2 * Bignum routines for RSA and DH and stuff.
12 #if defined __GNUC__ && defined __i386__
13 typedef unsigned long BignumInt
;
14 typedef unsigned long long BignumDblInt
;
15 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
16 #define BIGNUM_TOP_BIT 0x80000000UL
17 #define BIGNUM_INT_BITS 32
18 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
19 #define DIVMOD_WORD(q, r, hi, lo, w) \
21 "=d" (r), "=a" (q) : \
22 "r" (w), "d" (hi), "a" (lo))
23 #elif defined _MSC_VER && defined _M_IX86
24 typedef unsigned __int32 BignumInt
;
25 typedef unsigned __int64 BignumDblInt
;
26 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
27 #define BIGNUM_TOP_BIT 0x80000000UL
28 #define BIGNUM_INT_BITS 32
29 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
31 unsigned __int32 quot
;
32 unsigned __int32 remd
;
34 static __declspec(naked
) msvc_quorem __stdcall
msvc_divmod(
40 mov edx
, dword ptr
[esp
+4]
41 mov eax
, dword ptr
[esp
+8]
42 div dword ptr
[esp
+12]
46 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
47 const msvc_quorem qr = msvc_divmod((hi), (lo), (w)); \
48 (q) = qr.quot; (r) = qr.remd; \
51 typedef unsigned short BignumInt
;
52 typedef unsigned long BignumDblInt
;
53 #define BIGNUM_INT_MASK 0xFFFFU
54 #define BIGNUM_TOP_BIT 0x8000U
55 #define BIGNUM_INT_BITS 16
56 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
57 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
58 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
64 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
66 #define BIGNUM_INTERNAL
67 typedef BignumInt
*Bignum
;
71 BignumInt bnZero
[1] = { 0 };
72 BignumInt bnOne
[2] = { 1, 1 };
75 * The Bignum format is an array of `BignumInt'. The first
76 * element of the array counts the remaining elements. The
77 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
78 * significant digit first. (So it's trivial to extract the bit
79 * with value 2^n for any n.)
81 * All Bignums in this module are positive. Negative numbers must
82 * be dealt with outside it.
84 * INVARIANT: the most significant word of any Bignum must be
88 Bignum Zero
= bnZero
, One
= bnOne
;
90 static Bignum
newbn(int length
)
92 Bignum b
= snewn(length
+ 1, BignumInt
);
95 memset(b
, 0, (length
+ 1) * sizeof(*b
));
100 void bn_restore_invariant(Bignum b
)
102 while (b
[0] > 1 && b
[b
[0]] == 0)
106 Bignum
copybn(Bignum orig
)
108 Bignum b
= snewn(orig
[0] + 1, BignumInt
);
111 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
115 void freebn(Bignum b
)
118 * Burn the evidence, just in case.
120 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
124 Bignum
bn_power_2(int n
)
126 Bignum ret
= newbn(n
/ BIGNUM_INT_BITS
+ 1);
127 bignum_set_bit(ret
, n
, 1);
133 * Input is in the first len words of a and b.
134 * Result is returned in the first 2*len words of c.
136 static void internal_mul(BignumInt
*a
, BignumInt
*b
,
137 BignumInt
*c
, int len
)
142 for (j
= 0; j
< 2 * len
; j
++)
145 for (i
= len
- 1; i
>= 0; i
--) {
147 for (j
= len
- 1; j
>= 0; j
--) {
148 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
149 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
150 c
[i
+ j
+ 1] = (BignumInt
) t
;
151 t
= t
>> BIGNUM_INT_BITS
;
153 c
[i
] = (BignumInt
) t
;
157 static void internal_add_shifted(BignumInt
*number
,
158 unsigned n
, int shift
)
160 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
161 int bshift
= shift
% BIGNUM_INT_BITS
;
164 addend
= (BignumDblInt
)n
<< bshift
;
167 addend
+= number
[word
];
168 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
169 addend
>>= BIGNUM_INT_BITS
;
176 * Input in first alen words of a and first mlen words of m.
177 * Output in first alen words of a
178 * (of which first alen-mlen words will be zero).
179 * The MSW of m MUST have its high bit set.
180 * Quotient is accumulated in the `quotient' array, which is a Bignum
181 * rather than the internal bigendian format. Quotient parts are shifted
182 * left by `qshift' before adding into quot.
184 static void internal_mod(BignumInt
*a
, int alen
,
185 BignumInt
*m
, int mlen
,
186 BignumInt
*quot
, int qshift
)
198 for (i
= 0; i
<= alen
- mlen
; i
++) {
200 unsigned int q
, r
, c
, ai1
;
214 /* Find q = h:a[i] / m0 */
219 * To illustrate it, suppose a BignumInt is 8 bits, and
220 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
221 * our initial division will be 0xA123 / 0xA1, which
222 * will give a quotient of 0x100 and a divide overflow.
223 * However, the invariants in this division algorithm
224 * are not violated, since the full number A1:23:... is
225 * _less_ than the quotient prefix A1:B2:... and so the
226 * following correction loop would have sorted it out.
228 * In this situation we set q to be the largest
229 * quotient we _can_ stomach (0xFF, of course).
233 DIVMOD_WORD(q
, r
, h
, a
[i
], m0
);
235 /* Refine our estimate of q by looking at
236 h:a[i]:a[i+1] / m0:m1 */
238 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
241 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
242 if (r
>= (BignumDblInt
) m0
&&
243 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
247 /* Subtract q * m from a[i...] */
249 for (k
= mlen
- 1; k
>= 0; k
--) {
250 t
= MUL_WORD(q
, m
[k
]);
252 c
= t
>> BIGNUM_INT_BITS
;
253 if ((BignumInt
) t
> a
[i
+ k
])
255 a
[i
+ k
] -= (BignumInt
) t
;
258 /* Add back m in case of borrow */
261 for (k
= mlen
- 1; k
>= 0; k
--) {
264 a
[i
+ k
] = (BignumInt
) t
;
265 t
= t
>> BIGNUM_INT_BITS
;
270 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
275 * Compute (base ^ exp) % mod.
277 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
279 BignumInt
*a
, *b
, *n
, *m
;
285 * The most significant word of mod needs to be non-zero. It
286 * should already be, but let's make sure.
288 assert(mod
[mod
[0]] != 0);
291 * Make sure the base is smaller than the modulus, by reducing
292 * it modulo the modulus if not.
294 base
= bigmod(base_in
, mod
);
296 /* Allocate m of size mlen, copy mod to m */
297 /* We use big endian internally */
299 m
= snewn(mlen
, BignumInt
);
300 for (j
= 0; j
< mlen
; j
++)
301 m
[j
] = mod
[mod
[0] - j
];
303 /* Shift m left to make msb bit set */
304 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
305 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
308 for (i
= 0; i
< mlen
- 1; i
++)
309 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
310 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
313 /* Allocate n of size mlen, copy base to n */
314 n
= snewn(mlen
, BignumInt
);
316 for (j
= 0; j
< i
; j
++)
318 for (j
= 0; j
< base
[0]; j
++)
319 n
[i
+ j
] = base
[base
[0] - j
];
321 /* Allocate a and b of size 2*mlen. Set a = 1 */
322 a
= snewn(2 * mlen
, BignumInt
);
323 b
= snewn(2 * mlen
, BignumInt
);
324 for (i
= 0; i
< 2 * mlen
; i
++)
328 /* Skip leading zero bits of exp. */
330 j
= BIGNUM_INT_BITS
-1;
331 while (i
< exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
335 j
= BIGNUM_INT_BITS
-1;
339 /* Main computation */
342 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
343 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
344 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
345 internal_mul(b
+ mlen
, n
, a
, mlen
);
346 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
356 j
= BIGNUM_INT_BITS
-1;
359 /* Fixup result in case the modulus was shifted */
361 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
362 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
363 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
364 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
365 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
366 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
369 /* Copy result to buffer */
370 result
= newbn(mod
[0]);
371 for (i
= 0; i
< mlen
; i
++)
372 result
[result
[0] - i
] = a
[i
+ mlen
];
373 while (result
[0] > 1 && result
[result
[0]] == 0)
376 /* Free temporary arrays */
377 for (i
= 0; i
< 2 * mlen
; i
++)
380 for (i
= 0; i
< 2 * mlen
; i
++)
383 for (i
= 0; i
< mlen
; i
++)
386 for (i
= 0; i
< mlen
; i
++)
396 * Compute (p * q) % mod.
397 * The most significant word of mod MUST be non-zero.
398 * We assume that the result array is the same size as the mod array.
400 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
402 BignumInt
*a
, *n
, *m
, *o
;
404 int pqlen
, mlen
, rlen
, i
, j
;
407 /* Allocate m of size mlen, copy mod to m */
408 /* We use big endian internally */
410 m
= snewn(mlen
, BignumInt
);
411 for (j
= 0; j
< mlen
; j
++)
412 m
[j
] = mod
[mod
[0] - j
];
414 /* Shift m left to make msb bit set */
415 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
416 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
419 for (i
= 0; i
< mlen
- 1; i
++)
420 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
421 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
424 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
426 /* Allocate n of size pqlen, copy p to n */
427 n
= snewn(pqlen
, BignumInt
);
429 for (j
= 0; j
< i
; j
++)
431 for (j
= 0; j
< p
[0]; j
++)
432 n
[i
+ j
] = p
[p
[0] - j
];
434 /* Allocate o of size pqlen, copy q to o */
435 o
= snewn(pqlen
, BignumInt
);
437 for (j
= 0; j
< i
; j
++)
439 for (j
= 0; j
< q
[0]; j
++)
440 o
[i
+ j
] = q
[q
[0] - j
];
442 /* Allocate a of size 2*pqlen for result */
443 a
= snewn(2 * pqlen
, BignumInt
);
445 /* Main computation */
446 internal_mul(n
, o
, a
, pqlen
);
447 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
449 /* Fixup result in case the modulus was shifted */
451 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
452 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
453 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
454 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
455 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
456 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
459 /* Copy result to buffer */
460 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
461 result
= newbn(rlen
);
462 for (i
= 0; i
< rlen
; i
++)
463 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
464 while (result
[0] > 1 && result
[result
[0]] == 0)
467 /* Free temporary arrays */
468 for (i
= 0; i
< 2 * pqlen
; i
++)
471 for (i
= 0; i
< mlen
; i
++)
474 for (i
= 0; i
< pqlen
; i
++)
477 for (i
= 0; i
< pqlen
; i
++)
486 * The most significant word of mod MUST be non-zero.
487 * We assume that the result array is the same size as the mod array.
488 * We optionally write out a quotient if `quotient' is non-NULL.
489 * We can avoid writing out the result if `result' is NULL.
491 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
495 int plen
, mlen
, i
, j
;
497 /* Allocate m of size mlen, copy mod to m */
498 /* We use big endian internally */
500 m
= snewn(mlen
, BignumInt
);
501 for (j
= 0; j
< mlen
; j
++)
502 m
[j
] = mod
[mod
[0] - j
];
504 /* Shift m left to make msb bit set */
505 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
506 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
509 for (i
= 0; i
< mlen
- 1; i
++)
510 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
511 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
515 /* Ensure plen > mlen */
519 /* Allocate n of size plen, copy p to n */
520 n
= snewn(plen
, BignumInt
);
521 for (j
= 0; j
< plen
; j
++)
523 for (j
= 1; j
<= p
[0]; j
++)
526 /* Main computation */
527 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
529 /* Fixup result in case the modulus was shifted */
531 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
532 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
533 n
[plen
- 1] = n
[plen
- 1] << mshift
;
534 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
535 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
536 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
539 /* Copy result to buffer */
541 for (i
= 1; i
<= result
[0]; i
++) {
543 result
[i
] = j
>= 0 ? n
[j
] : 0;
547 /* Free temporary arrays */
548 for (i
= 0; i
< mlen
; i
++)
551 for (i
= 0; i
< plen
; i
++)
557 * Decrement a number.
559 void decbn(Bignum bn
)
562 while (i
< bn
[0] && bn
[i
] == 0)
563 bn
[i
++] = BIGNUM_INT_MASK
;
567 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
572 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
575 for (i
= 1; i
<= w
; i
++)
577 for (i
= nbytes
; i
--;) {
578 unsigned char byte
= *data
++;
579 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
582 while (result
[0] > 1 && result
[result
[0]] == 0)
588 * Read an SSH-1-format bignum from a data buffer. Return the number
589 * of bytes consumed, or -1 if there wasn't enough data.
591 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
593 const unsigned char *p
= data
;
601 for (i
= 0; i
< 2; i
++)
603 b
= (w
+ 7) / 8; /* bits -> bytes */
608 if (!result
) /* just return length */
611 *result
= bignum_from_bytes(p
, b
);
617 * Return the bit count of a bignum, for SSH-1 encoding.
619 int bignum_bitcount(Bignum bn
)
621 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
623 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
628 * Return the byte length of a bignum when SSH-1 encoded.
630 int ssh1_bignum_length(Bignum bn
)
632 return 2 + (bignum_bitcount(bn
) + 7) / 8;
636 * Return the byte length of a bignum when SSH-2 encoded.
638 int ssh2_bignum_length(Bignum bn
)
640 return 4 + (bignum_bitcount(bn
) + 8) / 8;
644 * Return a byte from a bignum; 0 is least significant, etc.
646 int bignum_byte(Bignum bn
, int i
)
648 if (i
>= BIGNUM_INT_BYTES
* bn
[0])
649 return 0; /* beyond the end */
651 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
652 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
656 * Return a bit from a bignum; 0 is least significant, etc.
658 int bignum_bit(Bignum bn
, int i
)
660 if (i
>= BIGNUM_INT_BITS
* bn
[0])
661 return 0; /* beyond the end */
663 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
667 * Set a bit in a bignum; 0 is least significant, etc.
669 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
671 if (bitnum
>= BIGNUM_INT_BITS
* bn
[0])
672 abort(); /* beyond the end */
674 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
675 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
684 * Write a SSH-1-format bignum into a buffer. It is assumed the
685 * buffer is big enough. Returns the number of bytes used.
687 int ssh1_write_bignum(void *data
, Bignum bn
)
689 unsigned char *p
= data
;
690 int len
= ssh1_bignum_length(bn
);
692 int bitc
= bignum_bitcount(bn
);
694 *p
++ = (bitc
>> 8) & 0xFF;
695 *p
++ = (bitc
) & 0xFF;
696 for (i
= len
- 2; i
--;)
697 *p
++ = bignum_byte(bn
, i
);
702 * Compare two bignums. Returns like strcmp.
704 int bignum_cmp(Bignum a
, Bignum b
)
706 int amax
= a
[0], bmax
= b
[0];
707 int i
= (amax
> bmax ? amax
: bmax
);
709 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
710 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
721 * Right-shift one bignum to form another.
723 Bignum
bignum_rshift(Bignum a
, int shift
)
726 int i
, shiftw
, shiftb
, shiftbb
, bits
;
729 bits
= bignum_bitcount(a
) - shift
;
730 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
733 shiftw
= shift
/ BIGNUM_INT_BITS
;
734 shiftb
= shift
% BIGNUM_INT_BITS
;
735 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
738 for (i
= 1; i
<= ret
[0]; i
++) {
740 ai1
= (i
+ shiftw
+ 1 <= a
[0] ? a
[i
+ shiftw
+ 1] : 0);
741 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
749 * Non-modular multiplication and addition.
751 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
753 int alen
= a
[0], blen
= b
[0];
754 int mlen
= (alen
> blen ? alen
: blen
);
755 int rlen
, i
, maxspot
;
756 BignumInt
*workspace
;
759 /* mlen space for a, mlen space for b, 2*mlen for result */
760 workspace
= snewn(mlen
* 4, BignumInt
);
761 for (i
= 0; i
< mlen
; i
++) {
762 workspace
[0 * mlen
+ i
] = (mlen
- i
<= a
[0] ? a
[mlen
- i
] : 0);
763 workspace
[1 * mlen
+ i
] = (mlen
- i
<= b
[0] ? b
[mlen
- i
] : 0);
766 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
767 workspace
+ 2 * mlen
, mlen
);
769 /* now just copy the result back */
770 rlen
= alen
+ blen
+ 1;
771 if (addend
&& rlen
<= addend
[0])
772 rlen
= addend
[0] + 1;
775 for (i
= 1; i
<= ret
[0]; i
++) {
776 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
782 /* now add in the addend, if any */
784 BignumDblInt carry
= 0;
785 for (i
= 1; i
<= rlen
; i
++) {
786 carry
+= (i
<= ret
[0] ? ret
[i
] : 0);
787 carry
+= (i
<= addend
[0] ? addend
[i
] : 0);
788 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
789 carry
>>= BIGNUM_INT_BITS
;
790 if (ret
[i
] != 0 && i
> maxspot
)
801 * Non-modular multiplication.
803 Bignum
bigmul(Bignum a
, Bignum b
)
805 return bigmuladd(a
, b
, NULL
);
809 * Create a bignum which is the bitmask covering another one. That
810 * is, the smallest integer which is >= N and is also one less than
813 Bignum
bignum_bitmask(Bignum n
)
815 Bignum ret
= copybn(n
);
820 while (n
[i
] == 0 && i
> 0)
823 return ret
; /* input was zero */
829 ret
[i
] = BIGNUM_INT_MASK
;
834 * Convert a (max 32-bit) long into a bignum.
836 Bignum
bignum_from_long(unsigned long nn
)
842 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
843 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
845 ret
[0] = (ret
[2] ?
2 : 1);
850 * Add a long to a bignum.
852 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
854 Bignum ret
= newbn(number
[0] + 1);
856 BignumDblInt carry
= 0, addend
= addendx
;
858 for (i
= 1; i
<= ret
[0]; i
++) {
859 carry
+= addend
& BIGNUM_INT_MASK
;
860 carry
+= (i
<= number
[0] ? number
[i
] : 0);
861 addend
>>= BIGNUM_INT_BITS
;
862 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
863 carry
>>= BIGNUM_INT_BITS
;
872 * Compute the residue of a bignum, modulo a (max 16-bit) short.
874 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
881 for (i
= number
[0]; i
> 0; i
--)
882 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
883 return (unsigned short) r
;
887 void diagbn(char *prefix
, Bignum md
)
889 int i
, nibbles
, morenibbles
;
890 static const char hex
[] = "0123456789ABCDEF";
892 debug(("%s0x", prefix ? prefix
: ""));
894 nibbles
= (3 + bignum_bitcount(md
)) / 4;
897 morenibbles
= 4 * md
[0] - nibbles
;
898 for (i
= 0; i
< morenibbles
; i
++)
900 for (i
= nibbles
; i
--;)
902 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
912 Bignum
bigdiv(Bignum a
, Bignum b
)
914 Bignum q
= newbn(a
[0]);
915 bigdivmod(a
, b
, NULL
, q
);
922 Bignum
bigmod(Bignum a
, Bignum b
)
924 Bignum r
= newbn(b
[0]);
925 bigdivmod(a
, b
, r
, NULL
);
930 * Greatest common divisor.
932 Bignum
biggcd(Bignum av
, Bignum bv
)
934 Bignum a
= copybn(av
);
935 Bignum b
= copybn(bv
);
937 while (bignum_cmp(b
, Zero
) != 0) {
938 Bignum t
= newbn(b
[0]);
939 bigdivmod(a
, b
, t
, NULL
);
940 while (t
[0] > 1 && t
[t
[0]] == 0)
952 * Modular inverse, using Euclid's extended algorithm.
954 Bignum
modinv(Bignum number
, Bignum modulus
)
956 Bignum a
= copybn(modulus
);
957 Bignum b
= copybn(number
);
958 Bignum xp
= copybn(Zero
);
959 Bignum x
= copybn(One
);
962 while (bignum_cmp(b
, One
) != 0) {
963 Bignum t
= newbn(b
[0]);
964 Bignum q
= newbn(a
[0]);
965 bigdivmod(a
, b
, t
, q
);
966 while (t
[0] > 1 && t
[t
[0]] == 0)
973 x
= bigmuladd(q
, xp
, t
);
983 /* now we know that sign * x == 1, and that x < modulus */
985 /* set a new x to be modulus - x */
986 Bignum newx
= newbn(modulus
[0]);
991 for (i
= 1; i
<= newx
[0]; i
++) {
992 BignumInt aword
= (i
<= modulus
[0] ? modulus
[i
] : 0);
993 BignumInt bword
= (i
<= x
[0] ? x
[i
] : 0);
994 newx
[i
] = aword
- bword
- carry
;
996 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1010 * Render a bignum into decimal. Return a malloced string holding
1011 * the decimal representation.
1013 char *bignum_decimal(Bignum x
)
1015 int ndigits
, ndigit
;
1019 BignumInt
*workspace
;
1022 * First, estimate the number of digits. Since log(10)/log(2)
1023 * is just greater than 93/28 (the joys of continued fraction
1024 * approximations...) we know that for every 93 bits, we need
1025 * at most 28 digits. This will tell us how much to malloc.
1027 * Formally: if x has i bits, that means x is strictly less
1028 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1029 * 10^(28i/93). We need an integer power of ten, so we must
1030 * round up (rounding down might make it less than x again).
1031 * Therefore if we multiply the bit count by 28/93, rounding
1032 * up, we will have enough digits.
1034 i
= bignum_bitcount(x
);
1035 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1036 ndigits
++; /* allow for trailing \0 */
1037 ret
= snewn(ndigits
, char);
1040 * Now allocate some workspace to hold the binary form as we
1041 * repeatedly divide it by ten. Initialise this to the
1042 * big-endian form of the number.
1044 workspace
= snewn(x
[0], BignumInt
);
1045 for (i
= 0; i
< x
[0]; i
++)
1046 workspace
[i
] = x
[x
[0] - i
];
1049 * Next, write the decimal number starting with the last digit.
1050 * We use ordinary short division, dividing 10 into the
1053 ndigit
= ndigits
- 1;
1058 for (i
= 0; i
< x
[0]; i
++) {
1059 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1060 workspace
[i
] = (BignumInt
) (carry
/ 10);
1065 ret
[--ndigit
] = (char) (carry
+ '0');
1069 * There's a chance we've fallen short of the start of the
1070 * string. Correct if so.
1073 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);