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 /* 64-bit architectures can do 32x32->64 chunks at a time */
56 typedef unsigned int BignumInt;
57 typedef unsigned long BignumDblInt;
58 #define BIGNUM_INT_MASK 0xFFFFFFFFU
59 #define BIGNUM_TOP_BIT 0x80000000U
60 #define BIGNUM_INT_BITS 32
61 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
62 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
63 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
68 /* 64-bit architectures in which unsigned long is 32 bits, not 64 */
69 typedef unsigned long BignumInt;
70 typedef unsigned long long BignumDblInt;
71 #define BIGNUM_INT_MASK 0xFFFFFFFFUL
72 #define BIGNUM_TOP_BIT 0x80000000UL
73 #define BIGNUM_INT_BITS 32
74 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
75 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
76 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
81 /* Fallback for all other cases */
82 typedef unsigned short BignumInt;
83 typedef unsigned long BignumDblInt;
84 #define BIGNUM_INT_MASK 0xFFFFU
85 #define BIGNUM_TOP_BIT 0x8000U
86 #define BIGNUM_INT_BITS 16
87 #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
88 #define DIVMOD_WORD(q, r, hi, lo, w) do { \
89 BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
95 #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
97 #define BIGNUM_INTERNAL
98 typedef BignumInt *Bignum;
102 BignumInt bnZero[1] = { 0 };
103 BignumInt bnOne[2] = { 1, 1 };
106 * The Bignum format is an array of `BignumInt'. The first
107 * element of the array counts the remaining elements. The
108 * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
109 * significant digit first. (So it's trivial to extract the bit
110 * with value 2^n for any n.)
112 * All Bignums in this module are positive. Negative numbers must
113 * be dealt with outside it.
115 * INVARIANT: the most significant word of any Bignum must be
119 Bignum Zero = bnZero, One = bnOne;
121 static Bignum newbn(int length)
123 Bignum b = snewn(length + 1, BignumInt);
126 memset(b, 0, (length + 1) * sizeof(*b));
131 void bn_restore_invariant(Bignum b)
133 while (b[0] > 1 && b[b[0]] == 0)
137 Bignum copybn(Bignum orig)
139 Bignum b = snewn(orig[0] + 1, BignumInt);
142 memcpy(b, orig, (orig[0] + 1) * sizeof(*b));
146 void freebn(Bignum b)
149 * Burn the evidence, just in case.
151 memset(b, 0, sizeof(b[0]) * (b[0] + 1));
155 Bignum bn_power_2(int n)
157 Bignum ret = newbn(n / BIGNUM_INT_BITS + 1);
158 bignum_set_bit(ret, n, 1);
164 * Input is in the first len words of a and b.
165 * Result is returned in the first 2*len words of c.
167 static void internal_mul(BignumInt *a, BignumInt *b,
168 BignumInt *c, int len)
173 for (j = 0; j < 2 * len; j++)
176 for (i = len - 1; i >= 0; i--) {
178 for (j = len - 1; j >= 0; j--) {
179 t += MUL_WORD(a[i], (BignumDblInt) b[j]);
180 t += (BignumDblInt) c[i + j + 1];
181 c[i + j + 1] = (BignumInt) t;
182 t = t >> BIGNUM_INT_BITS;
184 c[i] = (BignumInt) t;
188 static void internal_add_shifted(BignumInt *number,
189 unsigned n, int shift)
191 int word = 1 + (shift / BIGNUM_INT_BITS);
192 int bshift = shift % BIGNUM_INT_BITS;
195 addend = (BignumDblInt)n << bshift;
198 addend += number[word];
199 number[word] = (BignumInt) addend & BIGNUM_INT_MASK;
200 addend >>= BIGNUM_INT_BITS;
207 * Input in first alen words of a and first mlen words of m.
208 * Output in first alen words of a
209 * (of which first alen-mlen words will be zero).
210 * The MSW of m MUST have its high bit set.
211 * Quotient is accumulated in the `quotient' array, which is a Bignum
212 * rather than the internal bigendian format. Quotient parts are shifted
213 * left by `qshift' before adding into quot.
215 static void internal_mod(BignumInt *a, int alen,
216 BignumInt *m, int mlen,
217 BignumInt *quot, int qshift)
229 for (i = 0; i <= alen - mlen; i++) {
231 unsigned int q, r, c, ai1;
245 /* Find q = h:a[i] / m0 */
250 * To illustrate it, suppose a BignumInt is 8 bits, and
251 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
252 * our initial division will be 0xA123 / 0xA1, which
253 * will give a quotient of 0x100 and a divide overflow.
254 * However, the invariants in this division algorithm
255 * are not violated, since the full number A1:23:... is
256 * _less_ than the quotient prefix A1:B2:... and so the
257 * following correction loop would have sorted it out.
259 * In this situation we set q to be the largest
260 * quotient we _can_ stomach (0xFF, of course).
264 /* Macro doesn't want an array subscript expression passed
265 * into it (see definition), so use a temporary. */
266 BignumInt tmplo = a[i];
267 DIVMOD_WORD(q, r, h, tmplo, m0);
269 /* Refine our estimate of q by looking at
270 h:a[i]:a[i+1] / m0:m1 */
272 if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) {
275 r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */
276 if (r >= (BignumDblInt) m0 &&
277 t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--;
281 /* Subtract q * m from a[i...] */
283 for (k = mlen - 1; k >= 0; k--) {
284 t = MUL_WORD(q, m[k]);
286 c = (unsigned)(t >> BIGNUM_INT_BITS);
287 if ((BignumInt) t > a[i + k])
289 a[i + k] -= (BignumInt) t;
292 /* Add back m in case of borrow */
295 for (k = mlen - 1; k >= 0; k--) {
298 a[i + k] = (BignumInt) t;
299 t = t >> BIGNUM_INT_BITS;
304 internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i));
309 * Compute (base ^ exp) % mod.
311 Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
313 BignumInt *a, *b, *n, *m;
319 * The most significant word of mod needs to be non-zero. It
320 * should already be, but let's make sure.
322 assert(mod[mod[0]] != 0);
325 * Make sure the base is smaller than the modulus, by reducing
326 * it modulo the modulus if not.
328 base = bigmod(base_in, mod);
330 /* Allocate m of size mlen, copy mod to m */
331 /* We use big endian internally */
333 m = snewn(mlen, BignumInt);
334 for (j = 0; j < mlen; j++)
335 m[j] = mod[mod[0] - j];
337 /* Shift m left to make msb bit set */
338 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
339 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
342 for (i = 0; i < mlen - 1; i++)
343 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
344 m[mlen - 1] = m[mlen - 1] << mshift;
347 /* Allocate n of size mlen, copy base to n */
348 n = snewn(mlen, BignumInt);
350 for (j = 0; j < i; j++)
352 for (j = 0; j < (int)base[0]; j++)
353 n[i + j] = base[base[0] - j];
355 /* Allocate a and b of size 2*mlen. Set a = 1 */
356 a = snewn(2 * mlen, BignumInt);
357 b = snewn(2 * mlen, BignumInt);
358 for (i = 0; i < 2 * mlen; i++)
362 /* Skip leading zero bits of exp. */
364 j = BIGNUM_INT_BITS-1;
365 while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
369 j = BIGNUM_INT_BITS-1;
373 /* Main computation */
374 while (i < (int)exp[0]) {
376 internal_mul(a + mlen, a + mlen, b, mlen);
377 internal_mod(b, mlen * 2, m, mlen, NULL, 0);
378 if ((exp[exp[0] - i] & (1 << j)) != 0) {
379 internal_mul(b + mlen, n, a, mlen);
380 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
390 j = BIGNUM_INT_BITS-1;
393 /* Fixup result in case the modulus was shifted */
395 for (i = mlen - 1; i < 2 * mlen - 1; i++)
396 a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
397 a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;
398 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
399 for (i = 2 * mlen - 1; i >= mlen; i--)
400 a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
403 /* Copy result to buffer */
404 result = newbn(mod[0]);
405 for (i = 0; i < mlen; i++)
406 result[result[0] - i] = a[i + mlen];
407 while (result[0] > 1 && result[result[0]] == 0)
410 /* Free temporary arrays */
411 for (i = 0; i < 2 * mlen; i++)
414 for (i = 0; i < 2 * mlen; i++)
417 for (i = 0; i < mlen; i++)
420 for (i = 0; i < mlen; i++)
430 * Compute (p * q) % mod.
431 * The most significant word of mod MUST be non-zero.
432 * We assume that the result array is the same size as the mod array.
434 Bignum modmul(Bignum p, Bignum q, Bignum mod)
436 BignumInt *a, *n, *m, *o;
438 int pqlen, mlen, rlen, i, j;
441 /* Allocate m of size mlen, copy mod to m */
442 /* We use big endian internally */
444 m = snewn(mlen, BignumInt);
445 for (j = 0; j < mlen; j++)
446 m[j] = mod[mod[0] - j];
448 /* Shift m left to make msb bit set */
449 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
450 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
453 for (i = 0; i < mlen - 1; i++)
454 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
455 m[mlen - 1] = m[mlen - 1] << mshift;
458 pqlen = (p[0] > q[0] ? p[0] : q[0]);
460 /* Allocate n of size pqlen, copy p to n */
461 n = snewn(pqlen, BignumInt);
463 for (j = 0; j < i; j++)
465 for (j = 0; j < (int)p[0]; j++)
466 n[i + j] = p[p[0] - j];
468 /* Allocate o of size pqlen, copy q to o */
469 o = snewn(pqlen, BignumInt);
471 for (j = 0; j < i; j++)
473 for (j = 0; j < (int)q[0]; j++)
474 o[i + j] = q[q[0] - j];
476 /* Allocate a of size 2*pqlen for result */
477 a = snewn(2 * pqlen, BignumInt);
479 /* Main computation */
480 internal_mul(n, o, a, pqlen);
481 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
483 /* Fixup result in case the modulus was shifted */
485 for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
486 a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
487 a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
488 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
489 for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
490 a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
493 /* Copy result to buffer */
494 rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
495 result = newbn(rlen);
496 for (i = 0; i < rlen; i++)
497 result[result[0] - i] = a[i + 2 * pqlen - rlen];
498 while (result[0] > 1 && result[result[0]] == 0)
501 /* Free temporary arrays */
502 for (i = 0; i < 2 * pqlen; i++)
505 for (i = 0; i < mlen; i++)
508 for (i = 0; i < pqlen; i++)
511 for (i = 0; i < pqlen; i++)
520 * The most significant word of mod MUST be non-zero.
521 * We assume that the result array is the same size as the mod array.
522 * We optionally write out a quotient if `quotient' is non-NULL.
523 * We can avoid writing out the result if `result' is NULL.
525 static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
529 int plen, mlen, i, j;
531 /* Allocate m of size mlen, copy mod to m */
532 /* We use big endian internally */
534 m = snewn(mlen, BignumInt);
535 for (j = 0; j < mlen; j++)
536 m[j] = mod[mod[0] - j];
538 /* Shift m left to make msb bit set */
539 for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
540 if ((m[0] << mshift) & BIGNUM_TOP_BIT)
543 for (i = 0; i < mlen - 1; i++)
544 m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
545 m[mlen - 1] = m[mlen - 1] << mshift;
549 /* Ensure plen > mlen */
553 /* Allocate n of size plen, copy p to n */
554 n = snewn(plen, BignumInt);
555 for (j = 0; j < plen; j++)
557 for (j = 1; j <= (int)p[0]; j++)
560 /* Main computation */
561 internal_mod(n, plen, m, mlen, quotient, mshift);
563 /* Fixup result in case the modulus was shifted */
565 for (i = plen - mlen - 1; i < plen - 1; i++)
566 n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift));
567 n[plen - 1] = n[plen - 1] << mshift;
568 internal_mod(n, plen, m, mlen, quotient, 0);
569 for (i = plen - 1; i >= plen - mlen; i--)
570 n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift));
573 /* Copy result to buffer */
575 for (i = 1; i <= (int)result[0]; i++) {
577 result[i] = j >= 0 ? n[j] : 0;
581 /* Free temporary arrays */
582 for (i = 0; i < mlen; i++)
585 for (i = 0; i < plen; i++)
591 * Decrement a number.
593 void decbn(Bignum bn)
596 while (i < (int)bn[0] && bn[i] == 0)
597 bn[i++] = BIGNUM_INT_MASK;
601 Bignum bignum_from_bytes(const unsigned char *data, int nbytes)
606 w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */
609 for (i = 1; i <= w; i++)
611 for (i = nbytes; i--;) {
612 unsigned char byte = *data++;
613 result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS);
616 while (result[0] > 1 && result[result[0]] == 0)
622 * Read an SSH-1-format bignum from a data buffer. Return the number
623 * of bytes consumed, or -1 if there wasn't enough data.
625 int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result)
627 const unsigned char *p = data;
635 for (i = 0; i < 2; i++)
637 b = (w + 7) / 8; /* bits -> bytes */
642 if (!result) /* just return length */
645 *result = bignum_from_bytes(p, b);
651 * Return the bit count of a bignum, for SSH-1 encoding.
653 int bignum_bitcount(Bignum bn)
655 int bitcount = bn[0] * BIGNUM_INT_BITS - 1;
657 && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--;
662 * Return the byte length of a bignum when SSH-1 encoded.
664 int ssh1_bignum_length(Bignum bn)
666 return 2 + (bignum_bitcount(bn) + 7) / 8;
670 * Return the byte length of a bignum when SSH-2 encoded.
672 int ssh2_bignum_length(Bignum bn)
674 return 4 + (bignum_bitcount(bn) + 8) / 8;
678 * Return a byte from a bignum; 0 is least significant, etc.
680 int bignum_byte(Bignum bn, int i)
682 if (i >= (int)(BIGNUM_INT_BYTES * bn[0]))
683 return 0; /* beyond the end */
685 return (bn[i / BIGNUM_INT_BYTES + 1] >>
686 ((i % BIGNUM_INT_BYTES)*8)) & 0xFF;
690 * Return a bit from a bignum; 0 is least significant, etc.
692 int bignum_bit(Bignum bn, int i)
694 if (i >= (int)(BIGNUM_INT_BITS * bn[0]))
695 return 0; /* beyond the end */
697 return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1;
701 * Set a bit in a bignum; 0 is least significant, etc.
703 void bignum_set_bit(Bignum bn, int bitnum, int value)
705 if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0]))
706 abort(); /* beyond the end */
708 int v = bitnum / BIGNUM_INT_BITS + 1;
709 int mask = 1 << (bitnum % BIGNUM_INT_BITS);
718 * Write a SSH-1-format bignum into a buffer. It is assumed the
719 * buffer is big enough. Returns the number of bytes used.
721 int ssh1_write_bignum(void *data, Bignum bn)
723 unsigned char *p = data;
724 int len = ssh1_bignum_length(bn);
726 int bitc = bignum_bitcount(bn);
728 *p++ = (bitc >> 8) & 0xFF;
729 *p++ = (bitc) & 0xFF;
730 for (i = len - 2; i--;)
731 *p++ = bignum_byte(bn, i);
736 * Compare two bignums. Returns like strcmp.
738 int bignum_cmp(Bignum a, Bignum b)
740 int amax = a[0], bmax = b[0];
741 int i = (amax > bmax ? amax : bmax);
743 BignumInt aval = (i > amax ? 0 : a[i]);
744 BignumInt bval = (i > bmax ? 0 : b[i]);
755 * Right-shift one bignum to form another.
757 Bignum bignum_rshift(Bignum a, int shift)
760 int i, shiftw, shiftb, shiftbb, bits;
763 bits = bignum_bitcount(a) - shift;
764 ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);
767 shiftw = shift / BIGNUM_INT_BITS;
768 shiftb = shift % BIGNUM_INT_BITS;
769 shiftbb = BIGNUM_INT_BITS - shiftb;
772 for (i = 1; i <= (int)ret[0]; i++) {
774 ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0);
775 ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK;
783 * Non-modular multiplication and addition.
785 Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)
787 int alen = a[0], blen = b[0];
788 int mlen = (alen > blen ? alen : blen);
789 int rlen, i, maxspot;
790 BignumInt *workspace;
793 /* mlen space for a, mlen space for b, 2*mlen for result */
794 workspace = snewn(mlen * 4, BignumInt);
795 for (i = 0; i < mlen; i++) {
796 workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0);
797 workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0);
800 internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
801 workspace + 2 * mlen, mlen);
803 /* now just copy the result back */
804 rlen = alen + blen + 1;
805 if (addend && rlen <= (int)addend[0])
806 rlen = addend[0] + 1;
809 for (i = 1; i <= (int)ret[0]; i++) {
810 ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);
816 /* now add in the addend, if any */
818 BignumDblInt carry = 0;
819 for (i = 1; i <= rlen; i++) {
820 carry += (i <= (int)ret[0] ? ret[i] : 0);
821 carry += (i <= (int)addend[0] ? addend[i] : 0);
822 ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
823 carry >>= BIGNUM_INT_BITS;
824 if (ret[i] != 0 && i > maxspot)
835 * Non-modular multiplication.
837 Bignum bigmul(Bignum a, Bignum b)
839 return bigmuladd(a, b, NULL);
843 * Create a bignum which is the bitmask covering another one. That
844 * is, the smallest integer which is >= N and is also one less than
847 Bignum bignum_bitmask(Bignum n)
849 Bignum ret = copybn(n);
854 while (n[i] == 0 && i > 0)
857 return ret; /* input was zero */
863 ret[i] = BIGNUM_INT_MASK;
868 * Convert a (max 32-bit) long into a bignum.
870 Bignum bignum_from_long(unsigned long nn)
876 ret[1] = (BignumInt)(n & BIGNUM_INT_MASK);
877 ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK);
879 ret[0] = (ret[2] ? 2 : 1);
884 * Add a long to a bignum.
886 Bignum bignum_add_long(Bignum number, unsigned long addendx)
888 Bignum ret = newbn(number[0] + 1);
890 BignumDblInt carry = 0, addend = addendx;
892 for (i = 1; i <= (int)ret[0]; i++) {
893 carry += addend & BIGNUM_INT_MASK;
894 carry += (i <= (int)number[0] ? number[i] : 0);
895 addend >>= BIGNUM_INT_BITS;
896 ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
897 carry >>= BIGNUM_INT_BITS;
906 * Compute the residue of a bignum, modulo a (max 16-bit) short.
908 unsigned short bignum_mod_short(Bignum number, unsigned short modulus)
915 for (i = number[0]; i > 0; i--)
916 r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod;
917 return (unsigned short) r;
921 void diagbn(char *prefix, Bignum md)
923 int i, nibbles, morenibbles;
924 static const char hex[] = "0123456789ABCDEF";
926 debug(("%s0x", prefix ? prefix : ""));
928 nibbles = (3 + bignum_bitcount(md)) / 4;
931 morenibbles = 4 * md[0] - nibbles;
932 for (i = 0; i < morenibbles; i++)
934 for (i = nibbles; i--;)
936 hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));
946 Bignum bigdiv(Bignum a, Bignum b)
948 Bignum q = newbn(a[0]);
949 bigdivmod(a, b, NULL, q);
956 Bignum bigmod(Bignum a, Bignum b)
958 Bignum r = newbn(b[0]);
959 bigdivmod(a, b, r, NULL);
964 * Greatest common divisor.
966 Bignum biggcd(Bignum av, Bignum bv)
968 Bignum a = copybn(av);
969 Bignum b = copybn(bv);
971 while (bignum_cmp(b, Zero) != 0) {
972 Bignum t = newbn(b[0]);
973 bigdivmod(a, b, t, NULL);
974 while (t[0] > 1 && t[t[0]] == 0)
986 * Modular inverse, using Euclid's extended algorithm.
988 Bignum modinv(Bignum number, Bignum modulus)
990 Bignum a = copybn(modulus);
991 Bignum b = copybn(number);
992 Bignum xp = copybn(Zero);
993 Bignum x = copybn(One);
996 while (bignum_cmp(b, One) != 0) {
997 Bignum t = newbn(b[0]);
998 Bignum q = newbn(a[0]);
999 bigdivmod(a, b, t, q);
1000 while (t[0] > 1 && t[t[0]] == 0)
1007 x = bigmuladd(q, xp, t);
1017 /* now we know that sign * x == 1, and that x < modulus */
1019 /* set a new x to be modulus - x */
1020 Bignum newx = newbn(modulus[0]);
1021 BignumInt carry = 0;
1025 for (i = 1; i <= (int)newx[0]; i++) {
1026 BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0);
1027 BignumInt bword = (i <= (int)x[0] ? x[i] : 0);
1028 newx[i] = aword - bword - carry;
1030 carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
1044 * Render a bignum into decimal. Return a malloced string holding
1045 * the decimal representation.
1047 char *bignum_decimal(Bignum x)
1049 int ndigits, ndigit;
1053 BignumInt *workspace;
1056 * First, estimate the number of digits. Since log(10)/log(2)
1057 * is just greater than 93/28 (the joys of continued fraction
1058 * approximations...) we know that for every 93 bits, we need
1059 * at most 28 digits. This will tell us how much to malloc.
1061 * Formally: if x has i bits, that means x is strictly less
1062 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1063 * 10^(28i/93). We need an integer power of ten, so we must
1064 * round up (rounding down might make it less than x again).
1065 * Therefore if we multiply the bit count by 28/93, rounding
1066 * up, we will have enough digits.
1068 * i=0 (i.e., x=0) is an irritating special case.
1070 i = bignum_bitcount(x);
1072 ndigits = 1; /* x = 0 */
1074 ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */
1075 ndigits++; /* allow for trailing \0 */
1076 ret = snewn(ndigits, char);
1079 * Now allocate some workspace to hold the binary form as we
1080 * repeatedly divide it by ten. Initialise this to the
1081 * big-endian form of the number.
1083 workspace = snewn(x[0], BignumInt);
1084 for (i = 0; i < (int)x[0]; i++)
1085 workspace[i] = x[x[0] - i];
1088 * Next, write the decimal number starting with the last digit.
1089 * We use ordinary short division, dividing 10 into the
1092 ndigit = ndigits - 1;
1097 for (i = 0; i < (int)x[0]; i++) {
1098 carry = (carry << BIGNUM_INT_BITS) + workspace[i];
1099 workspace[i] = (BignumInt) (carry / 10);
1104 ret[--ndigit] = (char) (carry + '0');
1108 * There's a chance we've fallen short of the start of the
1109 * string. Correct if so.
1112 memmove(ret, ret + ndigit, ndigits - ndigit);