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 typedef unsigned short BignumInt
;
56 typedef unsigned long BignumDblInt
;
57 #define BIGNUM_INT_MASK 0xFFFFU
58 #define BIGNUM_TOP_BIT 0x8000U
59 #define BIGNUM_INT_BITS 16
60 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
61 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
62 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
68 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
70 #define BIGNUM_INTERNAL
71 typedef BignumInt
*Bignum
;
75 BignumInt bnZero
[1] = { 0 };
76 BignumInt bnOne
[2] = { 1, 1 };
79 * The Bignum format is an array of `BignumInt'. The first
80 * element of the array counts the remaining elements. The
81 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
82 * significant digit first. (So it's trivial to extract the bit
83 * with value 2^n for any n.)
85 * All Bignums in this module are positive. Negative numbers must
86 * be dealt with outside it.
88 * INVARIANT: the most significant word of any Bignum must be
92 Bignum Zero
= bnZero
, One
= bnOne
;
94 static Bignum
newbn(int length
)
96 Bignum b
= snewn(length
+ 1, BignumInt
);
99 memset(b
, 0, (length
+ 1) * sizeof(*b
));
104 void bn_restore_invariant(Bignum b
)
106 while (b
[0] > 1 && b
[b
[0]] == 0)
110 Bignum
copybn(Bignum orig
)
112 Bignum b
= snewn(orig
[0] + 1, BignumInt
);
115 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
119 void freebn(Bignum b
)
122 * Burn the evidence, just in case.
124 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
128 Bignum
bn_power_2(int n
)
130 Bignum ret
= newbn(n
/ BIGNUM_INT_BITS
+ 1);
131 bignum_set_bit(ret
, n
, 1);
137 * Input is in the first len words of a and b.
138 * Result is returned in the first 2*len words of c.
140 static void internal_mul(BignumInt
*a
, BignumInt
*b
,
141 BignumInt
*c
, int len
)
146 for (j
= 0; j
< 2 * len
; j
++)
149 for (i
= len
- 1; i
>= 0; i
--) {
151 for (j
= len
- 1; j
>= 0; j
--) {
152 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
153 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
154 c
[i
+ j
+ 1] = (BignumInt
) t
;
155 t
= t
>> BIGNUM_INT_BITS
;
157 c
[i
] = (BignumInt
) t
;
161 static void internal_add_shifted(BignumInt
*number
,
162 unsigned n
, int shift
)
164 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
165 int bshift
= shift
% BIGNUM_INT_BITS
;
168 addend
= (BignumDblInt
)n
<< bshift
;
171 addend
+= number
[word
];
172 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
173 addend
>>= BIGNUM_INT_BITS
;
180 * Input in first alen words of a and first mlen words of m.
181 * Output in first alen words of a
182 * (of which first alen-mlen words will be zero).
183 * The MSW of m MUST have its high bit set.
184 * Quotient is accumulated in the `quotient' array, which is a Bignum
185 * rather than the internal bigendian format. Quotient parts are shifted
186 * left by `qshift' before adding into quot.
188 static void internal_mod(BignumInt
*a
, int alen
,
189 BignumInt
*m
, int mlen
,
190 BignumInt
*quot
, int qshift
)
202 for (i
= 0; i
<= alen
- mlen
; i
++) {
204 unsigned int q
, r
, c
, ai1
;
218 /* Find q = h:a[i] / m0 */
223 * To illustrate it, suppose a BignumInt is 8 bits, and
224 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
225 * our initial division will be 0xA123 / 0xA1, which
226 * will give a quotient of 0x100 and a divide overflow.
227 * However, the invariants in this division algorithm
228 * are not violated, since the full number A1:23:... is
229 * _less_ than the quotient prefix A1:B2:... and so the
230 * following correction loop would have sorted it out.
232 * In this situation we set q to be the largest
233 * quotient we _can_ stomach (0xFF, of course).
237 /* Macro doesn't want an array subscript expression passed
238 * into it (see definition), so use a temporary. */
239 BignumInt tmplo
= a
[i
];
240 DIVMOD_WORD(q
, r
, h
, tmplo
, m0
);
242 /* Refine our estimate of q by looking at
243 h:a[i]:a[i+1] / m0:m1 */
245 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
248 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
249 if (r
>= (BignumDblInt
) m0
&&
250 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
254 /* Subtract q * m from a[i...] */
256 for (k
= mlen
- 1; k
>= 0; k
--) {
257 t
= MUL_WORD(q
, m
[k
]);
259 c
= (unsigned)(t
>> BIGNUM_INT_BITS
);
260 if ((BignumInt
) t
> a
[i
+ k
])
262 a
[i
+ k
] -= (BignumInt
) t
;
265 /* Add back m in case of borrow */
268 for (k
= mlen
- 1; k
>= 0; k
--) {
271 a
[i
+ k
] = (BignumInt
) t
;
272 t
= t
>> BIGNUM_INT_BITS
;
277 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
282 * Compute (base ^ exp) % mod.
284 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
286 BignumInt
*a
, *b
, *n
, *m
;
292 * The most significant word of mod needs to be non-zero. It
293 * should already be, but let's make sure.
295 assert(mod
[mod
[0]] != 0);
298 * Make sure the base is smaller than the modulus, by reducing
299 * it modulo the modulus if not.
301 base
= bigmod(base_in
, mod
);
303 /* Allocate m of size mlen, copy mod to m */
304 /* We use big endian internally */
306 m
= snewn(mlen
, BignumInt
);
307 for (j
= 0; j
< mlen
; j
++)
308 m
[j
] = mod
[mod
[0] - j
];
310 /* Shift m left to make msb bit set */
311 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
312 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
315 for (i
= 0; i
< mlen
- 1; i
++)
316 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
317 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
320 /* Allocate n of size mlen, copy base to n */
321 n
= snewn(mlen
, BignumInt
);
323 for (j
= 0; j
< i
; j
++)
325 for (j
= 0; j
< (int)base
[0]; j
++)
326 n
[i
+ j
] = base
[base
[0] - j
];
328 /* Allocate a and b of size 2*mlen. Set a = 1 */
329 a
= snewn(2 * mlen
, BignumInt
);
330 b
= snewn(2 * mlen
, BignumInt
);
331 for (i
= 0; i
< 2 * mlen
; i
++)
335 /* Skip leading zero bits of exp. */
337 j
= BIGNUM_INT_BITS
-1;
338 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
342 j
= BIGNUM_INT_BITS
-1;
346 /* Main computation */
347 while (i
< (int)exp
[0]) {
349 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
350 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
351 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
352 internal_mul(b
+ mlen
, n
, a
, mlen
);
353 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
363 j
= BIGNUM_INT_BITS
-1;
366 /* Fixup result in case the modulus was shifted */
368 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
369 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
370 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
371 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
372 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
373 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
376 /* Copy result to buffer */
377 result
= newbn(mod
[0]);
378 for (i
= 0; i
< mlen
; i
++)
379 result
[result
[0] - i
] = a
[i
+ mlen
];
380 while (result
[0] > 1 && result
[result
[0]] == 0)
383 /* Free temporary arrays */
384 for (i
= 0; i
< 2 * mlen
; i
++)
387 for (i
= 0; i
< 2 * mlen
; i
++)
390 for (i
= 0; i
< mlen
; i
++)
393 for (i
= 0; i
< mlen
; i
++)
403 * Compute (p * q) % mod.
404 * The most significant word of mod MUST be non-zero.
405 * We assume that the result array is the same size as the mod array.
407 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
409 BignumInt
*a
, *n
, *m
, *o
;
411 int pqlen
, mlen
, rlen
, i
, j
;
414 /* Allocate m of size mlen, copy mod to m */
415 /* We use big endian internally */
417 m
= snewn(mlen
, BignumInt
);
418 for (j
= 0; j
< mlen
; j
++)
419 m
[j
] = mod
[mod
[0] - j
];
421 /* Shift m left to make msb bit set */
422 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
423 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
426 for (i
= 0; i
< mlen
- 1; i
++)
427 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
428 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
431 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
433 /* Allocate n of size pqlen, copy p to n */
434 n
= snewn(pqlen
, BignumInt
);
436 for (j
= 0; j
< i
; j
++)
438 for (j
= 0; j
< (int)p
[0]; j
++)
439 n
[i
+ j
] = p
[p
[0] - j
];
441 /* Allocate o of size pqlen, copy q to o */
442 o
= snewn(pqlen
, BignumInt
);
444 for (j
= 0; j
< i
; j
++)
446 for (j
= 0; j
< (int)q
[0]; j
++)
447 o
[i
+ j
] = q
[q
[0] - j
];
449 /* Allocate a of size 2*pqlen for result */
450 a
= snewn(2 * pqlen
, BignumInt
);
452 /* Main computation */
453 internal_mul(n
, o
, a
, pqlen
);
454 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
456 /* Fixup result in case the modulus was shifted */
458 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
459 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
460 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
461 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
462 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
463 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
466 /* Copy result to buffer */
467 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
468 result
= newbn(rlen
);
469 for (i
= 0; i
< rlen
; i
++)
470 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
471 while (result
[0] > 1 && result
[result
[0]] == 0)
474 /* Free temporary arrays */
475 for (i
= 0; i
< 2 * pqlen
; i
++)
478 for (i
= 0; i
< mlen
; i
++)
481 for (i
= 0; i
< pqlen
; i
++)
484 for (i
= 0; i
< pqlen
; i
++)
493 * The most significant word of mod MUST be non-zero.
494 * We assume that the result array is the same size as the mod array.
495 * We optionally write out a quotient if `quotient' is non-NULL.
496 * We can avoid writing out the result if `result' is NULL.
498 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
502 int plen
, mlen
, i
, j
;
504 /* Allocate m of size mlen, copy mod to m */
505 /* We use big endian internally */
507 m
= snewn(mlen
, BignumInt
);
508 for (j
= 0; j
< mlen
; j
++)
509 m
[j
] = mod
[mod
[0] - j
];
511 /* Shift m left to make msb bit set */
512 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
513 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
516 for (i
= 0; i
< mlen
- 1; i
++)
517 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
518 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
522 /* Ensure plen > mlen */
526 /* Allocate n of size plen, copy p to n */
527 n
= snewn(plen
, BignumInt
);
528 for (j
= 0; j
< plen
; j
++)
530 for (j
= 1; j
<= (int)p
[0]; j
++)
533 /* Main computation */
534 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
536 /* Fixup result in case the modulus was shifted */
538 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
539 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
540 n
[plen
- 1] = n
[plen
- 1] << mshift
;
541 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
542 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
543 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
546 /* Copy result to buffer */
548 for (i
= 1; i
<= (int)result
[0]; i
++) {
550 result
[i
] = j
>= 0 ? n
[j
] : 0;
554 /* Free temporary arrays */
555 for (i
= 0; i
< mlen
; i
++)
558 for (i
= 0; i
< plen
; i
++)
564 * Decrement a number.
566 void decbn(Bignum bn
)
569 while (i
< (int)bn
[0] && bn
[i
] == 0)
570 bn
[i
++] = BIGNUM_INT_MASK
;
574 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
579 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
582 for (i
= 1; i
<= w
; i
++)
584 for (i
= nbytes
; i
--;) {
585 unsigned char byte
= *data
++;
586 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
589 while (result
[0] > 1 && result
[result
[0]] == 0)
595 * Read an SSH-1-format bignum from a data buffer. Return the number
596 * of bytes consumed, or -1 if there wasn't enough data.
598 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
600 const unsigned char *p
= data
;
608 for (i
= 0; i
< 2; i
++)
610 b
= (w
+ 7) / 8; /* bits -> bytes */
615 if (!result
) /* just return length */
618 *result
= bignum_from_bytes(p
, b
);
624 * Return the bit count of a bignum, for SSH-1 encoding.
626 int bignum_bitcount(Bignum bn
)
628 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
630 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
635 * Return the byte length of a bignum when SSH-1 encoded.
637 int ssh1_bignum_length(Bignum bn
)
639 return 2 + (bignum_bitcount(bn
) + 7) / 8;
643 * Return the byte length of a bignum when SSH-2 encoded.
645 int ssh2_bignum_length(Bignum bn
)
647 return 4 + (bignum_bitcount(bn
) + 8) / 8;
651 * Return a byte from a bignum; 0 is least significant, etc.
653 int bignum_byte(Bignum bn
, int i
)
655 if (i
>= (int)(BIGNUM_INT_BYTES
* bn
[0]))
656 return 0; /* beyond the end */
658 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
659 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
663 * Return a bit from a bignum; 0 is least significant, etc.
665 int bignum_bit(Bignum bn
, int i
)
667 if (i
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
668 return 0; /* beyond the end */
670 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
674 * Set a bit in a bignum; 0 is least significant, etc.
676 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
678 if (bitnum
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
679 abort(); /* beyond the end */
681 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
682 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
691 * Write a SSH-1-format bignum into a buffer. It is assumed the
692 * buffer is big enough. Returns the number of bytes used.
694 int ssh1_write_bignum(void *data
, Bignum bn
)
696 unsigned char *p
= data
;
697 int len
= ssh1_bignum_length(bn
);
699 int bitc
= bignum_bitcount(bn
);
701 *p
++ = (bitc
>> 8) & 0xFF;
702 *p
++ = (bitc
) & 0xFF;
703 for (i
= len
- 2; i
--;)
704 *p
++ = bignum_byte(bn
, i
);
709 * Compare two bignums. Returns like strcmp.
711 int bignum_cmp(Bignum a
, Bignum b
)
713 int amax
= a
[0], bmax
= b
[0];
714 int i
= (amax
> bmax ? amax
: bmax
);
716 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
717 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
728 * Right-shift one bignum to form another.
730 Bignum
bignum_rshift(Bignum a
, int shift
)
733 int i
, shiftw
, shiftb
, shiftbb
, bits
;
736 bits
= bignum_bitcount(a
) - shift
;
737 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
740 shiftw
= shift
/ BIGNUM_INT_BITS
;
741 shiftb
= shift
% BIGNUM_INT_BITS
;
742 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
745 for (i
= 1; i
<= (int)ret
[0]; i
++) {
747 ai1
= (i
+ shiftw
+ 1 <= (int)a
[0] ? a
[i
+ shiftw
+ 1] : 0);
748 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
756 * Non-modular multiplication and addition.
758 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
760 int alen
= a
[0], blen
= b
[0];
761 int mlen
= (alen
> blen ? alen
: blen
);
762 int rlen
, i
, maxspot
;
763 BignumInt
*workspace
;
766 /* mlen space for a, mlen space for b, 2*mlen for result */
767 workspace
= snewn(mlen
* 4, BignumInt
);
768 for (i
= 0; i
< mlen
; i
++) {
769 workspace
[0 * mlen
+ i
] = (mlen
- i
<= (int)a
[0] ? a
[mlen
- i
] : 0);
770 workspace
[1 * mlen
+ i
] = (mlen
- i
<= (int)b
[0] ? b
[mlen
- i
] : 0);
773 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
774 workspace
+ 2 * mlen
, mlen
);
776 /* now just copy the result back */
777 rlen
= alen
+ blen
+ 1;
778 if (addend
&& rlen
<= (int)addend
[0])
779 rlen
= addend
[0] + 1;
782 for (i
= 1; i
<= (int)ret
[0]; i
++) {
783 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
789 /* now add in the addend, if any */
791 BignumDblInt carry
= 0;
792 for (i
= 1; i
<= rlen
; i
++) {
793 carry
+= (i
<= (int)ret
[0] ? ret
[i
] : 0);
794 carry
+= (i
<= (int)addend
[0] ? addend
[i
] : 0);
795 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
796 carry
>>= BIGNUM_INT_BITS
;
797 if (ret
[i
] != 0 && i
> maxspot
)
808 * Non-modular multiplication.
810 Bignum
bigmul(Bignum a
, Bignum b
)
812 return bigmuladd(a
, b
, NULL
);
816 * Create a bignum which is the bitmask covering another one. That
817 * is, the smallest integer which is >= N and is also one less than
820 Bignum
bignum_bitmask(Bignum n
)
822 Bignum ret
= copybn(n
);
827 while (n
[i
] == 0 && i
> 0)
830 return ret
; /* input was zero */
836 ret
[i
] = BIGNUM_INT_MASK
;
841 * Convert a (max 32-bit) long into a bignum.
843 Bignum
bignum_from_long(unsigned long nn
)
849 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
850 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
852 ret
[0] = (ret
[2] ?
2 : 1);
857 * Add a long to a bignum.
859 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
861 Bignum ret
= newbn(number
[0] + 1);
863 BignumDblInt carry
= 0, addend
= addendx
;
865 for (i
= 1; i
<= (int)ret
[0]; i
++) {
866 carry
+= addend
& BIGNUM_INT_MASK
;
867 carry
+= (i
<= (int)number
[0] ? number
[i
] : 0);
868 addend
>>= BIGNUM_INT_BITS
;
869 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
870 carry
>>= BIGNUM_INT_BITS
;
879 * Compute the residue of a bignum, modulo a (max 16-bit) short.
881 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
888 for (i
= number
[0]; i
> 0; i
--)
889 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
890 return (unsigned short) r
;
894 void diagbn(char *prefix
, Bignum md
)
896 int i
, nibbles
, morenibbles
;
897 static const char hex
[] = "0123456789ABCDEF";
899 debug(("%s0x", prefix ? prefix
: ""));
901 nibbles
= (3 + bignum_bitcount(md
)) / 4;
904 morenibbles
= 4 * md
[0] - nibbles
;
905 for (i
= 0; i
< morenibbles
; i
++)
907 for (i
= nibbles
; i
--;)
909 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
919 Bignum
bigdiv(Bignum a
, Bignum b
)
921 Bignum q
= newbn(a
[0]);
922 bigdivmod(a
, b
, NULL
, q
);
929 Bignum
bigmod(Bignum a
, Bignum b
)
931 Bignum r
= newbn(b
[0]);
932 bigdivmod(a
, b
, r
, NULL
);
937 * Greatest common divisor.
939 Bignum
biggcd(Bignum av
, Bignum bv
)
941 Bignum a
= copybn(av
);
942 Bignum b
= copybn(bv
);
944 while (bignum_cmp(b
, Zero
) != 0) {
945 Bignum t
= newbn(b
[0]);
946 bigdivmod(a
, b
, t
, NULL
);
947 while (t
[0] > 1 && t
[t
[0]] == 0)
959 * Modular inverse, using Euclid's extended algorithm.
961 Bignum
modinv(Bignum number
, Bignum modulus
)
963 Bignum a
= copybn(modulus
);
964 Bignum b
= copybn(number
);
965 Bignum xp
= copybn(Zero
);
966 Bignum x
= copybn(One
);
969 while (bignum_cmp(b
, One
) != 0) {
970 Bignum t
= newbn(b
[0]);
971 Bignum q
= newbn(a
[0]);
972 bigdivmod(a
, b
, t
, q
);
973 while (t
[0] > 1 && t
[t
[0]] == 0)
980 x
= bigmuladd(q
, xp
, t
);
990 /* now we know that sign * x == 1, and that x < modulus */
992 /* set a new x to be modulus - x */
993 Bignum newx
= newbn(modulus
[0]);
998 for (i
= 1; i
<= (int)newx
[0]; i
++) {
999 BignumInt aword
= (i
<= (int)modulus
[0] ? modulus
[i
] : 0);
1000 BignumInt bword
= (i
<= (int)x
[0] ? x
[i
] : 0);
1001 newx
[i
] = aword
- bword
- carry
;
1003 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1017 * Render a bignum into decimal. Return a malloced string holding
1018 * the decimal representation.
1020 char *bignum_decimal(Bignum x
)
1022 int ndigits
, ndigit
;
1026 BignumInt
*workspace
;
1029 * First, estimate the number of digits. Since log(10)/log(2)
1030 * is just greater than 93/28 (the joys of continued fraction
1031 * approximations...) we know that for every 93 bits, we need
1032 * at most 28 digits. This will tell us how much to malloc.
1034 * Formally: if x has i bits, that means x is strictly less
1035 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1036 * 10^(28i/93). We need an integer power of ten, so we must
1037 * round up (rounding down might make it less than x again).
1038 * Therefore if we multiply the bit count by 28/93, rounding
1039 * up, we will have enough digits.
1041 * i=0 (i.e., x=0) is an irritating special case.
1043 i
= bignum_bitcount(x
);
1045 ndigits
= 1; /* x = 0 */
1047 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1048 ndigits
++; /* allow for trailing \0 */
1049 ret
= snewn(ndigits
, char);
1052 * Now allocate some workspace to hold the binary form as we
1053 * repeatedly divide it by ten. Initialise this to the
1054 * big-endian form of the number.
1056 workspace
= snewn(x
[0], BignumInt
);
1057 for (i
= 0; i
< (int)x
[0]; i
++)
1058 workspace
[i
] = x
[x
[0] - i
];
1061 * Next, write the decimal number starting with the last digit.
1062 * We use ordinary short division, dividing 10 into the
1065 ndigit
= ndigits
- 1;
1070 for (i
= 0; i
< (int)x
[0]; i
++) {
1071 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1072 workspace
[i
] = (BignumInt
) (carry
/ 10);
1077 ret
[--ndigit
] = (char) (carry
+ '0');
1081 * There's a chance we've fallen short of the start of the
1082 * string. Correct if so.
1085 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);