e5574168 |
1 | /* |
2 | * Bignum routines for RSA and DH and stuff. |
3 | */ |
4 | |
5 | #include <stdio.h> |
ed953b91 |
6 | #include <assert.h> |
e5574168 |
7 | #include <stdlib.h> |
8 | #include <string.h> |
9 | |
5c72ca61 |
10 | #include "misc.h" |
98ba26b9 |
11 | |
819a22b3 |
12 | /* |
13 | * Usage notes: |
14 | * * Do not call the DIVMOD_WORD macro with expressions such as array |
15 | * subscripts, as some implementations object to this (see below). |
16 | * * Note that none of the division methods below will cope if the |
17 | * quotient won't fit into BIGNUM_INT_BITS. Callers should be careful |
18 | * to avoid this case. |
19 | * If this condition occurs, in the case of the x86 DIV instruction, |
20 | * an overflow exception will occur, which (according to a correspondent) |
21 | * will manifest on Windows as something like |
22 | * 0xC0000095: Integer overflow |
23 | * The C variant won't give the right answer, either. |
24 | */ |
25 | |
a3412f52 |
26 | #if defined __GNUC__ && defined __i386__ |
27 | typedef unsigned long BignumInt; |
28 | typedef unsigned long long BignumDblInt; |
29 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
30 | #define BIGNUM_TOP_BIT 0x80000000UL |
31 | #define BIGNUM_INT_BITS 32 |
32 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
a47e8bba |
33 | #define DIVMOD_WORD(q, r, hi, lo, w) \ |
34 | __asm__("div %2" : \ |
35 | "=d" (r), "=a" (q) : \ |
36 | "r" (w), "d" (hi), "a" (lo)) |
036eddfb |
37 | #elif defined _MSC_VER && defined _M_IX86 |
38 | typedef unsigned __int32 BignumInt; |
39 | typedef unsigned __int64 BignumDblInt; |
40 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
41 | #define BIGNUM_TOP_BIT 0x80000000UL |
42 | #define BIGNUM_INT_BITS 32 |
43 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
819a22b3 |
44 | /* Note: MASM interprets array subscripts in the macro arguments as |
45 | * assembler syntax, which gives the wrong answer. Don't supply them. |
46 | * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */ |
036eddfb |
47 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
819a22b3 |
48 | __asm mov edx, hi \ |
49 | __asm mov eax, lo \ |
50 | __asm div w \ |
51 | __asm mov r, edx \ |
52 | __asm mov q, eax \ |
53 | } while(0) |
32e51f76 |
54 | #elif defined _LP64 |
55 | /* 64-bit architectures can do 32x32->64 chunks at a time */ |
56 | typedef unsigned int BignumInt; |
57 | typedef unsigned long BignumDblInt; |
58 | #define BIGNUM_INT_MASK 0xFFFFFFFFU |
59 | #define BIGNUM_TOP_BIT 0x80000000U |
60 | #define BIGNUM_INT_BITS 32 |
61 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
62 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
63 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
64 | q = n / w; \ |
65 | r = n % w; \ |
66 | } while (0) |
67 | #elif defined _LLP64 |
68 | /* 64-bit architectures in which unsigned long is 32 bits, not 64 */ |
69 | typedef unsigned long BignumInt; |
70 | typedef unsigned long long BignumDblInt; |
71 | #define BIGNUM_INT_MASK 0xFFFFFFFFUL |
72 | #define BIGNUM_TOP_BIT 0x80000000UL |
73 | #define BIGNUM_INT_BITS 32 |
74 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
75 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
76 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
77 | q = n / w; \ |
78 | r = n % w; \ |
79 | } while (0) |
a3412f52 |
80 | #else |
32e51f76 |
81 | /* Fallback for all other cases */ |
a3412f52 |
82 | typedef unsigned short BignumInt; |
83 | typedef unsigned long BignumDblInt; |
84 | #define BIGNUM_INT_MASK 0xFFFFU |
85 | #define BIGNUM_TOP_BIT 0x8000U |
86 | #define BIGNUM_INT_BITS 16 |
87 | #define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) |
a47e8bba |
88 | #define DIVMOD_WORD(q, r, hi, lo, w) do { \ |
89 | BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ |
90 | q = n / w; \ |
91 | r = n % w; \ |
92 | } while (0) |
a3412f52 |
93 | #endif |
94 | |
95 | #define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8) |
96 | |
3709bfe9 |
97 | #define BIGNUM_INTERNAL |
a3412f52 |
98 | typedef BignumInt *Bignum; |
3709bfe9 |
99 | |
e5574168 |
100 | #include "ssh.h" |
101 | |
a3412f52 |
102 | BignumInt bnZero[1] = { 0 }; |
103 | BignumInt bnOne[2] = { 1, 1 }; |
e5574168 |
104 | |
7d6ee6ff |
105 | /* |
a3412f52 |
106 | * The Bignum format is an array of `BignumInt'. The first |
7d6ee6ff |
107 | * element of the array counts the remaining elements. The |
a3412f52 |
108 | * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_ |
7d6ee6ff |
109 | * significant digit first. (So it's trivial to extract the bit |
110 | * with value 2^n for any n.) |
111 | * |
112 | * All Bignums in this module are positive. Negative numbers must |
113 | * be dealt with outside it. |
114 | * |
115 | * INVARIANT: the most significant word of any Bignum must be |
116 | * nonzero. |
117 | */ |
118 | |
7cca0d81 |
119 | Bignum Zero = bnZero, One = bnOne; |
e5574168 |
120 | |
32874aea |
121 | static Bignum newbn(int length) |
122 | { |
a3412f52 |
123 | Bignum b = snewn(length + 1, BignumInt); |
e5574168 |
124 | if (!b) |
125 | abort(); /* FIXME */ |
32874aea |
126 | memset(b, 0, (length + 1) * sizeof(*b)); |
e5574168 |
127 | b[0] = length; |
128 | return b; |
129 | } |
130 | |
32874aea |
131 | void bn_restore_invariant(Bignum b) |
132 | { |
133 | while (b[0] > 1 && b[b[0]] == 0) |
134 | b[0]--; |
3709bfe9 |
135 | } |
136 | |
32874aea |
137 | Bignum copybn(Bignum orig) |
138 | { |
a3412f52 |
139 | Bignum b = snewn(orig[0] + 1, BignumInt); |
7cca0d81 |
140 | if (!b) |
141 | abort(); /* FIXME */ |
32874aea |
142 | memcpy(b, orig, (orig[0] + 1) * sizeof(*b)); |
7cca0d81 |
143 | return b; |
144 | } |
145 | |
32874aea |
146 | void freebn(Bignum b) |
147 | { |
e5574168 |
148 | /* |
149 | * Burn the evidence, just in case. |
150 | */ |
dfb88efd |
151 | smemclr(b, sizeof(b[0]) * (b[0] + 1)); |
dcbde236 |
152 | sfree(b); |
e5574168 |
153 | } |
154 | |
32874aea |
155 | Bignum bn_power_2(int n) |
156 | { |
a3412f52 |
157 | Bignum ret = newbn(n / BIGNUM_INT_BITS + 1); |
3709bfe9 |
158 | bignum_set_bit(ret, n, 1); |
159 | return ret; |
160 | } |
161 | |
e5574168 |
162 | /* |
0c431b2f |
163 | * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all |
164 | * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried |
165 | * off the top. |
166 | */ |
167 | static BignumInt internal_add(const BignumInt *a, const BignumInt *b, |
168 | BignumInt *c, int len) |
169 | { |
170 | int i; |
171 | BignumDblInt carry = 0; |
172 | |
173 | for (i = len-1; i >= 0; i--) { |
174 | carry += (BignumDblInt)a[i] + b[i]; |
175 | c[i] = (BignumInt)carry; |
176 | carry >>= BIGNUM_INT_BITS; |
177 | } |
178 | |
179 | return (BignumInt)carry; |
180 | } |
181 | |
182 | /* |
183 | * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are |
184 | * all big-endian arrays of 'len' BignumInts. Any borrow from the top |
185 | * is ignored. |
186 | */ |
187 | static void internal_sub(const BignumInt *a, const BignumInt *b, |
188 | BignumInt *c, int len) |
189 | { |
190 | int i; |
191 | BignumDblInt carry = 1; |
192 | |
193 | for (i = len-1; i >= 0; i--) { |
194 | carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK); |
195 | c[i] = (BignumInt)carry; |
196 | carry >>= BIGNUM_INT_BITS; |
197 | } |
198 | } |
199 | |
200 | /* |
e5574168 |
201 | * Compute c = a * b. |
202 | * Input is in the first len words of a and b. |
203 | * Result is returned in the first 2*len words of c. |
5a502a19 |
204 | * |
205 | * 'scratch' must point to an array of BignumInt of size at least |
206 | * mul_compute_scratch(len). (This covers the needs of internal_mul |
207 | * and all its recursive calls to itself.) |
e5574168 |
208 | */ |
0c431b2f |
209 | #define KARATSUBA_THRESHOLD 50 |
5a502a19 |
210 | static int mul_compute_scratch(int len) |
211 | { |
212 | int ret = 0; |
213 | while (len > KARATSUBA_THRESHOLD) { |
214 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
215 | int midlen = botlen + 1; |
216 | ret += 4*midlen; |
217 | len = midlen; |
218 | } |
219 | return ret; |
220 | } |
132c534f |
221 | static void internal_mul(const BignumInt *a, const BignumInt *b, |
5a502a19 |
222 | BignumInt *c, int len, BignumInt *scratch) |
e5574168 |
223 | { |
0c431b2f |
224 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 |
225 | int i; |
0c431b2f |
226 | |
227 | /* |
228 | * Karatsuba divide-and-conquer algorithm. Cut each input in |
229 | * half, so that it's expressed as two big 'digits' in a giant |
230 | * base D: |
231 | * |
232 | * a = a_1 D + a_0 |
233 | * b = b_1 D + b_0 |
234 | * |
235 | * Then the product is of course |
236 | * |
237 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 |
238 | * |
239 | * and we compute the three coefficients by recursively |
240 | * calling ourself to do half-length multiplications. |
241 | * |
242 | * The clever bit that makes this worth doing is that we only |
243 | * need _one_ half-length multiplication for the central |
244 | * coefficient rather than the two that it obviouly looks |
245 | * like, because we can use a single multiplication to compute |
246 | * |
247 | * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0 |
248 | * |
249 | * and then we subtract the other two coefficients (a_1 b_1 |
250 | * and a_0 b_0) which we were computing anyway. |
251 | * |
252 | * Hence we get to multiply two numbers of length N in about |
253 | * three times as much work as it takes to multiply numbers of |
254 | * length N/2, which is obviously better than the four times |
255 | * as much work it would take if we just did a long |
256 | * conventional multiply. |
257 | */ |
258 | |
259 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
260 | int midlen = botlen + 1; |
0c431b2f |
261 | BignumDblInt carry; |
f3c29e34 |
262 | #ifdef KARA_DEBUG |
263 | int i; |
264 | #endif |
0c431b2f |
265 | |
266 | /* |
267 | * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping |
268 | * in the output array, so we can compute them immediately in |
269 | * place. |
270 | */ |
271 | |
f3c29e34 |
272 | #ifdef KARA_DEBUG |
273 | printf("a1,a0 = 0x"); |
274 | for (i = 0; i < len; i++) { |
275 | if (i == toplen) printf(", 0x"); |
276 | printf("%0*x", BIGNUM_INT_BITS/4, a[i]); |
277 | } |
278 | printf("\n"); |
279 | printf("b1,b0 = 0x"); |
280 | for (i = 0; i < len; i++) { |
281 | if (i == toplen) printf(", 0x"); |
282 | printf("%0*x", BIGNUM_INT_BITS/4, b[i]); |
283 | } |
284 | printf("\n"); |
285 | #endif |
286 | |
0c431b2f |
287 | /* a_1 b_1 */ |
5a502a19 |
288 | internal_mul(a, b, c, toplen, scratch); |
f3c29e34 |
289 | #ifdef KARA_DEBUG |
290 | printf("a1b1 = 0x"); |
291 | for (i = 0; i < 2*toplen; i++) { |
292 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); |
293 | } |
294 | printf("\n"); |
295 | #endif |
0c431b2f |
296 | |
297 | /* a_0 b_0 */ |
5a502a19 |
298 | internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch); |
f3c29e34 |
299 | #ifdef KARA_DEBUG |
300 | printf("a0b0 = 0x"); |
301 | for (i = 0; i < 2*botlen; i++) { |
302 | printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]); |
303 | } |
304 | printf("\n"); |
305 | #endif |
0c431b2f |
306 | |
0c431b2f |
307 | /* Zero padding. midlen exceeds toplen by at most 2, so just |
308 | * zero the first two words of each input and the rest will be |
309 | * copied over. */ |
310 | scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0; |
311 | |
757b0110 |
312 | for (i = 0; i < toplen; i++) { |
313 | scratch[midlen - toplen + i] = a[i]; /* a_1 */ |
314 | scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ |
0c431b2f |
315 | } |
316 | |
317 | /* compute a_1 + a_0 */ |
318 | scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); |
f3c29e34 |
319 | #ifdef KARA_DEBUG |
320 | printf("a1plusa0 = 0x"); |
321 | for (i = 0; i < midlen; i++) { |
322 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); |
323 | } |
324 | printf("\n"); |
325 | #endif |
0c431b2f |
326 | /* compute b_1 + b_0 */ |
327 | scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, |
328 | scratch+midlen+1, botlen); |
f3c29e34 |
329 | #ifdef KARA_DEBUG |
330 | printf("b1plusb0 = 0x"); |
331 | for (i = 0; i < midlen; i++) { |
332 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]); |
333 | } |
334 | printf("\n"); |
335 | #endif |
0c431b2f |
336 | |
337 | /* |
338 | * Now we can do the third multiplication. |
339 | */ |
5a502a19 |
340 | internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, |
341 | scratch + 4*midlen); |
f3c29e34 |
342 | #ifdef KARA_DEBUG |
343 | printf("a1plusa0timesb1plusb0 = 0x"); |
344 | for (i = 0; i < 2*midlen; i++) { |
345 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); |
346 | } |
347 | printf("\n"); |
348 | #endif |
0c431b2f |
349 | |
350 | /* |
351 | * Now we can reuse the first half of 'scratch' to compute the |
352 | * sum of the outer two coefficients, to subtract from that |
353 | * product to obtain the middle one. |
354 | */ |
355 | scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0; |
757b0110 |
356 | for (i = 0; i < 2*toplen; i++) |
357 | scratch[2*midlen - 2*toplen + i] = c[i]; |
0c431b2f |
358 | scratch[1] = internal_add(scratch+2, c + 2*toplen, |
359 | scratch+2, 2*botlen); |
f3c29e34 |
360 | #ifdef KARA_DEBUG |
361 | printf("a1b1plusa0b0 = 0x"); |
362 | for (i = 0; i < 2*midlen; i++) { |
363 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); |
364 | } |
365 | printf("\n"); |
366 | #endif |
0c431b2f |
367 | |
368 | internal_sub(scratch + 2*midlen, scratch, |
369 | scratch + 2*midlen, 2*midlen); |
f3c29e34 |
370 | #ifdef KARA_DEBUG |
371 | printf("a1b0plusa0b1 = 0x"); |
372 | for (i = 0; i < 2*midlen; i++) { |
373 | printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); |
374 | } |
375 | printf("\n"); |
376 | #endif |
0c431b2f |
377 | |
378 | /* |
379 | * And now all we need to do is to add that middle coefficient |
380 | * back into the output. We may have to propagate a carry |
381 | * further up the output, but we can be sure it won't |
382 | * propagate right the way off the top. |
383 | */ |
384 | carry = internal_add(c + 2*len - botlen - 2*midlen, |
385 | scratch + 2*midlen, |
386 | c + 2*len - botlen - 2*midlen, 2*midlen); |
757b0110 |
387 | i = 2*len - botlen - 2*midlen - 1; |
0c431b2f |
388 | while (carry) { |
757b0110 |
389 | assert(i >= 0); |
390 | carry += c[i]; |
391 | c[i] = (BignumInt)carry; |
0c431b2f |
392 | carry >>= BIGNUM_INT_BITS; |
757b0110 |
393 | i--; |
0c431b2f |
394 | } |
f3c29e34 |
395 | #ifdef KARA_DEBUG |
396 | printf("ab = 0x"); |
397 | for (i = 0; i < 2*len; i++) { |
398 | printf("%0*x", BIGNUM_INT_BITS/4, c[i]); |
399 | } |
400 | printf("\n"); |
401 | #endif |
0c431b2f |
402 | |
0c431b2f |
403 | } else { |
757b0110 |
404 | int i; |
405 | BignumInt carry; |
406 | BignumDblInt t; |
407 | const BignumInt *ap, *bp; |
408 | BignumInt *cp, *cps; |
0c431b2f |
409 | |
410 | /* |
411 | * Multiply in the ordinary O(N^2) way. |
412 | */ |
413 | |
757b0110 |
414 | for (i = 0; i < 2 * len; i++) |
415 | c[i] = 0; |
0c431b2f |
416 | |
757b0110 |
417 | for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { |
418 | carry = 0; |
419 | for (cp = cps, bp = b + len; cp--, bp-- > b ;) { |
420 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
421 | *cp = (BignumInt) t; |
08b5c9a2 |
422 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
0c431b2f |
423 | } |
757b0110 |
424 | *cp = carry; |
0c431b2f |
425 | } |
e5574168 |
426 | } |
427 | } |
428 | |
132c534f |
429 | /* |
430 | * Variant form of internal_mul used for the initial step of |
431 | * Montgomery reduction. Only bothers outputting 'len' words |
432 | * (everything above that is thrown away). |
433 | */ |
434 | static void internal_mul_low(const BignumInt *a, const BignumInt *b, |
5a502a19 |
435 | BignumInt *c, int len, BignumInt *scratch) |
132c534f |
436 | { |
132c534f |
437 | if (len > KARATSUBA_THRESHOLD) { |
757b0110 |
438 | int i; |
132c534f |
439 | |
440 | /* |
441 | * Karatsuba-aware version of internal_mul_low. As before, we |
442 | * express each input value as a shifted combination of two |
443 | * halves: |
444 | * |
445 | * a = a_1 D + a_0 |
446 | * b = b_1 D + b_0 |
447 | * |
448 | * Then the full product is, as before, |
449 | * |
450 | * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 |
451 | * |
452 | * Provided we choose D on the large side (so that a_0 and b_0 |
453 | * are _at least_ as long as a_1 and b_1), we don't need the |
454 | * topmost term at all, and we only need half of the middle |
455 | * term. So there's no point in doing the proper Karatsuba |
456 | * optimisation which computes the middle term using the top |
457 | * one, because we'd take as long computing the top one as |
458 | * just computing the middle one directly. |
459 | * |
460 | * So instead, we do a much more obvious thing: we call the |
461 | * fully optimised internal_mul to compute a_0 b_0, and we |
462 | * recursively call ourself to compute the _bottom halves_ of |
463 | * a_1 b_0 and a_0 b_1, each of which we add into the result |
464 | * in the obvious way. |
465 | * |
466 | * In other words, there's no actual Karatsuba _optimisation_ |
467 | * in this function; the only benefit in doing it this way is |
468 | * that we call internal_mul proper for a large part of the |
469 | * work, and _that_ can optimise its operation. |
470 | */ |
471 | |
472 | int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ |
132c534f |
473 | |
474 | /* |
5a502a19 |
475 | * Scratch space for the various bits and pieces we're going |
476 | * to be adding together: we need botlen*2 words for a_0 b_0 |
477 | * (though we may end up throwing away its topmost word), and |
478 | * toplen words for each of a_1 b_0 and a_0 b_1. That adds up |
479 | * to exactly 2*len. |
132c534f |
480 | */ |
132c534f |
481 | |
482 | /* a_0 b_0 */ |
5a502a19 |
483 | internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, |
484 | scratch + 2*len); |
132c534f |
485 | |
486 | /* a_1 b_0 */ |
5a502a19 |
487 | internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, |
488 | scratch + 2*len); |
132c534f |
489 | |
490 | /* a_0 b_1 */ |
5a502a19 |
491 | internal_mul_low(a + len - toplen, b, scratch, toplen, |
492 | scratch + 2*len); |
132c534f |
493 | |
494 | /* Copy the bottom half of the big coefficient into place */ |
757b0110 |
495 | for (i = 0; i < botlen; i++) |
496 | c[toplen + i] = scratch[2*toplen + botlen + i]; |
132c534f |
497 | |
498 | /* Add the two small coefficients, throwing away the returned carry */ |
499 | internal_add(scratch, scratch + toplen, scratch, toplen); |
500 | |
501 | /* And add that to the large coefficient, leaving the result in c. */ |
502 | internal_add(scratch, scratch + 2*toplen + botlen - toplen, |
503 | c, toplen); |
504 | |
132c534f |
505 | } else { |
757b0110 |
506 | int i; |
507 | BignumInt carry; |
508 | BignumDblInt t; |
509 | const BignumInt *ap, *bp; |
510 | BignumInt *cp, *cps; |
132c534f |
511 | |
757b0110 |
512 | /* |
513 | * Multiply in the ordinary O(N^2) way. |
514 | */ |
132c534f |
515 | |
757b0110 |
516 | for (i = 0; i < len; i++) |
517 | c[i] = 0; |
518 | |
519 | for (cps = c + len, ap = a + len; ap-- > a; cps--) { |
520 | carry = 0; |
521 | for (cp = cps, bp = b + len; bp--, cp-- > c ;) { |
522 | t = (MUL_WORD(*ap, *bp) + carry) + *cp; |
523 | *cp = (BignumInt) t; |
08b5c9a2 |
524 | carry = (BignumInt)(t >> BIGNUM_INT_BITS); |
132c534f |
525 | } |
526 | } |
132c534f |
527 | } |
528 | } |
529 | |
530 | /* |
531 | * Montgomery reduction. Expects x to be a big-endian array of 2*len |
532 | * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * |
533 | * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array |
534 | * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= |
535 | * x' < n. |
536 | * |
537 | * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts |
538 | * each, containing respectively n and the multiplicative inverse of |
539 | * -n mod r. |
540 | * |
5a502a19 |
541 | * 'tmp' is an array of BignumInt used as scratch space, of length at |
542 | * least 3*len + mul_compute_scratch(len). |
132c534f |
543 | */ |
544 | static void monty_reduce(BignumInt *x, const BignumInt *n, |
545 | const BignumInt *mninv, BignumInt *tmp, int len) |
546 | { |
547 | int i; |
548 | BignumInt carry; |
549 | |
550 | /* |
551 | * Multiply x by (-n)^{-1} mod r. This gives us a value m such |
552 | * that mn is congruent to -x mod r. Hence, mn+x is an exact |
553 | * multiple of r, and is also (obviously) congruent to x mod n. |
554 | */ |
5a502a19 |
555 | internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len); |
132c534f |
556 | |
557 | /* |
558 | * Compute t = (mn+x)/r in ordinary, non-modular, integer |
559 | * arithmetic. By construction this is exact, and is congruent mod |
560 | * n to x * r^{-1}, i.e. the answer we want. |
561 | * |
562 | * The following multiply leaves that answer in the _most_ |
563 | * significant half of the 'x' array, so then we must shift it |
564 | * down. |
565 | */ |
5a502a19 |
566 | internal_mul(tmp, n, tmp+len, len, tmp + 3*len); |
132c534f |
567 | carry = internal_add(x, tmp+len, x, 2*len); |
568 | for (i = 0; i < len; i++) |
569 | x[len + i] = x[i], x[i] = 0; |
570 | |
571 | /* |
572 | * Reduce t mod n. This doesn't require a full-on division by n, |
573 | * but merely a test and single optional subtraction, since we can |
574 | * show that 0 <= t < 2n. |
575 | * |
576 | * Proof: |
577 | * + we computed m mod r, so 0 <= m < r. |
578 | * + so 0 <= mn < rn, obviously |
579 | * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn |
580 | * + yielding 0 <= (mn+x)/r < 2n as required. |
581 | */ |
582 | if (!carry) { |
583 | for (i = 0; i < len; i++) |
584 | if (x[len + i] != n[i]) |
585 | break; |
586 | } |
587 | if (carry || i >= len || x[len + i] > n[i]) |
588 | internal_sub(x+len, n, x+len, len); |
589 | } |
590 | |
a3412f52 |
591 | static void internal_add_shifted(BignumInt *number, |
32874aea |
592 | unsigned n, int shift) |
593 | { |
a3412f52 |
594 | int word = 1 + (shift / BIGNUM_INT_BITS); |
595 | int bshift = shift % BIGNUM_INT_BITS; |
596 | BignumDblInt addend; |
9400cf6f |
597 | |
3014da2b |
598 | addend = (BignumDblInt)n << bshift; |
9400cf6f |
599 | |
600 | while (addend) { |
32874aea |
601 | addend += number[word]; |
a3412f52 |
602 | number[word] = (BignumInt) addend & BIGNUM_INT_MASK; |
603 | addend >>= BIGNUM_INT_BITS; |
32874aea |
604 | word++; |
9400cf6f |
605 | } |
606 | } |
607 | |
e5574168 |
608 | /* |
609 | * Compute a = a % m. |
9400cf6f |
610 | * Input in first alen words of a and first mlen words of m. |
611 | * Output in first alen words of a |
612 | * (of which first alen-mlen words will be zero). |
e5574168 |
613 | * The MSW of m MUST have its high bit set. |
9400cf6f |
614 | * Quotient is accumulated in the `quotient' array, which is a Bignum |
615 | * rather than the internal bigendian format. Quotient parts are shifted |
616 | * left by `qshift' before adding into quot. |
e5574168 |
617 | */ |
a3412f52 |
618 | static void internal_mod(BignumInt *a, int alen, |
619 | BignumInt *m, int mlen, |
620 | BignumInt *quot, int qshift) |
e5574168 |
621 | { |
a3412f52 |
622 | BignumInt m0, m1; |
e5574168 |
623 | unsigned int h; |
624 | int i, k; |
625 | |
e5574168 |
626 | m0 = m[0]; |
8bd9144b |
627 | assert(m0 >> (BIGNUM_INT_BITS-1) == 1); |
9400cf6f |
628 | if (mlen > 1) |
32874aea |
629 | m1 = m[1]; |
9400cf6f |
630 | else |
32874aea |
631 | m1 = 0; |
e5574168 |
632 | |
32874aea |
633 | for (i = 0; i <= alen - mlen; i++) { |
a3412f52 |
634 | BignumDblInt t; |
9400cf6f |
635 | unsigned int q, r, c, ai1; |
e5574168 |
636 | |
637 | if (i == 0) { |
638 | h = 0; |
639 | } else { |
32874aea |
640 | h = a[i - 1]; |
641 | a[i - 1] = 0; |
e5574168 |
642 | } |
643 | |
32874aea |
644 | if (i == alen - 1) |
645 | ai1 = 0; |
646 | else |
647 | ai1 = a[i + 1]; |
9400cf6f |
648 | |
e5574168 |
649 | /* Find q = h:a[i] / m0 */ |
62ef3d44 |
650 | if (h >= m0) { |
651 | /* |
652 | * Special case. |
653 | * |
654 | * To illustrate it, suppose a BignumInt is 8 bits, and |
655 | * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then |
656 | * our initial division will be 0xA123 / 0xA1, which |
657 | * will give a quotient of 0x100 and a divide overflow. |
658 | * However, the invariants in this division algorithm |
659 | * are not violated, since the full number A1:23:... is |
660 | * _less_ than the quotient prefix A1:B2:... and so the |
661 | * following correction loop would have sorted it out. |
662 | * |
663 | * In this situation we set q to be the largest |
664 | * quotient we _can_ stomach (0xFF, of course). |
665 | */ |
666 | q = BIGNUM_INT_MASK; |
667 | } else { |
819a22b3 |
668 | /* Macro doesn't want an array subscript expression passed |
669 | * into it (see definition), so use a temporary. */ |
670 | BignumInt tmplo = a[i]; |
671 | DIVMOD_WORD(q, r, h, tmplo, m0); |
62ef3d44 |
672 | |
673 | /* Refine our estimate of q by looking at |
674 | h:a[i]:a[i+1] / m0:m1 */ |
675 | t = MUL_WORD(m1, q); |
676 | if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { |
677 | q--; |
678 | t -= m1; |
679 | r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ |
680 | if (r >= (BignumDblInt) m0 && |
681 | t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; |
682 | } |
e5574168 |
683 | } |
684 | |
9400cf6f |
685 | /* Subtract q * m from a[i...] */ |
e5574168 |
686 | c = 0; |
9400cf6f |
687 | for (k = mlen - 1; k >= 0; k--) { |
a47e8bba |
688 | t = MUL_WORD(q, m[k]); |
e5574168 |
689 | t += c; |
62ddb51e |
690 | c = (unsigned)(t >> BIGNUM_INT_BITS); |
a3412f52 |
691 | if ((BignumInt) t > a[i + k]) |
32874aea |
692 | c++; |
a3412f52 |
693 | a[i + k] -= (BignumInt) t; |
e5574168 |
694 | } |
695 | |
696 | /* Add back m in case of borrow */ |
697 | if (c != h) { |
698 | t = 0; |
9400cf6f |
699 | for (k = mlen - 1; k >= 0; k--) { |
e5574168 |
700 | t += m[k]; |
32874aea |
701 | t += a[i + k]; |
a3412f52 |
702 | a[i + k] = (BignumInt) t; |
703 | t = t >> BIGNUM_INT_BITS; |
e5574168 |
704 | } |
32874aea |
705 | q--; |
e5574168 |
706 | } |
32874aea |
707 | if (quot) |
a3412f52 |
708 | internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); |
e5574168 |
709 | } |
710 | } |
711 | |
712 | /* |
09095ac5 |
713 | * Compute (base ^ exp) % mod, the pedestrian way. |
e5574168 |
714 | */ |
09095ac5 |
715 | Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) |
e5574168 |
716 | { |
5a502a19 |
717 | BignumInt *a, *b, *n, *m, *scratch; |
09095ac5 |
718 | int mshift; |
5a502a19 |
719 | int mlen, scratchlen, i, j; |
09095ac5 |
720 | Bignum base, result; |
ed953b91 |
721 | |
722 | /* |
723 | * The most significant word of mod needs to be non-zero. It |
724 | * should already be, but let's make sure. |
725 | */ |
726 | assert(mod[mod[0]] != 0); |
727 | |
728 | /* |
729 | * Make sure the base is smaller than the modulus, by reducing |
730 | * it modulo the modulus if not. |
731 | */ |
732 | base = bigmod(base_in, mod); |
e5574168 |
733 | |
09095ac5 |
734 | /* Allocate m of size mlen, copy mod to m */ |
735 | /* We use big endian internally */ |
736 | mlen = mod[0]; |
737 | m = snewn(mlen, BignumInt); |
738 | for (j = 0; j < mlen; j++) |
739 | m[j] = mod[mod[0] - j]; |
740 | |
741 | /* Shift m left to make msb bit set */ |
742 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
743 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
744 | break; |
745 | if (mshift) { |
746 | for (i = 0; i < mlen - 1; i++) |
747 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
748 | m[mlen - 1] = m[mlen - 1] << mshift; |
749 | } |
750 | |
751 | /* Allocate n of size mlen, copy base to n */ |
752 | n = snewn(mlen, BignumInt); |
753 | i = mlen - base[0]; |
754 | for (j = 0; j < i; j++) |
755 | n[j] = 0; |
756 | for (j = 0; j < (int)base[0]; j++) |
757 | n[i + j] = base[base[0] - j]; |
758 | |
759 | /* Allocate a and b of size 2*mlen. Set a = 1 */ |
760 | a = snewn(2 * mlen, BignumInt); |
761 | b = snewn(2 * mlen, BignumInt); |
762 | for (i = 0; i < 2 * mlen; i++) |
763 | a[i] = 0; |
764 | a[2 * mlen - 1] = 1; |
765 | |
5a502a19 |
766 | /* Scratch space for multiplies */ |
767 | scratchlen = mul_compute_scratch(mlen); |
768 | scratch = snewn(scratchlen, BignumInt); |
769 | |
09095ac5 |
770 | /* Skip leading zero bits of exp. */ |
771 | i = 0; |
772 | j = BIGNUM_INT_BITS-1; |
773 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
774 | j--; |
775 | if (j < 0) { |
776 | i++; |
777 | j = BIGNUM_INT_BITS-1; |
778 | } |
779 | } |
780 | |
781 | /* Main computation */ |
782 | while (i < (int)exp[0]) { |
783 | while (j >= 0) { |
5a502a19 |
784 | internal_mul(a + mlen, a + mlen, b, mlen, scratch); |
09095ac5 |
785 | internal_mod(b, mlen * 2, m, mlen, NULL, 0); |
786 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
5a502a19 |
787 | internal_mul(b + mlen, n, a, mlen, scratch); |
09095ac5 |
788 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
789 | } else { |
790 | BignumInt *t; |
791 | t = a; |
792 | a = b; |
793 | b = t; |
794 | } |
795 | j--; |
796 | } |
797 | i++; |
798 | j = BIGNUM_INT_BITS-1; |
799 | } |
800 | |
801 | /* Fixup result in case the modulus was shifted */ |
802 | if (mshift) { |
803 | for (i = mlen - 1; i < 2 * mlen - 1; i++) |
804 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
805 | a[2 * mlen - 1] = a[2 * mlen - 1] << mshift; |
806 | internal_mod(a, mlen * 2, m, mlen, NULL, 0); |
807 | for (i = 2 * mlen - 1; i >= mlen; i--) |
808 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); |
809 | } |
810 | |
811 | /* Copy result to buffer */ |
812 | result = newbn(mod[0]); |
813 | for (i = 0; i < mlen; i++) |
814 | result[result[0] - i] = a[i + mlen]; |
815 | while (result[0] > 1 && result[result[0]] == 0) |
816 | result[0]--; |
817 | |
818 | /* Free temporary arrays */ |
16430000 |
819 | smemclr(a, 2 * mlen * sizeof(*a)); |
09095ac5 |
820 | sfree(a); |
16430000 |
821 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
822 | sfree(scratch); |
16430000 |
823 | smemclr(b, 2 * mlen * sizeof(*b)); |
09095ac5 |
824 | sfree(b); |
16430000 |
825 | smemclr(m, mlen * sizeof(*m)); |
09095ac5 |
826 | sfree(m); |
16430000 |
827 | smemclr(n, mlen * sizeof(*n)); |
09095ac5 |
828 | sfree(n); |
829 | |
830 | freebn(base); |
831 | |
832 | return result; |
833 | } |
834 | |
835 | /* |
836 | * Compute (base ^ exp) % mod. Uses the Montgomery multiplication |
837 | * technique where possible, falling back to modpow_simple otherwise. |
838 | */ |
839 | Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) |
840 | { |
5a502a19 |
841 | BignumInt *a, *b, *x, *n, *mninv, *scratch; |
842 | int len, scratchlen, i, j; |
09095ac5 |
843 | Bignum base, base2, r, rn, inv, result; |
844 | |
845 | /* |
846 | * The most significant word of mod needs to be non-zero. It |
847 | * should already be, but let's make sure. |
848 | */ |
849 | assert(mod[mod[0]] != 0); |
850 | |
132c534f |
851 | /* |
852 | * mod had better be odd, or we can't do Montgomery multiplication |
853 | * using a power of two at all. |
854 | */ |
09095ac5 |
855 | if (!(mod[1] & 1)) |
856 | return modpow_simple(base_in, exp, mod); |
857 | |
858 | /* |
859 | * Make sure the base is smaller than the modulus, by reducing |
860 | * it modulo the modulus if not. |
861 | */ |
862 | base = bigmod(base_in, mod); |
e5574168 |
863 | |
132c534f |
864 | /* |
865 | * Compute the inverse of n mod r, for monty_reduce. (In fact we |
866 | * want the inverse of _minus_ n mod r, but we'll sort that out |
867 | * below.) |
868 | */ |
869 | len = mod[0]; |
870 | r = bn_power_2(BIGNUM_INT_BITS * len); |
871 | inv = modinv(mod, r); |
de81309d |
872 | assert(inv); /* cannot fail, since mod is odd and r is a power of 2 */ |
e5574168 |
873 | |
132c534f |
874 | /* |
875 | * Multiply the base by r mod n, to get it into Montgomery |
876 | * representation. |
877 | */ |
878 | base2 = modmul(base, r, mod); |
879 | freebn(base); |
880 | base = base2; |
881 | |
882 | rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ |
883 | |
884 | freebn(r); /* won't need this any more */ |
885 | |
886 | /* |
887 | * Set up internal arrays of the right lengths, in big-endian |
888 | * format, containing the base, the modulus, and the modulus's |
889 | * inverse. |
890 | */ |
891 | n = snewn(len, BignumInt); |
892 | for (j = 0; j < len; j++) |
893 | n[len - 1 - j] = mod[j + 1]; |
894 | |
895 | mninv = snewn(len, BignumInt); |
896 | for (j = 0; j < len; j++) |
08b5c9a2 |
897 | mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0); |
132c534f |
898 | freebn(inv); /* we don't need this copy of it any more */ |
899 | /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ |
900 | x = snewn(len, BignumInt); |
901 | for (j = 0; j < len; j++) |
902 | x[j] = 0; |
903 | internal_sub(x, mninv, mninv, len); |
904 | |
905 | /* x = snewn(len, BignumInt); */ /* already done above */ |
906 | for (j = 0; j < len; j++) |
08b5c9a2 |
907 | x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0); |
132c534f |
908 | freebn(base); /* we don't need this copy of it any more */ |
909 | |
910 | a = snewn(2*len, BignumInt); |
911 | b = snewn(2*len, BignumInt); |
912 | for (j = 0; j < len; j++) |
08b5c9a2 |
913 | a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0); |
132c534f |
914 | freebn(rn); |
915 | |
5a502a19 |
916 | /* Scratch space for multiplies */ |
917 | scratchlen = 3*len + mul_compute_scratch(len); |
918 | scratch = snewn(scratchlen, BignumInt); |
e5574168 |
919 | |
920 | /* Skip leading zero bits of exp. */ |
32874aea |
921 | i = 0; |
a3412f52 |
922 | j = BIGNUM_INT_BITS-1; |
62ddb51e |
923 | while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { |
e5574168 |
924 | j--; |
32874aea |
925 | if (j < 0) { |
926 | i++; |
a3412f52 |
927 | j = BIGNUM_INT_BITS-1; |
32874aea |
928 | } |
e5574168 |
929 | } |
930 | |
931 | /* Main computation */ |
62ddb51e |
932 | while (i < (int)exp[0]) { |
e5574168 |
933 | while (j >= 0) { |
5a502a19 |
934 | internal_mul(a + len, a + len, b, len, scratch); |
935 | monty_reduce(b, n, mninv, scratch, len); |
e5574168 |
936 | if ((exp[exp[0] - i] & (1 << j)) != 0) { |
5a502a19 |
937 | internal_mul(b + len, x, a, len, scratch); |
938 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 |
939 | } else { |
a3412f52 |
940 | BignumInt *t; |
32874aea |
941 | t = a; |
942 | a = b; |
943 | b = t; |
e5574168 |
944 | } |
945 | j--; |
946 | } |
32874aea |
947 | i++; |
a3412f52 |
948 | j = BIGNUM_INT_BITS-1; |
e5574168 |
949 | } |
950 | |
132c534f |
951 | /* |
952 | * Final monty_reduce to get back from the adjusted Montgomery |
953 | * representation. |
954 | */ |
5a502a19 |
955 | monty_reduce(a, n, mninv, scratch, len); |
e5574168 |
956 | |
957 | /* Copy result to buffer */ |
59600f67 |
958 | result = newbn(mod[0]); |
132c534f |
959 | for (i = 0; i < len; i++) |
960 | result[result[0] - i] = a[i + len]; |
32874aea |
961 | while (result[0] > 1 && result[result[0]] == 0) |
962 | result[0]--; |
e5574168 |
963 | |
964 | /* Free temporary arrays */ |
16430000 |
965 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
966 | sfree(scratch); |
16430000 |
967 | smemclr(a, 2 * len * sizeof(*a)); |
32874aea |
968 | sfree(a); |
16430000 |
969 | smemclr(b, 2 * len * sizeof(*b)); |
32874aea |
970 | sfree(b); |
16430000 |
971 | smemclr(mninv, len * sizeof(*mninv)); |
132c534f |
972 | sfree(mninv); |
16430000 |
973 | smemclr(n, len * sizeof(*n)); |
32874aea |
974 | sfree(n); |
16430000 |
975 | smemclr(x, len * sizeof(*x)); |
132c534f |
976 | sfree(x); |
ed953b91 |
977 | |
59600f67 |
978 | return result; |
e5574168 |
979 | } |
7cca0d81 |
980 | |
981 | /* |
982 | * Compute (p * q) % mod. |
983 | * The most significant word of mod MUST be non-zero. |
984 | * We assume that the result array is the same size as the mod array. |
985 | */ |
59600f67 |
986 | Bignum modmul(Bignum p, Bignum q, Bignum mod) |
7cca0d81 |
987 | { |
5a502a19 |
988 | BignumInt *a, *n, *m, *o, *scratch; |
989 | int mshift, scratchlen; |
80b10571 |
990 | int pqlen, mlen, rlen, i, j; |
59600f67 |
991 | Bignum result; |
7cca0d81 |
992 | |
8bd9144b |
993 | /* |
994 | * The most significant word of mod needs to be non-zero. It |
995 | * should already be, but let's make sure. |
996 | */ |
997 | assert(mod[mod[0]] != 0); |
998 | |
7cca0d81 |
999 | /* Allocate m of size mlen, copy mod to m */ |
1000 | /* We use big endian internally */ |
1001 | mlen = mod[0]; |
a3412f52 |
1002 | m = snewn(mlen, BignumInt); |
32874aea |
1003 | for (j = 0; j < mlen; j++) |
1004 | m[j] = mod[mod[0] - j]; |
7cca0d81 |
1005 | |
1006 | /* Shift m left to make msb bit set */ |
a3412f52 |
1007 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1008 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
32874aea |
1009 | break; |
7cca0d81 |
1010 | if (mshift) { |
1011 | for (i = 0; i < mlen - 1; i++) |
a3412f52 |
1012 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1013 | m[mlen - 1] = m[mlen - 1] << mshift; |
7cca0d81 |
1014 | } |
1015 | |
1016 | pqlen = (p[0] > q[0] ? p[0] : q[0]); |
1017 | |
5064e5e6 |
1018 | /* |
1019 | * Make sure that we're allowing enough space. The shifting below |
1020 | * will underflow the vectors we allocate if pqlen is too small. |
1021 | */ |
1022 | if (2*pqlen <= mlen) |
1023 | pqlen = mlen/2 + 1; |
1024 | |
7cca0d81 |
1025 | /* Allocate n of size pqlen, copy p to n */ |
a3412f52 |
1026 | n = snewn(pqlen, BignumInt); |
7cca0d81 |
1027 | i = pqlen - p[0]; |
32874aea |
1028 | for (j = 0; j < i; j++) |
1029 | n[j] = 0; |
62ddb51e |
1030 | for (j = 0; j < (int)p[0]; j++) |
32874aea |
1031 | n[i + j] = p[p[0] - j]; |
7cca0d81 |
1032 | |
1033 | /* Allocate o of size pqlen, copy q to o */ |
a3412f52 |
1034 | o = snewn(pqlen, BignumInt); |
7cca0d81 |
1035 | i = pqlen - q[0]; |
32874aea |
1036 | for (j = 0; j < i; j++) |
1037 | o[j] = 0; |
62ddb51e |
1038 | for (j = 0; j < (int)q[0]; j++) |
32874aea |
1039 | o[i + j] = q[q[0] - j]; |
7cca0d81 |
1040 | |
1041 | /* Allocate a of size 2*pqlen for result */ |
a3412f52 |
1042 | a = snewn(2 * pqlen, BignumInt); |
7cca0d81 |
1043 | |
5a502a19 |
1044 | /* Scratch space for multiplies */ |
1045 | scratchlen = mul_compute_scratch(pqlen); |
1046 | scratch = snewn(scratchlen, BignumInt); |
1047 | |
7cca0d81 |
1048 | /* Main computation */ |
5a502a19 |
1049 | internal_mul(n, o, a, pqlen, scratch); |
32874aea |
1050 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
7cca0d81 |
1051 | |
1052 | /* Fixup result in case the modulus was shifted */ |
1053 | if (mshift) { |
32874aea |
1054 | for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++) |
a3412f52 |
1055 | a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1056 | a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift; |
1057 | internal_mod(a, pqlen * 2, m, mlen, NULL, 0); |
1058 | for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--) |
a3412f52 |
1059 | a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); |
7cca0d81 |
1060 | } |
1061 | |
1062 | /* Copy result to buffer */ |
32874aea |
1063 | rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); |
80b10571 |
1064 | result = newbn(rlen); |
1065 | for (i = 0; i < rlen; i++) |
32874aea |
1066 | result[result[0] - i] = a[i + 2 * pqlen - rlen]; |
1067 | while (result[0] > 1 && result[result[0]] == 0) |
1068 | result[0]--; |
7cca0d81 |
1069 | |
1070 | /* Free temporary arrays */ |
16430000 |
1071 | smemclr(scratch, scratchlen * sizeof(*scratch)); |
5a502a19 |
1072 | sfree(scratch); |
16430000 |
1073 | smemclr(a, 2 * pqlen * sizeof(*a)); |
32874aea |
1074 | sfree(a); |
16430000 |
1075 | smemclr(m, mlen * sizeof(*m)); |
32874aea |
1076 | sfree(m); |
16430000 |
1077 | smemclr(n, pqlen * sizeof(*n)); |
32874aea |
1078 | sfree(n); |
16430000 |
1079 | smemclr(o, pqlen * sizeof(*o)); |
32874aea |
1080 | sfree(o); |
59600f67 |
1081 | |
1082 | return result; |
7cca0d81 |
1083 | } |
1084 | |
1085 | /* |
9400cf6f |
1086 | * Compute p % mod. |
1087 | * The most significant word of mod MUST be non-zero. |
1088 | * We assume that the result array is the same size as the mod array. |
5c72ca61 |
1089 | * We optionally write out a quotient if `quotient' is non-NULL. |
1090 | * We can avoid writing out the result if `result' is NULL. |
9400cf6f |
1091 | */ |
f28753ab |
1092 | static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) |
9400cf6f |
1093 | { |
a3412f52 |
1094 | BignumInt *n, *m; |
9400cf6f |
1095 | int mshift; |
1096 | int plen, mlen, i, j; |
1097 | |
8bd9144b |
1098 | /* |
1099 | * The most significant word of mod needs to be non-zero. It |
1100 | * should already be, but let's make sure. |
1101 | */ |
1102 | assert(mod[mod[0]] != 0); |
1103 | |
9400cf6f |
1104 | /* Allocate m of size mlen, copy mod to m */ |
1105 | /* We use big endian internally */ |
1106 | mlen = mod[0]; |
a3412f52 |
1107 | m = snewn(mlen, BignumInt); |
32874aea |
1108 | for (j = 0; j < mlen; j++) |
1109 | m[j] = mod[mod[0] - j]; |
9400cf6f |
1110 | |
1111 | /* Shift m left to make msb bit set */ |
a3412f52 |
1112 | for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) |
1113 | if ((m[0] << mshift) & BIGNUM_TOP_BIT) |
32874aea |
1114 | break; |
9400cf6f |
1115 | if (mshift) { |
1116 | for (i = 0; i < mlen - 1; i++) |
a3412f52 |
1117 | m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1118 | m[mlen - 1] = m[mlen - 1] << mshift; |
9400cf6f |
1119 | } |
1120 | |
1121 | plen = p[0]; |
1122 | /* Ensure plen > mlen */ |
32874aea |
1123 | if (plen <= mlen) |
1124 | plen = mlen + 1; |
9400cf6f |
1125 | |
1126 | /* Allocate n of size plen, copy p to n */ |
a3412f52 |
1127 | n = snewn(plen, BignumInt); |
32874aea |
1128 | for (j = 0; j < plen; j++) |
1129 | n[j] = 0; |
62ddb51e |
1130 | for (j = 1; j <= (int)p[0]; j++) |
32874aea |
1131 | n[plen - j] = p[j]; |
9400cf6f |
1132 | |
1133 | /* Main computation */ |
1134 | internal_mod(n, plen, m, mlen, quotient, mshift); |
1135 | |
1136 | /* Fixup result in case the modulus was shifted */ |
1137 | if (mshift) { |
1138 | for (i = plen - mlen - 1; i < plen - 1; i++) |
a3412f52 |
1139 | n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); |
32874aea |
1140 | n[plen - 1] = n[plen - 1] << mshift; |
9400cf6f |
1141 | internal_mod(n, plen, m, mlen, quotient, 0); |
1142 | for (i = plen - 1; i >= plen - mlen; i--) |
a3412f52 |
1143 | n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); |
9400cf6f |
1144 | } |
1145 | |
1146 | /* Copy result to buffer */ |
5c72ca61 |
1147 | if (result) { |
62ddb51e |
1148 | for (i = 1; i <= (int)result[0]; i++) { |
5c72ca61 |
1149 | int j = plen - i; |
1150 | result[i] = j >= 0 ? n[j] : 0; |
1151 | } |
9400cf6f |
1152 | } |
1153 | |
1154 | /* Free temporary arrays */ |
16430000 |
1155 | smemclr(m, mlen * sizeof(*m)); |
32874aea |
1156 | sfree(m); |
16430000 |
1157 | smemclr(n, plen * sizeof(*n)); |
32874aea |
1158 | sfree(n); |
9400cf6f |
1159 | } |
1160 | |
1161 | /* |
7cca0d81 |
1162 | * Decrement a number. |
1163 | */ |
32874aea |
1164 | void decbn(Bignum bn) |
1165 | { |
7cca0d81 |
1166 | int i = 1; |
62ddb51e |
1167 | while (i < (int)bn[0] && bn[i] == 0) |
a3412f52 |
1168 | bn[i++] = BIGNUM_INT_MASK; |
7cca0d81 |
1169 | bn[i]--; |
1170 | } |
1171 | |
27cd7fc2 |
1172 | Bignum bignum_from_bytes(const unsigned char *data, int nbytes) |
32874aea |
1173 | { |
3709bfe9 |
1174 | Bignum result; |
1175 | int w, i; |
1176 | |
a3412f52 |
1177 | w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ |
3709bfe9 |
1178 | |
1179 | result = newbn(w); |
32874aea |
1180 | for (i = 1; i <= w; i++) |
1181 | result[i] = 0; |
1182 | for (i = nbytes; i--;) { |
1183 | unsigned char byte = *data++; |
a3412f52 |
1184 | result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); |
3709bfe9 |
1185 | } |
1186 | |
32874aea |
1187 | while (result[0] > 1 && result[result[0]] == 0) |
1188 | result[0]--; |
3709bfe9 |
1189 | return result; |
1190 | } |
1191 | |
7cca0d81 |
1192 | /* |
2e85c969 |
1193 | * Read an SSH-1-format bignum from a data buffer. Return the number |
0016d70b |
1194 | * of bytes consumed, or -1 if there wasn't enough data. |
7cca0d81 |
1195 | */ |
0016d70b |
1196 | int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) |
32874aea |
1197 | { |
27cd7fc2 |
1198 | const unsigned char *p = data; |
7cca0d81 |
1199 | int i; |
1200 | int w, b; |
1201 | |
0016d70b |
1202 | if (len < 2) |
1203 | return -1; |
1204 | |
7cca0d81 |
1205 | w = 0; |
32874aea |
1206 | for (i = 0; i < 2; i++) |
1207 | w = (w << 8) + *p++; |
1208 | b = (w + 7) / 8; /* bits -> bytes */ |
7cca0d81 |
1209 | |
0016d70b |
1210 | if (len < b+2) |
1211 | return -1; |
1212 | |
32874aea |
1213 | if (!result) /* just return length */ |
1214 | return b + 2; |
a52f067e |
1215 | |
3709bfe9 |
1216 | *result = bignum_from_bytes(p, b); |
7cca0d81 |
1217 | |
3709bfe9 |
1218 | return p + b - data; |
7cca0d81 |
1219 | } |
5c58ad2d |
1220 | |
1221 | /* |
2e85c969 |
1222 | * Return the bit count of a bignum, for SSH-1 encoding. |
5c58ad2d |
1223 | */ |
32874aea |
1224 | int bignum_bitcount(Bignum bn) |
1225 | { |
a3412f52 |
1226 | int bitcount = bn[0] * BIGNUM_INT_BITS - 1; |
32874aea |
1227 | while (bitcount >= 0 |
a3412f52 |
1228 | && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; |
5c58ad2d |
1229 | return bitcount + 1; |
1230 | } |
1231 | |
1232 | /* |
2e85c969 |
1233 | * Return the byte length of a bignum when SSH-1 encoded. |
5c58ad2d |
1234 | */ |
32874aea |
1235 | int ssh1_bignum_length(Bignum bn) |
1236 | { |
1237 | return 2 + (bignum_bitcount(bn) + 7) / 8; |
ddecd643 |
1238 | } |
1239 | |
1240 | /* |
2e85c969 |
1241 | * Return the byte length of a bignum when SSH-2 encoded. |
ddecd643 |
1242 | */ |
32874aea |
1243 | int ssh2_bignum_length(Bignum bn) |
1244 | { |
1245 | return 4 + (bignum_bitcount(bn) + 8) / 8; |
5c58ad2d |
1246 | } |
1247 | |
1248 | /* |
1249 | * Return a byte from a bignum; 0 is least significant, etc. |
1250 | */ |
32874aea |
1251 | int bignum_byte(Bignum bn, int i) |
1252 | { |
62ddb51e |
1253 | if (i >= (int)(BIGNUM_INT_BYTES * bn[0])) |
32874aea |
1254 | return 0; /* beyond the end */ |
5c58ad2d |
1255 | else |
a3412f52 |
1256 | return (bn[i / BIGNUM_INT_BYTES + 1] >> |
1257 | ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; |
5c58ad2d |
1258 | } |
1259 | |
1260 | /* |
9400cf6f |
1261 | * Return a bit from a bignum; 0 is least significant, etc. |
1262 | */ |
32874aea |
1263 | int bignum_bit(Bignum bn, int i) |
1264 | { |
62ddb51e |
1265 | if (i >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea |
1266 | return 0; /* beyond the end */ |
9400cf6f |
1267 | else |
a3412f52 |
1268 | return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; |
9400cf6f |
1269 | } |
1270 | |
1271 | /* |
1272 | * Set a bit in a bignum; 0 is least significant, etc. |
1273 | */ |
32874aea |
1274 | void bignum_set_bit(Bignum bn, int bitnum, int value) |
1275 | { |
62ddb51e |
1276 | if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) |
32874aea |
1277 | abort(); /* beyond the end */ |
9400cf6f |
1278 | else { |
a3412f52 |
1279 | int v = bitnum / BIGNUM_INT_BITS + 1; |
1280 | int mask = 1 << (bitnum % BIGNUM_INT_BITS); |
32874aea |
1281 | if (value) |
1282 | bn[v] |= mask; |
1283 | else |
1284 | bn[v] &= ~mask; |
9400cf6f |
1285 | } |
1286 | } |
1287 | |
1288 | /* |
2e85c969 |
1289 | * Write a SSH-1-format bignum into a buffer. It is assumed the |
5c58ad2d |
1290 | * buffer is big enough. Returns the number of bytes used. |
1291 | */ |
32874aea |
1292 | int ssh1_write_bignum(void *data, Bignum bn) |
1293 | { |
5c58ad2d |
1294 | unsigned char *p = data; |
1295 | int len = ssh1_bignum_length(bn); |
1296 | int i; |
ddecd643 |
1297 | int bitc = bignum_bitcount(bn); |
5c58ad2d |
1298 | |
1299 | *p++ = (bitc >> 8) & 0xFF; |
32874aea |
1300 | *p++ = (bitc) & 0xFF; |
1301 | for (i = len - 2; i--;) |
1302 | *p++ = bignum_byte(bn, i); |
5c58ad2d |
1303 | return len; |
1304 | } |
9400cf6f |
1305 | |
1306 | /* |
1307 | * Compare two bignums. Returns like strcmp. |
1308 | */ |
32874aea |
1309 | int bignum_cmp(Bignum a, Bignum b) |
1310 | { |
9400cf6f |
1311 | int amax = a[0], bmax = b[0]; |
1312 | int i = (amax > bmax ? amax : bmax); |
1313 | while (i) { |
a3412f52 |
1314 | BignumInt aval = (i > amax ? 0 : a[i]); |
1315 | BignumInt bval = (i > bmax ? 0 : b[i]); |
32874aea |
1316 | if (aval < bval) |
1317 | return -1; |
1318 | if (aval > bval) |
1319 | return +1; |
1320 | i--; |
9400cf6f |
1321 | } |
1322 | return 0; |
1323 | } |
1324 | |
1325 | /* |
1326 | * Right-shift one bignum to form another. |
1327 | */ |
32874aea |
1328 | Bignum bignum_rshift(Bignum a, int shift) |
1329 | { |
9400cf6f |
1330 | Bignum ret; |
1331 | int i, shiftw, shiftb, shiftbb, bits; |
a3412f52 |
1332 | BignumInt ai, ai1; |
9400cf6f |
1333 | |
ddecd643 |
1334 | bits = bignum_bitcount(a) - shift; |
a3412f52 |
1335 | ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); |
9400cf6f |
1336 | |
1337 | if (ret) { |
a3412f52 |
1338 | shiftw = shift / BIGNUM_INT_BITS; |
1339 | shiftb = shift % BIGNUM_INT_BITS; |
1340 | shiftbb = BIGNUM_INT_BITS - shiftb; |
32874aea |
1341 | |
1342 | ai1 = a[shiftw + 1]; |
62ddb51e |
1343 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea |
1344 | ai = ai1; |
62ddb51e |
1345 | ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); |
a3412f52 |
1346 | ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; |
32874aea |
1347 | } |
9400cf6f |
1348 | } |
1349 | |
1350 | return ret; |
1351 | } |
1352 | |
1353 | /* |
1354 | * Non-modular multiplication and addition. |
1355 | */ |
32874aea |
1356 | Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) |
1357 | { |
9400cf6f |
1358 | int alen = a[0], blen = b[0]; |
1359 | int mlen = (alen > blen ? alen : blen); |
1360 | int rlen, i, maxspot; |
5a502a19 |
1361 | int wslen; |
a3412f52 |
1362 | BignumInt *workspace; |
9400cf6f |
1363 | Bignum ret; |
1364 | |
5a502a19 |
1365 | /* mlen space for a, mlen space for b, 2*mlen for result, |
1366 | * plus scratch space for multiplication */ |
1367 | wslen = mlen * 4 + mul_compute_scratch(mlen); |
1368 | workspace = snewn(wslen, BignumInt); |
9400cf6f |
1369 | for (i = 0; i < mlen; i++) { |
62ddb51e |
1370 | workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0); |
1371 | workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0); |
9400cf6f |
1372 | } |
1373 | |
32874aea |
1374 | internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, |
5a502a19 |
1375 | workspace + 2 * mlen, mlen, workspace + 4 * mlen); |
9400cf6f |
1376 | |
1377 | /* now just copy the result back */ |
1378 | rlen = alen + blen + 1; |
62ddb51e |
1379 | if (addend && rlen <= (int)addend[0]) |
32874aea |
1380 | rlen = addend[0] + 1; |
9400cf6f |
1381 | ret = newbn(rlen); |
1382 | maxspot = 0; |
62ddb51e |
1383 | for (i = 1; i <= (int)ret[0]; i++) { |
32874aea |
1384 | ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); |
1385 | if (ret[i] != 0) |
1386 | maxspot = i; |
9400cf6f |
1387 | } |
1388 | ret[0] = maxspot; |
1389 | |
1390 | /* now add in the addend, if any */ |
1391 | if (addend) { |
a3412f52 |
1392 | BignumDblInt carry = 0; |
32874aea |
1393 | for (i = 1; i <= rlen; i++) { |
62ddb51e |
1394 | carry += (i <= (int)ret[0] ? ret[i] : 0); |
1395 | carry += (i <= (int)addend[0] ? addend[i] : 0); |
a3412f52 |
1396 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1397 | carry >>= BIGNUM_INT_BITS; |
32874aea |
1398 | if (ret[i] != 0 && i > maxspot) |
1399 | maxspot = i; |
1400 | } |
9400cf6f |
1401 | } |
1402 | ret[0] = maxspot; |
1403 | |
16430000 |
1404 | smemclr(workspace, wslen * sizeof(*workspace)); |
c523f55f |
1405 | sfree(workspace); |
9400cf6f |
1406 | return ret; |
1407 | } |
1408 | |
1409 | /* |
1410 | * Non-modular multiplication. |
1411 | */ |
32874aea |
1412 | Bignum bigmul(Bignum a, Bignum b) |
1413 | { |
9400cf6f |
1414 | return bigmuladd(a, b, NULL); |
1415 | } |
1416 | |
1417 | /* |
d737853b |
1418 | * Simple addition. |
1419 | */ |
1420 | Bignum bigadd(Bignum a, Bignum b) |
1421 | { |
1422 | int alen = a[0], blen = b[0]; |
1423 | int rlen = (alen > blen ? alen : blen) + 1; |
1424 | int i, maxspot; |
1425 | Bignum ret; |
1426 | BignumDblInt carry; |
1427 | |
1428 | ret = newbn(rlen); |
1429 | |
1430 | carry = 0; |
1431 | maxspot = 0; |
1432 | for (i = 1; i <= rlen; i++) { |
1433 | carry += (i <= (int)a[0] ? a[i] : 0); |
1434 | carry += (i <= (int)b[0] ? b[i] : 0); |
1435 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1436 | carry >>= BIGNUM_INT_BITS; |
1437 | if (ret[i] != 0 && i > maxspot) |
1438 | maxspot = i; |
1439 | } |
1440 | ret[0] = maxspot; |
1441 | |
1442 | return ret; |
1443 | } |
1444 | |
1445 | /* |
1446 | * Subtraction. Returns a-b, or NULL if the result would come out |
1447 | * negative (recall that this entire bignum module only handles |
1448 | * positive numbers). |
1449 | */ |
1450 | Bignum bigsub(Bignum a, Bignum b) |
1451 | { |
1452 | int alen = a[0], blen = b[0]; |
1453 | int rlen = (alen > blen ? alen : blen); |
1454 | int i, maxspot; |
1455 | Bignum ret; |
1456 | BignumDblInt carry; |
1457 | |
1458 | ret = newbn(rlen); |
1459 | |
1460 | carry = 1; |
1461 | maxspot = 0; |
1462 | for (i = 1; i <= rlen; i++) { |
1463 | carry += (i <= (int)a[0] ? a[i] : 0); |
1464 | carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); |
1465 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1466 | carry >>= BIGNUM_INT_BITS; |
1467 | if (ret[i] != 0 && i > maxspot) |
1468 | maxspot = i; |
1469 | } |
1470 | ret[0] = maxspot; |
1471 | |
1472 | if (!carry) { |
1473 | freebn(ret); |
1474 | return NULL; |
1475 | } |
1476 | |
1477 | return ret; |
1478 | } |
1479 | |
1480 | /* |
3709bfe9 |
1481 | * Create a bignum which is the bitmask covering another one. That |
1482 | * is, the smallest integer which is >= N and is also one less than |
1483 | * a power of two. |
1484 | */ |
32874aea |
1485 | Bignum bignum_bitmask(Bignum n) |
1486 | { |
3709bfe9 |
1487 | Bignum ret = copybn(n); |
1488 | int i; |
a3412f52 |
1489 | BignumInt j; |
3709bfe9 |
1490 | |
1491 | i = ret[0]; |
1492 | while (n[i] == 0 && i > 0) |
32874aea |
1493 | i--; |
3709bfe9 |
1494 | if (i <= 0) |
32874aea |
1495 | return ret; /* input was zero */ |
3709bfe9 |
1496 | j = 1; |
1497 | while (j < n[i]) |
32874aea |
1498 | j = 2 * j + 1; |
3709bfe9 |
1499 | ret[i] = j; |
1500 | while (--i > 0) |
a3412f52 |
1501 | ret[i] = BIGNUM_INT_MASK; |
3709bfe9 |
1502 | return ret; |
1503 | } |
1504 | |
1505 | /* |
5c72ca61 |
1506 | * Convert a (max 32-bit) long into a bignum. |
9400cf6f |
1507 | */ |
a3412f52 |
1508 | Bignum bignum_from_long(unsigned long nn) |
32874aea |
1509 | { |
9400cf6f |
1510 | Bignum ret; |
a3412f52 |
1511 | BignumDblInt n = nn; |
9400cf6f |
1512 | |
5c72ca61 |
1513 | ret = newbn(3); |
a3412f52 |
1514 | ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); |
1515 | ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); |
5c72ca61 |
1516 | ret[3] = 0; |
1517 | ret[0] = (ret[2] ? 2 : 1); |
32874aea |
1518 | return ret; |
9400cf6f |
1519 | } |
1520 | |
1521 | /* |
1522 | * Add a long to a bignum. |
1523 | */ |
a3412f52 |
1524 | Bignum bignum_add_long(Bignum number, unsigned long addendx) |
32874aea |
1525 | { |
1526 | Bignum ret = newbn(number[0] + 1); |
9400cf6f |
1527 | int i, maxspot = 0; |
a3412f52 |
1528 | BignumDblInt carry = 0, addend = addendx; |
9400cf6f |
1529 | |
62ddb51e |
1530 | for (i = 1; i <= (int)ret[0]; i++) { |
a3412f52 |
1531 | carry += addend & BIGNUM_INT_MASK; |
62ddb51e |
1532 | carry += (i <= (int)number[0] ? number[i] : 0); |
a3412f52 |
1533 | addend >>= BIGNUM_INT_BITS; |
1534 | ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; |
1535 | carry >>= BIGNUM_INT_BITS; |
32874aea |
1536 | if (ret[i] != 0) |
1537 | maxspot = i; |
9400cf6f |
1538 | } |
1539 | ret[0] = maxspot; |
1540 | return ret; |
1541 | } |
1542 | |
1543 | /* |
1544 | * Compute the residue of a bignum, modulo a (max 16-bit) short. |
1545 | */ |
32874aea |
1546 | unsigned short bignum_mod_short(Bignum number, unsigned short modulus) |
1547 | { |
a3412f52 |
1548 | BignumDblInt mod, r; |
9400cf6f |
1549 | int i; |
1550 | |
1551 | r = 0; |
1552 | mod = modulus; |
1553 | for (i = number[0]; i > 0; i--) |
736cc6d1 |
1554 | r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; |
6e522441 |
1555 | return (unsigned short) r; |
9400cf6f |
1556 | } |
1557 | |
a3412f52 |
1558 | #ifdef DEBUG |
32874aea |
1559 | void diagbn(char *prefix, Bignum md) |
1560 | { |
9400cf6f |
1561 | int i, nibbles, morenibbles; |
1562 | static const char hex[] = "0123456789ABCDEF"; |
1563 | |
5c72ca61 |
1564 | debug(("%s0x", prefix ? prefix : "")); |
9400cf6f |
1565 | |
32874aea |
1566 | nibbles = (3 + bignum_bitcount(md)) / 4; |
1567 | if (nibbles < 1) |
1568 | nibbles = 1; |
1569 | morenibbles = 4 * md[0] - nibbles; |
1570 | for (i = 0; i < morenibbles; i++) |
5c72ca61 |
1571 | debug(("-")); |
32874aea |
1572 | for (i = nibbles; i--;) |
5c72ca61 |
1573 | debug(("%c", |
1574 | hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF])); |
9400cf6f |
1575 | |
32874aea |
1576 | if (prefix) |
5c72ca61 |
1577 | debug(("\n")); |
1578 | } |
f28753ab |
1579 | #endif |
5c72ca61 |
1580 | |
1581 | /* |
1582 | * Simple division. |
1583 | */ |
1584 | Bignum bigdiv(Bignum a, Bignum b) |
1585 | { |
1586 | Bignum q = newbn(a[0]); |
1587 | bigdivmod(a, b, NULL, q); |
1588 | return q; |
1589 | } |
1590 | |
1591 | /* |
1592 | * Simple remainder. |
1593 | */ |
1594 | Bignum bigmod(Bignum a, Bignum b) |
1595 | { |
1596 | Bignum r = newbn(b[0]); |
1597 | bigdivmod(a, b, r, NULL); |
1598 | return r; |
9400cf6f |
1599 | } |
1600 | |
1601 | /* |
1602 | * Greatest common divisor. |
1603 | */ |
32874aea |
1604 | Bignum biggcd(Bignum av, Bignum bv) |
1605 | { |
9400cf6f |
1606 | Bignum a = copybn(av); |
1607 | Bignum b = copybn(bv); |
1608 | |
9400cf6f |
1609 | while (bignum_cmp(b, Zero) != 0) { |
32874aea |
1610 | Bignum t = newbn(b[0]); |
5c72ca61 |
1611 | bigdivmod(a, b, t, NULL); |
32874aea |
1612 | while (t[0] > 1 && t[t[0]] == 0) |
1613 | t[0]--; |
1614 | freebn(a); |
1615 | a = b; |
1616 | b = t; |
9400cf6f |
1617 | } |
1618 | |
1619 | freebn(b); |
1620 | return a; |
1621 | } |
1622 | |
1623 | /* |
1624 | * Modular inverse, using Euclid's extended algorithm. |
1625 | */ |
32874aea |
1626 | Bignum modinv(Bignum number, Bignum modulus) |
1627 | { |
9400cf6f |
1628 | Bignum a = copybn(modulus); |
1629 | Bignum b = copybn(number); |
1630 | Bignum xp = copybn(Zero); |
1631 | Bignum x = copybn(One); |
1632 | int sign = +1; |
1633 | |
8bd9144b |
1634 | assert(number[number[0]] != 0); |
1635 | assert(modulus[modulus[0]] != 0); |
1636 | |
9400cf6f |
1637 | while (bignum_cmp(b, One) != 0) { |
de81309d |
1638 | Bignum t, q; |
1639 | |
1640 | if (bignum_cmp(b, Zero) == 0) { |
1641 | /* |
1642 | * Found a common factor between the inputs, so we cannot |
1643 | * return a modular inverse at all. |
1644 | */ |
c6456dca |
1645 | freebn(b); |
1646 | freebn(a); |
1647 | freebn(xp); |
1648 | freebn(x); |
de81309d |
1649 | return NULL; |
1650 | } |
1651 | |
1652 | t = newbn(b[0]); |
1653 | q = newbn(a[0]); |
5c72ca61 |
1654 | bigdivmod(a, b, t, q); |
32874aea |
1655 | while (t[0] > 1 && t[t[0]] == 0) |
1656 | t[0]--; |
1657 | freebn(a); |
1658 | a = b; |
1659 | b = t; |
1660 | t = xp; |
1661 | xp = x; |
1662 | x = bigmuladd(q, xp, t); |
1663 | sign = -sign; |
1664 | freebn(t); |
75374b2f |
1665 | freebn(q); |
9400cf6f |
1666 | } |
1667 | |
1668 | freebn(b); |
1669 | freebn(a); |
1670 | freebn(xp); |
1671 | |
1672 | /* now we know that sign * x == 1, and that x < modulus */ |
1673 | if (sign < 0) { |
32874aea |
1674 | /* set a new x to be modulus - x */ |
1675 | Bignum newx = newbn(modulus[0]); |
a3412f52 |
1676 | BignumInt carry = 0; |
32874aea |
1677 | int maxspot = 1; |
1678 | int i; |
1679 | |
62ddb51e |
1680 | for (i = 1; i <= (int)newx[0]; i++) { |
1681 | BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); |
1682 | BignumInt bword = (i <= (int)x[0] ? x[i] : 0); |
32874aea |
1683 | newx[i] = aword - bword - carry; |
1684 | bword = ~bword; |
1685 | carry = carry ? (newx[i] >= bword) : (newx[i] > bword); |
1686 | if (newx[i] != 0) |
1687 | maxspot = i; |
1688 | } |
1689 | newx[0] = maxspot; |
1690 | freebn(x); |
1691 | x = newx; |
9400cf6f |
1692 | } |
1693 | |
1694 | /* and return. */ |
1695 | return x; |
1696 | } |
6e522441 |
1697 | |
1698 | /* |
1699 | * Render a bignum into decimal. Return a malloced string holding |
1700 | * the decimal representation. |
1701 | */ |
32874aea |
1702 | char *bignum_decimal(Bignum x) |
1703 | { |
6e522441 |
1704 | int ndigits, ndigit; |
1705 | int i, iszero; |
a3412f52 |
1706 | BignumDblInt carry; |
6e522441 |
1707 | char *ret; |
a3412f52 |
1708 | BignumInt *workspace; |
6e522441 |
1709 | |
1710 | /* |
1711 | * First, estimate the number of digits. Since log(10)/log(2) |
1712 | * is just greater than 93/28 (the joys of continued fraction |
1713 | * approximations...) we know that for every 93 bits, we need |
1714 | * at most 28 digits. This will tell us how much to malloc. |
1715 | * |
1716 | * Formally: if x has i bits, that means x is strictly less |
1717 | * than 2^i. Since 2 is less than 10^(28/93), this is less than |
1718 | * 10^(28i/93). We need an integer power of ten, so we must |
1719 | * round up (rounding down might make it less than x again). |
1720 | * Therefore if we multiply the bit count by 28/93, rounding |
1721 | * up, we will have enough digits. |
74c79ce8 |
1722 | * |
1723 | * i=0 (i.e., x=0) is an irritating special case. |
6e522441 |
1724 | */ |
ddecd643 |
1725 | i = bignum_bitcount(x); |
74c79ce8 |
1726 | if (!i) |
1727 | ndigits = 1; /* x = 0 */ |
1728 | else |
1729 | ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ |
32874aea |
1730 | ndigits++; /* allow for trailing \0 */ |
3d88e64d |
1731 | ret = snewn(ndigits, char); |
6e522441 |
1732 | |
1733 | /* |
1734 | * Now allocate some workspace to hold the binary form as we |
1735 | * repeatedly divide it by ten. Initialise this to the |
1736 | * big-endian form of the number. |
1737 | */ |
a3412f52 |
1738 | workspace = snewn(x[0], BignumInt); |
62ddb51e |
1739 | for (i = 0; i < (int)x[0]; i++) |
32874aea |
1740 | workspace[i] = x[x[0] - i]; |
6e522441 |
1741 | |
1742 | /* |
1743 | * Next, write the decimal number starting with the last digit. |
1744 | * We use ordinary short division, dividing 10 into the |
1745 | * workspace. |
1746 | */ |
32874aea |
1747 | ndigit = ndigits - 1; |
6e522441 |
1748 | ret[ndigit] = '\0'; |
1749 | do { |
32874aea |
1750 | iszero = 1; |
1751 | carry = 0; |
62ddb51e |
1752 | for (i = 0; i < (int)x[0]; i++) { |
a3412f52 |
1753 | carry = (carry << BIGNUM_INT_BITS) + workspace[i]; |
1754 | workspace[i] = (BignumInt) (carry / 10); |
32874aea |
1755 | if (workspace[i]) |
1756 | iszero = 0; |
1757 | carry %= 10; |
1758 | } |
1759 | ret[--ndigit] = (char) (carry + '0'); |
6e522441 |
1760 | } while (!iszero); |
1761 | |
1762 | /* |
1763 | * There's a chance we've fallen short of the start of the |
1764 | * string. Correct if so. |
1765 | */ |
1766 | if (ndigit > 0) |
32874aea |
1767 | memmove(ret, ret + ndigit, ndigits - ndigit); |
6e522441 |
1768 | |
1769 | /* |
1770 | * Done. |
1771 | */ |
16430000 |
1772 | smemclr(workspace, x[0] * sizeof(*workspace)); |
c523f55f |
1773 | sfree(workspace); |
6e522441 |
1774 | return ret; |
1775 | } |
f3c29e34 |
1776 | |
1777 | #ifdef TESTBN |
1778 | |
1779 | #include <stdio.h> |
1780 | #include <stdlib.h> |
1781 | #include <ctype.h> |
1782 | |
1783 | /* |
4800a5e5 |
1784 | * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset |
f84f1e46 |
1785 | * |
1786 | * Then feed to this program's standard input the output of |
1787 | * testdata/bignum.py . |
f3c29e34 |
1788 | */ |
1789 | |
1790 | void modalfatalbox(char *p, ...) |
1791 | { |
1792 | va_list ap; |
1793 | fprintf(stderr, "FATAL ERROR: "); |
1794 | va_start(ap, p); |
1795 | vfprintf(stderr, p, ap); |
1796 | va_end(ap); |
1797 | fputc('\n', stderr); |
1798 | exit(1); |
1799 | } |
1800 | |
1801 | #define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) |
1802 | |
1803 | int main(int argc, char **argv) |
1804 | { |
1805 | char *buf; |
1806 | int line = 0; |
1807 | int passes = 0, fails = 0; |
1808 | |
1809 | while ((buf = fgetline(stdin)) != NULL) { |
1810 | int maxlen = strlen(buf); |
1811 | unsigned char *data = snewn(maxlen, unsigned char); |
f84f1e46 |
1812 | unsigned char *ptrs[5], *q; |
f3c29e34 |
1813 | int ptrnum; |
1814 | char *bufp = buf; |
1815 | |
1816 | line++; |
1817 | |
1818 | q = data; |
1819 | ptrnum = 0; |
1820 | |
f84f1e46 |
1821 | while (*bufp && !isspace((unsigned char)*bufp)) |
1822 | bufp++; |
1823 | if (bufp) |
1824 | *bufp++ = '\0'; |
1825 | |
f3c29e34 |
1826 | while (*bufp) { |
1827 | char *start, *end; |
1828 | int i; |
1829 | |
1830 | while (*bufp && !isxdigit((unsigned char)*bufp)) |
1831 | bufp++; |
1832 | start = bufp; |
1833 | |
1834 | if (!*bufp) |
1835 | break; |
1836 | |
1837 | while (*bufp && isxdigit((unsigned char)*bufp)) |
1838 | bufp++; |
1839 | end = bufp; |
1840 | |
1841 | if (ptrnum >= lenof(ptrs)) |
1842 | break; |
1843 | ptrs[ptrnum++] = q; |
1844 | |
1845 | for (i = -((end - start) & 1); i < end-start; i += 2) { |
1846 | unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); |
1847 | val = val * 16 + fromxdigit(start[i+1]); |
1848 | *q++ = val; |
1849 | } |
1850 | |
1851 | ptrs[ptrnum] = q; |
1852 | } |
1853 | |
f84f1e46 |
1854 | if (!strcmp(buf, "mul")) { |
1855 | Bignum a, b, c, p; |
1856 | |
1857 | if (ptrnum != 3) { |
f6939e2b |
1858 | printf("%d: mul with %d parameters, expected 3\n", line, ptrnum); |
f84f1e46 |
1859 | exit(1); |
1860 | } |
1861 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1862 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1863 | c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1864 | p = bigmul(a, b); |
f3c29e34 |
1865 | |
1866 | if (bignum_cmp(c, p) == 0) { |
1867 | passes++; |
1868 | } else { |
1869 | char *as = bignum_decimal(a); |
1870 | char *bs = bignum_decimal(b); |
1871 | char *cs = bignum_decimal(c); |
1872 | char *ps = bignum_decimal(p); |
1873 | |
1874 | printf("%d: fail: %s * %s gave %s expected %s\n", |
1875 | line, as, bs, ps, cs); |
1876 | fails++; |
1877 | |
1878 | sfree(as); |
1879 | sfree(bs); |
1880 | sfree(cs); |
1881 | sfree(ps); |
1882 | } |
1883 | freebn(a); |
1884 | freebn(b); |
1885 | freebn(c); |
1886 | freebn(p); |
5064e5e6 |
1887 | } else if (!strcmp(buf, "modmul")) { |
1888 | Bignum a, b, m, c, p; |
1889 | |
1890 | if (ptrnum != 4) { |
1891 | printf("%d: modmul with %d parameters, expected 4\n", |
1892 | line, ptrnum); |
1893 | exit(1); |
1894 | } |
1895 | a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1896 | b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1897 | m = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1898 | c = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); |
1899 | p = modmul(a, b, m); |
1900 | |
1901 | if (bignum_cmp(c, p) == 0) { |
1902 | passes++; |
1903 | } else { |
1904 | char *as = bignum_decimal(a); |
1905 | char *bs = bignum_decimal(b); |
1906 | char *ms = bignum_decimal(m); |
1907 | char *cs = bignum_decimal(c); |
1908 | char *ps = bignum_decimal(p); |
1909 | |
1910 | printf("%d: fail: %s * %s mod %s gave %s expected %s\n", |
1911 | line, as, bs, ms, ps, cs); |
1912 | fails++; |
1913 | |
1914 | sfree(as); |
1915 | sfree(bs); |
1916 | sfree(ms); |
1917 | sfree(cs); |
1918 | sfree(ps); |
1919 | } |
1920 | freebn(a); |
1921 | freebn(b); |
1922 | freebn(m); |
1923 | freebn(c); |
1924 | freebn(p); |
f84f1e46 |
1925 | } else if (!strcmp(buf, "pow")) { |
1926 | Bignum base, expt, modulus, expected, answer; |
1927 | |
1928 | if (ptrnum != 4) { |
f6939e2b |
1929 | printf("%d: mul with %d parameters, expected 4\n", line, ptrnum); |
f84f1e46 |
1930 | exit(1); |
1931 | } |
1932 | |
1933 | base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); |
1934 | expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); |
1935 | modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); |
1936 | expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); |
1937 | answer = modpow(base, expt, modulus); |
1938 | |
1939 | if (bignum_cmp(expected, answer) == 0) { |
1940 | passes++; |
1941 | } else { |
1942 | char *as = bignum_decimal(base); |
1943 | char *bs = bignum_decimal(expt); |
1944 | char *cs = bignum_decimal(modulus); |
1945 | char *ds = bignum_decimal(answer); |
1946 | char *ps = bignum_decimal(expected); |
1947 | |
1948 | printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", |
1949 | line, as, bs, cs, ds, ps); |
1950 | fails++; |
1951 | |
1952 | sfree(as); |
1953 | sfree(bs); |
1954 | sfree(cs); |
1955 | sfree(ds); |
1956 | sfree(ps); |
1957 | } |
1958 | freebn(base); |
1959 | freebn(expt); |
1960 | freebn(modulus); |
1961 | freebn(expected); |
1962 | freebn(answer); |
1963 | } else { |
1964 | printf("%d: unrecognised test keyword: '%s'\n", line, buf); |
1965 | exit(1); |
f3c29e34 |
1966 | } |
f84f1e46 |
1967 | |
f3c29e34 |
1968 | sfree(buf); |
1969 | sfree(data); |
1970 | } |
1971 | |
1972 | printf("passed %d failed %d total %d\n", passes, fails, passes+fails); |
1973 | return fails != 0; |
1974 | } |
1975 | |
1976 | #endif |