2 * Bignum routines for RSA and DH and stuff.
11 unsigned short bnZero
[1] = { 0 };
12 unsigned short bnOne
[2] = { 1, 1 };
14 Bignum Zero
= bnZero
, One
= bnOne
;
16 Bignum
newbn(int length
) {
17 Bignum b
= malloc((length
+1)*sizeof(unsigned short));
20 memset(b
, 0, (length
+1)*sizeof(*b
));
25 Bignum
copybn(Bignum orig
) {
26 Bignum b
= malloc((orig
[0]+1)*sizeof(unsigned short));
29 memcpy(b
, orig
, (orig
[0]+1)*sizeof(*b
));
33 void freebn(Bignum b
) {
35 * Burn the evidence, just in case.
37 memset(b
, 0, sizeof(b
[0]) * (b
[0] + 1));
43 * Input is in the first len words of a and b.
44 * Result is returned in the first 2*len words of c.
46 static void internal_mul(unsigned short *a
, unsigned short *b
,
47 unsigned short *c
, int len
)
52 for (j
= 0; j
< 2*len
; j
++)
55 for (i
= len
- 1; i
>= 0; i
--) {
58 for (j
= len
- 1; j
>= 0; j
--) {
59 t
+= ai
* (unsigned long) b
[j
];
60 t
+= (unsigned long) c
[i
+j
+1];
61 c
[i
+j
+1] = (unsigned short)t
;
64 c
[i
] = (unsigned short)t
;
68 static void internal_add_shifted(unsigned short *number
,
69 unsigned n
, int shift
) {
70 int word
= 1 + (shift
/ 16);
71 int bshift
= shift
% 16;
77 addend
+= number
[word
];
78 number
[word
] = (unsigned short) addend
& 0xFFFF;
86 * Input in first alen words of a and first mlen words of m.
87 * Output in first alen words of a
88 * (of which first alen-mlen words will be zero).
89 * The MSW of m MUST have its high bit set.
90 * Quotient is accumulated in the `quotient' array, which is a Bignum
91 * rather than the internal bigendian format. Quotient parts are shifted
92 * left by `qshift' before adding into quot.
94 static void internal_mod(unsigned short *a
, int alen
,
95 unsigned short *m
, int mlen
,
96 unsigned short *quot
, int qshift
)
98 unsigned short m0
, m1
;
108 for (i
= 0; i
<= alen
-mlen
; i
++) {
110 unsigned int q
, r
, c
, ai1
;
124 /* Find q = h:a[i] / m0 */
125 t
= ((unsigned long) h
<< 16) + a
[i
];
129 /* Refine our estimate of q by looking at
130 h:a[i]:a[i+1] / m0:m1 */
131 t
= (long) m1
* (long) q
;
132 if (t
> ((unsigned long) r
<< 16) + ai1
) {
135 r
= (r
+ m0
) & 0xffff; /* overflow? */
136 if (r
>= (unsigned long)m0
&&
137 t
> ((unsigned long) r
<< 16) + ai1
)
141 /* Subtract q * m from a[i...] */
143 for (k
= mlen
- 1; k
>= 0; k
--) {
144 t
= (long) q
* (long) m
[k
];
147 if ((unsigned short) t
> a
[i
+k
]) c
++;
148 a
[i
+k
] -= (unsigned short) t
;
151 /* Add back m in case of borrow */
154 for (k
= mlen
- 1; k
>= 0; k
--) {
157 a
[i
+k
] = (unsigned short)t
;
163 internal_add_shifted(quot
, q
, qshift
+ 16 * (alen
-mlen
-i
));
168 * Compute (base ^ exp) % mod.
169 * The base MUST be smaller than the modulus.
170 * The most significant word of mod MUST be non-zero.
171 * We assume that the result array is the same size as the mod array.
173 void modpow(Bignum base
, Bignum exp
, Bignum mod
, Bignum result
)
175 unsigned short *a
, *b
, *n
, *m
;
179 /* Allocate m of size mlen, copy mod to m */
180 /* We use big endian internally */
182 m
= malloc(mlen
* sizeof(unsigned short));
183 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
185 /* Shift m left to make msb bit set */
186 for (mshift
= 0; mshift
< 15; mshift
++)
187 if ((m
[0] << mshift
) & 0x8000) break;
189 for (i
= 0; i
< mlen
- 1; i
++)
190 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
191 m
[mlen
-1] = m
[mlen
-1] << mshift
;
194 /* Allocate n of size mlen, copy base to n */
195 n
= malloc(mlen
* sizeof(unsigned short));
197 for (j
= 0; j
< i
; j
++) n
[j
] = 0;
198 for (j
= 0; j
< base
[0]; j
++) n
[i
+j
] = base
[base
[0] - j
];
200 /* Allocate a and b of size 2*mlen. Set a = 1 */
201 a
= malloc(2 * mlen
* sizeof(unsigned short));
202 b
= malloc(2 * mlen
* sizeof(unsigned short));
203 for (i
= 0; i
< 2*mlen
; i
++) a
[i
] = 0;
206 /* Skip leading zero bits of exp. */
208 while (i
< exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
210 if (j
< 0) { i
++; j
= 15; }
213 /* Main computation */
216 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
);
217 internal_mod(b
, mlen
*2, m
, mlen
, NULL
, 0);
218 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
219 internal_mul(b
+ mlen
, n
, a
, mlen
);
220 internal_mod(a
, mlen
*2, m
, mlen
, NULL
, 0);
230 /* Fixup result in case the modulus was shifted */
232 for (i
= mlen
- 1; i
< 2*mlen
- 1; i
++)
233 a
[i
] = (a
[i
] << mshift
) | (a
[i
+1] >> (16-mshift
));
234 a
[2*mlen
-1] = a
[2*mlen
-1] << mshift
;
235 internal_mod(a
, mlen
*2, m
, mlen
, NULL
, 0);
236 for (i
= 2*mlen
- 1; i
>= mlen
; i
--)
237 a
[i
] = (a
[i
] >> mshift
) | (a
[i
-1] << (16-mshift
));
240 /* Copy result to buffer */
241 for (i
= 0; i
< mlen
; i
++)
242 result
[result
[0] - i
] = a
[i
+mlen
];
244 /* Free temporary arrays */
245 for (i
= 0; i
< 2*mlen
; i
++) a
[i
] = 0; free(a
);
246 for (i
= 0; i
< 2*mlen
; i
++) b
[i
] = 0; free(b
);
247 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; free(m
);
248 for (i
= 0; i
< mlen
; i
++) n
[i
] = 0; free(n
);
252 * Compute (p * q) % mod.
253 * The most significant word of mod MUST be non-zero.
254 * We assume that the result array is the same size as the mod array.
256 void modmul(Bignum p
, Bignum q
, Bignum mod
, Bignum result
)
258 unsigned short *a
, *n
, *m
, *o
;
260 int pqlen
, mlen
, i
, j
;
262 /* Allocate m of size mlen, copy mod to m */
263 /* We use big endian internally */
265 m
= malloc(mlen
* sizeof(unsigned short));
266 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
268 /* Shift m left to make msb bit set */
269 for (mshift
= 0; mshift
< 15; mshift
++)
270 if ((m
[0] << mshift
) & 0x8000) break;
272 for (i
= 0; i
< mlen
- 1; i
++)
273 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
274 m
[mlen
-1] = m
[mlen
-1] << mshift
;
277 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
279 /* Allocate n of size pqlen, copy p to n */
280 n
= malloc(pqlen
* sizeof(unsigned short));
282 for (j
= 0; j
< i
; j
++) n
[j
] = 0;
283 for (j
= 0; j
< p
[0]; j
++) n
[i
+j
] = p
[p
[0] - j
];
285 /* Allocate o of size pqlen, copy q to o */
286 o
= malloc(pqlen
* sizeof(unsigned short));
288 for (j
= 0; j
< i
; j
++) o
[j
] = 0;
289 for (j
= 0; j
< q
[0]; j
++) o
[i
+j
] = q
[q
[0] - j
];
291 /* Allocate a of size 2*pqlen for result */
292 a
= malloc(2 * pqlen
* sizeof(unsigned short));
294 /* Main computation */
295 internal_mul(n
, o
, a
, pqlen
);
296 internal_mod(a
, pqlen
*2, m
, mlen
, NULL
, 0);
298 /* Fixup result in case the modulus was shifted */
300 for (i
= 2*pqlen
- mlen
- 1; i
< 2*pqlen
- 1; i
++)
301 a
[i
] = (a
[i
] << mshift
) | (a
[i
+1] >> (16-mshift
));
302 a
[2*pqlen
-1] = a
[2*pqlen
-1] << mshift
;
303 internal_mod(a
, pqlen
*2, m
, mlen
, NULL
, 0);
304 for (i
= 2*pqlen
- 1; i
>= 2*pqlen
- mlen
; i
--)
305 a
[i
] = (a
[i
] >> mshift
) | (a
[i
-1] << (16-mshift
));
308 /* Copy result to buffer */
309 for (i
= 0; i
< mlen
; i
++)
310 result
[result
[0] - i
] = a
[i
+2*pqlen
-mlen
];
312 /* Free temporary arrays */
313 for (i
= 0; i
< 2*pqlen
; i
++) a
[i
] = 0; free(a
);
314 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; free(m
);
315 for (i
= 0; i
< pqlen
; i
++) n
[i
] = 0; free(n
);
316 for (i
= 0; i
< pqlen
; i
++) o
[i
] = 0; free(o
);
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.
323 * We optionally write out a quotient.
325 void bigmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
327 unsigned short *n
, *m
;
329 int plen
, mlen
, i
, j
;
331 /* Allocate m of size mlen, copy mod to m */
332 /* We use big endian internally */
334 m
= malloc(mlen
* sizeof(unsigned short));
335 for (j
= 0; j
< mlen
; j
++) m
[j
] = mod
[mod
[0] - j
];
337 /* Shift m left to make msb bit set */
338 for (mshift
= 0; mshift
< 15; mshift
++)
339 if ((m
[0] << mshift
) & 0x8000) break;
341 for (i
= 0; i
< mlen
- 1; i
++)
342 m
[i
] = (m
[i
] << mshift
) | (m
[i
+1] >> (16-mshift
));
343 m
[mlen
-1] = m
[mlen
-1] << mshift
;
347 /* Ensure plen > mlen */
348 if (plen
<= mlen
) plen
= mlen
+1;
350 /* Allocate n of size plen, copy p to n */
351 n
= malloc(plen
* sizeof(unsigned short));
352 for (j
= 0; j
< plen
; j
++) n
[j
] = 0;
353 for (j
= 1; j
<= p
[0]; j
++) n
[plen
-j
] = p
[j
];
355 /* Main computation */
356 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
358 /* Fixup result in case the modulus was shifted */
360 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
361 n
[i
] = (n
[i
] << mshift
) | (n
[i
+1] >> (16-mshift
));
362 n
[plen
-1] = n
[plen
-1] << mshift
;
363 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
364 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
365 n
[i
] = (n
[i
] >> mshift
) | (n
[i
-1] << (16-mshift
));
368 /* Copy result to buffer */
369 for (i
= 1; i
<= result
[0]; i
++) {
371 result
[i
] = j
>=0 ? n
[j
] : 0;
374 /* Free temporary arrays */
375 for (i
= 0; i
< mlen
; i
++) m
[i
] = 0; free(m
);
376 for (i
= 0; i
< plen
; i
++) n
[i
] = 0; free(n
);
380 * Decrement a number.
382 void decbn(Bignum bn
) {
384 while (i
< bn
[0] && bn
[i
] == 0)
390 * Read an ssh1-format bignum from a data buffer. Return the number
393 int ssh1_read_bignum(unsigned char *data
, Bignum
*result
) {
394 unsigned char *p
= data
;
403 b
= (w
+7)/8; /* bits -> bytes */
404 w
= (w
+15)/16; /* bits -> words */
406 if (!result
) /* just return length */
414 unsigned char byte
= *p
++;
416 bn
[1+i
/2] |= byte
<<8;
427 * Return the bit count of a bignum, for ssh1 encoding.
429 int ssh1_bignum_bitcount(Bignum bn
) {
430 int bitcount
= bn
[0] * 16 - 1;
432 while (bitcount
>= 0 && (bn
[bitcount
/16+1] >> (bitcount
% 16)) == 0)
438 * Return the byte length of a bignum when ssh1 encoded.
440 int ssh1_bignum_length(Bignum bn
) {
441 return 2 + (ssh1_bignum_bitcount(bn
)+7)/8;
445 * Return a byte from a bignum; 0 is least significant, etc.
447 int bignum_byte(Bignum bn
, int i
) {
449 return 0; /* beyond the end */
451 return (bn
[i
/2+1] >> 8) & 0xFF;
453 return (bn
[i
/2+1] ) & 0xFF;
457 * Return a bit from a bignum; 0 is least significant, etc.
459 int bignum_bit(Bignum bn
, int i
) {
461 return 0; /* beyond the end */
463 return (bn
[i
/16+1] >> (i
%16)) & 1;
467 * Set a bit in a bignum; 0 is least significant, etc.
469 void bignum_set_bit(Bignum bn
, int bitnum
, int value
) {
470 if (bitnum
>= 16*bn
[0])
471 abort(); /* beyond the end */
474 int mask
= 1 << (bitnum
%16);
483 * Write a ssh1-format bignum into a buffer. It is assumed the
484 * buffer is big enough. Returns the number of bytes used.
486 int ssh1_write_bignum(void *data
, Bignum bn
) {
487 unsigned char *p
= data
;
488 int len
= ssh1_bignum_length(bn
);
490 int bitc
= ssh1_bignum_bitcount(bn
);
492 *p
++ = (bitc
>> 8) & 0xFF;
493 *p
++ = (bitc
) & 0xFF;
494 for (i
= len
-2; i
-- ;)
495 *p
++ = bignum_byte(bn
, i
);
500 * Compare two bignums. Returns like strcmp.
502 int bignum_cmp(Bignum a
, Bignum b
) {
503 int amax
= a
[0], bmax
= b
[0];
504 int i
= (amax
> bmax ? amax
: bmax
);
506 unsigned short aval
= (i
> amax ?
0 : a
[i
]);
507 unsigned short bval
= (i
> bmax ?
0 : b
[i
]);
508 if (aval
< bval
) return -1;
509 if (aval
> bval
) return +1;
516 * Right-shift one bignum to form another.
518 Bignum
bignum_rshift(Bignum a
, int shift
) {
520 int i
, shiftw
, shiftb
, shiftbb
, bits
;
521 unsigned short ai
, ai1
;
523 bits
= ssh1_bignum_bitcount(a
) - shift
;
524 ret
= newbn((bits
+15)/16);
529 shiftbb
= 16 - shiftb
;
532 for (i
= 1; i
<= ret
[0]; i
++) {
534 ai1
= (i
+shiftw
+1 <= a
[0] ? a
[i
+shiftw
+1] : 0);
535 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & 0xFFFF;
543 * Non-modular multiplication and addition.
545 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
) {
546 int alen
= a
[0], blen
= b
[0];
547 int mlen
= (alen
> blen ? alen
: blen
);
548 int rlen
, i
, maxspot
;
549 unsigned short *workspace
;
552 /* mlen space for a, mlen space for b, 2*mlen for result */
553 workspace
= malloc(mlen
* 4 * sizeof(unsigned short));
554 for (i
= 0; i
< mlen
; i
++) {
555 workspace
[0*mlen
+ i
] = (mlen
-i
<= a
[0] ? a
[mlen
-i
] : 0);
556 workspace
[1*mlen
+ i
] = (mlen
-i
<= b
[0] ? b
[mlen
-i
] : 0);
559 internal_mul(workspace
+0*mlen
, workspace
+1*mlen
, workspace
+2*mlen
, mlen
);
561 /* now just copy the result back */
562 rlen
= alen
+ blen
+ 1;
563 if (addend
&& rlen
<= addend
[0])
564 rlen
= addend
[0] + 1;
567 for (i
= 1; i
<= ret
[0]; i
++) {
568 ret
[i
] = (i
<= 2*mlen ? workspace
[4*mlen
- i
] : 0);
574 /* now add in the addend, if any */
576 unsigned long carry
= 0;
577 for (i
= 1; i
<= rlen
; i
++) {
578 carry
+= (i
<= ret
[0] ? ret
[i
] : 0);
579 carry
+= (i
<= addend
[0] ? addend
[i
] : 0);
580 ret
[i
] = (unsigned short) carry
& 0xFFFF;
582 if (ret
[i
] != 0 && i
> maxspot
)
592 * Non-modular multiplication.
594 Bignum
bigmul(Bignum a
, Bignum b
) {
595 return bigmuladd(a
, b
, NULL
);
599 * Convert a (max 16-bit) short into a bignum.
601 Bignum
bignum_from_short(unsigned short n
) {
606 ret
[2] = (n
>> 16) & 0xFFFF;
607 ret
[0] = (ret
[2] ?
2 : 1);
612 * Add a long to a bignum.
614 Bignum
bignum_add_long(Bignum number
, unsigned long addend
) {
615 Bignum ret
= newbn(number
[0]+1);
617 unsigned long carry
= 0;
619 for (i
= 1; i
<= ret
[0]; i
++) {
620 carry
+= addend
& 0xFFFF;
621 carry
+= (i
<= number
[0] ? number
[i
] : 0);
623 ret
[i
] = (unsigned short) carry
& 0xFFFF;
633 * Compute the residue of a bignum, modulo a (max 16-bit) short.
635 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
) {
636 unsigned long mod
, r
;
641 for (i
= number
[0]; i
> 0; i
--)
642 r
= (r
* 65536 + number
[i
]) % mod
;
643 return (unsigned short) r
;
646 static void diagbn(char *prefix
, Bignum md
) {
647 int i
, nibbles
, morenibbles
;
648 static const char hex
[] = "0123456789ABCDEF";
650 printf("%s0x", prefix ? prefix
: "");
652 nibbles
= (3 + ssh1_bignum_bitcount(md
))/4; if (nibbles
<1) nibbles
=1;
653 morenibbles
= 4*md
[0] - nibbles
;
654 for (i
=0; i
<morenibbles
; i
++) putchar('-');
655 for (i
=nibbles
; i
-- ;)
656 putchar(hex
[(bignum_byte(md
, i
/2) >> (4*(i
%2))) & 0xF]);
658 if (prefix
) putchar('\n');
662 * Greatest common divisor.
664 Bignum
biggcd(Bignum av
, Bignum bv
) {
665 Bignum a
= copybn(av
);
666 Bignum b
= copybn(bv
);
670 while (bignum_cmp(b
, Zero
) != 0) {
671 Bignum t
= newbn(b
[0]);
672 bigmod(a
, b
, t
, NULL
);
674 while (t
[0] > 1 && t
[t
[0]] == 0) t
[0]--;
685 * Modular inverse, using Euclid's extended algorithm.
687 Bignum
modinv(Bignum number
, Bignum modulus
) {
688 Bignum a
= copybn(modulus
);
689 Bignum b
= copybn(number
);
690 Bignum xp
= copybn(Zero
);
691 Bignum x
= copybn(One
);
694 while (bignum_cmp(b
, One
) != 0) {
695 Bignum t
= newbn(b
[0]);
696 Bignum q
= newbn(a
[0]);
698 while (t
[0] > 1 && t
[t
[0]] == 0) t
[0]--;
704 x
= bigmuladd(q
, xp
, t
);
713 /* now we know that sign * x == 1, and that x < modulus */
715 /* set a new x to be modulus - x */
716 Bignum newx
= newbn(modulus
[0]);
717 unsigned short carry
= 0;
721 for (i
= 1; i
<= newx
[0]; i
++) {
722 unsigned short aword
= (i
<= modulus
[0] ? modulus
[i
] : 0);
723 unsigned short bword
= (i
<= x
[0] ? x
[i
] : 0);
724 newx
[i
] = aword
- bword
- carry
;
726 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
740 * Render a bignum into decimal. Return a malloced string holding
741 * the decimal representation.
743 char *bignum_decimal(Bignum x
) {
748 unsigned short *workspace
;
751 * First, estimate the number of digits. Since log(10)/log(2)
752 * is just greater than 93/28 (the joys of continued fraction
753 * approximations...) we know that for every 93 bits, we need
754 * at most 28 digits. This will tell us how much to malloc.
756 * Formally: if x has i bits, that means x is strictly less
757 * than 2^i. Since 2 is less than 10^(28/93), this is less than
758 * 10^(28i/93). We need an integer power of ten, so we must
759 * round up (rounding down might make it less than x again).
760 * Therefore if we multiply the bit count by 28/93, rounding
761 * up, we will have enough digits.
763 i
= ssh1_bignum_bitcount(x
);
764 ndigits
= (28*i
+ 92)/93; /* multiply by 28/93 and round up */
765 ndigits
++; /* allow for trailing \0 */
766 ret
= malloc(ndigits
);
769 * Now allocate some workspace to hold the binary form as we
770 * repeatedly divide it by ten. Initialise this to the
771 * big-endian form of the number.
773 workspace
= malloc(sizeof(unsigned short) * x
[0]);
774 for (i
= 0; i
< x
[0]; i
++)
775 workspace
[i
] = x
[x
[0] - i
];
778 * Next, write the decimal number starting with the last digit.
779 * We use ordinary short division, dividing 10 into the
787 for (i
= 0; i
< x
[0]; i
++) {
788 carry
= (carry
<< 16) + workspace
[i
];
789 workspace
[i
] = (unsigned short) (carry
/ 10);
794 ret
[--ndigit
] = (char)(carry
+ '0');
798 * There's a chance we've fallen short of the start of the
799 * string. Correct if so.
802 memmove(ret
, ret
+ndigit
, ndigits
-ndigit
);