Commit | Line | Data |
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e5574168 | 1 | /* |
2 | * Bignum routines for RSA and DH and stuff. | |
3 | */ | |
4 | ||
5 | #include <stdio.h> | |
ed953b91 | 6 | #include <assert.h> |
e5574168 | 7 | #include <stdlib.h> |
8 | #include <string.h> | |
9 | ||
5c72ca61 | 10 | #include "misc.h" |
98ba26b9 | 11 | |
819a22b3 | 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 | ||
a3412f52 | 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) | |
a47e8bba | 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)) | |
036eddfb | 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) | |
819a22b3 | 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> */ | |
036eddfb | 47 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
819a22b3 | 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) | |
32e51f76 | 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) | |
a3412f52 | 80 | #else |
32e51f76 | 81 | /* Fallback for all other cases */ |
a3412f52 | 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) | |
a47e8bba | 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) | |
a3412f52 | 93 | #endif |
94 | ||
95 | #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8) | |
96 | ||
3709bfe9 | 97 | #define BIGNUM_INTERNAL |
a3412f52 | 98 | typedef BignumInt *Bignum; |
3709bfe9 | 99 | |
e5574168 | 100 | #include "ssh.h" |
101 | ||
a3412f52 | 102 | BignumInt bnZero[1] = { 0 }; |
103 | BignumInt bnOne[2] = { 1, 1 }; | |
e5574168 | 104 | |
7d6ee6ff | 105 | /* |
a3412f52 | 106 | * The Bignum format is an array of `BignumInt'. The first |
7d6ee6ff | 107 | * element of the array counts the remaining elements. The |
a3412f52 | 108 | * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_ |
7d6ee6ff | 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 | ||
7cca0d81 | 119 | Bignum Zero = bnZero, One = bnOne; |
e5574168 | 120 | |
32874aea | 121 | static Bignum newbn(int length) |
122 | { | |
a3412f52 | 123 | Bignum b = snewn(length + 1, BignumInt); |
e5574168 | 124 | if (!b) |
125 | abort(); /* FIXME */ | |
32874aea | 126 | memset(b, 0, (length + 1) * sizeof(*b)); |
e5574168 | 127 | b[0] = length; |
128 | return b; | |
129 | } | |
130 | ||
32874aea | 131 | void bn_restore_invariant(Bignum b) |
132 | { | |
133 | while (b[0] > 1 && b[b[0]] == 0) | |
134 | b[0]--; | |
3709bfe9 | 135 | } |
136 | ||
32874aea | 137 | Bignum copybn(Bignum orig) |
138 | { | |
a3412f52 | 139 | Bignum b = snewn(orig[0] + 1, BignumInt); |
7cca0d81 | 140 | if (!b) |
141 | abort(); /* FIXME */ | |
32874aea | 142 | memcpy(b, orig, (orig[0] + 1) * sizeof(*b)); |
7cca0d81 | 143 | return b; |
144 | } | |
145 | ||
32874aea | 146 | void freebn(Bignum b) |
147 | { | |
e5574168 | 148 | /* |
149 | * Burn the evidence, just in case. | |
150 | */ | |
dfb88efd | 151 | smemclr(b, sizeof(b[0]) * (b[0] + 1)); |
dcbde236 | 152 | sfree(b); |
e5574168 | 153 | } |
154 | ||
32874aea | 155 | Bignum bn_power_2(int n) |
156 | { | |
a3412f52 | 157 | Bignum ret = newbn(n / BIGNUM_INT_BITS + 1); |
3709bfe9 | 158 | bignum_set_bit(ret, n, 1); |
159 | return ret; | |
160 | } | |
161 | ||
e5574168 | 162 | /* |
0c431b2f | 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 | /* | |
e5574168 | 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. | |
5a502a19 | 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.) | |
e5574168 | 208 | */ |
0c431b2f | 209 | #define KARATSUBA_THRESHOLD 50 |
5a502a19 | 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 | } | |
132c534f | 221 | static void internal_mul(const BignumInt *a, const BignumInt *b, |
5a502a19 | 222 | BignumInt *c, int len, BignumInt *scratch) |
e5574168 | 223 | { |
0c431b2f | 224 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 | 225 | int i; |
0c431b2f | 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; | |
0c431b2f | 261 | BignumDblInt carry; |
262 | ||
263 | /* | |
264 | * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping | |
265 | * in the output array, so we can compute them immediately in | |
266 | * place. | |
267 | */ | |
268 | ||
f3c29e34 | 269 | #ifdef KARA_DEBUG |
270 | printf("a1,a0 = 0x"); | |
271 | for (i = 0; i < len; i++) { | |
272 | if (i == toplen) printf(", 0x"); | |
273 | printf("%0*x", BIGNUM_INT_BITS/4, a[i]); | |
274 | } | |
275 | printf("\n"); | |
276 | printf("b1,b0 = 0x"); | |
277 | for (i = 0; i < len; i++) { | |
278 | if (i == toplen) printf(", 0x"); | |
279 | printf("%0*x", BIGNUM_INT_BITS/4, b[i]); | |
280 | } | |
281 | printf("\n"); | |
282 | #endif | |
283 | ||
0c431b2f | 284 | /* a_1 b_1 */ |
5a502a19 | 285 | internal_mul(a, b, c, toplen, scratch); |
f3c29e34 | 286 | #ifdef KARA_DEBUG |
287 | printf("a1b1 = 0x"); | |
288 | for (i = 0; i < 2*toplen; i++) { | |
289 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); | |
290 | } | |
291 | printf("\n"); | |
292 | #endif | |
0c431b2f | 293 | |
294 | /* a_0 b_0 */ | |
5a502a19 | 295 | internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch); |
f3c29e34 | 296 | #ifdef KARA_DEBUG |
297 | printf("a0b0 = 0x"); | |
298 | for (i = 0; i < 2*botlen; i++) { | |
299 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]); | |
300 | } | |
301 | printf("\n"); | |
302 | #endif | |
0c431b2f | 303 | |
0c431b2f | 304 | /* Zero padding. midlen exceeds toplen by at most 2, so just |
305 | * zero the first two words of each input and the rest will be | |
306 | * copied over. */ | |
307 | scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0; | |
308 | ||
757b0110 | 309 | for (i = 0; i < toplen; i++) { |
310 | scratch[midlen - toplen + i] = a[i]; /* a_1 */ | |
311 | scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ | |
0c431b2f | 312 | } |
313 | ||
314 | /* compute a_1 + a_0 */ | |
315 | scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); | |
f3c29e34 | 316 | #ifdef KARA_DEBUG |
317 | printf("a1plusa0 = 0x"); | |
318 | for (i = 0; i < midlen; i++) { | |
319 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); | |
320 | } | |
321 | printf("\n"); | |
322 | #endif | |
0c431b2f | 323 | /* compute b_1 + b_0 */ |
324 | scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, | |
325 | scratch+midlen+1, botlen); | |
f3c29e34 | 326 | #ifdef KARA_DEBUG |
327 | printf("b1plusb0 = 0x"); | |
328 | for (i = 0; i < midlen; i++) { | |
329 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]); | |
330 | } | |
331 | printf("\n"); | |
332 | #endif | |
0c431b2f | 333 | |
334 | /* | |
335 | * Now we can do the third multiplication. | |
336 | */ | |
5a502a19 | 337 | internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, |
338 | scratch + 4*midlen); | |
f3c29e34 | 339 | #ifdef KARA_DEBUG |
340 | printf("a1plusa0timesb1plusb0 = 0x"); | |
341 | for (i = 0; i < 2*midlen; i++) { | |
342 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); | |
343 | } | |
344 | printf("\n"); | |
345 | #endif | |
0c431b2f | 346 | |
347 | /* | |
348 | * Now we can reuse the first half of 'scratch' to compute the | |
349 | * sum of the outer two coefficients, to subtract from that | |
350 | * product to obtain the middle one. | |
351 | */ | |
352 | scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0; | |
757b0110 | 353 | for (i = 0; i < 2*toplen; i++) |
354 | scratch[2*midlen - 2*toplen + i] = c[i]; | |
0c431b2f | 355 | scratch[1] = internal_add(scratch+2, c + 2*toplen, |
356 | scratch+2, 2*botlen); | |
f3c29e34 | 357 | #ifdef KARA_DEBUG |
358 | printf("a1b1plusa0b0 = 0x"); | |
359 | for (i = 0; i < 2*midlen; i++) { | |
360 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); | |
361 | } | |
362 | printf("\n"); | |
363 | #endif | |
0c431b2f | 364 | |
365 | internal_sub(scratch + 2*midlen, scratch, | |
366 | scratch + 2*midlen, 2*midlen); | |
f3c29e34 | 367 | #ifdef KARA_DEBUG |
368 | printf("a1b0plusa0b1 = 0x"); | |
369 | for (i = 0; i < 2*midlen; i++) { | |
370 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); | |
371 | } | |
372 | printf("\n"); | |
373 | #endif | |
0c431b2f | 374 | |
375 | /* | |
376 | * And now all we need to do is to add that middle coefficient | |
377 | * back into the output. We may have to propagate a carry | |
378 | * further up the output, but we can be sure it won't | |
379 | * propagate right the way off the top. | |
380 | */ | |
381 | carry = internal_add(c + 2*len - botlen - 2*midlen, | |
382 | scratch + 2*midlen, | |
383 | c + 2*len - botlen - 2*midlen, 2*midlen); | |
757b0110 | 384 | i = 2*len - botlen - 2*midlen - 1; |
0c431b2f | 385 | while (carry) { |
757b0110 | 386 | assert(i >= 0); |
387 | carry += c[i]; | |
388 | c[i] = (BignumInt)carry; | |
0c431b2f | 389 | carry >>= BIGNUM_INT_BITS; |
757b0110 | 390 | i--; |
0c431b2f | 391 | } |
f3c29e34 | 392 | #ifdef KARA_DEBUG |
393 | printf("ab = 0x"); | |
394 | for (i = 0; i < 2*len; i++) { | |
395 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); | |
396 | } | |
397 | printf("\n"); | |
398 | #endif | |
0c431b2f | 399 | |
0c431b2f | 400 | } else { |
757b0110 | 401 | int i; |
402 | BignumInt carry; | |
403 | BignumDblInt t; | |
404 | const BignumInt *ap, *bp; | |
405 | BignumInt *cp, *cps; | |
0c431b2f | 406 | |
407 | /* | |
408 | * Multiply in the ordinary O(N^2) way. | |
409 | */ | |
410 | ||
757b0110 | 411 | for (i = 0; i < 2 * len; i++) |
412 | c[i] = 0; | |
0c431b2f | 413 | |
757b0110 | 414 | for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { |
415 | carry = 0; | |
416 | for (cp = cps, bp = b + len; cp--, bp-- > b ;) { | |
417 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; | |
418 | *cp = (BignumInt) t; | |
08b5c9a2 | 419 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
0c431b2f | 420 | } |
757b0110 | 421 | *cp = carry; |
0c431b2f | 422 | } |
e5574168 | 423 | } |
424 | } | |
425 | ||
132c534f | 426 | /* |
427 | * Variant form of internal_mul used for the initial step of | |
428 | * Montgomery reduction. Only bothers outputting 'len' words | |
429 | * (everything above that is thrown away). | |
430 | */ | |
431 | static void internal_mul_low(const BignumInt *a, const BignumInt *b, | |
5a502a19 | 432 | BignumInt *c, int len, BignumInt *scratch) |
132c534f | 433 | { |
132c534f | 434 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 | 435 | int i; |
132c534f | 436 | |
437 | /* | |
438 | * Karatsuba-aware version of internal_mul_low. As before, we | |
439 | * express each input value as a shifted combination of two | |
440 | * halves: | |
441 | * | |
442 | * a = a_1 D + a_0 | |
443 | * b = b_1 D + b_0 | |
444 | * | |
445 | * Then the full product is, as before, | |
446 | * | |
447 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 | |
448 | * | |
449 | * Provided we choose D on the large side (so that a_0 and b_0 | |
450 | * are _at least_ as long as a_1 and b_1), we don't need the | |
451 | * topmost term at all, and we only need half of the middle | |
452 | * term. So there's no point in doing the proper Karatsuba | |
453 | * optimisation which computes the middle term using the top | |
454 | * one, because we'd take as long computing the top one as | |
455 | * just computing the middle one directly. | |
456 | * | |
457 | * So instead, we do a much more obvious thing: we call the | |
458 | * fully optimised internal_mul to compute a_0 b_0, and we | |
459 | * recursively call ourself to compute the _bottom halves_ of | |
460 | * a_1 b_0 and a_0 b_1, each of which we add into the result | |
461 | * in the obvious way. | |
462 | * | |
463 | * In other words, there's no actual Karatsuba _optimisation_ | |
464 | * in this function; the only benefit in doing it this way is | |
465 | * that we call internal_mul proper for a large part of the | |
466 | * work, and _that_ can optimise its operation. | |
467 | */ | |
468 | ||
469 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ | |
132c534f | 470 | |
471 | /* | |
5a502a19 | 472 | * Scratch space for the various bits and pieces we're going |
473 | * to be adding together: we need botlen*2 words for a_0 b_0 | |
474 | * (though we may end up throwing away its topmost word), and | |
475 | * toplen words for each of a_1 b_0 and a_0 b_1. That adds up | |
476 | * to exactly 2*len. | |
132c534f | 477 | */ |
132c534f | 478 | |
479 | /* a_0 b_0 */ | |
5a502a19 | 480 | internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, |
481 | scratch + 2*len); | |
132c534f | 482 | |
483 | /* a_1 b_0 */ | |
5a502a19 | 484 | internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, |
485 | scratch + 2*len); | |
132c534f | 486 | |
487 | /* a_0 b_1 */ | |
5a502a19 | 488 | internal_mul_low(a + len - toplen, b, scratch, toplen, |
489 | scratch + 2*len); | |
132c534f | 490 | |
491 | /* Copy the bottom half of the big coefficient into place */ | |
757b0110 | 492 | for (i = 0; i < botlen; i++) |
493 | c[toplen + i] = scratch[2*toplen + botlen + i]; | |
132c534f | 494 | |
495 | /* Add the two small coefficients, throwing away the returned carry */ | |
496 | internal_add(scratch, scratch + toplen, scratch, toplen); | |
497 | ||
498 | /* And add that to the large coefficient, leaving the result in c. */ | |
499 | internal_add(scratch, scratch + 2*toplen + botlen - toplen, | |
500 | c, toplen); | |
501 | ||
132c534f | 502 | } else { |
757b0110 | 503 | int i; |
504 | BignumInt carry; | |
505 | BignumDblInt t; | |
506 | const BignumInt *ap, *bp; | |
507 | BignumInt *cp, *cps; | |
132c534f | 508 | |
757b0110 | 509 | /* |
510 | * Multiply in the ordinary O(N^2) way. | |
511 | */ | |
132c534f | 512 | |
757b0110 | 513 | for (i = 0; i < len; i++) |
514 | c[i] = 0; | |
515 | ||
516 | for (cps = c + len, ap = a + len; ap-- > a; cps--) { | |
517 | carry = 0; | |
518 | for (cp = cps, bp = b + len; bp--, cp-- > c ;) { | |
519 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; | |
520 | *cp = (BignumInt) t; | |
08b5c9a2 | 521 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
132c534f | 522 | } |
523 | } | |
132c534f | 524 | } |
525 | } | |
526 | ||
527 | /* | |
528 | * Montgomery reduction. Expects x to be a big-endian array of 2*len | |
529 | * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * | |
530 | * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array | |
531 | * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= | |
532 | * x' < n. | |
533 | * | |
534 | * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts | |
535 | * each, containing respectively n and the multiplicative inverse of | |
536 | * -n mod r. | |
537 | * | |
5a502a19 | 538 | * 'tmp' is an array of BignumInt used as scratch space, of length at |
539 | * least 3*len + mul_compute_scratch(len). | |
132c534f | 540 | */ |
541 | static void monty_reduce(BignumInt *x, const BignumInt *n, | |
542 | const BignumInt *mninv, BignumInt *tmp, int len) | |
543 | { | |
544 | int i; | |
545 | BignumInt carry; | |
546 | ||
547 | /* | |
548 | * Multiply x by (-n)^{-1} mod r. This gives us a value m such | |
549 | * that mn is congruent to -x mod r. Hence, mn+x is an exact | |
550 | * multiple of r, and is also (obviously) congruent to x mod n. | |
551 | */ | |
5a502a19 | 552 | internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len); |
132c534f | 553 | |
554 | /* | |
555 | * Compute t = (mn+x)/r in ordinary, non-modular, integer | |
556 | * arithmetic. By construction this is exact, and is congruent mod | |
557 | * n to x * r^{-1}, i.e. the answer we want. | |
558 | * | |
559 | * The following multiply leaves that answer in the _most_ | |
560 | * significant half of the 'x' array, so then we must shift it | |
561 | * down. | |
562 | */ | |
5a502a19 | 563 | internal_mul(tmp, n, tmp+len, len, tmp + 3*len); |
132c534f | 564 | carry = internal_add(x, tmp+len, x, 2*len); |
565 | for (i = 0; i < len; i++) | |
566 | x[len + i] = x[i], x[i] = 0; | |
567 | ||
568 | /* | |
569 | * Reduce t mod n. This doesn't require a full-on division by n, | |
570 | * but merely a test and single optional subtraction, since we can | |
571 | * show that 0 <= t < 2n. | |
572 | * | |
573 | * Proof: | |
574 | * + we computed m mod r, so 0 <= m < r. | |
575 | * + so 0 <= mn < rn, obviously | |
576 | * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn | |
577 | * + yielding 0 <= (mn+x)/r < 2n as required. | |
578 | */ | |
579 | if (!carry) { | |
580 | for (i = 0; i < len; i++) | |
581 | if (x[len + i] != n[i]) | |
582 | break; | |
583 | } | |
584 | if (carry || i >= len || x[len + i] > n[i]) | |
585 | internal_sub(x+len, n, x+len, len); | |
586 | } | |
587 | ||
a3412f52 | 588 | static void internal_add_shifted(BignumInt *number, |
32874aea | 589 | unsigned n, int shift) |
590 | { | |
a3412f52 | 591 | int word = 1 + (shift / BIGNUM_INT_BITS); |
592 | int bshift = shift % BIGNUM_INT_BITS; | |
593 | BignumDblInt addend; | |
9400cf6f | 594 | |
3014da2b | 595 | addend = (BignumDblInt)n << bshift; |
9400cf6f | 596 | |
597 | while (addend) { | |
32874aea | 598 | addend += number[word]; |
a3412f52 | 599 | number[word] = (BignumInt) addend & BIGNUM_INT_MASK; |
600 | addend >>= BIGNUM_INT_BITS; | |
32874aea | 601 | word++; |
9400cf6f | 602 | } |
603 | } | |
604 | ||
e5574168 | 605 | /* |
606 | * Compute a = a % m. | |
9400cf6f | 607 | * Input in first alen words of a and first mlen words of m. |
608 | * Output in first alen words of a | |
609 | * (of which first alen-mlen words will be zero). | |
e5574168 | 610 | * The MSW of m MUST have its high bit set. |
9400cf6f | 611 | * Quotient is accumulated in the `quotient' array, which is a Bignum |
612 | * rather than the internal bigendian format. Quotient parts are shifted | |
613 | * left by `qshift' before adding into quot. | |
e5574168 | 614 | */ |
a3412f52 | 615 | static void internal_mod(BignumInt *a, int alen, |
616 | BignumInt *m, int mlen, | |
617 | BignumInt *quot, int qshift) | |
e5574168 | 618 | { |
a3412f52 | 619 | BignumInt m0, m1; |
e5574168 | 620 | unsigned int h; |
621 | int i, k; | |
622 | ||
e5574168 | 623 | m0 = m[0]; |
9400cf6f | 624 | if (mlen > 1) |
32874aea | 625 | m1 = m[1]; |
9400cf6f | 626 | else |
32874aea | 627 | m1 = 0; |
e5574168 | 628 | |
32874aea | 629 | for (i = 0; i <= alen - mlen; i++) { |
a3412f52 | 630 | BignumDblInt t; |
9400cf6f | 631 | unsigned int q, r, c, ai1; |
e5574168 | 632 | |
633 | if (i == 0) { | |
634 | h = 0; | |
635 | } else { | |
32874aea | 636 | h = a[i - 1]; |
637 | a[i - 1] = 0; | |
e5574168 | 638 | } |
639 | ||
32874aea | 640 | if (i == alen - 1) |
641 | ai1 = 0; | |
642 | else | |
643 | ai1 = a[i + 1]; | |
9400cf6f | 644 | |
e5574168 | 645 | /* Find q = h:a[i] / m0 */ |
62ef3d44 | 646 | if (h >= m0) { |
647 | /* | |
648 | * Special case. | |
649 | * | |
650 | * To illustrate it, suppose a BignumInt is 8 bits, and | |
651 | * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then | |
652 | * our initial division will be 0xA123 / 0xA1, which | |
653 | * will give a quotient of 0x100 and a divide overflow. | |
654 | * However, the invariants in this division algorithm | |
655 | * are not violated, since the full number A1:23:... is | |
656 | * _less_ than the quotient prefix A1:B2:... and so the | |
657 | * following correction loop would have sorted it out. | |
658 | * | |
659 | * In this situation we set q to be the largest | |
660 | * quotient we _can_ stomach (0xFF, of course). | |
661 | */ | |
662 | q = BIGNUM_INT_MASK; | |
663 | } else { | |
819a22b3 | 664 | /* Macro doesn't want an array subscript expression passed |
665 | * into it (see definition), so use a temporary. */ | |
666 | BignumInt tmplo = a[i]; | |
667 | DIVMOD_WORD(q, r, h, tmplo, m0); | |
62ef3d44 | 668 | |
669 | /* Refine our estimate of q by looking at | |
670 | h:a[i]:a[i+1] / m0:m1 */ | |
671 | t = MUL_WORD(m1, q); | |
672 | if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { | |
673 | q--; | |
674 | t -= m1; | |
675 | r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ | |
676 | if (r >= (BignumDblInt) m0 && | |
677 | t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; | |
678 | } | |
e5574168 | 679 | } |
680 | ||
9400cf6f | 681 | /* Subtract q * m from a[i...] */ |
e5574168 | 682 | c = 0; |
9400cf6f | 683 | for (k = mlen - 1; k >= 0; k--) { |
a47e8bba | 684 | t = MUL_WORD(q, m[k]); |
e5574168 | 685 | t += c; |
62ddb51e | 686 | c = (unsigned)(t >> BIGNUM_INT_BITS); |
a3412f52 | 687 | if ((BignumInt) t > a[i + k]) |
32874aea | 688 | c++; |
a3412f52 | 689 | a[i + k] -= (BignumInt) t; |
e5574168 | 690 | } |
691 | ||
692 | /* Add back m in case of borrow */ | |
693 | if (c != h) { | |
694 | t = 0; | |
9400cf6f | 695 | for (k = mlen - 1; k >= 0; k--) { |
e5574168 | 696 | t += m[k]; |
32874aea | 697 | t += a[i + k]; |
a3412f52 | 698 | a[i + k] = (BignumInt) t; |
699 | t = t >> BIGNUM_INT_BITS; | |
e5574168 | 700 | } |
32874aea | 701 | q--; |
e5574168 | 702 | } |
32874aea | 703 | if (quot) |
a3412f52 | 704 | internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); |
e5574168 | 705 | } |
706 | } | |
707 | ||
708 | /* | |
09095ac5 | 709 | * Compute (base ^ exp) % mod, the pedestrian way. |
e5574168 | 710 | */ |
09095ac5 | 711 | Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) |
e5574168 | 712 | { |
5a502a19 | 713 | BignumInt *a, *b, *n, *m, *scratch; |
09095ac5 | 714 | int mshift; |
5a502a19 | 715 | int mlen, scratchlen, i, j; |
09095ac5 | 716 | Bignum base, result; |
ed953b91 | 717 | |
718 | /* | |
719 | * The most significant word of mod needs to be non-zero. It | |
720 | * should already be, but let's make sure. | |
721 | */ | |
722 | assert(mod[mod[0]] != 0); | |
723 | ||
724 | /* | |
725 | * Make sure the base is smaller than the modulus, by reducing | |
726 | * it modulo the modulus if not. | |
727 | */ | |
728 | base = bigmod(base_in, mod); | |
e5574168 | 729 | |
09095ac5 | 730 | /* Allocate m of size mlen, copy mod to m */ |
731 | /* We use big endian internally */ | |
732 | mlen = mod[0]; | |
733 | m = snewn(mlen, BignumInt); | |
734 | for (j = 0; j < mlen; j++) | |
735 | m[j] = mod[mod[0] - j]; | |
736 | ||
737 | /* Shift m left to make msb bit set */ | |
738 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) | |
739 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
740 | break; | |
741 | if (mshift) { | |
742 | for (i = 0; i < mlen - 1; i++) | |
743 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
744 | m[mlen - 1] = m[mlen - 1] << mshift; | |
745 | } | |
746 | ||
747 | /* Allocate n of size mlen, copy base to n */ | |
748 | n = snewn(mlen, BignumInt); | |
749 | i = mlen - base[0]; | |
750 | for (j = 0; j < i; j++) | |
751 | n[j] = 0; | |
752 | for (j = 0; j < (int)base[0]; j++) | |
753 | n[i + j] = base[base[0] - j]; | |
754 | ||
755 | /* Allocate a and b of size 2*mlen. Set a = 1 */ | |
756 | a = snewn(2 * mlen, BignumInt); | |
757 | b = snewn(2 * mlen, BignumInt); | |
758 | for (i = 0; i < 2 * mlen; i++) | |
759 | a[i] = 0; | |
760 | a[2 * mlen - 1] = 1; | |
761 | ||
5a502a19 | 762 | /* Scratch space for multiplies */ |
763 | scratchlen = mul_compute_scratch(mlen); | |
764 | scratch = snewn(scratchlen, BignumInt); | |
765 | ||
09095ac5 | 766 | /* Skip leading zero bits of exp. */ |
767 | i = 0; | |
768 | j = BIGNUM_INT_BITS-1; | |
769 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { | |
770 | j--; | |
771 | if (j < 0) { | |
772 | i++; | |
773 | j = BIGNUM_INT_BITS-1; | |
774 | } | |
775 | } | |
776 | ||
777 | /* Main computation */ | |
778 | while (i < (int)exp[0]) { | |
779 | while (j >= 0) { | |
5a502a19 | 780 | internal_mul(a + mlen, a + mlen, b, mlen, scratch); |
09095ac5 | 781 | internal_mod(b, mlen * 2, m, mlen, NULL, 0); |
782 | if ((exp[exp[0] - i] & (1 << j)) != 0) { | |
5a502a19 | 783 | internal_mul(b + mlen, n, a, mlen, scratch); |
09095ac5 | 784 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
785 | } else { | |
786 | BignumInt *t; | |
787 | t = a; | |
788 | a = b; | |
789 | b = t; | |
790 | } | |
791 | j--; | |
792 | } | |
793 | i++; | |
794 | j = BIGNUM_INT_BITS-1; | |
795 | } | |
796 | ||
797 | /* Fixup result in case the modulus was shifted */ | |
798 | if (mshift) { | |
799 | for (i = mlen - 1; i < 2 * mlen - 1; i++) | |
800 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); | |
801 | a[2 * mlen - 1] = a[2 * mlen - 1] << mshift; | |
802 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); | |
803 | for (i = 2 * mlen - 1; i >= mlen; i--) | |
804 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); | |
805 | } | |
806 | ||
807 | /* Copy result to buffer */ | |
808 | result = newbn(mod[0]); | |
809 | for (i = 0; i < mlen; i++) | |
810 | result[result[0] - i] = a[i + mlen]; | |
811 | while (result[0] > 1 && result[result[0]] == 0) | |
812 | result[0]--; | |
813 | ||
814 | /* Free temporary arrays */ | |
815 | for (i = 0; i < 2 * mlen; i++) | |
816 | a[i] = 0; | |
817 | sfree(a); | |
5a502a19 | 818 | for (i = 0; i < scratchlen; i++) |
819 | scratch[i] = 0; | |
820 | sfree(scratch); | |
09095ac5 | 821 | for (i = 0; i < 2 * mlen; i++) |
822 | b[i] = 0; | |
823 | sfree(b); | |
824 | for (i = 0; i < mlen; i++) | |
825 | m[i] = 0; | |
826 | sfree(m); | |
827 | for (i = 0; i < mlen; i++) | |
828 | n[i] = 0; | |
829 | sfree(n); | |
830 | ||
831 | freebn(base); | |
832 | ||
833 | return result; | |
834 | } | |
835 | ||
836 | /* | |
837 | * Compute (base ^ exp) % mod. Uses the Montgomery multiplication | |
838 | * technique where possible, falling back to modpow_simple otherwise. | |
839 | */ | |
840 | Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) | |
841 | { | |
5a502a19 | 842 | BignumInt *a, *b, *x, *n, *mninv, *scratch; |
843 | int len, scratchlen, i, j; | |
09095ac5 | 844 | Bignum base, base2, r, rn, inv, result; |
845 | ||
846 | /* | |
847 | * The most significant word of mod needs to be non-zero. It | |
848 | * should already be, but let's make sure. | |
849 | */ | |
850 | assert(mod[mod[0]] != 0); | |
851 | ||
132c534f | 852 | /* |
853 | * mod had better be odd, or we can't do Montgomery multiplication | |
854 | * using a power of two at all. | |
855 | */ | |
09095ac5 | 856 | if (!(mod[1] & 1)) |
857 | return modpow_simple(base_in, exp, mod); | |
858 | ||
859 | /* | |
860 | * Make sure the base is smaller than the modulus, by reducing | |
861 | * it modulo the modulus if not. | |
862 | */ | |
863 | base = bigmod(base_in, mod); | |
e5574168 | 864 | |
132c534f | 865 | /* |
866 | * Compute the inverse of n mod r, for monty_reduce. (In fact we | |
867 | * want the inverse of _minus_ n mod r, but we'll sort that out | |
868 | * below.) | |
869 | */ | |
870 | len = mod[0]; | |
871 | r = bn_power_2(BIGNUM_INT_BITS * len); | |
872 | inv = modinv(mod, r); | |
e5574168 | 873 | |
132c534f | 874 | /* |
875 | * Multiply the base by r mod n, to get it into Montgomery | |
876 | * representation. | |
877 | */ | |
878 | base2 = modmul(base, r, mod); | |
879 | freebn(base); | |
880 | base = base2; | |
881 | ||
882 | rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ | |
883 | ||
884 | freebn(r); /* won't need this any more */ | |
885 | ||
886 | /* | |
887 | * Set up internal arrays of the right lengths, in big-endian | |
888 | * format, containing the base, the modulus, and the modulus's | |
889 | * inverse. | |
890 | */ | |
891 | n = snewn(len, BignumInt); | |
892 | for (j = 0; j < len; j++) | |
893 | n[len - 1 - j] = mod[j + 1]; | |
894 | ||
895 | mninv = snewn(len, BignumInt); | |
896 | for (j = 0; j < len; j++) | |
08b5c9a2 | 897 | mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0); |
132c534f | 898 | freebn(inv); /* we don't need this copy of it any more */ |
899 | /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ | |
900 | x = snewn(len, BignumInt); | |
901 | for (j = 0; j < len; j++) | |
902 | x[j] = 0; | |
903 | internal_sub(x, mninv, mninv, len); | |
904 | ||
905 | /* x = snewn(len, BignumInt); */ /* already done above */ | |
906 | for (j = 0; j < len; j++) | |
08b5c9a2 | 907 | x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0); |
132c534f | 908 | freebn(base); /* we don't need this copy of it any more */ |
909 | ||
910 | a = snewn(2*len, BignumInt); | |
911 | b = snewn(2*len, BignumInt); | |
912 | for (j = 0; j < len; j++) | |
08b5c9a2 | 913 | a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0); |
132c534f | 914 | freebn(rn); |
915 | ||
5a502a19 | 916 | /* Scratch space for multiplies */ |
917 | scratchlen = 3*len + mul_compute_scratch(len); | |
918 | scratch = snewn(scratchlen, BignumInt); | |
e5574168 | 919 | |
920 | /* Skip leading zero bits of exp. */ | |
32874aea | 921 | i = 0; |
a3412f52 | 922 | j = BIGNUM_INT_BITS-1; |
62ddb51e | 923 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
e5574168 | 924 | j--; |
32874aea | 925 | if (j < 0) { |
926 | i++; | |
a3412f52 | 927 | j = BIGNUM_INT_BITS-1; |
32874aea | 928 | } |
e5574168 | 929 | } |
930 | ||
931 | /* Main computation */ | |
62ddb51e | 932 | while (i < (int)exp[0]) { |
e5574168 | 933 | while (j >= 0) { |
5a502a19 | 934 | internal_mul(a + len, a + len, b, len, scratch); |
935 | monty_reduce(b, n, mninv, scratch, len); | |
e5574168 | 936 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
5a502a19 | 937 | internal_mul(b + len, x, a, len, scratch); |
938 | monty_reduce(a, n, mninv, scratch, len); | |
e5574168 | 939 | } else { |
a3412f52 | 940 | BignumInt *t; |
32874aea | 941 | t = a; |
942 | a = b; | |
943 | b = t; | |
e5574168 | 944 | } |
945 | j--; | |
946 | } | |
32874aea | 947 | i++; |
a3412f52 | 948 | j = BIGNUM_INT_BITS-1; |
e5574168 | 949 | } |
950 | ||
132c534f | 951 | /* |
952 | * Final monty_reduce to get back from the adjusted Montgomery | |
953 | * representation. | |
954 | */ | |
5a502a19 | 955 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 | 956 | |
957 | /* Copy result to buffer */ | |
59600f67 | 958 | result = newbn(mod[0]); |
132c534f | 959 | for (i = 0; i < len; i++) |
960 | result[result[0] - i] = a[i + len]; | |
32874aea | 961 | while (result[0] > 1 && result[result[0]] == 0) |
962 | result[0]--; | |
e5574168 | 963 | |
964 | /* Free temporary arrays */ | |
5a502a19 | 965 | for (i = 0; i < scratchlen; i++) |
966 | scratch[i] = 0; | |
967 | sfree(scratch); | |
132c534f | 968 | for (i = 0; i < 2 * len; i++) |
32874aea | 969 | a[i] = 0; |
970 | sfree(a); | |
132c534f | 971 | for (i = 0; i < 2 * len; i++) |
32874aea | 972 | b[i] = 0; |
973 | sfree(b); | |
132c534f | 974 | for (i = 0; i < len; i++) |
975 | mninv[i] = 0; | |
976 | sfree(mninv); | |
977 | for (i = 0; i < len; i++) | |
32874aea | 978 | n[i] = 0; |
979 | sfree(n); | |
132c534f | 980 | for (i = 0; i < len; i++) |
981 | x[i] = 0; | |
982 | sfree(x); | |
ed953b91 | 983 | |
59600f67 | 984 | return result; |
e5574168 | 985 | } |
7cca0d81 | 986 | |
987 | /* | |
988 | * Compute (p * q) % mod. | |
989 | * The most significant word of mod MUST be non-zero. | |
990 | * We assume that the result array is the same size as the mod array. | |
991 | */ | |
59600f67 | 992 | Bignum modmul(Bignum p, Bignum q, Bignum mod) |
7cca0d81 | 993 | { |
5a502a19 | 994 | BignumInt *a, *n, *m, *o, *scratch; |
995 | int mshift, scratchlen; | |
80b10571 | 996 | int pqlen, mlen, rlen, i, j; |
59600f67 | 997 | Bignum result; |
7cca0d81 | 998 | |
999 | /* Allocate m of size mlen, copy mod to m */ | |
1000 | /* We use big endian internally */ | |
1001 | mlen = mod[0]; | |
a3412f52 | 1002 | m = snewn(mlen, BignumInt); |
32874aea | 1003 | for (j = 0; j < mlen; j++) |
1004 | m[j] = mod[mod[0] - j]; | |
7cca0d81 | 1005 | |
1006 | /* Shift m left to make msb bit set */ | |
a3412f52 | 1007 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1008 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
32874aea | 1009 | break; |
7cca0d81 | 1010 | if (mshift) { |
1011 | for (i = 0; i < mlen - 1; i++) | |
a3412f52 | 1012 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea | 1013 | m[mlen - 1] = m[mlen - 1] << mshift; |
7cca0d81 | 1014 | } |
1015 | ||
1016 | pqlen = (p[0] > q[0] ? p[0] : q[0]); | |
1017 | ||
aca5132b MW |
1018 | /* Make sure that we're allowing enough space. The shifting below will |
1019 | * underflow the vectors we allocate if `pqlen' is too small. | |
1020 | */ | |
1021 | if (2*pqlen <= mlen) | |
1022 | pqlen = mlen/2 + 1; | |
1023 | ||
7cca0d81 | 1024 | /* Allocate n of size pqlen, copy p to n */ |
a3412f52 | 1025 | n = snewn(pqlen, BignumInt); |
7cca0d81 | 1026 | i = pqlen - p[0]; |
32874aea | 1027 | for (j = 0; j < i; j++) |
1028 | n[j] = 0; | |
62ddb51e | 1029 | for (j = 0; j < (int)p[0]; j++) |
32874aea | 1030 | n[i + j] = p[p[0] - j]; |
7cca0d81 | 1031 | |
1032 | /* Allocate o of size pqlen, copy q to o */ | |
a3412f52 | 1033 | o = snewn(pqlen, BignumInt); |
7cca0d81 | 1034 | i = pqlen - q[0]; |
32874aea | 1035 | for (j = 0; j < i; j++) |
1036 | o[j] = 0; | |
62ddb51e | 1037 | for (j = 0; j < (int)q[0]; j++) |
32874aea | 1038 | o[i + j] = q[q[0] - j]; |
7cca0d81 | 1039 | |
1040 | /* Allocate a of size 2*pqlen for result */ | |
a3412f52 | 1041 | a = snewn(2 * pqlen, BignumInt); |
7cca0d81 | 1042 | |
5a502a19 | 1043 | /* Scratch space for multiplies */ |
1044 | scratchlen = mul_compute_scratch(pqlen); | |
1045 | scratch = snewn(scratchlen, BignumInt); | |
1046 | ||
7cca0d81 | 1047 | /* Main computation */ |
5a502a19 | 1048 | internal_mul(n, o, a, pqlen, scratch); |
32874aea | 1049 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
7cca0d81 | 1050 | |
1051 | /* Fixup result in case the modulus was shifted */ | |
1052 | if (mshift) { | |
32874aea | 1053 | for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++) |
a3412f52 | 1054 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea | 1055 | a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift; |
1056 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); | |
1057 | for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--) | |
a3412f52 | 1058 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); |
7cca0d81 | 1059 | } |
1060 | ||
1061 | /* Copy result to buffer */ | |
32874aea | 1062 | rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); |
80b10571 | 1063 | result = newbn(rlen); |
1064 | for (i = 0; i < rlen; i++) | |
32874aea | 1065 | result[result[0] - i] = a[i + 2 * pqlen - rlen]; |
1066 | while (result[0] > 1 && result[result[0]] == 0) | |
1067 | result[0]--; | |
7cca0d81 | 1068 | |
1069 | /* Free temporary arrays */ | |
5a502a19 | 1070 | for (i = 0; i < scratchlen; i++) |
1071 | scratch[i] = 0; | |
1072 | sfree(scratch); | |
32874aea | 1073 | for (i = 0; i < 2 * pqlen; i++) |
1074 | a[i] = 0; | |
1075 | sfree(a); | |
1076 | for (i = 0; i < mlen; i++) | |
1077 | m[i] = 0; | |
1078 | sfree(m); | |
1079 | for (i = 0; i < pqlen; i++) | |
1080 | n[i] = 0; | |
1081 | sfree(n); | |
1082 | for (i = 0; i < pqlen; i++) | |
1083 | o[i] = 0; | |
1084 | sfree(o); | |
59600f67 | 1085 | |
1086 | return result; | |
7cca0d81 | 1087 | } |
1088 | ||
1089 | /* | |
9400cf6f | 1090 | * Compute p % mod. |
1091 | * The most significant word of mod MUST be non-zero. | |
1092 | * We assume that the result array is the same size as the mod array. | |
5c72ca61 | 1093 | * We optionally write out a quotient if `quotient' is non-NULL. |
1094 | * We can avoid writing out the result if `result' is NULL. | |
9400cf6f | 1095 | */ |
f28753ab | 1096 | static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) |
9400cf6f | 1097 | { |
a3412f52 | 1098 | BignumInt *n, *m; |
9400cf6f | 1099 | int mshift; |
1100 | int plen, mlen, i, j; | |
1101 | ||
1102 | /* Allocate m of size mlen, copy mod to m */ | |
1103 | /* We use big endian internally */ | |
1104 | mlen = mod[0]; | |
a3412f52 | 1105 | m = snewn(mlen, BignumInt); |
32874aea | 1106 | for (j = 0; j < mlen; j++) |
1107 | m[j] = mod[mod[0] - j]; | |
9400cf6f | 1108 | |
1109 | /* Shift m left to make msb bit set */ | |
a3412f52 | 1110 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1111 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) | |
32874aea | 1112 | break; |
9400cf6f | 1113 | if (mshift) { |
1114 | for (i = 0; i < mlen - 1; i++) | |
a3412f52 | 1115 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea | 1116 | m[mlen - 1] = m[mlen - 1] << mshift; |
9400cf6f | 1117 | } |
1118 | ||
1119 | plen = p[0]; | |
1120 | /* Ensure plen > mlen */ | |
32874aea | 1121 | if (plen <= mlen) |
1122 | plen = mlen + 1; | |
9400cf6f | 1123 | |
1124 | /* Allocate n of size plen, copy p to n */ | |
a3412f52 | 1125 | n = snewn(plen, BignumInt); |
32874aea | 1126 | for (j = 0; j < plen; j++) |
1127 | n[j] = 0; | |
62ddb51e | 1128 | for (j = 1; j <= (int)p[0]; j++) |
32874aea | 1129 | n[plen - j] = p[j]; |
9400cf6f | 1130 | |
1131 | /* Main computation */ | |
1132 | internal_mod(n, plen, m, mlen, quotient, mshift); | |
1133 | ||
1134 | /* Fixup result in case the modulus was shifted */ | |
1135 | if (mshift) { | |
1136 | for (i = plen - mlen - 1; i < plen - 1; i++) | |
a3412f52 | 1137 | n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea | 1138 | n[plen - 1] = n[plen - 1] << mshift; |
9400cf6f | 1139 | internal_mod(n, plen, m, mlen, quotient, 0); |
1140 | for (i = plen - 1; i >= plen - mlen; i--) | |
a3412f52 | 1141 | n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); |
9400cf6f | 1142 | } |
1143 | ||
1144 | /* Copy result to buffer */ | |
5c72ca61 | 1145 | if (result) { |
62ddb51e | 1146 | for (i = 1; i <= (int)result[0]; i++) { |
5c72ca61 | 1147 | int j = plen - i; |
1148 | result[i] = j >= 0 ? n[j] : 0; | |
1149 | } | |
9400cf6f | 1150 | } |
1151 | ||
1152 | /* Free temporary arrays */ | |
32874aea | 1153 | for (i = 0; i < mlen; i++) |
1154 | m[i] = 0; | |
1155 | sfree(m); | |
1156 | for (i = 0; i < plen; i++) | |
1157 | n[i] = 0; | |
1158 | sfree(n); | |
9400cf6f | 1159 | } |
1160 | ||
1161 | /* | |
7cca0d81 | 1162 | * Decrement a number. |
1163 | */ | |
32874aea | 1164 | void decbn(Bignum bn) |
1165 | { | |
7cca0d81 | 1166 | int i = 1; |
62ddb51e | 1167 | while (i < (int)bn[0] && bn[i] == 0) |
a3412f52 | 1168 | bn[i++] = BIGNUM_INT_MASK; |
7cca0d81 | 1169 | bn[i]--; |
1170 | } | |
1171 | ||
27cd7fc2 | 1172 | Bignum bignum_from_bytes(const unsigned char *data, int nbytes) |
32874aea | 1173 | { |
3709bfe9 | 1174 | Bignum result; |
1175 | int w, i; | |
1176 | ||
a3412f52 | 1177 | w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ |
3709bfe9 | 1178 | |
1179 | result = newbn(w); | |
32874aea | 1180 | for (i = 1; i <= w; i++) |
1181 | result[i] = 0; | |
1182 | for (i = nbytes; i--;) { | |
1183 | unsigned char byte = *data++; | |
a3412f52 | 1184 | result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); |
3709bfe9 | 1185 | } |
1186 | ||
32874aea | 1187 | while (result[0] > 1 && result[result[0]] == 0) |
1188 | result[0]--; | |
3709bfe9 | 1189 | return result; |
1190 | } | |
1191 | ||
7cca0d81 | 1192 | /* |
2e85c969 | 1193 | * Read an SSH-1-format bignum from a data buffer. Return the number |
0016d70b | 1194 | * of bytes consumed, or -1 if there wasn't enough data. |
7cca0d81 | 1195 | */ |
0016d70b | 1196 | int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) |
32874aea | 1197 | { |
27cd7fc2 | 1198 | const unsigned char *p = data; |
7cca0d81 | 1199 | int i; |
1200 | int w, b; | |
1201 | ||
0016d70b | 1202 | if (len < 2) |
1203 | return -1; | |
1204 | ||
7cca0d81 | 1205 | w = 0; |
32874aea | 1206 | for (i = 0; i < 2; i++) |
1207 | w = (w << 8) + *p++; | |
1208 | b = (w + 7) / 8; /* bits -> bytes */ | |
7cca0d81 | 1209 | |
0016d70b | 1210 | if (len < b+2) |
1211 | return -1; | |
1212 | ||
32874aea | 1213 | if (!result) /* just return length */ |
1214 | return b + 2; | |
a52f067e | 1215 | |
3709bfe9 | 1216 | *result = bignum_from_bytes(p, b); |
7cca0d81 | 1217 | |
3709bfe9 | 1218 | return p + b - data; |
7cca0d81 | 1219 | } |
5c58ad2d | 1220 | |
1221 | /* | |
2e85c969 | 1222 | * Return the bit count of a bignum, for SSH-1 encoding. |
5c58ad2d | 1223 | */ |
32874aea | 1224 | int bignum_bitcount(Bignum bn) |
1225 | { | |
a3412f52 | 1226 | int bitcount = bn[0] * BIGNUM_INT_BITS - 1; |
32874aea | 1227 | while (bitcount >= 0 |
a3412f52 | 1228 | && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; |
5c58ad2d | 1229 | return bitcount + 1; |
1230 | } | |
1231 | ||
1232 | /* | |
2e85c969 | 1233 | * Return the byte length of a bignum when SSH-1 encoded. |
5c58ad2d | 1234 | */ |
32874aea | 1235 | int ssh1_bignum_length(Bignum bn) |
1236 | { | |
1237 | return 2 + (bignum_bitcount(bn) + 7) / 8; | |
ddecd643 | 1238 | } |
1239 | ||
1240 | /* | |
2e85c969 | 1241 | * Return the byte length of a bignum when SSH-2 encoded. |
ddecd643 | 1242 | */ |
32874aea | 1243 | int ssh2_bignum_length(Bignum bn) |
1244 | { | |
1245 | return 4 + (bignum_bitcount(bn) + 8) / 8; | |
5c58ad2d | 1246 | } |
1247 | ||
1248 | /* | |
1249 | * Return a byte from a bignum; 0 is least significant, etc. | |
1250 | */ | |
32874aea | 1251 | int bignum_byte(Bignum bn, int i) |
1252 | { | |
62ddb51e | 1253 | if (i >= (int)(BIGNUM_INT_BYTES * bn[0])) |
32874aea | 1254 | return 0; /* beyond the end */ |
5c58ad2d | 1255 | else |
a3412f52 | 1256 | return (bn[i / BIGNUM_INT_BYTES + 1] >> |
1257 | ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; | |
5c58ad2d | 1258 | } |
1259 | ||
1260 | /* | |
9400cf6f | 1261 | * Return a bit from a bignum; 0 is least significant, etc. |
1262 | */ | |
32874aea | 1263 | int bignum_bit(Bignum bn, int i) |
1264 | { | |
62ddb51e | 1265 | if (i >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea | 1266 | return 0; /* beyond the end */ |
9400cf6f | 1267 | else |
a3412f52 | 1268 | return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; |
9400cf6f | 1269 | } |
1270 | ||
1271 | /* | |
1272 | * Set a bit in a bignum; 0 is least significant, etc. | |
1273 | */ | |
32874aea | 1274 | void bignum_set_bit(Bignum bn, int bitnum, int value) |
1275 | { | |
62ddb51e | 1276 | if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea | 1277 | abort(); /* beyond the end */ |
9400cf6f | 1278 | else { |
a3412f52 | 1279 | int v = bitnum / BIGNUM_INT_BITS + 1; |
1280 | int mask = 1 << (bitnum % BIGNUM_INT_BITS); | |
32874aea | 1281 | if (value) |
1282 | bn[v] |= mask; | |
1283 | else | |
1284 | bn[v] &= ~mask; | |
9400cf6f | 1285 | } |
1286 | } | |
1287 | ||
1288 | /* | |
2e85c969 | 1289 | * Write a SSH-1-format bignum into a buffer. It is assumed the |
5c58ad2d | 1290 | * buffer is big enough. Returns the number of bytes used. |
1291 | */ | |
32874aea | 1292 | int ssh1_write_bignum(void *data, Bignum bn) |
1293 | { | |
5c58ad2d | 1294 | unsigned char *p = data; |
1295 | int len = ssh1_bignum_length(bn); | |
1296 | int i; | |
ddecd643 | 1297 | int bitc = bignum_bitcount(bn); |
5c58ad2d | 1298 | |
1299 | *p++ = (bitc >> 8) & 0xFF; | |
32874aea | 1300 | *p++ = (bitc) & 0xFF; |
1301 | for (i = len - 2; i--;) | |
1302 | *p++ = bignum_byte(bn, i); | |
5c58ad2d | 1303 | return len; |
1304 | } | |
9400cf6f | 1305 | |
1306 | /* | |
1307 | * Compare two bignums. Returns like strcmp. | |
1308 | */ | |
32874aea | 1309 | int bignum_cmp(Bignum a, Bignum b) |
1310 | { | |
9400cf6f | 1311 | int amax = a[0], bmax = b[0]; |
1312 | int i = (amax > bmax ? amax : bmax); | |
1313 | while (i) { | |
a3412f52 | 1314 | BignumInt aval = (i > amax ? 0 : a[i]); |
1315 | BignumInt bval = (i > bmax ? 0 : b[i]); | |
32874aea | 1316 | if (aval < bval) |
1317 | return -1; | |
1318 | if (aval > bval) | |
1319 | return +1; | |
1320 | i--; | |
9400cf6f | 1321 | } |
1322 | return 0; | |
1323 | } | |
1324 | ||
1325 | /* | |
1326 | * Right-shift one bignum to form another. | |
1327 | */ | |
32874aea | 1328 | Bignum bignum_rshift(Bignum a, int shift) |
1329 | { | |
9400cf6f | 1330 | Bignum ret; |
1331 | int i, shiftw, shiftb, shiftbb, bits; | |
a3412f52 | 1332 | BignumInt ai, ai1; |
9400cf6f | 1333 | |
ddecd643 | 1334 | bits = bignum_bitcount(a) - shift; |
a3412f52 | 1335 | ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); |
9400cf6f | 1336 | |
1337 | if (ret) { | |
a3412f52 | 1338 | shiftw = shift / BIGNUM_INT_BITS; |
1339 | shiftb = shift % BIGNUM_INT_BITS; | |
1340 | shiftbb = BIGNUM_INT_BITS - shiftb; | |
32874aea | 1341 | |
1342 | ai1 = a[shiftw + 1]; | |
62ddb51e | 1343 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea | 1344 | ai = ai1; |
62ddb51e | 1345 | ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); |
a3412f52 | 1346 | ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; |
32874aea | 1347 | } |
9400cf6f | 1348 | } |
1349 | ||
1350 | return ret; | |
1351 | } | |
1352 | ||
1353 | /* | |
1354 | * Non-modular multiplication and addition. | |
1355 | */ | |
32874aea | 1356 | Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) |
1357 | { | |
9400cf6f | 1358 | int alen = a[0], blen = b[0]; |
1359 | int mlen = (alen > blen ? alen : blen); | |
1360 | int rlen, i, maxspot; | |
5a502a19 | 1361 | int wslen; |
a3412f52 | 1362 | BignumInt *workspace; |
9400cf6f | 1363 | Bignum ret; |
1364 | ||
5a502a19 | 1365 | /* mlen space for a, mlen space for b, 2*mlen for result, |
1366 | * plus scratch space for multiplication */ | |
1367 | wslen = mlen * 4 + mul_compute_scratch(mlen); | |
1368 | workspace = snewn(wslen, BignumInt); | |
9400cf6f | 1369 | for (i = 0; i < mlen; i++) { |
62ddb51e | 1370 | workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0); |
1371 | workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0); | |
9400cf6f | 1372 | } |
1373 | ||
32874aea | 1374 | internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, |
5a502a19 | 1375 | workspace + 2 * mlen, mlen, workspace + 4 * mlen); |
9400cf6f | 1376 | |
1377 | /* now just copy the result back */ | |
1378 | rlen = alen + blen + 1; | |
62ddb51e | 1379 | if (addend && rlen <= (int)addend[0]) |
32874aea | 1380 | rlen = addend[0] + 1; |
9400cf6f | 1381 | ret = newbn(rlen); |
1382 | maxspot = 0; | |
62ddb51e | 1383 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea | 1384 | ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); |
1385 | if (ret[i] != 0) | |
1386 | maxspot = i; | |
9400cf6f | 1387 | } |
1388 | ret[0] = maxspot; | |
1389 | ||
1390 | /* now add in the addend, if any */ | |
1391 | if (addend) { | |
a3412f52 | 1392 | BignumDblInt carry = 0; |
32874aea | 1393 | for (i = 1; i <= rlen; i++) { |
62ddb51e | 1394 | carry += (i <= (int)ret[0] ? ret[i] : 0); |
1395 | carry += (i <= (int)addend[0] ? addend[i] : 0); | |
a3412f52 | 1396 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1397 | carry >>= BIGNUM_INT_BITS; | |
32874aea | 1398 | if (ret[i] != 0 && i > maxspot) |
1399 | maxspot = i; | |
1400 | } | |
9400cf6f | 1401 | } |
1402 | ret[0] = maxspot; | |
1403 | ||
5a502a19 | 1404 | for (i = 0; i < wslen; i++) |
1405 | workspace[i] = 0; | |
c523f55f | 1406 | sfree(workspace); |
9400cf6f | 1407 | return ret; |
1408 | } | |
1409 | ||
1410 | /* | |
1411 | * Non-modular multiplication. | |
1412 | */ | |
32874aea | 1413 | Bignum bigmul(Bignum a, Bignum b) |
1414 | { | |
9400cf6f | 1415 | return bigmuladd(a, b, NULL); |
1416 | } | |
1417 | ||
1418 | /* | |
d737853b | 1419 | * Simple addition. |
1420 | */ | |
1421 | Bignum bigadd(Bignum a, Bignum b) | |
1422 | { | |
1423 | int alen = a[0], blen = b[0]; | |
1424 | int rlen = (alen > blen ? alen : blen) + 1; | |
1425 | int i, maxspot; | |
1426 | Bignum ret; | |
1427 | BignumDblInt carry; | |
1428 | ||
1429 | ret = newbn(rlen); | |
1430 | ||
1431 | carry = 0; | |
1432 | maxspot = 0; | |
1433 | for (i = 1; i <= rlen; i++) { | |
1434 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1435 | carry += (i <= (int)b[0] ? b[i] : 0); | |
1436 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1437 | carry >>= BIGNUM_INT_BITS; | |
1438 | if (ret[i] != 0 && i > maxspot) | |
1439 | maxspot = i; | |
1440 | } | |
1441 | ret[0] = maxspot; | |
1442 | ||
1443 | return ret; | |
1444 | } | |
1445 | ||
1446 | /* | |
1447 | * Subtraction. Returns a-b, or NULL if the result would come out | |
1448 | * negative (recall that this entire bignum module only handles | |
1449 | * positive numbers). | |
1450 | */ | |
1451 | Bignum bigsub(Bignum a, Bignum b) | |
1452 | { | |
1453 | int alen = a[0], blen = b[0]; | |
1454 | int rlen = (alen > blen ? alen : blen); | |
1455 | int i, maxspot; | |
1456 | Bignum ret; | |
1457 | BignumDblInt carry; | |
1458 | ||
1459 | ret = newbn(rlen); | |
1460 | ||
1461 | carry = 1; | |
1462 | maxspot = 0; | |
1463 | for (i = 1; i <= rlen; i++) { | |
1464 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1465 | carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); | |
1466 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1467 | carry >>= BIGNUM_INT_BITS; | |
1468 | if (ret[i] != 0 && i > maxspot) | |
1469 | maxspot = i; | |
1470 | } | |
1471 | ret[0] = maxspot; | |
1472 | ||
1473 | if (!carry) { | |
1474 | freebn(ret); | |
1475 | return NULL; | |
1476 | } | |
1477 | ||
1478 | return ret; | |
1479 | } | |
1480 | ||
1481 | /* | |
3709bfe9 | 1482 | * Create a bignum which is the bitmask covering another one. That |
1483 | * is, the smallest integer which is >= N and is also one less than | |
1484 | * a power of two. | |
1485 | */ | |
32874aea | 1486 | Bignum bignum_bitmask(Bignum n) |
1487 | { | |
3709bfe9 | 1488 | Bignum ret = copybn(n); |
1489 | int i; | |
a3412f52 | 1490 | BignumInt j; |
3709bfe9 | 1491 | |
1492 | i = ret[0]; | |
1493 | while (n[i] == 0 && i > 0) | |
32874aea | 1494 | i--; |
3709bfe9 | 1495 | if (i <= 0) |
32874aea | 1496 | return ret; /* input was zero */ |
3709bfe9 | 1497 | j = 1; |
1498 | while (j < n[i]) | |
32874aea | 1499 | j = 2 * j + 1; |
3709bfe9 | 1500 | ret[i] = j; |
1501 | while (--i > 0) | |
a3412f52 | 1502 | ret[i] = BIGNUM_INT_MASK; |
3709bfe9 | 1503 | return ret; |
1504 | } | |
1505 | ||
1506 | /* | |
5c72ca61 | 1507 | * Convert a (max 32-bit) long into a bignum. |
9400cf6f | 1508 | */ |
a3412f52 | 1509 | Bignum bignum_from_long(unsigned long nn) |
32874aea | 1510 | { |
9400cf6f | 1511 | Bignum ret; |
a3412f52 | 1512 | BignumDblInt n = nn; |
9400cf6f | 1513 | |
5c72ca61 | 1514 | ret = newbn(3); |
a3412f52 | 1515 | ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); |
1516 | ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); | |
5c72ca61 | 1517 | ret[3] = 0; |
1518 | ret[0] = (ret[2] ? 2 : 1); | |
32874aea | 1519 | return ret; |
9400cf6f | 1520 | } |
1521 | ||
1522 | /* | |
1523 | * Add a long to a bignum. | |
1524 | */ | |
a3412f52 | 1525 | Bignum bignum_add_long(Bignum number, unsigned long addendx) |
32874aea | 1526 | { |
1527 | Bignum ret = newbn(number[0] + 1); | |
9400cf6f | 1528 | int i, maxspot = 0; |
a3412f52 | 1529 | BignumDblInt carry = 0, addend = addendx; |
9400cf6f | 1530 | |
62ddb51e | 1531 | for (i = 1; i <= (int)ret[0]; i++) { |
a3412f52 | 1532 | carry += addend & BIGNUM_INT_MASK; |
62ddb51e | 1533 | carry += (i <= (int)number[0] ? number[i] : 0); |
a3412f52 | 1534 | addend >>= BIGNUM_INT_BITS; |
1535 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1536 | carry >>= BIGNUM_INT_BITS; | |
32874aea | 1537 | if (ret[i] != 0) |
1538 | maxspot = i; | |
9400cf6f | 1539 | } |
1540 | ret[0] = maxspot; | |
1541 | return ret; | |
1542 | } | |
1543 | ||
1544 | /* | |
1545 | * Compute the residue of a bignum, modulo a (max 16-bit) short. | |
1546 | */ | |
32874aea | 1547 | unsigned short bignum_mod_short(Bignum number, unsigned short modulus) |
1548 | { | |
a3412f52 | 1549 | BignumDblInt mod, r; |
9400cf6f | 1550 | int i; |
1551 | ||
1552 | r = 0; | |
1553 | mod = modulus; | |
1554 | for (i = number[0]; i > 0; i--) | |
736cc6d1 | 1555 | r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; |
6e522441 | 1556 | return (unsigned short) r; |
9400cf6f | 1557 | } |
1558 | ||
a3412f52 | 1559 | #ifdef DEBUG |
32874aea | 1560 | void diagbn(char *prefix, Bignum md) |
1561 | { | |
9400cf6f | 1562 | int i, nibbles, morenibbles; |
1563 | static const char hex[] = "0123456789ABCDEF"; | |
1564 | ||
5c72ca61 | 1565 | debug(("%s0x", prefix ? prefix : "")); |
9400cf6f | 1566 | |
32874aea | 1567 | nibbles = (3 + bignum_bitcount(md)) / 4; |
1568 | if (nibbles < 1) | |
1569 | nibbles = 1; | |
1570 | morenibbles = 4 * md[0] - nibbles; | |
1571 | for (i = 0; i < morenibbles; i++) | |
5c72ca61 | 1572 | debug(("-")); |
32874aea | 1573 | for (i = nibbles; i--;) |
5c72ca61 | 1574 | debug(("%c", |
1575 | hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF])); | |
9400cf6f | 1576 | |
32874aea | 1577 | if (prefix) |
5c72ca61 | 1578 | debug(("\n")); |
1579 | } | |
f28753ab | 1580 | #endif |
5c72ca61 | 1581 | |
1582 | /* | |
1583 | * Simple division. | |
1584 | */ | |
1585 | Bignum bigdiv(Bignum a, Bignum b) | |
1586 | { | |
1587 | Bignum q = newbn(a[0]); | |
1588 | bigdivmod(a, b, NULL, q); | |
1589 | return q; | |
1590 | } | |
1591 | ||
1592 | /* | |
1593 | * Simple remainder. | |
1594 | */ | |
1595 | Bignum bigmod(Bignum a, Bignum b) | |
1596 | { | |
1597 | Bignum r = newbn(b[0]); | |
1598 | bigdivmod(a, b, r, NULL); | |
1599 | return r; | |
9400cf6f | 1600 | } |
1601 | ||
1602 | /* | |
1603 | * Greatest common divisor. | |
1604 | */ | |
32874aea | 1605 | Bignum biggcd(Bignum av, Bignum bv) |
1606 | { | |
9400cf6f | 1607 | Bignum a = copybn(av); |
1608 | Bignum b = copybn(bv); | |
1609 | ||
9400cf6f | 1610 | while (bignum_cmp(b, Zero) != 0) { |
32874aea | 1611 | Bignum t = newbn(b[0]); |
5c72ca61 | 1612 | bigdivmod(a, b, t, NULL); |
32874aea | 1613 | while (t[0] > 1 && t[t[0]] == 0) |
1614 | t[0]--; | |
1615 | freebn(a); | |
1616 | a = b; | |
1617 | b = t; | |
9400cf6f | 1618 | } |
1619 | ||
1620 | freebn(b); | |
1621 | return a; | |
1622 | } | |
1623 | ||
1624 | /* | |
1625 | * Modular inverse, using Euclid's extended algorithm. | |
1626 | */ | |
32874aea | 1627 | Bignum modinv(Bignum number, Bignum modulus) |
1628 | { | |
9400cf6f | 1629 | Bignum a = copybn(modulus); |
1630 | Bignum b = copybn(number); | |
1631 | Bignum xp = copybn(Zero); | |
1632 | Bignum x = copybn(One); | |
1633 | int sign = +1; | |
1634 | ||
1635 | while (bignum_cmp(b, One) != 0) { | |
32874aea | 1636 | Bignum t = newbn(b[0]); |
1637 | Bignum q = newbn(a[0]); | |
5c72ca61 | 1638 | bigdivmod(a, b, t, q); |
32874aea | 1639 | while (t[0] > 1 && t[t[0]] == 0) |
1640 | t[0]--; | |
1641 | freebn(a); | |
1642 | a = b; | |
1643 | b = t; | |
1644 | t = xp; | |
1645 | xp = x; | |
1646 | x = bigmuladd(q, xp, t); | |
1647 | sign = -sign; | |
1648 | freebn(t); | |
75374b2f | 1649 | freebn(q); |
9400cf6f | 1650 | } |
1651 | ||
1652 | freebn(b); | |
1653 | freebn(a); | |
1654 | freebn(xp); | |
1655 | ||
1656 | /* now we know that sign * x == 1, and that x < modulus */ | |
1657 | if (sign < 0) { | |
32874aea | 1658 | /* set a new x to be modulus - x */ |
1659 | Bignum newx = newbn(modulus[0]); | |
a3412f52 | 1660 | BignumInt carry = 0; |
32874aea | 1661 | int maxspot = 1; |
1662 | int i; | |
1663 | ||
62ddb51e | 1664 | for (i = 1; i <= (int)newx[0]; i++) { |
1665 | BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); | |
1666 | BignumInt bword = (i <= (int)x[0] ? x[i] : 0); | |
32874aea | 1667 | newx[i] = aword - bword - carry; |
1668 | bword = ~bword; | |
1669 | carry = carry ? (newx[i] >= bword) : (newx[i] > bword); | |
1670 | if (newx[i] != 0) | |
1671 | maxspot = i; | |
1672 | } | |
1673 | newx[0] = maxspot; | |
1674 | freebn(x); | |
1675 | x = newx; | |
9400cf6f | 1676 | } |
1677 | ||
1678 | /* and return. */ | |
1679 | return x; | |
1680 | } | |
6e522441 | 1681 | |
1682 | /* | |
1683 | * Render a bignum into decimal. Return a malloced string holding | |
1684 | * the decimal representation. | |
1685 | */ | |
32874aea | 1686 | char *bignum_decimal(Bignum x) |
1687 | { | |
6e522441 | 1688 | int ndigits, ndigit; |
1689 | int i, iszero; | |
a3412f52 | 1690 | BignumDblInt carry; |
6e522441 | 1691 | char *ret; |
a3412f52 | 1692 | BignumInt *workspace; |
6e522441 | 1693 | |
1694 | /* | |
1695 | * First, estimate the number of digits. Since log(10)/log(2) | |
1696 | * is just greater than 93/28 (the joys of continued fraction | |
1697 | * approximations...) we know that for every 93 bits, we need | |
1698 | * at most 28 digits. This will tell us how much to malloc. | |
1699 | * | |
1700 | * Formally: if x has i bits, that means x is strictly less | |
1701 | * than 2^i. Since 2 is less than 10^(28/93), this is less than | |
1702 | * 10^(28i/93). We need an integer power of ten, so we must | |
1703 | * round up (rounding down might make it less than x again). | |
1704 | * Therefore if we multiply the bit count by 28/93, rounding | |
1705 | * up, we will have enough digits. | |
74c79ce8 | 1706 | * |
1707 | * i=0 (i.e., x=0) is an irritating special case. | |
6e522441 | 1708 | */ |
ddecd643 | 1709 | i = bignum_bitcount(x); |
74c79ce8 | 1710 | if (!i) |
1711 | ndigits = 1; /* x = 0 */ | |
1712 | else | |
1713 | ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ | |
32874aea | 1714 | ndigits++; /* allow for trailing \0 */ |
3d88e64d | 1715 | ret = snewn(ndigits, char); |
6e522441 | 1716 | |
1717 | /* | |
1718 | * Now allocate some workspace to hold the binary form as we | |
1719 | * repeatedly divide it by ten. Initialise this to the | |
1720 | * big-endian form of the number. | |
1721 | */ | |
a3412f52 | 1722 | workspace = snewn(x[0], BignumInt); |
62ddb51e | 1723 | for (i = 0; i < (int)x[0]; i++) |
32874aea | 1724 | workspace[i] = x[x[0] - i]; |
6e522441 | 1725 | |
1726 | /* | |
1727 | * Next, write the decimal number starting with the last digit. | |
1728 | * We use ordinary short division, dividing 10 into the | |
1729 | * workspace. | |
1730 | */ | |
32874aea | 1731 | ndigit = ndigits - 1; |
6e522441 | 1732 | ret[ndigit] = '\0'; |
1733 | do { | |
32874aea | 1734 | iszero = 1; |
1735 | carry = 0; | |
62ddb51e | 1736 | for (i = 0; i < (int)x[0]; i++) { |
a3412f52 | 1737 | carry = (carry << BIGNUM_INT_BITS) + workspace[i]; |
1738 | workspace[i] = (BignumInt) (carry / 10); | |
32874aea | 1739 | if (workspace[i]) |
1740 | iszero = 0; | |
1741 | carry %= 10; | |
1742 | } | |
1743 | ret[--ndigit] = (char) (carry + '0'); | |
6e522441 | 1744 | } while (!iszero); |
1745 | ||
1746 | /* | |
1747 | * There's a chance we've fallen short of the start of the | |
1748 | * string. Correct if so. | |
1749 | */ | |
1750 | if (ndigit > 0) | |
32874aea | 1751 | memmove(ret, ret + ndigit, ndigits - ndigit); |
6e522441 | 1752 | |
1753 | /* | |
1754 | * Done. | |
1755 | */ | |
c523f55f | 1756 | sfree(workspace); |
6e522441 | 1757 | return ret; |
1758 | } | |
f3c29e34 | 1759 | |
1760 | #ifdef TESTBN | |
1761 | ||
1762 | #include <stdio.h> | |
1763 | #include <stdlib.h> | |
1764 | #include <ctype.h> | |
1765 | ||
1766 | /* | |
4800a5e5 | 1767 | * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset |
f84f1e46 | 1768 | * |
1769 | * Then feed to this program's standard input the output of | |
1770 | * testdata/bignum.py . | |
f3c29e34 | 1771 | */ |
1772 | ||
1773 | void modalfatalbox(char *p, ...) | |
1774 | { | |
1775 | va_list ap; | |
1776 | fprintf(stderr, "FATAL ERROR: "); | |
1777 | va_start(ap, p); | |
1778 | vfprintf(stderr, p, ap); | |
1779 | va_end(ap); | |
1780 | fputc('\n', stderr); | |
1781 | exit(1); | |
1782 | } | |
1783 | ||
1784 | #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) | |
1785 | ||
1786 | int main(int argc, char **argv) | |
1787 | { | |
1788 | char *buf; | |
1789 | int line = 0; | |
1790 | int passes = 0, fails = 0; | |
1791 | ||
1792 | while ((buf = fgetline(stdin)) != NULL) { | |
1793 | int maxlen = strlen(buf); | |
1794 | unsigned char *data = snewn(maxlen, unsigned char); | |
f84f1e46 | 1795 | unsigned char *ptrs[5], *q; |
f3c29e34 | 1796 | int ptrnum; |
1797 | char *bufp = buf; | |
1798 | ||
1799 | line++; | |
1800 | ||
1801 | q = data; | |
1802 | ptrnum = 0; | |
1803 | ||
f84f1e46 | 1804 | while (*bufp && !isspace((unsigned char)*bufp)) |
1805 | bufp++; | |
1806 | if (bufp) | |
1807 | *bufp++ = '\0'; | |
1808 | ||
f3c29e34 | 1809 | while (*bufp) { |
1810 | char *start, *end; | |
1811 | int i; | |
1812 | ||
1813 | while (*bufp && !isxdigit((unsigned char)*bufp)) | |
1814 | bufp++; | |
1815 | start = bufp; | |
1816 | ||
1817 | if (!*bufp) | |
1818 | break; | |
1819 | ||
1820 | while (*bufp && isxdigit((unsigned char)*bufp)) | |
1821 | bufp++; | |
1822 | end = bufp; | |
1823 | ||
1824 | if (ptrnum >= lenof(ptrs)) | |
1825 | break; | |
1826 | ptrs[ptrnum++] = q; | |
1827 | ||
1828 | for (i = -((end - start) & 1); i < end-start; i += 2) { | |
1829 | unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); | |
1830 | val = val * 16 + fromxdigit(start[i+1]); | |
1831 | *q++ = val; | |
1832 | } | |
1833 | ||
1834 | ptrs[ptrnum] = q; | |
1835 | } | |
1836 | ||
f84f1e46 | 1837 | if (!strcmp(buf, "mul")) { |
1838 | Bignum a, b, c, p; | |
1839 | ||
1840 | if (ptrnum != 3) { | |
f6939e2b | 1841 | printf("%d: mul with %d parameters, expected 3\n", line, ptrnum); |
f84f1e46 | 1842 | exit(1); |
1843 | } | |
1844 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1845 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1846 | c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1847 | p = bigmul(a, b); | |
f3c29e34 | 1848 | |
1849 | if (bignum_cmp(c, p) == 0) { | |
1850 | passes++; | |
1851 | } else { | |
1852 | char *as = bignum_decimal(a); | |
1853 | char *bs = bignum_decimal(b); | |
1854 | char *cs = bignum_decimal(c); | |
1855 | char *ps = bignum_decimal(p); | |
1856 | ||
1857 | printf("%d: fail: %s * %s gave %s expected %s\n", | |
1858 | line, as, bs, ps, cs); | |
1859 | fails++; | |
1860 | ||
1861 | sfree(as); | |
1862 | sfree(bs); | |
1863 | sfree(cs); | |
1864 | sfree(ps); | |
1865 | } | |
1866 | freebn(a); | |
1867 | freebn(b); | |
1868 | freebn(c); | |
1869 | freebn(p); | |
f84f1e46 | 1870 | } else if (!strcmp(buf, "pow")) { |
1871 | Bignum base, expt, modulus, expected, answer; | |
1872 | ||
1873 | if (ptrnum != 4) { | |
f6939e2b | 1874 | printf("%d: mul with %d parameters, expected 4\n", line, ptrnum); |
f84f1e46 | 1875 | exit(1); |
1876 | } | |
1877 | ||
1878 | base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1879 | expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1880 | modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1881 | expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); | |
1882 | answer = modpow(base, expt, modulus); | |
1883 | ||
1884 | if (bignum_cmp(expected, answer) == 0) { | |
1885 | passes++; | |
1886 | } else { | |
1887 | char *as = bignum_decimal(base); | |
1888 | char *bs = bignum_decimal(expt); | |
1889 | char *cs = bignum_decimal(modulus); | |
1890 | char *ds = bignum_decimal(answer); | |
1891 | char *ps = bignum_decimal(expected); | |
1892 | ||
1893 | printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", | |
1894 | line, as, bs, cs, ds, ps); | |
1895 | fails++; | |
1896 | ||
1897 | sfree(as); | |
1898 | sfree(bs); | |
1899 | sfree(cs); | |
1900 | sfree(ds); | |
1901 | sfree(ps); | |
1902 | } | |
1903 | freebn(base); | |
1904 | freebn(expt); | |
1905 | freebn(modulus); | |
1906 | freebn(expected); | |
1907 | freebn(answer); | |
1908 | } else { | |
1909 | printf("%d: unrecognised test keyword: '%s'\n", line, buf); | |
1910 | exit(1); | |
f3c29e34 | 1911 | } |
f84f1e46 | 1912 | |
f3c29e34 | 1913 | sfree(buf); |
1914 | sfree(data); | |
1915 | } | |
1916 | ||
1917 | printf("passed %d failed %d total %d\n", passes, fails, passes+fails); | |
1918 | return fails != 0; | |
1919 | } | |
1920 | ||
1921 | #endif |