2 * Bignum routines for RSA and DH and stuff.
15 * * Do not call the DIVMOD_WORD macro with expressions such as array
16 * subscripts, as some implementations object to this (see below).
17 * * Note that none of the division methods below will cope if the
18 * quotient won't fit into BIGNUM_INT_BITS. Callers should be careful
20 * If this condition occurs, in the case of the x86 DIV instruction,
21 * an overflow exception will occur, which (according to a correspondent)
22 * will manifest on Windows as something like
23 * 0xC0000095: Integer overflow
24 * The C variant won't give the right answer, either.
27 #if defined __GNUC__ && defined __i386__
28 typedef unsigned long BignumInt
;
29 typedef unsigned long long BignumDblInt
;
30 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
31 #define BIGNUM_TOP_BIT 0x80000000UL
32 #define BIGNUM_INT_BITS 32
33 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
34 #define DIVMOD_WORD(q, r, hi, lo, w) \
36 "=d" (r), "=a" (q) : \
37 "r" (w), "d" (hi), "a" (lo))
38 #elif defined _MSC_VER && defined _M_IX86
39 typedef unsigned __int32 BignumInt
;
40 typedef unsigned __int64 BignumDblInt
;
41 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
42 #define BIGNUM_TOP_BIT 0x80000000UL
43 #define BIGNUM_INT_BITS 32
44 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
45 /* Note: MASM interprets array subscripts in the macro arguments as
46 * assembler syntax, which gives the wrong answer. Don't supply them.
47 * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */
48 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
56 /* 64-bit architectures can do 32x32->64 chunks at a time */
57 typedef unsigned int BignumInt
;
58 typedef unsigned long BignumDblInt
;
59 #define BIGNUM_INT_MASK 0xFFFFFFFFU
60 #define BIGNUM_TOP_BIT 0x80000000U
61 #define BIGNUM_INT_BITS 32
62 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
63 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
64 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
69 /* 64-bit architectures in which unsigned long is 32 bits, not 64 */
70 typedef unsigned long BignumInt
;
71 typedef unsigned long long BignumDblInt
;
72 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
73 #define BIGNUM_TOP_BIT 0x80000000UL
74 #define BIGNUM_INT_BITS 32
75 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
76 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
77 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
82 /* Fallback for all other cases */
83 typedef unsigned short BignumInt
;
84 typedef unsigned long BignumDblInt
;
85 #define BIGNUM_INT_MASK 0xFFFFU
86 #define BIGNUM_TOP_BIT 0x8000U
87 #define BIGNUM_INT_BITS 16
88 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
89 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
90 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
96 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
98 #define BIGNUM_INTERNAL
99 typedef BignumInt
*Bignum
;
103 BignumInt bnZero
[1] = { 0 };
104 BignumInt bnOne
[2] = { 1, 1 };
107 * The Bignum format is an array of `BignumInt'. The first
108 * element of the array counts the remaining elements. The
109 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
110 * significant digit first. (So it's trivial to extract the bit
111 * with value 2^n for any n.)
113 * All Bignums in this module are positive. Negative numbers must
114 * be dealt with outside it.
116 * INVARIANT: the most significant word of any Bignum must be
120 Bignum Zero
= bnZero
, One
= bnOne
;
122 static Bignum
newbn(int length
)
126 assert(length
>= 0 && length
< INT_MAX
/ BIGNUM_INT_BITS
);
128 b
= snewn(length
+ 1, BignumInt
);
131 memset(b
, 0, (length
+ 1) * sizeof(*b
));
136 void bn_restore_invariant(Bignum b
)
138 while (b
[0] > 1 && b
[b
[0]] == 0)
142 Bignum
copybn(Bignum orig
)
144 Bignum b
= snewn(orig
[0] + 1, BignumInt
);
147 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
151 void freebn(Bignum b
)
154 * Burn the evidence, just in case.
156 smemclr(b
, sizeof(b
[0]) * (b
[0] + 1));
160 Bignum
bn_power_2(int n
)
166 ret
= newbn(n
/ BIGNUM_INT_BITS
+ 1);
167 bignum_set_bit(ret
, n
, 1);
172 * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all
173 * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried
176 static BignumInt
internal_add(const BignumInt
*a
, const BignumInt
*b
,
177 BignumInt
*c
, int len
)
180 BignumDblInt carry
= 0;
182 for (i
= len
-1; i
>= 0; i
--) {
183 carry
+= (BignumDblInt
)a
[i
] + b
[i
];
184 c
[i
] = (BignumInt
)carry
;
185 carry
>>= BIGNUM_INT_BITS
;
188 return (BignumInt
)carry
;
192 * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are
193 * all big-endian arrays of 'len' BignumInts. Any borrow from the top
196 static void internal_sub(const BignumInt
*a
, const BignumInt
*b
,
197 BignumInt
*c
, int len
)
200 BignumDblInt carry
= 1;
202 for (i
= len
-1; i
>= 0; i
--) {
203 carry
+= (BignumDblInt
)a
[i
] + (b
[i
] ^ BIGNUM_INT_MASK
);
204 c
[i
] = (BignumInt
)carry
;
205 carry
>>= BIGNUM_INT_BITS
;
211 * Input is in the first len words of a and b.
212 * Result is returned in the first 2*len words of c.
214 * 'scratch' must point to an array of BignumInt of size at least
215 * mul_compute_scratch(len). (This covers the needs of internal_mul
216 * and all its recursive calls to itself.)
218 #define KARATSUBA_THRESHOLD 50
219 static int mul_compute_scratch(int len
)
222 while (len
> KARATSUBA_THRESHOLD
) {
223 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
224 int midlen
= botlen
+ 1;
230 static void internal_mul(const BignumInt
*a
, const BignumInt
*b
,
231 BignumInt
*c
, int len
, BignumInt
*scratch
)
233 if (len
> KARATSUBA_THRESHOLD
) {
237 * Karatsuba divide-and-conquer algorithm. Cut each input in
238 * half, so that it's expressed as two big 'digits' in a giant
244 * Then the product is of course
246 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
248 * and we compute the three coefficients by recursively
249 * calling ourself to do half-length multiplications.
251 * The clever bit that makes this worth doing is that we only
252 * need _one_ half-length multiplication for the central
253 * coefficient rather than the two that it obviouly looks
254 * like, because we can use a single multiplication to compute
256 * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
258 * and then we subtract the other two coefficients (a_1 b_1
259 * and a_0 b_0) which we were computing anyway.
261 * Hence we get to multiply two numbers of length N in about
262 * three times as much work as it takes to multiply numbers of
263 * length N/2, which is obviously better than the four times
264 * as much work it would take if we just did a long
265 * conventional multiply.
268 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
269 int midlen
= botlen
+ 1;
276 * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
277 * in the output array, so we can compute them immediately in
282 printf("a1,a0 = 0x");
283 for (i
= 0; i
< len
; i
++) {
284 if (i
== toplen
) printf(", 0x");
285 printf("%0*x", BIGNUM_INT_BITS
/4, a
[i
]);
288 printf("b1,b0 = 0x");
289 for (i
= 0; i
< len
; i
++) {
290 if (i
== toplen
) printf(", 0x");
291 printf("%0*x", BIGNUM_INT_BITS
/4, b
[i
]);
297 internal_mul(a
, b
, c
, toplen
, scratch
);
300 for (i
= 0; i
< 2*toplen
; i
++) {
301 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
307 internal_mul(a
+ toplen
, b
+ toplen
, c
+ 2*toplen
, botlen
, scratch
);
310 for (i
= 0; i
< 2*botlen
; i
++) {
311 printf("%0*x", BIGNUM_INT_BITS
/4, c
[2*toplen
+i
]);
316 /* Zero padding. midlen exceeds toplen by at most 2, so just
317 * zero the first two words of each input and the rest will be
319 scratch
[0] = scratch
[1] = scratch
[midlen
] = scratch
[midlen
+1] = 0;
321 for (i
= 0; i
< toplen
; i
++) {
322 scratch
[midlen
- toplen
+ i
] = a
[i
]; /* a_1 */
323 scratch
[2*midlen
- toplen
+ i
] = b
[i
]; /* b_1 */
326 /* compute a_1 + a_0 */
327 scratch
[0] = internal_add(scratch
+1, a
+toplen
, scratch
+1, botlen
);
329 printf("a1plusa0 = 0x");
330 for (i
= 0; i
< midlen
; i
++) {
331 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
335 /* compute b_1 + b_0 */
336 scratch
[midlen
] = internal_add(scratch
+midlen
+1, b
+toplen
,
337 scratch
+midlen
+1, botlen
);
339 printf("b1plusb0 = 0x");
340 for (i
= 0; i
< midlen
; i
++) {
341 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[midlen
+i
]);
347 * Now we can do the third multiplication.
349 internal_mul(scratch
, scratch
+ midlen
, scratch
+ 2*midlen
, midlen
,
352 printf("a1plusa0timesb1plusb0 = 0x");
353 for (i
= 0; i
< 2*midlen
; i
++) {
354 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
360 * Now we can reuse the first half of 'scratch' to compute the
361 * sum of the outer two coefficients, to subtract from that
362 * product to obtain the middle one.
364 scratch
[0] = scratch
[1] = scratch
[2] = scratch
[3] = 0;
365 for (i
= 0; i
< 2*toplen
; i
++)
366 scratch
[2*midlen
- 2*toplen
+ i
] = c
[i
];
367 scratch
[1] = internal_add(scratch
+2, c
+ 2*toplen
,
368 scratch
+2, 2*botlen
);
370 printf("a1b1plusa0b0 = 0x");
371 for (i
= 0; i
< 2*midlen
; i
++) {
372 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
377 internal_sub(scratch
+ 2*midlen
, scratch
,
378 scratch
+ 2*midlen
, 2*midlen
);
380 printf("a1b0plusa0b1 = 0x");
381 for (i
= 0; i
< 2*midlen
; i
++) {
382 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
388 * And now all we need to do is to add that middle coefficient
389 * back into the output. We may have to propagate a carry
390 * further up the output, but we can be sure it won't
391 * propagate right the way off the top.
393 carry
= internal_add(c
+ 2*len
- botlen
- 2*midlen
,
395 c
+ 2*len
- botlen
- 2*midlen
, 2*midlen
);
396 i
= 2*len
- botlen
- 2*midlen
- 1;
400 c
[i
] = (BignumInt
)carry
;
401 carry
>>= BIGNUM_INT_BITS
;
406 for (i
= 0; i
< 2*len
; i
++) {
407 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
416 const BignumInt
*ap
, *bp
;
420 * Multiply in the ordinary O(N^2) way.
423 for (i
= 0; i
< 2 * len
; i
++)
426 for (cps
= c
+ 2*len
, ap
= a
+ len
; ap
-- > a
; cps
--) {
428 for (cp
= cps
, bp
= b
+ len
; cp
--, bp
-- > b
;) {
429 t
= (MUL_WORD(*ap
, *bp
) + carry
) + *cp
;
431 carry
= (BignumInt
)(t
>> BIGNUM_INT_BITS
);
439 * Variant form of internal_mul used for the initial step of
440 * Montgomery reduction. Only bothers outputting 'len' words
441 * (everything above that is thrown away).
443 static void internal_mul_low(const BignumInt
*a
, const BignumInt
*b
,
444 BignumInt
*c
, int len
, BignumInt
*scratch
)
446 if (len
> KARATSUBA_THRESHOLD
) {
450 * Karatsuba-aware version of internal_mul_low. As before, we
451 * express each input value as a shifted combination of two
457 * Then the full product is, as before,
459 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
461 * Provided we choose D on the large side (so that a_0 and b_0
462 * are _at least_ as long as a_1 and b_1), we don't need the
463 * topmost term at all, and we only need half of the middle
464 * term. So there's no point in doing the proper Karatsuba
465 * optimisation which computes the middle term using the top
466 * one, because we'd take as long computing the top one as
467 * just computing the middle one directly.
469 * So instead, we do a much more obvious thing: we call the
470 * fully optimised internal_mul to compute a_0 b_0, and we
471 * recursively call ourself to compute the _bottom halves_ of
472 * a_1 b_0 and a_0 b_1, each of which we add into the result
473 * in the obvious way.
475 * In other words, there's no actual Karatsuba _optimisation_
476 * in this function; the only benefit in doing it this way is
477 * that we call internal_mul proper for a large part of the
478 * work, and _that_ can optimise its operation.
481 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
484 * Scratch space for the various bits and pieces we're going
485 * to be adding together: we need botlen*2 words for a_0 b_0
486 * (though we may end up throwing away its topmost word), and
487 * toplen words for each of a_1 b_0 and a_0 b_1. That adds up
492 internal_mul(a
+ toplen
, b
+ toplen
, scratch
+ 2*toplen
, botlen
,
496 internal_mul_low(a
, b
+ len
- toplen
, scratch
+ toplen
, toplen
,
500 internal_mul_low(a
+ len
- toplen
, b
, scratch
, toplen
,
503 /* Copy the bottom half of the big coefficient into place */
504 for (i
= 0; i
< botlen
; i
++)
505 c
[toplen
+ i
] = scratch
[2*toplen
+ botlen
+ i
];
507 /* Add the two small coefficients, throwing away the returned carry */
508 internal_add(scratch
, scratch
+ toplen
, scratch
, toplen
);
510 /* And add that to the large coefficient, leaving the result in c. */
511 internal_add(scratch
, scratch
+ 2*toplen
+ botlen
- toplen
,
518 const BignumInt
*ap
, *bp
;
522 * Multiply in the ordinary O(N^2) way.
525 for (i
= 0; i
< len
; i
++)
528 for (cps
= c
+ len
, ap
= a
+ len
; ap
-- > a
; cps
--) {
530 for (cp
= cps
, bp
= b
+ len
; bp
--, cp
-- > c
;) {
531 t
= (MUL_WORD(*ap
, *bp
) + carry
) + *cp
;
533 carry
= (BignumInt
)(t
>> BIGNUM_INT_BITS
);
540 * Montgomery reduction. Expects x to be a big-endian array of 2*len
541 * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
542 * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
543 * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
546 * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
547 * each, containing respectively n and the multiplicative inverse of
550 * 'tmp' is an array of BignumInt used as scratch space, of length at
551 * least 3*len + mul_compute_scratch(len).
553 static void monty_reduce(BignumInt
*x
, const BignumInt
*n
,
554 const BignumInt
*mninv
, BignumInt
*tmp
, int len
)
560 * Multiply x by (-n)^{-1} mod r. This gives us a value m such
561 * that mn is congruent to -x mod r. Hence, mn+x is an exact
562 * multiple of r, and is also (obviously) congruent to x mod n.
564 internal_mul_low(x
+ len
, mninv
, tmp
, len
, tmp
+ 3*len
);
567 * Compute t = (mn+x)/r in ordinary, non-modular, integer
568 * arithmetic. By construction this is exact, and is congruent mod
569 * n to x * r^{-1}, i.e. the answer we want.
571 * The following multiply leaves that answer in the _most_
572 * significant half of the 'x' array, so then we must shift it
575 internal_mul(tmp
, n
, tmp
+len
, len
, tmp
+ 3*len
);
576 carry
= internal_add(x
, tmp
+len
, x
, 2*len
);
577 for (i
= 0; i
< len
; i
++)
578 x
[len
+ i
] = x
[i
], x
[i
] = 0;
581 * Reduce t mod n. This doesn't require a full-on division by n,
582 * but merely a test and single optional subtraction, since we can
583 * show that 0 <= t < 2n.
586 * + we computed m mod r, so 0 <= m < r.
587 * + so 0 <= mn < rn, obviously
588 * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
589 * + yielding 0 <= (mn+x)/r < 2n as required.
592 for (i
= 0; i
< len
; i
++)
593 if (x
[len
+ i
] != n
[i
])
596 if (carry
|| i
>= len
|| x
[len
+ i
] > n
[i
])
597 internal_sub(x
+len
, n
, x
+len
, len
);
600 static void internal_add_shifted(BignumInt
*number
,
601 unsigned n
, int shift
)
603 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
604 int bshift
= shift
% BIGNUM_INT_BITS
;
607 addend
= (BignumDblInt
)n
<< bshift
;
610 assert(word
<= number
[0]);
611 addend
+= number
[word
];
612 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
613 addend
>>= BIGNUM_INT_BITS
;
620 * Input in first alen words of a and first mlen words of m.
621 * Output in first alen words of a
622 * (of which first alen-mlen words will be zero).
623 * The MSW of m MUST have its high bit set.
624 * Quotient is accumulated in the `quotient' array, which is a Bignum
625 * rather than the internal bigendian format. Quotient parts are shifted
626 * left by `qshift' before adding into quot.
628 static void internal_mod(BignumInt
*a
, int alen
,
629 BignumInt
*m
, int mlen
,
630 BignumInt
*quot
, int qshift
)
637 assert(m0
>> (BIGNUM_INT_BITS
-1) == 1);
643 for (i
= 0; i
<= alen
- mlen
; i
++) {
645 unsigned int q
, r
, c
, ai1
;
659 /* Find q = h:a[i] / m0 */
664 * To illustrate it, suppose a BignumInt is 8 bits, and
665 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
666 * our initial division will be 0xA123 / 0xA1, which
667 * will give a quotient of 0x100 and a divide overflow.
668 * However, the invariants in this division algorithm
669 * are not violated, since the full number A1:23:... is
670 * _less_ than the quotient prefix A1:B2:... and so the
671 * following correction loop would have sorted it out.
673 * In this situation we set q to be the largest
674 * quotient we _can_ stomach (0xFF, of course).
678 /* Macro doesn't want an array subscript expression passed
679 * into it (see definition), so use a temporary. */
680 BignumInt tmplo
= a
[i
];
681 DIVMOD_WORD(q
, r
, h
, tmplo
, m0
);
683 /* Refine our estimate of q by looking at
684 h:a[i]:a[i+1] / m0:m1 */
686 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
689 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
690 if (r
>= (BignumDblInt
) m0
&&
691 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
695 /* Subtract q * m from a[i...] */
697 for (k
= mlen
- 1; k
>= 0; k
--) {
698 t
= MUL_WORD(q
, m
[k
]);
700 c
= (unsigned)(t
>> BIGNUM_INT_BITS
);
701 if ((BignumInt
) t
> a
[i
+ k
])
703 a
[i
+ k
] -= (BignumInt
) t
;
706 /* Add back m in case of borrow */
709 for (k
= mlen
- 1; k
>= 0; k
--) {
712 a
[i
+ k
] = (BignumInt
) t
;
713 t
= t
>> BIGNUM_INT_BITS
;
718 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
723 * Compute (base ^ exp) % mod, the pedestrian way.
725 Bignum
modpow_simple(Bignum base_in
, Bignum exp
, Bignum mod
)
727 BignumInt
*a
, *b
, *n
, *m
, *scratch
;
729 int mlen
, scratchlen
, i
, j
;
733 * The most significant word of mod needs to be non-zero. It
734 * should already be, but let's make sure.
736 assert(mod
[mod
[0]] != 0);
739 * Make sure the base is smaller than the modulus, by reducing
740 * it modulo the modulus if not.
742 base
= bigmod(base_in
, mod
);
744 /* Allocate m of size mlen, copy mod to m */
745 /* We use big endian internally */
747 m
= snewn(mlen
, BignumInt
);
748 for (j
= 0; j
< mlen
; j
++)
749 m
[j
] = mod
[mod
[0] - j
];
751 /* Shift m left to make msb bit set */
752 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
753 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
756 for (i
= 0; i
< mlen
- 1; i
++)
757 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
758 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
761 /* Allocate n of size mlen, copy base to n */
762 n
= snewn(mlen
, BignumInt
);
764 for (j
= 0; j
< i
; j
++)
766 for (j
= 0; j
< (int)base
[0]; j
++)
767 n
[i
+ j
] = base
[base
[0] - j
];
769 /* Allocate a and b of size 2*mlen. Set a = 1 */
770 a
= snewn(2 * mlen
, BignumInt
);
771 b
= snewn(2 * mlen
, BignumInt
);
772 for (i
= 0; i
< 2 * mlen
; i
++)
776 /* Scratch space for multiplies */
777 scratchlen
= mul_compute_scratch(mlen
);
778 scratch
= snewn(scratchlen
, BignumInt
);
780 /* Skip leading zero bits of exp. */
782 j
= BIGNUM_INT_BITS
-1;
783 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
787 j
= BIGNUM_INT_BITS
-1;
791 /* Main computation */
792 while (i
< (int)exp
[0]) {
794 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
, scratch
);
795 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
796 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
797 internal_mul(b
+ mlen
, n
, a
, mlen
, scratch
);
798 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
808 j
= BIGNUM_INT_BITS
-1;
811 /* Fixup result in case the modulus was shifted */
813 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
814 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
815 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
816 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
817 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
818 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
821 /* Copy result to buffer */
822 result
= newbn(mod
[0]);
823 for (i
= 0; i
< mlen
; i
++)
824 result
[result
[0] - i
] = a
[i
+ mlen
];
825 while (result
[0] > 1 && result
[result
[0]] == 0)
828 /* Free temporary arrays */
829 smemclr(a
, 2 * mlen
* sizeof(*a
));
831 smemclr(scratch
, scratchlen
* sizeof(*scratch
));
833 smemclr(b
, 2 * mlen
* sizeof(*b
));
835 smemclr(m
, mlen
* sizeof(*m
));
837 smemclr(n
, mlen
* sizeof(*n
));
846 * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
847 * technique where possible, falling back to modpow_simple otherwise.
849 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
851 BignumInt
*a
, *b
, *x
, *n
, *mninv
, *scratch
;
852 int len
, scratchlen
, i
, j
;
853 Bignum base
, base2
, r
, rn
, inv
, result
;
856 * The most significant word of mod needs to be non-zero. It
857 * should already be, but let's make sure.
859 assert(mod
[mod
[0]] != 0);
862 * mod had better be odd, or we can't do Montgomery multiplication
863 * using a power of two at all.
866 return modpow_simple(base_in
, exp
, mod
);
869 * Make sure the base is smaller than the modulus, by reducing
870 * it modulo the modulus if not.
872 base
= bigmod(base_in
, mod
);
875 * Compute the inverse of n mod r, for monty_reduce. (In fact we
876 * want the inverse of _minus_ n mod r, but we'll sort that out
880 r
= bn_power_2(BIGNUM_INT_BITS
* len
);
881 inv
= modinv(mod
, r
);
882 assert(inv
); /* cannot fail, since mod is odd and r is a power of 2 */
885 * Multiply the base by r mod n, to get it into Montgomery
888 base2
= modmul(base
, r
, mod
);
892 rn
= bigmod(r
, mod
); /* r mod n, i.e. Montgomerified 1 */
894 freebn(r
); /* won't need this any more */
897 * Set up internal arrays of the right lengths, in big-endian
898 * format, containing the base, the modulus, and the modulus's
901 n
= snewn(len
, BignumInt
);
902 for (j
= 0; j
< len
; j
++)
903 n
[len
- 1 - j
] = mod
[j
+ 1];
905 mninv
= snewn(len
, BignumInt
);
906 for (j
= 0; j
< len
; j
++)
907 mninv
[len
- 1 - j
] = (j
< (int)inv
[0] ? inv
[j
+ 1] : 0);
908 freebn(inv
); /* we don't need this copy of it any more */
909 /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
910 x
= snewn(len
, BignumInt
);
911 for (j
= 0; j
< len
; j
++)
913 internal_sub(x
, mninv
, mninv
, len
);
915 /* x = snewn(len, BignumInt); */ /* already done above */
916 for (j
= 0; j
< len
; j
++)
917 x
[len
- 1 - j
] = (j
< (int)base
[0] ? base
[j
+ 1] : 0);
918 freebn(base
); /* we don't need this copy of it any more */
920 a
= snewn(2*len
, BignumInt
);
921 b
= snewn(2*len
, BignumInt
);
922 for (j
= 0; j
< len
; j
++)
923 a
[2*len
- 1 - j
] = (j
< (int)rn
[0] ? rn
[j
+ 1] : 0);
926 /* Scratch space for multiplies */
927 scratchlen
= 3*len
+ mul_compute_scratch(len
);
928 scratch
= snewn(scratchlen
, BignumInt
);
930 /* Skip leading zero bits of exp. */
932 j
= BIGNUM_INT_BITS
-1;
933 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
937 j
= BIGNUM_INT_BITS
-1;
941 /* Main computation */
942 while (i
< (int)exp
[0]) {
944 internal_mul(a
+ len
, a
+ len
, b
, len
, scratch
);
945 monty_reduce(b
, n
, mninv
, scratch
, len
);
946 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
947 internal_mul(b
+ len
, x
, a
, len
, scratch
);
948 monty_reduce(a
, n
, mninv
, scratch
, len
);
958 j
= BIGNUM_INT_BITS
-1;
962 * Final monty_reduce to get back from the adjusted Montgomery
965 monty_reduce(a
, n
, mninv
, scratch
, len
);
967 /* Copy result to buffer */
968 result
= newbn(mod
[0]);
969 for (i
= 0; i
< len
; i
++)
970 result
[result
[0] - i
] = a
[i
+ len
];
971 while (result
[0] > 1 && result
[result
[0]] == 0)
974 /* Free temporary arrays */
975 smemclr(scratch
, scratchlen
* sizeof(*scratch
));
977 smemclr(a
, 2 * len
* sizeof(*a
));
979 smemclr(b
, 2 * len
* sizeof(*b
));
981 smemclr(mninv
, len
* sizeof(*mninv
));
983 smemclr(n
, len
* sizeof(*n
));
985 smemclr(x
, len
* sizeof(*x
));
992 * Compute (p * q) % mod.
993 * The most significant word of mod MUST be non-zero.
994 * We assume that the result array is the same size as the mod array.
996 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
998 BignumInt
*a
, *n
, *m
, *o
, *scratch
;
999 int mshift
, scratchlen
;
1000 int pqlen
, mlen
, rlen
, i
, j
;
1004 * The most significant word of mod needs to be non-zero. It
1005 * should already be, but let's make sure.
1007 assert(mod
[mod
[0]] != 0);
1009 /* Allocate m of size mlen, copy mod to m */
1010 /* We use big endian internally */
1012 m
= snewn(mlen
, BignumInt
);
1013 for (j
= 0; j
< mlen
; j
++)
1014 m
[j
] = mod
[mod
[0] - j
];
1016 /* Shift m left to make msb bit set */
1017 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
1018 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
1021 for (i
= 0; i
< mlen
- 1; i
++)
1022 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1023 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
1026 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
1029 * Make sure that we're allowing enough space. The shifting below
1030 * will underflow the vectors we allocate if pqlen is too small.
1032 if (2*pqlen
<= mlen
)
1035 /* Allocate n of size pqlen, copy p to n */
1036 n
= snewn(pqlen
, BignumInt
);
1038 for (j
= 0; j
< i
; j
++)
1040 for (j
= 0; j
< (int)p
[0]; j
++)
1041 n
[i
+ j
] = p
[p
[0] - j
];
1043 /* Allocate o of size pqlen, copy q to o */
1044 o
= snewn(pqlen
, BignumInt
);
1046 for (j
= 0; j
< i
; j
++)
1048 for (j
= 0; j
< (int)q
[0]; j
++)
1049 o
[i
+ j
] = q
[q
[0] - j
];
1051 /* Allocate a of size 2*pqlen for result */
1052 a
= snewn(2 * pqlen
, BignumInt
);
1054 /* Scratch space for multiplies */
1055 scratchlen
= mul_compute_scratch(pqlen
);
1056 scratch
= snewn(scratchlen
, BignumInt
);
1058 /* Main computation */
1059 internal_mul(n
, o
, a
, pqlen
, scratch
);
1060 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
1062 /* Fixup result in case the modulus was shifted */
1064 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
1065 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1066 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
1067 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
1068 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
1069 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
1072 /* Copy result to buffer */
1073 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
1074 result
= newbn(rlen
);
1075 for (i
= 0; i
< rlen
; i
++)
1076 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
1077 while (result
[0] > 1 && result
[result
[0]] == 0)
1080 /* Free temporary arrays */
1081 smemclr(scratch
, scratchlen
* sizeof(*scratch
));
1083 smemclr(a
, 2 * pqlen
* sizeof(*a
));
1085 smemclr(m
, mlen
* sizeof(*m
));
1087 smemclr(n
, pqlen
* sizeof(*n
));
1089 smemclr(o
, pqlen
* sizeof(*o
));
1097 * The most significant word of mod MUST be non-zero.
1098 * We assume that the result array is the same size as the mod array.
1099 * We optionally write out a quotient if `quotient' is non-NULL.
1100 * We can avoid writing out the result if `result' is NULL.
1102 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
1106 int plen
, mlen
, i
, j
;
1109 * The most significant word of mod needs to be non-zero. It
1110 * should already be, but let's make sure.
1112 assert(mod
[mod
[0]] != 0);
1114 /* Allocate m of size mlen, copy mod to m */
1115 /* We use big endian internally */
1117 m
= snewn(mlen
, BignumInt
);
1118 for (j
= 0; j
< mlen
; j
++)
1119 m
[j
] = mod
[mod
[0] - j
];
1121 /* Shift m left to make msb bit set */
1122 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
1123 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
1126 for (i
= 0; i
< mlen
- 1; i
++)
1127 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1128 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
1132 /* Ensure plen > mlen */
1136 /* Allocate n of size plen, copy p to n */
1137 n
= snewn(plen
, BignumInt
);
1138 for (j
= 0; j
< plen
; j
++)
1140 for (j
= 1; j
<= (int)p
[0]; j
++)
1143 /* Main computation */
1144 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
1146 /* Fixup result in case the modulus was shifted */
1148 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
1149 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1150 n
[plen
- 1] = n
[plen
- 1] << mshift
;
1151 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
1152 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
1153 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
1156 /* Copy result to buffer */
1158 for (i
= 1; i
<= (int)result
[0]; i
++) {
1160 result
[i
] = j
>= 0 ? n
[j
] : 0;
1164 /* Free temporary arrays */
1165 smemclr(m
, mlen
* sizeof(*m
));
1167 smemclr(n
, plen
* sizeof(*n
));
1172 * Decrement a number.
1174 void decbn(Bignum bn
)
1177 while (i
< (int)bn
[0] && bn
[i
] == 0)
1178 bn
[i
++] = BIGNUM_INT_MASK
;
1182 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
1187 assert(nbytes
>= 0 && nbytes
< INT_MAX
/8);
1189 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
1192 for (i
= 1; i
<= w
; i
++)
1194 for (i
= nbytes
; i
--;) {
1195 unsigned char byte
= *data
++;
1196 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
1199 while (result
[0] > 1 && result
[result
[0]] == 0)
1205 * Read an SSH-1-format bignum from a data buffer. Return the number
1206 * of bytes consumed, or -1 if there wasn't enough data.
1208 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
1210 const unsigned char *p
= data
;
1218 for (i
= 0; i
< 2; i
++)
1219 w
= (w
<< 8) + *p
++;
1220 b
= (w
+ 7) / 8; /* bits -> bytes */
1225 if (!result
) /* just return length */
1228 *result
= bignum_from_bytes(p
, b
);
1230 return p
+ b
- data
;
1234 * Return the bit count of a bignum, for SSH-1 encoding.
1236 int bignum_bitcount(Bignum bn
)
1238 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
1239 while (bitcount
>= 0
1240 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
1241 return bitcount
+ 1;
1245 * Return the byte length of a bignum when SSH-1 encoded.
1247 int ssh1_bignum_length(Bignum bn
)
1249 return 2 + (bignum_bitcount(bn
) + 7) / 8;
1253 * Return the byte length of a bignum when SSH-2 encoded.
1255 int ssh2_bignum_length(Bignum bn
)
1257 return 4 + (bignum_bitcount(bn
) + 8) / 8;
1261 * Return a byte from a bignum; 0 is least significant, etc.
1263 int bignum_byte(Bignum bn
, int i
)
1265 if (i
< 0 || i
>= (int)(BIGNUM_INT_BYTES
* bn
[0]))
1266 return 0; /* beyond the end */
1268 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
1269 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
1273 * Return a bit from a bignum; 0 is least significant, etc.
1275 int bignum_bit(Bignum bn
, int i
)
1277 if (i
< 0 || i
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1278 return 0; /* beyond the end */
1280 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
1284 * Set a bit in a bignum; 0 is least significant, etc.
1286 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
1288 if (bitnum
< 0 || bitnum
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1289 abort(); /* beyond the end */
1291 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
1292 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
1301 * Write a SSH-1-format bignum into a buffer. It is assumed the
1302 * buffer is big enough. Returns the number of bytes used.
1304 int ssh1_write_bignum(void *data
, Bignum bn
)
1306 unsigned char *p
= data
;
1307 int len
= ssh1_bignum_length(bn
);
1309 int bitc
= bignum_bitcount(bn
);
1311 *p
++ = (bitc
>> 8) & 0xFF;
1312 *p
++ = (bitc
) & 0xFF;
1313 for (i
= len
- 2; i
--;)
1314 *p
++ = bignum_byte(bn
, i
);
1319 * Compare two bignums. Returns like strcmp.
1321 int bignum_cmp(Bignum a
, Bignum b
)
1323 int amax
= a
[0], bmax
= b
[0];
1326 /* Annoyingly we have two representations of zero */
1327 if (amax
== 1 && a
[amax
] == 0)
1329 if (bmax
== 1 && b
[bmax
] == 0)
1332 assert(amax
== 0 || a
[amax
] != 0);
1333 assert(bmax
== 0 || b
[bmax
] != 0);
1335 i
= (amax
> bmax ? amax
: bmax
);
1337 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
1338 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
1349 * Right-shift one bignum to form another.
1351 Bignum
bignum_rshift(Bignum a
, int shift
)
1354 int i
, shiftw
, shiftb
, shiftbb
, bits
;
1359 bits
= bignum_bitcount(a
) - shift
;
1360 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
1363 shiftw
= shift
/ BIGNUM_INT_BITS
;
1364 shiftb
= shift
% BIGNUM_INT_BITS
;
1365 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
1367 ai1
= a
[shiftw
+ 1];
1368 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1370 ai1
= (i
+ shiftw
+ 1 <= (int)a
[0] ? a
[i
+ shiftw
+ 1] : 0);
1371 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
1379 * Non-modular multiplication and addition.
1381 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
1383 int alen
= a
[0], blen
= b
[0];
1384 int mlen
= (alen
> blen ? alen
: blen
);
1385 int rlen
, i
, maxspot
;
1387 BignumInt
*workspace
;
1390 /* mlen space for a, mlen space for b, 2*mlen for result,
1391 * plus scratch space for multiplication */
1392 wslen
= mlen
* 4 + mul_compute_scratch(mlen
);
1393 workspace
= snewn(wslen
, BignumInt
);
1394 for (i
= 0; i
< mlen
; i
++) {
1395 workspace
[0 * mlen
+ i
] = (mlen
- i
<= (int)a
[0] ? a
[mlen
- i
] : 0);
1396 workspace
[1 * mlen
+ i
] = (mlen
- i
<= (int)b
[0] ? b
[mlen
- i
] : 0);
1399 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
1400 workspace
+ 2 * mlen
, mlen
, workspace
+ 4 * mlen
);
1402 /* now just copy the result back */
1403 rlen
= alen
+ blen
+ 1;
1404 if (addend
&& rlen
<= (int)addend
[0])
1405 rlen
= addend
[0] + 1;
1408 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1409 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
1415 /* now add in the addend, if any */
1417 BignumDblInt carry
= 0;
1418 for (i
= 1; i
<= rlen
; i
++) {
1419 carry
+= (i
<= (int)ret
[0] ? ret
[i
] : 0);
1420 carry
+= (i
<= (int)addend
[0] ? addend
[i
] : 0);
1421 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1422 carry
>>= BIGNUM_INT_BITS
;
1423 if (ret
[i
] != 0 && i
> maxspot
)
1429 smemclr(workspace
, wslen
* sizeof(*workspace
));
1435 * Non-modular multiplication.
1437 Bignum
bigmul(Bignum a
, Bignum b
)
1439 return bigmuladd(a
, b
, NULL
);
1445 Bignum
bigadd(Bignum a
, Bignum b
)
1447 int alen
= a
[0], blen
= b
[0];
1448 int rlen
= (alen
> blen ? alen
: blen
) + 1;
1457 for (i
= 1; i
<= rlen
; i
++) {
1458 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1459 carry
+= (i
<= (int)b
[0] ? b
[i
] : 0);
1460 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1461 carry
>>= BIGNUM_INT_BITS
;
1462 if (ret
[i
] != 0 && i
> maxspot
)
1471 * Subtraction. Returns a-b, or NULL if the result would come out
1472 * negative (recall that this entire bignum module only handles
1473 * positive numbers).
1475 Bignum
bigsub(Bignum a
, Bignum b
)
1477 int alen
= a
[0], blen
= b
[0];
1478 int rlen
= (alen
> blen ? alen
: blen
);
1487 for (i
= 1; i
<= rlen
; i
++) {
1488 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1489 carry
+= (i
<= (int)b
[0] ? b
[i
] ^ BIGNUM_INT_MASK
: BIGNUM_INT_MASK
);
1490 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1491 carry
>>= BIGNUM_INT_BITS
;
1492 if (ret
[i
] != 0 && i
> maxspot
)
1506 * Create a bignum which is the bitmask covering another one. That
1507 * is, the smallest integer which is >= N and is also one less than
1510 Bignum
bignum_bitmask(Bignum n
)
1512 Bignum ret
= copybn(n
);
1517 while (n
[i
] == 0 && i
> 0)
1520 return ret
; /* input was zero */
1526 ret
[i
] = BIGNUM_INT_MASK
;
1531 * Convert a (max 32-bit) long into a bignum.
1533 Bignum
bignum_from_long(unsigned long nn
)
1536 BignumDblInt n
= nn
;
1539 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
1540 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
1542 ret
[0] = (ret
[2] ?
2 : 1);
1547 * Add a long to a bignum.
1549 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
1551 Bignum ret
= newbn(number
[0] + 1);
1553 BignumDblInt carry
= 0, addend
= addendx
;
1555 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1556 carry
+= addend
& BIGNUM_INT_MASK
;
1557 carry
+= (i
<= (int)number
[0] ? number
[i
] : 0);
1558 addend
>>= BIGNUM_INT_BITS
;
1559 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1560 carry
>>= BIGNUM_INT_BITS
;
1569 * Compute the residue of a bignum, modulo a (max 16-bit) short.
1571 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
1573 BignumDblInt mod
, r
;
1578 for (i
= number
[0]; i
> 0; i
--)
1579 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
1580 return (unsigned short) r
;
1584 void diagbn(char *prefix
, Bignum md
)
1586 int i
, nibbles
, morenibbles
;
1587 static const char hex
[] = "0123456789ABCDEF";
1589 debug(("%s0x", prefix ? prefix
: ""));
1591 nibbles
= (3 + bignum_bitcount(md
)) / 4;
1594 morenibbles
= 4 * md
[0] - nibbles
;
1595 for (i
= 0; i
< morenibbles
; i
++)
1597 for (i
= nibbles
; i
--;)
1599 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
1609 Bignum
bigdiv(Bignum a
, Bignum b
)
1611 Bignum q
= newbn(a
[0]);
1612 bigdivmod(a
, b
, NULL
, q
);
1619 Bignum
bigmod(Bignum a
, Bignum b
)
1621 Bignum r
= newbn(b
[0]);
1622 bigdivmod(a
, b
, r
, NULL
);
1627 * Greatest common divisor.
1629 Bignum
biggcd(Bignum av
, Bignum bv
)
1631 Bignum a
= copybn(av
);
1632 Bignum b
= copybn(bv
);
1634 while (bignum_cmp(b
, Zero
) != 0) {
1635 Bignum t
= newbn(b
[0]);
1636 bigdivmod(a
, b
, t
, NULL
);
1637 while (t
[0] > 1 && t
[t
[0]] == 0)
1649 * Modular inverse, using Euclid's extended algorithm.
1651 Bignum
modinv(Bignum number
, Bignum modulus
)
1653 Bignum a
= copybn(modulus
);
1654 Bignum b
= copybn(number
);
1655 Bignum xp
= copybn(Zero
);
1656 Bignum x
= copybn(One
);
1659 assert(number
[number
[0]] != 0);
1660 assert(modulus
[modulus
[0]] != 0);
1662 while (bignum_cmp(b
, One
) != 0) {
1665 if (bignum_cmp(b
, Zero
) == 0) {
1667 * Found a common factor between the inputs, so we cannot
1668 * return a modular inverse at all.
1679 bigdivmod(a
, b
, t
, q
);
1680 while (t
[0] > 1 && t
[t
[0]] == 0)
1687 x
= bigmuladd(q
, xp
, t
);
1697 /* now we know that sign * x == 1, and that x < modulus */
1699 /* set a new x to be modulus - x */
1700 Bignum newx
= newbn(modulus
[0]);
1701 BignumInt carry
= 0;
1705 for (i
= 1; i
<= (int)newx
[0]; i
++) {
1706 BignumInt aword
= (i
<= (int)modulus
[0] ? modulus
[i
] : 0);
1707 BignumInt bword
= (i
<= (int)x
[0] ? x
[i
] : 0);
1708 newx
[i
] = aword
- bword
- carry
;
1710 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1724 * Render a bignum into decimal. Return a malloced string holding
1725 * the decimal representation.
1727 char *bignum_decimal(Bignum x
)
1729 int ndigits
, ndigit
;
1733 BignumInt
*workspace
;
1736 * First, estimate the number of digits. Since log(10)/log(2)
1737 * is just greater than 93/28 (the joys of continued fraction
1738 * approximations...) we know that for every 93 bits, we need
1739 * at most 28 digits. This will tell us how much to malloc.
1741 * Formally: if x has i bits, that means x is strictly less
1742 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1743 * 10^(28i/93). We need an integer power of ten, so we must
1744 * round up (rounding down might make it less than x again).
1745 * Therefore if we multiply the bit count by 28/93, rounding
1746 * up, we will have enough digits.
1748 * i=0 (i.e., x=0) is an irritating special case.
1750 i
= bignum_bitcount(x
);
1752 ndigits
= 1; /* x = 0 */
1754 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1755 ndigits
++; /* allow for trailing \0 */
1756 ret
= snewn(ndigits
, char);
1759 * Now allocate some workspace to hold the binary form as we
1760 * repeatedly divide it by ten. Initialise this to the
1761 * big-endian form of the number.
1763 workspace
= snewn(x
[0], BignumInt
);
1764 for (i
= 0; i
< (int)x
[0]; i
++)
1765 workspace
[i
] = x
[x
[0] - i
];
1768 * Next, write the decimal number starting with the last digit.
1769 * We use ordinary short division, dividing 10 into the
1772 ndigit
= ndigits
- 1;
1777 for (i
= 0; i
< (int)x
[0]; i
++) {
1778 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1779 workspace
[i
] = (BignumInt
) (carry
/ 10);
1784 ret
[--ndigit
] = (char) (carry
+ '0');
1788 * There's a chance we've fallen short of the start of the
1789 * string. Correct if so.
1792 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);
1797 smemclr(workspace
, x
[0] * sizeof(*workspace
));
1809 * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset
1811 * Then feed to this program's standard input the output of
1812 * testdata/bignum.py .
1815 void modalfatalbox(char *p
, ...)
1818 fprintf(stderr
, "FATAL ERROR: ");
1820 vfprintf(stderr
, p
, ap
);
1822 fputc('\n', stderr
);
1826 #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
1828 int main(int argc
, char **argv
)
1832 int passes
= 0, fails
= 0;
1834 while ((buf
= fgetline(stdin
)) != NULL
) {
1835 int maxlen
= strlen(buf
);
1836 unsigned char *data
= snewn(maxlen
, unsigned char);
1837 unsigned char *ptrs
[5], *q
;
1846 while (*bufp
&& !isspace((unsigned char)*bufp
))
1855 while (*bufp
&& !isxdigit((unsigned char)*bufp
))
1862 while (*bufp
&& isxdigit((unsigned char)*bufp
))
1866 if (ptrnum
>= lenof(ptrs
))
1870 for (i
= -((end
- start
) & 1); i
< end
-start
; i
+= 2) {
1871 unsigned char val
= (i
< 0 ?
0 : fromxdigit(start
[i
]));
1872 val
= val
* 16 + fromxdigit(start
[i
+1]);
1879 if (!strcmp(buf
, "mul")) {
1883 printf("%d: mul with %d parameters, expected 3\n", line
, ptrnum
);
1886 a
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1887 b
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1888 c
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1891 if (bignum_cmp(c
, p
) == 0) {
1894 char *as
= bignum_decimal(a
);
1895 char *bs
= bignum_decimal(b
);
1896 char *cs
= bignum_decimal(c
);
1897 char *ps
= bignum_decimal(p
);
1899 printf("%d: fail: %s * %s gave %s expected %s\n",
1900 line
, as
, bs
, ps
, cs
);
1912 } else if (!strcmp(buf
, "modmul")) {
1913 Bignum a
, b
, m
, c
, p
;
1916 printf("%d: modmul with %d parameters, expected 4\n",
1920 a
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1921 b
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1922 m
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1923 c
= bignum_from_bytes(ptrs
[3], ptrs
[4]-ptrs
[3]);
1924 p
= modmul(a
, b
, m
);
1926 if (bignum_cmp(c
, p
) == 0) {
1929 char *as
= bignum_decimal(a
);
1930 char *bs
= bignum_decimal(b
);
1931 char *ms
= bignum_decimal(m
);
1932 char *cs
= bignum_decimal(c
);
1933 char *ps
= bignum_decimal(p
);
1935 printf("%d: fail: %s * %s mod %s gave %s expected %s\n",
1936 line
, as
, bs
, ms
, ps
, cs
);
1950 } else if (!strcmp(buf
, "pow")) {
1951 Bignum base
, expt
, modulus
, expected
, answer
;
1954 printf("%d: mul with %d parameters, expected 4\n", line
, ptrnum
);
1958 base
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1959 expt
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1960 modulus
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1961 expected
= bignum_from_bytes(ptrs
[3], ptrs
[4]-ptrs
[3]);
1962 answer
= modpow(base
, expt
, modulus
);
1964 if (bignum_cmp(expected
, answer
) == 0) {
1967 char *as
= bignum_decimal(base
);
1968 char *bs
= bignum_decimal(expt
);
1969 char *cs
= bignum_decimal(modulus
);
1970 char *ds
= bignum_decimal(answer
);
1971 char *ps
= bignum_decimal(expected
);
1973 printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",
1974 line
, as
, bs
, cs
, ds
, ps
);
1989 printf("%d: unrecognised test keyword: '%s'\n", line
, buf
);
1997 printf("passed %d failed %d total %d\n", passes
, fails
, passes
+fails
);