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> |
551a4acb |
9 | #include <limits.h> |
e5574168 |
10 | |
5c72ca61 |
11 | #include "misc.h" |
98ba26b9 |
12 | |
819a22b3 |
13 | /* |
14 | * Usage notes: |
15 | * * Do not call the DIVMOD_WORD macro with expressions such as array |
16 | * subscripts, as some implementations object to this (see below). |
17 | * * Note that none of the division methods below will cope if the |
18 | * quotient won't fit into BIGNUM_INT_BITS. Callers should be careful |
19 | * to avoid this case. |
20 | * If this condition occurs, in the case of the x86 DIV instruction, |
21 | * an overflow exception will occur, which (according to a correspondent) |
22 | * will manifest on Windows as something like |
23 | * 0xC0000095: Integer overflow |
24 | * The C variant won't give the right answer, either. |
25 | */ |
26 | |
a3412f52 |
27 | #if defined __GNUC__ && defined __i386__ |
28 | typedef unsigned long BignumInt; |
29 | typedef unsigned long long BignumDblInt; |
30 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
31 | #define BIGNUM_TOP_BIT 0x80000000UL |
32 | #define BIGNUM_INT_BITS 32 |
33 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
a47e8bba |
34 | #define DIVMOD_WORD(q, r, hi, lo, w) \ |
35 | __asm__("div %2" : \ |
36 | "=d" (r), "=a" (q) : \ |
37 | "r" (w), "d" (hi), "a" (lo)) |
036eddfb |
38 | #elif defined _MSC_VER && defined _M_IX86 |
39 | typedef unsigned __int32 BignumInt; |
40 | typedef unsigned __int64 BignumDblInt; |
41 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
42 | #define BIGNUM_TOP_BIT 0x80000000UL |
43 | #define BIGNUM_INT_BITS 32 |
44 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
819a22b3 |
45 | /* Note: MASM interprets array subscripts in the macro arguments as |
46 | * assembler syntax, which gives the wrong answer. Don't supply them. |
47 | * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */ |
036eddfb |
48 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
819a22b3 |
49 | __asm mov edx, hi \ |
50 | __asm mov eax, lo \ |
51 | __asm div w \ |
52 | __asm mov r, edx \ |
53 | __asm mov q, eax \ |
54 | } while(0) |
32e51f76 |
55 | #elif defined _LP64 |
56 | /* 64-bit architectures can do 32x32->64 chunks at a time */ |
57 | typedef unsigned int BignumInt; |
58 | typedef unsigned long BignumDblInt; |
59 | #define BIGNUM_INT_MASK 0xFFFFFFFFU |
60 | #define BIGNUM_TOP_BIT 0x80000000U |
61 | #define BIGNUM_INT_BITS 32 |
62 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
63 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
64 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
65 | q = n / w; \ |
66 | r = n % w; \ |
67 | } while (0) |
68 | #elif defined _LLP64 |
69 | /* 64-bit architectures in which unsigned long is 32 bits, not 64 */ |
70 | typedef unsigned long BignumInt; |
71 | typedef unsigned long long BignumDblInt; |
72 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
73 | #define BIGNUM_TOP_BIT 0x80000000UL |
74 | #define BIGNUM_INT_BITS 32 |
75 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
76 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
77 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
78 | q = n / w; \ |
79 | r = n % w; \ |
80 | } while (0) |
a3412f52 |
81 | #else |
32e51f76 |
82 | /* Fallback for all other cases */ |
a3412f52 |
83 | typedef unsigned short BignumInt; |
84 | typedef unsigned long BignumDblInt; |
85 | #define BIGNUM_INT_MASK 0xFFFFU |
86 | #define BIGNUM_TOP_BIT 0x8000U |
87 | #define BIGNUM_INT_BITS 16 |
88 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
a47e8bba |
89 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
90 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
91 | q = n / w; \ |
92 | r = n % w; \ |
93 | } while (0) |
a3412f52 |
94 | #endif |
95 | |
96 | #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8) |
97 | |
3709bfe9 |
98 | #define BIGNUM_INTERNAL |
a3412f52 |
99 | typedef BignumInt *Bignum; |
3709bfe9 |
100 | |
e5574168 |
101 | #include "ssh.h" |
102 | |
a3412f52 |
103 | BignumInt bnZero[1] = { 0 }; |
104 | BignumInt bnOne[2] = { 1, 1 }; |
e5574168 |
105 | |
7d6ee6ff |
106 | /* |
a3412f52 |
107 | * The Bignum format is an array of `BignumInt'. The first |
7d6ee6ff |
108 | * element of the array counts the remaining elements. The |
a3412f52 |
109 | * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_ |
7d6ee6ff |
110 | * significant digit first. (So it's trivial to extract the bit |
111 | * with value 2^n for any n.) |
112 | * |
113 | * All Bignums in this module are positive. Negative numbers must |
114 | * be dealt with outside it. |
115 | * |
116 | * INVARIANT: the most significant word of any Bignum must be |
117 | * nonzero. |
118 | */ |
119 | |
7cca0d81 |
120 | Bignum Zero = bnZero, One = bnOne; |
e5574168 |
121 | |
32874aea |
122 | static Bignum newbn(int length) |
123 | { |
551a4acb |
124 | Bignum b; |
125 | |
126 | assert(length >= 0 && length < INT_MAX / BIGNUM_INT_BITS); |
127 | |
128 | b = snewn(length + 1, BignumInt); |
e5574168 |
129 | if (!b) |
130 | abort(); /* FIXME */ |
32874aea |
131 | memset(b, 0, (length + 1) * sizeof(*b)); |
e5574168 |
132 | b[0] = length; |
133 | return b; |
134 | } |
135 | |
32874aea |
136 | void bn_restore_invariant(Bignum b) |
137 | { |
138 | while (b[0] > 1 && b[b[0]] == 0) |
139 | b[0]--; |
3709bfe9 |
140 | } |
141 | |
32874aea |
142 | Bignum copybn(Bignum orig) |
143 | { |
a3412f52 |
144 | Bignum b = snewn(orig[0] + 1, BignumInt); |
7cca0d81 |
145 | if (!b) |
146 | abort(); /* FIXME */ |
32874aea |
147 | memcpy(b, orig, (orig[0] + 1) * sizeof(*b)); |
7cca0d81 |
148 | return b; |
149 | } |
150 | |
32874aea |
151 | void freebn(Bignum b) |
152 | { |
e5574168 |
153 | /* |
154 | * Burn the evidence, just in case. |
155 | */ |
dfb88efd |
156 | smemclr(b, sizeof(b[0]) * (b[0] + 1)); |
dcbde236 |
157 | sfree(b); |
e5574168 |
158 | } |
159 | |
32874aea |
160 | Bignum bn_power_2(int n) |
161 | { |
551a4acb |
162 | Bignum ret; |
163 | |
164 | assert(n >= 0); |
165 | |
166 | ret = newbn(n / BIGNUM_INT_BITS + 1); |
3709bfe9 |
167 | bignum_set_bit(ret, n, 1); |
168 | return ret; |
169 | } |
170 | |
e5574168 |
171 | /* |
0c431b2f |
172 | * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all |
173 | * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried |
174 | * off the top. |
175 | */ |
176 | static BignumInt internal_add(const BignumInt *a, const BignumInt *b, |
177 | BignumInt *c, int len) |
178 | { |
179 | int i; |
180 | BignumDblInt carry = 0; |
181 | |
182 | for (i = len-1; i >= 0; i--) { |
183 | carry += (BignumDblInt)a[i] + b[i]; |
184 | c[i] = (BignumInt)carry; |
185 | carry >>= BIGNUM_INT_BITS; |
186 | } |
187 | |
188 | return (BignumInt)carry; |
189 | } |
190 | |
191 | /* |
192 | * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are |
193 | * all big-endian arrays of 'len' BignumInts. Any borrow from the top |
194 | * is ignored. |
195 | */ |
196 | static void internal_sub(const BignumInt *a, const BignumInt *b, |
197 | BignumInt *c, int len) |
198 | { |
199 | int i; |
200 | BignumDblInt carry = 1; |
201 | |
202 | for (i = len-1; i >= 0; i--) { |
203 | carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK); |
204 | c[i] = (BignumInt)carry; |
205 | carry >>= BIGNUM_INT_BITS; |
206 | } |
207 | } |
208 | |
209 | /* |
e5574168 |
210 | * Compute c = a * b. |
211 | * Input is in the first len words of a and b. |
212 | * Result is returned in the first 2*len words of c. |
5a502a19 |
213 | * |
214 | * 'scratch' must point to an array of BignumInt of size at least |
215 | * mul_compute_scratch(len). (This covers the needs of internal_mul |
216 | * and all its recursive calls to itself.) |
e5574168 |
217 | */ |
0c431b2f |
218 | #define KARATSUBA_THRESHOLD 50 |
5a502a19 |
219 | static int mul_compute_scratch(int len) |
220 | { |
221 | int ret = 0; |
222 | while (len > KARATSUBA_THRESHOLD) { |
223 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
224 | int midlen = botlen + 1; |
225 | ret += 4*midlen; |
226 | len = midlen; |
227 | } |
228 | return ret; |
229 | } |
132c534f |
230 | static void internal_mul(const BignumInt *a, const BignumInt *b, |
5a502a19 |
231 | BignumInt *c, int len, BignumInt *scratch) |
e5574168 |
232 | { |
0c431b2f |
233 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 |
234 | int i; |
0c431b2f |
235 | |
236 | /* |
237 | * Karatsuba divide-and-conquer algorithm. Cut each input in |
238 | * half, so that it's expressed as two big 'digits' in a giant |
239 | * base D: |
240 | * |
241 | * a = a_1 D + a_0 |
242 | * b = b_1 D + b_0 |
243 | * |
244 | * Then the product is of course |
245 | * |
246 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 |
247 | * |
248 | * and we compute the three coefficients by recursively |
249 | * calling ourself to do half-length multiplications. |
250 | * |
251 | * The clever bit that makes this worth doing is that we only |
252 | * need _one_ half-length multiplication for the central |
253 | * coefficient rather than the two that it obviouly looks |
254 | * like, because we can use a single multiplication to compute |
255 | * |
256 | * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0 |
257 | * |
258 | * and then we subtract the other two coefficients (a_1 b_1 |
259 | * and a_0 b_0) which we were computing anyway. |
260 | * |
261 | * Hence we get to multiply two numbers of length N in about |
262 | * three times as much work as it takes to multiply numbers of |
263 | * length N/2, which is obviously better than the four times |
264 | * as much work it would take if we just did a long |
265 | * conventional multiply. |
266 | */ |
267 | |
268 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
269 | int midlen = botlen + 1; |
0c431b2f |
270 | BignumDblInt carry; |
f3c29e34 |
271 | #ifdef KARA_DEBUG |
272 | int i; |
273 | #endif |
0c431b2f |
274 | |
275 | /* |
276 | * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping |
277 | * in the output array, so we can compute them immediately in |
278 | * place. |
279 | */ |
280 | |
f3c29e34 |
281 | #ifdef KARA_DEBUG |
282 | printf("a1,a0 = 0x"); |
283 | for (i = 0; i < len; i++) { |
284 | if (i == toplen) printf(", 0x"); |
285 | printf("%0*x", BIGNUM_INT_BITS/4, a[i]); |
286 | } |
287 | printf("\n"); |
288 | printf("b1,b0 = 0x"); |
289 | for (i = 0; i < len; i++) { |
290 | if (i == toplen) printf(", 0x"); |
291 | printf("%0*x", BIGNUM_INT_BITS/4, b[i]); |
292 | } |
293 | printf("\n"); |
294 | #endif |
295 | |
0c431b2f |
296 | /* a_1 b_1 */ |
5a502a19 |
297 | internal_mul(a, b, c, toplen, scratch); |
f3c29e34 |
298 | #ifdef KARA_DEBUG |
299 | printf("a1b1 = 0x"); |
300 | for (i = 0; i < 2*toplen; i++) { |
301 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); |
302 | } |
303 | printf("\n"); |
304 | #endif |
0c431b2f |
305 | |
306 | /* a_0 b_0 */ |
5a502a19 |
307 | internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch); |
f3c29e34 |
308 | #ifdef KARA_DEBUG |
309 | printf("a0b0 = 0x"); |
310 | for (i = 0; i < 2*botlen; i++) { |
311 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]); |
312 | } |
313 | printf("\n"); |
314 | #endif |
0c431b2f |
315 | |
0c431b2f |
316 | /* Zero padding. midlen exceeds toplen by at most 2, so just |
317 | * zero the first two words of each input and the rest will be |
318 | * copied over. */ |
319 | scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0; |
320 | |
757b0110 |
321 | for (i = 0; i < toplen; i++) { |
322 | scratch[midlen - toplen + i] = a[i]; /* a_1 */ |
323 | scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ |
0c431b2f |
324 | } |
325 | |
326 | /* compute a_1 + a_0 */ |
327 | scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); |
f3c29e34 |
328 | #ifdef KARA_DEBUG |
329 | printf("a1plusa0 = 0x"); |
330 | for (i = 0; i < midlen; i++) { |
331 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); |
332 | } |
333 | printf("\n"); |
334 | #endif |
0c431b2f |
335 | /* compute b_1 + b_0 */ |
336 | scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, |
337 | scratch+midlen+1, botlen); |
f3c29e34 |
338 | #ifdef KARA_DEBUG |
339 | printf("b1plusb0 = 0x"); |
340 | for (i = 0; i < midlen; i++) { |
341 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]); |
342 | } |
343 | printf("\n"); |
344 | #endif |
0c431b2f |
345 | |
346 | /* |
347 | * Now we can do the third multiplication. |
348 | */ |
5a502a19 |
349 | internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, |
350 | scratch + 4*midlen); |
f3c29e34 |
351 | #ifdef KARA_DEBUG |
352 | printf("a1plusa0timesb1plusb0 = 0x"); |
353 | for (i = 0; i < 2*midlen; i++) { |
354 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); |
355 | } |
356 | printf("\n"); |
357 | #endif |
0c431b2f |
358 | |
359 | /* |
360 | * Now we can reuse the first half of 'scratch' to compute the |
361 | * sum of the outer two coefficients, to subtract from that |
362 | * product to obtain the middle one. |
363 | */ |
364 | scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0; |
757b0110 |
365 | for (i = 0; i < 2*toplen; i++) |
366 | scratch[2*midlen - 2*toplen + i] = c[i]; |
0c431b2f |
367 | scratch[1] = internal_add(scratch+2, c + 2*toplen, |
368 | scratch+2, 2*botlen); |
f3c29e34 |
369 | #ifdef KARA_DEBUG |
370 | printf("a1b1plusa0b0 = 0x"); |
371 | for (i = 0; i < 2*midlen; i++) { |
372 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); |
373 | } |
374 | printf("\n"); |
375 | #endif |
0c431b2f |
376 | |
377 | internal_sub(scratch + 2*midlen, scratch, |
378 | scratch + 2*midlen, 2*midlen); |
f3c29e34 |
379 | #ifdef KARA_DEBUG |
380 | printf("a1b0plusa0b1 = 0x"); |
381 | for (i = 0; i < 2*midlen; i++) { |
382 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); |
383 | } |
384 | printf("\n"); |
385 | #endif |
0c431b2f |
386 | |
387 | /* |
388 | * And now all we need to do is to add that middle coefficient |
389 | * back into the output. We may have to propagate a carry |
390 | * further up the output, but we can be sure it won't |
391 | * propagate right the way off the top. |
392 | */ |
393 | carry = internal_add(c + 2*len - botlen - 2*midlen, |
394 | scratch + 2*midlen, |
395 | c + 2*len - botlen - 2*midlen, 2*midlen); |
757b0110 |
396 | i = 2*len - botlen - 2*midlen - 1; |
0c431b2f |
397 | while (carry) { |
757b0110 |
398 | assert(i >= 0); |
399 | carry += c[i]; |
400 | c[i] = (BignumInt)carry; |
0c431b2f |
401 | carry >>= BIGNUM_INT_BITS; |
757b0110 |
402 | i--; |
0c431b2f |
403 | } |
f3c29e34 |
404 | #ifdef KARA_DEBUG |
405 | printf("ab = 0x"); |
406 | for (i = 0; i < 2*len; i++) { |
407 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); |
408 | } |
409 | printf("\n"); |
410 | #endif |
0c431b2f |
411 | |
0c431b2f |
412 | } else { |
757b0110 |
413 | int i; |
414 | BignumInt carry; |
415 | BignumDblInt t; |
416 | const BignumInt *ap, *bp; |
417 | BignumInt *cp, *cps; |
0c431b2f |
418 | |
419 | /* |
420 | * Multiply in the ordinary O(N^2) way. |
421 | */ |
422 | |
757b0110 |
423 | for (i = 0; i < 2 * len; i++) |
424 | c[i] = 0; |
0c431b2f |
425 | |
757b0110 |
426 | for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { |
427 | carry = 0; |
428 | for (cp = cps, bp = b + len; cp--, bp-- > b ;) { |
429 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
430 | *cp = (BignumInt) t; |
08b5c9a2 |
431 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
0c431b2f |
432 | } |
757b0110 |
433 | *cp = carry; |
0c431b2f |
434 | } |
e5574168 |
435 | } |
436 | } |
437 | |
132c534f |
438 | /* |
439 | * Variant form of internal_mul used for the initial step of |
440 | * Montgomery reduction. Only bothers outputting 'len' words |
441 | * (everything above that is thrown away). |
442 | */ |
443 | static void internal_mul_low(const BignumInt *a, const BignumInt *b, |
5a502a19 |
444 | BignumInt *c, int len, BignumInt *scratch) |
132c534f |
445 | { |
132c534f |
446 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 |
447 | int i; |
132c534f |
448 | |
449 | /* |
450 | * Karatsuba-aware version of internal_mul_low. As before, we |
451 | * express each input value as a shifted combination of two |
452 | * halves: |
453 | * |
454 | * a = a_1 D + a_0 |
455 | * b = b_1 D + b_0 |
456 | * |
457 | * Then the full product is, as before, |
458 | * |
459 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 |
460 | * |
461 | * Provided we choose D on the large side (so that a_0 and b_0 |
462 | * are _at least_ as long as a_1 and b_1), we don't need the |
463 | * topmost term at all, and we only need half of the middle |
464 | * term. So there's no point in doing the proper Karatsuba |
465 | * optimisation which computes the middle term using the top |
466 | * one, because we'd take as long computing the top one as |
467 | * just computing the middle one directly. |
468 | * |
469 | * So instead, we do a much more obvious thing: we call the |
470 | * fully optimised internal_mul to compute a_0 b_0, and we |
471 | * recursively call ourself to compute the _bottom halves_ of |
472 | * a_1 b_0 and a_0 b_1, each of which we add into the result |
473 | * in the obvious way. |
474 | * |
475 | * In other words, there's no actual Karatsuba _optimisation_ |
476 | * in this function; the only benefit in doing it this way is |
477 | * that we call internal_mul proper for a large part of the |
478 | * work, and _that_ can optimise its operation. |
479 | */ |
480 | |
481 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
132c534f |
482 | |
483 | /* |
5a502a19 |
484 | * Scratch space for the various bits and pieces we're going |
485 | * to be adding together: we need botlen*2 words for a_0 b_0 |
486 | * (though we may end up throwing away its topmost word), and |
487 | * toplen words for each of a_1 b_0 and a_0 b_1. That adds up |
488 | * to exactly 2*len. |
132c534f |
489 | */ |
132c534f |
490 | |
491 | /* a_0 b_0 */ |
5a502a19 |
492 | internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, |
493 | scratch + 2*len); |
132c534f |
494 | |
495 | /* a_1 b_0 */ |
5a502a19 |
496 | internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, |
497 | scratch + 2*len); |
132c534f |
498 | |
499 | /* a_0 b_1 */ |
5a502a19 |
500 | internal_mul_low(a + len - toplen, b, scratch, toplen, |
501 | scratch + 2*len); |
132c534f |
502 | |
503 | /* Copy the bottom half of the big coefficient into place */ |
757b0110 |
504 | for (i = 0; i < botlen; i++) |
505 | c[toplen + i] = scratch[2*toplen + botlen + i]; |
132c534f |
506 | |
507 | /* Add the two small coefficients, throwing away the returned carry */ |
508 | internal_add(scratch, scratch + toplen, scratch, toplen); |
509 | |
510 | /* And add that to the large coefficient, leaving the result in c. */ |
511 | internal_add(scratch, scratch + 2*toplen + botlen - toplen, |
512 | c, toplen); |
513 | |
132c534f |
514 | } else { |
757b0110 |
515 | int i; |
516 | BignumInt carry; |
517 | BignumDblInt t; |
518 | const BignumInt *ap, *bp; |
519 | BignumInt *cp, *cps; |
132c534f |
520 | |
757b0110 |
521 | /* |
522 | * Multiply in the ordinary O(N^2) way. |
523 | */ |
132c534f |
524 | |
757b0110 |
525 | for (i = 0; i < len; i++) |
526 | c[i] = 0; |
527 | |
528 | for (cps = c + len, ap = a + len; ap-- > a; cps--) { |
529 | carry = 0; |
530 | for (cp = cps, bp = b + len; bp--, cp-- > c ;) { |
531 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
532 | *cp = (BignumInt) t; |
08b5c9a2 |
533 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
132c534f |
534 | } |
535 | } |
132c534f |
536 | } |
537 | } |
538 | |
539 | /* |
540 | * Montgomery reduction. Expects x to be a big-endian array of 2*len |
541 | * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * |
542 | * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array |
543 | * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= |
544 | * x' < n. |
545 | * |
546 | * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts |
547 | * each, containing respectively n and the multiplicative inverse of |
548 | * -n mod r. |
549 | * |
5a502a19 |
550 | * 'tmp' is an array of BignumInt used as scratch space, of length at |
551 | * least 3*len + mul_compute_scratch(len). |
132c534f |
552 | */ |
553 | static void monty_reduce(BignumInt *x, const BignumInt *n, |
554 | const BignumInt *mninv, BignumInt *tmp, int len) |
555 | { |
556 | int i; |
557 | BignumInt carry; |
558 | |
559 | /* |
560 | * Multiply x by (-n)^{-1} mod r. This gives us a value m such |
561 | * that mn is congruent to -x mod r. Hence, mn+x is an exact |
562 | * multiple of r, and is also (obviously) congruent to x mod n. |
563 | */ |
5a502a19 |
564 | internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len); |
132c534f |
565 | |
566 | /* |
567 | * Compute t = (mn+x)/r in ordinary, non-modular, integer |
568 | * arithmetic. By construction this is exact, and is congruent mod |
569 | * n to x * r^{-1}, i.e. the answer we want. |
570 | * |
571 | * The following multiply leaves that answer in the _most_ |
572 | * significant half of the 'x' array, so then we must shift it |
573 | * down. |
574 | */ |
5a502a19 |
575 | internal_mul(tmp, n, tmp+len, len, tmp + 3*len); |
132c534f |
576 | carry = internal_add(x, tmp+len, x, 2*len); |
577 | for (i = 0; i < len; i++) |
578 | x[len + i] = x[i], x[i] = 0; |
579 | |
580 | /* |
581 | * Reduce t mod n. This doesn't require a full-on division by n, |
582 | * but merely a test and single optional subtraction, since we can |
583 | * show that 0 <= t < 2n. |
584 | * |
585 | * Proof: |
586 | * + we computed m mod r, so 0 <= m < r. |
587 | * + so 0 <= mn < rn, obviously |
588 | * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn |
589 | * + yielding 0 <= (mn+x)/r < 2n as required. |
590 | */ |
591 | if (!carry) { |
592 | for (i = 0; i < len; i++) |
593 | if (x[len + i] != n[i]) |
594 | break; |
595 | } |
596 | if (carry || i >= len || x[len + i] > n[i]) |
597 | internal_sub(x+len, n, x+len, len); |
598 | } |
599 | |
a3412f52 |
600 | static void internal_add_shifted(BignumInt *number, |
32874aea |
601 | unsigned n, int shift) |
602 | { |
a3412f52 |
603 | int word = 1 + (shift / BIGNUM_INT_BITS); |
604 | int bshift = shift % BIGNUM_INT_BITS; |
605 | BignumDblInt addend; |
9400cf6f |
606 | |
3014da2b |
607 | addend = (BignumDblInt)n << bshift; |
9400cf6f |
608 | |
609 | while (addend) { |
16bd1b88 |
610 | assert(word <= number[0]); |
32874aea |
611 | addend += number[word]; |
a3412f52 |
612 | number[word] = (BignumInt) addend & BIGNUM_INT_MASK; |
613 | addend >>= BIGNUM_INT_BITS; |
32874aea |
614 | word++; |
9400cf6f |
615 | } |
616 | } |
617 | |
e5574168 |
618 | /* |
619 | * Compute a = a % m. |
9400cf6f |
620 | * Input in first alen words of a and first mlen words of m. |
621 | * Output in first alen words of a |
622 | * (of which first alen-mlen words will be zero). |
e5574168 |
623 | * The MSW of m MUST have its high bit set. |
9400cf6f |
624 | * Quotient is accumulated in the `quotient' array, which is a Bignum |
625 | * rather than the internal bigendian format. Quotient parts are shifted |
626 | * left by `qshift' before adding into quot. |
e5574168 |
627 | */ |
a3412f52 |
628 | static void internal_mod(BignumInt *a, int alen, |
629 | BignumInt *m, int mlen, |
630 | BignumInt *quot, int qshift) |
e5574168 |
631 | { |
a3412f52 |
632 | BignumInt m0, m1; |
e5574168 |
633 | unsigned int h; |
634 | int i, k; |
635 | |
e5574168 |
636 | m0 = m[0]; |
8bd9144b |
637 | assert(m0 >> (BIGNUM_INT_BITS-1) == 1); |
9400cf6f |
638 | if (mlen > 1) |
32874aea |
639 | m1 = m[1]; |
9400cf6f |
640 | else |
32874aea |
641 | m1 = 0; |
e5574168 |
642 | |
32874aea |
643 | for (i = 0; i <= alen - mlen; i++) { |
a3412f52 |
644 | BignumDblInt t; |
9400cf6f |
645 | unsigned int q, r, c, ai1; |
e5574168 |
646 | |
647 | if (i == 0) { |
648 | h = 0; |
649 | } else { |
32874aea |
650 | h = a[i - 1]; |
651 | a[i - 1] = 0; |
e5574168 |
652 | } |
653 | |
32874aea |
654 | if (i == alen - 1) |
655 | ai1 = 0; |
656 | else |
657 | ai1 = a[i + 1]; |
9400cf6f |
658 | |
e5574168 |
659 | /* Find q = h:a[i] / m0 */ |
62ef3d44 |
660 | if (h >= m0) { |
661 | /* |
662 | * Special case. |
663 | * |
664 | * To illustrate it, suppose a BignumInt is 8 bits, and |
665 | * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then |
666 | * our initial division will be 0xA123 / 0xA1, which |
667 | * will give a quotient of 0x100 and a divide overflow. |
668 | * However, the invariants in this division algorithm |
669 | * are not violated, since the full number A1:23:... is |
670 | * _less_ than the quotient prefix A1:B2:... and so the |
671 | * following correction loop would have sorted it out. |
672 | * |
673 | * In this situation we set q to be the largest |
674 | * quotient we _can_ stomach (0xFF, of course). |
675 | */ |
676 | q = BIGNUM_INT_MASK; |
677 | } else { |
819a22b3 |
678 | /* Macro doesn't want an array subscript expression passed |
679 | * into it (see definition), so use a temporary. */ |
680 | BignumInt tmplo = a[i]; |
681 | DIVMOD_WORD(q, r, h, tmplo, m0); |
62ef3d44 |
682 | |
683 | /* Refine our estimate of q by looking at |
684 | h:a[i]:a[i+1] / m0:m1 */ |
685 | t = MUL_WORD(m1, q); |
686 | if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { |
687 | q--; |
688 | t -= m1; |
689 | r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ |
690 | if (r >= (BignumDblInt) m0 && |
691 | t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; |
692 | } |
e5574168 |
693 | } |
694 | |
9400cf6f |
695 | /* Subtract q * m from a[i...] */ |
e5574168 |
696 | c = 0; |
9400cf6f |
697 | for (k = mlen - 1; k >= 0; k--) { |
a47e8bba |
698 | t = MUL_WORD(q, m[k]); |
e5574168 |
699 | t += c; |
62ddb51e |
700 | c = (unsigned)(t >> BIGNUM_INT_BITS); |
a3412f52 |
701 | if ((BignumInt) t > a[i + k]) |
32874aea |
702 | c++; |
a3412f52 |
703 | a[i + k] -= (BignumInt) t; |
e5574168 |
704 | } |
705 | |
706 | /* Add back m in case of borrow */ |
707 | if (c != h) { |
708 | t = 0; |
9400cf6f |
709 | for (k = mlen - 1; k >= 0; k--) { |
e5574168 |
710 | t += m[k]; |
32874aea |
711 | t += a[i + k]; |
a3412f52 |
712 | a[i + k] = (BignumInt) t; |
713 | t = t >> BIGNUM_INT_BITS; |
e5574168 |
714 | } |
32874aea |
715 | q--; |
e5574168 |
716 | } |
32874aea |
717 | if (quot) |
a3412f52 |
718 | internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); |
e5574168 |
719 | } |
720 | } |
721 | |
722 | /* |
09095ac5 |
723 | * Compute (base ^ exp) % mod, the pedestrian way. |
e5574168 |
724 | */ |
09095ac5 |
725 | Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) |
e5574168 |
726 | { |
5a502a19 |
727 | BignumInt *a, *b, *n, *m, *scratch; |
09095ac5 |
728 | int mshift; |
5a502a19 |
729 | int mlen, scratchlen, i, j; |
09095ac5 |
730 | Bignum base, result; |
ed953b91 |
731 | |
732 | /* |
733 | * The most significant word of mod needs to be non-zero. It |
734 | * should already be, but let's make sure. |
735 | */ |
736 | assert(mod[mod[0]] != 0); |
737 | |
738 | /* |
739 | * Make sure the base is smaller than the modulus, by reducing |
740 | * it modulo the modulus if not. |
741 | */ |
742 | base = bigmod(base_in, mod); |
e5574168 |
743 | |
09095ac5 |
744 | /* Allocate m of size mlen, copy mod to m */ |
745 | /* We use big endian internally */ |
746 | mlen = mod[0]; |
747 | m = snewn(mlen, BignumInt); |
748 | for (j = 0; j < mlen; j++) |
749 | m[j] = mod[mod[0] - j]; |
750 | |
751 | /* Shift m left to make msb bit set */ |
752 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
753 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
754 | break; |
755 | if (mshift) { |
756 | for (i = 0; i < mlen - 1; i++) |
757 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
758 | m[mlen - 1] = m[mlen - 1] << mshift; |
759 | } |
760 | |
761 | /* Allocate n of size mlen, copy base to n */ |
762 | n = snewn(mlen, BignumInt); |
763 | i = mlen - base[0]; |
764 | for (j = 0; j < i; j++) |
765 | n[j] = 0; |
766 | for (j = 0; j < (int)base[0]; j++) |
767 | n[i + j] = base[base[0] - j]; |
768 | |
769 | /* Allocate a and b of size 2*mlen. Set a = 1 */ |
770 | a = snewn(2 * mlen, BignumInt); |
771 | b = snewn(2 * mlen, BignumInt); |
772 | for (i = 0; i < 2 * mlen; i++) |
773 | a[i] = 0; |
774 | a[2 * mlen - 1] = 1; |
775 | |
5a502a19 |
776 | /* Scratch space for multiplies */ |
777 | scratchlen = mul_compute_scratch(mlen); |
778 | scratch = snewn(scratchlen, BignumInt); |
779 | |
09095ac5 |
780 | /* Skip leading zero bits of exp. */ |
781 | i = 0; |
782 | j = BIGNUM_INT_BITS-1; |
783 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
784 | j--; |
785 | if (j < 0) { |
786 | i++; |
787 | j = BIGNUM_INT_BITS-1; |
788 | } |
789 | } |
790 | |
791 | /* Main computation */ |
792 | while (i < (int)exp[0]) { |
793 | while (j >= 0) { |
5a502a19 |
794 | internal_mul(a + mlen, a + mlen, b, mlen, scratch); |
09095ac5 |
795 | internal_mod(b, mlen * 2, m, mlen, NULL, 0); |
796 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
5a502a19 |
797 | internal_mul(b + mlen, n, a, mlen, scratch); |
09095ac5 |
798 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
799 | } else { |
800 | BignumInt *t; |
801 | t = a; |
802 | a = b; |
803 | b = t; |
804 | } |
805 | j--; |
806 | } |
807 | i++; |
808 | j = BIGNUM_INT_BITS-1; |
809 | } |
810 | |
811 | /* Fixup result in case the modulus was shifted */ |
812 | if (mshift) { |
813 | for (i = mlen - 1; i < 2 * mlen - 1; i++) |
814 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
815 | a[2 * mlen - 1] = a[2 * mlen - 1] << mshift; |
816 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
817 | for (i = 2 * mlen - 1; i >= mlen; i--) |
818 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); |
819 | } |
820 | |
821 | /* Copy result to buffer */ |
822 | result = newbn(mod[0]); |
823 | for (i = 0; i < mlen; i++) |
824 | result[result[0] - i] = a[i + mlen]; |
825 | while (result[0] > 1 && result[result[0]] == 0) |
826 | result[0]--; |
827 | |
828 | /* Free temporary arrays */ |
16430000 |
829 | smemclr(a, 2 * mlen * sizeof(*a)); |
09095ac5 |
830 | sfree(a); |
16430000 |
831 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
832 | sfree(scratch); |
16430000 |
833 | smemclr(b, 2 * mlen * sizeof(*b)); |
09095ac5 |
834 | sfree(b); |
16430000 |
835 | smemclr(m, mlen * sizeof(*m)); |
09095ac5 |
836 | sfree(m); |
16430000 |
837 | smemclr(n, mlen * sizeof(*n)); |
09095ac5 |
838 | sfree(n); |
839 | |
840 | freebn(base); |
841 | |
842 | return result; |
843 | } |
844 | |
845 | /* |
846 | * Compute (base ^ exp) % mod. Uses the Montgomery multiplication |
847 | * technique where possible, falling back to modpow_simple otherwise. |
848 | */ |
849 | Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) |
850 | { |
5a502a19 |
851 | BignumInt *a, *b, *x, *n, *mninv, *scratch; |
852 | int len, scratchlen, i, j; |
09095ac5 |
853 | Bignum base, base2, r, rn, inv, result; |
854 | |
855 | /* |
856 | * The most significant word of mod needs to be non-zero. It |
857 | * should already be, but let's make sure. |
858 | */ |
859 | assert(mod[mod[0]] != 0); |
860 | |
132c534f |
861 | /* |
862 | * mod had better be odd, or we can't do Montgomery multiplication |
863 | * using a power of two at all. |
864 | */ |
09095ac5 |
865 | if (!(mod[1] & 1)) |
866 | return modpow_simple(base_in, exp, mod); |
867 | |
868 | /* |
869 | * Make sure the base is smaller than the modulus, by reducing |
870 | * it modulo the modulus if not. |
871 | */ |
872 | base = bigmod(base_in, mod); |
e5574168 |
873 | |
132c534f |
874 | /* |
875 | * Compute the inverse of n mod r, for monty_reduce. (In fact we |
876 | * want the inverse of _minus_ n mod r, but we'll sort that out |
877 | * below.) |
878 | */ |
879 | len = mod[0]; |
880 | r = bn_power_2(BIGNUM_INT_BITS * len); |
881 | inv = modinv(mod, r); |
de81309d |
882 | assert(inv); /* cannot fail, since mod is odd and r is a power of 2 */ |
e5574168 |
883 | |
132c534f |
884 | /* |
885 | * Multiply the base by r mod n, to get it into Montgomery |
886 | * representation. |
887 | */ |
888 | base2 = modmul(base, r, mod); |
889 | freebn(base); |
890 | base = base2; |
891 | |
892 | rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ |
893 | |
894 | freebn(r); /* won't need this any more */ |
895 | |
896 | /* |
897 | * Set up internal arrays of the right lengths, in big-endian |
898 | * format, containing the base, the modulus, and the modulus's |
899 | * inverse. |
900 | */ |
901 | n = snewn(len, BignumInt); |
902 | for (j = 0; j < len; j++) |
903 | n[len - 1 - j] = mod[j + 1]; |
904 | |
905 | mninv = snewn(len, BignumInt); |
906 | for (j = 0; j < len; j++) |
08b5c9a2 |
907 | mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0); |
132c534f |
908 | freebn(inv); /* we don't need this copy of it any more */ |
909 | /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ |
910 | x = snewn(len, BignumInt); |
911 | for (j = 0; j < len; j++) |
912 | x[j] = 0; |
913 | internal_sub(x, mninv, mninv, len); |
914 | |
915 | /* x = snewn(len, BignumInt); */ /* already done above */ |
916 | for (j = 0; j < len; j++) |
08b5c9a2 |
917 | x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0); |
132c534f |
918 | freebn(base); /* we don't need this copy of it any more */ |
919 | |
920 | a = snewn(2*len, BignumInt); |
921 | b = snewn(2*len, BignumInt); |
922 | for (j = 0; j < len; j++) |
08b5c9a2 |
923 | a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0); |
132c534f |
924 | freebn(rn); |
925 | |
5a502a19 |
926 | /* Scratch space for multiplies */ |
927 | scratchlen = 3*len + mul_compute_scratch(len); |
928 | scratch = snewn(scratchlen, BignumInt); |
e5574168 |
929 | |
930 | /* Skip leading zero bits of exp. */ |
32874aea |
931 | i = 0; |
a3412f52 |
932 | j = BIGNUM_INT_BITS-1; |
62ddb51e |
933 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
e5574168 |
934 | j--; |
32874aea |
935 | if (j < 0) { |
936 | i++; |
a3412f52 |
937 | j = BIGNUM_INT_BITS-1; |
32874aea |
938 | } |
e5574168 |
939 | } |
940 | |
941 | /* Main computation */ |
62ddb51e |
942 | while (i < (int)exp[0]) { |
e5574168 |
943 | while (j >= 0) { |
5a502a19 |
944 | internal_mul(a + len, a + len, b, len, scratch); |
945 | monty_reduce(b, n, mninv, scratch, len); |
e5574168 |
946 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
5a502a19 |
947 | internal_mul(b + len, x, a, len, scratch); |
948 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 |
949 | } else { |
a3412f52 |
950 | BignumInt *t; |
32874aea |
951 | t = a; |
952 | a = b; |
953 | b = t; |
e5574168 |
954 | } |
955 | j--; |
956 | } |
32874aea |
957 | i++; |
a3412f52 |
958 | j = BIGNUM_INT_BITS-1; |
e5574168 |
959 | } |
960 | |
132c534f |
961 | /* |
962 | * Final monty_reduce to get back from the adjusted Montgomery |
963 | * representation. |
964 | */ |
5a502a19 |
965 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 |
966 | |
967 | /* Copy result to buffer */ |
59600f67 |
968 | result = newbn(mod[0]); |
132c534f |
969 | for (i = 0; i < len; i++) |
970 | result[result[0] - i] = a[i + len]; |
32874aea |
971 | while (result[0] > 1 && result[result[0]] == 0) |
972 | result[0]--; |
e5574168 |
973 | |
974 | /* Free temporary arrays */ |
16430000 |
975 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
976 | sfree(scratch); |
16430000 |
977 | smemclr(a, 2 * len * sizeof(*a)); |
32874aea |
978 | sfree(a); |
16430000 |
979 | smemclr(b, 2 * len * sizeof(*b)); |
32874aea |
980 | sfree(b); |
16430000 |
981 | smemclr(mninv, len * sizeof(*mninv)); |
132c534f |
982 | sfree(mninv); |
16430000 |
983 | smemclr(n, len * sizeof(*n)); |
32874aea |
984 | sfree(n); |
16430000 |
985 | smemclr(x, len * sizeof(*x)); |
132c534f |
986 | sfree(x); |
ed953b91 |
987 | |
59600f67 |
988 | return result; |
e5574168 |
989 | } |
7cca0d81 |
990 | |
991 | /* |
992 | * Compute (p * q) % mod. |
993 | * The most significant word of mod MUST be non-zero. |
994 | * We assume that the result array is the same size as the mod array. |
995 | */ |
59600f67 |
996 | Bignum modmul(Bignum p, Bignum q, Bignum mod) |
7cca0d81 |
997 | { |
5a502a19 |
998 | BignumInt *a, *n, *m, *o, *scratch; |
999 | int mshift, scratchlen; |
80b10571 |
1000 | int pqlen, mlen, rlen, i, j; |
59600f67 |
1001 | Bignum result; |
7cca0d81 |
1002 | |
8bd9144b |
1003 | /* |
1004 | * The most significant word of mod needs to be non-zero. It |
1005 | * should already be, but let's make sure. |
1006 | */ |
1007 | assert(mod[mod[0]] != 0); |
1008 | |
7cca0d81 |
1009 | /* Allocate m of size mlen, copy mod to m */ |
1010 | /* We use big endian internally */ |
1011 | mlen = mod[0]; |
a3412f52 |
1012 | m = snewn(mlen, BignumInt); |
32874aea |
1013 | for (j = 0; j < mlen; j++) |
1014 | m[j] = mod[mod[0] - j]; |
7cca0d81 |
1015 | |
1016 | /* Shift m left to make msb bit set */ |
a3412f52 |
1017 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1018 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
32874aea |
1019 | break; |
7cca0d81 |
1020 | if (mshift) { |
1021 | for (i = 0; i < mlen - 1; i++) |
a3412f52 |
1022 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1023 | m[mlen - 1] = m[mlen - 1] << mshift; |
7cca0d81 |
1024 | } |
1025 | |
1026 | pqlen = (p[0] > q[0] ? p[0] : q[0]); |
1027 | |
5064e5e6 |
1028 | /* |
1029 | * Make sure that we're allowing enough space. The shifting below |
1030 | * will underflow the vectors we allocate if pqlen is too small. |
1031 | */ |
1032 | if (2*pqlen <= mlen) |
1033 | pqlen = mlen/2 + 1; |
1034 | |
7cca0d81 |
1035 | /* Allocate n of size pqlen, copy p to n */ |
a3412f52 |
1036 | n = snewn(pqlen, BignumInt); |
7cca0d81 |
1037 | i = pqlen - p[0]; |
32874aea |
1038 | for (j = 0; j < i; j++) |
1039 | n[j] = 0; |
62ddb51e |
1040 | for (j = 0; j < (int)p[0]; j++) |
32874aea |
1041 | n[i + j] = p[p[0] - j]; |
7cca0d81 |
1042 | |
1043 | /* Allocate o of size pqlen, copy q to o */ |
a3412f52 |
1044 | o = snewn(pqlen, BignumInt); |
7cca0d81 |
1045 | i = pqlen - q[0]; |
32874aea |
1046 | for (j = 0; j < i; j++) |
1047 | o[j] = 0; |
62ddb51e |
1048 | for (j = 0; j < (int)q[0]; j++) |
32874aea |
1049 | o[i + j] = q[q[0] - j]; |
7cca0d81 |
1050 | |
1051 | /* Allocate a of size 2*pqlen for result */ |
a3412f52 |
1052 | a = snewn(2 * pqlen, BignumInt); |
7cca0d81 |
1053 | |
5a502a19 |
1054 | /* Scratch space for multiplies */ |
1055 | scratchlen = mul_compute_scratch(pqlen); |
1056 | scratch = snewn(scratchlen, BignumInt); |
1057 | |
7cca0d81 |
1058 | /* Main computation */ |
5a502a19 |
1059 | internal_mul(n, o, a, pqlen, scratch); |
32874aea |
1060 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
7cca0d81 |
1061 | |
1062 | /* Fixup result in case the modulus was shifted */ |
1063 | if (mshift) { |
32874aea |
1064 | for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++) |
a3412f52 |
1065 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1066 | a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift; |
1067 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
1068 | for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--) |
a3412f52 |
1069 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); |
7cca0d81 |
1070 | } |
1071 | |
1072 | /* Copy result to buffer */ |
32874aea |
1073 | rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); |
80b10571 |
1074 | result = newbn(rlen); |
1075 | for (i = 0; i < rlen; i++) |
32874aea |
1076 | result[result[0] - i] = a[i + 2 * pqlen - rlen]; |
1077 | while (result[0] > 1 && result[result[0]] == 0) |
1078 | result[0]--; |
7cca0d81 |
1079 | |
1080 | /* Free temporary arrays */ |
16430000 |
1081 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
1082 | sfree(scratch); |
16430000 |
1083 | smemclr(a, 2 * pqlen * sizeof(*a)); |
32874aea |
1084 | sfree(a); |
16430000 |
1085 | smemclr(m, mlen * sizeof(*m)); |
32874aea |
1086 | sfree(m); |
16430000 |
1087 | smemclr(n, pqlen * sizeof(*n)); |
32874aea |
1088 | sfree(n); |
16430000 |
1089 | smemclr(o, pqlen * sizeof(*o)); |
32874aea |
1090 | sfree(o); |
59600f67 |
1091 | |
1092 | return result; |
7cca0d81 |
1093 | } |
1094 | |
1095 | /* |
9400cf6f |
1096 | * Compute p % mod. |
1097 | * The most significant word of mod MUST be non-zero. |
1098 | * We assume that the result array is the same size as the mod array. |
5c72ca61 |
1099 | * We optionally write out a quotient if `quotient' is non-NULL. |
1100 | * We can avoid writing out the result if `result' is NULL. |
9400cf6f |
1101 | */ |
f28753ab |
1102 | static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) |
9400cf6f |
1103 | { |
a3412f52 |
1104 | BignumInt *n, *m; |
9400cf6f |
1105 | int mshift; |
1106 | int plen, mlen, i, j; |
1107 | |
8bd9144b |
1108 | /* |
1109 | * The most significant word of mod needs to be non-zero. It |
1110 | * should already be, but let's make sure. |
1111 | */ |
1112 | assert(mod[mod[0]] != 0); |
1113 | |
9400cf6f |
1114 | /* Allocate m of size mlen, copy mod to m */ |
1115 | /* We use big endian internally */ |
1116 | mlen = mod[0]; |
a3412f52 |
1117 | m = snewn(mlen, BignumInt); |
32874aea |
1118 | for (j = 0; j < mlen; j++) |
1119 | m[j] = mod[mod[0] - j]; |
9400cf6f |
1120 | |
1121 | /* Shift m left to make msb bit set */ |
a3412f52 |
1122 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1123 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
32874aea |
1124 | break; |
9400cf6f |
1125 | if (mshift) { |
1126 | for (i = 0; i < mlen - 1; i++) |
a3412f52 |
1127 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1128 | m[mlen - 1] = m[mlen - 1] << mshift; |
9400cf6f |
1129 | } |
1130 | |
1131 | plen = p[0]; |
1132 | /* Ensure plen > mlen */ |
32874aea |
1133 | if (plen <= mlen) |
1134 | plen = mlen + 1; |
9400cf6f |
1135 | |
1136 | /* Allocate n of size plen, copy p to n */ |
a3412f52 |
1137 | n = snewn(plen, BignumInt); |
32874aea |
1138 | for (j = 0; j < plen; j++) |
1139 | n[j] = 0; |
62ddb51e |
1140 | for (j = 1; j <= (int)p[0]; j++) |
32874aea |
1141 | n[plen - j] = p[j]; |
9400cf6f |
1142 | |
1143 | /* Main computation */ |
1144 | internal_mod(n, plen, m, mlen, quotient, mshift); |
1145 | |
1146 | /* Fixup result in case the modulus was shifted */ |
1147 | if (mshift) { |
1148 | for (i = plen - mlen - 1; i < plen - 1; i++) |
a3412f52 |
1149 | n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1150 | n[plen - 1] = n[plen - 1] << mshift; |
9400cf6f |
1151 | internal_mod(n, plen, m, mlen, quotient, 0); |
1152 | for (i = plen - 1; i >= plen - mlen; i--) |
a3412f52 |
1153 | n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); |
9400cf6f |
1154 | } |
1155 | |
1156 | /* Copy result to buffer */ |
5c72ca61 |
1157 | if (result) { |
62ddb51e |
1158 | for (i = 1; i <= (int)result[0]; i++) { |
5c72ca61 |
1159 | int j = plen - i; |
1160 | result[i] = j >= 0 ? n[j] : 0; |
1161 | } |
9400cf6f |
1162 | } |
1163 | |
1164 | /* Free temporary arrays */ |
16430000 |
1165 | smemclr(m, mlen * sizeof(*m)); |
32874aea |
1166 | sfree(m); |
16430000 |
1167 | smemclr(n, plen * sizeof(*n)); |
32874aea |
1168 | sfree(n); |
9400cf6f |
1169 | } |
1170 | |
1171 | /* |
7cca0d81 |
1172 | * Decrement a number. |
1173 | */ |
32874aea |
1174 | void decbn(Bignum bn) |
1175 | { |
7cca0d81 |
1176 | int i = 1; |
62ddb51e |
1177 | while (i < (int)bn[0] && bn[i] == 0) |
a3412f52 |
1178 | bn[i++] = BIGNUM_INT_MASK; |
7cca0d81 |
1179 | bn[i]--; |
1180 | } |
1181 | |
27cd7fc2 |
1182 | Bignum bignum_from_bytes(const unsigned char *data, int nbytes) |
32874aea |
1183 | { |
3709bfe9 |
1184 | Bignum result; |
1185 | int w, i; |
1186 | |
551a4acb |
1187 | assert(nbytes >= 0 && nbytes < INT_MAX/8); |
1188 | |
a3412f52 |
1189 | w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ |
3709bfe9 |
1190 | |
1191 | result = newbn(w); |
32874aea |
1192 | for (i = 1; i <= w; i++) |
1193 | result[i] = 0; |
1194 | for (i = nbytes; i--;) { |
1195 | unsigned char byte = *data++; |
a3412f52 |
1196 | result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); |
3709bfe9 |
1197 | } |
1198 | |
32874aea |
1199 | while (result[0] > 1 && result[result[0]] == 0) |
1200 | result[0]--; |
3709bfe9 |
1201 | return result; |
1202 | } |
1203 | |
7cca0d81 |
1204 | /* |
2e85c969 |
1205 | * Read an SSH-1-format bignum from a data buffer. Return the number |
0016d70b |
1206 | * of bytes consumed, or -1 if there wasn't enough data. |
7cca0d81 |
1207 | */ |
0016d70b |
1208 | int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) |
32874aea |
1209 | { |
27cd7fc2 |
1210 | const unsigned char *p = data; |
7cca0d81 |
1211 | int i; |
1212 | int w, b; |
1213 | |
0016d70b |
1214 | if (len < 2) |
1215 | return -1; |
1216 | |
7cca0d81 |
1217 | w = 0; |
32874aea |
1218 | for (i = 0; i < 2; i++) |
1219 | w = (w << 8) + *p++; |
1220 | b = (w + 7) / 8; /* bits -> bytes */ |
7cca0d81 |
1221 | |
0016d70b |
1222 | if (len < b+2) |
1223 | return -1; |
1224 | |
32874aea |
1225 | if (!result) /* just return length */ |
1226 | return b + 2; |
a52f067e |
1227 | |
3709bfe9 |
1228 | *result = bignum_from_bytes(p, b); |
7cca0d81 |
1229 | |
3709bfe9 |
1230 | return p + b - data; |
7cca0d81 |
1231 | } |
5c58ad2d |
1232 | |
1233 | /* |
2e85c969 |
1234 | * Return the bit count of a bignum, for SSH-1 encoding. |
5c58ad2d |
1235 | */ |
32874aea |
1236 | int bignum_bitcount(Bignum bn) |
1237 | { |
a3412f52 |
1238 | int bitcount = bn[0] * BIGNUM_INT_BITS - 1; |
32874aea |
1239 | while (bitcount >= 0 |
a3412f52 |
1240 | && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; |
5c58ad2d |
1241 | return bitcount + 1; |
1242 | } |
1243 | |
1244 | /* |
2e85c969 |
1245 | * Return the byte length of a bignum when SSH-1 encoded. |
5c58ad2d |
1246 | */ |
32874aea |
1247 | int ssh1_bignum_length(Bignum bn) |
1248 | { |
1249 | return 2 + (bignum_bitcount(bn) + 7) / 8; |
ddecd643 |
1250 | } |
1251 | |
1252 | /* |
2e85c969 |
1253 | * Return the byte length of a bignum when SSH-2 encoded. |
ddecd643 |
1254 | */ |
32874aea |
1255 | int ssh2_bignum_length(Bignum bn) |
1256 | { |
1257 | return 4 + (bignum_bitcount(bn) + 8) / 8; |
5c58ad2d |
1258 | } |
1259 | |
1260 | /* |
1261 | * Return a byte from a bignum; 0 is least significant, etc. |
1262 | */ |
32874aea |
1263 | int bignum_byte(Bignum bn, int i) |
1264 | { |
551a4acb |
1265 | if (i < 0 || i >= (int)(BIGNUM_INT_BYTES * bn[0])) |
32874aea |
1266 | return 0; /* beyond the end */ |
5c58ad2d |
1267 | else |
a3412f52 |
1268 | return (bn[i / BIGNUM_INT_BYTES + 1] >> |
1269 | ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; |
5c58ad2d |
1270 | } |
1271 | |
1272 | /* |
9400cf6f |
1273 | * Return a bit from a bignum; 0 is least significant, etc. |
1274 | */ |
32874aea |
1275 | int bignum_bit(Bignum bn, int i) |
1276 | { |
551a4acb |
1277 | if (i < 0 || i >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea |
1278 | return 0; /* beyond the end */ |
9400cf6f |
1279 | else |
a3412f52 |
1280 | return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; |
9400cf6f |
1281 | } |
1282 | |
1283 | /* |
1284 | * Set a bit in a bignum; 0 is least significant, etc. |
1285 | */ |
32874aea |
1286 | void bignum_set_bit(Bignum bn, int bitnum, int value) |
1287 | { |
551a4acb |
1288 | if (bitnum < 0 || bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea |
1289 | abort(); /* beyond the end */ |
9400cf6f |
1290 | else { |
a3412f52 |
1291 | int v = bitnum / BIGNUM_INT_BITS + 1; |
1292 | int mask = 1 << (bitnum % BIGNUM_INT_BITS); |
32874aea |
1293 | if (value) |
1294 | bn[v] |= mask; |
1295 | else |
1296 | bn[v] &= ~mask; |
9400cf6f |
1297 | } |
1298 | } |
1299 | |
1300 | /* |
2e85c969 |
1301 | * Write a SSH-1-format bignum into a buffer. It is assumed the |
5c58ad2d |
1302 | * buffer is big enough. Returns the number of bytes used. |
1303 | */ |
32874aea |
1304 | int ssh1_write_bignum(void *data, Bignum bn) |
1305 | { |
5c58ad2d |
1306 | unsigned char *p = data; |
1307 | int len = ssh1_bignum_length(bn); |
1308 | int i; |
ddecd643 |
1309 | int bitc = bignum_bitcount(bn); |
5c58ad2d |
1310 | |
1311 | *p++ = (bitc >> 8) & 0xFF; |
32874aea |
1312 | *p++ = (bitc) & 0xFF; |
1313 | for (i = len - 2; i--;) |
1314 | *p++ = bignum_byte(bn, i); |
5c58ad2d |
1315 | return len; |
1316 | } |
9400cf6f |
1317 | |
1318 | /* |
1319 | * Compare two bignums. Returns like strcmp. |
1320 | */ |
32874aea |
1321 | int bignum_cmp(Bignum a, Bignum b) |
1322 | { |
9400cf6f |
1323 | int amax = a[0], bmax = b[0]; |
551a4acb |
1324 | int i; |
1325 | |
434a1d60 |
1326 | /* Annoyingly we have two representations of zero */ |
1327 | if (amax == 1 && a[amax] == 0) |
1328 | amax = 0; |
1329 | if (bmax == 1 && b[bmax] == 0) |
1330 | bmax = 0; |
1331 | |
551a4acb |
1332 | assert(amax == 0 || a[amax] != 0); |
1333 | assert(bmax == 0 || b[bmax] != 0); |
1334 | |
1335 | i = (amax > bmax ? amax : bmax); |
9400cf6f |
1336 | while (i) { |
a3412f52 |
1337 | BignumInt aval = (i > amax ? 0 : a[i]); |
1338 | BignumInt bval = (i > bmax ? 0 : b[i]); |
32874aea |
1339 | if (aval < bval) |
1340 | return -1; |
1341 | if (aval > bval) |
1342 | return +1; |
1343 | i--; |
9400cf6f |
1344 | } |
1345 | return 0; |
1346 | } |
1347 | |
1348 | /* |
1349 | * Right-shift one bignum to form another. |
1350 | */ |
32874aea |
1351 | Bignum bignum_rshift(Bignum a, int shift) |
1352 | { |
9400cf6f |
1353 | Bignum ret; |
1354 | int i, shiftw, shiftb, shiftbb, bits; |
a3412f52 |
1355 | BignumInt ai, ai1; |
9400cf6f |
1356 | |
551a4acb |
1357 | assert(shift >= 0); |
1358 | |
ddecd643 |
1359 | bits = bignum_bitcount(a) - shift; |
a3412f52 |
1360 | ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); |
9400cf6f |
1361 | |
1362 | if (ret) { |
a3412f52 |
1363 | shiftw = shift / BIGNUM_INT_BITS; |
1364 | shiftb = shift % BIGNUM_INT_BITS; |
1365 | shiftbb = BIGNUM_INT_BITS - shiftb; |
32874aea |
1366 | |
1367 | ai1 = a[shiftw + 1]; |
62ddb51e |
1368 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea |
1369 | ai = ai1; |
62ddb51e |
1370 | ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); |
a3412f52 |
1371 | ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; |
32874aea |
1372 | } |
9400cf6f |
1373 | } |
1374 | |
1375 | return ret; |
1376 | } |
1377 | |
1378 | /* |
1379 | * Non-modular multiplication and addition. |
1380 | */ |
32874aea |
1381 | Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) |
1382 | { |
9400cf6f |
1383 | int alen = a[0], blen = b[0]; |
1384 | int mlen = (alen > blen ? alen : blen); |
1385 | int rlen, i, maxspot; |
5a502a19 |
1386 | int wslen; |
a3412f52 |
1387 | BignumInt *workspace; |
9400cf6f |
1388 | Bignum ret; |
1389 | |
5a502a19 |
1390 | /* mlen space for a, mlen space for b, 2*mlen for result, |
1391 | * plus scratch space for multiplication */ |
1392 | wslen = mlen * 4 + mul_compute_scratch(mlen); |
1393 | workspace = snewn(wslen, BignumInt); |
9400cf6f |
1394 | for (i = 0; i < mlen; i++) { |
62ddb51e |
1395 | workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0); |
1396 | workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0); |
9400cf6f |
1397 | } |
1398 | |
32874aea |
1399 | internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, |
5a502a19 |
1400 | workspace + 2 * mlen, mlen, workspace + 4 * mlen); |
9400cf6f |
1401 | |
1402 | /* now just copy the result back */ |
1403 | rlen = alen + blen + 1; |
62ddb51e |
1404 | if (addend && rlen <= (int)addend[0]) |
32874aea |
1405 | rlen = addend[0] + 1; |
9400cf6f |
1406 | ret = newbn(rlen); |
1407 | maxspot = 0; |
62ddb51e |
1408 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea |
1409 | ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); |
1410 | if (ret[i] != 0) |
1411 | maxspot = i; |
9400cf6f |
1412 | } |
1413 | ret[0] = maxspot; |
1414 | |
1415 | /* now add in the addend, if any */ |
1416 | if (addend) { |
a3412f52 |
1417 | BignumDblInt carry = 0; |
32874aea |
1418 | for (i = 1; i <= rlen; i++) { |
62ddb51e |
1419 | carry += (i <= (int)ret[0] ? ret[i] : 0); |
1420 | carry += (i <= (int)addend[0] ? addend[i] : 0); |
a3412f52 |
1421 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1422 | carry >>= BIGNUM_INT_BITS; |
32874aea |
1423 | if (ret[i] != 0 && i > maxspot) |
1424 | maxspot = i; |
1425 | } |
9400cf6f |
1426 | } |
1427 | ret[0] = maxspot; |
1428 | |
16430000 |
1429 | smemclr(workspace, wslen * sizeof(*workspace)); |
c523f55f |
1430 | sfree(workspace); |
9400cf6f |
1431 | return ret; |
1432 | } |
1433 | |
1434 | /* |
1435 | * Non-modular multiplication. |
1436 | */ |
32874aea |
1437 | Bignum bigmul(Bignum a, Bignum b) |
1438 | { |
9400cf6f |
1439 | return bigmuladd(a, b, NULL); |
1440 | } |
1441 | |
1442 | /* |
d737853b |
1443 | * Simple addition. |
1444 | */ |
1445 | Bignum bigadd(Bignum a, Bignum b) |
1446 | { |
1447 | int alen = a[0], blen = b[0]; |
1448 | int rlen = (alen > blen ? alen : blen) + 1; |
1449 | int i, maxspot; |
1450 | Bignum ret; |
1451 | BignumDblInt carry; |
1452 | |
1453 | ret = newbn(rlen); |
1454 | |
1455 | carry = 0; |
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] : 0); |
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 | return ret; |
1468 | } |
1469 | |
1470 | /* |
1471 | * Subtraction. Returns a-b, or NULL if the result would come out |
1472 | * negative (recall that this entire bignum module only handles |
1473 | * positive numbers). |
1474 | */ |
1475 | Bignum bigsub(Bignum a, Bignum b) |
1476 | { |
1477 | int alen = a[0], blen = b[0]; |
1478 | int rlen = (alen > blen ? alen : blen); |
1479 | int i, maxspot; |
1480 | Bignum ret; |
1481 | BignumDblInt carry; |
1482 | |
1483 | ret = newbn(rlen); |
1484 | |
1485 | carry = 1; |
1486 | maxspot = 0; |
1487 | for (i = 1; i <= rlen; i++) { |
1488 | carry += (i <= (int)a[0] ? a[i] : 0); |
1489 | carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); |
1490 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1491 | carry >>= BIGNUM_INT_BITS; |
1492 | if (ret[i] != 0 && i > maxspot) |
1493 | maxspot = i; |
1494 | } |
1495 | ret[0] = maxspot; |
1496 | |
1497 | if (!carry) { |
1498 | freebn(ret); |
1499 | return NULL; |
1500 | } |
1501 | |
1502 | return ret; |
1503 | } |
1504 | |
1505 | /* |
3709bfe9 |
1506 | * Create a bignum which is the bitmask covering another one. That |
1507 | * is, the smallest integer which is >= N and is also one less than |
1508 | * a power of two. |
1509 | */ |
32874aea |
1510 | Bignum bignum_bitmask(Bignum n) |
1511 | { |
3709bfe9 |
1512 | Bignum ret = copybn(n); |
1513 | int i; |
a3412f52 |
1514 | BignumInt j; |
3709bfe9 |
1515 | |
1516 | i = ret[0]; |
1517 | while (n[i] == 0 && i > 0) |
32874aea |
1518 | i--; |
3709bfe9 |
1519 | if (i <= 0) |
32874aea |
1520 | return ret; /* input was zero */ |
3709bfe9 |
1521 | j = 1; |
1522 | while (j < n[i]) |
32874aea |
1523 | j = 2 * j + 1; |
3709bfe9 |
1524 | ret[i] = j; |
1525 | while (--i > 0) |
a3412f52 |
1526 | ret[i] = BIGNUM_INT_MASK; |
3709bfe9 |
1527 | return ret; |
1528 | } |
1529 | |
1530 | /* |
5c72ca61 |
1531 | * Convert a (max 32-bit) long into a bignum. |
9400cf6f |
1532 | */ |
a3412f52 |
1533 | Bignum bignum_from_long(unsigned long nn) |
32874aea |
1534 | { |
9400cf6f |
1535 | Bignum ret; |
a3412f52 |
1536 | BignumDblInt n = nn; |
9400cf6f |
1537 | |
5c72ca61 |
1538 | ret = newbn(3); |
a3412f52 |
1539 | ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); |
1540 | ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); |
5c72ca61 |
1541 | ret[3] = 0; |
1542 | ret[0] = (ret[2] ? 2 : 1); |
32874aea |
1543 | return ret; |
9400cf6f |
1544 | } |
1545 | |
1546 | /* |
1547 | * Add a long to a bignum. |
1548 | */ |
a3412f52 |
1549 | Bignum bignum_add_long(Bignum number, unsigned long addendx) |
32874aea |
1550 | { |
1551 | Bignum ret = newbn(number[0] + 1); |
9400cf6f |
1552 | int i, maxspot = 0; |
a3412f52 |
1553 | BignumDblInt carry = 0, addend = addendx; |
9400cf6f |
1554 | |
62ddb51e |
1555 | for (i = 1; i <= (int)ret[0]; i++) { |
a3412f52 |
1556 | carry += addend & BIGNUM_INT_MASK; |
62ddb51e |
1557 | carry += (i <= (int)number[0] ? number[i] : 0); |
a3412f52 |
1558 | addend >>= BIGNUM_INT_BITS; |
1559 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1560 | carry >>= BIGNUM_INT_BITS; |
32874aea |
1561 | if (ret[i] != 0) |
1562 | maxspot = i; |
9400cf6f |
1563 | } |
1564 | ret[0] = maxspot; |
1565 | return ret; |
1566 | } |
1567 | |
1568 | /* |
1569 | * Compute the residue of a bignum, modulo a (max 16-bit) short. |
1570 | */ |
32874aea |
1571 | unsigned short bignum_mod_short(Bignum number, unsigned short modulus) |
1572 | { |
a3412f52 |
1573 | BignumDblInt mod, r; |
9400cf6f |
1574 | int i; |
1575 | |
1576 | r = 0; |
1577 | mod = modulus; |
1578 | for (i = number[0]; i > 0; i--) |
736cc6d1 |
1579 | r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; |
6e522441 |
1580 | return (unsigned short) r; |
9400cf6f |
1581 | } |
1582 | |
a3412f52 |
1583 | #ifdef DEBUG |
32874aea |
1584 | void diagbn(char *prefix, Bignum md) |
1585 | { |
9400cf6f |
1586 | int i, nibbles, morenibbles; |
1587 | static const char hex[] = "0123456789ABCDEF"; |
1588 | |
5c72ca61 |
1589 | debug(("%s0x", prefix ? prefix : "")); |
9400cf6f |
1590 | |
32874aea |
1591 | nibbles = (3 + bignum_bitcount(md)) / 4; |
1592 | if (nibbles < 1) |
1593 | nibbles = 1; |
1594 | morenibbles = 4 * md[0] - nibbles; |
1595 | for (i = 0; i < morenibbles; i++) |
5c72ca61 |
1596 | debug(("-")); |
32874aea |
1597 | for (i = nibbles; i--;) |
5c72ca61 |
1598 | debug(("%c", |
1599 | hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF])); |
9400cf6f |
1600 | |
32874aea |
1601 | if (prefix) |
5c72ca61 |
1602 | debug(("\n")); |
1603 | } |
f28753ab |
1604 | #endif |
5c72ca61 |
1605 | |
1606 | /* |
1607 | * Simple division. |
1608 | */ |
1609 | Bignum bigdiv(Bignum a, Bignum b) |
1610 | { |
1611 | Bignum q = newbn(a[0]); |
1612 | bigdivmod(a, b, NULL, q); |
1613 | return q; |
1614 | } |
1615 | |
1616 | /* |
1617 | * Simple remainder. |
1618 | */ |
1619 | Bignum bigmod(Bignum a, Bignum b) |
1620 | { |
1621 | Bignum r = newbn(b[0]); |
1622 | bigdivmod(a, b, r, NULL); |
1623 | return r; |
9400cf6f |
1624 | } |
1625 | |
1626 | /* |
1627 | * Greatest common divisor. |
1628 | */ |
32874aea |
1629 | Bignum biggcd(Bignum av, Bignum bv) |
1630 | { |
9400cf6f |
1631 | Bignum a = copybn(av); |
1632 | Bignum b = copybn(bv); |
1633 | |
9400cf6f |
1634 | while (bignum_cmp(b, Zero) != 0) { |
32874aea |
1635 | Bignum t = newbn(b[0]); |
5c72ca61 |
1636 | bigdivmod(a, b, t, NULL); |
32874aea |
1637 | while (t[0] > 1 && t[t[0]] == 0) |
1638 | t[0]--; |
1639 | freebn(a); |
1640 | a = b; |
1641 | b = t; |
9400cf6f |
1642 | } |
1643 | |
1644 | freebn(b); |
1645 | return a; |
1646 | } |
1647 | |
1648 | /* |
1649 | * Modular inverse, using Euclid's extended algorithm. |
1650 | */ |
32874aea |
1651 | Bignum modinv(Bignum number, Bignum modulus) |
1652 | { |
9400cf6f |
1653 | Bignum a = copybn(modulus); |
1654 | Bignum b = copybn(number); |
1655 | Bignum xp = copybn(Zero); |
1656 | Bignum x = copybn(One); |
1657 | int sign = +1; |
1658 | |
8bd9144b |
1659 | assert(number[number[0]] != 0); |
1660 | assert(modulus[modulus[0]] != 0); |
1661 | |
9400cf6f |
1662 | while (bignum_cmp(b, One) != 0) { |
de81309d |
1663 | Bignum t, q; |
1664 | |
1665 | if (bignum_cmp(b, Zero) == 0) { |
1666 | /* |
1667 | * Found a common factor between the inputs, so we cannot |
1668 | * return a modular inverse at all. |
1669 | */ |
c6456dca |
1670 | freebn(b); |
1671 | freebn(a); |
1672 | freebn(xp); |
1673 | freebn(x); |
de81309d |
1674 | return NULL; |
1675 | } |
1676 | |
1677 | t = newbn(b[0]); |
1678 | q = newbn(a[0]); |
5c72ca61 |
1679 | bigdivmod(a, b, t, q); |
32874aea |
1680 | while (t[0] > 1 && t[t[0]] == 0) |
1681 | t[0]--; |
1682 | freebn(a); |
1683 | a = b; |
1684 | b = t; |
1685 | t = xp; |
1686 | xp = x; |
1687 | x = bigmuladd(q, xp, t); |
1688 | sign = -sign; |
1689 | freebn(t); |
75374b2f |
1690 | freebn(q); |
9400cf6f |
1691 | } |
1692 | |
1693 | freebn(b); |
1694 | freebn(a); |
1695 | freebn(xp); |
1696 | |
1697 | /* now we know that sign * x == 1, and that x < modulus */ |
1698 | if (sign < 0) { |
32874aea |
1699 | /* set a new x to be modulus - x */ |
1700 | Bignum newx = newbn(modulus[0]); |
a3412f52 |
1701 | BignumInt carry = 0; |
32874aea |
1702 | int maxspot = 1; |
1703 | int i; |
1704 | |
62ddb51e |
1705 | for (i = 1; i <= (int)newx[0]; i++) { |
1706 | BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); |
1707 | BignumInt bword = (i <= (int)x[0] ? x[i] : 0); |
32874aea |
1708 | newx[i] = aword - bword - carry; |
1709 | bword = ~bword; |
1710 | carry = carry ? (newx[i] >= bword) : (newx[i] > bword); |
1711 | if (newx[i] != 0) |
1712 | maxspot = i; |
1713 | } |
1714 | newx[0] = maxspot; |
1715 | freebn(x); |
1716 | x = newx; |
9400cf6f |
1717 | } |
1718 | |
1719 | /* and return. */ |
1720 | return x; |
1721 | } |
6e522441 |
1722 | |
1723 | /* |
1724 | * Render a bignum into decimal. Return a malloced string holding |
1725 | * the decimal representation. |
1726 | */ |
32874aea |
1727 | char *bignum_decimal(Bignum x) |
1728 | { |
6e522441 |
1729 | int ndigits, ndigit; |
1730 | int i, iszero; |
a3412f52 |
1731 | BignumDblInt carry; |
6e522441 |
1732 | char *ret; |
a3412f52 |
1733 | BignumInt *workspace; |
6e522441 |
1734 | |
1735 | /* |
1736 | * First, estimate the number of digits. Since log(10)/log(2) |
1737 | * is just greater than 93/28 (the joys of continued fraction |
1738 | * approximations...) we know that for every 93 bits, we need |
1739 | * at most 28 digits. This will tell us how much to malloc. |
1740 | * |
1741 | * Formally: if x has i bits, that means x is strictly less |
1742 | * than 2^i. Since 2 is less than 10^(28/93), this is less than |
1743 | * 10^(28i/93). We need an integer power of ten, so we must |
1744 | * round up (rounding down might make it less than x again). |
1745 | * Therefore if we multiply the bit count by 28/93, rounding |
1746 | * up, we will have enough digits. |
74c79ce8 |
1747 | * |
1748 | * i=0 (i.e., x=0) is an irritating special case. |
6e522441 |
1749 | */ |
ddecd643 |
1750 | i = bignum_bitcount(x); |
74c79ce8 |
1751 | if (!i) |
1752 | ndigits = 1; /* x = 0 */ |
1753 | else |
1754 | ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ |
32874aea |
1755 | ndigits++; /* allow for trailing \0 */ |
3d88e64d |
1756 | ret = snewn(ndigits, char); |
6e522441 |
1757 | |
1758 | /* |
1759 | * Now allocate some workspace to hold the binary form as we |
1760 | * repeatedly divide it by ten. Initialise this to the |
1761 | * big-endian form of the number. |
1762 | */ |
a3412f52 |
1763 | workspace = snewn(x[0], BignumInt); |
62ddb51e |
1764 | for (i = 0; i < (int)x[0]; i++) |
32874aea |
1765 | workspace[i] = x[x[0] - i]; |
6e522441 |
1766 | |
1767 | /* |
1768 | * Next, write the decimal number starting with the last digit. |
1769 | * We use ordinary short division, dividing 10 into the |
1770 | * workspace. |
1771 | */ |
32874aea |
1772 | ndigit = ndigits - 1; |
6e522441 |
1773 | ret[ndigit] = '\0'; |
1774 | do { |
32874aea |
1775 | iszero = 1; |
1776 | carry = 0; |
62ddb51e |
1777 | for (i = 0; i < (int)x[0]; i++) { |
a3412f52 |
1778 | carry = (carry << BIGNUM_INT_BITS) + workspace[i]; |
1779 | workspace[i] = (BignumInt) (carry / 10); |
32874aea |
1780 | if (workspace[i]) |
1781 | iszero = 0; |
1782 | carry %= 10; |
1783 | } |
1784 | ret[--ndigit] = (char) (carry + '0'); |
6e522441 |
1785 | } while (!iszero); |
1786 | |
1787 | /* |
1788 | * There's a chance we've fallen short of the start of the |
1789 | * string. Correct if so. |
1790 | */ |
1791 | if (ndigit > 0) |
32874aea |
1792 | memmove(ret, ret + ndigit, ndigits - ndigit); |
6e522441 |
1793 | |
1794 | /* |
1795 | * Done. |
1796 | */ |
16430000 |
1797 | smemclr(workspace, x[0] * sizeof(*workspace)); |
c523f55f |
1798 | sfree(workspace); |
6e522441 |
1799 | return ret; |
1800 | } |
f3c29e34 |
1801 | |
1802 | #ifdef TESTBN |
1803 | |
1804 | #include <stdio.h> |
1805 | #include <stdlib.h> |
1806 | #include <ctype.h> |
1807 | |
1808 | /* |
4800a5e5 |
1809 | * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset |
f84f1e46 |
1810 | * |
1811 | * Then feed to this program's standard input the output of |
1812 | * testdata/bignum.py . |
f3c29e34 |
1813 | */ |
1814 | |
1815 | void modalfatalbox(char *p, ...) |
1816 | { |
1817 | va_list ap; |
1818 | fprintf(stderr, "FATAL ERROR: "); |
1819 | va_start(ap, p); |
1820 | vfprintf(stderr, p, ap); |
1821 | va_end(ap); |
1822 | fputc('\n', stderr); |
1823 | exit(1); |
1824 | } |
1825 | |
1826 | #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) |
1827 | |
1828 | int main(int argc, char **argv) |
1829 | { |
1830 | char *buf; |
1831 | int line = 0; |
1832 | int passes = 0, fails = 0; |
1833 | |
1834 | while ((buf = fgetline(stdin)) != NULL) { |
1835 | int maxlen = strlen(buf); |
1836 | unsigned char *data = snewn(maxlen, unsigned char); |
f84f1e46 |
1837 | unsigned char *ptrs[5], *q; |
f3c29e34 |
1838 | int ptrnum; |
1839 | char *bufp = buf; |
1840 | |
1841 | line++; |
1842 | |
1843 | q = data; |
1844 | ptrnum = 0; |
1845 | |
f84f1e46 |
1846 | while (*bufp && !isspace((unsigned char)*bufp)) |
1847 | bufp++; |
1848 | if (bufp) |
1849 | *bufp++ = '\0'; |
1850 | |
f3c29e34 |
1851 | while (*bufp) { |
1852 | char *start, *end; |
1853 | int i; |
1854 | |
1855 | while (*bufp && !isxdigit((unsigned char)*bufp)) |
1856 | bufp++; |
1857 | start = bufp; |
1858 | |
1859 | if (!*bufp) |
1860 | break; |
1861 | |
1862 | while (*bufp && isxdigit((unsigned char)*bufp)) |
1863 | bufp++; |
1864 | end = bufp; |
1865 | |
1866 | if (ptrnum >= lenof(ptrs)) |
1867 | break; |
1868 | ptrs[ptrnum++] = q; |
1869 | |
1870 | for (i = -((end - start) & 1); i < end-start; i += 2) { |
1871 | unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); |
1872 | val = val * 16 + fromxdigit(start[i+1]); |
1873 | *q++ = val; |
1874 | } |
1875 | |
1876 | ptrs[ptrnum] = q; |
1877 | } |
1878 | |
f84f1e46 |
1879 | if (!strcmp(buf, "mul")) { |
1880 | Bignum a, b, c, p; |
1881 | |
1882 | if (ptrnum != 3) { |
f6939e2b |
1883 | printf("%d: mul with %d parameters, expected 3\n", line, ptrnum); |
f84f1e46 |
1884 | exit(1); |
1885 | } |
1886 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1887 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1888 | c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1889 | p = bigmul(a, b); |
f3c29e34 |
1890 | |
1891 | if (bignum_cmp(c, p) == 0) { |
1892 | passes++; |
1893 | } else { |
1894 | char *as = bignum_decimal(a); |
1895 | char *bs = bignum_decimal(b); |
1896 | char *cs = bignum_decimal(c); |
1897 | char *ps = bignum_decimal(p); |
1898 | |
1899 | printf("%d: fail: %s * %s gave %s expected %s\n", |
1900 | line, as, bs, ps, cs); |
1901 | fails++; |
1902 | |
1903 | sfree(as); |
1904 | sfree(bs); |
1905 | sfree(cs); |
1906 | sfree(ps); |
1907 | } |
1908 | freebn(a); |
1909 | freebn(b); |
1910 | freebn(c); |
1911 | freebn(p); |
5064e5e6 |
1912 | } else if (!strcmp(buf, "modmul")) { |
1913 | Bignum a, b, m, c, p; |
1914 | |
1915 | if (ptrnum != 4) { |
1916 | printf("%d: modmul with %d parameters, expected 4\n", |
1917 | line, ptrnum); |
1918 | exit(1); |
1919 | } |
1920 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1921 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1922 | m = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1923 | c = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); |
1924 | p = modmul(a, b, m); |
1925 | |
1926 | if (bignum_cmp(c, p) == 0) { |
1927 | passes++; |
1928 | } else { |
1929 | char *as = bignum_decimal(a); |
1930 | char *bs = bignum_decimal(b); |
1931 | char *ms = bignum_decimal(m); |
1932 | char *cs = bignum_decimal(c); |
1933 | char *ps = bignum_decimal(p); |
1934 | |
1935 | printf("%d: fail: %s * %s mod %s gave %s expected %s\n", |
1936 | line, as, bs, ms, ps, cs); |
1937 | fails++; |
1938 | |
1939 | sfree(as); |
1940 | sfree(bs); |
1941 | sfree(ms); |
1942 | sfree(cs); |
1943 | sfree(ps); |
1944 | } |
1945 | freebn(a); |
1946 | freebn(b); |
1947 | freebn(m); |
1948 | freebn(c); |
1949 | freebn(p); |
f84f1e46 |
1950 | } else if (!strcmp(buf, "pow")) { |
1951 | Bignum base, expt, modulus, expected, answer; |
1952 | |
1953 | if (ptrnum != 4) { |
f6939e2b |
1954 | printf("%d: mul with %d parameters, expected 4\n", line, ptrnum); |
f84f1e46 |
1955 | exit(1); |
1956 | } |
1957 | |
1958 | base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1959 | expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1960 | modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1961 | expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); |
1962 | answer = modpow(base, expt, modulus); |
1963 | |
1964 | if (bignum_cmp(expected, answer) == 0) { |
1965 | passes++; |
1966 | } else { |
1967 | char *as = bignum_decimal(base); |
1968 | char *bs = bignum_decimal(expt); |
1969 | char *cs = bignum_decimal(modulus); |
1970 | char *ds = bignum_decimal(answer); |
1971 | char *ps = bignum_decimal(expected); |
1972 | |
1973 | printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", |
1974 | line, as, bs, cs, ds, ps); |
1975 | fails++; |
1976 | |
1977 | sfree(as); |
1978 | sfree(bs); |
1979 | sfree(cs); |
1980 | sfree(ds); |
1981 | sfree(ps); |
1982 | } |
1983 | freebn(base); |
1984 | freebn(expt); |
1985 | freebn(modulus); |
1986 | freebn(expected); |
1987 | freebn(answer); |
1988 | } else { |
1989 | printf("%d: unrecognised test keyword: '%s'\n", line, buf); |
1990 | exit(1); |
f3c29e34 |
1991 | } |
f84f1e46 |
1992 | |
f3c29e34 |
1993 | sfree(buf); |
1994 | sfree(data); |
1995 | } |
1996 | |
1997 | printf("passed %d failed %d total %d\n", passes, fails, passes+fails); |
1998 | return fails != 0; |
1999 | } |
2000 | |
2001 | #endif |