2 * Bignum routines for RSA and DH and stuff.
11 unsigned short bnZero[1] = { 0 };
12 unsigned short bnOne[2] = { 1, 1 };
15 * The Bignum format is an array of `unsigned short'. The first
16 * element of the array counts the remaining elements. The
17 * remaining elements express the actual number, base 2^16, _least_
18 * significant digit first. (So it's trivial to extract the bit
19 * with value 2^n for any n.)
21 * All Bignums in this module are positive. Negative numbers must
22 * be dealt with outside it.
24 * INVARIANT: the most significant word of any Bignum must be
28 Bignum Zero = bnZero, One = bnOne;
30 Bignum newbn(int length) {
31 Bignum b = malloc((length+1)*sizeof(unsigned short));
34 memset(b, 0, (length+1)*sizeof(*b));
39 Bignum copybn(Bignum orig) {
40 Bignum b = malloc((orig[0]+1)*sizeof(unsigned short));
43 memcpy(b, orig, (orig[0]+1)*sizeof(*b));
47 void freebn(Bignum b) {
49 * Burn the evidence, just in case.
51 memset(b, 0, sizeof(b[0]) * (b[0] + 1));
57 * Input is in the first len words of a and b.
58 * Result is returned in the first 2*len words of c.
60 static void internal_mul(unsigned short *a, unsigned short *b,
61 unsigned short *c, int len)
66 for (j = 0; j < 2*len; j++)
69 for (i = len - 1; i >= 0; i--) {
72 for (j = len - 1; j >= 0; j--) {
73 t += ai * (unsigned long) b[j];
74 t += (unsigned long) c[i+j+1];
75 c[i+j+1] = (unsigned short)t;
78 c[i] = (unsigned short)t;
82 static void internal_add_shifted(unsigned short *number,
83 unsigned n, int shift) {
84 int word = 1 + (shift / 16);
85 int bshift = shift % 16;
91 addend += number[word];
92 number[word] = (unsigned short) addend & 0xFFFF;
100 * Input in first alen words of a and first mlen words of m.
101 * Output in first alen words of a
102 * (of which first alen-mlen words will be zero).
103 * The MSW of m MUST have its high bit set.
104 * Quotient is accumulated in the `quotient' array, which is a Bignum
105 * rather than the internal bigendian format. Quotient parts are shifted
106 * left by `qshift' before adding into quot.
108 static void internal_mod(unsigned short *a, int alen,
109 unsigned short *m, int mlen,
110 unsigned short *quot, int qshift)
112 unsigned short m0, m1;
122 for (i = 0; i <= alen-mlen; i++) {
124 unsigned int q, r, c, ai1;
138 /* Find q = h:a[i] / m0 */
139 t = ((unsigned long) h << 16) + a[i];
143 /* Refine our estimate of q by looking at
144 h:a[i]:a[i+1] / m0:m1 */
145 t = (long) m1 * (long) q;
146 if (t > ((unsigned long) r << 16) + ai1) {
149 r = (r + m0) & 0xffff; /* overflow? */
150 if (r >= (unsigned long)m0 &&
151 t > ((unsigned long) r << 16) + ai1)
155 /* Subtract q * m from a[i...] */
157 for (k = mlen - 1; k >= 0; k--) {
158 t = (long) q * (long) m[k];
161 if ((unsigned short) t > a[i+k]) c++;
162 a[i+k] -= (unsigned short) t;
165 /* Add back m in case of borrow */
168 for (k = mlen - 1; k >= 0; k--) {
171 a[i+k] = (unsigned short)t;
177 internal_add_shifted(quot, q, qshift + 16 * (alen-mlen-i));
182 * Compute (base ^ exp) % mod.
183 * The base MUST be smaller than the modulus.
184 * The most significant word of mod MUST be non-zero.
185 * We assume that the result array is the same size as the mod array.
187 void modpow(Bignum base, Bignum exp, Bignum mod, Bignum result)
189 unsigned short *a, *b, *n, *m;
193 /* Allocate m of size mlen, copy mod to m */
194 /* We use big endian internally */
196 m = malloc(mlen * sizeof(unsigned short));
197 for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j];
199 /* Shift m left to make msb bit set */
200 for (mshift = 0; mshift < 15; mshift++)
201 if ((m[0] << mshift) & 0x8000) break;
203 for (i = 0; i < mlen - 1; i++)
204 m[i] = (m[i] << mshift) | (m[i+1] >> (16-mshift));
205 m[mlen-1] = m[mlen-1] << mshift;
208 /* Allocate n of size mlen, copy base to n */
209 n = malloc(mlen * sizeof(unsigned short));
211 for (j = 0; j < i; j++) n[j] = 0;
212 for (j = 0; j < base[0]; j++) n[i+j] = base[base[0] - j];
214 /* Allocate a and b of size 2*mlen. Set a = 1 */
215 a = malloc(2 * mlen * sizeof(unsigned short));
216 b = malloc(2 * mlen * sizeof(unsigned short));
217 for (i = 0; i < 2*mlen; i++) a[i] = 0;
220 /* Skip leading zero bits of exp. */
222 while (i < exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
224 if (j < 0) { i++; j = 15; }
227 /* Main computation */
230 internal_mul(a + mlen, a + mlen, b, mlen);
231 internal_mod(b, mlen*2, m, mlen, NULL, 0);
232 if ((exp[exp[0] - i] & (1 << j)) != 0) {
233 internal_mul(b + mlen, n, a, mlen);
234 internal_mod(a, mlen*2, m, mlen, NULL, 0);
244 /* Fixup result in case the modulus was shifted */
246 for (i = mlen - 1; i < 2*mlen - 1; i++)
247 a[i] = (a[i] << mshift) | (a[i+1] >> (16-mshift));
248 a[2*mlen-1] = a[2*mlen-1] << mshift;
249 internal_mod(a, mlen*2, m, mlen, NULL, 0);
250 for (i = 2*mlen - 1; i >= mlen; i--)
251 a[i] = (a[i] >> mshift) | (a[i-1] << (16-mshift));
254 /* Copy result to buffer */
255 for (i = 0; i < mlen; i++)
256 result[result[0] - i] = a[i+mlen];
258 /* Free temporary arrays */
259 for (i = 0; i < 2*mlen; i++) a[i] = 0; free(a);
260 for (i = 0; i < 2*mlen; i++) b[i] = 0; free(b);
261 for (i = 0; i < mlen; i++) m[i] = 0; free(m);
262 for (i = 0; i < mlen; i++) n[i] = 0; free(n);
266 * Compute (p * q) % mod.
267 * The most significant word of mod MUST be non-zero.
268 * We assume that the result array is the same size as the mod array.
270 void modmul(Bignum p, Bignum q, Bignum mod, Bignum result)
272 unsigned short *a, *n, *m, *o;
274 int pqlen, mlen, i, j;
276 /* Allocate m of size mlen, copy mod to m */
277 /* We use big endian internally */
279 m = malloc(mlen * sizeof(unsigned short));
280 for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j];
282 /* Shift m left to make msb bit set */
283 for (mshift = 0; mshift < 15; mshift++)
284 if ((m[0] << mshift) & 0x8000) break;
286 for (i = 0; i < mlen - 1; i++)
287 m[i] = (m[i] << mshift) | (m[i+1] >> (16-mshift));
288 m[mlen-1] = m[mlen-1] << mshift;
291 pqlen = (p[0] > q[0] ? p[0] : q[0]);
293 /* Allocate n of size pqlen, copy p to n */
294 n = malloc(pqlen * sizeof(unsigned short));
296 for (j = 0; j < i; j++) n[j] = 0;
297 for (j = 0; j < p[0]; j++) n[i+j] = p[p[0] - j];
299 /* Allocate o of size pqlen, copy q to o */
300 o = malloc(pqlen * sizeof(unsigned short));
302 for (j = 0; j < i; j++) o[j] = 0;
303 for (j = 0; j < q[0]; j++) o[i+j] = q[q[0] - j];
305 /* Allocate a of size 2*pqlen for result */
306 a = malloc(2 * pqlen * sizeof(unsigned short));
308 /* Main computation */
309 internal_mul(n, o, a, pqlen);
310 internal_mod(a, pqlen*2, m, mlen, NULL, 0);
312 /* Fixup result in case the modulus was shifted */
314 for (i = 2*pqlen - mlen - 1; i < 2*pqlen - 1; i++)
315 a[i] = (a[i] << mshift) | (a[i+1] >> (16-mshift));
316 a[2*pqlen-1] = a[2*pqlen-1] << mshift;
317 internal_mod(a, pqlen*2, m, mlen, NULL, 0);
318 for (i = 2*pqlen - 1; i >= 2*pqlen - mlen; i--)
319 a[i] = (a[i] >> mshift) | (a[i-1] << (16-mshift));
322 /* Copy result to buffer */
323 for (i = 0; i < mlen; i++)
324 result[result[0] - i] = a[i+2*pqlen-mlen];
326 /* Free temporary arrays */
327 for (i = 0; i < 2*pqlen; i++) a[i] = 0; free(a);
328 for (i = 0; i < mlen; i++) m[i] = 0; free(m);
329 for (i = 0; i < pqlen; i++) n[i] = 0; free(n);
330 for (i = 0; i < pqlen; i++) o[i] = 0; free(o);
335 * The most significant word of mod MUST be non-zero.
336 * We assume that the result array is the same size as the mod array.
337 * We optionally write out a quotient.
339 void bigmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
341 unsigned short *n, *m;
343 int plen, mlen, i, j;
345 /* Allocate m of size mlen, copy mod to m */
346 /* We use big endian internally */
348 m = malloc(mlen * sizeof(unsigned short));
349 for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j];
351 /* Shift m left to make msb bit set */
352 for (mshift = 0; mshift < 15; mshift++)
353 if ((m[0] << mshift) & 0x8000) break;
355 for (i = 0; i < mlen - 1; i++)
356 m[i] = (m[i] << mshift) | (m[i+1] >> (16-mshift));
357 m[mlen-1] = m[mlen-1] << mshift;
361 /* Ensure plen > mlen */
362 if (plen <= mlen) plen = mlen+1;
364 /* Allocate n of size plen, copy p to n */
365 n = malloc(plen * sizeof(unsigned short));
366 for (j = 0; j < plen; j++) n[j] = 0;
367 for (j = 1; j <= p[0]; j++) n[plen-j] = p[j];
369 /* Main computation */
370 internal_mod(n, plen, m, mlen, quotient, mshift);
372 /* Fixup result in case the modulus was shifted */
374 for (i = plen - mlen - 1; i < plen - 1; i++)
375 n[i] = (n[i] << mshift) | (n[i+1] >> (16-mshift));
376 n[plen-1] = n[plen-1] << mshift;
377 internal_mod(n, plen, m, mlen, quotient, 0);
378 for (i = plen - 1; i >= plen - mlen; i--)
379 n[i] = (n[i] >> mshift) | (n[i-1] << (16-mshift));
382 /* Copy result to buffer */
383 for (i = 1; i <= result[0]; i++) {
385 result[i] = j>=0 ? n[j] : 0;
388 /* Free temporary arrays */
389 for (i = 0; i < mlen; i++) m[i] = 0; free(m);
390 for (i = 0; i < plen; i++) n[i] = 0; free(n);
394 * Decrement a number.
396 void decbn(Bignum bn) {
398 while (i < bn[0] && bn[i] == 0)
404 * Read an ssh1-format bignum from a data buffer. Return the number
407 int ssh1_read_bignum(unsigned char *data, Bignum *result) {
408 unsigned char *p = data;
417 b = (w+7)/8; /* bits -> bytes */
418 w = (w+15)/16; /* bits -> words */
420 if (!result) /* just return length */
428 unsigned char byte = *p++;
430 bn[1+i/2] |= byte<<8;
441 * Return the bit count of a bignum, for ssh1 encoding.
443 int ssh1_bignum_bitcount(Bignum bn) {
444 int bitcount = bn[0] * 16 - 1;
446 while (bitcount >= 0 && (bn[bitcount/16+1] >> (bitcount % 16)) == 0)
452 * Return the byte length of a bignum when ssh1 encoded.
454 int ssh1_bignum_length(Bignum bn) {
455 return 2 + (ssh1_bignum_bitcount(bn)+7)/8;
459 * Return a byte from a bignum; 0 is least significant, etc.
461 int bignum_byte(Bignum bn, int i) {
463 return 0; /* beyond the end */
465 return (bn[i/2+1] >> 8) & 0xFF;
467 return (bn[i/2+1] ) & 0xFF;
471 * Return a bit from a bignum; 0 is least significant, etc.
473 int bignum_bit(Bignum bn, int i) {
475 return 0; /* beyond the end */
477 return (bn[i/16+1] >> (i%16)) & 1;
481 * Set a bit in a bignum; 0 is least significant, etc.
483 void bignum_set_bit(Bignum bn, int bitnum, int value) {
484 if (bitnum >= 16*bn[0])
485 abort(); /* beyond the end */
488 int mask = 1 << (bitnum%16);
497 * Write a ssh1-format bignum into a buffer. It is assumed the
498 * buffer is big enough. Returns the number of bytes used.
500 int ssh1_write_bignum(void *data, Bignum bn) {
501 unsigned char *p = data;
502 int len = ssh1_bignum_length(bn);
504 int bitc = ssh1_bignum_bitcount(bn);
506 *p++ = (bitc >> 8) & 0xFF;
507 *p++ = (bitc ) & 0xFF;
508 for (i = len-2; i-- ;)
509 *p++ = bignum_byte(bn, i);
514 * Compare two bignums. Returns like strcmp.
516 int bignum_cmp(Bignum a, Bignum b) {
517 int amax = a[0], bmax = b[0];
518 int i = (amax > bmax ? amax : bmax);
520 unsigned short aval = (i > amax ? 0 : a[i]);
521 unsigned short bval = (i > bmax ? 0 : b[i]);
522 if (aval < bval) return -1;
523 if (aval > bval) return +1;
530 * Right-shift one bignum to form another.
532 Bignum bignum_rshift(Bignum a, int shift) {
534 int i, shiftw, shiftb, shiftbb, bits;
535 unsigned short ai, ai1;
537 bits = ssh1_bignum_bitcount(a) - shift;
538 ret = newbn((bits+15)/16);
543 shiftbb = 16 - shiftb;
546 for (i = 1; i <= ret[0]; i++) {
548 ai1 = (i+shiftw+1 <= a[0] ? a[i+shiftw+1] : 0);
549 ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & 0xFFFF;
557 * Non-modular multiplication and addition.
559 Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) {
560 int alen = a[0], blen = b[0];
561 int mlen = (alen > blen ? alen : blen);
562 int rlen, i, maxspot;
563 unsigned short *workspace;
566 /* mlen space for a, mlen space for b, 2*mlen for result */
567 workspace = malloc(mlen * 4 * sizeof(unsigned short));
568 for (i = 0; i < mlen; i++) {
569 workspace[0*mlen + i] = (mlen-i <= a[0] ? a[mlen-i] : 0);
570 workspace[1*mlen + i] = (mlen-i <= b[0] ? b[mlen-i] : 0);
573 internal_mul(workspace+0*mlen, workspace+1*mlen, workspace+2*mlen, mlen);
575 /* now just copy the result back */
576 rlen = alen + blen + 1;
577 if (addend && rlen <= addend[0])
578 rlen = addend[0] + 1;
581 for (i = 1; i <= ret[0]; i++) {
582 ret[i] = (i <= 2*mlen ? workspace[4*mlen - i] : 0);
588 /* now add in the addend, if any */
590 unsigned long carry = 0;
591 for (i = 1; i <= rlen; i++) {
592 carry += (i <= ret[0] ? ret[i] : 0);
593 carry += (i <= addend[0] ? addend[i] : 0);
594 ret[i] = (unsigned short) carry & 0xFFFF;
596 if (ret[i] != 0 && i > maxspot)
606 * Non-modular multiplication.
608 Bignum bigmul(Bignum a, Bignum b) {
609 return bigmuladd(a, b, NULL);
613 * Convert a (max 16-bit) short into a bignum.
615 Bignum bignum_from_short(unsigned short n) {
620 ret[2] = (n >> 16) & 0xFFFF;
621 ret[0] = (ret[2] ? 2 : 1);
626 * Add a long to a bignum.
628 Bignum bignum_add_long(Bignum number, unsigned long addend) {
629 Bignum ret = newbn(number[0]+1);
631 unsigned long carry = 0;
633 for (i = 1; i <= ret[0]; i++) {
634 carry += addend & 0xFFFF;
635 carry += (i <= number[0] ? number[i] : 0);
637 ret[i] = (unsigned short) carry & 0xFFFF;
647 * Compute the residue of a bignum, modulo a (max 16-bit) short.
649 unsigned short bignum_mod_short(Bignum number, unsigned short modulus) {
650 unsigned long mod, r;
655 for (i = number[0]; i > 0; i--)
656 r = (r * 65536 + number[i]) % mod;
657 return (unsigned short) r;
660 static void diagbn(char *prefix, Bignum md) {
661 int i, nibbles, morenibbles;
662 static const char hex[] = "0123456789ABCDEF";
664 printf("%s0x", prefix ? prefix : "");
666 nibbles = (3 + ssh1_bignum_bitcount(md))/4; if (nibbles<1) nibbles=1;
667 morenibbles = 4*md[0] - nibbles;
668 for (i=0; i<morenibbles; i++) putchar('-');
669 for (i=nibbles; i-- ;)
670 putchar(hex[(bignum_byte(md, i/2) >> (4*(i%2))) & 0xF]);
672 if (prefix) putchar('\n');
676 * Greatest common divisor.
678 Bignum biggcd(Bignum av, Bignum bv) {
679 Bignum a = copybn(av);
680 Bignum b = copybn(bv);
684 while (bignum_cmp(b, Zero) != 0) {
685 Bignum t = newbn(b[0]);
686 bigmod(a, b, t, NULL);
688 while (t[0] > 1 && t[t[0]] == 0) t[0]--;
699 * Modular inverse, using Euclid's extended algorithm.
701 Bignum modinv(Bignum number, Bignum modulus) {
702 Bignum a = copybn(modulus);
703 Bignum b = copybn(number);
704 Bignum xp = copybn(Zero);
705 Bignum x = copybn(One);
708 while (bignum_cmp(b, One) != 0) {
709 Bignum t = newbn(b[0]);
710 Bignum q = newbn(a[0]);
712 while (t[0] > 1 && t[t[0]] == 0) t[0]--;
718 x = bigmuladd(q, xp, t);
727 /* now we know that sign * x == 1, and that x < modulus */
729 /* set a new x to be modulus - x */
730 Bignum newx = newbn(modulus[0]);
731 unsigned short carry = 0;
735 for (i = 1; i <= newx[0]; i++) {
736 unsigned short aword = (i <= modulus[0] ? modulus[i] : 0);
737 unsigned short bword = (i <= x[0] ? x[i] : 0);
738 newx[i] = aword - bword - carry;
740 carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
754 * Render a bignum into decimal. Return a malloced string holding
755 * the decimal representation.
757 char *bignum_decimal(Bignum x) {
762 unsigned short *workspace;
765 * First, estimate the number of digits. Since log(10)/log(2)
766 * is just greater than 93/28 (the joys of continued fraction
767 * approximations...) we know that for every 93 bits, we need
768 * at most 28 digits. This will tell us how much to malloc.
770 * Formally: if x has i bits, that means x is strictly less
771 * than 2^i. Since 2 is less than 10^(28/93), this is less than
772 * 10^(28i/93). We need an integer power of ten, so we must
773 * round up (rounding down might make it less than x again).
774 * Therefore if we multiply the bit count by 28/93, rounding
775 * up, we will have enough digits.
777 i = ssh1_bignum_bitcount(x);
778 ndigits = (28*i + 92)/93; /* multiply by 28/93 and round up */
779 ndigits++; /* allow for trailing \0 */
780 ret = malloc(ndigits);
783 * Now allocate some workspace to hold the binary form as we
784 * repeatedly divide it by ten. Initialise this to the
785 * big-endian form of the number.
787 workspace = malloc(sizeof(unsigned short) * x[0]);
788 for (i = 0; i < x[0]; i++)
789 workspace[i] = x[x[0] - i];
792 * Next, write the decimal number starting with the last digit.
793 * We use ordinary short division, dividing 10 into the
801 for (i = 0; i < x[0]; i++) {
802 carry = (carry << 16) + workspace[i];
803 workspace[i] = (unsigned short) (carry / 10);
808 ret[--ndigit] = (char)(carry + '0');
812 * There's a chance we've fallen short of the start of the
813 * string. Correct if so.
816 memmove(ret, ret+ndigit, ndigits-ndigit);