1 | /*********************************************** |
---|
2 | * Copyright (C) 2011 Sebastian Jambor * |
---|
3 | * sebastian@momo.math.rwth-aachen.de * |
---|
4 | ***********************************************/ |
---|
5 | |
---|
6 | |
---|
7 | #include<cmath> |
---|
8 | #include<Singular/mod2.h> |
---|
9 | |
---|
10 | #include "minpoly.h" |
---|
11 | |
---|
12 | // TODO: avoid code copying, use subclassing instead! |
---|
13 | |
---|
14 | LinearDependencyMatrix::LinearDependencyMatrix (unsigned n, unsigned long p) |
---|
15 | { |
---|
16 | this->n = n; |
---|
17 | this->p = p; |
---|
18 | |
---|
19 | matrix = new unsigned long *[n]; |
---|
20 | for(int i = 0; i < n; i++) |
---|
21 | { |
---|
22 | matrix[i] = new unsigned long[2 * n + 1]; |
---|
23 | } |
---|
24 | pivots = new unsigned[n]; |
---|
25 | tmprow = new unsigned long[2 * n + 1]; |
---|
26 | rows = 0; |
---|
27 | } |
---|
28 | |
---|
29 | LinearDependencyMatrix::~LinearDependencyMatrix () |
---|
30 | { |
---|
31 | delete[]tmprow; |
---|
32 | delete[]pivots; |
---|
33 | |
---|
34 | for(int i = 0; i < n; i++) |
---|
35 | { |
---|
36 | delete[](matrix[i]); |
---|
37 | } |
---|
38 | delete[]matrix; |
---|
39 | } |
---|
40 | |
---|
41 | void LinearDependencyMatrix::resetMatrix () |
---|
42 | { |
---|
43 | rows = 0; |
---|
44 | } |
---|
45 | |
---|
46 | int LinearDependencyMatrix::firstNonzeroEntry (unsigned long *row) |
---|
47 | { |
---|
48 | for(int i = 0; i < n; i++) |
---|
49 | if(row[i] != 0) |
---|
50 | return i; |
---|
51 | |
---|
52 | return -1; |
---|
53 | } |
---|
54 | |
---|
55 | void LinearDependencyMatrix::reduceTmpRow () |
---|
56 | { |
---|
57 | for(int i = 0; i < rows; i++) |
---|
58 | { |
---|
59 | unsigned piv = pivots[i]; |
---|
60 | unsigned long x = tmprow[piv] % p; |
---|
61 | // if the corresponding entry in the row is zero, there is nothing to do |
---|
62 | if(x != 0) |
---|
63 | { |
---|
64 | // subtract tmprow[i] times the i-th row |
---|
65 | for(int j = piv; j < n + rows + 1; j++) |
---|
66 | { |
---|
67 | unsigned long tmp = matrix[i][j] * x % p; |
---|
68 | tmp = p - tmp; |
---|
69 | tmprow[j] += tmp; |
---|
70 | // We don't normalize here, so remember to do it after all reductions. |
---|
71 | // tmprow[j] %= p; |
---|
72 | } |
---|
73 | } |
---|
74 | } |
---|
75 | |
---|
76 | // normalize the entries of tmprow. |
---|
77 | for(int i = 0; i < n + rows + 1; i++) |
---|
78 | { |
---|
79 | tmprow[i] %= p; |
---|
80 | } |
---|
81 | } |
---|
82 | |
---|
83 | |
---|
84 | void LinearDependencyMatrix::normalizeTmp (unsigned i) |
---|
85 | { |
---|
86 | unsigned long inv = modularInverse (tmprow[i], p); |
---|
87 | tmprow[i] = 1; |
---|
88 | for(int j = i + 1; j < 2 * n + 1; j++) |
---|
89 | tmprow[j] = (tmprow[j] * inv) % p; |
---|
90 | } |
---|
91 | |
---|
92 | bool LinearDependencyMatrix::findLinearDependency (unsigned long *newRow, |
---|
93 | unsigned long *dep) |
---|
94 | { |
---|
95 | // Copy newRow to tmprow (we need to add RHS) |
---|
96 | for(int i = 0; i < n; i++) |
---|
97 | { |
---|
98 | tmprow[i] = newRow[i]; |
---|
99 | tmprow[n + i] = 0; |
---|
100 | } |
---|
101 | tmprow[2 * n] = 0; |
---|
102 | tmprow[n + rows] = 1; |
---|
103 | |
---|
104 | reduceTmpRow (); |
---|
105 | |
---|
106 | // Is tmprow reduced to zero? Then we have found a linear dependence. |
---|
107 | // Otherwise, we just add tmprow to the matrix. |
---|
108 | unsigned newpivot = firstNonzeroEntry (tmprow); |
---|
109 | if(newpivot == -1) |
---|
110 | { |
---|
111 | for(int i = 0; i <= n; i++) |
---|
112 | { |
---|
113 | dep[i] = tmprow[n + i]; |
---|
114 | } |
---|
115 | |
---|
116 | return true; |
---|
117 | } |
---|
118 | else |
---|
119 | { |
---|
120 | normalizeTmp (newpivot); |
---|
121 | |
---|
122 | for(int i = 0; i < 2 * n + 1; i++) |
---|
123 | { |
---|
124 | matrix[rows][i] = tmprow[i]; |
---|
125 | } |
---|
126 | |
---|
127 | pivots[rows] = newpivot; |
---|
128 | rows++; |
---|
129 | |
---|
130 | return false; |
---|
131 | } |
---|
132 | } |
---|
133 | |
---|
134 | #if 0 |
---|
135 | std::ostream & operator<< (std::ostream & out, |
---|
136 | const LinearDependencyMatrix & mat) |
---|
137 | { |
---|
138 | int width = ((int) log10 (mat.p)) + 1; |
---|
139 | |
---|
140 | out << "Pivots: " << std::endl; |
---|
141 | for(int j = 0; j < mat.n; j++) |
---|
142 | { |
---|
143 | out << std::setw (width) << mat.pivots[j] << " "; |
---|
144 | } |
---|
145 | out << std::endl; |
---|
146 | out << "matrix:" << std::endl; |
---|
147 | for(int i = 0; i < mat.rows; i++) |
---|
148 | { |
---|
149 | for(int j = 0; j < mat.n; j++) |
---|
150 | { |
---|
151 | out << std::setw (width) << mat.matrix[i][j] << " "; |
---|
152 | } |
---|
153 | out << "| "; |
---|
154 | for(int j = mat.n; j <= 2 * mat.n; j++) |
---|
155 | { |
---|
156 | out << std::setw (width) << mat.matrix[i][j] << " "; |
---|
157 | } |
---|
158 | out << std::endl; |
---|
159 | } |
---|
160 | out << "tmprow: " << std::endl; |
---|
161 | for(int j = 0; j < mat.n; j++) |
---|
162 | { |
---|
163 | out << std::setw (width) << mat.tmprow[j] << " "; |
---|
164 | } |
---|
165 | out << "| "; |
---|
166 | for(int j = mat.n; j <= 2 * mat.n; j++) |
---|
167 | { |
---|
168 | out << std::setw (width) << mat.tmprow[j] << " "; |
---|
169 | } |
---|
170 | out << std::endl; |
---|
171 | |
---|
172 | return out; |
---|
173 | } |
---|
174 | #endif |
---|
175 | |
---|
176 | |
---|
177 | NewVectorMatrix::NewVectorMatrix (unsigned n, unsigned long p) |
---|
178 | { |
---|
179 | this->n = n; |
---|
180 | this->p = p; |
---|
181 | |
---|
182 | matrix = new unsigned long *[n]; |
---|
183 | for(int i = 0; i < n; i++) |
---|
184 | { |
---|
185 | matrix[i] = new unsigned long[n]; |
---|
186 | } |
---|
187 | |
---|
188 | pivots = new unsigned[n]; |
---|
189 | rows = 0; |
---|
190 | } |
---|
191 | |
---|
192 | NewVectorMatrix::~NewVectorMatrix () |
---|
193 | { |
---|
194 | delete pivots; |
---|
195 | |
---|
196 | for(int i = 0; i < n; i++) |
---|
197 | { |
---|
198 | delete[]matrix[i]; |
---|
199 | } |
---|
200 | delete matrix; |
---|
201 | } |
---|
202 | |
---|
203 | int NewVectorMatrix::firstNonzeroEntry (unsigned long *row) |
---|
204 | { |
---|
205 | for(int i = 0; i < n; i++) |
---|
206 | if(row[i] != 0) |
---|
207 | return i; |
---|
208 | |
---|
209 | return -1; |
---|
210 | } |
---|
211 | |
---|
212 | //void NewVectorMatrix::subtractIthRow(unsigned long* row, unsigned i) { |
---|
213 | // unsigned piv = pivots[i]; |
---|
214 | // unsigned long x = row[piv]; |
---|
215 | // for (int j = piv; j < n; j++) { |
---|
216 | // unsigned tmp = matrix[i][j]*x % p; |
---|
217 | // tmp = p - tmp; |
---|
218 | // row[j] += tmp; |
---|
219 | // row[j] %= p; |
---|
220 | // } |
---|
221 | //} |
---|
222 | // |
---|
223 | void NewVectorMatrix::normalizeRow (unsigned long *row, unsigned i) |
---|
224 | { |
---|
225 | unsigned long inv = modularInverse (row[i], p); |
---|
226 | row[i] = 1; |
---|
227 | |
---|
228 | for(int j = i + 1; j < n; j++) |
---|
229 | { |
---|
230 | row[j] = (row[j] * inv) % p; |
---|
231 | } |
---|
232 | } |
---|
233 | |
---|
234 | void NewVectorMatrix::insertRow (unsigned long *row) |
---|
235 | { |
---|
236 | for(int i = 0; i < rows; i++) |
---|
237 | { |
---|
238 | unsigned piv = pivots[i]; |
---|
239 | unsigned long x = row[piv] % p; |
---|
240 | // if the corresponding entry in the row is zero, there is nothing to do |
---|
241 | if(x != 0) |
---|
242 | { |
---|
243 | // subtract row[i] times the i-th row |
---|
244 | for(int j = piv; j < n; j++) |
---|
245 | { |
---|
246 | unsigned long tmp = matrix[i][j] * x % p; |
---|
247 | tmp = p - tmp; |
---|
248 | row[j] += tmp; |
---|
249 | // We don't normalize here, so remember to do it after all reductions. |
---|
250 | // row[j] %= p; |
---|
251 | } |
---|
252 | } |
---|
253 | } |
---|
254 | |
---|
255 | // normalize the entries of row. |
---|
256 | for(int i = 0; i < n + rows + 1; i++) |
---|
257 | { |
---|
258 | row[i] %= p; |
---|
259 | } |
---|
260 | |
---|
261 | unsigned piv = firstNonzeroEntry (row); |
---|
262 | |
---|
263 | if(piv != -1) |
---|
264 | { |
---|
265 | // normalize and insert row into the matrix |
---|
266 | normalizeRow (row, piv); |
---|
267 | for(int i = 0; i < n; i++) |
---|
268 | { |
---|
269 | matrix[rows][i] = row[i]; |
---|
270 | } |
---|
271 | |
---|
272 | pivots[rows] = piv; |
---|
273 | rows++; |
---|
274 | } |
---|
275 | } |
---|
276 | |
---|
277 | |
---|
278 | void NewVectorMatrix::insertMatrix (LinearDependencyMatrix & mat) |
---|
279 | { |
---|
280 | // The matrix in LinearDependencyMatrix is already in reduced form. |
---|
281 | // Thus, if the current matrix is empty, we can simply copy the other matrix. |
---|
282 | if(rows == 0) |
---|
283 | { |
---|
284 | for(int i = 0; i < mat.rows; i++) |
---|
285 | { |
---|
286 | for(int j = 0; j < n; j++) |
---|
287 | { |
---|
288 | matrix[i][j] = mat.matrix[i][j]; |
---|
289 | |
---|
290 | rows = mat.rows; |
---|
291 | for(int i = 0; i < rows; i++) |
---|
292 | { |
---|
293 | pivots[i] = mat.pivots[i]; |
---|
294 | } |
---|
295 | } |
---|
296 | } |
---|
297 | } |
---|
298 | else |
---|
299 | { |
---|
300 | for(int i = 0; i < mat.rows; i++) |
---|
301 | { |
---|
302 | insertRow (mat.matrix[i]); |
---|
303 | } |
---|
304 | } |
---|
305 | } |
---|
306 | |
---|
307 | int NewVectorMatrix::findSmallestNonpivot () |
---|
308 | { |
---|
309 | // This method isn't very efficient, but it is called at most a few times, so efficiency is not important. |
---|
310 | if(rows == n) |
---|
311 | return -1; |
---|
312 | |
---|
313 | for(int i = 0; i < n; i++) |
---|
314 | { |
---|
315 | bool isPivot = false; |
---|
316 | for(int j = 0; j < rows; j++) |
---|
317 | { |
---|
318 | if(pivots[j] == i) |
---|
319 | { |
---|
320 | isPivot = true; |
---|
321 | break; |
---|
322 | } |
---|
323 | } |
---|
324 | |
---|
325 | if(!isPivot) |
---|
326 | { |
---|
327 | return i; |
---|
328 | } |
---|
329 | } |
---|
330 | assume(0); |
---|
331 | return -1; // to make the compiler happy |
---|
332 | |
---|
333 | } |
---|
334 | |
---|
335 | |
---|
336 | void vectorMatrixMult (unsigned long *vec, unsigned long **mat, |
---|
337 | unsigned long *result, unsigned n, unsigned long p) |
---|
338 | { |
---|
339 | unsigned long tmp; |
---|
340 | for(int i = 0; i < n; i++) |
---|
341 | { |
---|
342 | result[i] = 0; |
---|
343 | for(int j = 0; j < n; j++) |
---|
344 | { |
---|
345 | tmp = vec[j] * mat[j][i] % p; |
---|
346 | result[i] += tmp; |
---|
347 | } |
---|
348 | // We can afford to reduce mod p only after all additions, since p < 2^31, but an unsigned long can store 2^64. |
---|
349 | // Thus the only possibility for an overflow would occurr for matrices with about 2^31 rows. |
---|
350 | result[i] %= p; |
---|
351 | } |
---|
352 | } |
---|
353 | |
---|
354 | #if 0 |
---|
355 | void printVec (unsigned long *vec, int n) |
---|
356 | { |
---|
357 | for(int i = 0; i < n; i++) |
---|
358 | { |
---|
359 | std::cout << vec[i] << ", "; |
---|
360 | } |
---|
361 | |
---|
362 | std::cout << std::endl; |
---|
363 | } |
---|
364 | #endif |
---|
365 | |
---|
366 | unsigned long *computeMinimalPolynomial (unsigned long **matrix, unsigned n, |
---|
367 | unsigned long p) |
---|
368 | { |
---|
369 | LinearDependencyMatrix lindepmat (n, p); |
---|
370 | NewVectorMatrix newvectormat (n, p); |
---|
371 | |
---|
372 | unsigned long *result = new unsigned long[n + 1]; |
---|
373 | unsigned long *mpvec = new unsigned long[n + 1]; |
---|
374 | unsigned long *tmp = new unsigned long[n + 1]; |
---|
375 | |
---|
376 | // initialize result = 1 |
---|
377 | for(int i = 0; i <= n; i++) |
---|
378 | { |
---|
379 | result[i] = 0; |
---|
380 | } |
---|
381 | result[0] = 1; |
---|
382 | |
---|
383 | int degresult = 0; |
---|
384 | |
---|
385 | |
---|
386 | int i = 0; |
---|
387 | |
---|
388 | unsigned long *vec = new unsigned long[n]; |
---|
389 | unsigned long *vecnew = new unsigned long[n]; |
---|
390 | |
---|
391 | while(i != -1) |
---|
392 | { |
---|
393 | for(int j = 0; j < n; j++) |
---|
394 | { |
---|
395 | vec[j] = 0; |
---|
396 | } |
---|
397 | vec[i] = 1; |
---|
398 | |
---|
399 | lindepmat.resetMatrix (); |
---|
400 | |
---|
401 | while(true) |
---|
402 | { |
---|
403 | bool ld = lindepmat.findLinearDependency (vec, mpvec); |
---|
404 | |
---|
405 | if(ld) |
---|
406 | { |
---|
407 | break; |
---|
408 | } |
---|
409 | |
---|
410 | vectorMatrixMult (vec, matrix, vecnew, n, p); |
---|
411 | unsigned long *swap = vec; |
---|
412 | vec = vecnew; |
---|
413 | vecnew = swap; |
---|
414 | } |
---|
415 | |
---|
416 | |
---|
417 | unsigned degmpvec = n; |
---|
418 | while(mpvec[degmpvec] == 0) |
---|
419 | { |
---|
420 | degmpvec--; |
---|
421 | } |
---|
422 | |
---|
423 | // if the dimension is already maximal, i.e., the minimal polynomial of vec has the highest possible degree, |
---|
424 | // then we are done. |
---|
425 | if(degmpvec == n) |
---|
426 | { |
---|
427 | unsigned long *swap = result; |
---|
428 | result = mpvec; |
---|
429 | mpvec = swap; |
---|
430 | i = -1; |
---|
431 | } |
---|
432 | else |
---|
433 | { |
---|
434 | // otherwise, we have to compute the lcm of mpvec and prior result |
---|
435 | |
---|
436 | // tmp = zeropol |
---|
437 | for(int j = 0; j <= n; j++) |
---|
438 | { |
---|
439 | tmp[j] = 0; |
---|
440 | } |
---|
441 | degresult = lcm (tmp, result, mpvec, p, degresult, degmpvec); |
---|
442 | unsigned long *swap = result; |
---|
443 | result = tmp; |
---|
444 | tmp = swap; |
---|
445 | |
---|
446 | if(degresult == n) |
---|
447 | { |
---|
448 | // minimal polynomial has highest possible degree, so we are done. |
---|
449 | i = -1; |
---|
450 | } |
---|
451 | else |
---|
452 | { |
---|
453 | newvectormat.insertMatrix (lindepmat); |
---|
454 | i = newvectormat.findSmallestNonpivot (); |
---|
455 | } |
---|
456 | } |
---|
457 | } |
---|
458 | |
---|
459 | // TODO: take lcms of the different monomials! |
---|
460 | |
---|
461 | delete[]vecnew; |
---|
462 | delete[]vec; |
---|
463 | delete[]tmp; |
---|
464 | delete[]mpvec; |
---|
465 | |
---|
466 | return result; |
---|
467 | } |
---|
468 | |
---|
469 | |
---|
470 | void rem (unsigned long *a, unsigned long *q, unsigned long p, int °a, |
---|
471 | int degq) |
---|
472 | { |
---|
473 | while(degq <= dega) |
---|
474 | { |
---|
475 | unsigned d = dega - degq; |
---|
476 | long factor = a[dega] * modularInverse (q[degq], p) % p; |
---|
477 | for(int i = degq; i >= 0; i--) |
---|
478 | { |
---|
479 | long tmp = p - (factor * q[i] % p); |
---|
480 | a[d + i] += tmp; |
---|
481 | a[d + i] %= p; |
---|
482 | } |
---|
483 | |
---|
484 | while(a[dega] == 0 && dega >= 0) |
---|
485 | { |
---|
486 | dega--; |
---|
487 | } |
---|
488 | } |
---|
489 | } |
---|
490 | |
---|
491 | |
---|
492 | void quo (unsigned long *a, unsigned long *q, unsigned long p, int °a, |
---|
493 | int degq) |
---|
494 | { |
---|
495 | unsigned degres = dega - degq; |
---|
496 | unsigned long *result = new unsigned long[degres + 1]; |
---|
497 | |
---|
498 | while(degq <= dega) |
---|
499 | { |
---|
500 | unsigned d = dega - degq; |
---|
501 | long inv = modularInverse (q[degq], p); |
---|
502 | result[d] = a[dega] * inv % p; |
---|
503 | for(int i = degq; i >= 0; i--) |
---|
504 | { |
---|
505 | long tmp = p - (result[d] * q[i] % p); |
---|
506 | a[d + i] += tmp; |
---|
507 | a[d + i] %= p; |
---|
508 | } |
---|
509 | |
---|
510 | while(a[dega] == 0 && dega >= 0) |
---|
511 | { |
---|
512 | dega--; |
---|
513 | } |
---|
514 | } |
---|
515 | |
---|
516 | // copy result into a |
---|
517 | for(int i = 0; i <= degres; i++) |
---|
518 | { |
---|
519 | a[i] = result[i]; |
---|
520 | } |
---|
521 | // set all following entries of a to zero |
---|
522 | for(int i = degres + 1; i <= degq + degres; i++) |
---|
523 | { |
---|
524 | a[i] = 0; |
---|
525 | } |
---|
526 | |
---|
527 | dega = degres; |
---|
528 | |
---|
529 | delete[]result; |
---|
530 | } |
---|
531 | |
---|
532 | |
---|
533 | void mult (unsigned long *result, unsigned long *a, unsigned long *b, |
---|
534 | unsigned long p, int dega, int degb) |
---|
535 | { |
---|
536 | // NOTE: we assume that every entry in result is preinitialized to zero! |
---|
537 | |
---|
538 | for(int i = 0; i <= dega; i++) |
---|
539 | { |
---|
540 | for(int j = 0; j <= degb; j++) |
---|
541 | { |
---|
542 | result[i + j] += (a[i] * b[j] % p); |
---|
543 | result[i + j] %= p; |
---|
544 | } |
---|
545 | } |
---|
546 | } |
---|
547 | |
---|
548 | |
---|
549 | int gcd (unsigned long *g, unsigned long *a, unsigned long *b, |
---|
550 | unsigned long p, int dega, int degb) |
---|
551 | { |
---|
552 | unsigned long *tmp1 = new unsigned long[dega + 1]; |
---|
553 | unsigned long *tmp2 = new unsigned long[degb + 1]; |
---|
554 | for(int i = 0; i <= dega; i++) |
---|
555 | { |
---|
556 | tmp1[i] = a[i]; |
---|
557 | } |
---|
558 | for(int i = 0; i <= degb; i++) |
---|
559 | { |
---|
560 | tmp2[i] = b[i]; |
---|
561 | } |
---|
562 | int degtmp1 = dega; |
---|
563 | int degtmp2 = degb; |
---|
564 | |
---|
565 | unsigned long *swappol; |
---|
566 | int swapdeg; |
---|
567 | |
---|
568 | while(degtmp2 >= 0) |
---|
569 | { |
---|
570 | rem (tmp1, tmp2, p, degtmp1, degtmp2); |
---|
571 | swappol = tmp1; |
---|
572 | tmp1 = tmp2; |
---|
573 | tmp2 = swappol; |
---|
574 | |
---|
575 | swapdeg = degtmp1; |
---|
576 | degtmp1 = degtmp2; |
---|
577 | degtmp2 = swapdeg; |
---|
578 | } |
---|
579 | |
---|
580 | for(int i = 0; i <= degtmp1; i++) |
---|
581 | { |
---|
582 | g[i] = tmp1[i]; |
---|
583 | } |
---|
584 | |
---|
585 | delete[]tmp1; |
---|
586 | delete[]tmp2; |
---|
587 | |
---|
588 | return degtmp1; |
---|
589 | } |
---|
590 | |
---|
591 | |
---|
592 | int lcm (unsigned long *l, unsigned long *a, unsigned long *b, |
---|
593 | unsigned long p, int dega, int degb) |
---|
594 | { |
---|
595 | unsigned long *g = new unsigned long[dega + 1]; |
---|
596 | // initialize entries of g to zero |
---|
597 | for(int i = 0; i <= dega; i++) |
---|
598 | { |
---|
599 | g[i] = 0; |
---|
600 | } |
---|
601 | |
---|
602 | int degg = gcd (g, a, b, p, dega, degb); |
---|
603 | |
---|
604 | if(degg > 0) |
---|
605 | { |
---|
606 | // non-trivial gcd, so compute a = (a/g) |
---|
607 | quo (a, g, p, dega, degg); |
---|
608 | } |
---|
609 | mult (l, a, b, p, dega, degb); |
---|
610 | |
---|
611 | // normalize |
---|
612 | if(l[dega + degb + 1] != 1) |
---|
613 | { |
---|
614 | unsigned long inv = modularInverse (l[dega + degb], p); |
---|
615 | for(int i = 0; i <= dega + degb; i++) |
---|
616 | { |
---|
617 | l[i] *= inv; |
---|
618 | l[i] %= p; |
---|
619 | } |
---|
620 | } |
---|
621 | |
---|
622 | return dega + degb; |
---|
623 | } |
---|
624 | |
---|
625 | |
---|
626 | // computes the gcd and the cofactors of u and v |
---|
627 | // gcd(u,v) = u3 = u*u1 + v*u2 |
---|
628 | long modularInverse (long x, long p) |
---|
629 | { |
---|
630 | long u1 = 1; |
---|
631 | long u2 = 0; |
---|
632 | long u3 = x; |
---|
633 | long v1 = 0; |
---|
634 | long v2 = 1; |
---|
635 | long v3 = p; |
---|
636 | |
---|
637 | long q, t1, t2, t3; |
---|
638 | while(v3 != 0) |
---|
639 | { |
---|
640 | q = u3 / v3; |
---|
641 | t1 = u1 - q * v1; |
---|
642 | t2 = u2 - q * v2; |
---|
643 | t3 = u3 - q * v3; |
---|
644 | u1 = v1; |
---|
645 | u2 = v2; |
---|
646 | u3 = v3; |
---|
647 | v1 = t1; |
---|
648 | v2 = t2; |
---|
649 | v3 = t3; |
---|
650 | } |
---|
651 | |
---|
652 | if(u1 < 0) |
---|
653 | { |
---|
654 | u1 += p; |
---|
655 | } |
---|
656 | |
---|
657 | return u1; |
---|
658 | } |
---|