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1 | /* | |
2 | * Bignum routines for RSA and DH and stuff. | |
3 | */ | |
4 | ||
5 | #include <stdio.h> | |
6 | #include <assert.h> | |
7 | #include <stdlib.h> | |
8 | #include <string.h> | |
9 | ||
10 | #include "misc.h" | |
11 | ||
12 | /* | |
13 | * Usage notes: | |
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 | |
18 | * to avoid this case. | |
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. | |
24 | */ | |
25 | ||
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) \ | |
34 | __asm__("div %2" : \ | |
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 { \ | |
48 | __asm mov edx, hi \ | |
49 | __asm mov eax, lo \ | |
50 | __asm div w \ | |
51 | __asm mov r, edx \ | |
52 | __asm mov q, eax \ | |
53 | } while(0) | |
54 | #elif defined _LP64 | |
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; \ | |
64 | q = n / w; \ | |
65 | r = n % w; \ | |
66 | } while (0) | |
67 | #elif defined _LLP64 | |
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; \ | |
77 | q = n / w; \ | |
78 | r = n % w; \ | |
79 | } while (0) | |
80 | #else | |
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; \ | |
90 | q = n / w; \ | |
91 | r = n % w; \ | |
92 | } while (0) | |
93 | #endif | |
94 | ||
95 | #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8) | |
96 | ||
97 | #define BIGNUM_INTERNAL | |
98 | typedef BignumInt *Bignum; | |
99 | ||
100 | #include "ssh.h" | |
101 | ||
102 | BignumInt bnZero[1] = { 0 }; | |
103 | BignumInt bnOne[2] = { 1, 1 }; | |
104 | ||
105 | /* | |
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.) | |
111 | * | |
112 | * All Bignums in this module are positive. Negative numbers must | |
113 | * be dealt with outside it. | |
114 | * | |
115 | * INVARIANT: the most significant word of any Bignum must be | |
116 | * nonzero. | |
117 | */ | |
118 | ||
119 | Bignum Zero = bnZero, One = bnOne; | |
120 | ||
121 | static Bignum newbn(int length) | |
122 | { | |
123 | Bignum b = snewn(length + 1, BignumInt); | |
124 | if (!b) | |
125 | abort(); /* FIXME */ | |
126 | memset(b, 0, (length + 1) * sizeof(*b)); | |
127 | b[0] = length; | |
128 | return b; | |
129 | } | |
130 | ||
131 | void bn_restore_invariant(Bignum b) | |
132 | { | |
133 | while (b[0] > 1 && b[b[0]] == 0) | |
134 | b[0]--; | |
135 | } | |
136 | ||
137 | Bignum copybn(Bignum orig) | |
138 | { | |
139 | Bignum b = snewn(orig[0] + 1, BignumInt); | |
140 | if (!b) | |
141 | abort(); /* FIXME */ | |
142 | memcpy(b, orig, (orig[0] + 1) * sizeof(*b)); | |
143 | return b; | |
144 | } | |
145 | ||
146 | void freebn(Bignum b) | |
147 | { | |
148 | /* | |
149 | * Burn the evidence, just in case. | |
150 | */ | |
151 | memset(b, 0, sizeof(b[0]) * (b[0] + 1)); | |
152 | sfree(b); | |
153 | } | |
154 | ||
155 | Bignum bn_power_2(int n) | |
156 | { | |
157 | Bignum ret = newbn(n / BIGNUM_INT_BITS + 1); | |
158 | bignum_set_bit(ret, n, 1); | |
159 | return ret; | |
160 | } | |
161 | ||
162 | /* | |
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 | |
165 | * off the top. | |
166 | */ | |
167 | static BignumInt internal_add(const BignumInt *a, const BignumInt *b, | |
168 | BignumInt *c, int len) | |
169 | { | |
170 | int i; | |
171 | BignumDblInt carry = 0; | |
172 | ||
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; | |
177 | } | |
178 | ||
179 | return (BignumInt)carry; | |
180 | } | |
181 | ||
182 | /* | |
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 | |
185 | * is ignored. | |
186 | */ | |
187 | static void internal_sub(const BignumInt *a, const BignumInt *b, | |
188 | BignumInt *c, int len) | |
189 | { | |
190 | int i; | |
191 | BignumDblInt carry = 1; | |
192 | ||
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; | |
197 | } | |
198 | } | |
199 | ||
200 | /* | |
201 | * Compute c = a * b. | |
202 | * Input is in the first len words of a and b. | |
203 | * Result is returned in the first 2*len words of c. | |
204 | * | |
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.) | |
208 | */ | |
209 | #define KARATSUBA_THRESHOLD 50 | |
210 | static int mul_compute_scratch(int len) | |
211 | { | |
212 | int ret = 0; | |
213 | while (len > KARATSUBA_THRESHOLD) { | |
214 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ | |
215 | int midlen = botlen + 1; | |
216 | ret += 4*midlen; | |
217 | len = midlen; | |
218 | } | |
219 | return ret; | |
220 | } | |
221 | static void internal_mul(const BignumInt *a, const BignumInt *b, | |
222 | BignumInt *c, int len, BignumInt *scratch) | |
223 | { | |
224 | if (len > KARATSUBA_THRESHOLD) { | |
225 | int i; | |
226 | ||
227 | /* | |
228 | * Karatsuba divide-and-conquer algorithm. Cut each input in | |
229 | * half, so that it's expressed as two big 'digits' in a giant | |
230 | * base D: | |
231 | * | |
232 | * a = a_1 D + a_0 | |
233 | * b = b_1 D + b_0 | |
234 | * | |
235 | * Then the product is of course | |
236 | * | |
237 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 | |
238 | * | |
239 | * and we compute the three coefficients by recursively | |
240 | * calling ourself to do half-length multiplications. | |
241 | * | |
242 | * The clever bit that makes this worth doing is that we only | |
243 | * need _one_ half-length multiplication for the central | |
244 | * coefficient rather than the two that it obviouly looks | |
245 | * like, because we can use a single multiplication to compute | |
246 | * | |
247 | * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0 | |
248 | * | |
249 | * and then we subtract the other two coefficients (a_1 b_1 | |
250 | * and a_0 b_0) which we were computing anyway. | |
251 | * | |
252 | * Hence we get to multiply two numbers of length N in about | |
253 | * three times as much work as it takes to multiply numbers of | |
254 | * length N/2, which is obviously better than the four times | |
255 | * as much work it would take if we just did a long | |
256 | * conventional multiply. | |
257 | */ | |
258 | ||
259 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ | |
260 | int midlen = botlen + 1; | |
261 | BignumDblInt carry; | |
262 | #ifdef KARA_DEBUG | |
263 | int i; | |
264 | #endif | |
265 | ||
266 | /* | |
267 | * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping | |
268 | * in the output array, so we can compute them immediately in | |
269 | * place. | |
270 | */ | |
271 | ||
272 | #ifdef KARA_DEBUG | |
273 | printf("a1,a0 = 0x"); | |
274 | for (i = 0; i < len; i++) { | |
275 | if (i == toplen) printf(", 0x"); | |
276 | printf("%0*x", BIGNUM_INT_BITS/4, a[i]); | |
277 | } | |
278 | printf("\n"); | |
279 | printf("b1,b0 = 0x"); | |
280 | for (i = 0; i < len; i++) { | |
281 | if (i == toplen) printf(", 0x"); | |
282 | printf("%0*x", BIGNUM_INT_BITS/4, b[i]); | |
283 | } | |
284 | printf("\n"); | |
285 | #endif | |
286 | ||
287 | /* a_1 b_1 */ | |
288 | internal_mul(a, b, c, toplen, scratch); | |
289 | #ifdef KARA_DEBUG | |
290 | printf("a1b1 = 0x"); | |
291 | for (i = 0; i < 2*toplen; i++) { | |
292 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); | |
293 | } | |
294 | printf("\n"); | |
295 | #endif | |
296 | ||
297 | /* a_0 b_0 */ | |
298 | internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch); | |
299 | #ifdef KARA_DEBUG | |
300 | printf("a0b0 = 0x"); | |
301 | for (i = 0; i < 2*botlen; i++) { | |
302 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]); | |
303 | } | |
304 | printf("\n"); | |
305 | #endif | |
306 | ||
307 | /* Zero padding. midlen exceeds toplen by at most 2, so just | |
308 | * zero the first two words of each input and the rest will be | |
309 | * copied over. */ | |
310 | scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0; | |
311 | ||
312 | for (i = 0; i < toplen; i++) { | |
313 | scratch[midlen - toplen + i] = a[i]; /* a_1 */ | |
314 | scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ | |
315 | } | |
316 | ||
317 | /* compute a_1 + a_0 */ | |
318 | scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); | |
319 | #ifdef KARA_DEBUG | |
320 | printf("a1plusa0 = 0x"); | |
321 | for (i = 0; i < midlen; i++) { | |
322 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); | |
323 | } | |
324 | printf("\n"); | |
325 | #endif | |
326 | /* compute b_1 + b_0 */ | |
327 | scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, | |
328 | scratch+midlen+1, botlen); | |
329 | #ifdef KARA_DEBUG | |
330 | printf("b1plusb0 = 0x"); | |
331 | for (i = 0; i < midlen; i++) { | |
332 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]); | |
333 | } | |
334 | printf("\n"); | |
335 | #endif | |
336 | ||
337 | /* | |
338 | * Now we can do the third multiplication. | |
339 | */ | |
340 | internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, | |
341 | scratch + 4*midlen); | |
342 | #ifdef KARA_DEBUG | |
343 | printf("a1plusa0timesb1plusb0 = 0x"); | |
344 | for (i = 0; i < 2*midlen; i++) { | |
345 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); | |
346 | } | |
347 | printf("\n"); | |
348 | #endif | |
349 | ||
350 | /* | |
351 | * Now we can reuse the first half of 'scratch' to compute the | |
352 | * sum of the outer two coefficients, to subtract from that | |
353 | * product to obtain the middle one. | |
354 | */ | |
355 | scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0; | |
356 | for (i = 0; i < 2*toplen; i++) | |
357 | scratch[2*midlen - 2*toplen + i] = c[i]; | |
358 | scratch[1] = internal_add(scratch+2, c + 2*toplen, | |
359 | scratch+2, 2*botlen); | |
360 | #ifdef KARA_DEBUG | |
361 | printf("a1b1plusa0b0 = 0x"); | |
362 | for (i = 0; i < 2*midlen; i++) { | |
363 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); | |
364 | } | |
365 | printf("\n"); | |
366 | #endif | |
367 | ||
368 | internal_sub(scratch + 2*midlen, scratch, | |
369 | scratch + 2*midlen, 2*midlen); | |
370 | #ifdef KARA_DEBUG | |
371 | printf("a1b0plusa0b1 = 0x"); | |
372 | for (i = 0; i < 2*midlen; i++) { | |
373 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); | |
374 | } | |
375 | printf("\n"); | |
376 | #endif | |
377 | ||
378 | /* | |
379 | * And now all we need to do is to add that middle coefficient | |
380 | * back into the output. We may have to propagate a carry | |
381 | * further up the output, but we can be sure it won't | |
382 | * propagate right the way off the top. | |
383 | */ | |
384 | carry = internal_add(c + 2*len - botlen - 2*midlen, | |
385 | scratch + 2*midlen, | |
386 | c + 2*len - botlen - 2*midlen, 2*midlen); | |
387 | i = 2*len - botlen - 2*midlen - 1; | |
388 | while (carry) { | |
389 | assert(i >= 0); | |
390 | carry += c[i]; | |
391 | c[i] = (BignumInt)carry; | |
392 | carry >>= BIGNUM_INT_BITS; | |
393 | i--; | |
394 | } | |
395 | #ifdef KARA_DEBUG | |
396 | printf("ab = 0x"); | |
397 | for (i = 0; i < 2*len; i++) { | |
398 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); | |
399 | } | |
400 | printf("\n"); | |
401 | #endif | |
402 | ||
403 | } else { | |
404 | int i; | |
405 | BignumInt carry; | |
406 | BignumDblInt t; | |
407 | const BignumInt *ap, *bp; | |
408 | BignumInt *cp, *cps; | |
409 | ||
410 | /* | |
411 | * Multiply in the ordinary O(N^2) way. | |
412 | */ | |
413 | ||
414 | for (i = 0; i < 2 * len; i++) | |
415 | c[i] = 0; | |
416 | ||
417 | for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { | |
418 | carry = 0; | |
419 | for (cp = cps, bp = b + len; cp--, bp-- > b ;) { | |
420 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; | |
421 | *cp = (BignumInt) t; | |
422 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); | |
423 | } | |
424 | *cp = carry; | |
425 | } | |
426 | } | |
427 | } | |
428 | ||
429 | /* | |
430 | * Variant form of internal_mul used for the initial step of | |
431 | * Montgomery reduction. Only bothers outputting 'len' words | |
432 | * (everything above that is thrown away). | |
433 | */ | |
434 | static void internal_mul_low(const BignumInt *a, const BignumInt *b, | |
435 | BignumInt *c, int len, BignumInt *scratch) | |
436 | { | |
437 | if (len > KARATSUBA_THRESHOLD) { | |
438 | int i; | |
439 | ||
440 | /* | |
441 | * Karatsuba-aware version of internal_mul_low. As before, we | |
442 | * express each input value as a shifted combination of two | |
443 | * halves: | |
444 | * | |
445 | * a = a_1 D + a_0 | |
446 | * b = b_1 D + b_0 | |
447 | * | |
448 | * Then the full product is, as before, | |
449 | * | |
450 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 | |
451 | * | |
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. | |
459 | * | |
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. | |
465 | * | |
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. | |
470 | */ | |
471 | ||
472 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ | |
473 | ||
474 | /* | |
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 | |
479 | * to exactly 2*len. | |
480 | */ | |
481 | ||
482 | /* a_0 b_0 */ | |
483 | internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, | |
484 | scratch + 2*len); | |
485 | ||
486 | /* a_1 b_0 */ | |
487 | internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, | |
488 | scratch + 2*len); | |
489 | ||
490 | /* a_0 b_1 */ | |
491 | internal_mul_low(a + len - toplen, b, scratch, toplen, | |
492 | scratch + 2*len); | |
493 | ||
494 | /* Copy the bottom half of the big coefficient into place */ | |
495 | for (i = 0; i < botlen; i++) | |
496 | c[toplen + i] = scratch[2*toplen + botlen + i]; | |
497 | ||
498 | /* Add the two small coefficients, throwing away the returned carry */ | |
499 | internal_add(scratch, scratch + toplen, scratch, toplen); | |
500 | ||
501 | /* And add that to the large coefficient, leaving the result in c. */ | |
502 | internal_add(scratch, scratch + 2*toplen + botlen - toplen, | |
503 | c, toplen); | |
504 | ||
505 | } else { | |
506 | int i; | |
507 | BignumInt carry; | |
508 | BignumDblInt t; | |
509 | const BignumInt *ap, *bp; | |
510 | BignumInt *cp, *cps; | |
511 | ||
512 | /* | |
513 | * Multiply in the ordinary O(N^2) way. | |
514 | */ | |
515 | ||
516 | for (i = 0; i < len; i++) | |
517 | c[i] = 0; | |
518 | ||
519 | for (cps = c + len, ap = a + len; ap-- > a; cps--) { | |
520 | carry = 0; | |
521 | for (cp = cps, bp = b + len; bp--, cp-- > c ;) { | |
522 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; | |
523 | *cp = (BignumInt) t; | |
524 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); | |
525 | } | |
526 | } | |
527 | } | |
528 | } | |
529 | ||
530 | /* | |
531 | * Montgomery reduction. Expects x to be a big-endian array of 2*len | |
532 | * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * | |
533 | * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array | |
534 | * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= | |
535 | * x' < n. | |
536 | * | |
537 | * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts | |
538 | * each, containing respectively n and the multiplicative inverse of | |
539 | * -n mod r. | |
540 | * | |
541 | * 'tmp' is an array of BignumInt used as scratch space, of length at | |
542 | * least 3*len + mul_compute_scratch(len). | |
543 | */ | |
544 | static void monty_reduce(BignumInt *x, const BignumInt *n, | |
545 | const BignumInt *mninv, BignumInt *tmp, int len) | |
546 | { | |
547 | int i; | |
548 | BignumInt carry; | |
549 | ||
550 | /* | |
551 | * Multiply x by (-n)^{-1} mod r. This gives us a value m such | |
552 | * that mn is congruent to -x mod r. Hence, mn+x is an exact | |
553 | * multiple of r, and is also (obviously) congruent to x mod n. | |
554 | */ | |
555 | internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len); | |
556 | ||
557 | /* | |
558 | * Compute t = (mn+x)/r in ordinary, non-modular, integer | |
559 | * arithmetic. By construction this is exact, and is congruent mod | |
560 | * n to x * r^{-1}, i.e. the answer we want. | |
561 | * | |
562 | * The following multiply leaves that answer in the _most_ | |
563 | * significant half of the 'x' array, so then we must shift it | |
564 | * down. | |
565 | */ | |
566 | internal_mul(tmp, n, tmp+len, len, tmp + 3*len); | |
567 | carry = internal_add(x, tmp+len, x, 2*len); | |
568 | for (i = 0; i < len; i++) | |
569 | x[len + i] = x[i], x[i] = 0; | |
570 | ||
571 | /* | |
572 | * Reduce t mod n. This doesn't require a full-on division by n, | |
573 | * but merely a test and single optional subtraction, since we can | |
574 | * show that 0 <= t < 2n. | |
575 | * | |
576 | * Proof: | |
577 | * + we computed m mod r, so 0 <= m < r. | |
578 | * + so 0 <= mn < rn, obviously | |
579 | * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn | |
580 | * + yielding 0 <= (mn+x)/r < 2n as required. | |
581 | */ | |
582 | if (!carry) { | |
583 | for (i = 0; i < len; i++) | |
584 | if (x[len + i] != n[i]) | |
585 | break; | |
586 | } | |
587 | if (carry || i >= len || x[len + i] > n[i]) | |
588 | internal_sub(x+len, n, x+len, len); | |
589 | } | |
590 | ||
591 | static void internal_add_shifted(BignumInt *number, | |
592 | unsigned n, int shift) | |
593 | { | |
594 | int word = 1 + (shift / BIGNUM_INT_BITS); | |
595 | int bshift = shift % BIGNUM_INT_BITS; | |
596 | BignumDblInt addend; | |
597 | ||
598 | addend = (BignumDblInt)n << bshift; | |
599 | ||
600 | while (addend) { | |
601 | addend += number[word]; | |
602 | number[word] = (BignumInt) addend & BIGNUM_INT_MASK; | |
603 | addend >>= BIGNUM_INT_BITS; | |
604 | word++; | |
605 | } | |
606 | } | |
607 | ||
608 | /* | |
609 | * Compute a = a % m. | |
610 | * Input in first alen words of a and first mlen words of m. | |
611 | * Output in first alen words of a | |
612 | * (of which first alen-mlen words will be zero). | |
613 | * The MSW of m MUST have its high bit set. | |
614 | * Quotient is accumulated in the `quotient' array, which is a Bignum | |
615 | * rather than the internal bigendian format. Quotient parts are shifted | |
616 | * left by `qshift' before adding into quot. | |
617 | */ | |
618 | static void internal_mod(BignumInt *a, int alen, | |
619 | BignumInt *m, int mlen, | |
620 | BignumInt *quot, int qshift) | |
621 | { | |
622 | BignumInt m0, m1; | |
623 | unsigned int h; | |
624 | int i, k; | |
625 | ||
626 | m0 = m[0]; | |
627 | if (mlen > 1) | |
628 | m1 = m[1]; | |
629 | else | |
630 | m1 = 0; | |
631 | ||
632 | for (i = 0; i <= alen - mlen; i++) { | |
633 | BignumDblInt t; | |
634 | unsigned int q, r, c, ai1; | |
635 | ||
636 | if (i == 0) { | |
637 | h = 0; | |
638 | } else { | |
639 | h = a[i - 1]; | |
640 | a[i - 1] = 0; | |
641 | } | |
642 | ||
643 | if (i == alen - 1) | |
644 | ai1 = 0; | |
645 | else | |
646 | ai1 = a[i + 1]; | |
647 | ||
648 | /* Find q = h:a[i] / m0 */ | |
649 | if (h >= m0) { | |
650 | /* | |
651 | * Special case. | |
652 | * | |
653 | * To illustrate it, suppose a BignumInt is 8 bits, and | |
654 | * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then | |
655 | * our initial division will be 0xA123 / 0xA1, which | |
656 | * will give a quotient of 0x100 and a divide overflow. | |
657 | * However, the invariants in this division algorithm | |
658 | * are not violated, since the full number A1:23:... is | |
659 | * _less_ than the quotient prefix A1:B2:... and so the | |
660 | * following correction loop would have sorted it out. | |
661 | * | |
662 | * In this situation we set q to be the largest | |
663 | * quotient we _can_ stomach (0xFF, of course). | |
664 | */ | |
665 | q = BIGNUM_INT_MASK; | |
666 | } else { | |
667 | /* Macro doesn't want an array subscript expression passed | |
668 | * into it (see definition), so use a temporary. */ | |
669 | BignumInt tmplo = a[i]; | |
670 | DIVMOD_WORD(q, r, h, tmplo, m0); | |
671 | ||
672 | /* Refine our estimate of q by looking at | |
673 | h:a[i]:a[i+1] / m0:m1 */ | |
674 | t = MUL_WORD(m1, q); | |
675 | if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { | |
676 | q--; | |
677 | t -= m1; | |
678 | r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ | |
679 | if (r >= (BignumDblInt) m0 && | |
680 | t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; | |
681 | } | |
682 | } | |
683 | ||
684 | /* Subtract q * m from a[i...] */ | |
685 | c = 0; | |
686 | for (k = mlen - 1; k >= 0; k--) { | |
687 | t = MUL_WORD(q, m[k]); | |
688 | t += c; | |
689 | c = (unsigned)(t >> BIGNUM_INT_BITS); | |
690 | if ((BignumInt) t > a[i + k]) | |
691 | c++; | |
692 | a[i + k] -= (BignumInt) t; | |
693 | } | |
694 | ||
695 | /* Add back m in case of borrow */ | |
696 | if (c != h) { | |
697 | t = 0; | |
698 | for (k = mlen - 1; k >= 0; k--) { | |
699 | t += m[k]; | |
700 | t += a[i + k]; | |
701 | a[i + k] = (BignumInt) t; | |
702 | t = t >> BIGNUM_INT_BITS; | |
703 | } | |
704 | q--; | |
705 | } | |
706 | if (quot) | |
707 | internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); | |
708 | } | |
709 | } | |
710 | ||
711 | /* | |
712 | * Compute (base ^ exp) % mod, the pedestrian way. | |
713 | */ | |
714 | Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) | |
715 | { | |
716 | BignumInt *a, *b, *n, *m, *scratch; | |
717 | int mshift; | |
718 | int mlen, scratchlen, i, j; | |
719 | Bignum base, result; | |
720 | ||
721 | /* | |
722 | * The most significant word of mod needs to be non-zero. It | |
723 | * should already be, but let's make sure. | |
724 | */ | |
725 | assert(mod[mod[0]] != 0); | |
726 | ||
727 | /* | |
728 | * Make sure the base is smaller than the modulus, by reducing | |
729 | * it modulo the modulus if not. | |
730 | */ | |
731 | base = bigmod(base_in, mod); | |
732 | ||
733 | /* Allocate m of size mlen, copy mod to m */ | |
734 | /* We use big endian internally */ | |
735 | mlen = mod[0]; | |
736 | m = snewn(mlen, BignumInt); | |
737 | for (j = 0; j < mlen; j++) | |
738 | m[j] = mod[mod[0] - j]; | |
739 | ||
740 | /* Shift m left to make msb bit set */ | |
741 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) | |
742 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
743 | break; | |
744 | if (mshift) { | |
745 | for (i = 0; i < mlen - 1; i++) | |
746 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
747 | m[mlen - 1] = m[mlen - 1] << mshift; | |
748 | } | |
749 | ||
750 | /* Allocate n of size mlen, copy base to n */ | |
751 | n = snewn(mlen, BignumInt); | |
752 | i = mlen - base[0]; | |
753 | for (j = 0; j < i; j++) | |
754 | n[j] = 0; | |
755 | for (j = 0; j < (int)base[0]; j++) | |
756 | n[i + j] = base[base[0] - j]; | |
757 | ||
758 | /* Allocate a and b of size 2*mlen. Set a = 1 */ | |
759 | a = snewn(2 * mlen, BignumInt); | |
760 | b = snewn(2 * mlen, BignumInt); | |
761 | for (i = 0; i < 2 * mlen; i++) | |
762 | a[i] = 0; | |
763 | a[2 * mlen - 1] = 1; | |
764 | ||
765 | /* Scratch space for multiplies */ | |
766 | scratchlen = mul_compute_scratch(mlen); | |
767 | scratch = snewn(scratchlen, BignumInt); | |
768 | ||
769 | /* Skip leading zero bits of exp. */ | |
770 | i = 0; | |
771 | j = BIGNUM_INT_BITS-1; | |
772 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { | |
773 | j--; | |
774 | if (j < 0) { | |
775 | i++; | |
776 | j = BIGNUM_INT_BITS-1; | |
777 | } | |
778 | } | |
779 | ||
780 | /* Main computation */ | |
781 | while (i < (int)exp[0]) { | |
782 | while (j >= 0) { | |
783 | internal_mul(a + mlen, a + mlen, b, mlen, scratch); | |
784 | internal_mod(b, mlen * 2, m, mlen, NULL, 0); | |
785 | if ((exp[exp[0] - i] & (1 << j)) != 0) { | |
786 | internal_mul(b + mlen, n, a, mlen, scratch); | |
787 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); | |
788 | } else { | |
789 | BignumInt *t; | |
790 | t = a; | |
791 | a = b; | |
792 | b = t; | |
793 | } | |
794 | j--; | |
795 | } | |
796 | i++; | |
797 | j = BIGNUM_INT_BITS-1; | |
798 | } | |
799 | ||
800 | /* Fixup result in case the modulus was shifted */ | |
801 | if (mshift) { | |
802 | for (i = mlen - 1; i < 2 * mlen - 1; i++) | |
803 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
804 | a[2 * mlen - 1] = a[2 * mlen - 1] << mshift; | |
805 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); | |
806 | for (i = 2 * mlen - 1; i >= mlen; i--) | |
807 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); | |
808 | } | |
809 | ||
810 | /* Copy result to buffer */ | |
811 | result = newbn(mod[0]); | |
812 | for (i = 0; i < mlen; i++) | |
813 | result[result[0] - i] = a[i + mlen]; | |
814 | while (result[0] > 1 && result[result[0]] == 0) | |
815 | result[0]--; | |
816 | ||
817 | /* Free temporary arrays */ | |
818 | for (i = 0; i < 2 * mlen; i++) | |
819 | a[i] = 0; | |
820 | sfree(a); | |
821 | for (i = 0; i < scratchlen; i++) | |
822 | scratch[i] = 0; | |
823 | sfree(scratch); | |
824 | for (i = 0; i < 2 * mlen; i++) | |
825 | b[i] = 0; | |
826 | sfree(b); | |
827 | for (i = 0; i < mlen; i++) | |
828 | m[i] = 0; | |
829 | sfree(m); | |
830 | for (i = 0; i < mlen; i++) | |
831 | n[i] = 0; | |
832 | sfree(n); | |
833 | ||
834 | freebn(base); | |
835 | ||
836 | return result; | |
837 | } | |
838 | ||
839 | /* | |
840 | * Compute (base ^ exp) % mod. Uses the Montgomery multiplication | |
841 | * technique where possible, falling back to modpow_simple otherwise. | |
842 | */ | |
843 | Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) | |
844 | { | |
845 | BignumInt *a, *b, *x, *n, *mninv, *scratch; | |
846 | int len, scratchlen, i, j; | |
847 | Bignum base, base2, r, rn, inv, result; | |
848 | ||
849 | /* | |
850 | * The most significant word of mod needs to be non-zero. It | |
851 | * should already be, but let's make sure. | |
852 | */ | |
853 | assert(mod[mod[0]] != 0); | |
854 | ||
855 | /* | |
856 | * mod had better be odd, or we can't do Montgomery multiplication | |
857 | * using a power of two at all. | |
858 | */ | |
859 | if (!(mod[1] & 1)) | |
860 | return modpow_simple(base_in, exp, mod); | |
861 | ||
862 | /* | |
863 | * Make sure the base is smaller than the modulus, by reducing | |
864 | * it modulo the modulus if not. | |
865 | */ | |
866 | base = bigmod(base_in, mod); | |
867 | ||
868 | /* | |
869 | * Compute the inverse of n mod r, for monty_reduce. (In fact we | |
870 | * want the inverse of _minus_ n mod r, but we'll sort that out | |
871 | * below.) | |
872 | */ | |
873 | len = mod[0]; | |
874 | r = bn_power_2(BIGNUM_INT_BITS * len); | |
875 | inv = modinv(mod, r); | |
876 | ||
877 | /* | |
878 | * Multiply the base by r mod n, to get it into Montgomery | |
879 | * representation. | |
880 | */ | |
881 | base2 = modmul(base, r, mod); | |
882 | freebn(base); | |
883 | base = base2; | |
884 | ||
885 | rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ | |
886 | ||
887 | freebn(r); /* won't need this any more */ | |
888 | ||
889 | /* | |
890 | * Set up internal arrays of the right lengths, in big-endian | |
891 | * format, containing the base, the modulus, and the modulus's | |
892 | * inverse. | |
893 | */ | |
894 | n = snewn(len, BignumInt); | |
895 | for (j = 0; j < len; j++) | |
896 | n[len - 1 - j] = mod[j + 1]; | |
897 | ||
898 | mninv = snewn(len, BignumInt); | |
899 | for (j = 0; j < len; j++) | |
900 | mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0); | |
901 | freebn(inv); /* we don't need this copy of it any more */ | |
902 | /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ | |
903 | x = snewn(len, BignumInt); | |
904 | for (j = 0; j < len; j++) | |
905 | x[j] = 0; | |
906 | internal_sub(x, mninv, mninv, len); | |
907 | ||
908 | /* x = snewn(len, BignumInt); */ /* already done above */ | |
909 | for (j = 0; j < len; j++) | |
910 | x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0); | |
911 | freebn(base); /* we don't need this copy of it any more */ | |
912 | ||
913 | a = snewn(2*len, BignumInt); | |
914 | b = snewn(2*len, BignumInt); | |
915 | for (j = 0; j < len; j++) | |
916 | a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0); | |
917 | freebn(rn); | |
918 | ||
919 | /* Scratch space for multiplies */ | |
920 | scratchlen = 3*len + mul_compute_scratch(len); | |
921 | scratch = snewn(scratchlen, BignumInt); | |
922 | ||
923 | /* Skip leading zero bits of exp. */ | |
924 | i = 0; | |
925 | j = BIGNUM_INT_BITS-1; | |
926 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { | |
927 | j--; | |
928 | if (j < 0) { | |
929 | i++; | |
930 | j = BIGNUM_INT_BITS-1; | |
931 | } | |
932 | } | |
933 | ||
934 | /* Main computation */ | |
935 | while (i < (int)exp[0]) { | |
936 | while (j >= 0) { | |
937 | internal_mul(a + len, a + len, b, len, scratch); | |
938 | monty_reduce(b, n, mninv, scratch, len); | |
939 | if ((exp[exp[0] - i] & (1 << j)) != 0) { | |
940 | internal_mul(b + len, x, a, len, scratch); | |
941 | monty_reduce(a, n, mninv, scratch, len); | |
942 | } else { | |
943 | BignumInt *t; | |
944 | t = a; | |
945 | a = b; | |
946 | b = t; | |
947 | } | |
948 | j--; | |
949 | } | |
950 | i++; | |
951 | j = BIGNUM_INT_BITS-1; | |
952 | } | |
953 | ||
954 | /* | |
955 | * Final monty_reduce to get back from the adjusted Montgomery | |
956 | * representation. | |
957 | */ | |
958 | monty_reduce(a, n, mninv, scratch, len); | |
959 | ||
960 | /* Copy result to buffer */ | |
961 | result = newbn(mod[0]); | |
962 | for (i = 0; i < len; i++) | |
963 | result[result[0] - i] = a[i + len]; | |
964 | while (result[0] > 1 && result[result[0]] == 0) | |
965 | result[0]--; | |
966 | ||
967 | /* Free temporary arrays */ | |
968 | for (i = 0; i < scratchlen; i++) | |
969 | scratch[i] = 0; | |
970 | sfree(scratch); | |
971 | for (i = 0; i < 2 * len; i++) | |
972 | a[i] = 0; | |
973 | sfree(a); | |
974 | for (i = 0; i < 2 * len; i++) | |
975 | b[i] = 0; | |
976 | sfree(b); | |
977 | for (i = 0; i < len; i++) | |
978 | mninv[i] = 0; | |
979 | sfree(mninv); | |
980 | for (i = 0; i < len; i++) | |
981 | n[i] = 0; | |
982 | sfree(n); | |
983 | for (i = 0; i < len; i++) | |
984 | x[i] = 0; | |
985 | sfree(x); | |
986 | ||
987 | return result; | |
988 | } | |
989 | ||
990 | /* | |
991 | * Compute (p * q) % mod. | |
992 | * The most significant word of mod MUST be non-zero. | |
993 | * We assume that the result array is the same size as the mod array. | |
994 | */ | |
995 | Bignum modmul(Bignum p, Bignum q, Bignum mod) | |
996 | { | |
997 | BignumInt *a, *n, *m, *o, *scratch; | |
998 | int mshift, scratchlen; | |
999 | int pqlen, mlen, rlen, i, j; | |
1000 | Bignum result; | |
1001 | ||
1002 | /* Allocate m of size mlen, copy mod to m */ | |
1003 | /* We use big endian internally */ | |
1004 | mlen = mod[0]; | |
1005 | m = snewn(mlen, BignumInt); | |
1006 | for (j = 0; j < mlen; j++) | |
1007 | m[j] = mod[mod[0] - j]; | |
1008 | ||
1009 | /* Shift m left to make msb bit set */ | |
1010 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) | |
1011 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
1012 | break; | |
1013 | if (mshift) { | |
1014 | for (i = 0; i < mlen - 1; i++) | |
1015 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
1016 | m[mlen - 1] = m[mlen - 1] << mshift; | |
1017 | } | |
1018 | ||
1019 | pqlen = (p[0] > q[0] ? p[0] : q[0]); | |
1020 | ||
1021 | /* Allocate n of size pqlen, copy p to n */ | |
1022 | n = snewn(pqlen, BignumInt); | |
1023 | i = pqlen - p[0]; | |
1024 | for (j = 0; j < i; j++) | |
1025 | n[j] = 0; | |
1026 | for (j = 0; j < (int)p[0]; j++) | |
1027 | n[i + j] = p[p[0] - j]; | |
1028 | ||
1029 | /* Allocate o of size pqlen, copy q to o */ | |
1030 | o = snewn(pqlen, BignumInt); | |
1031 | i = pqlen - q[0]; | |
1032 | for (j = 0; j < i; j++) | |
1033 | o[j] = 0; | |
1034 | for (j = 0; j < (int)q[0]; j++) | |
1035 | o[i + j] = q[q[0] - j]; | |
1036 | ||
1037 | /* Allocate a of size 2*pqlen for result */ | |
1038 | a = snewn(2 * pqlen, BignumInt); | |
1039 | ||
1040 | /* Scratch space for multiplies */ | |
1041 | scratchlen = mul_compute_scratch(pqlen); | |
1042 | scratch = snewn(scratchlen, BignumInt); | |
1043 | ||
1044 | /* Main computation */ | |
1045 | internal_mul(n, o, a, pqlen, scratch); | |
1046 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); | |
1047 | ||
1048 | /* Fixup result in case the modulus was shifted */ | |
1049 | if (mshift) { | |
1050 | for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++) | |
1051 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
1052 | a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift; | |
1053 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); | |
1054 | for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--) | |
1055 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); | |
1056 | } | |
1057 | ||
1058 | /* Copy result to buffer */ | |
1059 | rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); | |
1060 | result = newbn(rlen); | |
1061 | for (i = 0; i < rlen; i++) | |
1062 | result[result[0] - i] = a[i + 2 * pqlen - rlen]; | |
1063 | while (result[0] > 1 && result[result[0]] == 0) | |
1064 | result[0]--; | |
1065 | ||
1066 | /* Free temporary arrays */ | |
1067 | for (i = 0; i < scratchlen; i++) | |
1068 | scratch[i] = 0; | |
1069 | sfree(scratch); | |
1070 | for (i = 0; i < 2 * pqlen; i++) | |
1071 | a[i] = 0; | |
1072 | sfree(a); | |
1073 | for (i = 0; i < mlen; i++) | |
1074 | m[i] = 0; | |
1075 | sfree(m); | |
1076 | for (i = 0; i < pqlen; i++) | |
1077 | n[i] = 0; | |
1078 | sfree(n); | |
1079 | for (i = 0; i < pqlen; i++) | |
1080 | o[i] = 0; | |
1081 | sfree(o); | |
1082 | ||
1083 | return result; | |
1084 | } | |
1085 | ||
1086 | /* | |
1087 | * Compute p % mod. | |
1088 | * The most significant word of mod MUST be non-zero. | |
1089 | * We assume that the result array is the same size as the mod array. | |
1090 | * We optionally write out a quotient if `quotient' is non-NULL. | |
1091 | * We can avoid writing out the result if `result' is NULL. | |
1092 | */ | |
1093 | static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) | |
1094 | { | |
1095 | BignumInt *n, *m; | |
1096 | int mshift; | |
1097 | int plen, mlen, i, j; | |
1098 | ||
1099 | /* Allocate m of size mlen, copy mod to m */ | |
1100 | /* We use big endian internally */ | |
1101 | mlen = mod[0]; | |
1102 | m = snewn(mlen, BignumInt); | |
1103 | for (j = 0; j < mlen; j++) | |
1104 | m[j] = mod[mod[0] - j]; | |
1105 | ||
1106 | /* Shift m left to make msb bit set */ | |
1107 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) | |
1108 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
1109 | break; | |
1110 | if (mshift) { | |
1111 | for (i = 0; i < mlen - 1; i++) | |
1112 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
1113 | m[mlen - 1] = m[mlen - 1] << mshift; | |
1114 | } | |
1115 | ||
1116 | plen = p[0]; | |
1117 | /* Ensure plen > mlen */ | |
1118 | if (plen <= mlen) | |
1119 | plen = mlen + 1; | |
1120 | ||
1121 | /* Allocate n of size plen, copy p to n */ | |
1122 | n = snewn(plen, BignumInt); | |
1123 | for (j = 0; j < plen; j++) | |
1124 | n[j] = 0; | |
1125 | for (j = 1; j <= (int)p[0]; j++) | |
1126 | n[plen - j] = p[j]; | |
1127 | ||
1128 | /* Main computation */ | |
1129 | internal_mod(n, plen, m, mlen, quotient, mshift); | |
1130 | ||
1131 | /* Fixup result in case the modulus was shifted */ | |
1132 | if (mshift) { | |
1133 | for (i = plen - mlen - 1; i < plen - 1; i++) | |
1134 | n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
1135 | n[plen - 1] = n[plen - 1] << mshift; | |
1136 | internal_mod(n, plen, m, mlen, quotient, 0); | |
1137 | for (i = plen - 1; i >= plen - mlen; i--) | |
1138 | n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); | |
1139 | } | |
1140 | ||
1141 | /* Copy result to buffer */ | |
1142 | if (result) { | |
1143 | for (i = 1; i <= (int)result[0]; i++) { | |
1144 | int j = plen - i; | |
1145 | result[i] = j >= 0 ? n[j] : 0; | |
1146 | } | |
1147 | } | |
1148 | ||
1149 | /* Free temporary arrays */ | |
1150 | for (i = 0; i < mlen; i++) | |
1151 | m[i] = 0; | |
1152 | sfree(m); | |
1153 | for (i = 0; i < plen; i++) | |
1154 | n[i] = 0; | |
1155 | sfree(n); | |
1156 | } | |
1157 | ||
1158 | /* | |
1159 | * Decrement a number. | |
1160 | */ | |
1161 | void decbn(Bignum bn) | |
1162 | { | |
1163 | int i = 1; | |
1164 | while (i < (int)bn[0] && bn[i] == 0) | |
1165 | bn[i++] = BIGNUM_INT_MASK; | |
1166 | bn[i]--; | |
1167 | } | |
1168 | ||
1169 | Bignum bignum_from_bytes(const unsigned char *data, int nbytes) | |
1170 | { | |
1171 | Bignum result; | |
1172 | int w, i; | |
1173 | ||
1174 | w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ | |
1175 | ||
1176 | result = newbn(w); | |
1177 | for (i = 1; i <= w; i++) | |
1178 | result[i] = 0; | |
1179 | for (i = nbytes; i--;) { | |
1180 | unsigned char byte = *data++; | |
1181 | result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); | |
1182 | } | |
1183 | ||
1184 | while (result[0] > 1 && result[result[0]] == 0) | |
1185 | result[0]--; | |
1186 | return result; | |
1187 | } | |
1188 | ||
1189 | /* | |
1190 | * Read an SSH-1-format bignum from a data buffer. Return the number | |
1191 | * of bytes consumed, or -1 if there wasn't enough data. | |
1192 | */ | |
1193 | int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) | |
1194 | { | |
1195 | const unsigned char *p = data; | |
1196 | int i; | |
1197 | int w, b; | |
1198 | ||
1199 | if (len < 2) | |
1200 | return -1; | |
1201 | ||
1202 | w = 0; | |
1203 | for (i = 0; i < 2; i++) | |
1204 | w = (w << 8) + *p++; | |
1205 | b = (w + 7) / 8; /* bits -> bytes */ | |
1206 | ||
1207 | if (len < b+2) | |
1208 | return -1; | |
1209 | ||
1210 | if (!result) /* just return length */ | |
1211 | return b + 2; | |
1212 | ||
1213 | *result = bignum_from_bytes(p, b); | |
1214 | ||
1215 | return p + b - data; | |
1216 | } | |
1217 | ||
1218 | /* | |
1219 | * Return the bit count of a bignum, for SSH-1 encoding. | |
1220 | */ | |
1221 | int bignum_bitcount(Bignum bn) | |
1222 | { | |
1223 | int bitcount = bn[0] * BIGNUM_INT_BITS - 1; | |
1224 | while (bitcount >= 0 | |
1225 | && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; | |
1226 | return bitcount + 1; | |
1227 | } | |
1228 | ||
1229 | /* | |
1230 | * Return the byte length of a bignum when SSH-1 encoded. | |
1231 | */ | |
1232 | int ssh1_bignum_length(Bignum bn) | |
1233 | { | |
1234 | return 2 + (bignum_bitcount(bn) + 7) / 8; | |
1235 | } | |
1236 | ||
1237 | /* | |
1238 | * Return the byte length of a bignum when SSH-2 encoded. | |
1239 | */ | |
1240 | int ssh2_bignum_length(Bignum bn) | |
1241 | { | |
1242 | return 4 + (bignum_bitcount(bn) + 8) / 8; | |
1243 | } | |
1244 | ||
1245 | /* | |
1246 | * Return a byte from a bignum; 0 is least significant, etc. | |
1247 | */ | |
1248 | int bignum_byte(Bignum bn, int i) | |
1249 | { | |
1250 | if (i >= (int)(BIGNUM_INT_BYTES * bn[0])) | |
1251 | return 0; /* beyond the end */ | |
1252 | else | |
1253 | return (bn[i / BIGNUM_INT_BYTES + 1] >> | |
1254 | ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; | |
1255 | } | |
1256 | ||
1257 | /* | |
1258 | * Return a bit from a bignum; 0 is least significant, etc. | |
1259 | */ | |
1260 | int bignum_bit(Bignum bn, int i) | |
1261 | { | |
1262 | if (i >= (int)(BIGNUM_INT_BITS * bn[0])) | |
1263 | return 0; /* beyond the end */ | |
1264 | else | |
1265 | return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; | |
1266 | } | |
1267 | ||
1268 | /* | |
1269 | * Set a bit in a bignum; 0 is least significant, etc. | |
1270 | */ | |
1271 | void bignum_set_bit(Bignum bn, int bitnum, int value) | |
1272 | { | |
1273 | if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) | |
1274 | abort(); /* beyond the end */ | |
1275 | else { | |
1276 | int v = bitnum / BIGNUM_INT_BITS + 1; | |
1277 | int mask = 1 << (bitnum % BIGNUM_INT_BITS); | |
1278 | if (value) | |
1279 | bn[v] |= mask; | |
1280 | else | |
1281 | bn[v] &= ~mask; | |
1282 | } | |
1283 | } | |
1284 | ||
1285 | /* | |
1286 | * Write a SSH-1-format bignum into a buffer. It is assumed the | |
1287 | * buffer is big enough. Returns the number of bytes used. | |
1288 | */ | |
1289 | int ssh1_write_bignum(void *data, Bignum bn) | |
1290 | { | |
1291 | unsigned char *p = data; | |
1292 | int len = ssh1_bignum_length(bn); | |
1293 | int i; | |
1294 | int bitc = bignum_bitcount(bn); | |
1295 | ||
1296 | *p++ = (bitc >> 8) & 0xFF; | |
1297 | *p++ = (bitc) & 0xFF; | |
1298 | for (i = len - 2; i--;) | |
1299 | *p++ = bignum_byte(bn, i); | |
1300 | return len; | |
1301 | } | |
1302 | ||
1303 | /* | |
1304 | * Compare two bignums. Returns like strcmp. | |
1305 | */ | |
1306 | int bignum_cmp(Bignum a, Bignum b) | |
1307 | { | |
1308 | int amax = a[0], bmax = b[0]; | |
1309 | int i = (amax > bmax ? amax : bmax); | |
1310 | while (i) { | |
1311 | BignumInt aval = (i > amax ? 0 : a[i]); | |
1312 | BignumInt bval = (i > bmax ? 0 : b[i]); | |
1313 | if (aval < bval) | |
1314 | return -1; | |
1315 | if (aval > bval) | |
1316 | return +1; | |
1317 | i--; | |
1318 | } | |
1319 | return 0; | |
1320 | } | |
1321 | ||
1322 | /* | |
1323 | * Right-shift one bignum to form another. | |
1324 | */ | |
1325 | Bignum bignum_rshift(Bignum a, int shift) | |
1326 | { | |
1327 | Bignum ret; | |
1328 | int i, shiftw, shiftb, shiftbb, bits; | |
1329 | BignumInt ai, ai1; | |
1330 | ||
1331 | bits = bignum_bitcount(a) - shift; | |
1332 | ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); | |
1333 | ||
1334 | if (ret) { | |
1335 | shiftw = shift / BIGNUM_INT_BITS; | |
1336 | shiftb = shift % BIGNUM_INT_BITS; | |
1337 | shiftbb = BIGNUM_INT_BITS - shiftb; | |
1338 | ||
1339 | ai1 = a[shiftw + 1]; | |
1340 | for (i = 1; i <= (int)ret[0]; i++) { | |
1341 | ai = ai1; | |
1342 | ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); | |
1343 | ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; | |
1344 | } | |
1345 | } | |
1346 | ||
1347 | return ret; | |
1348 | } | |
1349 | ||
1350 | /* | |
1351 | * Non-modular multiplication and addition. | |
1352 | */ | |
1353 | Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) | |
1354 | { | |
1355 | int alen = a[0], blen = b[0]; | |
1356 | int mlen = (alen > blen ? alen : blen); | |
1357 | int rlen, i, maxspot; | |
1358 | int wslen; | |
1359 | BignumInt *workspace; | |
1360 | Bignum ret; | |
1361 | ||
1362 | /* mlen space for a, mlen space for b, 2*mlen for result, | |
1363 | * plus scratch space for multiplication */ | |
1364 | wslen = mlen * 4 + mul_compute_scratch(mlen); | |
1365 | workspace = snewn(wslen, BignumInt); | |
1366 | for (i = 0; i < mlen; i++) { | |
1367 | workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0); | |
1368 | workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0); | |
1369 | } | |
1370 | ||
1371 | internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, | |
1372 | workspace + 2 * mlen, mlen, workspace + 4 * mlen); | |
1373 | ||
1374 | /* now just copy the result back */ | |
1375 | rlen = alen + blen + 1; | |
1376 | if (addend && rlen <= (int)addend[0]) | |
1377 | rlen = addend[0] + 1; | |
1378 | ret = newbn(rlen); | |
1379 | maxspot = 0; | |
1380 | for (i = 1; i <= (int)ret[0]; i++) { | |
1381 | ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); | |
1382 | if (ret[i] != 0) | |
1383 | maxspot = i; | |
1384 | } | |
1385 | ret[0] = maxspot; | |
1386 | ||
1387 | /* now add in the addend, if any */ | |
1388 | if (addend) { | |
1389 | BignumDblInt carry = 0; | |
1390 | for (i = 1; i <= rlen; i++) { | |
1391 | carry += (i <= (int)ret[0] ? ret[i] : 0); | |
1392 | carry += (i <= (int)addend[0] ? addend[i] : 0); | |
1393 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1394 | carry >>= BIGNUM_INT_BITS; | |
1395 | if (ret[i] != 0 && i > maxspot) | |
1396 | maxspot = i; | |
1397 | } | |
1398 | } | |
1399 | ret[0] = maxspot; | |
1400 | ||
1401 | for (i = 0; i < wslen; i++) | |
1402 | workspace[i] = 0; | |
1403 | sfree(workspace); | |
1404 | return ret; | |
1405 | } | |
1406 | ||
1407 | /* | |
1408 | * Non-modular multiplication. | |
1409 | */ | |
1410 | Bignum bigmul(Bignum a, Bignum b) | |
1411 | { | |
1412 | return bigmuladd(a, b, NULL); | |
1413 | } | |
1414 | ||
1415 | /* | |
1416 | * Simple addition. | |
1417 | */ | |
1418 | Bignum bigadd(Bignum a, Bignum b) | |
1419 | { | |
1420 | int alen = a[0], blen = b[0]; | |
1421 | int rlen = (alen > blen ? alen : blen) + 1; | |
1422 | int i, maxspot; | |
1423 | Bignum ret; | |
1424 | BignumDblInt carry; | |
1425 | ||
1426 | ret = newbn(rlen); | |
1427 | ||
1428 | carry = 0; | |
1429 | maxspot = 0; | |
1430 | for (i = 1; i <= rlen; i++) { | |
1431 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1432 | carry += (i <= (int)b[0] ? b[i] : 0); | |
1433 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1434 | carry >>= BIGNUM_INT_BITS; | |
1435 | if (ret[i] != 0 && i > maxspot) | |
1436 | maxspot = i; | |
1437 | } | |
1438 | ret[0] = maxspot; | |
1439 | ||
1440 | return ret; | |
1441 | } | |
1442 | ||
1443 | /* | |
1444 | * Subtraction. Returns a-b, or NULL if the result would come out | |
1445 | * negative (recall that this entire bignum module only handles | |
1446 | * positive numbers). | |
1447 | */ | |
1448 | Bignum bigsub(Bignum a, Bignum b) | |
1449 | { | |
1450 | int alen = a[0], blen = b[0]; | |
1451 | int rlen = (alen > blen ? alen : blen); | |
1452 | int i, maxspot; | |
1453 | Bignum ret; | |
1454 | BignumDblInt carry; | |
1455 | ||
1456 | ret = newbn(rlen); | |
1457 | ||
1458 | carry = 1; | |
1459 | maxspot = 0; | |
1460 | for (i = 1; i <= rlen; i++) { | |
1461 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1462 | carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); | |
1463 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1464 | carry >>= BIGNUM_INT_BITS; | |
1465 | if (ret[i] != 0 && i > maxspot) | |
1466 | maxspot = i; | |
1467 | } | |
1468 | ret[0] = maxspot; | |
1469 | ||
1470 | if (!carry) { | |
1471 | freebn(ret); | |
1472 | return NULL; | |
1473 | } | |
1474 | ||
1475 | return ret; | |
1476 | } | |
1477 | ||
1478 | /* | |
1479 | * Create a bignum which is the bitmask covering another one. That | |
1480 | * is, the smallest integer which is >= N and is also one less than | |
1481 | * a power of two. | |
1482 | */ | |
1483 | Bignum bignum_bitmask(Bignum n) | |
1484 | { | |
1485 | Bignum ret = copybn(n); | |
1486 | int i; | |
1487 | BignumInt j; | |
1488 | ||
1489 | i = ret[0]; | |
1490 | while (n[i] == 0 && i > 0) | |
1491 | i--; | |
1492 | if (i <= 0) | |
1493 | return ret; /* input was zero */ | |
1494 | j = 1; | |
1495 | while (j < n[i]) | |
1496 | j = 2 * j + 1; | |
1497 | ret[i] = j; | |
1498 | while (--i > 0) | |
1499 | ret[i] = BIGNUM_INT_MASK; | |
1500 | return ret; | |
1501 | } | |
1502 | ||
1503 | /* | |
1504 | * Convert a (max 32-bit) long into a bignum. | |
1505 | */ | |
1506 | Bignum bignum_from_long(unsigned long nn) | |
1507 | { | |
1508 | Bignum ret; | |
1509 | BignumDblInt n = nn; | |
1510 | ||
1511 | ret = newbn(3); | |
1512 | ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); | |
1513 | ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); | |
1514 | ret[3] = 0; | |
1515 | ret[0] = (ret[2] ? 2 : 1); | |
1516 | return ret; | |
1517 | } | |
1518 | ||
1519 | /* | |
1520 | * Add a long to a bignum. | |
1521 | */ | |
1522 | Bignum bignum_add_long(Bignum number, unsigned long addendx) | |
1523 | { | |
1524 | Bignum ret = newbn(number[0] + 1); | |
1525 | int i, maxspot = 0; | |
1526 | BignumDblInt carry = 0, addend = addendx; | |
1527 | ||
1528 | for (i = 1; i <= (int)ret[0]; i++) { | |
1529 | carry += addend & BIGNUM_INT_MASK; | |
1530 | carry += (i <= (int)number[0] ? number[i] : 0); | |
1531 | addend >>= BIGNUM_INT_BITS; | |
1532 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1533 | carry >>= BIGNUM_INT_BITS; | |
1534 | if (ret[i] != 0) | |
1535 | maxspot = i; | |
1536 | } | |
1537 | ret[0] = maxspot; | |
1538 | return ret; | |
1539 | } | |
1540 | ||
1541 | /* | |
1542 | * Compute the residue of a bignum, modulo a (max 16-bit) short. | |
1543 | */ | |
1544 | unsigned short bignum_mod_short(Bignum number, unsigned short modulus) | |
1545 | { | |
1546 | BignumDblInt mod, r; | |
1547 | int i; | |
1548 | ||
1549 | r = 0; | |
1550 | mod = modulus; | |
1551 | for (i = number[0]; i > 0; i--) | |
1552 | r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; | |
1553 | return (unsigned short) r; | |
1554 | } | |
1555 | ||
1556 | #ifdef DEBUG | |
1557 | void diagbn(char *prefix, Bignum md) | |
1558 | { | |
1559 | int i, nibbles, morenibbles; | |
1560 | static const char hex[] = "0123456789ABCDEF"; | |
1561 | ||
1562 | debug(("%s0x", prefix ? prefix : "")); | |
1563 | ||
1564 | nibbles = (3 + bignum_bitcount(md)) / 4; | |
1565 | if (nibbles < 1) | |
1566 | nibbles = 1; | |
1567 | morenibbles = 4 * md[0] - nibbles; | |
1568 | for (i = 0; i < morenibbles; i++) | |
1569 | debug(("-")); | |
1570 | for (i = nibbles; i--;) | |
1571 | debug(("%c", | |
1572 | hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF])); | |
1573 | ||
1574 | if (prefix) | |
1575 | debug(("\n")); | |
1576 | } | |
1577 | #endif | |
1578 | ||
1579 | /* | |
1580 | * Simple division. | |
1581 | */ | |
1582 | Bignum bigdiv(Bignum a, Bignum b) | |
1583 | { | |
1584 | Bignum q = newbn(a[0]); | |
1585 | bigdivmod(a, b, NULL, q); | |
1586 | return q; | |
1587 | } | |
1588 | ||
1589 | /* | |
1590 | * Simple remainder. | |
1591 | */ | |
1592 | Bignum bigmod(Bignum a, Bignum b) | |
1593 | { | |
1594 | Bignum r = newbn(b[0]); | |
1595 | bigdivmod(a, b, r, NULL); | |
1596 | return r; | |
1597 | } | |
1598 | ||
1599 | /* | |
1600 | * Greatest common divisor. | |
1601 | */ | |
1602 | Bignum biggcd(Bignum av, Bignum bv) | |
1603 | { | |
1604 | Bignum a = copybn(av); | |
1605 | Bignum b = copybn(bv); | |
1606 | ||
1607 | while (bignum_cmp(b, Zero) != 0) { | |
1608 | Bignum t = newbn(b[0]); | |
1609 | bigdivmod(a, b, t, NULL); | |
1610 | while (t[0] > 1 && t[t[0]] == 0) | |
1611 | t[0]--; | |
1612 | freebn(a); | |
1613 | a = b; | |
1614 | b = t; | |
1615 | } | |
1616 | ||
1617 | freebn(b); | |
1618 | return a; | |
1619 | } | |
1620 | ||
1621 | /* | |
1622 | * Modular inverse, using Euclid's extended algorithm. | |
1623 | */ | |
1624 | Bignum modinv(Bignum number, Bignum modulus) | |
1625 | { | |
1626 | Bignum a = copybn(modulus); | |
1627 | Bignum b = copybn(number); | |
1628 | Bignum xp = copybn(Zero); | |
1629 | Bignum x = copybn(One); | |
1630 | int sign = +1; | |
1631 | ||
1632 | while (bignum_cmp(b, One) != 0) { | |
1633 | Bignum t = newbn(b[0]); | |
1634 | Bignum q = newbn(a[0]); | |
1635 | bigdivmod(a, b, t, q); | |
1636 | while (t[0] > 1 && t[t[0]] == 0) | |
1637 | t[0]--; | |
1638 | freebn(a); | |
1639 | a = b; | |
1640 | b = t; | |
1641 | t = xp; | |
1642 | xp = x; | |
1643 | x = bigmuladd(q, xp, t); | |
1644 | sign = -sign; | |
1645 | freebn(t); | |
1646 | freebn(q); | |
1647 | } | |
1648 | ||
1649 | freebn(b); | |
1650 | freebn(a); | |
1651 | freebn(xp); | |
1652 | ||
1653 | /* now we know that sign * x == 1, and that x < modulus */ | |
1654 | if (sign < 0) { | |
1655 | /* set a new x to be modulus - x */ | |
1656 | Bignum newx = newbn(modulus[0]); | |
1657 | BignumInt carry = 0; | |
1658 | int maxspot = 1; | |
1659 | int i; | |
1660 | ||
1661 | for (i = 1; i <= (int)newx[0]; i++) { | |
1662 | BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); | |
1663 | BignumInt bword = (i <= (int)x[0] ? x[i] : 0); | |
1664 | newx[i] = aword - bword - carry; | |
1665 | bword = ~bword; | |
1666 | carry = carry ? (newx[i] >= bword) : (newx[i] > bword); | |
1667 | if (newx[i] != 0) | |
1668 | maxspot = i; | |
1669 | } | |
1670 | newx[0] = maxspot; | |
1671 | freebn(x); | |
1672 | x = newx; | |
1673 | } | |
1674 | ||
1675 | /* and return. */ | |
1676 | return x; | |
1677 | } | |
1678 | ||
1679 | /* | |
1680 | * Render a bignum into decimal. Return a malloced string holding | |
1681 | * the decimal representation. | |
1682 | */ | |
1683 | char *bignum_decimal(Bignum x) | |
1684 | { | |
1685 | int ndigits, ndigit; | |
1686 | int i, iszero; | |
1687 | BignumDblInt carry; | |
1688 | char *ret; | |
1689 | BignumInt *workspace; | |
1690 | ||
1691 | /* | |
1692 | * First, estimate the number of digits. Since log(10)/log(2) | |
1693 | * is just greater than 93/28 (the joys of continued fraction | |
1694 | * approximations...) we know that for every 93 bits, we need | |
1695 | * at most 28 digits. This will tell us how much to malloc. | |
1696 | * | |
1697 | * Formally: if x has i bits, that means x is strictly less | |
1698 | * than 2^i. Since 2 is less than 10^(28/93), this is less than | |
1699 | * 10^(28i/93). We need an integer power of ten, so we must | |
1700 | * round up (rounding down might make it less than x again). | |
1701 | * Therefore if we multiply the bit count by 28/93, rounding | |
1702 | * up, we will have enough digits. | |
1703 | * | |
1704 | * i=0 (i.e., x=0) is an irritating special case. | |
1705 | */ | |
1706 | i = bignum_bitcount(x); | |
1707 | if (!i) | |
1708 | ndigits = 1; /* x = 0 */ | |
1709 | else | |
1710 | ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ | |
1711 | ndigits++; /* allow for trailing \0 */ | |
1712 | ret = snewn(ndigits, char); | |
1713 | ||
1714 | /* | |
1715 | * Now allocate some workspace to hold the binary form as we | |
1716 | * repeatedly divide it by ten. Initialise this to the | |
1717 | * big-endian form of the number. | |
1718 | */ | |
1719 | workspace = snewn(x[0], BignumInt); | |
1720 | for (i = 0; i < (int)x[0]; i++) | |
1721 | workspace[i] = x[x[0] - i]; | |
1722 | ||
1723 | /* | |
1724 | * Next, write the decimal number starting with the last digit. | |
1725 | * We use ordinary short division, dividing 10 into the | |
1726 | * workspace. | |
1727 | */ | |
1728 | ndigit = ndigits - 1; | |
1729 | ret[ndigit] = '\0'; | |
1730 | do { | |
1731 | iszero = 1; | |
1732 | carry = 0; | |
1733 | for (i = 0; i < (int)x[0]; i++) { | |
1734 | carry = (carry << BIGNUM_INT_BITS) + workspace[i]; | |
1735 | workspace[i] = (BignumInt) (carry / 10); | |
1736 | if (workspace[i]) | |
1737 | iszero = 0; | |
1738 | carry %= 10; | |
1739 | } | |
1740 | ret[--ndigit] = (char) (carry + '0'); | |
1741 | } while (!iszero); | |
1742 | ||
1743 | /* | |
1744 | * There's a chance we've fallen short of the start of the | |
1745 | * string. Correct if so. | |
1746 | */ | |
1747 | if (ndigit > 0) | |
1748 | memmove(ret, ret + ndigit, ndigits - ndigit); | |
1749 | ||
1750 | /* | |
1751 | * Done. | |
1752 | */ | |
1753 | sfree(workspace); | |
1754 | return ret; | |
1755 | } | |
1756 | ||
1757 | #ifdef TESTBN | |
1758 | ||
1759 | #include <stdio.h> | |
1760 | #include <stdlib.h> | |
1761 | #include <ctype.h> | |
1762 | ||
1763 | /* | |
1764 | * gcc -g -O0 -DTESTBN -o testbn sshbn.c misc.c -I unix -I charset | |
1765 | * | |
1766 | * Then feed to this program's standard input the output of | |
1767 | * testdata/bignum.py . | |
1768 | */ | |
1769 | ||
1770 | void modalfatalbox(char *p, ...) | |
1771 | { | |
1772 | va_list ap; | |
1773 | fprintf(stderr, "FATAL ERROR: "); | |
1774 | va_start(ap, p); | |
1775 | vfprintf(stderr, p, ap); | |
1776 | va_end(ap); | |
1777 | fputc('\n', stderr); | |
1778 | exit(1); | |
1779 | } | |
1780 | ||
1781 | #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) | |
1782 | ||
1783 | int main(int argc, char **argv) | |
1784 | { | |
1785 | char *buf; | |
1786 | int line = 0; | |
1787 | int passes = 0, fails = 0; | |
1788 | ||
1789 | while ((buf = fgetline(stdin)) != NULL) { | |
1790 | int maxlen = strlen(buf); | |
1791 | unsigned char *data = snewn(maxlen, unsigned char); | |
1792 | unsigned char *ptrs[5], *q; | |
1793 | int ptrnum; | |
1794 | char *bufp = buf; | |
1795 | ||
1796 | line++; | |
1797 | ||
1798 | q = data; | |
1799 | ptrnum = 0; | |
1800 | ||
1801 | while (*bufp && !isspace((unsigned char)*bufp)) | |
1802 | bufp++; | |
1803 | if (bufp) | |
1804 | *bufp++ = '\0'; | |
1805 | ||
1806 | while (*bufp) { | |
1807 | char *start, *end; | |
1808 | int i; | |
1809 | ||
1810 | while (*bufp && !isxdigit((unsigned char)*bufp)) | |
1811 | bufp++; | |
1812 | start = bufp; | |
1813 | ||
1814 | if (!*bufp) | |
1815 | break; | |
1816 | ||
1817 | while (*bufp && isxdigit((unsigned char)*bufp)) | |
1818 | bufp++; | |
1819 | end = bufp; | |
1820 | ||
1821 | if (ptrnum >= lenof(ptrs)) | |
1822 | break; | |
1823 | ptrs[ptrnum++] = q; | |
1824 | ||
1825 | for (i = -((end - start) & 1); i < end-start; i += 2) { | |
1826 | unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); | |
1827 | val = val * 16 + fromxdigit(start[i+1]); | |
1828 | *q++ = val; | |
1829 | } | |
1830 | ||
1831 | ptrs[ptrnum] = q; | |
1832 | } | |
1833 | ||
1834 | if (!strcmp(buf, "mul")) { | |
1835 | Bignum a, b, c, p; | |
1836 | ||
1837 | if (ptrnum != 3) { | |
1838 | printf("%d: mul with %d parameters, expected 3\n", line); | |
1839 | exit(1); | |
1840 | } | |
1841 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1842 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1843 | c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1844 | p = bigmul(a, b); | |
1845 | ||
1846 | if (bignum_cmp(c, p) == 0) { | |
1847 | passes++; | |
1848 | } else { | |
1849 | char *as = bignum_decimal(a); | |
1850 | char *bs = bignum_decimal(b); | |
1851 | char *cs = bignum_decimal(c); | |
1852 | char *ps = bignum_decimal(p); | |
1853 | ||
1854 | printf("%d: fail: %s * %s gave %s expected %s\n", | |
1855 | line, as, bs, ps, cs); | |
1856 | fails++; | |
1857 | ||
1858 | sfree(as); | |
1859 | sfree(bs); | |
1860 | sfree(cs); | |
1861 | sfree(ps); | |
1862 | } | |
1863 | freebn(a); | |
1864 | freebn(b); | |
1865 | freebn(c); | |
1866 | freebn(p); | |
1867 | } else if (!strcmp(buf, "pow")) { | |
1868 | Bignum base, expt, modulus, expected, answer; | |
1869 | ||
1870 | if (ptrnum != 4) { | |
1871 | printf("%d: mul with %d parameters, expected 3\n", line); | |
1872 | exit(1); | |
1873 | } | |
1874 | ||
1875 | base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1876 | expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1877 | modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1878 | expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); | |
1879 | answer = modpow(base, expt, modulus); | |
1880 | ||
1881 | if (bignum_cmp(expected, answer) == 0) { | |
1882 | passes++; | |
1883 | } else { | |
1884 | char *as = bignum_decimal(base); | |
1885 | char *bs = bignum_decimal(expt); | |
1886 | char *cs = bignum_decimal(modulus); | |
1887 | char *ds = bignum_decimal(answer); | |
1888 | char *ps = bignum_decimal(expected); | |
1889 | ||
1890 | printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", | |
1891 | line, as, bs, cs, ds, ps); | |
1892 | fails++; | |
1893 | ||
1894 | sfree(as); | |
1895 | sfree(bs); | |
1896 | sfree(cs); | |
1897 | sfree(ds); | |
1898 | sfree(ps); | |
1899 | } | |
1900 | freebn(base); | |
1901 | freebn(expt); | |
1902 | freebn(modulus); | |
1903 | freebn(expected); | |
1904 | freebn(answer); | |
1905 | } else { | |
1906 | printf("%d: unrecognised test keyword: '%s'\n", line, buf); | |
1907 | exit(1); | |
1908 | } | |
1909 | ||
1910 | sfree(buf); | |
1911 | sfree(data); | |
1912 | } | |
1913 | ||
1914 | printf("passed %d failed %d total %d\n", passes, fails, passes+fails); | |
1915 | return fails != 0; | |
1916 | } | |
1917 | ||
1918 | #endif |