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 |
c40be1ad | 164 | * little-endian arrays of 'len' BignumInts. Returns a BignumInt carried |
0c431b2f | 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 | ||
c40be1ad | 173 | for (i = 0; i < len; i++) { |
0c431b2f | 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 | |
c40be1ad | 184 | * all little-endian arrays of 'len' BignumInts. Any borrow from the top |
0c431b2f | 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 | ||
c40be1ad | 193 | for (i = 0; i < len; i++) { |
0c431b2f | 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"); | |
c40be1ad | 273 | printf("%0*x", BIGNUM_INT_BITS/4, a[len - 1 - i]); |
f3c29e34 | 274 | } |
275 | printf("\n"); | |
276 | printf("b1,b0 = 0x"); | |
277 | for (i = 0; i < len; i++) { | |
278 | if (i == toplen) printf(", 0x"); | |
c40be1ad | 279 | printf("%0*x", BIGNUM_INT_BITS/4, b[len - 1 - i]); |
f3c29e34 | 280 | } |
281 | printf("\n"); | |
282 | #endif | |
283 | ||
0c431b2f | 284 | /* a_1 b_1 */ |
c40be1ad | 285 | internal_mul(a + botlen, b + botlen, c + 2*botlen, toplen, scratch); |
f3c29e34 | 286 | #ifdef KARA_DEBUG |
287 | printf("a1b1 = 0x"); | |
288 | for (i = 0; i < 2*toplen; i++) { | |
c40be1ad | 289 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - 1 - i]); |
f3c29e34 | 290 | } |
291 | printf("\n"); | |
292 | #endif | |
0c431b2f | 293 | |
294 | /* a_0 b_0 */ | |
c40be1ad | 295 | internal_mul(a, b, c, botlen, scratch); |
f3c29e34 | 296 | #ifdef KARA_DEBUG |
297 | printf("a0b0 = 0x"); | |
298 | for (i = 0; i < 2*botlen; i++) { | |
c40be1ad | 299 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*botlen - 1 - i]); |
f3c29e34 | 300 | } |
301 | printf("\n"); | |
302 | #endif | |
0c431b2f | 303 | |
c40be1ad MW |
304 | /* Zero padding. botlen exceeds toplen by at most 1, and we'll set |
305 | * the extra carry explicitly below, so we only need to zero at most | |
306 | * one of the top words here. | |
307 | */ | |
308 | scratch[midlen - 2] = scratch[2*midlen - 2] = 0; | |
0c431b2f | 309 | |
757b0110 | 310 | for (i = 0; i < toplen; i++) { |
c40be1ad MW |
311 | scratch[i] = a[i + botlen]; /* a_1 */ |
312 | scratch[midlen + i] = b[i + botlen]; /* b_1 */ | |
0c431b2f | 313 | } |
314 | ||
315 | /* compute a_1 + a_0 */ | |
c40be1ad | 316 | scratch[midlen - 1] = internal_add(scratch, a, scratch, botlen); |
f3c29e34 | 317 | #ifdef KARA_DEBUG |
318 | printf("a1plusa0 = 0x"); | |
319 | for (i = 0; i < midlen; i++) { | |
c40be1ad | 320 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen - 1 - i]); |
f3c29e34 | 321 | } |
322 | printf("\n"); | |
323 | #endif | |
0c431b2f | 324 | /* compute b_1 + b_0 */ |
c40be1ad MW |
325 | scratch[2*midlen - 1] = internal_add(scratch+midlen, b, |
326 | scratch+midlen, botlen); | |
f3c29e34 | 327 | #ifdef KARA_DEBUG |
328 | printf("b1plusb0 = 0x"); | |
329 | for (i = 0; i < midlen; i++) { | |
c40be1ad | 330 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]); |
f3c29e34 | 331 | } |
332 | printf("\n"); | |
333 | #endif | |
0c431b2f | 334 | |
335 | /* | |
336 | * Now we can do the third multiplication. | |
337 | */ | |
5a502a19 | 338 | internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, |
339 | scratch + 4*midlen); | |
f3c29e34 | 340 | #ifdef KARA_DEBUG |
341 | printf("a1plusa0timesb1plusb0 = 0x"); | |
342 | for (i = 0; i < 2*midlen; i++) { | |
c40be1ad | 343 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]); |
f3c29e34 | 344 | } |
345 | printf("\n"); | |
346 | #endif | |
0c431b2f | 347 | |
348 | /* | |
349 | * Now we can reuse the first half of 'scratch' to compute the | |
350 | * sum of the outer two coefficients, to subtract from that | |
351 | * product to obtain the middle one. | |
352 | */ | |
c40be1ad | 353 | scratch[2*botlen - 2] = scratch[2*botlen - 1] = 0; |
757b0110 | 354 | for (i = 0; i < 2*toplen; i++) |
c40be1ad MW |
355 | scratch[i] = c[2*botlen + i]; |
356 | scratch[2*botlen] = internal_add(scratch, c, scratch, 2*botlen); | |
357 | scratch[2*botlen + 1] = 0; | |
f3c29e34 | 358 | #ifdef KARA_DEBUG |
359 | printf("a1b1plusa0b0 = 0x"); | |
360 | for (i = 0; i < 2*midlen; i++) { | |
c40be1ad | 361 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]); |
f3c29e34 | 362 | } |
363 | printf("\n"); | |
364 | #endif | |
0c431b2f | 365 | |
c40be1ad | 366 | internal_sub(scratch + 2*midlen, scratch, scratch, 2*midlen); |
f3c29e34 | 367 | #ifdef KARA_DEBUG |
368 | printf("a1b0plusa0b1 = 0x"); | |
369 | for (i = 0; i < 2*midlen; i++) { | |
c40be1ad | 370 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]); |
f3c29e34 | 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 | */ | |
c40be1ad MW |
381 | carry = internal_add(c + botlen, scratch, c + botlen, 2*midlen); |
382 | i = botlen + 2*midlen; | |
0c431b2f | 383 | while (carry) { |
c40be1ad | 384 | assert(i <= 2*len); |
757b0110 | 385 | carry += c[i]; |
386 | c[i] = (BignumInt)carry; | |
0c431b2f | 387 | carry >>= BIGNUM_INT_BITS; |
c40be1ad | 388 | i++; |
0c431b2f | 389 | } |
f3c29e34 | 390 | #ifdef KARA_DEBUG |
391 | printf("ab = 0x"); | |
392 | for (i = 0; i < 2*len; i++) { | |
c40be1ad | 393 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - i]); |
f3c29e34 | 394 | } |
395 | printf("\n"); | |
396 | #endif | |
0c431b2f | 397 | |
0c431b2f | 398 | } else { |
757b0110 | 399 | int i; |
400 | BignumInt carry; | |
401 | BignumDblInt t; | |
c40be1ad | 402 | const BignumInt *ap, *alim = a + len, *bp, *blim = b + len; |
757b0110 | 403 | BignumInt *cp, *cps; |
0c431b2f | 404 | |
405 | /* | |
406 | * Multiply in the ordinary O(N^2) way. | |
407 | */ | |
408 | ||
757b0110 | 409 | for (i = 0; i < 2 * len; i++) |
410 | c[i] = 0; | |
0c431b2f | 411 | |
c40be1ad | 412 | for (cps = c, ap = a; ap < alim; ap++, cps++) { |
757b0110 | 413 | carry = 0; |
c40be1ad | 414 | for (cp = cps, bp = b, i = blim - bp; i--; bp++, cp++) { |
757b0110 | 415 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
416 | *cp = (BignumInt) t; | |
08b5c9a2 | 417 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
0c431b2f | 418 | } |
757b0110 | 419 | *cp = carry; |
0c431b2f | 420 | } |
e5574168 | 421 | } |
422 | } | |
423 | ||
132c534f | 424 | /* |
425 | * Variant form of internal_mul used for the initial step of | |
426 | * Montgomery reduction. Only bothers outputting 'len' words | |
427 | * (everything above that is thrown away). | |
428 | */ | |
429 | static void internal_mul_low(const BignumInt *a, const BignumInt *b, | |
5a502a19 | 430 | BignumInt *c, int len, BignumInt *scratch) |
132c534f | 431 | { |
132c534f | 432 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 | 433 | int i; |
132c534f | 434 | |
435 | /* | |
436 | * Karatsuba-aware version of internal_mul_low. As before, we | |
437 | * express each input value as a shifted combination of two | |
438 | * halves: | |
439 | * | |
440 | * a = a_1 D + a_0 | |
441 | * b = b_1 D + b_0 | |
442 | * | |
443 | * Then the full product is, as before, | |
444 | * | |
445 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 | |
446 | * | |
447 | * Provided we choose D on the large side (so that a_0 and b_0 | |
448 | * are _at least_ as long as a_1 and b_1), we don't need the | |
449 | * topmost term at all, and we only need half of the middle | |
450 | * term. So there's no point in doing the proper Karatsuba | |
451 | * optimisation which computes the middle term using the top | |
452 | * one, because we'd take as long computing the top one as | |
453 | * just computing the middle one directly. | |
454 | * | |
455 | * So instead, we do a much more obvious thing: we call the | |
456 | * fully optimised internal_mul to compute a_0 b_0, and we | |
457 | * recursively call ourself to compute the _bottom halves_ of | |
458 | * a_1 b_0 and a_0 b_1, each of which we add into the result | |
459 | * in the obvious way. | |
460 | * | |
461 | * In other words, there's no actual Karatsuba _optimisation_ | |
462 | * in this function; the only benefit in doing it this way is | |
463 | * that we call internal_mul proper for a large part of the | |
464 | * work, and _that_ can optimise its operation. | |
465 | */ | |
466 | ||
467 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ | |
132c534f | 468 | |
469 | /* | |
5a502a19 | 470 | * Scratch space for the various bits and pieces we're going |
471 | * to be adding together: we need botlen*2 words for a_0 b_0 | |
472 | * (though we may end up throwing away its topmost word), and | |
473 | * toplen words for each of a_1 b_0 and a_0 b_1. That adds up | |
474 | * to exactly 2*len. | |
132c534f | 475 | */ |
132c534f | 476 | |
477 | /* a_0 b_0 */ | |
c40be1ad | 478 | internal_mul(a, b, scratch + 2*toplen, botlen, scratch + 2*len); |
132c534f | 479 | |
480 | /* a_1 b_0 */ | |
c40be1ad | 481 | internal_mul_low(a + botlen, b, scratch + toplen, toplen, |
5a502a19 | 482 | scratch + 2*len); |
132c534f | 483 | |
484 | /* a_0 b_1 */ | |
c40be1ad | 485 | internal_mul_low(a, b + botlen, scratch, toplen, scratch + 2*len); |
132c534f | 486 | |
487 | /* Copy the bottom half of the big coefficient into place */ | |
757b0110 | 488 | for (i = 0; i < botlen; i++) |
c40be1ad | 489 | c[i] = scratch[2*toplen + i]; |
132c534f | 490 | |
491 | /* Add the two small coefficients, throwing away the returned carry */ | |
492 | internal_add(scratch, scratch + toplen, scratch, toplen); | |
493 | ||
494 | /* And add that to the large coefficient, leaving the result in c. */ | |
c40be1ad MW |
495 | internal_add(scratch, scratch + 2*toplen + botlen, |
496 | c + botlen, toplen); | |
132c534f | 497 | |
132c534f | 498 | } else { |
757b0110 | 499 | int i; |
500 | BignumInt carry; | |
501 | BignumDblInt t; | |
c40be1ad MW |
502 | const BignumInt *ap, *alim = a + len, *bp; |
503 | BignumInt *cp, *cps, *clim = c + len; | |
132c534f | 504 | |
757b0110 | 505 | /* |
506 | * Multiply in the ordinary O(N^2) way. | |
507 | */ | |
132c534f | 508 | |
757b0110 | 509 | for (i = 0; i < len; i++) |
510 | c[i] = 0; | |
511 | ||
c40be1ad | 512 | for (cps = c, ap = a; ap < alim; ap++, cps++) { |
757b0110 | 513 | carry = 0; |
c40be1ad | 514 | for (cp = cps, bp = b, i = clim - cp; i--; bp++, cp++) { |
757b0110 | 515 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
516 | *cp = (BignumInt) t; | |
08b5c9a2 | 517 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
132c534f | 518 | } |
519 | } | |
132c534f | 520 | } |
521 | } | |
522 | ||
523 | /* | |
c40be1ad | 524 | * Montgomery reduction. Expects x to be a little-endian array of 2*len |
132c534f | 525 | * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * |
526 | * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array | |
527 | * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= | |
528 | * x' < n. | |
529 | * | |
c40be1ad | 530 | * 'n' and 'mninv' should be little-endian arrays of 'len' BignumInts |
132c534f | 531 | * each, containing respectively n and the multiplicative inverse of |
532 | * -n mod r. | |
533 | * | |
5a502a19 | 534 | * 'tmp' is an array of BignumInt used as scratch space, of length at |
535 | * least 3*len + mul_compute_scratch(len). | |
132c534f | 536 | */ |
537 | static void monty_reduce(BignumInt *x, const BignumInt *n, | |
538 | const BignumInt *mninv, BignumInt *tmp, int len) | |
539 | { | |
540 | int i; | |
541 | BignumInt carry; | |
542 | ||
543 | /* | |
544 | * Multiply x by (-n)^{-1} mod r. This gives us a value m such | |
545 | * that mn is congruent to -x mod r. Hence, mn+x is an exact | |
546 | * multiple of r, and is also (obviously) congruent to x mod n. | |
547 | */ | |
c40be1ad | 548 | internal_mul_low(x, mninv, tmp, len, tmp + 3*len); |
132c534f | 549 | |
550 | /* | |
551 | * Compute t = (mn+x)/r in ordinary, non-modular, integer | |
552 | * arithmetic. By construction this is exact, and is congruent mod | |
553 | * n to x * r^{-1}, i.e. the answer we want. | |
554 | * | |
555 | * The following multiply leaves that answer in the _most_ | |
556 | * significant half of the 'x' array, so then we must shift it | |
557 | * down. | |
558 | */ | |
5a502a19 | 559 | internal_mul(tmp, n, tmp+len, len, tmp + 3*len); |
132c534f | 560 | carry = internal_add(x, tmp+len, x, 2*len); |
561 | for (i = 0; i < len; i++) | |
c40be1ad | 562 | x[i] = x[len + i], x[len + i] = 0; |
132c534f | 563 | |
564 | /* | |
565 | * Reduce t mod n. This doesn't require a full-on division by n, | |
566 | * but merely a test and single optional subtraction, since we can | |
567 | * show that 0 <= t < 2n. | |
568 | * | |
569 | * Proof: | |
570 | * + we computed m mod r, so 0 <= m < r. | |
571 | * + so 0 <= mn < rn, obviously | |
572 | * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn | |
573 | * + yielding 0 <= (mn+x)/r < 2n as required. | |
574 | */ | |
575 | if (!carry) { | |
c40be1ad MW |
576 | for (i = len; i-- > 0; ) |
577 | if (x[i] != n[i]) | |
132c534f | 578 | break; |
579 | } | |
c40be1ad MW |
580 | if (carry || i < 0 || x[i] > n[i]) |
581 | internal_sub(x, n, x, len); | |
132c534f | 582 | } |
583 | ||
a3412f52 | 584 | static void internal_add_shifted(BignumInt *number, |
32874aea | 585 | unsigned n, int shift) |
586 | { | |
a3412f52 | 587 | int word = 1 + (shift / BIGNUM_INT_BITS); |
588 | int bshift = shift % BIGNUM_INT_BITS; | |
589 | BignumDblInt addend; | |
9400cf6f | 590 | |
3014da2b | 591 | addend = (BignumDblInt)n << bshift; |
9400cf6f | 592 | |
593 | while (addend) { | |
32874aea | 594 | addend += number[word]; |
a3412f52 | 595 | number[word] = (BignumInt) addend & BIGNUM_INT_MASK; |
596 | addend >>= BIGNUM_INT_BITS; | |
32874aea | 597 | word++; |
9400cf6f | 598 | } |
599 | } | |
600 | ||
e5574168 | 601 | /* |
602 | * Compute a = a % m. | |
9400cf6f | 603 | * Input in first alen words of a and first mlen words of m. |
604 | * Output in first alen words of a | |
c40be1ad | 605 | * (of which last alen-mlen words will be zero). |
e5574168 | 606 | * The MSW of m MUST have its high bit set. |
c40be1ad MW |
607 | * Quotient is accumulated in the `quotient' array. Quotient parts |
608 | * are shifted left by `qshift' before adding into quot. | |
e5574168 | 609 | */ |
a3412f52 | 610 | static void internal_mod(BignumInt *a, int alen, |
611 | BignumInt *m, int mlen, | |
612 | BignumInt *quot, int qshift) | |
e5574168 | 613 | { |
a3412f52 | 614 | BignumInt m0, m1; |
e5574168 | 615 | unsigned int h; |
c40be1ad | 616 | int i, j, k; |
e5574168 | 617 | |
c40be1ad | 618 | m0 = m[mlen - 1]; |
9400cf6f | 619 | if (mlen > 1) |
c40be1ad | 620 | m1 = m[mlen - 2]; |
9400cf6f | 621 | else |
32874aea | 622 | m1 = 0; |
e5574168 | 623 | |
c40be1ad | 624 | for (i = alen, h = 0; i-- >= mlen; ) { |
a3412f52 | 625 | BignumDblInt t; |
9400cf6f | 626 | unsigned int q, r, c, ai1; |
e5574168 | 627 | |
c40be1ad MW |
628 | if (i) |
629 | ai1 = a[i - 1]; | |
630 | else | |
631 | ai1 = 0; | |
9400cf6f | 632 | |
e5574168 | 633 | /* Find q = h:a[i] / m0 */ |
62ef3d44 | 634 | if (h >= m0) { |
635 | /* | |
636 | * Special case. | |
637 | * | |
638 | * To illustrate it, suppose a BignumInt is 8 bits, and | |
639 | * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then | |
640 | * our initial division will be 0xA123 / 0xA1, which | |
641 | * will give a quotient of 0x100 and a divide overflow. | |
642 | * However, the invariants in this division algorithm | |
643 | * are not violated, since the full number A1:23:... is | |
644 | * _less_ than the quotient prefix A1:B2:... and so the | |
645 | * following correction loop would have sorted it out. | |
646 | * | |
647 | * In this situation we set q to be the largest | |
648 | * quotient we _can_ stomach (0xFF, of course). | |
649 | */ | |
650 | q = BIGNUM_INT_MASK; | |
651 | } else { | |
819a22b3 | 652 | /* Macro doesn't want an array subscript expression passed |
653 | * into it (see definition), so use a temporary. */ | |
654 | BignumInt tmplo = a[i]; | |
655 | DIVMOD_WORD(q, r, h, tmplo, m0); | |
62ef3d44 | 656 | |
657 | /* Refine our estimate of q by looking at | |
c40be1ad | 658 | h:a[i]:a[i-1] / m0:m1 */ |
62ef3d44 | 659 | t = MUL_WORD(m1, q); |
660 | if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { | |
661 | q--; | |
662 | t -= m1; | |
663 | r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ | |
664 | if (r >= (BignumDblInt) m0 && | |
665 | t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; | |
666 | } | |
e5574168 | 667 | } |
668 | ||
c40be1ad MW |
669 | j = i + 1 - mlen; |
670 | ||
9400cf6f | 671 | /* Subtract q * m from a[i...] */ |
e5574168 | 672 | c = 0; |
c40be1ad | 673 | for (k = 0; k < mlen; k++) { |
a47e8bba | 674 | t = MUL_WORD(q, m[k]); |
e5574168 | 675 | t += c; |
62ddb51e | 676 | c = (unsigned)(t >> BIGNUM_INT_BITS); |
c40be1ad | 677 | if ((BignumInt) t > a[j + k]) |
32874aea | 678 | c++; |
c40be1ad | 679 | a[j + k] -= (BignumInt) t; |
e5574168 | 680 | } |
681 | ||
682 | /* Add back m in case of borrow */ | |
683 | if (c != h) { | |
684 | t = 0; | |
c40be1ad | 685 | for (k = 0; k < mlen; k++) { |
e5574168 | 686 | t += m[k]; |
c40be1ad MW |
687 | t += a[j + k]; |
688 | a[j + k] = (BignumInt) t; | |
a3412f52 | 689 | t = t >> BIGNUM_INT_BITS; |
e5574168 | 690 | } |
32874aea | 691 | q--; |
e5574168 | 692 | } |
c40be1ad | 693 | |
32874aea | 694 | if (quot) |
c40be1ad MW |
695 | internal_add_shifted(quot, q, |
696 | qshift + BIGNUM_INT_BITS * (i + 1 - mlen)); | |
697 | ||
698 | if (i >= mlen) { | |
699 | h = a[i]; | |
700 | a[i] = 0; | |
701 | } | |
e5574168 | 702 | } |
703 | } | |
704 | ||
c40be1ad MW |
705 | static void shift_left(BignumInt *x, int xlen, int shift) |
706 | { | |
707 | int i; | |
708 | ||
709 | if (!shift) | |
710 | return; | |
711 | for (i = xlen; --i > 0; ) | |
712 | x[i] = (x[i] << shift) | (x[i - 1] >> (BIGNUM_INT_BITS - shift)); | |
713 | x[0] = x[0] << shift; | |
714 | } | |
715 | ||
716 | static void shift_right(BignumInt *x, int xlen, int shift) | |
717 | { | |
718 | int i; | |
719 | ||
720 | if (!shift || !xlen) | |
721 | return; | |
722 | xlen--; | |
723 | for (i = 0; i < xlen; i++) | |
724 | x[i] = (x[i] >> shift) | (x[i + 1] << (BIGNUM_INT_BITS - shift)); | |
725 | x[i] = x[i] >> shift; | |
726 | } | |
727 | ||
e5574168 | 728 | /* |
09095ac5 | 729 | * Compute (base ^ exp) % mod, the pedestrian way. |
e5574168 | 730 | */ |
09095ac5 | 731 | Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) |
e5574168 | 732 | { |
5a502a19 | 733 | BignumInt *a, *b, *n, *m, *scratch; |
09095ac5 | 734 | int mshift; |
5a502a19 | 735 | int mlen, scratchlen, i, j; |
09095ac5 | 736 | Bignum base, result; |
ed953b91 | 737 | |
738 | /* | |
739 | * The most significant word of mod needs to be non-zero. It | |
740 | * should already be, but let's make sure. | |
741 | */ | |
742 | assert(mod[mod[0]] != 0); | |
743 | ||
744 | /* | |
745 | * Make sure the base is smaller than the modulus, by reducing | |
746 | * it modulo the modulus if not. | |
747 | */ | |
748 | base = bigmod(base_in, mod); | |
e5574168 | 749 | |
09095ac5 | 750 | /* Allocate m of size mlen, copy mod to m */ |
09095ac5 | 751 | mlen = mod[0]; |
752 | m = snewn(mlen, BignumInt); | |
753 | for (j = 0; j < mlen; j++) | |
c40be1ad | 754 | m[j] = mod[j + 1]; |
09095ac5 | 755 | |
756 | /* Shift m left to make msb bit set */ | |
757 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) | |
c40be1ad | 758 | if ((m[mlen - 1] << mshift) & BIGNUM_TOP_BIT) |
09095ac5 | 759 | break; |
c40be1ad MW |
760 | if (mshift) |
761 | shift_left(m, mlen, mshift); | |
09095ac5 | 762 | |
763 | /* Allocate n of size mlen, copy base to n */ | |
764 | n = snewn(mlen, BignumInt); | |
c40be1ad MW |
765 | for (i = 0; i < (int)base[0]; i++) |
766 | n[i] = base[i + 1]; | |
767 | for (; i < mlen; i++) | |
768 | n[i] = 0; | |
09095ac5 | 769 | |
770 | /* Allocate a and b of size 2*mlen. Set a = 1 */ | |
771 | a = snewn(2 * mlen, BignumInt); | |
772 | b = snewn(2 * mlen, BignumInt); | |
c40be1ad MW |
773 | a[0] = 1; |
774 | for (i = 1; i < 2 * mlen; i++) | |
09095ac5 | 775 | a[i] = 0; |
09095ac5 | 776 | |
5a502a19 | 777 | /* Scratch space for multiplies */ |
778 | scratchlen = mul_compute_scratch(mlen); | |
779 | scratch = snewn(scratchlen, BignumInt); | |
780 | ||
09095ac5 | 781 | /* Skip leading zero bits of exp. */ |
782 | i = 0; | |
783 | j = BIGNUM_INT_BITS-1; | |
784 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { | |
785 | j--; | |
786 | if (j < 0) { | |
787 | i++; | |
788 | j = BIGNUM_INT_BITS-1; | |
789 | } | |
790 | } | |
791 | ||
792 | /* Main computation */ | |
793 | while (i < (int)exp[0]) { | |
794 | while (j >= 0) { | |
c40be1ad | 795 | internal_mul(a, a, b, mlen, scratch); |
09095ac5 | 796 | internal_mod(b, mlen * 2, m, mlen, NULL, 0); |
797 | if ((exp[exp[0] - i] & (1 << j)) != 0) { | |
c40be1ad | 798 | internal_mul(b, n, a, mlen, scratch); |
09095ac5 | 799 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
800 | } else { | |
801 | BignumInt *t; | |
802 | t = a; | |
803 | a = b; | |
804 | b = t; | |
805 | } | |
806 | j--; | |
807 | } | |
808 | i++; | |
809 | j = BIGNUM_INT_BITS-1; | |
810 | } | |
811 | ||
812 | /* Fixup result in case the modulus was shifted */ | |
813 | if (mshift) { | |
c40be1ad MW |
814 | shift_left(a, mlen + 1, mshift); |
815 | internal_mod(a, mlen + 1, m, mlen, NULL, 0); | |
816 | shift_right(a, mlen, mshift); | |
09095ac5 | 817 | } |
818 | ||
819 | /* Copy result to buffer */ | |
820 | result = newbn(mod[0]); | |
821 | for (i = 0; i < mlen; i++) | |
c40be1ad | 822 | result[i + 1] = a[i]; |
09095ac5 | 823 | while (result[0] > 1 && result[result[0]] == 0) |
824 | result[0]--; | |
825 | ||
826 | /* Free temporary arrays */ | |
827 | for (i = 0; i < 2 * mlen; i++) | |
828 | a[i] = 0; | |
829 | sfree(a); | |
5a502a19 | 830 | for (i = 0; i < scratchlen; i++) |
831 | scratch[i] = 0; | |
832 | sfree(scratch); | |
09095ac5 | 833 | for (i = 0; i < 2 * mlen; i++) |
834 | b[i] = 0; | |
835 | sfree(b); | |
836 | for (i = 0; i < mlen; i++) | |
837 | m[i] = 0; | |
838 | sfree(m); | |
839 | for (i = 0; i < mlen; i++) | |
840 | n[i] = 0; | |
841 | sfree(n); | |
842 | ||
843 | freebn(base); | |
844 | ||
845 | return result; | |
846 | } | |
847 | ||
848 | /* | |
849 | * Compute (base ^ exp) % mod. Uses the Montgomery multiplication | |
850 | * technique where possible, falling back to modpow_simple otherwise. | |
851 | */ | |
852 | Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) | |
853 | { | |
5a502a19 | 854 | BignumInt *a, *b, *x, *n, *mninv, *scratch; |
855 | int len, scratchlen, i, j; | |
09095ac5 | 856 | Bignum base, base2, r, rn, inv, result; |
857 | ||
858 | /* | |
859 | * The most significant word of mod needs to be non-zero. It | |
860 | * should already be, but let's make sure. | |
861 | */ | |
862 | assert(mod[mod[0]] != 0); | |
863 | ||
132c534f | 864 | /* |
865 | * mod had better be odd, or we can't do Montgomery multiplication | |
866 | * using a power of two at all. | |
867 | */ | |
09095ac5 | 868 | if (!(mod[1] & 1)) |
869 | return modpow_simple(base_in, exp, mod); | |
870 | ||
871 | /* | |
872 | * Make sure the base is smaller than the modulus, by reducing | |
873 | * it modulo the modulus if not. | |
874 | */ | |
875 | base = bigmod(base_in, mod); | |
e5574168 | 876 | |
132c534f | 877 | /* |
878 | * Compute the inverse of n mod r, for monty_reduce. (In fact we | |
879 | * want the inverse of _minus_ n mod r, but we'll sort that out | |
880 | * below.) | |
881 | */ | |
882 | len = mod[0]; | |
883 | r = bn_power_2(BIGNUM_INT_BITS * len); | |
884 | inv = modinv(mod, r); | |
e5574168 | 885 | |
132c534f | 886 | /* |
887 | * Multiply the base by r mod n, to get it into Montgomery | |
888 | * representation. | |
889 | */ | |
890 | base2 = modmul(base, r, mod); | |
891 | freebn(base); | |
892 | base = base2; | |
893 | ||
894 | rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ | |
895 | ||
896 | freebn(r); /* won't need this any more */ | |
897 | ||
898 | /* | |
c40be1ad MW |
899 | * Set up internal arrays of the right lengths containing the base, |
900 | * the modulus, and the modulus's inverse. | |
132c534f | 901 | */ |
902 | n = snewn(len, BignumInt); | |
903 | for (j = 0; j < len; j++) | |
c40be1ad | 904 | n[j] = mod[j + 1]; |
132c534f | 905 | |
906 | mninv = snewn(len, BignumInt); | |
907 | for (j = 0; j < len; j++) | |
c40be1ad | 908 | mninv[j] = (j < (int)inv[0] ? inv[j + 1] : 0); |
132c534f | 909 | freebn(inv); /* we don't need this copy of it any more */ |
910 | /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ | |
911 | x = snewn(len, BignumInt); | |
912 | for (j = 0; j < len; j++) | |
913 | x[j] = 0; | |
914 | internal_sub(x, mninv, mninv, len); | |
915 | ||
916 | /* x = snewn(len, BignumInt); */ /* already done above */ | |
917 | for (j = 0; j < len; j++) | |
c40be1ad | 918 | x[j] = (j < (int)base[0] ? base[j + 1] : 0); |
132c534f | 919 | freebn(base); /* we don't need this copy of it any more */ |
920 | ||
921 | a = snewn(2*len, BignumInt); | |
922 | b = snewn(2*len, BignumInt); | |
923 | for (j = 0; j < len; j++) | |
c40be1ad | 924 | a[j] = (j < (int)rn[0] ? rn[j + 1] : 0); |
132c534f | 925 | freebn(rn); |
926 | ||
5a502a19 | 927 | /* Scratch space for multiplies */ |
928 | scratchlen = 3*len + mul_compute_scratch(len); | |
929 | scratch = snewn(scratchlen, BignumInt); | |
e5574168 | 930 | |
931 | /* Skip leading zero bits of exp. */ | |
32874aea | 932 | i = 0; |
a3412f52 | 933 | j = BIGNUM_INT_BITS-1; |
62ddb51e | 934 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
e5574168 | 935 | j--; |
32874aea | 936 | if (j < 0) { |
937 | i++; | |
a3412f52 | 938 | j = BIGNUM_INT_BITS-1; |
32874aea | 939 | } |
e5574168 | 940 | } |
941 | ||
942 | /* Main computation */ | |
62ddb51e | 943 | while (i < (int)exp[0]) { |
e5574168 | 944 | while (j >= 0) { |
c40be1ad | 945 | internal_mul(a, a, b, len, scratch); |
5a502a19 | 946 | monty_reduce(b, n, mninv, scratch, len); |
e5574168 | 947 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
c40be1ad | 948 | internal_mul(b, x, a, len, scratch); |
5a502a19 | 949 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 | 950 | } else { |
a3412f52 | 951 | BignumInt *t; |
32874aea | 952 | t = a; |
953 | a = b; | |
954 | b = t; | |
e5574168 | 955 | } |
956 | j--; | |
957 | } | |
32874aea | 958 | i++; |
a3412f52 | 959 | j = BIGNUM_INT_BITS-1; |
e5574168 | 960 | } |
961 | ||
132c534f | 962 | /* |
963 | * Final monty_reduce to get back from the adjusted Montgomery | |
964 | * representation. | |
965 | */ | |
5a502a19 | 966 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 | 967 | |
968 | /* Copy result to buffer */ | |
59600f67 | 969 | result = newbn(mod[0]); |
132c534f | 970 | for (i = 0; i < len; i++) |
c40be1ad | 971 | result[i + 1] = a[i]; |
32874aea | 972 | while (result[0] > 1 && result[result[0]] == 0) |
973 | result[0]--; | |
e5574168 | 974 | |
975 | /* Free temporary arrays */ | |
5a502a19 | 976 | for (i = 0; i < scratchlen; i++) |
977 | scratch[i] = 0; | |
978 | sfree(scratch); | |
132c534f | 979 | for (i = 0; i < 2 * len; i++) |
32874aea | 980 | a[i] = 0; |
981 | sfree(a); | |
132c534f | 982 | for (i = 0; i < 2 * len; i++) |
32874aea | 983 | b[i] = 0; |
984 | sfree(b); | |
132c534f | 985 | for (i = 0; i < len; i++) |
986 | mninv[i] = 0; | |
987 | sfree(mninv); | |
988 | for (i = 0; i < len; i++) | |
32874aea | 989 | n[i] = 0; |
990 | sfree(n); | |
132c534f | 991 | for (i = 0; i < len; i++) |
992 | x[i] = 0; | |
993 | sfree(x); | |
ed953b91 | 994 | |
59600f67 | 995 | return result; |
e5574168 | 996 | } |
7cca0d81 | 997 | |
998 | /* | |
999 | * Compute (p * q) % mod. | |
1000 | * The most significant word of mod MUST be non-zero. | |
1001 | * We assume that the result array is the same size as the mod array. | |
1002 | */ | |
59600f67 | 1003 | Bignum modmul(Bignum p, Bignum q, Bignum mod) |
7cca0d81 | 1004 | { |
5a502a19 | 1005 | BignumInt *a, *n, *m, *o, *scratch; |
1006 | int mshift, scratchlen; | |
80b10571 | 1007 | int pqlen, mlen, rlen, i, j; |
59600f67 | 1008 | Bignum result; |
7cca0d81 | 1009 | |
1010 | /* Allocate m of size mlen, copy mod to m */ | |
7cca0d81 | 1011 | mlen = mod[0]; |
a3412f52 | 1012 | m = snewn(mlen, BignumInt); |
32874aea | 1013 | for (j = 0; j < mlen; j++) |
c40be1ad | 1014 | m[j] = mod[j + 1]; |
7cca0d81 | 1015 | |
1016 | /* Shift m left to make msb bit set */ | |
a3412f52 | 1017 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
c40be1ad | 1018 | if ((m[mlen - 1] << mshift) & BIGNUM_TOP_BIT) |
32874aea | 1019 | break; |
c40be1ad MW |
1020 | if (mshift) |
1021 | shift_left(m, mlen, mshift); | |
7cca0d81 | 1022 | |
1023 | pqlen = (p[0] > q[0] ? p[0] : q[0]); | |
1024 | ||
aca5132b MW |
1025 | /* Make sure that we're allowing enough space. The shifting below will |
1026 | * underflow the vectors we allocate if `pqlen' is too small. | |
1027 | */ | |
1028 | if (2*pqlen <= mlen) | |
1029 | pqlen = mlen/2 + 1; | |
1030 | ||
7cca0d81 | 1031 | /* Allocate n of size pqlen, copy p to n */ |
a3412f52 | 1032 | n = snewn(pqlen, BignumInt); |
c40be1ad MW |
1033 | for (i = 0; i < (int)p[0]; i++) |
1034 | n[i] = p[i + 1]; | |
1035 | for (; i < pqlen; i++) | |
1036 | n[i] = 0; | |
7cca0d81 | 1037 | |
1038 | /* Allocate o of size pqlen, copy q to o */ | |
a3412f52 | 1039 | o = snewn(pqlen, BignumInt); |
c40be1ad MW |
1040 | for (i = 0; i < (int)q[0]; i++) |
1041 | o[i] = q[i + 1]; | |
1042 | for (; i < pqlen; i++) | |
1043 | o[i] = 0; | |
7cca0d81 | 1044 | |
1045 | /* Allocate a of size 2*pqlen for result */ | |
a3412f52 | 1046 | a = snewn(2 * pqlen, BignumInt); |
7cca0d81 | 1047 | |
5a502a19 | 1048 | /* Scratch space for multiplies */ |
1049 | scratchlen = mul_compute_scratch(pqlen); | |
1050 | scratch = snewn(scratchlen, BignumInt); | |
1051 | ||
7cca0d81 | 1052 | /* Main computation */ |
5a502a19 | 1053 | internal_mul(n, o, a, pqlen, scratch); |
32874aea | 1054 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
7cca0d81 | 1055 | |
1056 | /* Fixup result in case the modulus was shifted */ | |
1057 | if (mshift) { | |
c40be1ad MW |
1058 | shift_left(a, mlen + 1, mshift); |
1059 | internal_mod(a, mlen + 1, m, mlen, NULL, 0); | |
1060 | shift_right(a, mlen, mshift); | |
7cca0d81 | 1061 | } |
1062 | ||
1063 | /* Copy result to buffer */ | |
32874aea | 1064 | rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); |
80b10571 | 1065 | result = newbn(rlen); |
1066 | for (i = 0; i < rlen; i++) | |
c40be1ad | 1067 | result[i + 1] = a[i]; |
32874aea | 1068 | while (result[0] > 1 && result[result[0]] == 0) |
1069 | result[0]--; | |
7cca0d81 | 1070 | |
1071 | /* Free temporary arrays */ | |
5a502a19 | 1072 | for (i = 0; i < scratchlen; i++) |
1073 | scratch[i] = 0; | |
1074 | sfree(scratch); | |
32874aea | 1075 | for (i = 0; i < 2 * pqlen; i++) |
1076 | a[i] = 0; | |
1077 | sfree(a); | |
1078 | for (i = 0; i < mlen; i++) | |
1079 | m[i] = 0; | |
1080 | sfree(m); | |
1081 | for (i = 0; i < pqlen; i++) | |
1082 | n[i] = 0; | |
1083 | sfree(n); | |
1084 | for (i = 0; i < pqlen; i++) | |
1085 | o[i] = 0; | |
1086 | sfree(o); | |
59600f67 | 1087 | |
1088 | return result; | |
7cca0d81 | 1089 | } |
1090 | ||
1091 | /* | |
9400cf6f | 1092 | * Compute p % mod. |
1093 | * The most significant word of mod MUST be non-zero. | |
1094 | * We assume that the result array is the same size as the mod array. | |
5c72ca61 | 1095 | * We optionally write out a quotient if `quotient' is non-NULL. |
1096 | * We can avoid writing out the result if `result' is NULL. | |
9400cf6f | 1097 | */ |
f28753ab | 1098 | static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) |
9400cf6f | 1099 | { |
a3412f52 | 1100 | BignumInt *n, *m; |
9400cf6f | 1101 | int mshift; |
1102 | int plen, mlen, i, j; | |
1103 | ||
1104 | /* Allocate m of size mlen, copy mod to m */ | |
9400cf6f | 1105 | mlen = mod[0]; |
a3412f52 | 1106 | m = snewn(mlen, BignumInt); |
32874aea | 1107 | for (j = 0; j < mlen; j++) |
c40be1ad | 1108 | m[j] = mod[j + 1]; |
9400cf6f | 1109 | |
1110 | /* Shift m left to make msb bit set */ | |
a3412f52 | 1111 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
c40be1ad | 1112 | if ((m[mlen - 1] << mshift) & BIGNUM_TOP_BIT) |
32874aea | 1113 | break; |
c40be1ad MW |
1114 | if (mshift) |
1115 | shift_left(m, mlen, mshift); | |
9400cf6f | 1116 | |
1117 | plen = p[0]; | |
1118 | /* Ensure plen > mlen */ | |
32874aea | 1119 | if (plen <= mlen) |
1120 | plen = mlen + 1; | |
9400cf6f | 1121 | |
1122 | /* Allocate n of size plen, copy p to n */ | |
a3412f52 | 1123 | n = snewn(plen, BignumInt); |
c40be1ad MW |
1124 | for (i = 0; i < (int)p[0]; i++) |
1125 | n[i] = p[i + 1]; | |
1126 | for (; i < plen; i++) | |
1127 | n[i] = 0; | |
9400cf6f | 1128 | |
1129 | /* Main computation */ | |
1130 | internal_mod(n, plen, m, mlen, quotient, mshift); | |
1131 | ||
1132 | /* Fixup result in case the modulus was shifted */ | |
1133 | if (mshift) { | |
c40be1ad | 1134 | shift_left(n, mlen + 1, mshift); |
9400cf6f | 1135 | internal_mod(n, plen, m, mlen, quotient, 0); |
c40be1ad | 1136 | shift_right(n, mlen, mshift); |
9400cf6f | 1137 | } |
1138 | ||
1139 | /* Copy result to buffer */ | |
5c72ca61 | 1140 | if (result) { |
c40be1ad MW |
1141 | for (i = 0; i < (int)result[0]; i++) |
1142 | result[i + 1] = i < plen ? n[i] : 0; | |
1143 | bn_restore_invariant(result); | |
9400cf6f | 1144 | } |
1145 | ||
1146 | /* Free temporary arrays */ | |
32874aea | 1147 | for (i = 0; i < mlen; i++) |
1148 | m[i] = 0; | |
1149 | sfree(m); | |
1150 | for (i = 0; i < plen; i++) | |
1151 | n[i] = 0; | |
1152 | sfree(n); | |
9400cf6f | 1153 | } |
1154 | ||
1155 | /* | |
7cca0d81 | 1156 | * Decrement a number. |
1157 | */ | |
32874aea | 1158 | void decbn(Bignum bn) |
1159 | { | |
7cca0d81 | 1160 | int i = 1; |
62ddb51e | 1161 | while (i < (int)bn[0] && bn[i] == 0) |
a3412f52 | 1162 | bn[i++] = BIGNUM_INT_MASK; |
7cca0d81 | 1163 | bn[i]--; |
1164 | } | |
1165 | ||
27cd7fc2 | 1166 | Bignum bignum_from_bytes(const unsigned char *data, int nbytes) |
32874aea | 1167 | { |
3709bfe9 | 1168 | Bignum result; |
1169 | int w, i; | |
1170 | ||
a3412f52 | 1171 | w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ |
3709bfe9 | 1172 | |
1173 | result = newbn(w); | |
32874aea | 1174 | for (i = 1; i <= w; i++) |
1175 | result[i] = 0; | |
1176 | for (i = nbytes; i--;) { | |
1177 | unsigned char byte = *data++; | |
a3412f52 | 1178 | result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); |
3709bfe9 | 1179 | } |
1180 | ||
32874aea | 1181 | while (result[0] > 1 && result[result[0]] == 0) |
1182 | result[0]--; | |
3709bfe9 | 1183 | return result; |
1184 | } | |
1185 | ||
7cca0d81 | 1186 | /* |
2e85c969 | 1187 | * Read an SSH-1-format bignum from a data buffer. Return the number |
0016d70b | 1188 | * of bytes consumed, or -1 if there wasn't enough data. |
7cca0d81 | 1189 | */ |
0016d70b | 1190 | int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) |
32874aea | 1191 | { |
27cd7fc2 | 1192 | const unsigned char *p = data; |
7cca0d81 | 1193 | int i; |
1194 | int w, b; | |
1195 | ||
0016d70b | 1196 | if (len < 2) |
1197 | return -1; | |
1198 | ||
7cca0d81 | 1199 | w = 0; |
32874aea | 1200 | for (i = 0; i < 2; i++) |
1201 | w = (w << 8) + *p++; | |
1202 | b = (w + 7) / 8; /* bits -> bytes */ | |
7cca0d81 | 1203 | |
0016d70b | 1204 | if (len < b+2) |
1205 | return -1; | |
1206 | ||
32874aea | 1207 | if (!result) /* just return length */ |
1208 | return b + 2; | |
a52f067e | 1209 | |
3709bfe9 | 1210 | *result = bignum_from_bytes(p, b); |
7cca0d81 | 1211 | |
3709bfe9 | 1212 | return p + b - data; |
7cca0d81 | 1213 | } |
5c58ad2d | 1214 | |
1215 | /* | |
2e85c969 | 1216 | * Return the bit count of a bignum, for SSH-1 encoding. |
5c58ad2d | 1217 | */ |
32874aea | 1218 | int bignum_bitcount(Bignum bn) |
1219 | { | |
a3412f52 | 1220 | int bitcount = bn[0] * BIGNUM_INT_BITS - 1; |
32874aea | 1221 | while (bitcount >= 0 |
a3412f52 | 1222 | && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; |
5c58ad2d | 1223 | return bitcount + 1; |
1224 | } | |
1225 | ||
1226 | /* | |
2e85c969 | 1227 | * Return the byte length of a bignum when SSH-1 encoded. |
5c58ad2d | 1228 | */ |
32874aea | 1229 | int ssh1_bignum_length(Bignum bn) |
1230 | { | |
1231 | return 2 + (bignum_bitcount(bn) + 7) / 8; | |
ddecd643 | 1232 | } |
1233 | ||
1234 | /* | |
2e85c969 | 1235 | * Return the byte length of a bignum when SSH-2 encoded. |
ddecd643 | 1236 | */ |
32874aea | 1237 | int ssh2_bignum_length(Bignum bn) |
1238 | { | |
1239 | return 4 + (bignum_bitcount(bn) + 8) / 8; | |
5c58ad2d | 1240 | } |
1241 | ||
1242 | /* | |
1243 | * Return a byte from a bignum; 0 is least significant, etc. | |
1244 | */ | |
32874aea | 1245 | int bignum_byte(Bignum bn, int i) |
1246 | { | |
62ddb51e | 1247 | if (i >= (int)(BIGNUM_INT_BYTES * bn[0])) |
32874aea | 1248 | return 0; /* beyond the end */ |
5c58ad2d | 1249 | else |
a3412f52 | 1250 | return (bn[i / BIGNUM_INT_BYTES + 1] >> |
1251 | ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; | |
5c58ad2d | 1252 | } |
1253 | ||
1254 | /* | |
9400cf6f | 1255 | * Return a bit from a bignum; 0 is least significant, etc. |
1256 | */ | |
32874aea | 1257 | int bignum_bit(Bignum bn, int i) |
1258 | { | |
62ddb51e | 1259 | if (i >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea | 1260 | return 0; /* beyond the end */ |
9400cf6f | 1261 | else |
a3412f52 | 1262 | return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; |
9400cf6f | 1263 | } |
1264 | ||
1265 | /* | |
1266 | * Set a bit in a bignum; 0 is least significant, etc. | |
1267 | */ | |
32874aea | 1268 | void bignum_set_bit(Bignum bn, int bitnum, int value) |
1269 | { | |
62ddb51e | 1270 | if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea | 1271 | abort(); /* beyond the end */ |
9400cf6f | 1272 | else { |
a3412f52 | 1273 | int v = bitnum / BIGNUM_INT_BITS + 1; |
1274 | int mask = 1 << (bitnum % BIGNUM_INT_BITS); | |
32874aea | 1275 | if (value) |
1276 | bn[v] |= mask; | |
1277 | else | |
1278 | bn[v] &= ~mask; | |
9400cf6f | 1279 | } |
1280 | } | |
1281 | ||
1282 | /* | |
2e85c969 | 1283 | * Write a SSH-1-format bignum into a buffer. It is assumed the |
5c58ad2d | 1284 | * buffer is big enough. Returns the number of bytes used. |
1285 | */ | |
32874aea | 1286 | int ssh1_write_bignum(void *data, Bignum bn) |
1287 | { | |
5c58ad2d | 1288 | unsigned char *p = data; |
1289 | int len = ssh1_bignum_length(bn); | |
1290 | int i; | |
ddecd643 | 1291 | int bitc = bignum_bitcount(bn); |
5c58ad2d | 1292 | |
1293 | *p++ = (bitc >> 8) & 0xFF; | |
32874aea | 1294 | *p++ = (bitc) & 0xFF; |
1295 | for (i = len - 2; i--;) | |
1296 | *p++ = bignum_byte(bn, i); | |
5c58ad2d | 1297 | return len; |
1298 | } | |
9400cf6f | 1299 | |
1300 | /* | |
1301 | * Compare two bignums. Returns like strcmp. | |
1302 | */ | |
32874aea | 1303 | int bignum_cmp(Bignum a, Bignum b) |
1304 | { | |
9400cf6f | 1305 | int amax = a[0], bmax = b[0]; |
1306 | int i = (amax > bmax ? amax : bmax); | |
1307 | while (i) { | |
a3412f52 | 1308 | BignumInt aval = (i > amax ? 0 : a[i]); |
1309 | BignumInt bval = (i > bmax ? 0 : b[i]); | |
32874aea | 1310 | if (aval < bval) |
1311 | return -1; | |
1312 | if (aval > bval) | |
1313 | return +1; | |
1314 | i--; | |
9400cf6f | 1315 | } |
1316 | return 0; | |
1317 | } | |
1318 | ||
1319 | /* | |
1320 | * Right-shift one bignum to form another. | |
1321 | */ | |
32874aea | 1322 | Bignum bignum_rshift(Bignum a, int shift) |
1323 | { | |
9400cf6f | 1324 | Bignum ret; |
1325 | int i, shiftw, shiftb, shiftbb, bits; | |
a3412f52 | 1326 | BignumInt ai, ai1; |
9400cf6f | 1327 | |
ddecd643 | 1328 | bits = bignum_bitcount(a) - shift; |
a3412f52 | 1329 | ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); |
9400cf6f | 1330 | |
1331 | if (ret) { | |
a3412f52 | 1332 | shiftw = shift / BIGNUM_INT_BITS; |
1333 | shiftb = shift % BIGNUM_INT_BITS; | |
1334 | shiftbb = BIGNUM_INT_BITS - shiftb; | |
32874aea | 1335 | |
1336 | ai1 = a[shiftw + 1]; | |
62ddb51e | 1337 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea | 1338 | ai = ai1; |
62ddb51e | 1339 | ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); |
a3412f52 | 1340 | ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; |
32874aea | 1341 | } |
9400cf6f | 1342 | } |
1343 | ||
1344 | return ret; | |
1345 | } | |
1346 | ||
1347 | /* | |
1348 | * Non-modular multiplication and addition. | |
1349 | */ | |
32874aea | 1350 | Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) |
1351 | { | |
9400cf6f | 1352 | int alen = a[0], blen = b[0]; |
1353 | int mlen = (alen > blen ? alen : blen); | |
1354 | int rlen, i, maxspot; | |
5a502a19 | 1355 | int wslen; |
a3412f52 | 1356 | BignumInt *workspace; |
9400cf6f | 1357 | Bignum ret; |
1358 | ||
5a502a19 | 1359 | /* mlen space for a, mlen space for b, 2*mlen for result, |
1360 | * plus scratch space for multiplication */ | |
1361 | wslen = mlen * 4 + mul_compute_scratch(mlen); | |
1362 | workspace = snewn(wslen, BignumInt); | |
9400cf6f | 1363 | for (i = 0; i < mlen; i++) { |
c40be1ad MW |
1364 | workspace[0 * mlen + i] = i < (int)a[0] ? a[i + 1] : 0; |
1365 | workspace[1 * mlen + i] = i < (int)b[0] ? b[i + 1] : 0; | |
9400cf6f | 1366 | } |
1367 | ||
32874aea | 1368 | internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, |
5a502a19 | 1369 | workspace + 2 * mlen, mlen, workspace + 4 * mlen); |
9400cf6f | 1370 | |
1371 | /* now just copy the result back */ | |
1372 | rlen = alen + blen + 1; | |
62ddb51e | 1373 | if (addend && rlen <= (int)addend[0]) |
32874aea | 1374 | rlen = addend[0] + 1; |
9400cf6f | 1375 | ret = newbn(rlen); |
1376 | maxspot = 0; | |
c40be1ad MW |
1377 | for (i = 0; i < (int)ret[0]; i++) { |
1378 | ret[i + 1] = (i < 2 * mlen ? workspace[2 * mlen + i] : 0); | |
1379 | if (ret[i + 1] != 0) | |
1380 | maxspot = i + 1; | |
9400cf6f | 1381 | } |
1382 | ret[0] = maxspot; | |
1383 | ||
1384 | /* now add in the addend, if any */ | |
1385 | if (addend) { | |
a3412f52 | 1386 | BignumDblInt carry = 0; |
32874aea | 1387 | for (i = 1; i <= rlen; i++) { |
62ddb51e | 1388 | carry += (i <= (int)ret[0] ? ret[i] : 0); |
1389 | carry += (i <= (int)addend[0] ? addend[i] : 0); | |
a3412f52 | 1390 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1391 | carry >>= BIGNUM_INT_BITS; | |
32874aea | 1392 | if (ret[i] != 0 && i > maxspot) |
1393 | maxspot = i; | |
1394 | } | |
9400cf6f | 1395 | } |
1396 | ret[0] = maxspot; | |
1397 | ||
5a502a19 | 1398 | for (i = 0; i < wslen; i++) |
1399 | workspace[i] = 0; | |
c523f55f | 1400 | sfree(workspace); |
9400cf6f | 1401 | return ret; |
1402 | } | |
1403 | ||
1404 | /* | |
1405 | * Non-modular multiplication. | |
1406 | */ | |
32874aea | 1407 | Bignum bigmul(Bignum a, Bignum b) |
1408 | { | |
9400cf6f | 1409 | return bigmuladd(a, b, NULL); |
1410 | } | |
1411 | ||
1412 | /* | |
d737853b | 1413 | * Simple addition. |
1414 | */ | |
1415 | Bignum bigadd(Bignum a, Bignum b) | |
1416 | { | |
1417 | int alen = a[0], blen = b[0]; | |
1418 | int rlen = (alen > blen ? alen : blen) + 1; | |
1419 | int i, maxspot; | |
1420 | Bignum ret; | |
1421 | BignumDblInt carry; | |
1422 | ||
1423 | ret = newbn(rlen); | |
1424 | ||
1425 | carry = 0; | |
1426 | maxspot = 0; | |
1427 | for (i = 1; i <= rlen; i++) { | |
1428 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1429 | carry += (i <= (int)b[0] ? b[i] : 0); | |
1430 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1431 | carry >>= BIGNUM_INT_BITS; | |
1432 | if (ret[i] != 0 && i > maxspot) | |
1433 | maxspot = i; | |
1434 | } | |
1435 | ret[0] = maxspot; | |
1436 | ||
1437 | return ret; | |
1438 | } | |
1439 | ||
1440 | /* | |
1441 | * Subtraction. Returns a-b, or NULL if the result would come out | |
1442 | * negative (recall that this entire bignum module only handles | |
1443 | * positive numbers). | |
1444 | */ | |
1445 | Bignum bigsub(Bignum a, Bignum b) | |
1446 | { | |
1447 | int alen = a[0], blen = b[0]; | |
1448 | int rlen = (alen > blen ? alen : blen); | |
1449 | int i, maxspot; | |
1450 | Bignum ret; | |
1451 | BignumDblInt carry; | |
1452 | ||
1453 | ret = newbn(rlen); | |
1454 | ||
1455 | carry = 1; | |
1456 | maxspot = 0; | |
1457 | for (i = 1; i <= rlen; i++) { | |
1458 | carry += (i <= (int)a[0] ? a[i] : 0); | |
1459 | carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); | |
1460 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1461 | carry >>= BIGNUM_INT_BITS; | |
1462 | if (ret[i] != 0 && i > maxspot) | |
1463 | maxspot = i; | |
1464 | } | |
1465 | ret[0] = maxspot; | |
1466 | ||
1467 | if (!carry) { | |
1468 | freebn(ret); | |
1469 | return NULL; | |
1470 | } | |
1471 | ||
1472 | return ret; | |
1473 | } | |
1474 | ||
1475 | /* | |
3709bfe9 | 1476 | * Create a bignum which is the bitmask covering another one. That |
1477 | * is, the smallest integer which is >= N and is also one less than | |
1478 | * a power of two. | |
1479 | */ | |
32874aea | 1480 | Bignum bignum_bitmask(Bignum n) |
1481 | { | |
3709bfe9 | 1482 | Bignum ret = copybn(n); |
1483 | int i; | |
a3412f52 | 1484 | BignumInt j; |
3709bfe9 | 1485 | |
1486 | i = ret[0]; | |
1487 | while (n[i] == 0 && i > 0) | |
32874aea | 1488 | i--; |
3709bfe9 | 1489 | if (i <= 0) |
32874aea | 1490 | return ret; /* input was zero */ |
3709bfe9 | 1491 | j = 1; |
1492 | while (j < n[i]) | |
32874aea | 1493 | j = 2 * j + 1; |
3709bfe9 | 1494 | ret[i] = j; |
1495 | while (--i > 0) | |
a3412f52 | 1496 | ret[i] = BIGNUM_INT_MASK; |
3709bfe9 | 1497 | return ret; |
1498 | } | |
1499 | ||
1500 | /* | |
5c72ca61 | 1501 | * Convert a (max 32-bit) long into a bignum. |
9400cf6f | 1502 | */ |
a3412f52 | 1503 | Bignum bignum_from_long(unsigned long nn) |
32874aea | 1504 | { |
9400cf6f | 1505 | Bignum ret; |
a3412f52 | 1506 | BignumDblInt n = nn; |
9400cf6f | 1507 | |
5c72ca61 | 1508 | ret = newbn(3); |
a3412f52 | 1509 | ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); |
1510 | ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); | |
5c72ca61 | 1511 | ret[3] = 0; |
1512 | ret[0] = (ret[2] ? 2 : 1); | |
32874aea | 1513 | return ret; |
9400cf6f | 1514 | } |
1515 | ||
1516 | /* | |
1517 | * Add a long to a bignum. | |
1518 | */ | |
a3412f52 | 1519 | Bignum bignum_add_long(Bignum number, unsigned long addendx) |
32874aea | 1520 | { |
1521 | Bignum ret = newbn(number[0] + 1); | |
9400cf6f | 1522 | int i, maxspot = 0; |
a3412f52 | 1523 | BignumDblInt carry = 0, addend = addendx; |
9400cf6f | 1524 | |
62ddb51e | 1525 | for (i = 1; i <= (int)ret[0]; i++) { |
a3412f52 | 1526 | carry += addend & BIGNUM_INT_MASK; |
62ddb51e | 1527 | carry += (i <= (int)number[0] ? number[i] : 0); |
a3412f52 | 1528 | addend >>= BIGNUM_INT_BITS; |
1529 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; | |
1530 | carry >>= BIGNUM_INT_BITS; | |
32874aea | 1531 | if (ret[i] != 0) |
1532 | maxspot = i; | |
9400cf6f | 1533 | } |
1534 | ret[0] = maxspot; | |
1535 | return ret; | |
1536 | } | |
1537 | ||
1538 | /* | |
1539 | * Compute the residue of a bignum, modulo a (max 16-bit) short. | |
1540 | */ | |
32874aea | 1541 | unsigned short bignum_mod_short(Bignum number, unsigned short modulus) |
1542 | { | |
a3412f52 | 1543 | BignumDblInt mod, r; |
9400cf6f | 1544 | int i; |
1545 | ||
1546 | r = 0; | |
1547 | mod = modulus; | |
1548 | for (i = number[0]; i > 0; i--) | |
736cc6d1 | 1549 | r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; |
6e522441 | 1550 | return (unsigned short) r; |
9400cf6f | 1551 | } |
1552 | ||
a3412f52 | 1553 | #ifdef DEBUG |
32874aea | 1554 | void diagbn(char *prefix, Bignum md) |
1555 | { | |
9400cf6f | 1556 | int i, nibbles, morenibbles; |
1557 | static const char hex[] = "0123456789ABCDEF"; | |
1558 | ||
5c72ca61 | 1559 | debug(("%s0x", prefix ? prefix : "")); |
9400cf6f | 1560 | |
32874aea | 1561 | nibbles = (3 + bignum_bitcount(md)) / 4; |
1562 | if (nibbles < 1) | |
1563 | nibbles = 1; | |
1564 | morenibbles = 4 * md[0] - nibbles; | |
1565 | for (i = 0; i < morenibbles; i++) | |
5c72ca61 | 1566 | debug(("-")); |
32874aea | 1567 | for (i = nibbles; i--;) |
5c72ca61 | 1568 | debug(("%c", |
1569 | hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF])); | |
9400cf6f | 1570 | |
32874aea | 1571 | if (prefix) |
5c72ca61 | 1572 | debug(("\n")); |
1573 | } | |
f28753ab | 1574 | #endif |
5c72ca61 | 1575 | |
1576 | /* | |
1577 | * Simple division. | |
1578 | */ | |
1579 | Bignum bigdiv(Bignum a, Bignum b) | |
1580 | { | |
1581 | Bignum q = newbn(a[0]); | |
1582 | bigdivmod(a, b, NULL, q); | |
1583 | return q; | |
1584 | } | |
1585 | ||
1586 | /* | |
1587 | * Simple remainder. | |
1588 | */ | |
1589 | Bignum bigmod(Bignum a, Bignum b) | |
1590 | { | |
1591 | Bignum r = newbn(b[0]); | |
1592 | bigdivmod(a, b, r, NULL); | |
1593 | return r; | |
9400cf6f | 1594 | } |
1595 | ||
1596 | /* | |
1597 | * Greatest common divisor. | |
1598 | */ | |
32874aea | 1599 | Bignum biggcd(Bignum av, Bignum bv) |
1600 | { | |
9400cf6f | 1601 | Bignum a = copybn(av); |
1602 | Bignum b = copybn(bv); | |
1603 | ||
9400cf6f | 1604 | while (bignum_cmp(b, Zero) != 0) { |
32874aea | 1605 | Bignum t = newbn(b[0]); |
5c72ca61 | 1606 | bigdivmod(a, b, t, NULL); |
32874aea | 1607 | while (t[0] > 1 && t[t[0]] == 0) |
1608 | t[0]--; | |
1609 | freebn(a); | |
1610 | a = b; | |
1611 | b = t; | |
9400cf6f | 1612 | } |
1613 | ||
1614 | freebn(b); | |
1615 | return a; | |
1616 | } | |
1617 | ||
1618 | /* | |
1619 | * Modular inverse, using Euclid's extended algorithm. | |
1620 | */ | |
32874aea | 1621 | Bignum modinv(Bignum number, Bignum modulus) |
1622 | { | |
9400cf6f | 1623 | Bignum a = copybn(modulus); |
1624 | Bignum b = copybn(number); | |
1625 | Bignum xp = copybn(Zero); | |
1626 | Bignum x = copybn(One); | |
1627 | int sign = +1; | |
1628 | ||
1629 | while (bignum_cmp(b, One) != 0) { | |
32874aea | 1630 | Bignum t = newbn(b[0]); |
1631 | Bignum q = newbn(a[0]); | |
5c72ca61 | 1632 | bigdivmod(a, b, t, q); |
32874aea | 1633 | while (t[0] > 1 && t[t[0]] == 0) |
1634 | t[0]--; | |
1635 | freebn(a); | |
1636 | a = b; | |
1637 | b = t; | |
1638 | t = xp; | |
1639 | xp = x; | |
1640 | x = bigmuladd(q, xp, t); | |
1641 | sign = -sign; | |
1642 | freebn(t); | |
75374b2f | 1643 | freebn(q); |
9400cf6f | 1644 | } |
1645 | ||
1646 | freebn(b); | |
1647 | freebn(a); | |
1648 | freebn(xp); | |
1649 | ||
1650 | /* now we know that sign * x == 1, and that x < modulus */ | |
1651 | if (sign < 0) { | |
32874aea | 1652 | /* set a new x to be modulus - x */ |
1653 | Bignum newx = newbn(modulus[0]); | |
a3412f52 | 1654 | BignumInt carry = 0; |
32874aea | 1655 | int maxspot = 1; |
1656 | int i; | |
1657 | ||
62ddb51e | 1658 | for (i = 1; i <= (int)newx[0]; i++) { |
1659 | BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); | |
1660 | BignumInt bword = (i <= (int)x[0] ? x[i] : 0); | |
32874aea | 1661 | newx[i] = aword - bword - carry; |
1662 | bword = ~bword; | |
1663 | carry = carry ? (newx[i] >= bword) : (newx[i] > bword); | |
1664 | if (newx[i] != 0) | |
1665 | maxspot = i; | |
1666 | } | |
1667 | newx[0] = maxspot; | |
1668 | freebn(x); | |
1669 | x = newx; | |
9400cf6f | 1670 | } |
1671 | ||
1672 | /* and return. */ | |
1673 | return x; | |
1674 | } | |
6e522441 | 1675 | |
1676 | /* | |
1677 | * Render a bignum into decimal. Return a malloced string holding | |
1678 | * the decimal representation. | |
1679 | */ | |
32874aea | 1680 | char *bignum_decimal(Bignum x) |
1681 | { | |
6e522441 | 1682 | int ndigits, ndigit; |
1683 | int i, iszero; | |
a3412f52 | 1684 | BignumDblInt carry; |
6e522441 | 1685 | char *ret; |
a3412f52 | 1686 | BignumInt *workspace; |
6e522441 | 1687 | |
1688 | /* | |
1689 | * First, estimate the number of digits. Since log(10)/log(2) | |
1690 | * is just greater than 93/28 (the joys of continued fraction | |
1691 | * approximations...) we know that for every 93 bits, we need | |
1692 | * at most 28 digits. This will tell us how much to malloc. | |
1693 | * | |
1694 | * Formally: if x has i bits, that means x is strictly less | |
1695 | * than 2^i. Since 2 is less than 10^(28/93), this is less than | |
1696 | * 10^(28i/93). We need an integer power of ten, so we must | |
1697 | * round up (rounding down might make it less than x again). | |
1698 | * Therefore if we multiply the bit count by 28/93, rounding | |
1699 | * up, we will have enough digits. | |
74c79ce8 | 1700 | * |
1701 | * i=0 (i.e., x=0) is an irritating special case. | |
6e522441 | 1702 | */ |
ddecd643 | 1703 | i = bignum_bitcount(x); |
74c79ce8 | 1704 | if (!i) |
1705 | ndigits = 1; /* x = 0 */ | |
1706 | else | |
1707 | ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ | |
32874aea | 1708 | ndigits++; /* allow for trailing \0 */ |
3d88e64d | 1709 | ret = snewn(ndigits, char); |
6e522441 | 1710 | |
1711 | /* | |
1712 | * Now allocate some workspace to hold the binary form as we | |
1713 | * repeatedly divide it by ten. Initialise this to the | |
1714 | * big-endian form of the number. | |
1715 | */ | |
a3412f52 | 1716 | workspace = snewn(x[0], BignumInt); |
62ddb51e | 1717 | for (i = 0; i < (int)x[0]; i++) |
32874aea | 1718 | workspace[i] = x[x[0] - i]; |
6e522441 | 1719 | |
1720 | /* | |
1721 | * Next, write the decimal number starting with the last digit. | |
1722 | * We use ordinary short division, dividing 10 into the | |
1723 | * workspace. | |
1724 | */ | |
32874aea | 1725 | ndigit = ndigits - 1; |
6e522441 | 1726 | ret[ndigit] = '\0'; |
1727 | do { | |
32874aea | 1728 | iszero = 1; |
1729 | carry = 0; | |
62ddb51e | 1730 | for (i = 0; i < (int)x[0]; i++) { |
a3412f52 | 1731 | carry = (carry << BIGNUM_INT_BITS) + workspace[i]; |
1732 | workspace[i] = (BignumInt) (carry / 10); | |
32874aea | 1733 | if (workspace[i]) |
1734 | iszero = 0; | |
1735 | carry %= 10; | |
1736 | } | |
1737 | ret[--ndigit] = (char) (carry + '0'); | |
6e522441 | 1738 | } while (!iszero); |
1739 | ||
1740 | /* | |
1741 | * There's a chance we've fallen short of the start of the | |
1742 | * string. Correct if so. | |
1743 | */ | |
1744 | if (ndigit > 0) | |
32874aea | 1745 | memmove(ret, ret + ndigit, ndigits - ndigit); |
6e522441 | 1746 | |
1747 | /* | |
1748 | * Done. | |
1749 | */ | |
c523f55f | 1750 | sfree(workspace); |
6e522441 | 1751 | return ret; |
1752 | } | |
f3c29e34 | 1753 | |
1754 | #ifdef TESTBN | |
1755 | ||
1756 | #include <stdio.h> | |
1757 | #include <stdlib.h> | |
1758 | #include <ctype.h> | |
1759 | ||
1760 | /* | |
4800a5e5 | 1761 | * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset |
f84f1e46 | 1762 | * |
1763 | * Then feed to this program's standard input the output of | |
1764 | * testdata/bignum.py . | |
f3c29e34 | 1765 | */ |
1766 | ||
1767 | void modalfatalbox(char *p, ...) | |
1768 | { | |
1769 | va_list ap; | |
1770 | fprintf(stderr, "FATAL ERROR: "); | |
1771 | va_start(ap, p); | |
1772 | vfprintf(stderr, p, ap); | |
1773 | va_end(ap); | |
1774 | fputc('\n', stderr); | |
1775 | exit(1); | |
1776 | } | |
1777 | ||
1778 | #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) | |
1779 | ||
1780 | int main(int argc, char **argv) | |
1781 | { | |
1782 | char *buf; | |
1783 | int line = 0; | |
1784 | int passes = 0, fails = 0; | |
1785 | ||
1786 | while ((buf = fgetline(stdin)) != NULL) { | |
1787 | int maxlen = strlen(buf); | |
1788 | unsigned char *data = snewn(maxlen, unsigned char); | |
f84f1e46 | 1789 | unsigned char *ptrs[5], *q; |
f3c29e34 | 1790 | int ptrnum; |
1791 | char *bufp = buf; | |
1792 | ||
1793 | line++; | |
1794 | ||
1795 | q = data; | |
1796 | ptrnum = 0; | |
1797 | ||
f84f1e46 | 1798 | while (*bufp && !isspace((unsigned char)*bufp)) |
1799 | bufp++; | |
1800 | if (bufp) | |
1801 | *bufp++ = '\0'; | |
1802 | ||
f3c29e34 | 1803 | while (*bufp) { |
1804 | char *start, *end; | |
1805 | int i; | |
1806 | ||
1807 | while (*bufp && !isxdigit((unsigned char)*bufp)) | |
1808 | bufp++; | |
1809 | start = bufp; | |
1810 | ||
1811 | if (!*bufp) | |
1812 | break; | |
1813 | ||
1814 | while (*bufp && isxdigit((unsigned char)*bufp)) | |
1815 | bufp++; | |
1816 | end = bufp; | |
1817 | ||
1818 | if (ptrnum >= lenof(ptrs)) | |
1819 | break; | |
1820 | ptrs[ptrnum++] = q; | |
1821 | ||
1822 | for (i = -((end - start) & 1); i < end-start; i += 2) { | |
1823 | unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); | |
1824 | val = val * 16 + fromxdigit(start[i+1]); | |
1825 | *q++ = val; | |
1826 | } | |
1827 | ||
1828 | ptrs[ptrnum] = q; | |
1829 | } | |
1830 | ||
f84f1e46 | 1831 | if (!strcmp(buf, "mul")) { |
1832 | Bignum a, b, c, p; | |
1833 | ||
1834 | if (ptrnum != 3) { | |
f6939e2b | 1835 | printf("%d: mul with %d parameters, expected 3\n", line, ptrnum); |
f84f1e46 | 1836 | exit(1); |
1837 | } | |
1838 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1839 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1840 | c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1841 | p = bigmul(a, b); | |
f3c29e34 | 1842 | |
1843 | if (bignum_cmp(c, p) == 0) { | |
1844 | passes++; | |
1845 | } else { | |
1846 | char *as = bignum_decimal(a); | |
1847 | char *bs = bignum_decimal(b); | |
1848 | char *cs = bignum_decimal(c); | |
1849 | char *ps = bignum_decimal(p); | |
1850 | ||
1851 | printf("%d: fail: %s * %s gave %s expected %s\n", | |
1852 | line, as, bs, ps, cs); | |
1853 | fails++; | |
1854 | ||
1855 | sfree(as); | |
1856 | sfree(bs); | |
1857 | sfree(cs); | |
1858 | sfree(ps); | |
1859 | } | |
1860 | freebn(a); | |
1861 | freebn(b); | |
1862 | freebn(c); | |
1863 | freebn(p); | |
f84f1e46 | 1864 | } else if (!strcmp(buf, "pow")) { |
1865 | Bignum base, expt, modulus, expected, answer; | |
1866 | ||
1867 | if (ptrnum != 4) { | |
f6939e2b | 1868 | printf("%d: mul with %d parameters, expected 4\n", line, ptrnum); |
f84f1e46 | 1869 | exit(1); |
1870 | } | |
1871 | ||
1872 | base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); | |
1873 | expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); | |
1874 | modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); | |
1875 | expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); | |
1876 | answer = modpow(base, expt, modulus); | |
1877 | ||
1878 | if (bignum_cmp(expected, answer) == 0) { | |
1879 | passes++; | |
1880 | } else { | |
1881 | char *as = bignum_decimal(base); | |
1882 | char *bs = bignum_decimal(expt); | |
1883 | char *cs = bignum_decimal(modulus); | |
1884 | char *ds = bignum_decimal(answer); | |
1885 | char *ps = bignum_decimal(expected); | |
1886 | ||
1887 | printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", | |
1888 | line, as, bs, cs, ds, ps); | |
1889 | fails++; | |
1890 | ||
1891 | sfree(as); | |
1892 | sfree(bs); | |
1893 | sfree(cs); | |
1894 | sfree(ds); | |
1895 | sfree(ps); | |
1896 | } | |
1897 | freebn(base); | |
1898 | freebn(expt); | |
1899 | freebn(modulus); | |
1900 | freebn(expected); | |
1901 | freebn(answer); | |
1902 | } else { | |
1903 | printf("%d: unrecognised test keyword: '%s'\n", line, buf); | |
1904 | exit(1); | |
f3c29e34 | 1905 | } |
f84f1e46 | 1906 | |
f3c29e34 | 1907 | sfree(buf); |
1908 | sfree(data); | |
1909 | } | |
1910 | ||
1911 | printf("passed %d failed %d total %d\n", passes, fails, passes+fails); | |
1912 | return fails != 0; | |
1913 | } | |
1914 | ||
1915 | #endif |