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)
43 Bignum b = smalloc((length + 1) * sizeof(unsigned short));
46 memset(b, 0, (length + 1) * sizeof(*b));
51 void bn_restore_invariant(Bignum b)
53 while (b[0] > 1 && b[b[0]] == 0)
57 Bignum copybn(Bignum orig)
59 Bignum b = smalloc((orig[0] + 1) * sizeof(unsigned short));
62 memcpy(b, orig, (orig[0] + 1) * sizeof(*b));
69 * Burn the evidence, just in case.
71 memset(b, 0, sizeof(b[0]) * (b[0] + 1));
75 Bignum bn_power_2(int n)
77 Bignum ret = newbn(n / 16 + 1);
78 bignum_set_bit(ret, n, 1);
84 * Input is in the first len words of a and b.
85 * Result is returned in the first 2*len words of c.
87 static void internal_mul(unsigned short *a, unsigned short *b,
88 unsigned short *c, int len)
93 for (j = 0; j < 2 * len; j++)
96 for (i = len - 1; i >= 0; i--) {
99 for (j = len - 1; j >= 0; j--) {
100 t += ai * (unsigned long) b[j];
101 t += (unsigned long) c[i + j + 1];
102 c[i + j + 1] = (unsigned short) t;
105 c[i] = (unsigned short) t;
109 static void internal_add_shifted(unsigned short *number,
110 unsigned n, int shift)
112 int word = 1 + (shift / 16);
113 int bshift = shift % 16;
114 unsigned long addend;
116 addend = n << bshift;
119 addend += number[word];
120 number[word] = (unsigned short) addend & 0xFFFF;
128 * Input in first alen words of a and first mlen words of m.
129 * Output in first alen words of a
130 * (of which first alen-mlen words will be zero).
131 * The MSW of m MUST have its high bit set.
132 * Quotient is accumulated in the `quotient' array, which is a Bignum
133 * rather than the internal bigendian format. Quotient parts are shifted
134 * left by `qshift' before adding into quot.
136 static void internal_mod(unsigned short *a, int alen,
137 unsigned short *m, int mlen,
138 unsigned short *quot, int qshift)
140 unsigned short m0, m1;
150 for (i = 0; i <= alen - mlen; i++) {
152 unsigned int q, r, c, ai1;
166 /* Find q = h:a[i] / m0 */
167 t = ((unsigned long) h << 16) + a[i];
171 /* Refine our estimate of q by looking at
172 h:a[i]:a[i+1] / m0:m1 */
173 t = (long) m1 *(long) q;
174 if (t > ((unsigned long) r << 16) + ai1) {
177 r = (r + m0) & 0xffff; /* overflow? */
178 if (r >= (unsigned long) m0 &&
179 t > ((unsigned long) r << 16) + ai1) q--;
182 /* Subtract q * m from a[i...] */
184 for (k = mlen - 1; k >= 0; k--) {
185 t = (long) q *(long) m[k];
188 if ((unsigned short) t > a[i + k])
190 a[i + k] -= (unsigned short) t;
193 /* Add back m in case of borrow */
196 for (k = mlen - 1; k >= 0; k--) {
199 a[i + k] = (unsigned short) t;
205 internal_add_shifted(quot, q, qshift + 16 * (alen - mlen - i));
210 * Compute (base ^ exp) % mod.
211 * The base MUST be smaller than the modulus.
212 * The most significant word of mod MUST be non-zero.
213 * We assume that the result array is the same size as the mod array.
215 Bignum modpow(Bignum base, Bignum exp, Bignum mod)
217 unsigned short *a, *b, *n, *m;
222 /* Allocate m of size mlen, copy mod to m */
223 /* We use big endian internally */
225 m = smalloc(mlen * sizeof(unsigned short));
226 for (j = 0; j < mlen; j++)
227 m[j] = mod[mod[0] - j];
229 /* Shift m left to make msb bit set */
230 for (mshift = 0; mshift < 15; mshift++)
231 if ((m[0] << mshift) & 0x8000)
234 for (i = 0; i < mlen - 1; i++)
235 m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - mshift));
236 m[mlen - 1] = m[mlen - 1] << mshift;
239 /* Allocate n of size mlen, copy base to n */
240 n = smalloc(mlen * sizeof(unsigned short));
242 for (j = 0; j < i; j++)
244 for (j = 0; j < base[0]; j++)
245 n[i + j] = base[base[0] - j];
247 /* Allocate a and b of size 2*mlen. Set a = 1 */
248 a = smalloc(2 * mlen * sizeof(unsigned short));
249 b = smalloc(2 * mlen * sizeof(unsigned short));
250 for (i = 0; i < 2 * mlen; i++)
254 /* Skip leading zero bits of exp. */
257 while (i < exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
265 /* Main computation */
268 internal_mul(a + mlen, a + mlen, b, mlen);
269 internal_mod(b, mlen * 2, m, mlen, NULL, 0);
270 if ((exp[exp[0] - i] & (1 << j)) != 0) {
271 internal_mul(b + mlen, n, a, mlen);
272 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
285 /* Fixup result in case the modulus was shifted */
287 for (i = mlen - 1; i < 2 * mlen - 1; i++)
288 a[i] = (a[i] << mshift) | (a[i + 1] >> (16 - mshift));
289 a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;
290 internal_mod(a, mlen * 2, m, mlen, NULL, 0);
291 for (i = 2 * mlen - 1; i >= mlen; i--)
292 a[i] = (a[i] >> mshift) | (a[i - 1] << (16 - mshift));
295 /* Copy result to buffer */
296 result = newbn(mod[0]);
297 for (i = 0; i < mlen; i++)
298 result[result[0] - i] = a[i + mlen];
299 while (result[0] > 1 && result[result[0]] == 0)
302 /* Free temporary arrays */
303 for (i = 0; i < 2 * mlen; i++)
306 for (i = 0; i < 2 * mlen; i++)
309 for (i = 0; i < mlen; i++)
312 for (i = 0; i < mlen; i++)
320 * Compute (p * q) % mod.
321 * The most significant word of mod MUST be non-zero.
322 * We assume that the result array is the same size as the mod array.
324 Bignum modmul(Bignum p, Bignum q, Bignum mod)
326 unsigned short *a, *n, *m, *o;
328 int pqlen, mlen, rlen, i, j;
331 /* Allocate m of size mlen, copy mod to m */
332 /* We use big endian internally */
334 m = smalloc(mlen * sizeof(unsigned short));
335 for (j = 0; j < mlen; j++)
336 m[j] = mod[mod[0] - j];
338 /* Shift m left to make msb bit set */
339 for (mshift = 0; mshift < 15; mshift++)
340 if ((m[0] << mshift) & 0x8000)
343 for (i = 0; i < mlen - 1; i++)
344 m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - mshift));
345 m[mlen - 1] = m[mlen - 1] << mshift;
348 pqlen = (p[0] > q[0] ? p[0] : q[0]);
350 /* Allocate n of size pqlen, copy p to n */
351 n = smalloc(pqlen * sizeof(unsigned short));
353 for (j = 0; j < i; j++)
355 for (j = 0; j < p[0]; j++)
356 n[i + j] = p[p[0] - j];
358 /* Allocate o of size pqlen, copy q to o */
359 o = smalloc(pqlen * sizeof(unsigned short));
361 for (j = 0; j < i; j++)
363 for (j = 0; j < q[0]; j++)
364 o[i + j] = q[q[0] - j];
366 /* Allocate a of size 2*pqlen for result */
367 a = smalloc(2 * pqlen * sizeof(unsigned short));
369 /* Main computation */
370 internal_mul(n, o, a, pqlen);
371 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
373 /* Fixup result in case the modulus was shifted */
375 for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
376 a[i] = (a[i] << mshift) | (a[i + 1] >> (16 - mshift));
377 a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
378 internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
379 for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
380 a[i] = (a[i] >> mshift) | (a[i - 1] << (16 - mshift));
383 /* Copy result to buffer */
384 rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
385 result = newbn(rlen);
386 for (i = 0; i < rlen; i++)
387 result[result[0] - i] = a[i + 2 * pqlen - rlen];
388 while (result[0] > 1 && result[result[0]] == 0)
391 /* Free temporary arrays */
392 for (i = 0; i < 2 * pqlen; i++)
395 for (i = 0; i < mlen; i++)
398 for (i = 0; i < pqlen; i++)
401 for (i = 0; i < pqlen; i++)
410 * The most significant word of mod MUST be non-zero.
411 * We assume that the result array is the same size as the mod array.
412 * We optionally write out a quotient.
414 void bigmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
416 unsigned short *n, *m;
418 int plen, mlen, i, j;
420 /* Allocate m of size mlen, copy mod to m */
421 /* We use big endian internally */
423 m = smalloc(mlen * sizeof(unsigned short));
424 for (j = 0; j < mlen; j++)
425 m[j] = mod[mod[0] - j];
427 /* Shift m left to make msb bit set */
428 for (mshift = 0; mshift < 15; mshift++)
429 if ((m[0] << mshift) & 0x8000)
432 for (i = 0; i < mlen - 1; i++)
433 m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - mshift));
434 m[mlen - 1] = m[mlen - 1] << mshift;
438 /* Ensure plen > mlen */
442 /* Allocate n of size plen, copy p to n */
443 n = smalloc(plen * sizeof(unsigned short));
444 for (j = 0; j < plen; j++)
446 for (j = 1; j <= p[0]; j++)
449 /* Main computation */
450 internal_mod(n, plen, m, mlen, quotient, mshift);
452 /* Fixup result in case the modulus was shifted */
454 for (i = plen - mlen - 1; i < plen - 1; i++)
455 n[i] = (n[i] << mshift) | (n[i + 1] >> (16 - mshift));
456 n[plen - 1] = n[plen - 1] << mshift;
457 internal_mod(n, plen, m, mlen, quotient, 0);
458 for (i = plen - 1; i >= plen - mlen; i--)
459 n[i] = (n[i] >> mshift) | (n[i - 1] << (16 - mshift));
462 /* Copy result to buffer */
463 for (i = 1; i <= result[0]; i++) {
465 result[i] = j >= 0 ? n[j] : 0;
468 /* Free temporary arrays */
469 for (i = 0; i < mlen; i++)
472 for (i = 0; i < plen; i++)
478 * Decrement a number.
480 void decbn(Bignum bn)
483 while (i < bn[0] && bn[i] == 0)
488 Bignum bignum_from_bytes(unsigned char *data, int nbytes)
493 w = (nbytes + 1) / 2; /* bytes -> words */
496 for (i = 1; i <= w; i++)
498 for (i = nbytes; i--;) {
499 unsigned char byte = *data++;
501 result[1 + i / 2] |= byte << 8;
503 result[1 + i / 2] |= byte;
506 while (result[0] > 1 && result[result[0]] == 0)
512 * Read an ssh1-format bignum from a data buffer. Return the number
515 int ssh1_read_bignum(unsigned char *data, Bignum * result)
517 unsigned char *p = data;
522 for (i = 0; i < 2; i++)
524 b = (w + 7) / 8; /* bits -> bytes */
526 if (!result) /* just return length */
529 *result = bignum_from_bytes(p, b);
535 * Return the bit count of a bignum, for ssh1 encoding.
537 int bignum_bitcount(Bignum bn)
539 int bitcount = bn[0] * 16 - 1;
541 && (bn[bitcount / 16 + 1] >> (bitcount % 16)) == 0) bitcount--;
546 * Return the byte length of a bignum when ssh1 encoded.
548 int ssh1_bignum_length(Bignum bn)
550 return 2 + (bignum_bitcount(bn) + 7) / 8;
554 * Return the byte length of a bignum when ssh2 encoded.
556 int ssh2_bignum_length(Bignum bn)
558 return 4 + (bignum_bitcount(bn) + 8) / 8;
562 * Return a byte from a bignum; 0 is least significant, etc.
564 int bignum_byte(Bignum bn, int i)
567 return 0; /* beyond the end */
569 return (bn[i / 2 + 1] >> 8) & 0xFF;
571 return (bn[i / 2 + 1]) & 0xFF;
575 * Return a bit from a bignum; 0 is least significant, etc.
577 int bignum_bit(Bignum bn, int i)
580 return 0; /* beyond the end */
582 return (bn[i / 16 + 1] >> (i % 16)) & 1;
586 * Set a bit in a bignum; 0 is least significant, etc.
588 void bignum_set_bit(Bignum bn, int bitnum, int value)
590 if (bitnum >= 16 * bn[0])
591 abort(); /* beyond the end */
593 int v = bitnum / 16 + 1;
594 int mask = 1 << (bitnum % 16);
603 * Write a ssh1-format bignum into a buffer. It is assumed the
604 * buffer is big enough. Returns the number of bytes used.
606 int ssh1_write_bignum(void *data, Bignum bn)
608 unsigned char *p = data;
609 int len = ssh1_bignum_length(bn);
611 int bitc = bignum_bitcount(bn);
613 *p++ = (bitc >> 8) & 0xFF;
614 *p++ = (bitc) & 0xFF;
615 for (i = len - 2; i--;)
616 *p++ = bignum_byte(bn, i);
621 * Compare two bignums. Returns like strcmp.
623 int bignum_cmp(Bignum a, Bignum b)
625 int amax = a[0], bmax = b[0];
626 int i = (amax > bmax ? amax : bmax);
628 unsigned short aval = (i > amax ? 0 : a[i]);
629 unsigned short bval = (i > bmax ? 0 : b[i]);
640 * Right-shift one bignum to form another.
642 Bignum bignum_rshift(Bignum a, int shift)
645 int i, shiftw, shiftb, shiftbb, bits;
646 unsigned short ai, ai1;
648 bits = bignum_bitcount(a) - shift;
649 ret = newbn((bits + 15) / 16);
654 shiftbb = 16 - shiftb;
657 for (i = 1; i <= ret[0]; i++) {
659 ai1 = (i + shiftw + 1 <= a[0] ? a[i + shiftw + 1] : 0);
660 ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & 0xFFFF;
668 * Non-modular multiplication and addition.
670 Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)
672 int alen = a[0], blen = b[0];
673 int mlen = (alen > blen ? alen : blen);
674 int rlen, i, maxspot;
675 unsigned short *workspace;
678 /* mlen space for a, mlen space for b, 2*mlen for result */
679 workspace = smalloc(mlen * 4 * sizeof(unsigned short));
680 for (i = 0; i < mlen; i++) {
681 workspace[0 * mlen + i] = (mlen - i <= a[0] ? a[mlen - i] : 0);
682 workspace[1 * mlen + i] = (mlen - i <= b[0] ? b[mlen - i] : 0);
685 internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
686 workspace + 2 * mlen, mlen);
688 /* now just copy the result back */
689 rlen = alen + blen + 1;
690 if (addend && rlen <= addend[0])
691 rlen = addend[0] + 1;
694 for (i = 1; i <= ret[0]; i++) {
695 ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);
701 /* now add in the addend, if any */
703 unsigned long carry = 0;
704 for (i = 1; i <= rlen; i++) {
705 carry += (i <= ret[0] ? ret[i] : 0);
706 carry += (i <= addend[0] ? addend[i] : 0);
707 ret[i] = (unsigned short) carry & 0xFFFF;
709 if (ret[i] != 0 && i > maxspot)
719 * Non-modular multiplication.
721 Bignum bigmul(Bignum a, Bignum b)
723 return bigmuladd(a, b, NULL);
727 * Create a bignum which is the bitmask covering another one. That
728 * is, the smallest integer which is >= N and is also one less than
731 Bignum bignum_bitmask(Bignum n)
733 Bignum ret = copybn(n);
738 while (n[i] == 0 && i > 0)
741 return ret; /* input was zero */
752 * Convert a (max 16-bit) short into a bignum.
754 Bignum bignum_from_short(unsigned short n)
760 ret[2] = (n >> 16) & 0xFFFF;
761 ret[0] = (ret[2] ? 2 : 1);
766 * Add a long to a bignum.
768 Bignum bignum_add_long(Bignum number, unsigned long addend)
770 Bignum ret = newbn(number[0] + 1);
772 unsigned long carry = 0;
774 for (i = 1; i <= ret[0]; i++) {
775 carry += addend & 0xFFFF;
776 carry += (i <= number[0] ? number[i] : 0);
778 ret[i] = (unsigned short) carry & 0xFFFF;
788 * Compute the residue of a bignum, modulo a (max 16-bit) short.
790 unsigned short bignum_mod_short(Bignum number, unsigned short modulus)
792 unsigned long mod, r;
797 for (i = number[0]; i > 0; i--)
798 r = (r * 65536 + number[i]) % mod;
799 return (unsigned short) r;
802 void diagbn(char *prefix, Bignum md)
804 int i, nibbles, morenibbles;
805 static const char hex[] = "0123456789ABCDEF";
807 debugprint(("%s0x", prefix ? prefix : ""));
809 nibbles = (3 + bignum_bitcount(md)) / 4;
812 morenibbles = 4 * md[0] - nibbles;
813 for (i = 0; i < morenibbles; i++)
815 for (i = nibbles; i--;)
818 hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));
825 * Greatest common divisor.
827 Bignum biggcd(Bignum av, Bignum bv)
829 Bignum a = copybn(av);
830 Bignum b = copybn(bv);
834 while (bignum_cmp(b, Zero) != 0) {
835 Bignum t = newbn(b[0]);
836 bigmod(a, b, t, NULL);
838 while (t[0] > 1 && t[t[0]] == 0)
850 * Modular inverse, using Euclid's extended algorithm.
852 Bignum modinv(Bignum number, Bignum modulus)
854 Bignum a = copybn(modulus);
855 Bignum b = copybn(number);
856 Bignum xp = copybn(Zero);
857 Bignum x = copybn(One);
860 while (bignum_cmp(b, One) != 0) {
861 Bignum t = newbn(b[0]);
862 Bignum q = newbn(a[0]);
864 while (t[0] > 1 && t[t[0]] == 0)
871 x = bigmuladd(q, xp, t);
880 /* now we know that sign * x == 1, and that x < modulus */
882 /* set a new x to be modulus - x */
883 Bignum newx = newbn(modulus[0]);
884 unsigned short carry = 0;
888 for (i = 1; i <= newx[0]; i++) {
889 unsigned short aword = (i <= modulus[0] ? modulus[i] : 0);
890 unsigned short bword = (i <= x[0] ? x[i] : 0);
891 newx[i] = aword - bword - carry;
893 carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
907 * Render a bignum into decimal. Return a malloced string holding
908 * the decimal representation.
910 char *bignum_decimal(Bignum x)
916 unsigned short *workspace;
919 * First, estimate the number of digits. Since log(10)/log(2)
920 * is just greater than 93/28 (the joys of continued fraction
921 * approximations...) we know that for every 93 bits, we need
922 * at most 28 digits. This will tell us how much to malloc.
924 * Formally: if x has i bits, that means x is strictly less
925 * than 2^i. Since 2 is less than 10^(28/93), this is less than
926 * 10^(28i/93). We need an integer power of ten, so we must
927 * round up (rounding down might make it less than x again).
928 * Therefore if we multiply the bit count by 28/93, rounding
929 * up, we will have enough digits.
931 i = bignum_bitcount(x);
932 ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */
933 ndigits++; /* allow for trailing \0 */
934 ret = smalloc(ndigits);
937 * Now allocate some workspace to hold the binary form as we
938 * repeatedly divide it by ten. Initialise this to the
939 * big-endian form of the number.
941 workspace = smalloc(sizeof(unsigned short) * x[0]);
942 for (i = 0; i < x[0]; i++)
943 workspace[i] = x[x[0] - i];
946 * Next, write the decimal number starting with the last digit.
947 * We use ordinary short division, dividing 10 into the
950 ndigit = ndigits - 1;
955 for (i = 0; i < x[0]; i++) {
956 carry = (carry << 16) + workspace[i];
957 workspace[i] = (unsigned short) (carry / 10);
962 ret[--ndigit] = (char) (carry + '0');
966 * There's a chance we've fallen short of the start of the
967 * string. Correct if so.
970 memmove(ret, ret + ndigit, ndigits - ndigit);