2 * Bignum routines for RSA and DH and stuff.
11 #define BIGNUM_INTERNAL
12 typedef unsigned short *Bignum
;
16 unsigned short bnZero
[1] = { 0 };
17 unsigned short bnOne
[2] = { 1, 1 };
20 * The Bignum format is an array of `unsigned short'. The first
21 * element of the array counts the remaining elements. The
22 * remaining elements express the actual number, base 2^16, _least_
23 * significant digit first. (So it's trivial to extract the bit
24 * with value 2^n for any n.)
26 * All Bignums in this module are positive. Negative numbers must
27 * be dealt with outside it.
29 * INVARIANT: the most significant word of any Bignum must be
33 Bignum Zero
= bnZero
, One
= bnOne
;
35 static Bignum
newbn(int length
)
37 Bignum b
= smalloc((length
+ 1) * sizeof(unsigned short));
40 memset(b
, 0, (length
+ 1) * sizeof(*b
));
45 void bn_restore_invariant(Bignum b
)
47 while (b
[0] > 1 && b
[b
[0]] == 0)
51 Bignum
copybn(Bignum orig
)
53 Bignum b
= smalloc((orig
[0] + 1) * sizeof(unsigned short));
56 memcpy(b
, orig
, (orig
[0] + 1) * sizeof(*b
));
63 * Burn the evidence, just in case.
65 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
69 Bignum
bn_power_2(int n
)
71 Bignum ret
= newbn(n
/ 16 + 1);
72 bignum_set_bit(ret
, n
, 1);
78 * Input is in the first len words of a and b.
79 * Result is returned in the first 2*len words of c.
81 static void internal_mul(unsigned short *a
, unsigned short *b
,
82 unsigned short *c
, int len
)
87 for (j
= 0; j
< 2 * len
; j
++)
90 for (i
= len
- 1; i
>= 0; i
--) {
93 for (j
= len
- 1; j
>= 0; j
--) {
94 t
+= ai
* (unsigned long) b
[j
];
95 t
+= (unsigned long) c
[i
+ j
+ 1];
96 c
[i
+ j
+ 1] = (unsigned short) t
;
99 c
[i
] = (unsigned short) t
;
103 static void internal_add_shifted(unsigned short *number
,
104 unsigned n
, int shift
)
106 int word
= 1 + (shift
/ 16);
107 int bshift
= shift
% 16;
108 unsigned long addend
;
110 addend
= n
<< bshift
;
113 addend
+= number
[word
];
114 number
[word
] = (unsigned short) addend
& 0xFFFF;
122 * Input in first alen words of a and first mlen words of m.
123 * Output in first alen words of a
124 * (of which first alen-mlen words will be zero).
125 * The MSW of m MUST have its high bit set.
126 * Quotient is accumulated in the `quotient' array, which is a Bignum
127 * rather than the internal bigendian format. Quotient parts are shifted
128 * left by `qshift' before adding into quot.
130 static void internal_mod(unsigned short *a
, int alen
,
131 unsigned short *m
, int mlen
,
132 unsigned short *quot
, int qshift
)
134 unsigned short m0
, m1
;
144 for (i
= 0; i
<= alen
- mlen
; i
++) {
146 unsigned int q
, r
, c
, ai1
;
160 /* Find q = h:a[i] / m0 */
161 t
= ((unsigned long) h
<< 16) + a
[i
];
165 /* Refine our estimate of q by looking at
166 h:a[i]:a[i+1] / m0:m1 */
167 t
= (long) m1
*(long) q
;
168 if (t
> ((unsigned long) r
<< 16) + ai1
) {
171 r
= (r
+ m0
) & 0xffff; /* overflow? */
172 if (r
>= (unsigned long) m0
&&
173 t
> ((unsigned long) r
<< 16) + ai1
) q
--;
176 /* Subtract q * m from a[i...] */
178 for (k
= mlen
- 1; k
>= 0; k
--) {
179 t
= (long) q
*(long) m
[k
];
182 if ((unsigned short) t
> a
[i
+ k
])
184 a
[i
+ k
] -= (unsigned short) t
;
187 /* Add back m in case of borrow */
190 for (k
= mlen
- 1; k
>= 0; k
--) {
193 a
[i
+ k
] = (unsigned short) t
;
199 internal_add_shifted(quot
, q
, qshift
+ 16 * (alen
- mlen
- i
));
204 * Compute (base ^ exp) % mod.
205 * The base MUST be smaller than the modulus.
206 * The most significant word of mod MUST be non-zero.
207 * We assume that the result array is the same size as the mod array.
209 Bignum
modpow(Bignum base
, Bignum exp
, Bignum mod
)
211 unsigned short *a
, *b
, *n
, *m
;
216 /* Allocate m of size mlen, copy mod to m */
217 /* We use big endian internally */
219 m
= smalloc(mlen
* sizeof(unsigned short));
220 for (j
= 0; j
< mlen
; j
++)
221 m
[j
] = mod
[mod
[0] - j
];
223 /* Shift m left to make msb bit set */
224 for (mshift
= 0; mshift
< 15; mshift
++)
225 if ((m
[0] << mshift
) & 0x8000)
228 for (i
= 0; i
< mlen
- 1; i
++)
229 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (16 - mshift
));
230 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
233 /* Allocate n of size mlen, copy base to n */
234 n
= smalloc(mlen
* sizeof(unsigned short));
236 for (j
= 0; j
< i
; j
++)
238 for (j
= 0; j
< base
[0]; j
++)
239 n
[i
+ j
] = base
[base
[0] - j
];
241 /* Allocate a and b of size 2*mlen. Set a = 1 */
242 a
= smalloc(2 * mlen
* sizeof(unsigned short));
243 b
= smalloc(2 * mlen
* sizeof(unsigned short));
244 for (i
= 0; i
< 2 * mlen
; i
++)
248 /* Skip leading zero bits of exp. */
251 while (i
< exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
259 /* Main computation */
262 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
263 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
264 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
265 internal_mul(b
+ mlen
, n
, a
, mlen
);
266 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
279 /* Fixup result in case the modulus was shifted */
281 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
282 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (16 - mshift
));
283 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
284 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
285 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
286 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (16 - mshift
));
289 /* Copy result to buffer */
290 result
= newbn(mod
[0]);
291 for (i
= 0; i
< mlen
; i
++)
292 result
[result
[0] - i
] = a
[i
+ mlen
];
293 while (result
[0] > 1 && result
[result
[0]] == 0)
296 /* Free temporary arrays */
297 for (i
= 0; i
< 2 * mlen
; i
++)
300 for (i
= 0; i
< 2 * mlen
; i
++)
303 for (i
= 0; i
< mlen
; i
++)
306 for (i
= 0; i
< mlen
; i
++)
314 * Compute (p * q) % mod.
315 * The most significant word of mod MUST be non-zero.
316 * We assume that the result array is the same size as the mod array.
318 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
320 unsigned short *a
, *n
, *m
, *o
;
322 int pqlen
, mlen
, rlen
, i
, j
;
325 /* Allocate m of size mlen, copy mod to m */
326 /* We use big endian internally */
328 m
= smalloc(mlen
* sizeof(unsigned short));
329 for (j
= 0; j
< mlen
; j
++)
330 m
[j
] = mod
[mod
[0] - j
];
332 /* Shift m left to make msb bit set */
333 for (mshift
= 0; mshift
< 15; mshift
++)
334 if ((m
[0] << mshift
) & 0x8000)
337 for (i
= 0; i
< mlen
- 1; i
++)
338 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (16 - mshift
));
339 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
342 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
344 /* Allocate n of size pqlen, copy p to n */
345 n
= smalloc(pqlen
* sizeof(unsigned short));
347 for (j
= 0; j
< i
; j
++)
349 for (j
= 0; j
< p
[0]; j
++)
350 n
[i
+ j
] = p
[p
[0] - j
];
352 /* Allocate o of size pqlen, copy q to o */
353 o
= smalloc(pqlen
* sizeof(unsigned short));
355 for (j
= 0; j
< i
; j
++)
357 for (j
= 0; j
< q
[0]; j
++)
358 o
[i
+ j
] = q
[q
[0] - j
];
360 /* Allocate a of size 2*pqlen for result */
361 a
= smalloc(2 * pqlen
* sizeof(unsigned short));
363 /* Main computation */
364 internal_mul(n
, o
, a
, pqlen
);
365 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
367 /* Fixup result in case the modulus was shifted */
369 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
370 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (16 - mshift
));
371 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
372 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
373 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
374 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (16 - mshift
));
377 /* Copy result to buffer */
378 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
379 result
= newbn(rlen
);
380 for (i
= 0; i
< rlen
; i
++)
381 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
382 while (result
[0] > 1 && result
[result
[0]] == 0)
385 /* Free temporary arrays */
386 for (i
= 0; i
< 2 * pqlen
; i
++)
389 for (i
= 0; i
< mlen
; i
++)
392 for (i
= 0; i
< pqlen
; i
++)
395 for (i
= 0; i
< pqlen
; i
++)
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.
406 * We optionally write out a quotient if `quotient' is non-NULL.
407 * We can avoid writing out the result if `result' is NULL.
409 void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
411 unsigned short *n
, *m
;
413 int plen
, mlen
, i
, j
;
415 /* Allocate m of size mlen, copy mod to m */
416 /* We use big endian internally */
418 m
= smalloc(mlen
* sizeof(unsigned short));
419 for (j
= 0; j
< mlen
; j
++)
420 m
[j
] = mod
[mod
[0] - j
];
422 /* Shift m left to make msb bit set */
423 for (mshift
= 0; mshift
< 15; mshift
++)
424 if ((m
[0] << mshift
) & 0x8000)
427 for (i
= 0; i
< mlen
- 1; i
++)
428 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (16 - mshift
));
429 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
433 /* Ensure plen > mlen */
437 /* Allocate n of size plen, copy p to n */
438 n
= smalloc(plen
* sizeof(unsigned short));
439 for (j
= 0; j
< plen
; j
++)
441 for (j
= 1; j
<= p
[0]; j
++)
444 /* Main computation */
445 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
447 /* Fixup result in case the modulus was shifted */
449 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
450 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (16 - mshift
));
451 n
[plen
- 1] = n
[plen
- 1] << mshift
;
452 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
453 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
454 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (16 - mshift
));
457 /* Copy result to buffer */
459 for (i
= 1; i
<= result
[0]; i
++) {
461 result
[i
] = j
>= 0 ? n
[j
] : 0;
465 /* Free temporary arrays */
466 for (i
= 0; i
< mlen
; i
++)
469 for (i
= 0; i
< plen
; i
++)
475 * Decrement a number.
477 void decbn(Bignum bn
)
480 while (i
< bn
[0] && bn
[i
] == 0)
485 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
490 w
= (nbytes
+ 1) / 2; /* bytes -> words */
493 for (i
= 1; i
<= w
; i
++)
495 for (i
= nbytes
; i
--;) {
496 unsigned char byte
= *data
++;
498 result
[1 + i
/ 2] |= byte
<< 8;
500 result
[1 + i
/ 2] |= byte
;
503 while (result
[0] > 1 && result
[result
[0]] == 0)
509 * Read an ssh1-format bignum from a data buffer. Return the number
512 int ssh1_read_bignum(const unsigned char *data
, Bignum
* result
)
514 const unsigned char *p
= data
;
519 for (i
= 0; i
< 2; i
++)
521 b
= (w
+ 7) / 8; /* bits -> bytes */
523 if (!result
) /* just return length */
526 *result
= bignum_from_bytes(p
, b
);
532 * Return the bit count of a bignum, for ssh1 encoding.
534 int bignum_bitcount(Bignum bn
)
536 int bitcount
= bn
[0] * 16 - 1;
538 && (bn
[bitcount
/ 16 + 1] >> (bitcount
% 16)) == 0) bitcount
--;
543 * Return the byte length of a bignum when ssh1 encoded.
545 int ssh1_bignum_length(Bignum bn
)
547 return 2 + (bignum_bitcount(bn
) + 7) / 8;
551 * Return the byte length of a bignum when ssh2 encoded.
553 int ssh2_bignum_length(Bignum bn
)
555 return 4 + (bignum_bitcount(bn
) + 8) / 8;
559 * Return a byte from a bignum; 0 is least significant, etc.
561 int bignum_byte(Bignum bn
, int i
)
564 return 0; /* beyond the end */
566 return (bn
[i
/ 2 + 1] >> 8) & 0xFF;
568 return (bn
[i
/ 2 + 1]) & 0xFF;
572 * Return a bit from a bignum; 0 is least significant, etc.
574 int bignum_bit(Bignum bn
, int i
)
577 return 0; /* beyond the end */
579 return (bn
[i
/ 16 + 1] >> (i
% 16)) & 1;
583 * Set a bit in a bignum; 0 is least significant, etc.
585 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
587 if (bitnum
>= 16 * bn
[0])
588 abort(); /* beyond the end */
590 int v
= bitnum
/ 16 + 1;
591 int mask
= 1 << (bitnum
% 16);
600 * Write a ssh1-format bignum into a buffer. It is assumed the
601 * buffer is big enough. Returns the number of bytes used.
603 int ssh1_write_bignum(void *data
, Bignum bn
)
605 unsigned char *p
= data
;
606 int len
= ssh1_bignum_length(bn
);
608 int bitc
= bignum_bitcount(bn
);
610 *p
++ = (bitc
>> 8) & 0xFF;
611 *p
++ = (bitc
) & 0xFF;
612 for (i
= len
- 2; i
--;)
613 *p
++ = bignum_byte(bn
, i
);
618 * Compare two bignums. Returns like strcmp.
620 int bignum_cmp(Bignum a
, Bignum b
)
622 int amax
= a
[0], bmax
= b
[0];
623 int i
= (amax
> bmax ? amax
: bmax
);
625 unsigned short aval
= (i
> amax ?
0 : a
[i
]);
626 unsigned short bval
= (i
> bmax ?
0 : b
[i
]);
637 * Right-shift one bignum to form another.
639 Bignum
bignum_rshift(Bignum a
, int shift
)
642 int i
, shiftw
, shiftb
, shiftbb
, bits
;
643 unsigned short ai
, ai1
;
645 bits
= bignum_bitcount(a
) - shift
;
646 ret
= newbn((bits
+ 15) / 16);
651 shiftbb
= 16 - shiftb
;
654 for (i
= 1; i
<= ret
[0]; i
++) {
656 ai1
= (i
+ shiftw
+ 1 <= a
[0] ? a
[i
+ shiftw
+ 1] : 0);
657 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & 0xFFFF;
665 * Non-modular multiplication and addition.
667 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
669 int alen
= a
[0], blen
= b
[0];
670 int mlen
= (alen
> blen ? alen
: blen
);
671 int rlen
, i
, maxspot
;
672 unsigned short *workspace
;
675 /* mlen space for a, mlen space for b, 2*mlen for result */
676 workspace
= smalloc(mlen
* 4 * sizeof(unsigned short));
677 for (i
= 0; i
< mlen
; i
++) {
678 workspace
[0 * mlen
+ i
] = (mlen
- i
<= a
[0] ? a
[mlen
- i
] : 0);
679 workspace
[1 * mlen
+ i
] = (mlen
- i
<= b
[0] ? b
[mlen
- i
] : 0);
682 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
683 workspace
+ 2 * mlen
, mlen
);
685 /* now just copy the result back */
686 rlen
= alen
+ blen
+ 1;
687 if (addend
&& rlen
<= addend
[0])
688 rlen
= addend
[0] + 1;
691 for (i
= 1; i
<= ret
[0]; i
++) {
692 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
698 /* now add in the addend, if any */
700 unsigned long carry
= 0;
701 for (i
= 1; i
<= rlen
; i
++) {
702 carry
+= (i
<= ret
[0] ? ret
[i
] : 0);
703 carry
+= (i
<= addend
[0] ? addend
[i
] : 0);
704 ret
[i
] = (unsigned short) carry
& 0xFFFF;
706 if (ret
[i
] != 0 && i
> maxspot
)
716 * Non-modular multiplication.
718 Bignum
bigmul(Bignum a
, Bignum b
)
720 return bigmuladd(a
, b
, NULL
);
724 * Create a bignum which is the bitmask covering another one. That
725 * is, the smallest integer which is >= N and is also one less than
728 Bignum
bignum_bitmask(Bignum n
)
730 Bignum ret
= copybn(n
);
735 while (n
[i
] == 0 && i
> 0)
738 return ret
; /* input was zero */
749 * Convert a (max 32-bit) long into a bignum.
751 Bignum
bignum_from_long(unsigned long n
)
756 ret
[1] = (unsigned short)(n
& 0xFFFF);
757 ret
[2] = (unsigned short)((n
>> 16) & 0xFFFF);
759 ret
[0] = (ret
[2] ?
2 : 1);
764 * Add a long to a bignum.
766 Bignum
bignum_add_long(Bignum number
, unsigned long addend
)
768 Bignum ret
= newbn(number
[0] + 1);
770 unsigned long carry
= 0;
772 for (i
= 1; i
<= ret
[0]; i
++) {
773 carry
+= addend
& 0xFFFF;
774 carry
+= (i
<= number
[0] ? number
[i
] : 0);
776 ret
[i
] = (unsigned short) carry
& 0xFFFF;
786 * Compute the residue of a bignum, modulo a (max 16-bit) short.
788 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
790 unsigned long mod
, r
;
795 for (i
= number
[0]; i
> 0; i
--)
796 r
= (r
* 65536 + number
[i
]) % mod
;
797 return (unsigned short) r
;
800 void diagbn(char *prefix
, Bignum md
)
803 int i
, nibbles
, morenibbles
;
804 static const char hex
[] = "0123456789ABCDEF";
806 debug(("%s0x", prefix ? prefix
: ""));
808 nibbles
= (3 + bignum_bitcount(md
)) / 4;
811 morenibbles
= 4 * md
[0] - nibbles
;
812 for (i
= 0; i
< morenibbles
; i
++)
814 for (i
= nibbles
; i
--;)
816 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
826 Bignum
bigdiv(Bignum a
, Bignum b
)
828 Bignum q
= newbn(a
[0]);
829 bigdivmod(a
, b
, NULL
, q
);
836 Bignum
bigmod(Bignum a
, Bignum b
)
838 Bignum r
= newbn(b
[0]);
839 bigdivmod(a
, b
, r
, NULL
);
844 * Greatest common divisor.
846 Bignum
biggcd(Bignum av
, Bignum bv
)
848 Bignum a
= copybn(av
);
849 Bignum b
= copybn(bv
);
851 while (bignum_cmp(b
, Zero
) != 0) {
852 Bignum t
= newbn(b
[0]);
853 bigdivmod(a
, b
, t
, NULL
);
854 while (t
[0] > 1 && t
[t
[0]] == 0)
866 * Modular inverse, using Euclid's extended algorithm.
868 Bignum
modinv(Bignum number
, Bignum modulus
)
870 Bignum a
= copybn(modulus
);
871 Bignum b
= copybn(number
);
872 Bignum xp
= copybn(Zero
);
873 Bignum x
= copybn(One
);
876 while (bignum_cmp(b
, One
) != 0) {
877 Bignum t
= newbn(b
[0]);
878 Bignum q
= newbn(a
[0]);
879 bigdivmod(a
, b
, t
, q
);
880 while (t
[0] > 1 && t
[t
[0]] == 0)
887 x
= bigmuladd(q
, xp
, t
);
896 /* now we know that sign * x == 1, and that x < modulus */
898 /* set a new x to be modulus - x */
899 Bignum newx
= newbn(modulus
[0]);
900 unsigned short carry
= 0;
904 for (i
= 1; i
<= newx
[0]; i
++) {
905 unsigned short aword
= (i
<= modulus
[0] ? modulus
[i
] : 0);
906 unsigned short bword
= (i
<= x
[0] ? x
[i
] : 0);
907 newx
[i
] = aword
- bword
- carry
;
909 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
923 * Render a bignum into decimal. Return a malloced string holding
924 * the decimal representation.
926 char *bignum_decimal(Bignum x
)
932 unsigned short *workspace
;
935 * First, estimate the number of digits. Since log(10)/log(2)
936 * is just greater than 93/28 (the joys of continued fraction
937 * approximations...) we know that for every 93 bits, we need
938 * at most 28 digits. This will tell us how much to malloc.
940 * Formally: if x has i bits, that means x is strictly less
941 * than 2^i. Since 2 is less than 10^(28/93), this is less than
942 * 10^(28i/93). We need an integer power of ten, so we must
943 * round up (rounding down might make it less than x again).
944 * Therefore if we multiply the bit count by 28/93, rounding
945 * up, we will have enough digits.
947 i
= bignum_bitcount(x
);
948 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
949 ndigits
++; /* allow for trailing \0 */
950 ret
= smalloc(ndigits
);
953 * Now allocate some workspace to hold the binary form as we
954 * repeatedly divide it by ten. Initialise this to the
955 * big-endian form of the number.
957 workspace
= smalloc(sizeof(unsigned short) * x
[0]);
958 for (i
= 0; i
< x
[0]; i
++)
959 workspace
[i
] = x
[x
[0] - i
];
962 * Next, write the decimal number starting with the last digit.
963 * We use ordinary short division, dividing 10 into the
966 ndigit
= ndigits
- 1;
971 for (i
= 0; i
< x
[0]; i
++) {
972 carry
= (carry
<< 16) + workspace
[i
];
973 workspace
[i
] = (unsigned short) (carry
/ 10);
978 ret
[--ndigit
] = (char) (carry
+ '0');
982 * There's a chance we've fallen short of the start of the
983 * string. Correct if so.
986 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);