2 * Bignum routines for RSA and DH and stuff.
14 * * Do not call the DIVMOD_WORD macro with expressions such as array
15 * subscripts, as some implementations object to this (see below).
16 * * Note that none of the division methods below will cope if the
17 * quotient won't fit into BIGNUM_INT_BITS. Callers should be careful
19 * If this condition occurs, in the case of the x86 DIV instruction,
20 * an overflow exception will occur, which (according to a correspondent)
21 * will manifest on Windows as something like
22 * 0xC0000095: Integer overflow
23 * The C variant won't give the right answer, either.
26 #if defined __GNUC__ && defined __i386__
27 typedef unsigned long BignumInt
;
28 typedef unsigned long long BignumDblInt
;
29 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
30 #define BIGNUM_TOP_BIT 0x80000000UL
31 #define BIGNUM_INT_BITS 32
32 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
33 #define DIVMOD_WORD(q, r, hi, lo, w) \
35 "=d" (r), "=a" (q) : \
36 "r" (w), "d" (hi), "a" (lo))
37 #elif defined _MSC_VER && defined _M_IX86
38 typedef unsigned __int32 BignumInt
;
39 typedef unsigned __int64 BignumDblInt
;
40 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
41 #define BIGNUM_TOP_BIT 0x80000000UL
42 #define BIGNUM_INT_BITS 32
43 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
44 /* Note: MASM interprets array subscripts in the macro arguments as
45 * assembler syntax, which gives the wrong answer. Don't supply them.
46 * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */
47 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
55 /* 64-bit architectures can do 32x32->64 chunks at a time */
56 typedef unsigned int BignumInt
;
57 typedef unsigned long BignumDblInt
;
58 #define BIGNUM_INT_MASK 0xFFFFFFFFU
59 #define BIGNUM_TOP_BIT 0x80000000U
60 #define BIGNUM_INT_BITS 32
61 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
62 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
63 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
68 /* 64-bit architectures in which unsigned long is 32 bits, not 64 */
69 typedef unsigned long BignumInt
;
70 typedef unsigned long long BignumDblInt
;
71 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
72 #define BIGNUM_TOP_BIT 0x80000000UL
73 #define BIGNUM_INT_BITS 32
74 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
75 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
76 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
81 /* Fallback for all other cases */
82 typedef unsigned short BignumInt
;
83 typedef unsigned long BignumDblInt
;
84 #define BIGNUM_INT_MASK 0xFFFFU
85 #define BIGNUM_TOP_BIT 0x8000U
86 #define BIGNUM_INT_BITS 16
87 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
88 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
89 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
95 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
97 #define BIGNUM_INTERNAL
98 typedef BignumInt
*Bignum
;
102 BignumInt bnZero
[1] = { 0 };
103 BignumInt bnOne
[2] = { 1, 1 };
106 * The Bignum format is an array of `BignumInt'. The first
107 * element of the array counts the remaining elements. The
108 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
109 * significant digit first. (So it's trivial to extract the bit
110 * with value 2^n for any n.)
112 * All Bignums in this module are positive. Negative numbers must
113 * be dealt with outside it.
115 * INVARIANT: the most significant word of any Bignum must be
119 Bignum Zero
= bnZero
, One
= bnOne
;
121 static Bignum
newbn(int length
)
123 Bignum b
= snewn(length
+ 1, BignumInt
);
126 memset(b
, 0, (length
+ 1) * sizeof(*b
));
131 void bn_restore_invariant(Bignum b
)
133 while (b
[0] > 1 && b
[b
[0]] == 0)
137 Bignum
copybn(Bignum orig
)
139 Bignum b
= snewn(orig
[0] + 1, BignumInt
);
142 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
146 void freebn(Bignum b
)
149 * Burn the evidence, just in case.
151 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
155 Bignum
bn_power_2(int n
)
157 Bignum ret
= newbn(n
/ BIGNUM_INT_BITS
+ 1);
158 bignum_set_bit(ret
, n
, 1);
163 * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all
164 * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried
167 static BignumInt
internal_add(const BignumInt
*a
, const BignumInt
*b
,
168 BignumInt
*c
, int len
)
171 BignumDblInt carry
= 0;
173 for (i
= len
-1; i
>= 0; i
--) {
174 carry
+= (BignumDblInt
)a
[i
] + b
[i
];
175 c
[i
] = (BignumInt
)carry
;
176 carry
>>= BIGNUM_INT_BITS
;
179 return (BignumInt
)carry
;
183 * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are
184 * all big-endian arrays of 'len' BignumInts. Any borrow from the top
187 static void internal_sub(const BignumInt
*a
, const BignumInt
*b
,
188 BignumInt
*c
, int len
)
191 BignumDblInt carry
= 1;
193 for (i
= len
-1; i
>= 0; i
--) {
194 carry
+= (BignumDblInt
)a
[i
] + (b
[i
] ^ BIGNUM_INT_MASK
);
195 c
[i
] = (BignumInt
)carry
;
196 carry
>>= BIGNUM_INT_BITS
;
202 * Input is in the first len words of a and b.
203 * Result is returned in the first 2*len words of c.
205 #define KARATSUBA_THRESHOLD 50
206 static void internal_mul(const BignumInt
*a
, const BignumInt
*b
,
207 BignumInt
*c
, int len
)
212 if (len
> KARATSUBA_THRESHOLD
) {
215 * Karatsuba divide-and-conquer algorithm. Cut each input in
216 * half, so that it's expressed as two big 'digits' in a giant
222 * Then the product is of course
224 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
226 * and we compute the three coefficients by recursively
227 * calling ourself to do half-length multiplications.
229 * The clever bit that makes this worth doing is that we only
230 * need _one_ half-length multiplication for the central
231 * coefficient rather than the two that it obviouly looks
232 * like, because we can use a single multiplication to compute
234 * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
236 * and then we subtract the other two coefficients (a_1 b_1
237 * and a_0 b_0) which we were computing anyway.
239 * Hence we get to multiply two numbers of length N in about
240 * three times as much work as it takes to multiply numbers of
241 * length N/2, which is obviously better than the four times
242 * as much work it would take if we just did a long
243 * conventional multiply.
246 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
247 int midlen
= botlen
+ 1;
255 * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
256 * in the output array, so we can compute them immediately in
261 printf("a1,a0 = 0x");
262 for (i
= 0; i
< len
; i
++) {
263 if (i
== toplen
) printf(", 0x");
264 printf("%0*x", BIGNUM_INT_BITS
/4, a
[i
]);
267 printf("b1,b0 = 0x");
268 for (i
= 0; i
< len
; i
++) {
269 if (i
== toplen
) printf(", 0x");
270 printf("%0*x", BIGNUM_INT_BITS
/4, b
[i
]);
276 internal_mul(a
, b
, c
, toplen
);
279 for (i
= 0; i
< 2*toplen
; i
++) {
280 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
286 internal_mul(a
+ toplen
, b
+ toplen
, c
+ 2*toplen
, botlen
);
289 for (i
= 0; i
< 2*botlen
; i
++) {
290 printf("%0*x", BIGNUM_INT_BITS
/4, c
[2*toplen
+i
]);
296 * We must allocate scratch space for the central coefficient,
297 * and also for the two input values that we multiply when
298 * computing it. Since either or both may carry into the
299 * (botlen+1)th word, we must use a slightly longer length
302 scratch
= snewn(4 * midlen
, BignumInt
);
304 /* Zero padding. midlen exceeds toplen by at most 2, so just
305 * zero the first two words of each input and the rest will be
307 scratch
[0] = scratch
[1] = scratch
[midlen
] = scratch
[midlen
+1] = 0;
309 for (j
= 0; j
< toplen
; j
++) {
310 scratch
[midlen
- toplen
+ j
] = a
[j
]; /* a_1 */
311 scratch
[2*midlen
- toplen
+ j
] = b
[j
]; /* b_1 */
314 /* compute a_1 + a_0 */
315 scratch
[0] = internal_add(scratch
+1, a
+toplen
, scratch
+1, botlen
);
317 printf("a1plusa0 = 0x");
318 for (i
= 0; i
< midlen
; i
++) {
319 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
323 /* compute b_1 + b_0 */
324 scratch
[midlen
] = internal_add(scratch
+midlen
+1, b
+toplen
,
325 scratch
+midlen
+1, botlen
);
327 printf("b1plusb0 = 0x");
328 for (i
= 0; i
< midlen
; i
++) {
329 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[midlen
+i
]);
335 * Now we can do the third multiplication.
337 internal_mul(scratch
, scratch
+ midlen
, scratch
+ 2*midlen
, midlen
);
339 printf("a1plusa0timesb1plusb0 = 0x");
340 for (i
= 0; i
< 2*midlen
; i
++) {
341 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
347 * Now we can reuse the first half of 'scratch' to compute the
348 * sum of the outer two coefficients, to subtract from that
349 * product to obtain the middle one.
351 scratch
[0] = scratch
[1] = scratch
[2] = scratch
[3] = 0;
352 for (j
= 0; j
< 2*toplen
; j
++)
353 scratch
[2*midlen
- 2*toplen
+ j
] = c
[j
];
354 scratch
[1] = internal_add(scratch
+2, c
+ 2*toplen
,
355 scratch
+2, 2*botlen
);
357 printf("a1b1plusa0b0 = 0x");
358 for (i
= 0; i
< 2*midlen
; i
++) {
359 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
364 internal_sub(scratch
+ 2*midlen
, scratch
,
365 scratch
+ 2*midlen
, 2*midlen
);
367 printf("a1b0plusa0b1 = 0x");
368 for (i
= 0; i
< 2*midlen
; i
++) {
369 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
375 * And now all we need to do is to add that middle coefficient
376 * back into the output. We may have to propagate a carry
377 * further up the output, but we can be sure it won't
378 * propagate right the way off the top.
380 carry
= internal_add(c
+ 2*len
- botlen
- 2*midlen
,
382 c
+ 2*len
- botlen
- 2*midlen
, 2*midlen
);
383 j
= 2*len
- botlen
- 2*midlen
- 1;
387 c
[j
] = (BignumInt
)carry
;
388 carry
>>= BIGNUM_INT_BITS
;
392 for (i
= 0; i
< 2*len
; i
++) {
393 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
399 for (j
= 0; j
< 4 * midlen
; j
++)
406 * Multiply in the ordinary O(N^2) way.
409 for (j
= 0; j
< 2 * len
; j
++)
412 for (i
= len
- 1; i
>= 0; i
--) {
414 for (j
= len
- 1; j
>= 0; j
--) {
415 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
416 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
417 c
[i
+ j
+ 1] = (BignumInt
) t
;
418 t
= t
>> BIGNUM_INT_BITS
;
420 c
[i
] = (BignumInt
) t
;
426 * Variant form of internal_mul used for the initial step of
427 * Montgomery reduction. Only bothers outputting 'len' words
428 * (everything above that is thrown away).
430 static void internal_mul_low(const BignumInt
*a
, const BignumInt
*b
,
431 BignumInt
*c
, int len
)
436 if (len
> KARATSUBA_THRESHOLD
) {
439 * Karatsuba-aware version of internal_mul_low. As before, we
440 * express each input value as a shifted combination of two
446 * Then the full product is, as before,
448 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
450 * Provided we choose D on the large side (so that a_0 and b_0
451 * are _at least_ as long as a_1 and b_1), we don't need the
452 * topmost term at all, and we only need half of the middle
453 * term. So there's no point in doing the proper Karatsuba
454 * optimisation which computes the middle term using the top
455 * one, because we'd take as long computing the top one as
456 * just computing the middle one directly.
458 * So instead, we do a much more obvious thing: we call the
459 * fully optimised internal_mul to compute a_0 b_0, and we
460 * recursively call ourself to compute the _bottom halves_ of
461 * a_1 b_0 and a_0 b_1, each of which we add into the result
462 * in the obvious way.
464 * In other words, there's no actual Karatsuba _optimisation_
465 * in this function; the only benefit in doing it this way is
466 * that we call internal_mul proper for a large part of the
467 * work, and _that_ can optimise its operation.
470 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
474 * Allocate scratch space for the various bits and pieces
475 * we're going to be adding together. We need botlen*2 words
476 * for a_0 b_0 (though we may end up throwing away its topmost
477 * word), and toplen words for each of a_1 b_0 and a_0 b_1.
478 * That adds up to exactly 2*len.
480 scratch
= snewn(len
*2, BignumInt
);
483 internal_mul(a
+ toplen
, b
+ toplen
, scratch
+ 2*toplen
, botlen
);
486 internal_mul_low(a
, b
+ len
- toplen
, scratch
+ toplen
, toplen
);
489 internal_mul_low(a
+ len
- toplen
, b
, scratch
, toplen
);
491 /* Copy the bottom half of the big coefficient into place */
492 for (j
= 0; j
< botlen
; j
++)
493 c
[toplen
+ j
] = scratch
[2*toplen
+ botlen
+ j
];
495 /* Add the two small coefficients, throwing away the returned carry */
496 internal_add(scratch
, scratch
+ toplen
, scratch
, toplen
);
498 /* And add that to the large coefficient, leaving the result in c. */
499 internal_add(scratch
, scratch
+ 2*toplen
+ botlen
- toplen
,
503 for (j
= 0; j
< len
*2; j
++)
509 for (j
= 0; j
< len
; j
++)
512 for (i
= len
- 1; i
>= 0; i
--) {
514 for (j
= len
- 1; j
>= len
- i
- 1; j
--) {
515 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
516 t
+= (BignumDblInt
) c
[i
+ j
+ 1 - len
];
517 c
[i
+ j
+ 1 - len
] = (BignumInt
) t
;
518 t
= t
>> BIGNUM_INT_BITS
;
526 * Montgomery reduction. Expects x to be a big-endian array of 2*len
527 * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
528 * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
529 * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
532 * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
533 * each, containing respectively n and the multiplicative inverse of
536 * 'tmp' is an array of at least '3*len' BignumInts used as scratch
539 static void monty_reduce(BignumInt
*x
, const BignumInt
*n
,
540 const BignumInt
*mninv
, BignumInt
*tmp
, int len
)
546 * Multiply x by (-n)^{-1} mod r. This gives us a value m such
547 * that mn is congruent to -x mod r. Hence, mn+x is an exact
548 * multiple of r, and is also (obviously) congruent to x mod n.
550 internal_mul_low(x
+ len
, mninv
, tmp
, len
);
553 * Compute t = (mn+x)/r in ordinary, non-modular, integer
554 * arithmetic. By construction this is exact, and is congruent mod
555 * n to x * r^{-1}, i.e. the answer we want.
557 * The following multiply leaves that answer in the _most_
558 * significant half of the 'x' array, so then we must shift it
561 internal_mul(tmp
, n
, tmp
+len
, len
);
562 carry
= internal_add(x
, tmp
+len
, x
, 2*len
);
563 for (i
= 0; i
< len
; i
++)
564 x
[len
+ i
] = x
[i
], x
[i
] = 0;
567 * Reduce t mod n. This doesn't require a full-on division by n,
568 * but merely a test and single optional subtraction, since we can
569 * show that 0 <= t < 2n.
572 * + we computed m mod r, so 0 <= m < r.
573 * + so 0 <= mn < rn, obviously
574 * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
575 * + yielding 0 <= (mn+x)/r < 2n as required.
578 for (i
= 0; i
< len
; i
++)
579 if (x
[len
+ i
] != n
[i
])
582 if (carry
|| i
>= len
|| x
[len
+ i
] > n
[i
])
583 internal_sub(x
+len
, n
, x
+len
, len
);
586 static void internal_add_shifted(BignumInt
*number
,
587 unsigned n
, int shift
)
589 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
590 int bshift
= shift
% BIGNUM_INT_BITS
;
593 addend
= (BignumDblInt
)n
<< bshift
;
596 addend
+= number
[word
];
597 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
598 addend
>>= BIGNUM_INT_BITS
;
605 * Input in first alen words of a and first mlen words of m.
606 * Output in first alen words of a
607 * (of which first alen-mlen words will be zero).
608 * The MSW of m MUST have its high bit set.
609 * Quotient is accumulated in the `quotient' array, which is a Bignum
610 * rather than the internal bigendian format. Quotient parts are shifted
611 * left by `qshift' before adding into quot.
613 static void internal_mod(BignumInt
*a
, int alen
,
614 BignumInt
*m
, int mlen
,
615 BignumInt
*quot
, int qshift
)
627 for (i
= 0; i
<= alen
- mlen
; i
++) {
629 unsigned int q
, r
, c
, ai1
;
643 /* Find q = h:a[i] / m0 */
648 * To illustrate it, suppose a BignumInt is 8 bits, and
649 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
650 * our initial division will be 0xA123 / 0xA1, which
651 * will give a quotient of 0x100 and a divide overflow.
652 * However, the invariants in this division algorithm
653 * are not violated, since the full number A1:23:... is
654 * _less_ than the quotient prefix A1:B2:... and so the
655 * following correction loop would have sorted it out.
657 * In this situation we set q to be the largest
658 * quotient we _can_ stomach (0xFF, of course).
662 /* Macro doesn't want an array subscript expression passed
663 * into it (see definition), so use a temporary. */
664 BignumInt tmplo
= a
[i
];
665 DIVMOD_WORD(q
, r
, h
, tmplo
, m0
);
667 /* Refine our estimate of q by looking at
668 h:a[i]:a[i+1] / m0:m1 */
670 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
673 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
674 if (r
>= (BignumDblInt
) m0
&&
675 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
679 /* Subtract q * m from a[i...] */
681 for (k
= mlen
- 1; k
>= 0; k
--) {
682 t
= MUL_WORD(q
, m
[k
]);
684 c
= (unsigned)(t
>> BIGNUM_INT_BITS
);
685 if ((BignumInt
) t
> a
[i
+ k
])
687 a
[i
+ k
] -= (BignumInt
) t
;
690 /* Add back m in case of borrow */
693 for (k
= mlen
- 1; k
>= 0; k
--) {
696 a
[i
+ k
] = (BignumInt
) t
;
697 t
= t
>> BIGNUM_INT_BITS
;
702 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
707 * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
710 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
712 BignumInt
*a
, *b
, *x
, *n
, *mninv
, *tmp
;
714 Bignum base
, base2
, r
, rn
, inv
, result
;
717 * The most significant word of mod needs to be non-zero. It
718 * should already be, but let's make sure.
720 assert(mod
[mod
[0]] != 0);
723 * Make sure the base is smaller than the modulus, by reducing
724 * it modulo the modulus if not.
726 base
= bigmod(base_in
, mod
);
729 * mod had better be odd, or we can't do Montgomery multiplication
730 * using a power of two at all.
735 * Compute the inverse of n mod r, for monty_reduce. (In fact we
736 * want the inverse of _minus_ n mod r, but we'll sort that out
740 r
= bn_power_2(BIGNUM_INT_BITS
* len
);
741 inv
= modinv(mod
, r
);
744 * Multiply the base by r mod n, to get it into Montgomery
747 base2
= modmul(base
, r
, mod
);
751 rn
= bigmod(r
, mod
); /* r mod n, i.e. Montgomerified 1 */
753 freebn(r
); /* won't need this any more */
756 * Set up internal arrays of the right lengths, in big-endian
757 * format, containing the base, the modulus, and the modulus's
760 n
= snewn(len
, BignumInt
);
761 for (j
= 0; j
< len
; j
++)
762 n
[len
- 1 - j
] = mod
[j
+ 1];
764 mninv
= snewn(len
, BignumInt
);
765 for (j
= 0; j
< len
; j
++)
766 mninv
[len
- 1 - j
] = (j
< inv
[0] ? inv
[j
+ 1] : 0);
767 freebn(inv
); /* we don't need this copy of it any more */
768 /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
769 x
= snewn(len
, BignumInt
);
770 for (j
= 0; j
< len
; j
++)
772 internal_sub(x
, mninv
, mninv
, len
);
774 /* x = snewn(len, BignumInt); */ /* already done above */
775 for (j
= 0; j
< len
; j
++)
776 x
[len
- 1 - j
] = (j
< base
[0] ? base
[j
+ 1] : 0);
777 freebn(base
); /* we don't need this copy of it any more */
779 a
= snewn(2*len
, BignumInt
);
780 b
= snewn(2*len
, BignumInt
);
781 for (j
= 0; j
< len
; j
++)
782 a
[2*len
- 1 - j
] = (j
< rn
[0] ? rn
[j
+ 1] : 0);
785 tmp
= snewn(3*len
, BignumInt
);
787 /* Skip leading zero bits of exp. */
789 j
= BIGNUM_INT_BITS
-1;
790 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
794 j
= BIGNUM_INT_BITS
-1;
798 /* Main computation */
799 while (i
< (int)exp
[0]) {
801 internal_mul(a
+ len
, a
+ len
, b
, len
);
802 monty_reduce(b
, n
, mninv
, tmp
, len
);
803 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
804 internal_mul(b
+ len
, x
, a
, len
);
805 monty_reduce(a
, n
, mninv
, tmp
, len
);
815 j
= BIGNUM_INT_BITS
-1;
819 * Final monty_reduce to get back from the adjusted Montgomery
822 monty_reduce(a
, n
, mninv
, tmp
, len
);
824 /* Copy result to buffer */
825 result
= newbn(mod
[0]);
826 for (i
= 0; i
< len
; i
++)
827 result
[result
[0] - i
] = a
[i
+ len
];
828 while (result
[0] > 1 && result
[result
[0]] == 0)
831 /* Free temporary arrays */
832 for (i
= 0; i
< 3 * len
; i
++)
835 for (i
= 0; i
< 2 * len
; i
++)
838 for (i
= 0; i
< 2 * len
; i
++)
841 for (i
= 0; i
< len
; i
++)
844 for (i
= 0; i
< len
; i
++)
847 for (i
= 0; i
< len
; i
++)
855 * Compute (p * q) % mod.
856 * The most significant word of mod MUST be non-zero.
857 * We assume that the result array is the same size as the mod array.
859 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
861 BignumInt
*a
, *n
, *m
, *o
;
863 int pqlen
, mlen
, rlen
, i
, j
;
866 /* Allocate m of size mlen, copy mod to m */
867 /* We use big endian internally */
869 m
= snewn(mlen
, BignumInt
);
870 for (j
= 0; j
< mlen
; j
++)
871 m
[j
] = mod
[mod
[0] - j
];
873 /* Shift m left to make msb bit set */
874 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
875 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
878 for (i
= 0; i
< mlen
- 1; i
++)
879 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
880 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
883 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
885 /* Allocate n of size pqlen, copy p to n */
886 n
= snewn(pqlen
, BignumInt
);
888 for (j
= 0; j
< i
; j
++)
890 for (j
= 0; j
< (int)p
[0]; j
++)
891 n
[i
+ j
] = p
[p
[0] - j
];
893 /* Allocate o of size pqlen, copy q to o */
894 o
= snewn(pqlen
, BignumInt
);
896 for (j
= 0; j
< i
; j
++)
898 for (j
= 0; j
< (int)q
[0]; j
++)
899 o
[i
+ j
] = q
[q
[0] - j
];
901 /* Allocate a of size 2*pqlen for result */
902 a
= snewn(2 * pqlen
, BignumInt
);
904 /* Main computation */
905 internal_mul(n
, o
, a
, pqlen
);
906 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
908 /* Fixup result in case the modulus was shifted */
910 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
911 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
912 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
913 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
914 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
915 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
918 /* Copy result to buffer */
919 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
920 result
= newbn(rlen
);
921 for (i
= 0; i
< rlen
; i
++)
922 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
923 while (result
[0] > 1 && result
[result
[0]] == 0)
926 /* Free temporary arrays */
927 for (i
= 0; i
< 2 * pqlen
; i
++)
930 for (i
= 0; i
< mlen
; i
++)
933 for (i
= 0; i
< pqlen
; i
++)
936 for (i
= 0; i
< pqlen
; i
++)
945 * The most significant word of mod MUST be non-zero.
946 * We assume that the result array is the same size as the mod array.
947 * We optionally write out a quotient if `quotient' is non-NULL.
948 * We can avoid writing out the result if `result' is NULL.
950 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
954 int plen
, mlen
, i
, j
;
956 /* Allocate m of size mlen, copy mod to m */
957 /* We use big endian internally */
959 m
= snewn(mlen
, BignumInt
);
960 for (j
= 0; j
< mlen
; j
++)
961 m
[j
] = mod
[mod
[0] - j
];
963 /* Shift m left to make msb bit set */
964 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
965 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
968 for (i
= 0; i
< mlen
- 1; i
++)
969 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
970 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
974 /* Ensure plen > mlen */
978 /* Allocate n of size plen, copy p to n */
979 n
= snewn(plen
, BignumInt
);
980 for (j
= 0; j
< plen
; j
++)
982 for (j
= 1; j
<= (int)p
[0]; j
++)
985 /* Main computation */
986 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
988 /* Fixup result in case the modulus was shifted */
990 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
991 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
992 n
[plen
- 1] = n
[plen
- 1] << mshift
;
993 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
994 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
995 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
998 /* Copy result to buffer */
1000 for (i
= 1; i
<= (int)result
[0]; i
++) {
1002 result
[i
] = j
>= 0 ? n
[j
] : 0;
1006 /* Free temporary arrays */
1007 for (i
= 0; i
< mlen
; i
++)
1010 for (i
= 0; i
< plen
; i
++)
1016 * Decrement a number.
1018 void decbn(Bignum bn
)
1021 while (i
< (int)bn
[0] && bn
[i
] == 0)
1022 bn
[i
++] = BIGNUM_INT_MASK
;
1026 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
1031 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
1034 for (i
= 1; i
<= w
; i
++)
1036 for (i
= nbytes
; i
--;) {
1037 unsigned char byte
= *data
++;
1038 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
1041 while (result
[0] > 1 && result
[result
[0]] == 0)
1047 * Read an SSH-1-format bignum from a data buffer. Return the number
1048 * of bytes consumed, or -1 if there wasn't enough data.
1050 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
1052 const unsigned char *p
= data
;
1060 for (i
= 0; i
< 2; i
++)
1061 w
= (w
<< 8) + *p
++;
1062 b
= (w
+ 7) / 8; /* bits -> bytes */
1067 if (!result
) /* just return length */
1070 *result
= bignum_from_bytes(p
, b
);
1072 return p
+ b
- data
;
1076 * Return the bit count of a bignum, for SSH-1 encoding.
1078 int bignum_bitcount(Bignum bn
)
1080 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
1081 while (bitcount
>= 0
1082 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
1083 return bitcount
+ 1;
1087 * Return the byte length of a bignum when SSH-1 encoded.
1089 int ssh1_bignum_length(Bignum bn
)
1091 return 2 + (bignum_bitcount(bn
) + 7) / 8;
1095 * Return the byte length of a bignum when SSH-2 encoded.
1097 int ssh2_bignum_length(Bignum bn
)
1099 return 4 + (bignum_bitcount(bn
) + 8) / 8;
1103 * Return a byte from a bignum; 0 is least significant, etc.
1105 int bignum_byte(Bignum bn
, int i
)
1107 if (i
>= (int)(BIGNUM_INT_BYTES
* bn
[0]))
1108 return 0; /* beyond the end */
1110 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
1111 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
1115 * Return a bit from a bignum; 0 is least significant, etc.
1117 int bignum_bit(Bignum bn
, int i
)
1119 if (i
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1120 return 0; /* beyond the end */
1122 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
1126 * Set a bit in a bignum; 0 is least significant, etc.
1128 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
1130 if (bitnum
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1131 abort(); /* beyond the end */
1133 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
1134 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
1143 * Write a SSH-1-format bignum into a buffer. It is assumed the
1144 * buffer is big enough. Returns the number of bytes used.
1146 int ssh1_write_bignum(void *data
, Bignum bn
)
1148 unsigned char *p
= data
;
1149 int len
= ssh1_bignum_length(bn
);
1151 int bitc
= bignum_bitcount(bn
);
1153 *p
++ = (bitc
>> 8) & 0xFF;
1154 *p
++ = (bitc
) & 0xFF;
1155 for (i
= len
- 2; i
--;)
1156 *p
++ = bignum_byte(bn
, i
);
1161 * Compare two bignums. Returns like strcmp.
1163 int bignum_cmp(Bignum a
, Bignum b
)
1165 int amax
= a
[0], bmax
= b
[0];
1166 int i
= (amax
> bmax ? amax
: bmax
);
1168 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
1169 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
1180 * Right-shift one bignum to form another.
1182 Bignum
bignum_rshift(Bignum a
, int shift
)
1185 int i
, shiftw
, shiftb
, shiftbb
, bits
;
1188 bits
= bignum_bitcount(a
) - shift
;
1189 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
1192 shiftw
= shift
/ BIGNUM_INT_BITS
;
1193 shiftb
= shift
% BIGNUM_INT_BITS
;
1194 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
1196 ai1
= a
[shiftw
+ 1];
1197 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1199 ai1
= (i
+ shiftw
+ 1 <= (int)a
[0] ? a
[i
+ shiftw
+ 1] : 0);
1200 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
1208 * Non-modular multiplication and addition.
1210 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
1212 int alen
= a
[0], blen
= b
[0];
1213 int mlen
= (alen
> blen ? alen
: blen
);
1214 int rlen
, i
, maxspot
;
1215 BignumInt
*workspace
;
1218 /* mlen space for a, mlen space for b, 2*mlen for result */
1219 workspace
= snewn(mlen
* 4, BignumInt
);
1220 for (i
= 0; i
< mlen
; i
++) {
1221 workspace
[0 * mlen
+ i
] = (mlen
- i
<= (int)a
[0] ? a
[mlen
- i
] : 0);
1222 workspace
[1 * mlen
+ i
] = (mlen
- i
<= (int)b
[0] ? b
[mlen
- i
] : 0);
1225 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
1226 workspace
+ 2 * mlen
, mlen
);
1228 /* now just copy the result back */
1229 rlen
= alen
+ blen
+ 1;
1230 if (addend
&& rlen
<= (int)addend
[0])
1231 rlen
= addend
[0] + 1;
1234 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1235 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
1241 /* now add in the addend, if any */
1243 BignumDblInt carry
= 0;
1244 for (i
= 1; i
<= rlen
; i
++) {
1245 carry
+= (i
<= (int)ret
[0] ? ret
[i
] : 0);
1246 carry
+= (i
<= (int)addend
[0] ? addend
[i
] : 0);
1247 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1248 carry
>>= BIGNUM_INT_BITS
;
1249 if (ret
[i
] != 0 && i
> maxspot
)
1260 * Non-modular multiplication.
1262 Bignum
bigmul(Bignum a
, Bignum b
)
1264 return bigmuladd(a
, b
, NULL
);
1270 Bignum
bigadd(Bignum a
, Bignum b
)
1272 int alen
= a
[0], blen
= b
[0];
1273 int rlen
= (alen
> blen ? alen
: blen
) + 1;
1282 for (i
= 1; i
<= rlen
; i
++) {
1283 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1284 carry
+= (i
<= (int)b
[0] ? b
[i
] : 0);
1285 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1286 carry
>>= BIGNUM_INT_BITS
;
1287 if (ret
[i
] != 0 && i
> maxspot
)
1296 * Subtraction. Returns a-b, or NULL if the result would come out
1297 * negative (recall that this entire bignum module only handles
1298 * positive numbers).
1300 Bignum
bigsub(Bignum a
, Bignum b
)
1302 int alen
= a
[0], blen
= b
[0];
1303 int rlen
= (alen
> blen ? alen
: blen
);
1312 for (i
= 1; i
<= rlen
; i
++) {
1313 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1314 carry
+= (i
<= (int)b
[0] ? b
[i
] ^ BIGNUM_INT_MASK
: BIGNUM_INT_MASK
);
1315 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1316 carry
>>= BIGNUM_INT_BITS
;
1317 if (ret
[i
] != 0 && i
> maxspot
)
1331 * Create a bignum which is the bitmask covering another one. That
1332 * is, the smallest integer which is >= N and is also one less than
1335 Bignum
bignum_bitmask(Bignum n
)
1337 Bignum ret
= copybn(n
);
1342 while (n
[i
] == 0 && i
> 0)
1345 return ret
; /* input was zero */
1351 ret
[i
] = BIGNUM_INT_MASK
;
1356 * Convert a (max 32-bit) long into a bignum.
1358 Bignum
bignum_from_long(unsigned long nn
)
1361 BignumDblInt n
= nn
;
1364 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
1365 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
1367 ret
[0] = (ret
[2] ?
2 : 1);
1372 * Add a long to a bignum.
1374 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
1376 Bignum ret
= newbn(number
[0] + 1);
1378 BignumDblInt carry
= 0, addend
= addendx
;
1380 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1381 carry
+= addend
& BIGNUM_INT_MASK
;
1382 carry
+= (i
<= (int)number
[0] ? number
[i
] : 0);
1383 addend
>>= BIGNUM_INT_BITS
;
1384 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1385 carry
>>= BIGNUM_INT_BITS
;
1394 * Compute the residue of a bignum, modulo a (max 16-bit) short.
1396 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
1398 BignumDblInt mod
, r
;
1403 for (i
= number
[0]; i
> 0; i
--)
1404 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
1405 return (unsigned short) r
;
1409 void diagbn(char *prefix
, Bignum md
)
1411 int i
, nibbles
, morenibbles
;
1412 static const char hex
[] = "0123456789ABCDEF";
1414 debug(("%s0x", prefix ? prefix
: ""));
1416 nibbles
= (3 + bignum_bitcount(md
)) / 4;
1419 morenibbles
= 4 * md
[0] - nibbles
;
1420 for (i
= 0; i
< morenibbles
; i
++)
1422 for (i
= nibbles
; i
--;)
1424 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
1434 Bignum
bigdiv(Bignum a
, Bignum b
)
1436 Bignum q
= newbn(a
[0]);
1437 bigdivmod(a
, b
, NULL
, q
);
1444 Bignum
bigmod(Bignum a
, Bignum b
)
1446 Bignum r
= newbn(b
[0]);
1447 bigdivmod(a
, b
, r
, NULL
);
1452 * Greatest common divisor.
1454 Bignum
biggcd(Bignum av
, Bignum bv
)
1456 Bignum a
= copybn(av
);
1457 Bignum b
= copybn(bv
);
1459 while (bignum_cmp(b
, Zero
) != 0) {
1460 Bignum t
= newbn(b
[0]);
1461 bigdivmod(a
, b
, t
, NULL
);
1462 while (t
[0] > 1 && t
[t
[0]] == 0)
1474 * Modular inverse, using Euclid's extended algorithm.
1476 Bignum
modinv(Bignum number
, Bignum modulus
)
1478 Bignum a
= copybn(modulus
);
1479 Bignum b
= copybn(number
);
1480 Bignum xp
= copybn(Zero
);
1481 Bignum x
= copybn(One
);
1484 while (bignum_cmp(b
, One
) != 0) {
1485 Bignum t
= newbn(b
[0]);
1486 Bignum q
= newbn(a
[0]);
1487 bigdivmod(a
, b
, t
, q
);
1488 while (t
[0] > 1 && t
[t
[0]] == 0)
1495 x
= bigmuladd(q
, xp
, t
);
1505 /* now we know that sign * x == 1, and that x < modulus */
1507 /* set a new x to be modulus - x */
1508 Bignum newx
= newbn(modulus
[0]);
1509 BignumInt carry
= 0;
1513 for (i
= 1; i
<= (int)newx
[0]; i
++) {
1514 BignumInt aword
= (i
<= (int)modulus
[0] ? modulus
[i
] : 0);
1515 BignumInt bword
= (i
<= (int)x
[0] ? x
[i
] : 0);
1516 newx
[i
] = aword
- bword
- carry
;
1518 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1532 * Render a bignum into decimal. Return a malloced string holding
1533 * the decimal representation.
1535 char *bignum_decimal(Bignum x
)
1537 int ndigits
, ndigit
;
1541 BignumInt
*workspace
;
1544 * First, estimate the number of digits. Since log(10)/log(2)
1545 * is just greater than 93/28 (the joys of continued fraction
1546 * approximations...) we know that for every 93 bits, we need
1547 * at most 28 digits. This will tell us how much to malloc.
1549 * Formally: if x has i bits, that means x is strictly less
1550 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1551 * 10^(28i/93). We need an integer power of ten, so we must
1552 * round up (rounding down might make it less than x again).
1553 * Therefore if we multiply the bit count by 28/93, rounding
1554 * up, we will have enough digits.
1556 * i=0 (i.e., x=0) is an irritating special case.
1558 i
= bignum_bitcount(x
);
1560 ndigits
= 1; /* x = 0 */
1562 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1563 ndigits
++; /* allow for trailing \0 */
1564 ret
= snewn(ndigits
, char);
1567 * Now allocate some workspace to hold the binary form as we
1568 * repeatedly divide it by ten. Initialise this to the
1569 * big-endian form of the number.
1571 workspace
= snewn(x
[0], BignumInt
);
1572 for (i
= 0; i
< (int)x
[0]; i
++)
1573 workspace
[i
] = x
[x
[0] - i
];
1576 * Next, write the decimal number starting with the last digit.
1577 * We use ordinary short division, dividing 10 into the
1580 ndigit
= ndigits
- 1;
1585 for (i
= 0; i
< (int)x
[0]; i
++) {
1586 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1587 workspace
[i
] = (BignumInt
) (carry
/ 10);
1592 ret
[--ndigit
] = (char) (carry
+ '0');
1596 * There's a chance we've fallen short of the start of the
1597 * string. Correct if so.
1600 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);
1616 * gcc -g -O0 -DTESTBN -o testbn sshbn.c misc.c -I unix -I charset
1619 void modalfatalbox(char *p
, ...)
1622 fprintf(stderr
, "FATAL ERROR: ");
1624 vfprintf(stderr
, p
, ap
);
1626 fputc('\n', stderr
);
1630 #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
1632 int main(int argc
, char **argv
)
1636 int passes
= 0, fails
= 0;
1638 while ((buf
= fgetline(stdin
)) != NULL
) {
1639 int maxlen
= strlen(buf
);
1640 unsigned char *data
= snewn(maxlen
, unsigned char);
1641 unsigned char *ptrs
[4], *q
;
1654 while (*bufp
&& !isxdigit((unsigned char)*bufp
))
1661 while (*bufp
&& isxdigit((unsigned char)*bufp
))
1665 if (ptrnum
>= lenof(ptrs
))
1669 for (i
= -((end
- start
) & 1); i
< end
-start
; i
+= 2) {
1670 unsigned char val
= (i
< 0 ?
0 : fromxdigit(start
[i
]));
1671 val
= val
* 16 + fromxdigit(start
[i
+1]);
1679 Bignum a
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1680 Bignum b
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1681 Bignum c
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1682 Bignum p
= bigmul(a
, b
);
1684 if (bignum_cmp(c
, p
) == 0) {
1687 char *as
= bignum_decimal(a
);
1688 char *bs
= bignum_decimal(b
);
1689 char *cs
= bignum_decimal(c
);
1690 char *ps
= bignum_decimal(p
);
1692 printf("%d: fail: %s * %s gave %s expected %s\n",
1693 line
, as
, bs
, ps
, cs
);
1710 printf("passed %d failed %d total %d\n", passes
, fails
, passes
+fails
);