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))
24 typedef unsigned short BignumInt
;
25 typedef unsigned long BignumDblInt
;
26 #define BIGNUM_INT_MASK 0xFFFFU
27 #define BIGNUM_TOP_BIT 0x8000U
28 #define BIGNUM_INT_BITS 16
29 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
30 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
31 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
37 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
39 #define BIGNUM_INTERNAL
40 typedef BignumInt
*Bignum
;
44 BignumInt bnZero
[1] = { 0 };
45 BignumInt bnOne
[2] = { 1, 1 };
48 * The Bignum format is an array of `BignumInt'. The first
49 * element of the array counts the remaining elements. The
50 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
51 * significant digit first. (So it's trivial to extract the bit
52 * with value 2^n for any n.)
54 * All Bignums in this module are positive. Negative numbers must
55 * be dealt with outside it.
57 * INVARIANT: the most significant word of any Bignum must be
61 Bignum Zero
= bnZero
, One
= bnOne
;
63 static Bignum
newbn(int length
)
65 Bignum b
= snewn(length
+ 1, BignumInt
);
68 memset(b
, 0, (length
+ 1) * sizeof(*b
));
73 void bn_restore_invariant(Bignum b
)
75 while (b
[0] > 1 && b
[b
[0]] == 0)
79 Bignum
copybn(Bignum orig
)
81 Bignum b
= snewn(orig
[0] + 1, BignumInt
);
84 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
91 * Burn the evidence, just in case.
93 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
97 Bignum
bn_power_2(int n
)
99 Bignum ret
= newbn(n
/ BIGNUM_INT_BITS
+ 1);
100 bignum_set_bit(ret
, n
, 1);
106 * Input is in the first len words of a and b.
107 * Result is returned in the first 2*len words of c.
109 static void internal_mul(BignumInt
*a
, BignumInt
*b
,
110 BignumInt
*c
, int len
)
115 for (j
= 0; j
< 2 * len
; j
++)
118 for (i
= len
- 1; i
>= 0; i
--) {
120 for (j
= len
- 1; j
>= 0; j
--) {
121 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
122 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
123 c
[i
+ j
+ 1] = (BignumInt
) t
;
124 t
= t
>> BIGNUM_INT_BITS
;
126 c
[i
] = (BignumInt
) t
;
130 static void internal_add_shifted(BignumInt
*number
,
131 unsigned n
, int shift
)
133 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
134 int bshift
= shift
% BIGNUM_INT_BITS
;
137 addend
= (BignumDblInt
)n
<< bshift
;
140 addend
+= number
[word
];
141 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
142 addend
>>= BIGNUM_INT_BITS
;
149 * Input in first alen words of a and first mlen words of m.
150 * Output in first alen words of a
151 * (of which first alen-mlen words will be zero).
152 * The MSW of m MUST have its high bit set.
153 * Quotient is accumulated in the `quotient' array, which is a Bignum
154 * rather than the internal bigendian format. Quotient parts are shifted
155 * left by `qshift' before adding into quot.
157 static void internal_mod(BignumInt
*a
, int alen
,
158 BignumInt
*m
, int mlen
,
159 BignumInt
*quot
, int qshift
)
171 for (i
= 0; i
<= alen
- mlen
; i
++) {
173 unsigned int q
, r
, c
, ai1
;
187 /* Find q = h:a[i] / m0 */
192 * To illustrate it, suppose a BignumInt is 8 bits, and
193 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
194 * our initial division will be 0xA123 / 0xA1, which
195 * will give a quotient of 0x100 and a divide overflow.
196 * However, the invariants in this division algorithm
197 * are not violated, since the full number A1:23:... is
198 * _less_ than the quotient prefix A1:B2:... and so the
199 * following correction loop would have sorted it out.
201 * In this situation we set q to be the largest
202 * quotient we _can_ stomach (0xFF, of course).
206 DIVMOD_WORD(q
, r
, h
, a
[i
], m0
);
208 /* Refine our estimate of q by looking at
209 h:a[i]:a[i+1] / m0:m1 */
211 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
214 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
215 if (r
>= (BignumDblInt
) m0
&&
216 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
220 /* Subtract q * m from a[i...] */
222 for (k
= mlen
- 1; k
>= 0; k
--) {
223 t
= MUL_WORD(q
, m
[k
]);
225 c
= t
>> BIGNUM_INT_BITS
;
226 if ((BignumInt
) t
> a
[i
+ k
])
228 a
[i
+ k
] -= (BignumInt
) t
;
231 /* Add back m in case of borrow */
234 for (k
= mlen
- 1; k
>= 0; k
--) {
237 a
[i
+ k
] = (BignumInt
) t
;
238 t
= t
>> BIGNUM_INT_BITS
;
243 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
248 * Compute (base ^ exp) % mod.
250 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
252 BignumInt
*a
, *b
, *n
, *m
;
258 * The most significant word of mod needs to be non-zero. It
259 * should already be, but let's make sure.
261 assert(mod
[mod
[0]] != 0);
264 * Make sure the base is smaller than the modulus, by reducing
265 * it modulo the modulus if not.
267 base
= bigmod(base_in
, mod
);
269 /* Allocate m of size mlen, copy mod to m */
270 /* We use big endian internally */
272 m
= snewn(mlen
, BignumInt
);
273 for (j
= 0; j
< mlen
; j
++)
274 m
[j
] = mod
[mod
[0] - j
];
276 /* Shift m left to make msb bit set */
277 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
278 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
281 for (i
= 0; i
< mlen
- 1; i
++)
282 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
283 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
286 /* Allocate n of size mlen, copy base to n */
287 n
= snewn(mlen
, BignumInt
);
289 for (j
= 0; j
< i
; j
++)
291 for (j
= 0; j
< base
[0]; j
++)
292 n
[i
+ j
] = base
[base
[0] - j
];
294 /* Allocate a and b of size 2*mlen. Set a = 1 */
295 a
= snewn(2 * mlen
, BignumInt
);
296 b
= snewn(2 * mlen
, BignumInt
);
297 for (i
= 0; i
< 2 * mlen
; i
++)
301 /* Skip leading zero bits of exp. */
303 j
= BIGNUM_INT_BITS
-1;
304 while (i
< exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
308 j
= BIGNUM_INT_BITS
-1;
312 /* Main computation */
315 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
316 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
317 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
318 internal_mul(b
+ mlen
, n
, a
, mlen
);
319 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
329 j
= BIGNUM_INT_BITS
-1;
332 /* Fixup result in case the modulus was shifted */
334 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
335 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
336 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
337 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
338 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
339 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
342 /* Copy result to buffer */
343 result
= newbn(mod
[0]);
344 for (i
= 0; i
< mlen
; i
++)
345 result
[result
[0] - i
] = a
[i
+ mlen
];
346 while (result
[0] > 1 && result
[result
[0]] == 0)
349 /* Free temporary arrays */
350 for (i
= 0; i
< 2 * mlen
; i
++)
353 for (i
= 0; i
< 2 * mlen
; i
++)
356 for (i
= 0; i
< mlen
; i
++)
359 for (i
= 0; i
< mlen
; i
++)
369 * Compute (p * q) % mod.
370 * The most significant word of mod MUST be non-zero.
371 * We assume that the result array is the same size as the mod array.
373 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
375 BignumInt
*a
, *n
, *m
, *o
;
377 int pqlen
, mlen
, rlen
, i
, j
;
380 /* Allocate m of size mlen, copy mod to m */
381 /* We use big endian internally */
383 m
= snewn(mlen
, BignumInt
);
384 for (j
= 0; j
< mlen
; j
++)
385 m
[j
] = mod
[mod
[0] - j
];
387 /* Shift m left to make msb bit set */
388 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
389 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
392 for (i
= 0; i
< mlen
- 1; i
++)
393 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
394 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
397 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
399 /* Allocate n of size pqlen, copy p to n */
400 n
= snewn(pqlen
, BignumInt
);
402 for (j
= 0; j
< i
; j
++)
404 for (j
= 0; j
< p
[0]; j
++)
405 n
[i
+ j
] = p
[p
[0] - j
];
407 /* Allocate o of size pqlen, copy q to o */
408 o
= snewn(pqlen
, BignumInt
);
410 for (j
= 0; j
< i
; j
++)
412 for (j
= 0; j
< q
[0]; j
++)
413 o
[i
+ j
] = q
[q
[0] - j
];
415 /* Allocate a of size 2*pqlen for result */
416 a
= snewn(2 * pqlen
, BignumInt
);
418 /* Main computation */
419 internal_mul(n
, o
, a
, pqlen
);
420 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
422 /* Fixup result in case the modulus was shifted */
424 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
425 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
426 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
427 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
428 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
429 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
432 /* Copy result to buffer */
433 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
434 result
= newbn(rlen
);
435 for (i
= 0; i
< rlen
; i
++)
436 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
437 while (result
[0] > 1 && result
[result
[0]] == 0)
440 /* Free temporary arrays */
441 for (i
= 0; i
< 2 * pqlen
; i
++)
444 for (i
= 0; i
< mlen
; i
++)
447 for (i
= 0; i
< pqlen
; i
++)
450 for (i
= 0; i
< pqlen
; i
++)
459 * The most significant word of mod MUST be non-zero.
460 * We assume that the result array is the same size as the mod array.
461 * We optionally write out a quotient if `quotient' is non-NULL.
462 * We can avoid writing out the result if `result' is NULL.
464 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
468 int plen
, mlen
, i
, j
;
470 /* Allocate m of size mlen, copy mod to m */
471 /* We use big endian internally */
473 m
= snewn(mlen
, BignumInt
);
474 for (j
= 0; j
< mlen
; j
++)
475 m
[j
] = mod
[mod
[0] - j
];
477 /* Shift m left to make msb bit set */
478 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
479 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
482 for (i
= 0; i
< mlen
- 1; i
++)
483 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
484 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
488 /* Ensure plen > mlen */
492 /* Allocate n of size plen, copy p to n */
493 n
= snewn(plen
, BignumInt
);
494 for (j
= 0; j
< plen
; j
++)
496 for (j
= 1; j
<= p
[0]; j
++)
499 /* Main computation */
500 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
502 /* Fixup result in case the modulus was shifted */
504 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
505 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
506 n
[plen
- 1] = n
[plen
- 1] << mshift
;
507 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
508 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
509 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
512 /* Copy result to buffer */
514 for (i
= 1; i
<= result
[0]; i
++) {
516 result
[i
] = j
>= 0 ? n
[j
] : 0;
520 /* Free temporary arrays */
521 for (i
= 0; i
< mlen
; i
++)
524 for (i
= 0; i
< plen
; i
++)
530 * Decrement a number.
532 void decbn(Bignum bn
)
535 while (i
< bn
[0] && bn
[i
] == 0)
536 bn
[i
++] = BIGNUM_INT_MASK
;
540 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
545 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
548 for (i
= 1; i
<= w
; i
++)
550 for (i
= nbytes
; i
--;) {
551 unsigned char byte
= *data
++;
552 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
555 while (result
[0] > 1 && result
[result
[0]] == 0)
561 * Read an SSH-1-format bignum from a data buffer. Return the number
562 * of bytes consumed, or -1 if there wasn't enough data.
564 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
566 const unsigned char *p
= data
;
574 for (i
= 0; i
< 2; i
++)
576 b
= (w
+ 7) / 8; /* bits -> bytes */
581 if (!result
) /* just return length */
584 *result
= bignum_from_bytes(p
, b
);
590 * Return the bit count of a bignum, for SSH-1 encoding.
592 int bignum_bitcount(Bignum bn
)
594 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
596 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
601 * Return the byte length of a bignum when SSH-1 encoded.
603 int ssh1_bignum_length(Bignum bn
)
605 return 2 + (bignum_bitcount(bn
) + 7) / 8;
609 * Return the byte length of a bignum when SSH-2 encoded.
611 int ssh2_bignum_length(Bignum bn
)
613 return 4 + (bignum_bitcount(bn
) + 8) / 8;
617 * Return a byte from a bignum; 0 is least significant, etc.
619 int bignum_byte(Bignum bn
, int i
)
621 if (i
>= BIGNUM_INT_BYTES
* bn
[0])
622 return 0; /* beyond the end */
624 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
625 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
629 * Return a bit from a bignum; 0 is least significant, etc.
631 int bignum_bit(Bignum bn
, int i
)
633 if (i
>= BIGNUM_INT_BITS
* bn
[0])
634 return 0; /* beyond the end */
636 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
640 * Set a bit in a bignum; 0 is least significant, etc.
642 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
644 if (bitnum
>= BIGNUM_INT_BITS
* bn
[0])
645 abort(); /* beyond the end */
647 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
648 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
657 * Write a SSH-1-format bignum into a buffer. It is assumed the
658 * buffer is big enough. Returns the number of bytes used.
660 int ssh1_write_bignum(void *data
, Bignum bn
)
662 unsigned char *p
= data
;
663 int len
= ssh1_bignum_length(bn
);
665 int bitc
= bignum_bitcount(bn
);
667 *p
++ = (bitc
>> 8) & 0xFF;
668 *p
++ = (bitc
) & 0xFF;
669 for (i
= len
- 2; i
--;)
670 *p
++ = bignum_byte(bn
, i
);
675 * Compare two bignums. Returns like strcmp.
677 int bignum_cmp(Bignum a
, Bignum b
)
679 int amax
= a
[0], bmax
= b
[0];
680 int i
= (amax
> bmax ? amax
: bmax
);
682 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
683 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
694 * Right-shift one bignum to form another.
696 Bignum
bignum_rshift(Bignum a
, int shift
)
699 int i
, shiftw
, shiftb
, shiftbb
, bits
;
702 bits
= bignum_bitcount(a
) - shift
;
703 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
706 shiftw
= shift
/ BIGNUM_INT_BITS
;
707 shiftb
= shift
% BIGNUM_INT_BITS
;
708 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
711 for (i
= 1; i
<= ret
[0]; i
++) {
713 ai1
= (i
+ shiftw
+ 1 <= a
[0] ? a
[i
+ shiftw
+ 1] : 0);
714 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
722 * Non-modular multiplication and addition.
724 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
726 int alen
= a
[0], blen
= b
[0];
727 int mlen
= (alen
> blen ? alen
: blen
);
728 int rlen
, i
, maxspot
;
729 BignumInt
*workspace
;
732 /* mlen space for a, mlen space for b, 2*mlen for result */
733 workspace
= snewn(mlen
* 4, BignumInt
);
734 for (i
= 0; i
< mlen
; i
++) {
735 workspace
[0 * mlen
+ i
] = (mlen
- i
<= a
[0] ? a
[mlen
- i
] : 0);
736 workspace
[1 * mlen
+ i
] = (mlen
- i
<= b
[0] ? b
[mlen
- i
] : 0);
739 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
740 workspace
+ 2 * mlen
, mlen
);
742 /* now just copy the result back */
743 rlen
= alen
+ blen
+ 1;
744 if (addend
&& rlen
<= addend
[0])
745 rlen
= addend
[0] + 1;
748 for (i
= 1; i
<= ret
[0]; i
++) {
749 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
755 /* now add in the addend, if any */
757 BignumDblInt carry
= 0;
758 for (i
= 1; i
<= rlen
; i
++) {
759 carry
+= (i
<= ret
[0] ? ret
[i
] : 0);
760 carry
+= (i
<= addend
[0] ? addend
[i
] : 0);
761 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
762 carry
>>= BIGNUM_INT_BITS
;
763 if (ret
[i
] != 0 && i
> maxspot
)
774 * Non-modular multiplication.
776 Bignum
bigmul(Bignum a
, Bignum b
)
778 return bigmuladd(a
, b
, NULL
);
782 * Create a bignum which is the bitmask covering another one. That
783 * is, the smallest integer which is >= N and is also one less than
786 Bignum
bignum_bitmask(Bignum n
)
788 Bignum ret
= copybn(n
);
793 while (n
[i
] == 0 && i
> 0)
796 return ret
; /* input was zero */
802 ret
[i
] = BIGNUM_INT_MASK
;
807 * Convert a (max 32-bit) long into a bignum.
809 Bignum
bignum_from_long(unsigned long nn
)
815 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
816 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
818 ret
[0] = (ret
[2] ?
2 : 1);
823 * Add a long to a bignum.
825 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
827 Bignum ret
= newbn(number
[0] + 1);
829 BignumDblInt carry
= 0, addend
= addendx
;
831 for (i
= 1; i
<= ret
[0]; i
++) {
832 carry
+= addend
& BIGNUM_INT_MASK
;
833 carry
+= (i
<= number
[0] ? number
[i
] : 0);
834 addend
>>= BIGNUM_INT_BITS
;
835 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
836 carry
>>= BIGNUM_INT_BITS
;
845 * Compute the residue of a bignum, modulo a (max 16-bit) short.
847 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
854 for (i
= number
[0]; i
> 0; i
--)
855 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
856 return (unsigned short) r
;
860 void diagbn(char *prefix
, Bignum md
)
862 int i
, nibbles
, morenibbles
;
863 static const char hex
[] = "0123456789ABCDEF";
865 debug(("%s0x", prefix ? prefix
: ""));
867 nibbles
= (3 + bignum_bitcount(md
)) / 4;
870 morenibbles
= 4 * md
[0] - nibbles
;
871 for (i
= 0; i
< morenibbles
; i
++)
873 for (i
= nibbles
; i
--;)
875 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
885 Bignum
bigdiv(Bignum a
, Bignum b
)
887 Bignum q
= newbn(a
[0]);
888 bigdivmod(a
, b
, NULL
, q
);
895 Bignum
bigmod(Bignum a
, Bignum b
)
897 Bignum r
= newbn(b
[0]);
898 bigdivmod(a
, b
, r
, NULL
);
903 * Greatest common divisor.
905 Bignum
biggcd(Bignum av
, Bignum bv
)
907 Bignum a
= copybn(av
);
908 Bignum b
= copybn(bv
);
910 while (bignum_cmp(b
, Zero
) != 0) {
911 Bignum t
= newbn(b
[0]);
912 bigdivmod(a
, b
, t
, NULL
);
913 while (t
[0] > 1 && t
[t
[0]] == 0)
925 * Modular inverse, using Euclid's extended algorithm.
927 Bignum
modinv(Bignum number
, Bignum modulus
)
929 Bignum a
= copybn(modulus
);
930 Bignum b
= copybn(number
);
931 Bignum xp
= copybn(Zero
);
932 Bignum x
= copybn(One
);
935 while (bignum_cmp(b
, One
) != 0) {
936 Bignum t
= newbn(b
[0]);
937 Bignum q
= newbn(a
[0]);
938 bigdivmod(a
, b
, t
, q
);
939 while (t
[0] > 1 && t
[t
[0]] == 0)
946 x
= bigmuladd(q
, xp
, t
);
956 /* now we know that sign * x == 1, and that x < modulus */
958 /* set a new x to be modulus - x */
959 Bignum newx
= newbn(modulus
[0]);
964 for (i
= 1; i
<= newx
[0]; i
++) {
965 BignumInt aword
= (i
<= modulus
[0] ? modulus
[i
] : 0);
966 BignumInt bword
= (i
<= x
[0] ? x
[i
] : 0);
967 newx
[i
] = aword
- bword
- carry
;
969 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
983 * Render a bignum into decimal. Return a malloced string holding
984 * the decimal representation.
986 char *bignum_decimal(Bignum x
)
992 BignumInt
*workspace
;
995 * First, estimate the number of digits. Since log(10)/log(2)
996 * is just greater than 93/28 (the joys of continued fraction
997 * approximations...) we know that for every 93 bits, we need
998 * at most 28 digits. This will tell us how much to malloc.
1000 * Formally: if x has i bits, that means x is strictly less
1001 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1002 * 10^(28i/93). We need an integer power of ten, so we must
1003 * round up (rounding down might make it less than x again).
1004 * Therefore if we multiply the bit count by 28/93, rounding
1005 * up, we will have enough digits.
1007 i
= bignum_bitcount(x
);
1008 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1009 ndigits
++; /* allow for trailing \0 */
1010 ret
= snewn(ndigits
, char);
1013 * Now allocate some workspace to hold the binary form as we
1014 * repeatedly divide it by ten. Initialise this to the
1015 * big-endian form of the number.
1017 workspace
= snewn(x
[0], BignumInt
);
1018 for (i
= 0; i
< x
[0]; i
++)
1019 workspace
[i
] = x
[x
[0] - i
];
1022 * Next, write the decimal number starting with the last digit.
1023 * We use ordinary short division, dividing 10 into the
1026 ndigit
= ndigits
- 1;
1031 for (i
= 0; i
< x
[0]; i
++) {
1032 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1033 workspace
[i
] = (BignumInt
) (carry
/ 10);
1038 ret
[--ndigit
] = (char) (carry
+ '0');
1042 * There's a chance we've fallen short of the start of the
1043 * string. Correct if so.
1046 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);