2 * Bignum routines for RSA and DH and stuff.
9 #if 0 // use PuTTY main debugging for diagbn()
12 #define debugprint debug
14 #define debugprint(x) printf x
17 #define BIGNUM_INTERNAL
18 typedef unsigned short *Bignum
;
22 unsigned short bnZero
[1] = { 0 };
23 unsigned short bnOne
[2] = { 1, 1 };
26 * The Bignum format is an array of `unsigned short'. The first
27 * element of the array counts the remaining elements. The
28 * remaining elements express the actual number, base 2^16, _least_
29 * significant digit first. (So it's trivial to extract the bit
30 * with value 2^n for any n.)
32 * All Bignums in this module are positive. Negative numbers must
33 * be dealt with outside it.
35 * INVARIANT: the most significant word of any Bignum must be
39 Bignum Zero
= bnZero
, One
= bnOne
;
41 static Bignum
newbn(int length
) {
42 Bignum b
= smalloc((length
+1)*sizeof(unsigned short));
45 memset(b
, 0, (length
+1)*sizeof(*b
));
50 void bn_restore_invariant(Bignum b
) {
51 while (b
[0] > 1 && b
[b
[0]] == 0) b
[0]--;
54 Bignum
copybn(Bignum orig
) {
55 Bignum b
= smalloc((orig
[0]+1)*sizeof(unsigned short));
58 memcpy(b
, orig
, (orig
[0]+1)*sizeof(*b
));
62 void freebn(Bignum b
) {
64 * Burn the evidence, just in case.
66 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
70 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
) {
105 int word
= 1 + (shift
/ 16);
106 int bshift
= shift
% 16;
107 unsigned long addend
;
109 addend
= n
<< bshift
;
112 addend
+= number
[word
];
113 number
[word
] = (unsigned short) addend
& 0xFFFF;
121 * Input in first alen words of a and first mlen words of m.
122 * Output in first alen words of a
123 * (of which first alen-mlen words will be zero).
124 * The MSW of m MUST have its high bit set.
125 * Quotient is accumulated in the `quotient' array, which is a Bignum
126 * rather than the internal bigendian format. Quotient parts are shifted
127 * left by `qshift' before adding into quot.
129 static void internal_mod(unsigned short *a
, int alen
,
130 unsigned short *m
, int mlen
,
131 unsigned short *quot
, int qshift
)
133 unsigned short m0
, m1
;
143 for (i
= 0; i
<= alen
-mlen
; i
++) {
145 unsigned int q
, r
, c
, ai1
;
159 /* Find q = h:a[i] / m0 */
160 t
= ((unsigned long) h
<< 16) + a
[i
];
164 /* Refine our estimate of q by looking at
165 h:a[i]:a[i+1] / m0:m1 */
166 t
= (long) m1
* (long) q
;
167 if (t
> ((unsigned long) r
<< 16) + ai1
) {
170 r
= (r
+ m0
) & 0xffff; /* overflow? */
171 if (r
>= (unsigned long)m0
&&
172 t
> ((unsigned long) r
<< 16) + ai1
)
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
]) c
++;
183 a
[i
+k
] -= (unsigned short) t
;
186 /* Add back m in case of borrow */
189 for (k
= mlen
- 1; k
>= 0; k
--) {
192 a
[i
+k
] = (unsigned short)t
;
198 internal_add_shifted(quot
, q
, qshift
+ 16 * (alen
-mlen
-i
));
203 * Compute (base ^ exp) % mod.
204 * The base MUST be smaller than the modulus.
205 * The most significant word of mod MUST be non-zero.
206 * We assume that the result array is the same size as the mod array.
208 Bignum
modpow(Bignum base
, Bignum exp
, Bignum mod
)
210 unsigned short *a
, *b
, *n
, *m
;
215 /* Allocate m of size mlen, copy mod to m */
216 /* We use big endian internally */
218 m
= smalloc(mlen
* sizeof(unsigned short));
219 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
221 /* Shift m left to make msb bit set */
222 for (mshift
= 0; mshift
< 15; mshift
++)
223 if ((m
[0] << mshift
) & 0x8000) break;
225 for (i
= 0; i
< mlen
- 1; i
++)
226 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
227 m
[mlen
-1] = m
[mlen
-1] << mshift
;
230 /* Allocate n of size mlen, copy base to n */
231 n
= smalloc(mlen
* sizeof(unsigned short));
233 for (j
= 0; j
< i
; j
++) n
[j
] = 0;
234 for (j
= 0; j
< base
[0]; j
++) n
[i
+j
] = base
[base
[0] - j
];
236 /* Allocate a and b of size 2*mlen. Set a = 1 */
237 a
= smalloc(2 * mlen
* sizeof(unsigned short));
238 b
= smalloc(2 * mlen
* sizeof(unsigned short));
239 for (i
= 0; i
< 2*mlen
; i
++) a
[i
] = 0;
242 /* Skip leading zero bits of exp. */
244 while (i
< exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
246 if (j
< 0) { i
++; j
= 15; }
249 /* Main computation */
252 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
253 internal_mod(b
, mlen
*2, m
, mlen
, NULL
, 0);
254 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
255 internal_mul(b
+ mlen
, n
, a
, mlen
);
256 internal_mod(a
, mlen
*2, m
, mlen
, NULL
, 0);
266 /* Fixup result in case the modulus was shifted */
268 for (i
= mlen
- 1; i
< 2*mlen
- 1; i
++)
269 a
[i
] = (a
[i
] << mshift
) | (a
[i
+1] >> (16-mshift
));
270 a
[2*mlen
-1] = a
[2*mlen
-1] << mshift
;
271 internal_mod(a
, mlen
*2, m
, mlen
, NULL
, 0);
272 for (i
= 2*mlen
- 1; i
>= mlen
; i
--)
273 a
[i
] = (a
[i
] >> mshift
) | (a
[i
-1] << (16-mshift
));
276 /* Copy result to buffer */
277 result
= newbn(mod
[0]);
278 for (i
= 0; i
< mlen
; i
++)
279 result
[result
[0] - i
] = a
[i
+mlen
];
280 while (result
[0] > 1 && result
[result
[0]] == 0) result
[0]--;
282 /* Free temporary arrays */
283 for (i
= 0; i
< 2*mlen
; i
++) a
[i
] = 0; sfree(a
);
284 for (i
= 0; i
< 2*mlen
; i
++) b
[i
] = 0; sfree(b
);
285 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; sfree(m
);
286 for (i
= 0; i
< mlen
; i
++) n
[i
] = 0; sfree(n
);
292 * Compute (p * q) % mod.
293 * The most significant word of mod MUST be non-zero.
294 * We assume that the result array is the same size as the mod array.
296 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
298 unsigned short *a
, *n
, *m
, *o
;
300 int pqlen
, mlen
, rlen
, i
, j
;
303 /* Allocate m of size mlen, copy mod to m */
304 /* We use big endian internally */
306 m
= smalloc(mlen
* sizeof(unsigned short));
307 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
309 /* Shift m left to make msb bit set */
310 for (mshift
= 0; mshift
< 15; mshift
++)
311 if ((m
[0] << mshift
) & 0x8000) break;
313 for (i
= 0; i
< mlen
- 1; i
++)
314 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
315 m
[mlen
-1] = m
[mlen
-1] << mshift
;
318 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
320 /* Allocate n of size pqlen, copy p to n */
321 n
= smalloc(pqlen
* sizeof(unsigned short));
323 for (j
= 0; j
< i
; j
++) n
[j
] = 0;
324 for (j
= 0; j
< p
[0]; j
++) n
[i
+j
] = p
[p
[0] - j
];
326 /* Allocate o of size pqlen, copy q to o */
327 o
= smalloc(pqlen
* sizeof(unsigned short));
329 for (j
= 0; j
< i
; j
++) o
[j
] = 0;
330 for (j
= 0; j
< q
[0]; j
++) o
[i
+j
] = q
[q
[0] - j
];
332 /* Allocate a of size 2*pqlen for result */
333 a
= smalloc(2 * pqlen
* sizeof(unsigned short));
335 /* Main computation */
336 internal_mul(n
, o
, a
, pqlen
);
337 internal_mod(a
, pqlen
*2, m
, mlen
, NULL
, 0);
339 /* Fixup result in case the modulus was shifted */
341 for (i
= 2*pqlen
- mlen
- 1; i
< 2*pqlen
- 1; i
++)
342 a
[i
] = (a
[i
] << mshift
) | (a
[i
+1] >> (16-mshift
));
343 a
[2*pqlen
-1] = a
[2*pqlen
-1] << mshift
;
344 internal_mod(a
, pqlen
*2, m
, mlen
, NULL
, 0);
345 for (i
= 2*pqlen
- 1; i
>= 2*pqlen
- mlen
; i
--)
346 a
[i
] = (a
[i
] >> mshift
) | (a
[i
-1] << (16-mshift
));
349 /* Copy result to buffer */
350 rlen
= (mlen
< pqlen
*2 ? mlen
: pqlen
*2);
351 result
= newbn(rlen
);
352 for (i
= 0; i
< rlen
; i
++)
353 result
[result
[0] - i
] = a
[i
+2*pqlen
-rlen
];
354 while (result
[0] > 1 && result
[result
[0]] == 0) result
[0]--;
356 /* Free temporary arrays */
357 for (i
= 0; i
< 2*pqlen
; i
++) a
[i
] = 0; sfree(a
);
358 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; sfree(m
);
359 for (i
= 0; i
< pqlen
; i
++) n
[i
] = 0; sfree(n
);
360 for (i
= 0; i
< pqlen
; i
++) o
[i
] = 0; sfree(o
);
367 * The most significant word of mod MUST be non-zero.
368 * We assume that the result array is the same size as the mod array.
369 * We optionally write out a quotient.
371 void bigmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
373 unsigned short *n
, *m
;
375 int plen
, mlen
, i
, j
;
377 /* Allocate m of size mlen, copy mod to m */
378 /* We use big endian internally */
380 m
= smalloc(mlen
* sizeof(unsigned short));
381 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
383 /* Shift m left to make msb bit set */
384 for (mshift
= 0; mshift
< 15; mshift
++)
385 if ((m
[0] << mshift
) & 0x8000) break;
387 for (i
= 0; i
< mlen
- 1; i
++)
388 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
389 m
[mlen
-1] = m
[mlen
-1] << mshift
;
393 /* Ensure plen > mlen */
394 if (plen
<= mlen
) plen
= mlen
+1;
396 /* Allocate n of size plen, copy p to n */
397 n
= smalloc(plen
* sizeof(unsigned short));
398 for (j
= 0; j
< plen
; j
++) n
[j
] = 0;
399 for (j
= 1; j
<= p
[0]; j
++) n
[plen
-j
] = p
[j
];
401 /* Main computation */
402 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
404 /* Fixup result in case the modulus was shifted */
406 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
407 n
[i
] = (n
[i
] << mshift
) | (n
[i
+1] >> (16-mshift
));
408 n
[plen
-1] = n
[plen
-1] << mshift
;
409 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
410 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
411 n
[i
] = (n
[i
] >> mshift
) | (n
[i
-1] << (16-mshift
));
414 /* Copy result to buffer */
415 for (i
= 1; i
<= result
[0]; i
++) {
417 result
[i
] = j
>=0 ? n
[j
] : 0;
420 /* Free temporary arrays */
421 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; sfree(m
);
422 for (i
= 0; i
< plen
; i
++) n
[i
] = 0; sfree(n
);
426 * Decrement a number.
428 void decbn(Bignum bn
) {
430 while (i
< bn
[0] && bn
[i
] == 0)
435 Bignum
bignum_from_bytes(unsigned char *data
, int nbytes
) {
439 w
= (nbytes
+1)/2; /* bytes -> words */
444 for (i
=nbytes
; i
-- ;) {
445 unsigned char byte
= *data
++;
447 result
[1+i
/2] |= byte
<<8;
449 result
[1+i
/2] |= byte
;
452 while (result
[0] > 1 && result
[result
[0]] == 0) result
[0]--;
457 * Read an ssh1-format bignum from a data buffer. Return the number
460 int ssh1_read_bignum(unsigned char *data
, Bignum
*result
) {
461 unsigned char *p
= data
;
468 b
= (w
+7)/8; /* bits -> bytes */
470 if (!result
) /* just return length */
473 *result
= bignum_from_bytes(p
, b
);
479 * Return the bit count of a bignum, for ssh1 encoding.
481 int bignum_bitcount(Bignum bn
) {
482 int bitcount
= bn
[0] * 16 - 1;
483 while (bitcount
>= 0 && (bn
[bitcount
/16+1] >> (bitcount
% 16)) == 0)
489 * Return the byte length of a bignum when ssh1 encoded.
491 int ssh1_bignum_length(Bignum bn
) {
492 return 2 + (bignum_bitcount(bn
)+7)/8;
496 * Return the byte length of a bignum when ssh2 encoded.
498 int ssh2_bignum_length(Bignum bn
) {
499 return 4 + (bignum_bitcount(bn
)+8)/8;
503 * Return a byte from a bignum; 0 is least significant, etc.
505 int bignum_byte(Bignum bn
, int i
) {
507 return 0; /* beyond the end */
509 return (bn
[i
/2+1] >> 8) & 0xFF;
511 return (bn
[i
/2+1] ) & 0xFF;
515 * Return a bit from a bignum; 0 is least significant, etc.
517 int bignum_bit(Bignum bn
, int i
) {
519 return 0; /* beyond the end */
521 return (bn
[i
/16+1] >> (i
%16)) & 1;
525 * Set a bit in a bignum; 0 is least significant, etc.
527 void bignum_set_bit(Bignum bn
, int bitnum
, int value
) {
528 if (bitnum
>= 16*bn
[0])
529 abort(); /* beyond the end */
532 int mask
= 1 << (bitnum
%16);
541 * Write a ssh1-format bignum into a buffer. It is assumed the
542 * buffer is big enough. Returns the number of bytes used.
544 int ssh1_write_bignum(void *data
, Bignum bn
) {
545 unsigned char *p
= data
;
546 int len
= ssh1_bignum_length(bn
);
548 int bitc
= bignum_bitcount(bn
);
550 *p
++ = (bitc
>> 8) & 0xFF;
551 *p
++ = (bitc
) & 0xFF;
552 for (i
= len
-2; i
-- ;)
553 *p
++ = bignum_byte(bn
, i
);
558 * Compare two bignums. Returns like strcmp.
560 int bignum_cmp(Bignum a
, Bignum b
) {
561 int amax
= a
[0], bmax
= b
[0];
562 int i
= (amax
> bmax ? amax
: bmax
);
564 unsigned short aval
= (i
> amax ?
0 : a
[i
]);
565 unsigned short bval
= (i
> bmax ?
0 : b
[i
]);
566 if (aval
< bval
) return -1;
567 if (aval
> bval
) return +1;
574 * Right-shift one bignum to form another.
576 Bignum
bignum_rshift(Bignum a
, int shift
) {
578 int i
, shiftw
, shiftb
, shiftbb
, bits
;
579 unsigned short ai
, ai1
;
581 bits
= bignum_bitcount(a
) - shift
;
582 ret
= newbn((bits
+15)/16);
587 shiftbb
= 16 - shiftb
;
590 for (i
= 1; i
<= ret
[0]; i
++) {
592 ai1
= (i
+shiftw
+1 <= a
[0] ? a
[i
+shiftw
+1] : 0);
593 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & 0xFFFF;
601 * Non-modular multiplication and addition.
603 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
) {
604 int alen
= a
[0], blen
= b
[0];
605 int mlen
= (alen
> blen ? alen
: blen
);
606 int rlen
, i
, maxspot
;
607 unsigned short *workspace
;
610 /* mlen space for a, mlen space for b, 2*mlen for result */
611 workspace
= smalloc(mlen
* 4 * sizeof(unsigned short));
612 for (i
= 0; i
< mlen
; i
++) {
613 workspace
[0*mlen
+ i
] = (mlen
-i
<= a
[0] ? a
[mlen
-i
] : 0);
614 workspace
[1*mlen
+ i
] = (mlen
-i
<= b
[0] ? b
[mlen
-i
] : 0);
617 internal_mul(workspace
+0*mlen
, workspace
+1*mlen
, workspace
+2*mlen
, mlen
);
619 /* now just copy the result back */
620 rlen
= alen
+ blen
+ 1;
621 if (addend
&& rlen
<= addend
[0])
622 rlen
= addend
[0] + 1;
625 for (i
= 1; i
<= ret
[0]; i
++) {
626 ret
[i
] = (i
<= 2*mlen ? workspace
[4*mlen
- i
] : 0);
632 /* now add in the addend, if any */
634 unsigned long carry
= 0;
635 for (i
= 1; i
<= rlen
; i
++) {
636 carry
+= (i
<= ret
[0] ? ret
[i
] : 0);
637 carry
+= (i
<= addend
[0] ? addend
[i
] : 0);
638 ret
[i
] = (unsigned short) carry
& 0xFFFF;
640 if (ret
[i
] != 0 && i
> maxspot
)
650 * Non-modular multiplication.
652 Bignum
bigmul(Bignum a
, Bignum b
) {
653 return bigmuladd(a
, b
, NULL
);
657 * Create a bignum which is the bitmask covering another one. That
658 * is, the smallest integer which is >= N and is also one less than
661 Bignum
bignum_bitmask(Bignum n
) {
662 Bignum ret
= copybn(n
);
667 while (n
[i
] == 0 && i
> 0)
670 return ret
; /* input was zero */
681 * Convert a (max 16-bit) short into a bignum.
683 Bignum
bignum_from_short(unsigned short n
) {
688 ret
[2] = (n
>> 16) & 0xFFFF;
689 ret
[0] = (ret
[2] ?
2 : 1);
694 * Add a long to a bignum.
696 Bignum
bignum_add_long(Bignum number
, unsigned long addend
) {
697 Bignum ret
= newbn(number
[0]+1);
699 unsigned long carry
= 0;
701 for (i
= 1; i
<= ret
[0]; i
++) {
702 carry
+= addend
& 0xFFFF;
703 carry
+= (i
<= number
[0] ? number
[i
] : 0);
705 ret
[i
] = (unsigned short) carry
& 0xFFFF;
715 * Compute the residue of a bignum, modulo a (max 16-bit) short.
717 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
) {
718 unsigned long mod
, r
;
723 for (i
= number
[0]; i
> 0; i
--)
724 r
= (r
* 65536 + number
[i
]) % mod
;
725 return (unsigned short) r
;
728 void diagbn(char *prefix
, Bignum md
) {
729 int i
, nibbles
, morenibbles
;
730 static const char hex
[] = "0123456789ABCDEF";
732 debugprint(("%s0x", prefix ? prefix
: ""));
734 nibbles
= (3 + bignum_bitcount(md
))/4; if (nibbles
<1) nibbles
=1;
735 morenibbles
= 4*md
[0] - nibbles
;
736 for (i
=0; i
<morenibbles
; i
++) debugprint(("-"));
737 for (i
=nibbles
; i
-- ;)
738 debugprint(("%c",hex
[(bignum_byte(md
, i
/2) >> (4*(i
%2))) & 0xF]));
740 if (prefix
) debugprint(("\n"));
744 * Greatest common divisor.
746 Bignum
biggcd(Bignum av
, Bignum bv
) {
747 Bignum a
= copybn(av
);
748 Bignum b
= copybn(bv
);
752 while (bignum_cmp(b
, Zero
) != 0) {
753 Bignum t
= newbn(b
[0]);
754 bigmod(a
, b
, t
, NULL
);
756 while (t
[0] > 1 && t
[t
[0]] == 0) t
[0]--;
767 * Modular inverse, using Euclid's extended algorithm.
769 Bignum
modinv(Bignum number
, Bignum modulus
) {
770 Bignum a
= copybn(modulus
);
771 Bignum b
= copybn(number
);
772 Bignum xp
= copybn(Zero
);
773 Bignum x
= copybn(One
);
776 while (bignum_cmp(b
, One
) != 0) {
777 Bignum t
= newbn(b
[0]);
778 Bignum q
= newbn(a
[0]);
780 while (t
[0] > 1 && t
[t
[0]] == 0) t
[0]--;
786 x
= bigmuladd(q
, xp
, t
);
795 /* now we know that sign * x == 1, and that x < modulus */
797 /* set a new x to be modulus - x */
798 Bignum newx
= newbn(modulus
[0]);
799 unsigned short carry
= 0;
803 for (i
= 1; i
<= newx
[0]; i
++) {
804 unsigned short aword
= (i
<= modulus
[0] ? modulus
[i
] : 0);
805 unsigned short bword
= (i
<= x
[0] ? x
[i
] : 0);
806 newx
[i
] = aword
- bword
- carry
;
808 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
822 * Render a bignum into decimal. Return a malloced string holding
823 * the decimal representation.
825 char *bignum_decimal(Bignum x
) {
830 unsigned short *workspace
;
833 * First, estimate the number of digits. Since log(10)/log(2)
834 * is just greater than 93/28 (the joys of continued fraction
835 * approximations...) we know that for every 93 bits, we need
836 * at most 28 digits. This will tell us how much to malloc.
838 * Formally: if x has i bits, that means x is strictly less
839 * than 2^i. Since 2 is less than 10^(28/93), this is less than
840 * 10^(28i/93). We need an integer power of ten, so we must
841 * round up (rounding down might make it less than x again).
842 * Therefore if we multiply the bit count by 28/93, rounding
843 * up, we will have enough digits.
845 i
= bignum_bitcount(x
);
846 ndigits
= (28*i
+ 92)/93; /* multiply by 28/93 and round up */
847 ndigits
++; /* allow for trailing \0 */
848 ret
= smalloc(ndigits
);
851 * Now allocate some workspace to hold the binary form as we
852 * repeatedly divide it by ten. Initialise this to the
853 * big-endian form of the number.
855 workspace
= smalloc(sizeof(unsigned short) * x
[0]);
856 for (i
= 0; i
< x
[0]; i
++)
857 workspace
[i
] = x
[x
[0] - i
];
860 * Next, write the decimal number starting with the last digit.
861 * We use ordinary short division, dividing 10 into the
869 for (i
= 0; i
< x
[0]; i
++) {
870 carry
= (carry
<< 16) + workspace
[i
];
871 workspace
[i
] = (unsigned short) (carry
/ 10);
876 ret
[--ndigit
] = (char)(carry
+ '0');
880 * There's a chance we've fallen short of the start of the
881 * string. Correct if so.
884 memmove(ret
, ret
+ndigit
, ndigits
-ndigit
);