2 * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
11 #include "internal/cryptlib.h"
14 #if defined(OPENSSL_NO_ASM) || !defined(OPENSSL_BN_ASM_PART_WORDS)
16 * Here follows specialised variants of bn_add_words() and bn_sub_words().
17 * They have the property performing operations on arrays of different sizes.
18 * The sizes of those arrays is expressed through cl, which is the common
19 * length ( basically, min(len(a),len(b)) ), and dl, which is the delta
20 * between the two lengths, calculated as len(a)-len(b). All lengths are the
21 * number of BN_ULONGs... For the operations that require a result array as
22 * parameter, it must have the length cl+abs(dl). These functions should
23 * probably end up in bn_asm.c as soon as there are assembler counterparts
24 * for the systems that use assembler files.
27 BN_ULONG
bn_sub_part_words(BN_ULONG
*r
,
28 const BN_ULONG
*a
, const BN_ULONG
*b
,
34 c
= bn_sub_words(r
, a
, b
, cl
);
46 r
[0] = (0 - t
- c
) & BN_MASK2
;
53 r
[1] = (0 - t
- c
) & BN_MASK2
;
60 r
[2] = (0 - t
- c
) & BN_MASK2
;
67 r
[3] = (0 - t
- c
) & BN_MASK2
;
80 r
[0] = (t
- c
) & BN_MASK2
;
87 r
[1] = (t
- c
) & BN_MASK2
;
94 r
[2] = (t
- c
) & BN_MASK2
;
101 r
[3] = (t
- c
) & BN_MASK2
;
113 switch (save_dl
- dl
) {
159 * Karatsuba recursive multiplication algorithm (cf. Knuth, The Art of
160 * Computer Programming, Vol. 2)
164 * r is 2*n2 words in size,
165 * a and b are both n2 words in size.
166 * n2 must be a power of 2.
167 * We multiply and return the result.
168 * t must be 2*n2 words in size
171 * a[0]*b[0]+a[1]*b[1]+(a[0]-a[1])*(b[1]-b[0])
174 /* dnX may not be positive, but n2/2+dnX has to be */
175 void bn_mul_recursive(BN_ULONG
*r
, BN_ULONG
*a
, BN_ULONG
*b
, int n2
,
176 int dna
, int dnb
, BN_ULONG
*t
)
178 int n
= n2
/ 2, c1
, c2
;
179 int tna
= n
+ dna
, tnb
= n
+ dnb
;
180 unsigned int neg
, zero
;
186 bn_mul_comba4(r
, a
, b
);
191 * Only call bn_mul_comba 8 if n2 == 8 and the two arrays are complete
194 if (n2
== 8 && dna
== 0 && dnb
== 0) {
195 bn_mul_comba8(r
, a
, b
);
198 # endif /* BN_MUL_COMBA */
199 /* Else do normal multiply */
200 if (n2
< BN_MUL_RECURSIVE_SIZE_NORMAL
) {
201 bn_mul_normal(r
, a
, n2
+ dna
, b
, n2
+ dnb
);
203 memset(&r
[2 * n2
+ dna
+ dnb
], 0,
204 sizeof(BN_ULONG
) * -(dna
+ dnb
));
207 /* r=(a[0]-a[1])*(b[1]-b[0]) */
208 c1
= bn_cmp_part_words(a
, &(a
[n
]), tna
, n
- tna
);
209 c2
= bn_cmp_part_words(&(b
[n
]), b
, tnb
, tnb
- n
);
211 switch (c1
* 3 + c2
) {
213 bn_sub_part_words(t
, &(a
[n
]), a
, tna
, tna
- n
); /* - */
214 bn_sub_part_words(&(t
[n
]), b
, &(b
[n
]), tnb
, n
- tnb
); /* - */
220 bn_sub_part_words(t
, &(a
[n
]), a
, tna
, tna
- n
); /* - */
221 bn_sub_part_words(&(t
[n
]), &(b
[n
]), b
, tnb
, tnb
- n
); /* + */
230 bn_sub_part_words(t
, a
, &(a
[n
]), tna
, n
- tna
); /* + */
231 bn_sub_part_words(&(t
[n
]), b
, &(b
[n
]), tnb
, n
- tnb
); /* - */
238 bn_sub_part_words(t
, a
, &(a
[n
]), tna
, n
- tna
);
239 bn_sub_part_words(&(t
[n
]), &(b
[n
]), b
, tnb
, tnb
- n
);
244 if (n
== 4 && dna
== 0 && dnb
== 0) { /* XXX: bn_mul_comba4 could take
245 * extra args to do this well */
247 bn_mul_comba4(&(t
[n2
]), t
, &(t
[n
]));
249 memset(&t
[n2
], 0, sizeof(*t
) * 8);
251 bn_mul_comba4(r
, a
, b
);
252 bn_mul_comba4(&(r
[n2
]), &(a
[n
]), &(b
[n
]));
253 } else if (n
== 8 && dna
== 0 && dnb
== 0) { /* XXX: bn_mul_comba8 could
254 * take extra args to do
257 bn_mul_comba8(&(t
[n2
]), t
, &(t
[n
]));
259 memset(&t
[n2
], 0, sizeof(*t
) * 16);
261 bn_mul_comba8(r
, a
, b
);
262 bn_mul_comba8(&(r
[n2
]), &(a
[n
]), &(b
[n
]));
264 # endif /* BN_MUL_COMBA */
268 bn_mul_recursive(&(t
[n2
]), t
, &(t
[n
]), n
, 0, 0, p
);
270 memset(&t
[n2
], 0, sizeof(*t
) * n2
);
271 bn_mul_recursive(r
, a
, b
, n
, 0, 0, p
);
272 bn_mul_recursive(&(r
[n2
]), &(a
[n
]), &(b
[n
]), n
, dna
, dnb
, p
);
276 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
277 * r[10] holds (a[0]*b[0])
278 * r[32] holds (b[1]*b[1])
281 c1
= (int)(bn_add_words(t
, r
, &(r
[n2
]), n2
));
283 if (neg
) { /* if t[32] is negative */
284 c1
-= (int)(bn_sub_words(&(t
[n2
]), t
, &(t
[n2
]), n2
));
286 /* Might have a carry */
287 c1
+= (int)(bn_add_words(&(t
[n2
]), &(t
[n2
]), t
, n2
));
291 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
292 * r[10] holds (a[0]*b[0])
293 * r[32] holds (b[1]*b[1])
294 * c1 holds the carry bits
296 c1
+= (int)(bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[n2
]), n2
));
300 ln
= (lo
+ c1
) & BN_MASK2
;
304 * The overflow will stop before we over write words we should not
307 if (ln
< (BN_ULONG
)c1
) {
311 ln
= (lo
+ 1) & BN_MASK2
;
319 * n+tn is the word length t needs to be n*4 is size, as does r
321 /* tnX may not be negative but less than n */
322 void bn_mul_part_recursive(BN_ULONG
*r
, BN_ULONG
*a
, BN_ULONG
*b
, int n
,
323 int tna
, int tnb
, BN_ULONG
*t
)
325 int i
, j
, n2
= n
* 2;
330 bn_mul_normal(r
, a
, n
+ tna
, b
, n
+ tnb
);
334 /* r=(a[0]-a[1])*(b[1]-b[0]) */
335 c1
= bn_cmp_part_words(a
, &(a
[n
]), tna
, n
- tna
);
336 c2
= bn_cmp_part_words(&(b
[n
]), b
, tnb
, tnb
- n
);
338 switch (c1
* 3 + c2
) {
340 bn_sub_part_words(t
, &(a
[n
]), a
, tna
, tna
- n
); /* - */
341 bn_sub_part_words(&(t
[n
]), b
, &(b
[n
]), tnb
, n
- tnb
); /* - */
345 bn_sub_part_words(t
, &(a
[n
]), a
, tna
, tna
- n
); /* - */
346 bn_sub_part_words(&(t
[n
]), &(b
[n
]), b
, tnb
, tnb
- n
); /* + */
353 bn_sub_part_words(t
, a
, &(a
[n
]), tna
, n
- tna
); /* + */
354 bn_sub_part_words(&(t
[n
]), b
, &(b
[n
]), tnb
, n
- tnb
); /* - */
359 bn_sub_part_words(t
, a
, &(a
[n
]), tna
, n
- tna
);
360 bn_sub_part_words(&(t
[n
]), &(b
[n
]), b
, tnb
, tnb
- n
);
364 * The zero case isn't yet implemented here. The speedup would probably
369 bn_mul_comba4(&(t
[n2
]), t
, &(t
[n
]));
370 bn_mul_comba4(r
, a
, b
);
371 bn_mul_normal(&(r
[n2
]), &(a
[n
]), tn
, &(b
[n
]), tn
);
372 memset(&r
[n2
+ tn
* 2], 0, sizeof(*r
) * (n2
- tn
* 2));
376 bn_mul_comba8(&(t
[n2
]), t
, &(t
[n
]));
377 bn_mul_comba8(r
, a
, b
);
378 bn_mul_normal(&(r
[n2
]), &(a
[n
]), tna
, &(b
[n
]), tnb
);
379 memset(&r
[n2
+ tna
+ tnb
], 0, sizeof(*r
) * (n2
- tna
- tnb
));
382 bn_mul_recursive(&(t
[n2
]), t
, &(t
[n
]), n
, 0, 0, p
);
383 bn_mul_recursive(r
, a
, b
, n
, 0, 0, p
);
386 * If there is only a bottom half to the number, just do it
393 bn_mul_recursive(&(r
[n2
]), &(a
[n
]), &(b
[n
]),
394 i
, tna
- i
, tnb
- i
, p
);
395 memset(&r
[n2
+ i
* 2], 0, sizeof(*r
) * (n2
- i
* 2));
396 } else if (j
> 0) { /* eg, n == 16, i == 8 and tn == 11 */
397 bn_mul_part_recursive(&(r
[n2
]), &(a
[n
]), &(b
[n
]),
398 i
, tna
- i
, tnb
- i
, p
);
399 memset(&(r
[n2
+ tna
+ tnb
]), 0,
400 sizeof(BN_ULONG
) * (n2
- tna
- tnb
));
401 } else { /* (j < 0) eg, n == 16, i == 8 and tn == 5 */
403 memset(&r
[n2
], 0, sizeof(*r
) * n2
);
404 if (tna
< BN_MUL_RECURSIVE_SIZE_NORMAL
405 && tnb
< BN_MUL_RECURSIVE_SIZE_NORMAL
) {
406 bn_mul_normal(&(r
[n2
]), &(a
[n
]), tna
, &(b
[n
]), tnb
);
411 * these simplified conditions work exclusively because
412 * difference between tna and tnb is 1 or 0
414 if (i
< tna
|| i
< tnb
) {
415 bn_mul_part_recursive(&(r
[n2
]),
417 i
, tna
- i
, tnb
- i
, p
);
419 } else if (i
== tna
|| i
== tnb
) {
420 bn_mul_recursive(&(r
[n2
]),
422 i
, tna
- i
, tnb
- i
, p
);
431 * t[32] holds (a[0]-a[1])*(b[1]-b[0]), c1 is the sign
432 * r[10] holds (a[0]*b[0])
433 * r[32] holds (b[1]*b[1])
436 c1
= (int)(bn_add_words(t
, r
, &(r
[n2
]), n2
));
438 if (neg
) { /* if t[32] is negative */
439 c1
-= (int)(bn_sub_words(&(t
[n2
]), t
, &(t
[n2
]), n2
));
441 /* Might have a carry */
442 c1
+= (int)(bn_add_words(&(t
[n2
]), &(t
[n2
]), t
, n2
));
446 * t[32] holds (a[0]-a[1])*(b[1]-b[0])+(a[0]*b[0])+(a[1]*b[1])
447 * r[10] holds (a[0]*b[0])
448 * r[32] holds (b[1]*b[1])
449 * c1 holds the carry bits
451 c1
+= (int)(bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[n2
]), n2
));
455 ln
= (lo
+ c1
) & BN_MASK2
;
459 * The overflow will stop before we over write words we should not
462 if (ln
< (BN_ULONG
)c1
) {
466 ln
= (lo
+ 1) & BN_MASK2
;
474 * a and b must be the same size, which is n2.
475 * r needs to be n2 words and t needs to be n2*2
477 void bn_mul_low_recursive(BN_ULONG
*r
, BN_ULONG
*a
, BN_ULONG
*b
, int n2
,
482 bn_mul_recursive(r
, a
, b
, n
, 0, 0, &(t
[0]));
483 if (n
>= BN_MUL_LOW_RECURSIVE_SIZE_NORMAL
) {
484 bn_mul_low_recursive(&(t
[0]), &(a
[0]), &(b
[n
]), n
, &(t
[n2
]));
485 bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[0]), n
);
486 bn_mul_low_recursive(&(t
[0]), &(a
[n
]), &(b
[0]), n
, &(t
[n2
]));
487 bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[0]), n
);
489 bn_mul_low_normal(&(t
[0]), &(a
[0]), &(b
[n
]), n
);
490 bn_mul_low_normal(&(t
[n
]), &(a
[n
]), &(b
[0]), n
);
491 bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[0]), n
);
492 bn_add_words(&(r
[n
]), &(r
[n
]), &(t
[n
]), n
);
495 #endif /* BN_RECURSION */
497 int BN_mul(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, BN_CTX
*ctx
)
499 int ret
= bn_mul_fixed_top(r
, a
, b
, ctx
);
507 int bn_mul_fixed_top(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, BN_CTX
*ctx
)
512 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
527 if ((al
== 0) || (bl
== 0)) {
534 if ((r
== a
) || (r
== b
)) {
535 if ((rr
= BN_CTX_get(ctx
)) == NULL
)
540 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
547 if (bn_wexpand(rr
, 8) == NULL
)
550 bn_mul_comba4(rr
->d
, a
->d
, b
->d
);
555 if (bn_wexpand(rr
, 16) == NULL
)
558 bn_mul_comba8(rr
->d
, a
->d
, b
->d
);
562 #endif /* BN_MUL_COMBA */
564 if ((al
>= BN_MULL_SIZE_NORMAL
) && (bl
>= BN_MULL_SIZE_NORMAL
)) {
565 if (i
>= -1 && i
<= 1) {
567 * Find out the power of two lower or equal to the longest of the
571 j
= BN_num_bits_word((BN_ULONG
)al
);
574 j
= BN_num_bits_word((BN_ULONG
)bl
);
577 assert(j
<= al
|| j
<= bl
);
582 if (al
> j
|| bl
> j
) {
583 if (bn_wexpand(t
, k
* 4) == NULL
)
585 if (bn_wexpand(rr
, k
* 4) == NULL
)
587 bn_mul_part_recursive(rr
->d
, a
->d
, b
->d
,
588 j
, al
- j
, bl
- j
, t
->d
);
589 } else { /* al <= j || bl <= j */
591 if (bn_wexpand(t
, k
* 2) == NULL
)
593 if (bn_wexpand(rr
, k
* 2) == NULL
)
595 bn_mul_recursive(rr
->d
, a
->d
, b
->d
, j
, al
- j
, bl
- j
, t
->d
);
601 #endif /* BN_RECURSION */
602 if (bn_wexpand(rr
, top
) == NULL
)
605 bn_mul_normal(rr
->d
, a
->d
, al
, b
->d
, bl
);
607 #if defined(BN_MUL_COMBA) || defined(BN_RECURSION)
610 rr
->neg
= a
->neg
^ b
->neg
;
611 rr
->flags
|= BN_FLG_FIXED_TOP
;
612 if (r
!= rr
&& BN_copy(r
, rr
) == NULL
)
622 void bn_mul_normal(BN_ULONG
*r
, BN_ULONG
*a
, int na
, BN_ULONG
*b
, int nb
)
640 (void)bn_mul_words(r
, a
, na
, 0);
643 rr
[0] = bn_mul_words(r
, a
, na
, b
[0]);
648 rr
[1] = bn_mul_add_words(&(r
[1]), a
, na
, b
[1]);
651 rr
[2] = bn_mul_add_words(&(r
[2]), a
, na
, b
[2]);
654 rr
[3] = bn_mul_add_words(&(r
[3]), a
, na
, b
[3]);
657 rr
[4] = bn_mul_add_words(&(r
[4]), a
, na
, b
[4]);
664 void bn_mul_low_normal(BN_ULONG
*r
, BN_ULONG
*a
, BN_ULONG
*b
, int n
)
666 bn_mul_words(r
, a
, n
, b
[0]);
671 bn_mul_add_words(&(r
[1]), a
, n
, b
[1]);
674 bn_mul_add_words(&(r
[2]), a
, n
, b
[2]);
677 bn_mul_add_words(&(r
[3]), a
, n
, b
[3]);
680 bn_mul_add_words(&(r
[4]), a
, n
, b
[4]);