[n]),tn);
604 memset(&(r[n2+tn*2]),0,sizeof(BN_ULONG)*(n2-tn*2));
610 bn_mul_comba8(&(t[n2]),t,&(t[n]));
611 bn_mul_comba8(r,a,b);
612 bn_mul_normal(&(r[n2]),&(a[n]),tn,&(b[n]),tn);
613 memset(&(r[n2+tn*2]),0,sizeof(BN_ULONG)*(n2-tn*2));
618 bn_mul_recursive(&(t[n2]),t,&(t[n]),n,p);
619 bn_mul_recursive(r,a,b,n,p);
621 /* If there is only a bottom half to the number,
626 bn_mul_recursive(&(r[n2]),&(a[n]),&(b[n]),i,p);
627 memset(&(r[n2+i*2]),0,sizeof(BN_ULONG)*(n2-i*2));
629 else if (j > 0) /* eg, n == 16, i == 8 and tn == 11 */
631 bn_mul_part_recursive(&(r[n2]),&(a[n]),&(b[n]),
633 memset(&(r[n2+tn*2]),0,
634 sizeof(BN_ULONG)*(n2-tn*2));
636 else /* (j < 0) eg, n == 16, i == 8 and tn == 5 */
638 memset(&(r[n2]),0,sizeof(BN_ULONG)*n2);
639 if (tn < BN_MUL_RECURSIVE_SIZE_NORMAL)
641 bn_mul_normal(&(r[n2]),&(a[n]),tn,&(b[n]),tn);
650 bn_mul_part_recursive(&(r[n2]),
657 bn_mul_recursive(&(r[n2]),
667 /* t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
668 * r[10] holds (a[0]*b[0])
669 * r[32] holds (b[1]*b[1])
672 c1=(int)(bn_add_words(t,r,&(r[n2]),n2));
674 if (neg) /* if t[32] is negative */
676 c1-=(int)(bn_sub_words(&(t[n2]),t,&(t[n2]),n2));
680 /* Might have a carry */
681 c1+=(int)(bn_add_words(&(t[n2]),&(t[n2]),t,n2));
684 /* t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
685 * r[10] holds (a[0]*b[0])
686 * r[32] holds (b[1]*b[1])
687 * c1 holds the carry bits
689 c1+=(int)(bn_add_words(&(r[n]),&(r[n]),&(t[n2]),n2));
697 /* The overflow will stop before we over write
698 * words we should not overwrite */
711 /* a and b must be the same size, which is n2.
712 * r needs to be n2 words and t needs to be n2*2
714 void bn_mul_low_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2,
720 fprintf(stderr," bn_mul_low_recursive %d * %d\n",n2,n2);
723 bn_mul_recursive(r,a,b,n,&(t[0]));
724 if (n >= BN_MUL_LOW_RECURSIVE_SIZE_NORMAL)
726 bn_mul_low_recursive(&(t[0]),&(a[0]),&(b[n]),n,&(t[n2]));
727 bn_add_words(&(r[n]),&(r[n]),&(t[0]),n);
728 bn_mul_low_recursive(&(t[0]),&(a[n]),&(b[0]),n,&(t[n2]));
729 bn_add_words(&(r[n]),&(r[n]),&(t[0]),n);
733 bn_mul_low_normal(&(t[0]),&(a[0]),&(b[n]),n);
734 bn_mul_low_normal(&(t[n]),&(a[n]),&(b[0]),n);
735 bn_add_words(&(r[n]),&(r[n]),&(t[0]),n);
736 bn_add_words(&(r[n]),&(r[n]),&(t[n]),n);
740 /* a and b must be the same size, which is n2.
741 * r needs to be n2 words and t needs to be n2*2
742 * l is the low words of the output.
745 void bn_mul_high(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, BN_ULONG *l, int n2,
751 BN_ULONG ll,lc,*lp,*mp;
754 fprintf(stderr," bn_mul_high %d * %d\n",n2,n2);
758 /* Calculate (al-ah)*(bh-bl) */
760 c1=bn_cmp_words(&(a[0]),&(a[n]),n);
761 c2=bn_cmp_words(&(b[n]),&(b[0]),n);
765 bn_sub_words(&(r[0]),&(a[n]),&(a[0]),n);
766 bn_sub_words(&(r[n]),&(b[0]),&(b[n]),n);
772 bn_sub_words(&(r[0]),&(a[n]),&(a[0]),n);
773 bn_sub_words(&(r[n]),&(b[n]),&(b[0]),n);
782 bn_sub_words(&(r[0]),&(a[0]),&(a[n]),n);
783 bn_sub_words(&(r[n]),&(b[0]),&(b[n]),n);
790 bn_sub_words(&(r[0]),&(a[0]),&(a[n]),n);
791 bn_sub_words(&(r[n]),&(b[n]),&(b[0]),n);
796 /* t[10] = (a[0]-a[1])*(b[1]-b[0]) */
797 /* r[10] = (a[1]*b[1]) */
801 bn_mul_comba8(&(t[0]),&(r[0]),&(r[n]));
802 bn_mul_comba8(r,&(a[n]),&(b[n]));
807 bn_mul_recursive(&(t[0]),&(r[0]),&(r[n]),n,&(t[n2]));
808 bn_mul_recursive(r,&(a[n]),&(b[n]),n,&(t[n2]));
812 * s1 == low(ah*bh)+low((al-ah)*(bh-bl))+low(al*bl)+high(al*bl)
813 * We know s0 and s1 so the only unknown is high(al*bl)
814 * high(al*bl) == s1 - low(ah*bh+s0+(al-ah)*(bh-bl))
815 * high(al*bl) == s1 - (r[0]+l[0]+t[0])
820 c1=(int)(bn_add_words(lp,&(r[0]),&(l[0]),n));
829 neg=(int)(bn_sub_words(&(t[n2]),lp,&(t[0]),n));
832 bn_add_words(&(t[n2]),lp,&(t[0]),n);
838 bn_sub_words(&(t[n2+n]),&(l[n]),&(t[n2]),n);
845 lp[i]=((~mp[i])+1)&BN_MASK2;
850 * t[10] = (a[0]-a[1])*(b[1]-b[0]) neg is the sign
851 * r[10] = (a[1]*b[1])
854 * R[21] = al*bl + ah*bh + (a[0]-a[1])*(b[1]-b[0])
857 /* R[1]=t[3]+l[0]+r[0](+-)t[0] (have carry/borrow)
858 * R[2]=r[0]+t[3]+r[1](+-)t[1] (have carry/borrow)
859 * R[3]=r[1]+(carry/borrow)
864 c1= (int)(bn_add_words(lp,&(t[n2+n]),&(l[0]),n));
871 c1+=(int)(bn_add_words(&(t[n2]),lp, &(r[0]),n));
873 c1-=(int)(bn_sub_words(&(t[n2]),&(t[n2]),&(t[0]),n));
875 c1+=(int)(bn_add_words(&(t[n2]),&(t[n2]),&(t[0]),n));
877 c2 =(int)(bn_add_words(&(r[0]),&(r[0]),&(t[n2+n]),n));
878 c2+=(int)(bn_add_words(&(r[0]),&(r[0]),&(r[n]),n));
880 c2-=(int)(bn_sub_words(&(r[0]),&(r[0]),&(t[n]),n));
882 c2+=(int)(bn_add_words(&(r[0]),&(r[0]),&(t[n]),n));
884 if (c1 != 0) /* Add starting at r[0], could be +ve or -ve */
891 ll=(r[i]+lc)&BN_MASK2;
901 r[i++]=(ll-lc)&BN_MASK2;
906 if (c2 != 0) /* Add starting at r[1] */
913 ll=(r[i]+lc)&BN_MASK2;
923 r[i++]=(ll-lc)&BN_MASK2;
929 #endif /* BN_RECURSION */
931 int BN_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
936 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
943 BIGNUM *free_a = NULL, *free_b = NULL;
946 fprintf(stderr,"BN_mul %d * %d\n",a->top,b->top);
956 if ((al == 0) || (bl == 0))
964 if ((r == a) || (r == b))
966 if ((rr = BN_CTX_get(ctx)) == NULL) goto err;
970 rr->neg=a->neg^b->neg;
972 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
981 if (bn_wexpand(rr,8) == NULL) goto err;
983 bn_mul_comba4(rr->d,a->d,b->d);
989 if (bn_wexpand(rr,16) == NULL) goto err;
991 bn_mul_comba8(rr->d,a->d,b->d);
995 #endif /* BN_MUL_COMBA */
997 if ((al >= BN_MULL_SIZE_NORMAL) && (bl >= BN_MULL_SIZE_NORMAL))
999 if (i == 1 && !BN_get_flags(b,BN_FLG_STATIC_DATA))
1001 BIGNUM *tmp_bn = (BIGNUM *)b;
1002 bn_wexpand(tmp_bn,al);
1007 else if (i == -1 && !BN_get_flags(a,BN_FLG_STATIC_DATA))
1009 BIGNUM *tmp_bn = (BIGNUM *)a;
1010 bn_wexpand(tmp_bn,bl);
1017 /* symmetric and > 4 */
1019 j=BN_num_bits_word((BN_ULONG)al);
1022 t = BN_CTX_get(ctx);
1023 if (al == j) /* exact multiple */
1027 bn_mul_recursive(rr->d,a->d,b->d,al,t->d);
1033 bn_mul_part_recursive(rr->d,a->d,b->d,al-j,j,t->d);
1039 #endif /* BN_RECURSION */
1040 if (bn_wexpand(rr,top) == NULL) goto err;
1042 bn_mul_normal(rr->d,a->d,al,b->d,bl);
1044 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
1048 if (r != rr) BN_copy(r,rr);
1051 if (free_a) BN_free(free_a);
1052 if (free_b) BN_free(free_b);
1057 void bn_mul_normal(BN_ULONG *r, BN_ULONG *a, int na, BN_ULONG *b, int nb)
1062 fprintf(stderr," bn_mul_normal %d * %d\n",na,nb);
1070 itmp=na; na=nb; nb=itmp;
1071 ltmp=a; a=b; b=ltmp;
1075 rr[0]=bn_mul_words(r,a,na,b[0]);
1079 if (--nb <= 0) return;
1080 rr[1]=bn_mul_add_words(&(r[1]),a,na,b[1]);
1081 if (--nb <= 0) return;
1082 rr[2]=bn_mul_add_words(&(r[2]),a,na,b[2]);
1083 if (--nb <= 0) return;
1084 rr[3]=bn_mul_add_words(&(r[3]),a,na,b[3]);
1085 if (--nb <= 0) return;
1086 rr[4]=bn_mul_add_words(&(r[4]),a,na,b[4]);
1093 void bn_mul_low_normal(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n)
1096 fprintf(stderr," bn_mul_low_normal %d * %d\n",n,n);
1098 bn_mul_words(r,a,n,b[0]);
1102 if (--n <= 0) return;
1103 bn_mul_add_words(&(r[1]),a,n,b[1]);
1104 if (--n <= 0) return;
1105 bn_mul_add_words(&(r[2]),a,n,b[2]);
1106 if (--n <= 0) return;
1107 bn_mul_add_words(&(r[3]),a,n,b[3]);
1108 if (--n <= 0) return;
1109 bn_mul_add_words(&(r[4]),a,n,b[4]);