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);
163 * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all
164 * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried
167 static BignumInt
internal_add(const BignumInt
*a
, const BignumInt
*b
,
168 BignumInt
*c
, int len
)
171 BignumDblInt carry
= 0;
173 for (i
= len
-1; i
>= 0; i
--) {
174 carry
+= (BignumDblInt
)a
[i
] + b
[i
];
175 c
[i
] = (BignumInt
)carry
;
176 carry
>>= BIGNUM_INT_BITS
;
179 return (BignumInt
)carry
;
183 * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are
184 * all big-endian arrays of 'len' BignumInts. Any borrow from the top
187 static void internal_sub(const BignumInt
*a
, const BignumInt
*b
,
188 BignumInt
*c
, int len
)
191 BignumDblInt carry
= 1;
193 for (i
= len
-1; i
>= 0; i
--) {
194 carry
+= (BignumDblInt
)a
[i
] + (b
[i
] ^ BIGNUM_INT_MASK
);
195 c
[i
] = (BignumInt
)carry
;
196 carry
>>= BIGNUM_INT_BITS
;
202 * Input is in the first len words of a and b.
203 * Result is returned in the first 2*len words of c.
205 #define KARATSUBA_THRESHOLD 50
206 static void internal_mul(const BignumInt
*a
, const BignumInt
*b
,
207 BignumInt
*c
, int len
)
212 if (len
> KARATSUBA_THRESHOLD
) {
215 * Karatsuba divide-and-conquer algorithm. Cut each input in
216 * half, so that it's expressed as two big 'digits' in a giant
222 * Then the product is of course
224 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
226 * and we compute the three coefficients by recursively
227 * calling ourself to do half-length multiplications.
229 * The clever bit that makes this worth doing is that we only
230 * need _one_ half-length multiplication for the central
231 * coefficient rather than the two that it obviouly looks
232 * like, because we can use a single multiplication to compute
234 * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
236 * and then we subtract the other two coefficients (a_1 b_1
237 * and a_0 b_0) which we were computing anyway.
239 * Hence we get to multiply two numbers of length N in about
240 * three times as much work as it takes to multiply numbers of
241 * length N/2, which is obviously better than the four times
242 * as much work it would take if we just did a long
243 * conventional multiply.
246 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
247 int midlen
= botlen
+ 1;
252 * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
253 * in the output array, so we can compute them immediately in
258 internal_mul(a
, b
, c
, toplen
);
261 internal_mul(a
+ toplen
, b
+ toplen
, c
+ 2*toplen
, botlen
);
264 * We must allocate scratch space for the central coefficient,
265 * and also for the two input values that we multiply when
266 * computing it. Since either or both may carry into the
267 * (botlen+1)th word, we must use a slightly longer length
270 scratch
= snewn(4 * midlen
, BignumInt
);
272 /* Zero padding. midlen exceeds toplen by at most 2, so just
273 * zero the first two words of each input and the rest will be
275 scratch
[0] = scratch
[1] = scratch
[midlen
] = scratch
[midlen
+1] = 0;
277 for (j
= 0; j
< toplen
; j
++) {
278 scratch
[midlen
- toplen
+ j
] = a
[j
]; /* a_1 */
279 scratch
[2*midlen
- toplen
+ j
] = b
[j
]; /* b_1 */
282 /* compute a_1 + a_0 */
283 scratch
[0] = internal_add(scratch
+1, a
+toplen
, scratch
+1, botlen
);
284 /* compute b_1 + b_0 */
285 scratch
[midlen
] = internal_add(scratch
+midlen
+1, b
+toplen
,
286 scratch
+midlen
+1, botlen
);
289 * Now we can do the third multiplication.
291 internal_mul(scratch
, scratch
+ midlen
, scratch
+ 2*midlen
, midlen
);
294 * Now we can reuse the first half of 'scratch' to compute the
295 * sum of the outer two coefficients, to subtract from that
296 * product to obtain the middle one.
298 scratch
[0] = scratch
[1] = scratch
[2] = scratch
[3] = 0;
299 for (j
= 0; j
< 2*toplen
; j
++)
300 scratch
[2*midlen
- 2*toplen
+ j
] = c
[j
];
301 scratch
[1] = internal_add(scratch
+2, c
+ 2*toplen
,
302 scratch
+2, 2*botlen
);
304 internal_sub(scratch
+ 2*midlen
, scratch
,
305 scratch
+ 2*midlen
, 2*midlen
);
308 * And now all we need to do is to add that middle coefficient
309 * back into the output. We may have to propagate a carry
310 * further up the output, but we can be sure it won't
311 * propagate right the way off the top.
313 carry
= internal_add(c
+ 2*len
- botlen
- 2*midlen
,
315 c
+ 2*len
- botlen
- 2*midlen
, 2*midlen
);
316 j
= 2*len
- botlen
- 2*midlen
- 1;
320 c
[j
] = (BignumInt
)carry
;
321 carry
>>= BIGNUM_INT_BITS
;
325 for (j
= 0; j
< 4 * midlen
; j
++)
332 * Multiply in the ordinary O(N^2) way.
335 for (j
= 0; j
< 2 * len
; j
++)
338 for (i
= len
- 1; i
>= 0; i
--) {
340 for (j
= len
- 1; j
>= 0; j
--) {
341 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
342 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
343 c
[i
+ j
+ 1] = (BignumInt
) t
;
344 t
= t
>> BIGNUM_INT_BITS
;
346 c
[i
] = (BignumInt
) t
;
352 * Variant form of internal_mul used for the initial step of
353 * Montgomery reduction. Only bothers outputting 'len' words
354 * (everything above that is thrown away).
356 static void internal_mul_low(const BignumInt
*a
, const BignumInt
*b
,
357 BignumInt
*c
, int len
)
362 if (len
> KARATSUBA_THRESHOLD
) {
365 * Karatsuba-aware version of internal_mul_low. As before, we
366 * express each input value as a shifted combination of two
372 * Then the full product is, as before,
374 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
376 * Provided we choose D on the large side (so that a_0 and b_0
377 * are _at least_ as long as a_1 and b_1), we don't need the
378 * topmost term at all, and we only need half of the middle
379 * term. So there's no point in doing the proper Karatsuba
380 * optimisation which computes the middle term using the top
381 * one, because we'd take as long computing the top one as
382 * just computing the middle one directly.
384 * So instead, we do a much more obvious thing: we call the
385 * fully optimised internal_mul to compute a_0 b_0, and we
386 * recursively call ourself to compute the _bottom halves_ of
387 * a_1 b_0 and a_0 b_1, each of which we add into the result
388 * in the obvious way.
390 * In other words, there's no actual Karatsuba _optimisation_
391 * in this function; the only benefit in doing it this way is
392 * that we call internal_mul proper for a large part of the
393 * work, and _that_ can optimise its operation.
396 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
400 * Allocate scratch space for the various bits and pieces
401 * we're going to be adding together. We need botlen*2 words
402 * for a_0 b_0 (though we may end up throwing away its topmost
403 * word), and toplen words for each of a_1 b_0 and a_0 b_1.
404 * That adds up to exactly 2*len.
406 scratch
= snewn(len
*2, BignumInt
);
409 internal_mul(a
+ toplen
, b
+ toplen
, scratch
+ 2*toplen
, botlen
);
412 internal_mul_low(a
, b
+ len
- toplen
, scratch
+ toplen
, toplen
);
415 internal_mul_low(a
+ len
- toplen
, b
, scratch
, toplen
);
417 /* Copy the bottom half of the big coefficient into place */
418 for (j
= 0; j
< botlen
; j
++)
419 c
[toplen
+ j
] = scratch
[2*toplen
+ botlen
+ j
];
421 /* Add the two small coefficients, throwing away the returned carry */
422 internal_add(scratch
, scratch
+ toplen
, scratch
, toplen
);
424 /* And add that to the large coefficient, leaving the result in c. */
425 internal_add(scratch
, scratch
+ 2*toplen
+ botlen
- toplen
,
429 for (j
= 0; j
< len
*2; j
++)
435 for (j
= 0; j
< len
; j
++)
438 for (i
= len
- 1; i
>= 0; i
--) {
440 for (j
= len
- 1; j
>= len
- i
- 1; j
--) {
441 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
442 t
+= (BignumDblInt
) c
[i
+ j
+ 1 - len
];
443 c
[i
+ j
+ 1 - len
] = (BignumInt
) t
;
444 t
= t
>> BIGNUM_INT_BITS
;
452 * Montgomery reduction. Expects x to be a big-endian array of 2*len
453 * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
454 * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
455 * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
458 * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
459 * each, containing respectively n and the multiplicative inverse of
462 * 'tmp' is an array of at least '3*len' BignumInts used as scratch
465 static void monty_reduce(BignumInt
*x
, const BignumInt
*n
,
466 const BignumInt
*mninv
, BignumInt
*tmp
, int len
)
472 * Multiply x by (-n)^{-1} mod r. This gives us a value m such
473 * that mn is congruent to -x mod r. Hence, mn+x is an exact
474 * multiple of r, and is also (obviously) congruent to x mod n.
476 internal_mul_low(x
+ len
, mninv
, tmp
, len
);
479 * Compute t = (mn+x)/r in ordinary, non-modular, integer
480 * arithmetic. By construction this is exact, and is congruent mod
481 * n to x * r^{-1}, i.e. the answer we want.
483 * The following multiply leaves that answer in the _most_
484 * significant half of the 'x' array, so then we must shift it
487 internal_mul(tmp
, n
, tmp
+len
, len
);
488 carry
= internal_add(x
, tmp
+len
, x
, 2*len
);
489 for (i
= 0; i
< len
; i
++)
490 x
[len
+ i
] = x
[i
], x
[i
] = 0;
493 * Reduce t mod n. This doesn't require a full-on division by n,
494 * but merely a test and single optional subtraction, since we can
495 * show that 0 <= t < 2n.
498 * + we computed m mod r, so 0 <= m < r.
499 * + so 0 <= mn < rn, obviously
500 * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
501 * + yielding 0 <= (mn+x)/r < 2n as required.
504 for (i
= 0; i
< len
; i
++)
505 if (x
[len
+ i
] != n
[i
])
508 if (carry
|| i
>= len
|| x
[len
+ i
] > n
[i
])
509 internal_sub(x
+len
, n
, x
+len
, len
);
512 static void internal_add_shifted(BignumInt
*number
,
513 unsigned n
, int shift
)
515 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
516 int bshift
= shift
% BIGNUM_INT_BITS
;
519 addend
= (BignumDblInt
)n
<< bshift
;
522 addend
+= number
[word
];
523 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
524 addend
>>= BIGNUM_INT_BITS
;
531 * Input in first alen words of a and first mlen words of m.
532 * Output in first alen words of a
533 * (of which first alen-mlen words will be zero).
534 * The MSW of m MUST have its high bit set.
535 * Quotient is accumulated in the `quotient' array, which is a Bignum
536 * rather than the internal bigendian format. Quotient parts are shifted
537 * left by `qshift' before adding into quot.
539 static void internal_mod(BignumInt
*a
, int alen
,
540 BignumInt
*m
, int mlen
,
541 BignumInt
*quot
, int qshift
)
553 for (i
= 0; i
<= alen
- mlen
; i
++) {
555 unsigned int q
, r
, c
, ai1
;
569 /* Find q = h:a[i] / m0 */
574 * To illustrate it, suppose a BignumInt is 8 bits, and
575 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
576 * our initial division will be 0xA123 / 0xA1, which
577 * will give a quotient of 0x100 and a divide overflow.
578 * However, the invariants in this division algorithm
579 * are not violated, since the full number A1:23:... is
580 * _less_ than the quotient prefix A1:B2:... and so the
581 * following correction loop would have sorted it out.
583 * In this situation we set q to be the largest
584 * quotient we _can_ stomach (0xFF, of course).
588 /* Macro doesn't want an array subscript expression passed
589 * into it (see definition), so use a temporary. */
590 BignumInt tmplo
= a
[i
];
591 DIVMOD_WORD(q
, r
, h
, tmplo
, m0
);
593 /* Refine our estimate of q by looking at
594 h:a[i]:a[i+1] / m0:m1 */
596 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
599 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
600 if (r
>= (BignumDblInt
) m0
&&
601 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
605 /* Subtract q * m from a[i...] */
607 for (k
= mlen
- 1; k
>= 0; k
--) {
608 t
= MUL_WORD(q
, m
[k
]);
610 c
= (unsigned)(t
>> BIGNUM_INT_BITS
);
611 if ((BignumInt
) t
> a
[i
+ k
])
613 a
[i
+ k
] -= (BignumInt
) t
;
616 /* Add back m in case of borrow */
619 for (k
= mlen
- 1; k
>= 0; k
--) {
622 a
[i
+ k
] = (BignumInt
) t
;
623 t
= t
>> BIGNUM_INT_BITS
;
628 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
633 * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
636 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
638 BignumInt
*a
, *b
, *x
, *n
, *mninv
, *tmp
;
640 Bignum base
, base2
, r
, rn
, inv
, result
;
643 * The most significant word of mod needs to be non-zero. It
644 * should already be, but let's make sure.
646 assert(mod
[mod
[0]] != 0);
649 * Make sure the base is smaller than the modulus, by reducing
650 * it modulo the modulus if not.
652 base
= bigmod(base_in
, mod
);
655 * mod had better be odd, or we can't do Montgomery multiplication
656 * using a power of two at all.
661 * Compute the inverse of n mod r, for monty_reduce. (In fact we
662 * want the inverse of _minus_ n mod r, but we'll sort that out
666 r
= bn_power_2(BIGNUM_INT_BITS
* len
);
667 inv
= modinv(mod
, r
);
670 * Multiply the base by r mod n, to get it into Montgomery
673 base2
= modmul(base
, r
, mod
);
677 rn
= bigmod(r
, mod
); /* r mod n, i.e. Montgomerified 1 */
679 freebn(r
); /* won't need this any more */
682 * Set up internal arrays of the right lengths, in big-endian
683 * format, containing the base, the modulus, and the modulus's
686 n
= snewn(len
, BignumInt
);
687 for (j
= 0; j
< len
; j
++)
688 n
[len
- 1 - j
] = mod
[j
+ 1];
690 mninv
= snewn(len
, BignumInt
);
691 for (j
= 0; j
< len
; j
++)
692 mninv
[len
- 1 - j
] = (j
< inv
[0] ? inv
[j
+ 1] : 0);
693 freebn(inv
); /* we don't need this copy of it any more */
694 /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
695 x
= snewn(len
, BignumInt
);
696 for (j
= 0; j
< len
; j
++)
698 internal_sub(x
, mninv
, mninv
, len
);
700 /* x = snewn(len, BignumInt); */ /* already done above */
701 for (j
= 0; j
< len
; j
++)
702 x
[len
- 1 - j
] = (j
< base
[0] ? base
[j
+ 1] : 0);
703 freebn(base
); /* we don't need this copy of it any more */
705 a
= snewn(2*len
, BignumInt
);
706 b
= snewn(2*len
, BignumInt
);
707 for (j
= 0; j
< len
; j
++)
708 a
[2*len
- 1 - j
] = (j
< rn
[0] ? rn
[j
+ 1] : 0);
711 tmp
= snewn(3*len
, BignumInt
);
713 /* Skip leading zero bits of exp. */
715 j
= BIGNUM_INT_BITS
-1;
716 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
720 j
= BIGNUM_INT_BITS
-1;
724 /* Main computation */
725 while (i
< (int)exp
[0]) {
727 internal_mul(a
+ len
, a
+ len
, b
, len
);
728 monty_reduce(b
, n
, mninv
, tmp
, len
);
729 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
730 internal_mul(b
+ len
, x
, a
, len
);
731 monty_reduce(a
, n
, mninv
, tmp
, len
);
741 j
= BIGNUM_INT_BITS
-1;
745 * Final monty_reduce to get back from the adjusted Montgomery
748 monty_reduce(a
, n
, mninv
, tmp
, len
);
750 /* Copy result to buffer */
751 result
= newbn(mod
[0]);
752 for (i
= 0; i
< len
; i
++)
753 result
[result
[0] - i
] = a
[i
+ len
];
754 while (result
[0] > 1 && result
[result
[0]] == 0)
757 /* Free temporary arrays */
758 for (i
= 0; i
< 3 * len
; i
++)
761 for (i
= 0; i
< 2 * len
; i
++)
764 for (i
= 0; i
< 2 * len
; i
++)
767 for (i
= 0; i
< len
; i
++)
770 for (i
= 0; i
< len
; i
++)
773 for (i
= 0; i
< len
; i
++)
781 * Compute (p * q) % mod.
782 * The most significant word of mod MUST be non-zero.
783 * We assume that the result array is the same size as the mod array.
785 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
787 BignumInt
*a
, *n
, *m
, *o
;
789 int pqlen
, mlen
, rlen
, i
, j
;
792 /* Allocate m of size mlen, copy mod to m */
793 /* We use big endian internally */
795 m
= snewn(mlen
, BignumInt
);
796 for (j
= 0; j
< mlen
; j
++)
797 m
[j
] = mod
[mod
[0] - j
];
799 /* Shift m left to make msb bit set */
800 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
801 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
804 for (i
= 0; i
< mlen
- 1; i
++)
805 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
806 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
809 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
811 /* Allocate n of size pqlen, copy p to n */
812 n
= snewn(pqlen
, BignumInt
);
814 for (j
= 0; j
< i
; j
++)
816 for (j
= 0; j
< (int)p
[0]; j
++)
817 n
[i
+ j
] = p
[p
[0] - j
];
819 /* Allocate o of size pqlen, copy q to o */
820 o
= snewn(pqlen
, BignumInt
);
822 for (j
= 0; j
< i
; j
++)
824 for (j
= 0; j
< (int)q
[0]; j
++)
825 o
[i
+ j
] = q
[q
[0] - j
];
827 /* Allocate a of size 2*pqlen for result */
828 a
= snewn(2 * pqlen
, BignumInt
);
830 /* Main computation */
831 internal_mul(n
, o
, a
, pqlen
);
832 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
834 /* Fixup result in case the modulus was shifted */
836 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
837 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
838 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
839 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
840 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
841 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
844 /* Copy result to buffer */
845 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
846 result
= newbn(rlen
);
847 for (i
= 0; i
< rlen
; i
++)
848 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
849 while (result
[0] > 1 && result
[result
[0]] == 0)
852 /* Free temporary arrays */
853 for (i
= 0; i
< 2 * pqlen
; i
++)
856 for (i
= 0; i
< mlen
; i
++)
859 for (i
= 0; i
< pqlen
; i
++)
862 for (i
= 0; i
< pqlen
; i
++)
871 * The most significant word of mod MUST be non-zero.
872 * We assume that the result array is the same size as the mod array.
873 * We optionally write out a quotient if `quotient' is non-NULL.
874 * We can avoid writing out the result if `result' is NULL.
876 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
880 int plen
, mlen
, i
, j
;
882 /* Allocate m of size mlen, copy mod to m */
883 /* We use big endian internally */
885 m
= snewn(mlen
, BignumInt
);
886 for (j
= 0; j
< mlen
; j
++)
887 m
[j
] = mod
[mod
[0] - j
];
889 /* Shift m left to make msb bit set */
890 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
891 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
894 for (i
= 0; i
< mlen
- 1; i
++)
895 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
896 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
900 /* Ensure plen > mlen */
904 /* Allocate n of size plen, copy p to n */
905 n
= snewn(plen
, BignumInt
);
906 for (j
= 0; j
< plen
; j
++)
908 for (j
= 1; j
<= (int)p
[0]; j
++)
911 /* Main computation */
912 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
914 /* Fixup result in case the modulus was shifted */
916 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
917 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
918 n
[plen
- 1] = n
[plen
- 1] << mshift
;
919 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
920 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
921 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
924 /* Copy result to buffer */
926 for (i
= 1; i
<= (int)result
[0]; i
++) {
928 result
[i
] = j
>= 0 ? n
[j
] : 0;
932 /* Free temporary arrays */
933 for (i
= 0; i
< mlen
; i
++)
936 for (i
= 0; i
< plen
; i
++)
942 * Decrement a number.
944 void decbn(Bignum bn
)
947 while (i
< (int)bn
[0] && bn
[i
] == 0)
948 bn
[i
++] = BIGNUM_INT_MASK
;
952 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
957 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
960 for (i
= 1; i
<= w
; i
++)
962 for (i
= nbytes
; i
--;) {
963 unsigned char byte
= *data
++;
964 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
967 while (result
[0] > 1 && result
[result
[0]] == 0)
973 * Read an SSH-1-format bignum from a data buffer. Return the number
974 * of bytes consumed, or -1 if there wasn't enough data.
976 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
978 const unsigned char *p
= data
;
986 for (i
= 0; i
< 2; i
++)
988 b
= (w
+ 7) / 8; /* bits -> bytes */
993 if (!result
) /* just return length */
996 *result
= bignum_from_bytes(p
, b
);
1002 * Return the bit count of a bignum, for SSH-1 encoding.
1004 int bignum_bitcount(Bignum bn
)
1006 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
1007 while (bitcount
>= 0
1008 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
1009 return bitcount
+ 1;
1013 * Return the byte length of a bignum when SSH-1 encoded.
1015 int ssh1_bignum_length(Bignum bn
)
1017 return 2 + (bignum_bitcount(bn
) + 7) / 8;
1021 * Return the byte length of a bignum when SSH-2 encoded.
1023 int ssh2_bignum_length(Bignum bn
)
1025 return 4 + (bignum_bitcount(bn
) + 8) / 8;
1029 * Return a byte from a bignum; 0 is least significant, etc.
1031 int bignum_byte(Bignum bn
, int i
)
1033 if (i
>= (int)(BIGNUM_INT_BYTES
* bn
[0]))
1034 return 0; /* beyond the end */
1036 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
1037 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
1041 * Return a bit from a bignum; 0 is least significant, etc.
1043 int bignum_bit(Bignum bn
, int i
)
1045 if (i
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1046 return 0; /* beyond the end */
1048 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
1052 * Set a bit in a bignum; 0 is least significant, etc.
1054 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
1056 if (bitnum
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1057 abort(); /* beyond the end */
1059 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
1060 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
1069 * Write a SSH-1-format bignum into a buffer. It is assumed the
1070 * buffer is big enough. Returns the number of bytes used.
1072 int ssh1_write_bignum(void *data
, Bignum bn
)
1074 unsigned char *p
= data
;
1075 int len
= ssh1_bignum_length(bn
);
1077 int bitc
= bignum_bitcount(bn
);
1079 *p
++ = (bitc
>> 8) & 0xFF;
1080 *p
++ = (bitc
) & 0xFF;
1081 for (i
= len
- 2; i
--;)
1082 *p
++ = bignum_byte(bn
, i
);
1087 * Compare two bignums. Returns like strcmp.
1089 int bignum_cmp(Bignum a
, Bignum b
)
1091 int amax
= a
[0], bmax
= b
[0];
1092 int i
= (amax
> bmax ? amax
: bmax
);
1094 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
1095 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
1106 * Right-shift one bignum to form another.
1108 Bignum
bignum_rshift(Bignum a
, int shift
)
1111 int i
, shiftw
, shiftb
, shiftbb
, bits
;
1114 bits
= bignum_bitcount(a
) - shift
;
1115 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
1118 shiftw
= shift
/ BIGNUM_INT_BITS
;
1119 shiftb
= shift
% BIGNUM_INT_BITS
;
1120 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
1122 ai1
= a
[shiftw
+ 1];
1123 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1125 ai1
= (i
+ shiftw
+ 1 <= (int)a
[0] ? a
[i
+ shiftw
+ 1] : 0);
1126 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
1134 * Non-modular multiplication and addition.
1136 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
1138 int alen
= a
[0], blen
= b
[0];
1139 int mlen
= (alen
> blen ? alen
: blen
);
1140 int rlen
, i
, maxspot
;
1141 BignumInt
*workspace
;
1144 /* mlen space for a, mlen space for b, 2*mlen for result */
1145 workspace
= snewn(mlen
* 4, BignumInt
);
1146 for (i
= 0; i
< mlen
; i
++) {
1147 workspace
[0 * mlen
+ i
] = (mlen
- i
<= (int)a
[0] ? a
[mlen
- i
] : 0);
1148 workspace
[1 * mlen
+ i
] = (mlen
- i
<= (int)b
[0] ? b
[mlen
- i
] : 0);
1151 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
1152 workspace
+ 2 * mlen
, mlen
);
1154 /* now just copy the result back */
1155 rlen
= alen
+ blen
+ 1;
1156 if (addend
&& rlen
<= (int)addend
[0])
1157 rlen
= addend
[0] + 1;
1160 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1161 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
1167 /* now add in the addend, if any */
1169 BignumDblInt carry
= 0;
1170 for (i
= 1; i
<= rlen
; i
++) {
1171 carry
+= (i
<= (int)ret
[0] ? ret
[i
] : 0);
1172 carry
+= (i
<= (int)addend
[0] ? addend
[i
] : 0);
1173 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1174 carry
>>= BIGNUM_INT_BITS
;
1175 if (ret
[i
] != 0 && i
> maxspot
)
1186 * Non-modular multiplication.
1188 Bignum
bigmul(Bignum a
, Bignum b
)
1190 return bigmuladd(a
, b
, NULL
);
1196 Bignum
bigadd(Bignum a
, Bignum b
)
1198 int alen
= a
[0], blen
= b
[0];
1199 int rlen
= (alen
> blen ? alen
: blen
) + 1;
1208 for (i
= 1; i
<= rlen
; i
++) {
1209 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1210 carry
+= (i
<= (int)b
[0] ? b
[i
] : 0);
1211 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1212 carry
>>= BIGNUM_INT_BITS
;
1213 if (ret
[i
] != 0 && i
> maxspot
)
1222 * Subtraction. Returns a-b, or NULL if the result would come out
1223 * negative (recall that this entire bignum module only handles
1224 * positive numbers).
1226 Bignum
bigsub(Bignum a
, Bignum b
)
1228 int alen
= a
[0], blen
= b
[0];
1229 int rlen
= (alen
> blen ? alen
: blen
);
1238 for (i
= 1; i
<= rlen
; i
++) {
1239 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1240 carry
+= (i
<= (int)b
[0] ? b
[i
] ^ BIGNUM_INT_MASK
: BIGNUM_INT_MASK
);
1241 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1242 carry
>>= BIGNUM_INT_BITS
;
1243 if (ret
[i
] != 0 && i
> maxspot
)
1257 * Create a bignum which is the bitmask covering another one. That
1258 * is, the smallest integer which is >= N and is also one less than
1261 Bignum
bignum_bitmask(Bignum n
)
1263 Bignum ret
= copybn(n
);
1268 while (n
[i
] == 0 && i
> 0)
1271 return ret
; /* input was zero */
1277 ret
[i
] = BIGNUM_INT_MASK
;
1282 * Convert a (max 32-bit) long into a bignum.
1284 Bignum
bignum_from_long(unsigned long nn
)
1287 BignumDblInt n
= nn
;
1290 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
1291 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
1293 ret
[0] = (ret
[2] ?
2 : 1);
1298 * Add a long to a bignum.
1300 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
1302 Bignum ret
= newbn(number
[0] + 1);
1304 BignumDblInt carry
= 0, addend
= addendx
;
1306 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1307 carry
+= addend
& BIGNUM_INT_MASK
;
1308 carry
+= (i
<= (int)number
[0] ? number
[i
] : 0);
1309 addend
>>= BIGNUM_INT_BITS
;
1310 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1311 carry
>>= BIGNUM_INT_BITS
;
1320 * Compute the residue of a bignum, modulo a (max 16-bit) short.
1322 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
1324 BignumDblInt mod
, r
;
1329 for (i
= number
[0]; i
> 0; i
--)
1330 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
1331 return (unsigned short) r
;
1335 void diagbn(char *prefix
, Bignum md
)
1337 int i
, nibbles
, morenibbles
;
1338 static const char hex
[] = "0123456789ABCDEF";
1340 debug(("%s0x", prefix ? prefix
: ""));
1342 nibbles
= (3 + bignum_bitcount(md
)) / 4;
1345 morenibbles
= 4 * md
[0] - nibbles
;
1346 for (i
= 0; i
< morenibbles
; i
++)
1348 for (i
= nibbles
; i
--;)
1350 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
1360 Bignum
bigdiv(Bignum a
, Bignum b
)
1362 Bignum q
= newbn(a
[0]);
1363 bigdivmod(a
, b
, NULL
, q
);
1370 Bignum
bigmod(Bignum a
, Bignum b
)
1372 Bignum r
= newbn(b
[0]);
1373 bigdivmod(a
, b
, r
, NULL
);
1378 * Greatest common divisor.
1380 Bignum
biggcd(Bignum av
, Bignum bv
)
1382 Bignum a
= copybn(av
);
1383 Bignum b
= copybn(bv
);
1385 while (bignum_cmp(b
, Zero
) != 0) {
1386 Bignum t
= newbn(b
[0]);
1387 bigdivmod(a
, b
, t
, NULL
);
1388 while (t
[0] > 1 && t
[t
[0]] == 0)
1400 * Modular inverse, using Euclid's extended algorithm.
1402 Bignum
modinv(Bignum number
, Bignum modulus
)
1404 Bignum a
= copybn(modulus
);
1405 Bignum b
= copybn(number
);
1406 Bignum xp
= copybn(Zero
);
1407 Bignum x
= copybn(One
);
1410 while (bignum_cmp(b
, One
) != 0) {
1411 Bignum t
= newbn(b
[0]);
1412 Bignum q
= newbn(a
[0]);
1413 bigdivmod(a
, b
, t
, q
);
1414 while (t
[0] > 1 && t
[t
[0]] == 0)
1421 x
= bigmuladd(q
, xp
, t
);
1431 /* now we know that sign * x == 1, and that x < modulus */
1433 /* set a new x to be modulus - x */
1434 Bignum newx
= newbn(modulus
[0]);
1435 BignumInt carry
= 0;
1439 for (i
= 1; i
<= (int)newx
[0]; i
++) {
1440 BignumInt aword
= (i
<= (int)modulus
[0] ? modulus
[i
] : 0);
1441 BignumInt bword
= (i
<= (int)x
[0] ? x
[i
] : 0);
1442 newx
[i
] = aword
- bword
- carry
;
1444 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1458 * Render a bignum into decimal. Return a malloced string holding
1459 * the decimal representation.
1461 char *bignum_decimal(Bignum x
)
1463 int ndigits
, ndigit
;
1467 BignumInt
*workspace
;
1470 * First, estimate the number of digits. Since log(10)/log(2)
1471 * is just greater than 93/28 (the joys of continued fraction
1472 * approximations...) we know that for every 93 bits, we need
1473 * at most 28 digits. This will tell us how much to malloc.
1475 * Formally: if x has i bits, that means x is strictly less
1476 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1477 * 10^(28i/93). We need an integer power of ten, so we must
1478 * round up (rounding down might make it less than x again).
1479 * Therefore if we multiply the bit count by 28/93, rounding
1480 * up, we will have enough digits.
1482 * i=0 (i.e., x=0) is an irritating special case.
1484 i
= bignum_bitcount(x
);
1486 ndigits
= 1; /* x = 0 */
1488 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1489 ndigits
++; /* allow for trailing \0 */
1490 ret
= snewn(ndigits
, char);
1493 * Now allocate some workspace to hold the binary form as we
1494 * repeatedly divide it by ten. Initialise this to the
1495 * big-endian form of the number.
1497 workspace
= snewn(x
[0], BignumInt
);
1498 for (i
= 0; i
< (int)x
[0]; i
++)
1499 workspace
[i
] = x
[x
[0] - i
];
1502 * Next, write the decimal number starting with the last digit.
1503 * We use ordinary short division, dividing 10 into the
1506 ndigit
= ndigits
- 1;
1511 for (i
= 0; i
< (int)x
[0]; i
++) {
1512 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1513 workspace
[i
] = (BignumInt
) (carry
/ 10);
1518 ret
[--ndigit
] = (char) (carry
+ '0');
1522 * There's a chance we've fallen short of the start of the
1523 * string. Correct if so.
1526 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);