2 * Bignum routines for RSA and DH and stuff.
11 #if defined __GNUC__ && defined __i386__
12 typedef unsigned long BignumInt;
13 typedef unsigned long long BignumDblInt;
14 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
15 #define BIGNUM_TOP_BIT 0x80000000UL
16 #define BIGNUM_INT_BITS 32
17 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
19 typedef unsigned short BignumInt;
20 typedef unsigned long BignumDblInt;
21 #define BIGNUM_INT_MASK 0xFFFFU
22 #define BIGNUM_TOP_BIT 0x8000U
23 #define BIGNUM_INT_BITS 16
24 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
27 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
29 #define BIGNUM_INTERNAL
30 typedef BignumInt *Bignum;
34 BignumInt bnZero[1] = { 0 };
35 BignumInt bnOne[2] = { 1, 1 };
38 * The Bignum format is an array of `BignumInt'. The first
39 * element of the array counts the remaining elements. The
40 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
41 * significant digit first. (So it's trivial to extract the bit
42 * with value 2^n for any n.)
44 * All Bignums in this module are positive. Negative numbers must
45 * be dealt with outside it.
47 * INVARIANT: the most significant word of any Bignum must be
51 Bignum Zero = bnZero, One = bnOne;
53 static Bignum newbn(int length)
55 Bignum b = snewn(length + 1, BignumInt);
58 memset(b, 0, (length + 1) * sizeof(*b));
63 void bn_restore_invariant(Bignum b)
65 while (b[0] > 1 && b[b[0]] == 0)
69 Bignum copybn(Bignum orig)
71 Bignum b = snewn(orig[0] + 1, BignumInt);
74 memcpy(b, orig, (orig[0] + 1) * sizeof(*b));
81 * Burn the evidence, just in case.
83 memset(b, 0, sizeof(b[0]) * (b[0] + 1));
87 Bignum bn_power_2(int n)
89 Bignum ret = newbn(n / BIGNUM_INT_BITS + 1);
90 bignum_set_bit(ret, n, 1);
96 * Input is in the first len words of a and b.
97 * Result is returned in the first 2*len words of c.
99 static void internal_mul(BignumInt *a, BignumInt *b,
100 BignumInt *c, int len)
105 for (j = 0; j < 2 * len; j++)
108 for (i = len - 1; i >= 0; i--) {
110 for (j = len - 1; j >= 0; j--) {
111 t += MUL_WORD(a[i], (BignumDblInt) b[j]);
112 t += (BignumDblInt) c[i + j + 1];
113 c[i + j + 1] = (BignumInt) t;
114 t = t >> BIGNUM_INT_BITS;
116 c[i] = (BignumInt) t;
120 static void internal_add_shifted(BignumInt *number,
121 unsigned n, int shift)
123 int word = 1 + (shift / BIGNUM_INT_BITS);
124 int bshift = shift % BIGNUM_INT_BITS;
127 addend = n << bshift;
130 addend += number[word];
131 number[word] = (BignumInt) addend & BIGNUM_INT_MASK;
132 addend >>= BIGNUM_INT_BITS;
139 * Input in first alen words of a and first mlen words of m.
140 * Output in first alen words of a
141 * (of which first alen-mlen words will be zero).
142 * The MSW of m MUST have its high bit set.
143 * Quotient is accumulated in the `quotient' array, which is a Bignum
144 * rather than the internal bigendian format. Quotient parts are shifted
145 * left by `qshift' before adding into quot.
147 static void internal_mod(BignumInt *a, int alen,
148 BignumInt *m, int mlen,
149 BignumInt *quot, int qshift)
161 for (i = 0; i <= alen - mlen; i++) {
163 unsigned int q, r, c, ai1;
177 /* Find q = h:a[i] / m0 */
178 t = ((BignumDblInt) h << BIGNUM_INT_BITS) + a[i];
182 /* Refine our estimate of q by looking at
183 h:a[i]:a[i+1] / m0:m1 */
184 t = (BignumDblInt) m1 * (BignumDblInt) q;
185 if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) {
188 r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */
189 if (r >= (BignumDblInt) m0 &&
190 t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--;
193 /* Subtract q * m from a[i...] */
195 for (k = mlen - 1; k >= 0; k--) {
196 t = (BignumDblInt) q * (BignumDblInt) m[k];
198 c = t >> BIGNUM_INT_BITS;
199 if ((BignumInt) t > a[i + k])
201 a[i + k] -= (BignumInt) t;
204 /* Add back m in case of borrow */
207 for (k = mlen - 1; k >= 0; k--) {
210 a[i + k] = (BignumInt) t;
211 t = t >> BIGNUM_INT_BITS;
216 internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i));
221 * Compute (base ^ exp) % mod.
222 * The base MUST be smaller than the modulus.
223 * The most significant word of mod MUST be non-zero.
224 * We assume that the result array is the same size as the mod array.
226 Bignum modpow(Bignum base, Bignum exp, Bignum mod)
228 BignumInt *a, *b, *n, *m;
233 /* Allocate m of size mlen, copy mod to m */
234 /* We use big endian internally */
236 m = snewn(mlen, BignumInt);
237 for (j = 0; j < mlen; j++)
238 m[j] = mod[mod[0] - j];
240 /* Shift m left to make msb bit set */
241 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
242 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
245 for (i = 0; i < mlen - 1; i++)
246 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
247 m[mlen - 1] = m[mlen - 1] << mshift;
250 /* Allocate n of size mlen, copy base to n */
251 n = snewn(mlen, BignumInt);
253 for (j = 0; j < i; j++)
255 for (j = 0; j < base[0]; j++)
256 n[i + j] = base[base[0] - j];
258 /* Allocate a and b of size 2*mlen. Set a = 1 */
259 a = snewn(2 * mlen, BignumInt);
260 b = snewn(2 * mlen, BignumInt);
261 for (i = 0; i < 2 * mlen; i++)
265 /* Skip leading zero bits of exp. */
267 j = BIGNUM_INT_BITS-1;
268 while (i < exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
272 j = BIGNUM_INT_BITS-1;
276 /* Main computation */
279 internal_mul(a + mlen, a + mlen, b, mlen);
280 internal_mod(b, mlen * 2, m, mlen, NULL, 0);
281 if ((exp[exp[0] - i] & (1 << j)) != 0) {
282 internal_mul(b + mlen, n, a, mlen);
283 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
293 j = BIGNUM_INT_BITS-1;
296 /* Fixup result in case the modulus was shifted */
298 for (i = mlen - 1; i < 2 * mlen - 1; i++)
299 a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
300 a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;
301 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
302 for (i = 2 * mlen - 1; i >= mlen; i--)
303 a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
306 /* Copy result to buffer */
307 result = newbn(mod[0]);
308 for (i = 0; i < mlen; i++)
309 result[result[0] - i] = a[i + mlen];
310 while (result[0] > 1 && result[result[0]] == 0)
313 /* Free temporary arrays */
314 for (i = 0; i < 2 * mlen; i++)
317 for (i = 0; i < 2 * mlen; i++)
320 for (i = 0; i < mlen; i++)
323 for (i = 0; i < mlen; i++)
331 * Compute (p * q) % mod.
332 * The most significant word of mod MUST be non-zero.
333 * We assume that the result array is the same size as the mod array.
335 Bignum modmul(Bignum p, Bignum q, Bignum mod)
337 BignumInt *a, *n, *m, *o;
339 int pqlen, mlen, rlen, i, j;
342 /* Allocate m of size mlen, copy mod to m */
343 /* We use big endian internally */
345 m = snewn(mlen, BignumInt);
346 for (j = 0; j < mlen; j++)
347 m[j] = mod[mod[0] - j];
349 /* Shift m left to make msb bit set */
350 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
351 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
354 for (i = 0; i < mlen - 1; i++)
355 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
356 m[mlen - 1] = m[mlen - 1] << mshift;
359 pqlen = (p[0] > q[0] ? p[0] : q[0]);
361 /* Allocate n of size pqlen, copy p to n */
362 n = snewn(pqlen, BignumInt);
364 for (j = 0; j < i; j++)
366 for (j = 0; j < p[0]; j++)
367 n[i + j] = p[p[0] - j];
369 /* Allocate o of size pqlen, copy q to o */
370 o = snewn(pqlen, BignumInt);
372 for (j = 0; j < i; j++)
374 for (j = 0; j < q[0]; j++)
375 o[i + j] = q[q[0] - j];
377 /* Allocate a of size 2*pqlen for result */
378 a = snewn(2 * pqlen, BignumInt);
380 /* Main computation */
381 internal_mul(n, o, a, pqlen);
382 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
384 /* Fixup result in case the modulus was shifted */
386 for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
387 a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
388 a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
389 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
390 for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
391 a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
394 /* Copy result to buffer */
395 rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
396 result = newbn(rlen);
397 for (i = 0; i < rlen; i++)
398 result[result[0] - i] = a[i + 2 * pqlen - rlen];
399 while (result[0] > 1 && result[result[0]] == 0)
402 /* Free temporary arrays */
403 for (i = 0; i < 2 * pqlen; i++)
406 for (i = 0; i < mlen; i++)
409 for (i = 0; i < pqlen; i++)
412 for (i = 0; i < pqlen; i++)
421 * The most significant word of mod MUST be non-zero.
422 * We assume that the result array is the same size as the mod array.
423 * We optionally write out a quotient if `quotient' is non-NULL.
424 * We can avoid writing out the result if `result' is NULL.
426 static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
430 int plen, mlen, i, j;
432 /* Allocate m of size mlen, copy mod to m */
433 /* We use big endian internally */
435 m = snewn(mlen, BignumInt);
436 for (j = 0; j < mlen; j++)
437 m[j] = mod[mod[0] - j];
439 /* Shift m left to make msb bit set */
440 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
441 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
444 for (i = 0; i < mlen - 1; i++)
445 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
446 m[mlen - 1] = m[mlen - 1] << mshift;
450 /* Ensure plen > mlen */
454 /* Allocate n of size plen, copy p to n */
455 n = snewn(plen, BignumInt);
456 for (j = 0; j < plen; j++)
458 for (j = 1; j <= p[0]; j++)
461 /* Main computation */
462 internal_mod(n, plen, m, mlen, quotient, mshift);
464 /* Fixup result in case the modulus was shifted */
466 for (i = plen - mlen - 1; i < plen - 1; i++)
467 n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift));
468 n[plen - 1] = n[plen - 1] << mshift;
469 internal_mod(n, plen, m, mlen, quotient, 0);
470 for (i = plen - 1; i >= plen - mlen; i--)
471 n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift));
474 /* Copy result to buffer */
476 for (i = 1; i <= result[0]; i++) {
478 result[i] = j >= 0 ? n[j] : 0;
482 /* Free temporary arrays */
483 for (i = 0; i < mlen; i++)
486 for (i = 0; i < plen; i++)
492 * Decrement a number.
494 void decbn(Bignum bn)
497 while (i < bn[0] && bn[i] == 0)
498 bn[i++] = BIGNUM_INT_MASK;
502 Bignum bignum_from_bytes(const unsigned char *data, int nbytes)
507 w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */
510 for (i = 1; i <= w; i++)
512 for (i = nbytes; i--;) {
513 unsigned char byte = *data++;
514 result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS);
517 while (result[0] > 1 && result[result[0]] == 0)
523 * Read an ssh1-format bignum from a data buffer. Return the number
526 int ssh1_read_bignum(const unsigned char *data, Bignum * result)
528 const unsigned char *p = data;
533 for (i = 0; i < 2; i++)
535 b = (w + 7) / 8; /* bits -> bytes */
537 if (!result) /* just return length */
540 *result = bignum_from_bytes(p, b);
546 * Return the bit count of a bignum, for ssh1 encoding.
548 int bignum_bitcount(Bignum bn)
550 int bitcount = bn[0] * BIGNUM_INT_BITS - 1;
552 && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--;
557 * Return the byte length of a bignum when ssh1 encoded.
559 int ssh1_bignum_length(Bignum bn)
561 return 2 + (bignum_bitcount(bn) + 7) / 8;
565 * Return the byte length of a bignum when ssh2 encoded.
567 int ssh2_bignum_length(Bignum bn)
569 return 4 + (bignum_bitcount(bn) + 8) / 8;
573 * Return a byte from a bignum; 0 is least significant, etc.
575 int bignum_byte(Bignum bn, int i)
577 if (i >= BIGNUM_INT_BYTES * bn[0])
578 return 0; /* beyond the end */
580 return (bn[i / BIGNUM_INT_BYTES + 1] >>
581 ((i % BIGNUM_INT_BYTES)*8)) & 0xFF;
585 * Return a bit from a bignum; 0 is least significant, etc.
587 int bignum_bit(Bignum bn, int i)
589 if (i >= BIGNUM_INT_BITS * bn[0])
590 return 0; /* beyond the end */
592 return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1;
596 * Set a bit in a bignum; 0 is least significant, etc.
598 void bignum_set_bit(Bignum bn, int bitnum, int value)
600 if (bitnum >= BIGNUM_INT_BITS * bn[0])
601 abort(); /* beyond the end */
603 int v = bitnum / BIGNUM_INT_BITS + 1;
604 int mask = 1 << (bitnum % BIGNUM_INT_BITS);
613 * Write a ssh1-format bignum into a buffer. It is assumed the
614 * buffer is big enough. Returns the number of bytes used.
616 int ssh1_write_bignum(void *data, Bignum bn)
618 unsigned char *p = data;
619 int len = ssh1_bignum_length(bn);
621 int bitc = bignum_bitcount(bn);
623 *p++ = (bitc >> 8) & 0xFF;
624 *p++ = (bitc) & 0xFF;
625 for (i = len - 2; i--;)
626 *p++ = bignum_byte(bn, i);
631 * Compare two bignums. Returns like strcmp.
633 int bignum_cmp(Bignum a, Bignum b)
635 int amax = a[0], bmax = b[0];
636 int i = (amax > bmax ? amax : bmax);
638 BignumInt aval = (i > amax ? 0 : a[i]);
639 BignumInt bval = (i > bmax ? 0 : b[i]);
650 * Right-shift one bignum to form another.
652 Bignum bignum_rshift(Bignum a, int shift)
655 int i, shiftw, shiftb, shiftbb, bits;
658 bits = bignum_bitcount(a) - shift;
659 ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);
662 shiftw = shift / BIGNUM_INT_BITS;
663 shiftb = shift % BIGNUM_INT_BITS;
664 shiftbb = BIGNUM_INT_BITS - shiftb;
667 for (i = 1; i <= ret[0]; i++) {
669 ai1 = (i + shiftw + 1 <= a[0] ? a[i + shiftw + 1] : 0);
670 ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK;
678 * Non-modular multiplication and addition.
680 Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)
682 int alen = a[0], blen = b[0];
683 int mlen = (alen > blen ? alen : blen);
684 int rlen, i, maxspot;
685 BignumInt *workspace;
688 /* mlen space for a, mlen space for b, 2*mlen for result */
689 workspace = snewn(mlen * 4, BignumInt);
690 for (i = 0; i < mlen; i++) {
691 workspace[0 * mlen + i] = (mlen - i <= a[0] ? a[mlen - i] : 0);
692 workspace[1 * mlen + i] = (mlen - i <= b[0] ? b[mlen - i] : 0);
695 internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
696 workspace + 2 * mlen, mlen);
698 /* now just copy the result back */
699 rlen = alen + blen + 1;
700 if (addend && rlen <= addend[0])
701 rlen = addend[0] + 1;
704 for (i = 1; i <= ret[0]; i++) {
705 ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);
711 /* now add in the addend, if any */
713 BignumDblInt carry = 0;
714 for (i = 1; i <= rlen; i++) {
715 carry += (i <= ret[0] ? ret[i] : 0);
716 carry += (i <= addend[0] ? addend[i] : 0);
717 ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
718 carry >>= BIGNUM_INT_BITS;
719 if (ret[i] != 0 && i > maxspot)
729 * Non-modular multiplication.
731 Bignum bigmul(Bignum a, Bignum b)
733 return bigmuladd(a, b, NULL);
737 * Create a bignum which is the bitmask covering another one. That
738 * is, the smallest integer which is >= N and is also one less than
741 Bignum bignum_bitmask(Bignum n)
743 Bignum ret = copybn(n);
748 while (n[i] == 0 && i > 0)
751 return ret; /* input was zero */
757 ret[i] = BIGNUM_INT_MASK;
762 * Convert a (max 32-bit) long into a bignum.
764 Bignum bignum_from_long(unsigned long nn)
770 ret[1] = (BignumInt)(n & BIGNUM_INT_MASK);
771 ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK);
773 ret[0] = (ret[2] ? 2 : 1);
778 * Add a long to a bignum.
780 Bignum bignum_add_long(Bignum number, unsigned long addendx)
782 Bignum ret = newbn(number[0] + 1);
784 BignumDblInt carry = 0, addend = addendx;
786 for (i = 1; i <= ret[0]; i++) {
787 carry += addend & BIGNUM_INT_MASK;
788 carry += (i <= number[0] ? number[i] : 0);
789 addend >>= BIGNUM_INT_BITS;
790 ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
791 carry >>= BIGNUM_INT_BITS;
800 * Compute the residue of a bignum, modulo a (max 16-bit) short.
802 unsigned short bignum_mod_short(Bignum number, unsigned short modulus)
809 for (i = number[0]; i > 0; i--)
810 r = (r * 65536 + number[i]) % mod;
811 return (unsigned short) r;
815 void diagbn(char *prefix, Bignum md)
817 int i, nibbles, morenibbles;
818 static const char hex[] = "0123456789ABCDEF";
820 debug(("%s0x", prefix ? prefix : ""));
822 nibbles = (3 + bignum_bitcount(md)) / 4;
825 morenibbles = 4 * md[0] - nibbles;
826 for (i = 0; i < morenibbles; i++)
828 for (i = nibbles; i--;)
830 hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));
840 Bignum bigdiv(Bignum a, Bignum b)
842 Bignum q = newbn(a[0]);
843 bigdivmod(a, b, NULL, q);
850 Bignum bigmod(Bignum a, Bignum b)
852 Bignum r = newbn(b[0]);
853 bigdivmod(a, b, r, NULL);
858 * Greatest common divisor.
860 Bignum biggcd(Bignum av, Bignum bv)
862 Bignum a = copybn(av);
863 Bignum b = copybn(bv);
865 while (bignum_cmp(b, Zero) != 0) {
866 Bignum t = newbn(b[0]);
867 bigdivmod(a, b, t, NULL);
868 while (t[0] > 1 && t[t[0]] == 0)
880 * Modular inverse, using Euclid's extended algorithm.
882 Bignum modinv(Bignum number, Bignum modulus)
884 Bignum a = copybn(modulus);
885 Bignum b = copybn(number);
886 Bignum xp = copybn(Zero);
887 Bignum x = copybn(One);
890 while (bignum_cmp(b, One) != 0) {
891 Bignum t = newbn(b[0]);
892 Bignum q = newbn(a[0]);
893 bigdivmod(a, b, t, q);
894 while (t[0] > 1 && t[t[0]] == 0)
901 x = bigmuladd(q, xp, t);
910 /* now we know that sign * x == 1, and that x < modulus */
912 /* set a new x to be modulus - x */
913 Bignum newx = newbn(modulus[0]);
918 for (i = 1; i <= newx[0]; i++) {
919 BignumInt aword = (i <= modulus[0] ? modulus[i] : 0);
920 BignumInt bword = (i <= x[0] ? x[i] : 0);
921 newx[i] = aword - bword - carry;
923 carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
937 * Render a bignum into decimal. Return a malloced string holding
938 * the decimal representation.
940 char *bignum_decimal(Bignum x)
946 BignumInt *workspace;
949 * First, estimate the number of digits. Since log(10)/log(2)
950 * is just greater than 93/28 (the joys of continued fraction
951 * approximations...) we know that for every 93 bits, we need
952 * at most 28 digits. This will tell us how much to malloc.
954 * Formally: if x has i bits, that means x is strictly less
955 * than 2^i. Since 2 is less than 10^(28/93), this is less than
956 * 10^(28i/93). We need an integer power of ten, so we must
957 * round up (rounding down might make it less than x again).
958 * Therefore if we multiply the bit count by 28/93, rounding
959 * up, we will have enough digits.
961 i = bignum_bitcount(x);
962 ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */
963 ndigits++; /* allow for trailing \0 */
964 ret = snewn(ndigits, char);
967 * Now allocate some workspace to hold the binary form as we
968 * repeatedly divide it by ten. Initialise this to the
969 * big-endian form of the number.
971 workspace = snewn(x[0], BignumInt);
972 for (i = 0; i < x[0]; i++)
973 workspace[i] = x[x[0] - i];
976 * Next, write the decimal number starting with the last digit.
977 * We use ordinary short division, dividing 10 into the
980 ndigit = ndigits - 1;
985 for (i = 0; i < x[0]; i++) {
986 carry = (carry << BIGNUM_INT_BITS) + workspace[i];
987 workspace[i] = (BignumInt) (carry / 10);
992 ret[--ndigit] = (char) (carry + '0');
996 * There's a chance we've fallen short of the start of the
997 * string. Correct if so.
1000 memmove(ret, ret + ndigit, ndigits - ndigit);