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 * 'scratch' must point to an array of BignumInt of size at least
206 * mul_compute_scratch(len). (This covers the needs of internal_mul
207 * and all its recursive calls to itself.)
209 #define KARATSUBA_THRESHOLD 50
210 static int mul_compute_scratch(int len
)
213 while (len
> KARATSUBA_THRESHOLD
) {
214 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
215 int midlen
= botlen
+ 1;
221 static void internal_mul(const BignumInt
*a
, const BignumInt
*b
,
222 BignumInt
*c
, int len
, BignumInt
*scratch
)
227 if (len
> KARATSUBA_THRESHOLD
) {
230 * Karatsuba divide-and-conquer algorithm. Cut each input in
231 * half, so that it's expressed as two big 'digits' in a giant
237 * Then the product is of course
239 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
241 * and we compute the three coefficients by recursively
242 * calling ourself to do half-length multiplications.
244 * The clever bit that makes this worth doing is that we only
245 * need _one_ half-length multiplication for the central
246 * coefficient rather than the two that it obviouly looks
247 * like, because we can use a single multiplication to compute
249 * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
251 * and then we subtract the other two coefficients (a_1 b_1
252 * and a_0 b_0) which we were computing anyway.
254 * Hence we get to multiply two numbers of length N in about
255 * three times as much work as it takes to multiply numbers of
256 * length N/2, which is obviously better than the four times
257 * as much work it would take if we just did a long
258 * conventional multiply.
261 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
262 int midlen
= botlen
+ 1;
269 * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
270 * in the output array, so we can compute them immediately in
275 printf("a1,a0 = 0x");
276 for (i
= 0; i
< len
; i
++) {
277 if (i
== toplen
) printf(", 0x");
278 printf("%0*x", BIGNUM_INT_BITS
/4, a
[i
]);
281 printf("b1,b0 = 0x");
282 for (i
= 0; i
< len
; i
++) {
283 if (i
== toplen
) printf(", 0x");
284 printf("%0*x", BIGNUM_INT_BITS
/4, b
[i
]);
290 internal_mul(a
, b
, c
, toplen
, scratch
);
293 for (i
= 0; i
< 2*toplen
; i
++) {
294 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
300 internal_mul(a
+ toplen
, b
+ toplen
, c
+ 2*toplen
, botlen
, scratch
);
303 for (i
= 0; i
< 2*botlen
; i
++) {
304 printf("%0*x", BIGNUM_INT_BITS
/4, c
[2*toplen
+i
]);
309 /* Zero padding. midlen exceeds toplen by at most 2, so just
310 * zero the first two words of each input and the rest will be
312 scratch
[0] = scratch
[1] = scratch
[midlen
] = scratch
[midlen
+1] = 0;
314 for (j
= 0; j
< toplen
; j
++) {
315 scratch
[midlen
- toplen
+ j
] = a
[j
]; /* a_1 */
316 scratch
[2*midlen
- toplen
+ j
] = b
[j
]; /* b_1 */
319 /* compute a_1 + a_0 */
320 scratch
[0] = internal_add(scratch
+1, a
+toplen
, scratch
+1, botlen
);
322 printf("a1plusa0 = 0x");
323 for (i
= 0; i
< midlen
; i
++) {
324 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
328 /* compute b_1 + b_0 */
329 scratch
[midlen
] = internal_add(scratch
+midlen
+1, b
+toplen
,
330 scratch
+midlen
+1, botlen
);
332 printf("b1plusb0 = 0x");
333 for (i
= 0; i
< midlen
; i
++) {
334 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[midlen
+i
]);
340 * Now we can do the third multiplication.
342 internal_mul(scratch
, scratch
+ midlen
, scratch
+ 2*midlen
, midlen
,
345 printf("a1plusa0timesb1plusb0 = 0x");
346 for (i
= 0; i
< 2*midlen
; i
++) {
347 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
353 * Now we can reuse the first half of 'scratch' to compute the
354 * sum of the outer two coefficients, to subtract from that
355 * product to obtain the middle one.
357 scratch
[0] = scratch
[1] = scratch
[2] = scratch
[3] = 0;
358 for (j
= 0; j
< 2*toplen
; j
++)
359 scratch
[2*midlen
- 2*toplen
+ j
] = c
[j
];
360 scratch
[1] = internal_add(scratch
+2, c
+ 2*toplen
,
361 scratch
+2, 2*botlen
);
363 printf("a1b1plusa0b0 = 0x");
364 for (i
= 0; i
< 2*midlen
; i
++) {
365 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[i
]);
370 internal_sub(scratch
+ 2*midlen
, scratch
,
371 scratch
+ 2*midlen
, 2*midlen
);
373 printf("a1b0plusa0b1 = 0x");
374 for (i
= 0; i
< 2*midlen
; i
++) {
375 printf("%0*x", BIGNUM_INT_BITS
/4, scratch
[2*midlen
+i
]);
381 * And now all we need to do is to add that middle coefficient
382 * back into the output. We may have to propagate a carry
383 * further up the output, but we can be sure it won't
384 * propagate right the way off the top.
386 carry
= internal_add(c
+ 2*len
- botlen
- 2*midlen
,
388 c
+ 2*len
- botlen
- 2*midlen
, 2*midlen
);
389 j
= 2*len
- botlen
- 2*midlen
- 1;
393 c
[j
] = (BignumInt
)carry
;
394 carry
>>= BIGNUM_INT_BITS
;
399 for (i
= 0; i
< 2*len
; i
++) {
400 printf("%0*x", BIGNUM_INT_BITS
/4, c
[i
]);
408 * Multiply in the ordinary O(N^2) way.
411 for (j
= 0; j
< 2 * len
; j
++)
414 for (i
= len
- 1; i
>= 0; i
--) {
416 for (j
= len
- 1; j
>= 0; j
--) {
417 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
418 t
+= (BignumDblInt
) c
[i
+ j
+ 1];
419 c
[i
+ j
+ 1] = (BignumInt
) t
;
420 t
= t
>> BIGNUM_INT_BITS
;
422 c
[i
] = (BignumInt
) t
;
428 * Variant form of internal_mul used for the initial step of
429 * Montgomery reduction. Only bothers outputting 'len' words
430 * (everything above that is thrown away).
432 static void internal_mul_low(const BignumInt
*a
, const BignumInt
*b
,
433 BignumInt
*c
, int len
, BignumInt
*scratch
)
438 if (len
> KARATSUBA_THRESHOLD
) {
441 * Karatsuba-aware version of internal_mul_low. As before, we
442 * express each input value as a shifted combination of two
448 * Then the full product is, as before,
450 * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
452 * Provided we choose D on the large side (so that a_0 and b_0
453 * are _at least_ as long as a_1 and b_1), we don't need the
454 * topmost term at all, and we only need half of the middle
455 * term. So there's no point in doing the proper Karatsuba
456 * optimisation which computes the middle term using the top
457 * one, because we'd take as long computing the top one as
458 * just computing the middle one directly.
460 * So instead, we do a much more obvious thing: we call the
461 * fully optimised internal_mul to compute a_0 b_0, and we
462 * recursively call ourself to compute the _bottom halves_ of
463 * a_1 b_0 and a_0 b_1, each of which we add into the result
464 * in the obvious way.
466 * In other words, there's no actual Karatsuba _optimisation_
467 * in this function; the only benefit in doing it this way is
468 * that we call internal_mul proper for a large part of the
469 * work, and _that_ can optimise its operation.
472 int toplen
= len
/2, botlen
= len
- toplen
; /* botlen is the bigger */
475 * Scratch space for the various bits and pieces we're going
476 * to be adding together: we need botlen*2 words for a_0 b_0
477 * (though we may end up throwing away its topmost word), and
478 * toplen words for each of a_1 b_0 and a_0 b_1. That adds up
483 internal_mul(a
+ toplen
, b
+ toplen
, scratch
+ 2*toplen
, botlen
,
487 internal_mul_low(a
, b
+ len
- toplen
, scratch
+ toplen
, toplen
,
491 internal_mul_low(a
+ len
- toplen
, b
, scratch
, toplen
,
494 /* Copy the bottom half of the big coefficient into place */
495 for (j
= 0; j
< botlen
; j
++)
496 c
[toplen
+ j
] = scratch
[2*toplen
+ botlen
+ j
];
498 /* Add the two small coefficients, throwing away the returned carry */
499 internal_add(scratch
, scratch
+ toplen
, scratch
, toplen
);
501 /* And add that to the large coefficient, leaving the result in c. */
502 internal_add(scratch
, scratch
+ 2*toplen
+ botlen
- toplen
,
507 for (j
= 0; j
< len
; j
++)
510 for (i
= len
- 1; i
>= 0; i
--) {
512 for (j
= len
- 1; j
>= len
- i
- 1; j
--) {
513 t
+= MUL_WORD(a
[i
], (BignumDblInt
) b
[j
]);
514 t
+= (BignumDblInt
) c
[i
+ j
+ 1 - len
];
515 c
[i
+ j
+ 1 - len
] = (BignumInt
) t
;
516 t
= t
>> BIGNUM_INT_BITS
;
524 * Montgomery reduction. Expects x to be a big-endian array of 2*len
525 * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
526 * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
527 * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
530 * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
531 * each, containing respectively n and the multiplicative inverse of
534 * 'tmp' is an array of BignumInt used as scratch space, of length at
535 * least 3*len + mul_compute_scratch(len).
537 static void monty_reduce(BignumInt
*x
, const BignumInt
*n
,
538 const BignumInt
*mninv
, BignumInt
*tmp
, int len
)
544 * Multiply x by (-n)^{-1} mod r. This gives us a value m such
545 * that mn is congruent to -x mod r. Hence, mn+x is an exact
546 * multiple of r, and is also (obviously) congruent to x mod n.
548 internal_mul_low(x
+ len
, mninv
, tmp
, len
, tmp
+ 3*len
);
551 * Compute t = (mn+x)/r in ordinary, non-modular, integer
552 * arithmetic. By construction this is exact, and is congruent mod
553 * n to x * r^{-1}, i.e. the answer we want.
555 * The following multiply leaves that answer in the _most_
556 * significant half of the 'x' array, so then we must shift it
559 internal_mul(tmp
, n
, tmp
+len
, len
, tmp
+ 3*len
);
560 carry
= internal_add(x
, tmp
+len
, x
, 2*len
);
561 for (i
= 0; i
< len
; i
++)
562 x
[len
+ i
] = x
[i
], x
[i
] = 0;
565 * Reduce t mod n. This doesn't require a full-on division by n,
566 * but merely a test and single optional subtraction, since we can
567 * show that 0 <= t < 2n.
570 * + we computed m mod r, so 0 <= m < r.
571 * + so 0 <= mn < rn, obviously
572 * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
573 * + yielding 0 <= (mn+x)/r < 2n as required.
576 for (i
= 0; i
< len
; i
++)
577 if (x
[len
+ i
] != n
[i
])
580 if (carry
|| i
>= len
|| x
[len
+ i
] > n
[i
])
581 internal_sub(x
+len
, n
, x
+len
, len
);
584 static void internal_add_shifted(BignumInt
*number
,
585 unsigned n
, int shift
)
587 int word
= 1 + (shift
/ BIGNUM_INT_BITS
);
588 int bshift
= shift
% BIGNUM_INT_BITS
;
591 addend
= (BignumDblInt
)n
<< bshift
;
594 addend
+= number
[word
];
595 number
[word
] = (BignumInt
) addend
& BIGNUM_INT_MASK
;
596 addend
>>= BIGNUM_INT_BITS
;
603 * Input in first alen words of a and first mlen words of m.
604 * Output in first alen words of a
605 * (of which first alen-mlen words will be zero).
606 * The MSW of m MUST have its high bit set.
607 * Quotient is accumulated in the `quotient' array, which is a Bignum
608 * rather than the internal bigendian format. Quotient parts are shifted
609 * left by `qshift' before adding into quot.
611 static void internal_mod(BignumInt
*a
, int alen
,
612 BignumInt
*m
, int mlen
,
613 BignumInt
*quot
, int qshift
)
625 for (i
= 0; i
<= alen
- mlen
; i
++) {
627 unsigned int q
, r
, c
, ai1
;
641 /* Find q = h:a[i] / m0 */
646 * To illustrate it, suppose a BignumInt is 8 bits, and
647 * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
648 * our initial division will be 0xA123 / 0xA1, which
649 * will give a quotient of 0x100 and a divide overflow.
650 * However, the invariants in this division algorithm
651 * are not violated, since the full number A1:23:... is
652 * _less_ than the quotient prefix A1:B2:... and so the
653 * following correction loop would have sorted it out.
655 * In this situation we set q to be the largest
656 * quotient we _can_ stomach (0xFF, of course).
660 /* Macro doesn't want an array subscript expression passed
661 * into it (see definition), so use a temporary. */
662 BignumInt tmplo
= a
[i
];
663 DIVMOD_WORD(q
, r
, h
, tmplo
, m0
);
665 /* Refine our estimate of q by looking at
666 h:a[i]:a[i+1] / m0:m1 */
668 if (t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) {
671 r
= (r
+ m0
) & BIGNUM_INT_MASK
; /* overflow? */
672 if (r
>= (BignumDblInt
) m0
&&
673 t
> ((BignumDblInt
) r
<< BIGNUM_INT_BITS
) + ai1
) q
--;
677 /* Subtract q * m from a[i...] */
679 for (k
= mlen
- 1; k
>= 0; k
--) {
680 t
= MUL_WORD(q
, m
[k
]);
682 c
= (unsigned)(t
>> BIGNUM_INT_BITS
);
683 if ((BignumInt
) t
> a
[i
+ k
])
685 a
[i
+ k
] -= (BignumInt
) t
;
688 /* Add back m in case of borrow */
691 for (k
= mlen
- 1; k
>= 0; k
--) {
694 a
[i
+ k
] = (BignumInt
) t
;
695 t
= t
>> BIGNUM_INT_BITS
;
700 internal_add_shifted(quot
, q
, qshift
+ BIGNUM_INT_BITS
* (alen
- mlen
- i
));
705 * Compute (base ^ exp) % mod, the pedestrian way.
707 Bignum
modpow_simple(Bignum base_in
, Bignum exp
, Bignum mod
)
709 BignumInt
*a
, *b
, *n
, *m
, *scratch
;
711 int mlen
, scratchlen
, i
, j
;
715 * The most significant word of mod needs to be non-zero. It
716 * should already be, but let's make sure.
718 assert(mod
[mod
[0]] != 0);
721 * Make sure the base is smaller than the modulus, by reducing
722 * it modulo the modulus if not.
724 base
= bigmod(base_in
, mod
);
726 /* Allocate m of size mlen, copy mod to m */
727 /* We use big endian internally */
729 m
= snewn(mlen
, BignumInt
);
730 for (j
= 0; j
< mlen
; j
++)
731 m
[j
] = mod
[mod
[0] - j
];
733 /* Shift m left to make msb bit set */
734 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
735 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
738 for (i
= 0; i
< mlen
- 1; i
++)
739 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
740 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
743 /* Allocate n of size mlen, copy base to n */
744 n
= snewn(mlen
, BignumInt
);
746 for (j
= 0; j
< i
; j
++)
748 for (j
= 0; j
< (int)base
[0]; j
++)
749 n
[i
+ j
] = base
[base
[0] - j
];
751 /* Allocate a and b of size 2*mlen. Set a = 1 */
752 a
= snewn(2 * mlen
, BignumInt
);
753 b
= snewn(2 * mlen
, BignumInt
);
754 for (i
= 0; i
< 2 * mlen
; i
++)
758 /* Scratch space for multiplies */
759 scratchlen
= mul_compute_scratch(mlen
);
760 scratch
= snewn(scratchlen
, BignumInt
);
762 /* Skip leading zero bits of exp. */
764 j
= BIGNUM_INT_BITS
-1;
765 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
769 j
= BIGNUM_INT_BITS
-1;
773 /* Main computation */
774 while (i
< (int)exp
[0]) {
776 internal_mul(a
+ mlen
, a
+ mlen
, b
, mlen
, scratch
);
777 internal_mod(b
, mlen
* 2, m
, mlen
, NULL
, 0);
778 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
779 internal_mul(b
+ mlen
, n
, a
, mlen
, scratch
);
780 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
790 j
= BIGNUM_INT_BITS
-1;
793 /* Fixup result in case the modulus was shifted */
795 for (i
= mlen
- 1; i
< 2 * mlen
- 1; i
++)
796 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
797 a
[2 * mlen
- 1] = a
[2 * mlen
- 1] << mshift
;
798 internal_mod(a
, mlen
* 2, m
, mlen
, NULL
, 0);
799 for (i
= 2 * mlen
- 1; i
>= mlen
; i
--)
800 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
803 /* Copy result to buffer */
804 result
= newbn(mod
[0]);
805 for (i
= 0; i
< mlen
; i
++)
806 result
[result
[0] - i
] = a
[i
+ mlen
];
807 while (result
[0] > 1 && result
[result
[0]] == 0)
810 /* Free temporary arrays */
811 for (i
= 0; i
< 2 * mlen
; i
++)
814 for (i
= 0; i
< scratchlen
; i
++)
817 for (i
= 0; i
< 2 * mlen
; i
++)
820 for (i
= 0; i
< mlen
; i
++)
823 for (i
= 0; i
< mlen
; i
++)
833 * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
834 * technique where possible, falling back to modpow_simple otherwise.
836 Bignum
modpow(Bignum base_in
, Bignum exp
, Bignum mod
)
838 BignumInt
*a
, *b
, *x
, *n
, *mninv
, *scratch
;
839 int len
, scratchlen
, i
, j
;
840 Bignum base
, base2
, r
, rn
, inv
, result
;
843 * The most significant word of mod needs to be non-zero. It
844 * should already be, but let's make sure.
846 assert(mod
[mod
[0]] != 0);
849 * mod had better be odd, or we can't do Montgomery multiplication
850 * using a power of two at all.
853 return modpow_simple(base_in
, exp
, mod
);
856 * Make sure the base is smaller than the modulus, by reducing
857 * it modulo the modulus if not.
859 base
= bigmod(base_in
, mod
);
862 * Compute the inverse of n mod r, for monty_reduce. (In fact we
863 * want the inverse of _minus_ n mod r, but we'll sort that out
867 r
= bn_power_2(BIGNUM_INT_BITS
* len
);
868 inv
= modinv(mod
, r
);
871 * Multiply the base by r mod n, to get it into Montgomery
874 base2
= modmul(base
, r
, mod
);
878 rn
= bigmod(r
, mod
); /* r mod n, i.e. Montgomerified 1 */
880 freebn(r
); /* won't need this any more */
883 * Set up internal arrays of the right lengths, in big-endian
884 * format, containing the base, the modulus, and the modulus's
887 n
= snewn(len
, BignumInt
);
888 for (j
= 0; j
< len
; j
++)
889 n
[len
- 1 - j
] = mod
[j
+ 1];
891 mninv
= snewn(len
, BignumInt
);
892 for (j
= 0; j
< len
; j
++)
893 mninv
[len
- 1 - j
] = (j
< inv
[0] ? inv
[j
+ 1] : 0);
894 freebn(inv
); /* we don't need this copy of it any more */
895 /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
896 x
= snewn(len
, BignumInt
);
897 for (j
= 0; j
< len
; j
++)
899 internal_sub(x
, mninv
, mninv
, len
);
901 /* x = snewn(len, BignumInt); */ /* already done above */
902 for (j
= 0; j
< len
; j
++)
903 x
[len
- 1 - j
] = (j
< base
[0] ? base
[j
+ 1] : 0);
904 freebn(base
); /* we don't need this copy of it any more */
906 a
= snewn(2*len
, BignumInt
);
907 b
= snewn(2*len
, BignumInt
);
908 for (j
= 0; j
< len
; j
++)
909 a
[2*len
- 1 - j
] = (j
< rn
[0] ? rn
[j
+ 1] : 0);
912 /* Scratch space for multiplies */
913 scratchlen
= 3*len
+ mul_compute_scratch(len
);
914 scratch
= snewn(scratchlen
, BignumInt
);
916 /* Skip leading zero bits of exp. */
918 j
= BIGNUM_INT_BITS
-1;
919 while (i
< (int)exp
[0] && (exp
[exp
[0] - i
] & (1 << j
)) == 0) {
923 j
= BIGNUM_INT_BITS
-1;
927 /* Main computation */
928 while (i
< (int)exp
[0]) {
930 internal_mul(a
+ len
, a
+ len
, b
, len
, scratch
);
931 monty_reduce(b
, n
, mninv
, scratch
, len
);
932 if ((exp
[exp
[0] - i
] & (1 << j
)) != 0) {
933 internal_mul(b
+ len
, x
, a
, len
, scratch
);
934 monty_reduce(a
, n
, mninv
, scratch
, len
);
944 j
= BIGNUM_INT_BITS
-1;
948 * Final monty_reduce to get back from the adjusted Montgomery
951 monty_reduce(a
, n
, mninv
, scratch
, len
);
953 /* Copy result to buffer */
954 result
= newbn(mod
[0]);
955 for (i
= 0; i
< len
; i
++)
956 result
[result
[0] - i
] = a
[i
+ len
];
957 while (result
[0] > 1 && result
[result
[0]] == 0)
960 /* Free temporary arrays */
961 for (i
= 0; i
< scratchlen
; i
++)
964 for (i
= 0; i
< 2 * len
; i
++)
967 for (i
= 0; i
< 2 * len
; i
++)
970 for (i
= 0; i
< len
; i
++)
973 for (i
= 0; i
< len
; i
++)
976 for (i
= 0; i
< len
; i
++)
984 * Compute (p * q) % mod.
985 * The most significant word of mod MUST be non-zero.
986 * We assume that the result array is the same size as the mod array.
988 Bignum
modmul(Bignum p
, Bignum q
, Bignum mod
)
990 BignumInt
*a
, *n
, *m
, *o
, *scratch
;
991 int mshift
, scratchlen
;
992 int pqlen
, mlen
, rlen
, i
, j
;
995 /* Allocate m of size mlen, copy mod to m */
996 /* We use big endian internally */
998 m
= snewn(mlen
, BignumInt
);
999 for (j
= 0; j
< mlen
; j
++)
1000 m
[j
] = mod
[mod
[0] - j
];
1002 /* Shift m left to make msb bit set */
1003 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
1004 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
1007 for (i
= 0; i
< mlen
- 1; i
++)
1008 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1009 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
1012 pqlen
= (p
[0] > q
[0] ? p
[0] : q
[0]);
1014 /* Allocate n of size pqlen, copy p to n */
1015 n
= snewn(pqlen
, BignumInt
);
1017 for (j
= 0; j
< i
; j
++)
1019 for (j
= 0; j
< (int)p
[0]; j
++)
1020 n
[i
+ j
] = p
[p
[0] - j
];
1022 /* Allocate o of size pqlen, copy q to o */
1023 o
= snewn(pqlen
, BignumInt
);
1025 for (j
= 0; j
< i
; j
++)
1027 for (j
= 0; j
< (int)q
[0]; j
++)
1028 o
[i
+ j
] = q
[q
[0] - j
];
1030 /* Allocate a of size 2*pqlen for result */
1031 a
= snewn(2 * pqlen
, BignumInt
);
1033 /* Scratch space for multiplies */
1034 scratchlen
= mul_compute_scratch(pqlen
);
1035 scratch
= snewn(scratchlen
, BignumInt
);
1037 /* Main computation */
1038 internal_mul(n
, o
, a
, pqlen
, scratch
);
1039 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
1041 /* Fixup result in case the modulus was shifted */
1043 for (i
= 2 * pqlen
- mlen
- 1; i
< 2 * pqlen
- 1; i
++)
1044 a
[i
] = (a
[i
] << mshift
) | (a
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1045 a
[2 * pqlen
- 1] = a
[2 * pqlen
- 1] << mshift
;
1046 internal_mod(a
, pqlen
* 2, m
, mlen
, NULL
, 0);
1047 for (i
= 2 * pqlen
- 1; i
>= 2 * pqlen
- mlen
; i
--)
1048 a
[i
] = (a
[i
] >> mshift
) | (a
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
1051 /* Copy result to buffer */
1052 rlen
= (mlen
< pqlen
* 2 ? mlen
: pqlen
* 2);
1053 result
= newbn(rlen
);
1054 for (i
= 0; i
< rlen
; i
++)
1055 result
[result
[0] - i
] = a
[i
+ 2 * pqlen
- rlen
];
1056 while (result
[0] > 1 && result
[result
[0]] == 0)
1059 /* Free temporary arrays */
1060 for (i
= 0; i
< scratchlen
; i
++)
1063 for (i
= 0; i
< 2 * pqlen
; i
++)
1066 for (i
= 0; i
< mlen
; i
++)
1069 for (i
= 0; i
< pqlen
; i
++)
1072 for (i
= 0; i
< pqlen
; i
++)
1081 * The most significant word of mod MUST be non-zero.
1082 * We assume that the result array is the same size as the mod array.
1083 * We optionally write out a quotient if `quotient' is non-NULL.
1084 * We can avoid writing out the result if `result' is NULL.
1086 static void bigdivmod(Bignum p
, Bignum mod
, Bignum result
, Bignum quotient
)
1090 int plen
, mlen
, i
, j
;
1092 /* Allocate m of size mlen, copy mod to m */
1093 /* We use big endian internally */
1095 m
= snewn(mlen
, BignumInt
);
1096 for (j
= 0; j
< mlen
; j
++)
1097 m
[j
] = mod
[mod
[0] - j
];
1099 /* Shift m left to make msb bit set */
1100 for (mshift
= 0; mshift
< BIGNUM_INT_BITS
-1; mshift
++)
1101 if ((m
[0] << mshift
) & BIGNUM_TOP_BIT
)
1104 for (i
= 0; i
< mlen
- 1; i
++)
1105 m
[i
] = (m
[i
] << mshift
) | (m
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1106 m
[mlen
- 1] = m
[mlen
- 1] << mshift
;
1110 /* Ensure plen > mlen */
1114 /* Allocate n of size plen, copy p to n */
1115 n
= snewn(plen
, BignumInt
);
1116 for (j
= 0; j
< plen
; j
++)
1118 for (j
= 1; j
<= (int)p
[0]; j
++)
1121 /* Main computation */
1122 internal_mod(n
, plen
, m
, mlen
, quotient
, mshift
);
1124 /* Fixup result in case the modulus was shifted */
1126 for (i
= plen
- mlen
- 1; i
< plen
- 1; i
++)
1127 n
[i
] = (n
[i
] << mshift
) | (n
[i
+ 1] >> (BIGNUM_INT_BITS
- mshift
));
1128 n
[plen
- 1] = n
[plen
- 1] << mshift
;
1129 internal_mod(n
, plen
, m
, mlen
, quotient
, 0);
1130 for (i
= plen
- 1; i
>= plen
- mlen
; i
--)
1131 n
[i
] = (n
[i
] >> mshift
) | (n
[i
- 1] << (BIGNUM_INT_BITS
- mshift
));
1134 /* Copy result to buffer */
1136 for (i
= 1; i
<= (int)result
[0]; i
++) {
1138 result
[i
] = j
>= 0 ? n
[j
] : 0;
1142 /* Free temporary arrays */
1143 for (i
= 0; i
< mlen
; i
++)
1146 for (i
= 0; i
< plen
; i
++)
1152 * Decrement a number.
1154 void decbn(Bignum bn
)
1157 while (i
< (int)bn
[0] && bn
[i
] == 0)
1158 bn
[i
++] = BIGNUM_INT_MASK
;
1162 Bignum
bignum_from_bytes(const unsigned char *data
, int nbytes
)
1167 w
= (nbytes
+ BIGNUM_INT_BYTES
- 1) / BIGNUM_INT_BYTES
; /* bytes->words */
1170 for (i
= 1; i
<= w
; i
++)
1172 for (i
= nbytes
; i
--;) {
1173 unsigned char byte
= *data
++;
1174 result
[1 + i
/ BIGNUM_INT_BYTES
] |= byte
<< (8*i
% BIGNUM_INT_BITS
);
1177 while (result
[0] > 1 && result
[result
[0]] == 0)
1183 * Read an SSH-1-format bignum from a data buffer. Return the number
1184 * of bytes consumed, or -1 if there wasn't enough data.
1186 int ssh1_read_bignum(const unsigned char *data
, int len
, Bignum
* result
)
1188 const unsigned char *p
= data
;
1196 for (i
= 0; i
< 2; i
++)
1197 w
= (w
<< 8) + *p
++;
1198 b
= (w
+ 7) / 8; /* bits -> bytes */
1203 if (!result
) /* just return length */
1206 *result
= bignum_from_bytes(p
, b
);
1208 return p
+ b
- data
;
1212 * Return the bit count of a bignum, for SSH-1 encoding.
1214 int bignum_bitcount(Bignum bn
)
1216 int bitcount
= bn
[0] * BIGNUM_INT_BITS
- 1;
1217 while (bitcount
>= 0
1218 && (bn
[bitcount
/ BIGNUM_INT_BITS
+ 1] >> (bitcount
% BIGNUM_INT_BITS
)) == 0) bitcount
--;
1219 return bitcount
+ 1;
1223 * Return the byte length of a bignum when SSH-1 encoded.
1225 int ssh1_bignum_length(Bignum bn
)
1227 return 2 + (bignum_bitcount(bn
) + 7) / 8;
1231 * Return the byte length of a bignum when SSH-2 encoded.
1233 int ssh2_bignum_length(Bignum bn
)
1235 return 4 + (bignum_bitcount(bn
) + 8) / 8;
1239 * Return a byte from a bignum; 0 is least significant, etc.
1241 int bignum_byte(Bignum bn
, int i
)
1243 if (i
>= (int)(BIGNUM_INT_BYTES
* bn
[0]))
1244 return 0; /* beyond the end */
1246 return (bn
[i
/ BIGNUM_INT_BYTES
+ 1] >>
1247 ((i
% BIGNUM_INT_BYTES
)*8)) & 0xFF;
1251 * Return a bit from a bignum; 0 is least significant, etc.
1253 int bignum_bit(Bignum bn
, int i
)
1255 if (i
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1256 return 0; /* beyond the end */
1258 return (bn
[i
/ BIGNUM_INT_BITS
+ 1] >> (i
% BIGNUM_INT_BITS
)) & 1;
1262 * Set a bit in a bignum; 0 is least significant, etc.
1264 void bignum_set_bit(Bignum bn
, int bitnum
, int value
)
1266 if (bitnum
>= (int)(BIGNUM_INT_BITS
* bn
[0]))
1267 abort(); /* beyond the end */
1269 int v
= bitnum
/ BIGNUM_INT_BITS
+ 1;
1270 int mask
= 1 << (bitnum
% BIGNUM_INT_BITS
);
1279 * Write a SSH-1-format bignum into a buffer. It is assumed the
1280 * buffer is big enough. Returns the number of bytes used.
1282 int ssh1_write_bignum(void *data
, Bignum bn
)
1284 unsigned char *p
= data
;
1285 int len
= ssh1_bignum_length(bn
);
1287 int bitc
= bignum_bitcount(bn
);
1289 *p
++ = (bitc
>> 8) & 0xFF;
1290 *p
++ = (bitc
) & 0xFF;
1291 for (i
= len
- 2; i
--;)
1292 *p
++ = bignum_byte(bn
, i
);
1297 * Compare two bignums. Returns like strcmp.
1299 int bignum_cmp(Bignum a
, Bignum b
)
1301 int amax
= a
[0], bmax
= b
[0];
1302 int i
= (amax
> bmax ? amax
: bmax
);
1304 BignumInt aval
= (i
> amax ?
0 : a
[i
]);
1305 BignumInt bval
= (i
> bmax ?
0 : b
[i
]);
1316 * Right-shift one bignum to form another.
1318 Bignum
bignum_rshift(Bignum a
, int shift
)
1321 int i
, shiftw
, shiftb
, shiftbb
, bits
;
1324 bits
= bignum_bitcount(a
) - shift
;
1325 ret
= newbn((bits
+ BIGNUM_INT_BITS
- 1) / BIGNUM_INT_BITS
);
1328 shiftw
= shift
/ BIGNUM_INT_BITS
;
1329 shiftb
= shift
% BIGNUM_INT_BITS
;
1330 shiftbb
= BIGNUM_INT_BITS
- shiftb
;
1332 ai1
= a
[shiftw
+ 1];
1333 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1335 ai1
= (i
+ shiftw
+ 1 <= (int)a
[0] ? a
[i
+ shiftw
+ 1] : 0);
1336 ret
[i
] = ((ai
>> shiftb
) | (ai1
<< shiftbb
)) & BIGNUM_INT_MASK
;
1344 * Non-modular multiplication and addition.
1346 Bignum
bigmuladd(Bignum a
, Bignum b
, Bignum addend
)
1348 int alen
= a
[0], blen
= b
[0];
1349 int mlen
= (alen
> blen ? alen
: blen
);
1350 int rlen
, i
, maxspot
;
1352 BignumInt
*workspace
;
1355 /* mlen space for a, mlen space for b, 2*mlen for result,
1356 * plus scratch space for multiplication */
1357 wslen
= mlen
* 4 + mul_compute_scratch(mlen
);
1358 workspace
= snewn(wslen
, BignumInt
);
1359 for (i
= 0; i
< mlen
; i
++) {
1360 workspace
[0 * mlen
+ i
] = (mlen
- i
<= (int)a
[0] ? a
[mlen
- i
] : 0);
1361 workspace
[1 * mlen
+ i
] = (mlen
- i
<= (int)b
[0] ? b
[mlen
- i
] : 0);
1364 internal_mul(workspace
+ 0 * mlen
, workspace
+ 1 * mlen
,
1365 workspace
+ 2 * mlen
, mlen
, workspace
+ 4 * mlen
);
1367 /* now just copy the result back */
1368 rlen
= alen
+ blen
+ 1;
1369 if (addend
&& rlen
<= (int)addend
[0])
1370 rlen
= addend
[0] + 1;
1373 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1374 ret
[i
] = (i
<= 2 * mlen ? workspace
[4 * mlen
- i
] : 0);
1380 /* now add in the addend, if any */
1382 BignumDblInt carry
= 0;
1383 for (i
= 1; i
<= rlen
; i
++) {
1384 carry
+= (i
<= (int)ret
[0] ? ret
[i
] : 0);
1385 carry
+= (i
<= (int)addend
[0] ? addend
[i
] : 0);
1386 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1387 carry
>>= BIGNUM_INT_BITS
;
1388 if (ret
[i
] != 0 && i
> maxspot
)
1394 for (i
= 0; i
< wslen
; i
++)
1401 * Non-modular multiplication.
1403 Bignum
bigmul(Bignum a
, Bignum b
)
1405 return bigmuladd(a
, b
, NULL
);
1411 Bignum
bigadd(Bignum a
, Bignum b
)
1413 int alen
= a
[0], blen
= b
[0];
1414 int rlen
= (alen
> blen ? alen
: blen
) + 1;
1423 for (i
= 1; i
<= rlen
; i
++) {
1424 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1425 carry
+= (i
<= (int)b
[0] ? b
[i
] : 0);
1426 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1427 carry
>>= BIGNUM_INT_BITS
;
1428 if (ret
[i
] != 0 && i
> maxspot
)
1437 * Subtraction. Returns a-b, or NULL if the result would come out
1438 * negative (recall that this entire bignum module only handles
1439 * positive numbers).
1441 Bignum
bigsub(Bignum a
, Bignum b
)
1443 int alen
= a
[0], blen
= b
[0];
1444 int rlen
= (alen
> blen ? alen
: blen
);
1453 for (i
= 1; i
<= rlen
; i
++) {
1454 carry
+= (i
<= (int)a
[0] ? a
[i
] : 0);
1455 carry
+= (i
<= (int)b
[0] ? b
[i
] ^ BIGNUM_INT_MASK
: BIGNUM_INT_MASK
);
1456 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1457 carry
>>= BIGNUM_INT_BITS
;
1458 if (ret
[i
] != 0 && i
> maxspot
)
1472 * Create a bignum which is the bitmask covering another one. That
1473 * is, the smallest integer which is >= N and is also one less than
1476 Bignum
bignum_bitmask(Bignum n
)
1478 Bignum ret
= copybn(n
);
1483 while (n
[i
] == 0 && i
> 0)
1486 return ret
; /* input was zero */
1492 ret
[i
] = BIGNUM_INT_MASK
;
1497 * Convert a (max 32-bit) long into a bignum.
1499 Bignum
bignum_from_long(unsigned long nn
)
1502 BignumDblInt n
= nn
;
1505 ret
[1] = (BignumInt
)(n
& BIGNUM_INT_MASK
);
1506 ret
[2] = (BignumInt
)((n
>> BIGNUM_INT_BITS
) & BIGNUM_INT_MASK
);
1508 ret
[0] = (ret
[2] ?
2 : 1);
1513 * Add a long to a bignum.
1515 Bignum
bignum_add_long(Bignum number
, unsigned long addendx
)
1517 Bignum ret
= newbn(number
[0] + 1);
1519 BignumDblInt carry
= 0, addend
= addendx
;
1521 for (i
= 1; i
<= (int)ret
[0]; i
++) {
1522 carry
+= addend
& BIGNUM_INT_MASK
;
1523 carry
+= (i
<= (int)number
[0] ? number
[i
] : 0);
1524 addend
>>= BIGNUM_INT_BITS
;
1525 ret
[i
] = (BignumInt
) carry
& BIGNUM_INT_MASK
;
1526 carry
>>= BIGNUM_INT_BITS
;
1535 * Compute the residue of a bignum, modulo a (max 16-bit) short.
1537 unsigned short bignum_mod_short(Bignum number
, unsigned short modulus
)
1539 BignumDblInt mod
, r
;
1544 for (i
= number
[0]; i
> 0; i
--)
1545 r
= (r
* (BIGNUM_TOP_BIT
% mod
) * 2 + number
[i
] % mod
) % mod
;
1546 return (unsigned short) r
;
1550 void diagbn(char *prefix
, Bignum md
)
1552 int i
, nibbles
, morenibbles
;
1553 static const char hex
[] = "0123456789ABCDEF";
1555 debug(("%s0x", prefix ? prefix
: ""));
1557 nibbles
= (3 + bignum_bitcount(md
)) / 4;
1560 morenibbles
= 4 * md
[0] - nibbles
;
1561 for (i
= 0; i
< morenibbles
; i
++)
1563 for (i
= nibbles
; i
--;)
1565 hex
[(bignum_byte(md
, i
/ 2) >> (4 * (i
% 2))) & 0xF]));
1575 Bignum
bigdiv(Bignum a
, Bignum b
)
1577 Bignum q
= newbn(a
[0]);
1578 bigdivmod(a
, b
, NULL
, q
);
1585 Bignum
bigmod(Bignum a
, Bignum b
)
1587 Bignum r
= newbn(b
[0]);
1588 bigdivmod(a
, b
, r
, NULL
);
1593 * Greatest common divisor.
1595 Bignum
biggcd(Bignum av
, Bignum bv
)
1597 Bignum a
= copybn(av
);
1598 Bignum b
= copybn(bv
);
1600 while (bignum_cmp(b
, Zero
) != 0) {
1601 Bignum t
= newbn(b
[0]);
1602 bigdivmod(a
, b
, t
, NULL
);
1603 while (t
[0] > 1 && t
[t
[0]] == 0)
1615 * Modular inverse, using Euclid's extended algorithm.
1617 Bignum
modinv(Bignum number
, Bignum modulus
)
1619 Bignum a
= copybn(modulus
);
1620 Bignum b
= copybn(number
);
1621 Bignum xp
= copybn(Zero
);
1622 Bignum x
= copybn(One
);
1625 while (bignum_cmp(b
, One
) != 0) {
1626 Bignum t
= newbn(b
[0]);
1627 Bignum q
= newbn(a
[0]);
1628 bigdivmod(a
, b
, t
, q
);
1629 while (t
[0] > 1 && t
[t
[0]] == 0)
1636 x
= bigmuladd(q
, xp
, t
);
1646 /* now we know that sign * x == 1, and that x < modulus */
1648 /* set a new x to be modulus - x */
1649 Bignum newx
= newbn(modulus
[0]);
1650 BignumInt carry
= 0;
1654 for (i
= 1; i
<= (int)newx
[0]; i
++) {
1655 BignumInt aword
= (i
<= (int)modulus
[0] ? modulus
[i
] : 0);
1656 BignumInt bword
= (i
<= (int)x
[0] ? x
[i
] : 0);
1657 newx
[i
] = aword
- bword
- carry
;
1659 carry
= carry ?
(newx
[i
] >= bword
) : (newx
[i
] > bword
);
1673 * Render a bignum into decimal. Return a malloced string holding
1674 * the decimal representation.
1676 char *bignum_decimal(Bignum x
)
1678 int ndigits
, ndigit
;
1682 BignumInt
*workspace
;
1685 * First, estimate the number of digits. Since log(10)/log(2)
1686 * is just greater than 93/28 (the joys of continued fraction
1687 * approximations...) we know that for every 93 bits, we need
1688 * at most 28 digits. This will tell us how much to malloc.
1690 * Formally: if x has i bits, that means x is strictly less
1691 * than 2^i. Since 2 is less than 10^(28/93), this is less than
1692 * 10^(28i/93). We need an integer power of ten, so we must
1693 * round up (rounding down might make it less than x again).
1694 * Therefore if we multiply the bit count by 28/93, rounding
1695 * up, we will have enough digits.
1697 * i=0 (i.e., x=0) is an irritating special case.
1699 i
= bignum_bitcount(x
);
1701 ndigits
= 1; /* x = 0 */
1703 ndigits
= (28 * i
+ 92) / 93; /* multiply by 28/93 and round up */
1704 ndigits
++; /* allow for trailing \0 */
1705 ret
= snewn(ndigits
, char);
1708 * Now allocate some workspace to hold the binary form as we
1709 * repeatedly divide it by ten. Initialise this to the
1710 * big-endian form of the number.
1712 workspace
= snewn(x
[0], BignumInt
);
1713 for (i
= 0; i
< (int)x
[0]; i
++)
1714 workspace
[i
] = x
[x
[0] - i
];
1717 * Next, write the decimal number starting with the last digit.
1718 * We use ordinary short division, dividing 10 into the
1721 ndigit
= ndigits
- 1;
1726 for (i
= 0; i
< (int)x
[0]; i
++) {
1727 carry
= (carry
<< BIGNUM_INT_BITS
) + workspace
[i
];
1728 workspace
[i
] = (BignumInt
) (carry
/ 10);
1733 ret
[--ndigit
] = (char) (carry
+ '0');
1737 * There's a chance we've fallen short of the start of the
1738 * string. Correct if so.
1741 memmove(ret
, ret
+ ndigit
, ndigits
- ndigit
);
1757 * gcc -g -O0 -DTESTBN -o testbn sshbn.c misc.c -I unix -I charset
1759 * Then feed to this program's standard input the output of
1760 * testdata/bignum.py .
1763 void modalfatalbox(char *p
, ...)
1766 fprintf(stderr
, "FATAL ERROR: ");
1768 vfprintf(stderr
, p
, ap
);
1770 fputc('\n', stderr
);
1774 #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
1776 int main(int argc
, char **argv
)
1780 int passes
= 0, fails
= 0;
1782 while ((buf
= fgetline(stdin
)) != NULL
) {
1783 int maxlen
= strlen(buf
);
1784 unsigned char *data
= snewn(maxlen
, unsigned char);
1785 unsigned char *ptrs
[5], *q
;
1794 while (*bufp
&& !isspace((unsigned char)*bufp
))
1803 while (*bufp
&& !isxdigit((unsigned char)*bufp
))
1810 while (*bufp
&& isxdigit((unsigned char)*bufp
))
1814 if (ptrnum
>= lenof(ptrs
))
1818 for (i
= -((end
- start
) & 1); i
< end
-start
; i
+= 2) {
1819 unsigned char val
= (i
< 0 ?
0 : fromxdigit(start
[i
]));
1820 val
= val
* 16 + fromxdigit(start
[i
+1]);
1827 if (!strcmp(buf
, "mul")) {
1831 printf("%d: mul with %d parameters, expected 3\n", line
);
1834 a
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1835 b
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1836 c
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1839 if (bignum_cmp(c
, p
) == 0) {
1842 char *as
= bignum_decimal(a
);
1843 char *bs
= bignum_decimal(b
);
1844 char *cs
= bignum_decimal(c
);
1845 char *ps
= bignum_decimal(p
);
1847 printf("%d: fail: %s * %s gave %s expected %s\n",
1848 line
, as
, bs
, ps
, cs
);
1860 } else if (!strcmp(buf
, "pow")) {
1861 Bignum base
, expt
, modulus
, expected
, answer
;
1864 printf("%d: mul with %d parameters, expected 3\n", line
);
1868 base
= bignum_from_bytes(ptrs
[0], ptrs
[1]-ptrs
[0]);
1869 expt
= bignum_from_bytes(ptrs
[1], ptrs
[2]-ptrs
[1]);
1870 modulus
= bignum_from_bytes(ptrs
[2], ptrs
[3]-ptrs
[2]);
1871 expected
= bignum_from_bytes(ptrs
[3], ptrs
[4]-ptrs
[3]);
1872 answer
= modpow(base
, expt
, modulus
);
1874 if (bignum_cmp(expected
, answer
) == 0) {
1877 char *as
= bignum_decimal(base
);
1878 char *bs
= bignum_decimal(expt
);
1879 char *cs
= bignum_decimal(modulus
);
1880 char *ds
= bignum_decimal(answer
);
1881 char *ps
= bignum_decimal(expected
);
1883 printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",
1884 line
, as
, bs
, cs
, ds
, ps
);
1899 printf("%d: unrecognised test keyword: '%s'\n", line
, buf
);
1907 printf("passed %d failed %d total %d\n", passes
, fails
, passes
+fails
);