1 | |
---|
2 | #include <NTL/LLL.h> |
---|
3 | #include <NTL/fileio.h> |
---|
4 | #include <NTL/vec_xdouble.h> |
---|
5 | #include <NTL/vec_double.h> |
---|
6 | |
---|
7 | #include <NTL/new.h> |
---|
8 | |
---|
9 | NTL_START_IMPL |
---|
10 | |
---|
11 | |
---|
12 | static void RowTransform(vec_ZZ& A, vec_ZZ& B, const ZZ& MU1) |
---|
13 | // x = x - y*MU |
---|
14 | { |
---|
15 | static ZZ T, MU; |
---|
16 | long k; |
---|
17 | |
---|
18 | long n = A.length(); |
---|
19 | long i; |
---|
20 | |
---|
21 | MU = MU1; |
---|
22 | |
---|
23 | if (MU == 1) { |
---|
24 | for (i = 1; i <= n; i++) |
---|
25 | sub(A(i), A(i), B(i)); |
---|
26 | |
---|
27 | return; |
---|
28 | } |
---|
29 | |
---|
30 | if (MU == -1) { |
---|
31 | for (i = 1; i <= n; i++) |
---|
32 | add(A(i), A(i), B(i)); |
---|
33 | |
---|
34 | return; |
---|
35 | } |
---|
36 | |
---|
37 | if (MU == 0) return; |
---|
38 | |
---|
39 | if (NumTwos(MU) >= NTL_ZZ_NBITS) |
---|
40 | k = MakeOdd(MU); |
---|
41 | else |
---|
42 | k = 0; |
---|
43 | |
---|
44 | |
---|
45 | if (MU.WideSinglePrecision()) { |
---|
46 | long mu1; |
---|
47 | conv(mu1, MU); |
---|
48 | |
---|
49 | if (k > 0) { |
---|
50 | |
---|
51 | for (i = 1; i <= n; i++) { |
---|
52 | mul(T, B(i), mu1); |
---|
53 | LeftShift(T, T, k); |
---|
54 | sub(A(i), A(i), T); |
---|
55 | } |
---|
56 | |
---|
57 | } |
---|
58 | else { |
---|
59 | |
---|
60 | for (i = 1; i <= n; i++) { |
---|
61 | MulSubFrom(A(i), B(i), mu1); |
---|
62 | } |
---|
63 | |
---|
64 | } |
---|
65 | } |
---|
66 | else { |
---|
67 | for (i = 1; i <= n; i++) { |
---|
68 | mul(T, B(i), MU); |
---|
69 | if (k > 0) LeftShift(T, T, k); |
---|
70 | sub(A(i), A(i), T); |
---|
71 | } |
---|
72 | } |
---|
73 | } |
---|
74 | |
---|
75 | |
---|
76 | static void RowTransform2(vec_ZZ& A, vec_ZZ& B, const ZZ& MU1) |
---|
77 | // x = x + y*MU |
---|
78 | { |
---|
79 | static ZZ T, MU; |
---|
80 | long k; |
---|
81 | |
---|
82 | long n = A.length(); |
---|
83 | long i; |
---|
84 | |
---|
85 | MU = MU1; |
---|
86 | |
---|
87 | if (MU == 1) { |
---|
88 | for (i = 1; i <= n; i++) |
---|
89 | add(A(i), A(i), B(i)); |
---|
90 | |
---|
91 | return; |
---|
92 | } |
---|
93 | |
---|
94 | if (MU == -1) { |
---|
95 | for (i = 1; i <= n; i++) |
---|
96 | sub(A(i), A(i), B(i)); |
---|
97 | |
---|
98 | return; |
---|
99 | } |
---|
100 | |
---|
101 | if (MU == 0) return; |
---|
102 | |
---|
103 | if (NumTwos(MU) >= NTL_ZZ_NBITS) |
---|
104 | k = MakeOdd(MU); |
---|
105 | else |
---|
106 | k = 0; |
---|
107 | |
---|
108 | if (MU.WideSinglePrecision()) { |
---|
109 | long mu1; |
---|
110 | conv(mu1, MU); |
---|
111 | |
---|
112 | for (i = 1; i <= n; i++) { |
---|
113 | mul(T, B(i), mu1); |
---|
114 | if (k > 0) LeftShift(T, T, k); |
---|
115 | add(A(i), A(i), T); |
---|
116 | } |
---|
117 | } |
---|
118 | else { |
---|
119 | for (i = 1; i <= n; i++) { |
---|
120 | mul(T, B(i), MU); |
---|
121 | if (k > 0) LeftShift(T, T, k); |
---|
122 | add(A(i), A(i), T); |
---|
123 | } |
---|
124 | } |
---|
125 | } |
---|
126 | |
---|
127 | |
---|
128 | class GivensCache_XD { |
---|
129 | public: |
---|
130 | GivensCache_XD(long m, long n); |
---|
131 | ~GivensCache_XD(); |
---|
132 | |
---|
133 | void flush(); |
---|
134 | void selective_flush(long l); |
---|
135 | void swap(long l); |
---|
136 | void swap(); |
---|
137 | void touch(); |
---|
138 | void incr(); |
---|
139 | |
---|
140 | long sz; |
---|
141 | |
---|
142 | xdouble **buf; |
---|
143 | long *bl; |
---|
144 | long *bv; |
---|
145 | long bp; |
---|
146 | }; |
---|
147 | |
---|
148 | |
---|
149 | GivensCache_XD::GivensCache_XD(long m, long n) |
---|
150 | { |
---|
151 | sz = min(m, n)/10; |
---|
152 | if (sz < 2) |
---|
153 | sz = 2; |
---|
154 | else if (sz > 20) |
---|
155 | sz = 20; |
---|
156 | |
---|
157 | typedef xdouble *xdoubleptr; |
---|
158 | |
---|
159 | long i; |
---|
160 | buf = NTL_NEW_OP xdoubleptr[sz]; |
---|
161 | if (!buf) Error("out of memory"); |
---|
162 | for (i = 0; i < sz; i++) |
---|
163 | if (!(buf[i] = NTL_NEW_OP xdouble[n+1])) Error("out of memory"); |
---|
164 | |
---|
165 | bl = NTL_NEW_OP long[sz]; |
---|
166 | if (!bl) Error("out of memory"); |
---|
167 | for (i = 0; i < sz; i++) bl[0] = 0; |
---|
168 | |
---|
169 | bv = NTL_NEW_OP long[sz]; |
---|
170 | if (!bv) Error("out of memory"); |
---|
171 | for (i = 0; i < sz; i++) bv[0] = 0; |
---|
172 | |
---|
173 | bp = 0; |
---|
174 | } |
---|
175 | |
---|
176 | GivensCache_XD::~GivensCache_XD() |
---|
177 | { |
---|
178 | long i; |
---|
179 | |
---|
180 | for (i = 0; i < sz; i++) delete [] buf[i]; |
---|
181 | delete [] buf; |
---|
182 | delete [] bl; |
---|
183 | delete [] bv; |
---|
184 | } |
---|
185 | |
---|
186 | void GivensCache_XD::flush() |
---|
187 | { |
---|
188 | long i; |
---|
189 | for (i = 0; i < sz; i++) bl[i] = 0; |
---|
190 | } |
---|
191 | |
---|
192 | void GivensCache_XD::selective_flush(long l) |
---|
193 | { |
---|
194 | long i; |
---|
195 | |
---|
196 | for (i = 0; i < sz; i++) |
---|
197 | if (bl[i] && bv[i] >= l) |
---|
198 | bl[i] = 0; |
---|
199 | } |
---|
200 | |
---|
201 | void GivensCache_XD::swap(long l) |
---|
202 | { |
---|
203 | long k = bl[bp]; |
---|
204 | long i; |
---|
205 | |
---|
206 | i = 0; |
---|
207 | while (i < sz && bl[i] != l) |
---|
208 | i++; |
---|
209 | |
---|
210 | if (i < sz) { |
---|
211 | bl[bp] = l; |
---|
212 | bl[i] = k; |
---|
213 | } |
---|
214 | else |
---|
215 | bl[bp] = l; |
---|
216 | |
---|
217 | selective_flush(l); |
---|
218 | } |
---|
219 | |
---|
220 | void GivensCache_XD::swap() |
---|
221 | { |
---|
222 | swap(bl[bp] - 1); |
---|
223 | } |
---|
224 | |
---|
225 | void GivensCache_XD::touch() |
---|
226 | { |
---|
227 | long k = bl[bp]; |
---|
228 | bl[bp] = 0; |
---|
229 | selective_flush(k); |
---|
230 | } |
---|
231 | |
---|
232 | void GivensCache_XD::incr() |
---|
233 | { |
---|
234 | long k = bl[bp]; |
---|
235 | long k1 = k+1; |
---|
236 | long i; |
---|
237 | |
---|
238 | i = 0; |
---|
239 | while (i < sz && bl[i] != k1) |
---|
240 | i++; |
---|
241 | |
---|
242 | if (i < sz) { |
---|
243 | bp = i; |
---|
244 | return; |
---|
245 | } |
---|
246 | |
---|
247 | i = 0; |
---|
248 | while (i < sz && bl[i] != 0) |
---|
249 | i++; |
---|
250 | |
---|
251 | if (i < sz) { |
---|
252 | bp = i; |
---|
253 | return; |
---|
254 | } |
---|
255 | |
---|
256 | long max_val = 0; |
---|
257 | long max_index = 0; |
---|
258 | for (i = 0; i < sz; i++) { |
---|
259 | long t = labs(bl[i]-k1); |
---|
260 | if (t > max_val) { |
---|
261 | max_val = t; |
---|
262 | max_index = i; |
---|
263 | } |
---|
264 | } |
---|
265 | |
---|
266 | bp = max_index; |
---|
267 | bl[max_index] = 0; |
---|
268 | } |
---|
269 | |
---|
270 | |
---|
271 | static |
---|
272 | void GivensComputeGS(xdouble **B1, xdouble **mu, xdouble **aux, long k, long n, |
---|
273 | GivensCache_XD& cache) |
---|
274 | { |
---|
275 | long i, j; |
---|
276 | |
---|
277 | xdouble c, s, a, b, t; |
---|
278 | |
---|
279 | xdouble *p = mu[k]; |
---|
280 | |
---|
281 | xdouble *pp = cache.buf[cache.bp]; |
---|
282 | |
---|
283 | if (!cache.bl[cache.bp]) { |
---|
284 | for (j = 1; j <= n; j++) |
---|
285 | pp[j] = B1[k][j]; |
---|
286 | |
---|
287 | long backoff; |
---|
288 | backoff = k/4; |
---|
289 | if (backoff < 2) |
---|
290 | backoff = 2; |
---|
291 | else if (backoff > cache.sz + 2) |
---|
292 | backoff = cache.sz + 2; |
---|
293 | |
---|
294 | long ub = k-(backoff-1); |
---|
295 | |
---|
296 | for (i = 1; i < ub; i++) { |
---|
297 | xdouble *cptr = mu[i]; |
---|
298 | xdouble *sptr = aux[i]; |
---|
299 | |
---|
300 | for (j = n; j > i; j--) { |
---|
301 | c = cptr[j]; |
---|
302 | s = sptr[j]; |
---|
303 | |
---|
304 | a = c*pp[j-1] - s*pp[j]; |
---|
305 | b = s*pp[j-1] + c*pp[j]; |
---|
306 | |
---|
307 | pp[j-1] = a; |
---|
308 | pp[j] = b; |
---|
309 | } |
---|
310 | |
---|
311 | pp[i] = pp[i]/mu[i][i]; |
---|
312 | } |
---|
313 | |
---|
314 | cache.bl[cache.bp] = k; |
---|
315 | cache.bv[cache.bp] = k-backoff; |
---|
316 | } |
---|
317 | |
---|
318 | for (j = 1; j <= n; j++) |
---|
319 | p[j] = pp[j]; |
---|
320 | |
---|
321 | for (i = max(cache.bv[cache.bp]+1, 1); i < k; i++) { |
---|
322 | xdouble *cptr = mu[i]; |
---|
323 | xdouble *sptr = aux[i]; |
---|
324 | |
---|
325 | for (j = n; j > i; j--) { |
---|
326 | c = cptr[j]; |
---|
327 | s = sptr[j]; |
---|
328 | |
---|
329 | a = c*p[j-1] - s*p[j]; |
---|
330 | b = s*p[j-1] + c*p[j]; |
---|
331 | |
---|
332 | p[j-1] = a; |
---|
333 | p[j] = b; |
---|
334 | } |
---|
335 | |
---|
336 | p[i] = p[i]/mu[i][i]; |
---|
337 | } |
---|
338 | |
---|
339 | for (j = n; j > k; j--) { |
---|
340 | a = p[j-1]; |
---|
341 | b = p[j]; |
---|
342 | |
---|
343 | if (b == 0) { |
---|
344 | c = 1; |
---|
345 | s = 0; |
---|
346 | } |
---|
347 | else if (fabs(b) > fabs(a)) { |
---|
348 | t = -a/b; |
---|
349 | s = 1/sqrt(1 + t*t); |
---|
350 | c = s*t; |
---|
351 | } |
---|
352 | else { |
---|
353 | t = -b/a; |
---|
354 | c = 1/sqrt(1 + t*t); |
---|
355 | s = c*t; |
---|
356 | } |
---|
357 | |
---|
358 | p[j-1] = c*a - s*b; |
---|
359 | p[j] = c; |
---|
360 | aux[k][j] = s; |
---|
361 | } |
---|
362 | |
---|
363 | if (k > n+1) Error("G_LLL_XD: internal error"); |
---|
364 | if (k > n) p[k] = 0; |
---|
365 | } |
---|
366 | |
---|
367 | static xdouble red_fudge = to_xdouble(0); |
---|
368 | static long log_red = 0; |
---|
369 | |
---|
370 | static void init_red_fudge() |
---|
371 | { |
---|
372 | long i; |
---|
373 | |
---|
374 | log_red = long(0.50*NTL_DOUBLE_PRECISION); |
---|
375 | red_fudge = 1; |
---|
376 | |
---|
377 | for (i = log_red; i > 0; i--) |
---|
378 | red_fudge = red_fudge*0.5; |
---|
379 | } |
---|
380 | |
---|
381 | static void inc_red_fudge() |
---|
382 | { |
---|
383 | |
---|
384 | red_fudge = red_fudge * 2; |
---|
385 | log_red--; |
---|
386 | |
---|
387 | //cerr << "G_LLL_XD: warning--relaxing reduction (" << log_red << ")\n"; |
---|
388 | |
---|
389 | if (log_red < 4) |
---|
390 | Error("G_LLL_XD: can not continue...sorry"); |
---|
391 | } |
---|
392 | |
---|
393 | |
---|
394 | |
---|
395 | static long verbose = 0; |
---|
396 | |
---|
397 | static unsigned long NumSwaps = 0; |
---|
398 | static double StartTime = 0; |
---|
399 | static double LastTime = 0; |
---|
400 | |
---|
401 | |
---|
402 | |
---|
403 | static |
---|
404 | long ll_G_LLL_XD(mat_ZZ& B, mat_ZZ* U, xdouble delta, long deep, |
---|
405 | LLLCheckFct check, xdouble **B1, xdouble **mu, |
---|
406 | xdouble **aux, |
---|
407 | long m, long init_k, long &quit, GivensCache_XD& cache) |
---|
408 | { |
---|
409 | long n = B.NumCols(); |
---|
410 | |
---|
411 | long i, j, k, Fc1; |
---|
412 | ZZ MU; |
---|
413 | xdouble mu1; |
---|
414 | |
---|
415 | xdouble t1; |
---|
416 | ZZ T1; |
---|
417 | xdouble *tp; |
---|
418 | |
---|
419 | |
---|
420 | xdouble half = to_xdouble(0.5); |
---|
421 | xdouble half_plus_fudge = 0.5 + red_fudge; |
---|
422 | |
---|
423 | quit = 0; |
---|
424 | k = init_k; |
---|
425 | |
---|
426 | long counter; |
---|
427 | |
---|
428 | long trigger_index; |
---|
429 | long small_trigger; |
---|
430 | long cnt; |
---|
431 | |
---|
432 | long max_k = 0; |
---|
433 | |
---|
434 | double tt; |
---|
435 | |
---|
436 | cache.flush(); |
---|
437 | |
---|
438 | while (k <= m) { |
---|
439 | |
---|
440 | if (k > max_k) { |
---|
441 | max_k = k; |
---|
442 | } |
---|
443 | |
---|
444 | GivensComputeGS(B1, mu, aux, k, n, cache); |
---|
445 | |
---|
446 | counter = 0; |
---|
447 | trigger_index = k; |
---|
448 | small_trigger = 0; |
---|
449 | cnt = 0; |
---|
450 | |
---|
451 | do { |
---|
452 | // size reduction |
---|
453 | |
---|
454 | counter++; |
---|
455 | if (counter > 10000) { |
---|
456 | Error("G_LLL_XD: warning--possible infinite loop\n"); |
---|
457 | counter = 0; |
---|
458 | } |
---|
459 | |
---|
460 | |
---|
461 | Fc1 = 0; |
---|
462 | |
---|
463 | for (j = k-1; j >= 1; j--) { |
---|
464 | t1 = fabs(mu[k][j]); |
---|
465 | if (t1 > half_plus_fudge) { |
---|
466 | |
---|
467 | if (!Fc1) { |
---|
468 | if (j > trigger_index || |
---|
469 | (j == trigger_index && small_trigger)) { |
---|
470 | |
---|
471 | cnt++; |
---|
472 | |
---|
473 | if (cnt > 10) { |
---|
474 | inc_red_fudge(); |
---|
475 | half_plus_fudge = 0.5 + red_fudge; |
---|
476 | cnt = 0; |
---|
477 | } |
---|
478 | } |
---|
479 | |
---|
480 | trigger_index = j; |
---|
481 | small_trigger = (t1 < 4); |
---|
482 | } |
---|
483 | |
---|
484 | |
---|
485 | Fc1 = 1; |
---|
486 | |
---|
487 | mu1 = mu[k][j]; |
---|
488 | if (mu1 >= 0) |
---|
489 | mu1 = ceil(mu1-half); |
---|
490 | else |
---|
491 | mu1 = floor(mu1+half); |
---|
492 | |
---|
493 | |
---|
494 | xdouble *mu_k = mu[k]; |
---|
495 | xdouble *mu_j = mu[j]; |
---|
496 | |
---|
497 | if (mu1 == 1) { |
---|
498 | for (i = 1; i <= j-1; i++) |
---|
499 | mu_k[i] -= mu_j[i]; |
---|
500 | } |
---|
501 | else if (mu1 == -1) { |
---|
502 | for (i = 1; i <= j-1; i++) |
---|
503 | mu_k[i] += mu_j[i]; |
---|
504 | } |
---|
505 | else { |
---|
506 | for (i = 1; i <= j-1; i++) |
---|
507 | MulSub(mu_k[i], mu_k[i], mu1, mu_j[i]); |
---|
508 | } |
---|
509 | |
---|
510 | mu_k[j] -= mu1; |
---|
511 | |
---|
512 | conv(MU, mu1); |
---|
513 | |
---|
514 | // cout << j << " " << MU << "\n"; |
---|
515 | |
---|
516 | RowTransform(B(k), B(j), MU); |
---|
517 | if (U) RowTransform((*U)(k), (*U)(j), MU); |
---|
518 | } |
---|
519 | } |
---|
520 | |
---|
521 | if (Fc1) { |
---|
522 | for (i = 1; i <= n; i++) |
---|
523 | conv(B1[k][i], B(k, i)); |
---|
524 | cache.touch(); |
---|
525 | GivensComputeGS(B1, mu, aux, k, n, cache); |
---|
526 | } |
---|
527 | } while (Fc1); |
---|
528 | |
---|
529 | if (check && (*check)(B(k))) |
---|
530 | quit = 1; |
---|
531 | |
---|
532 | if (IsZero(B(k))) { |
---|
533 | for (i = k; i < m; i++) { |
---|
534 | // swap i, i+1 |
---|
535 | swap(B(i), B(i+1)); |
---|
536 | tp = B1[i]; B1[i] = B1[i+1]; B1[i+1] = tp; |
---|
537 | if (U) swap((*U)(i), (*U)(i+1)); |
---|
538 | } |
---|
539 | |
---|
540 | cache.flush(); |
---|
541 | |
---|
542 | m--; |
---|
543 | if (quit) break; |
---|
544 | continue; |
---|
545 | } |
---|
546 | |
---|
547 | if (quit) break; |
---|
548 | |
---|
549 | if (deep > 0) { |
---|
550 | // deep insertions |
---|
551 | |
---|
552 | Error("sorry...deep insertions not implemented"); |
---|
553 | } // end deep insertions |
---|
554 | |
---|
555 | // test G_LLL reduction condition |
---|
556 | |
---|
557 | if (k > 1 && |
---|
558 | (delta - mu[k][k-1]*mu[k][k-1])*(mu[k-1][k-1])*(mu[k-1][k-1]) > |
---|
559 | (mu[k][k])*(mu[k][k])) { |
---|
560 | |
---|
561 | // swap rows k, k-1 |
---|
562 | swap(B(k), B(k-1)); |
---|
563 | tp = B1[k]; B1[k] = B1[k-1]; B1[k-1] = tp; |
---|
564 | if (U) swap((*U)(k), (*U)(k-1)); |
---|
565 | |
---|
566 | cache.swap(); |
---|
567 | |
---|
568 | k--; |
---|
569 | NumSwaps++; |
---|
570 | |
---|
571 | // cout << "- " << k << "\n"; |
---|
572 | } |
---|
573 | else { |
---|
574 | cache.incr(); |
---|
575 | k++; |
---|
576 | // cout << "+ " << k << "\n"; |
---|
577 | } |
---|
578 | } |
---|
579 | |
---|
580 | return m; |
---|
581 | } |
---|
582 | |
---|
583 | |
---|
584 | |
---|
585 | |
---|
586 | static |
---|
587 | long G_LLL_XD(mat_ZZ& B, mat_ZZ* U, xdouble delta, long deep, |
---|
588 | LLLCheckFct check) |
---|
589 | { |
---|
590 | long m = B.NumRows(); |
---|
591 | long n = B.NumCols(); |
---|
592 | |
---|
593 | long i, j; |
---|
594 | long new_m, dep, quit; |
---|
595 | xdouble s; |
---|
596 | ZZ MU; |
---|
597 | xdouble mu1; |
---|
598 | |
---|
599 | xdouble t1; |
---|
600 | ZZ T1; |
---|
601 | |
---|
602 | init_red_fudge(); |
---|
603 | |
---|
604 | if (U) ident(*U, m); |
---|
605 | |
---|
606 | xdouble **B1; // approximates B |
---|
607 | |
---|
608 | typedef xdouble *xdoubleptr; |
---|
609 | |
---|
610 | B1 = NTL_NEW_OP xdoubleptr[m+1]; |
---|
611 | if (!B1) Error("G_LLL_XD: out of memory"); |
---|
612 | |
---|
613 | for (i = 1; i <= m; i++) { |
---|
614 | B1[i] = NTL_NEW_OP xdouble[n+1]; |
---|
615 | if (!B1[i]) Error("G_LLL_XD: out of memory"); |
---|
616 | } |
---|
617 | |
---|
618 | xdouble **mu; |
---|
619 | mu = NTL_NEW_OP xdoubleptr[m+1]; |
---|
620 | if (!mu) Error("G_LLL_XD: out of memory"); |
---|
621 | |
---|
622 | for (i = 1; i <= m; i++) { |
---|
623 | mu[i] = NTL_NEW_OP xdouble[n+2]; |
---|
624 | if (!mu[i]) Error("G_LLL_XD: out of memory"); |
---|
625 | } |
---|
626 | |
---|
627 | xdouble **aux; |
---|
628 | aux = NTL_NEW_OP xdoubleptr[m+1]; |
---|
629 | if (!aux) Error("G_LLL_XD: out of memory"); |
---|
630 | |
---|
631 | for (i = 1; i <= m; i++) { |
---|
632 | aux[i] = NTL_NEW_OP xdouble[n+1]; |
---|
633 | if (!aux[i]) Error("G_LLL_XD: out of memory"); |
---|
634 | } |
---|
635 | |
---|
636 | for (i = 1; i <=m; i++) |
---|
637 | for (j = 1; j <= n; j++) |
---|
638 | conv(B1[i][j], B(i, j)); |
---|
639 | |
---|
640 | GivensCache_XD cache(m, n); |
---|
641 | |
---|
642 | new_m = |
---|
643 | ll_G_LLL_XD(B, U, delta, deep, check, B1, mu, aux, m, 1, quit, cache); |
---|
644 | |
---|
645 | dep = m - new_m; |
---|
646 | m = new_m; |
---|
647 | |
---|
648 | if (dep > 0) { |
---|
649 | // for consistency, we move all of the zero rows to the front |
---|
650 | |
---|
651 | for (i = 0; i < m; i++) { |
---|
652 | swap(B(m+dep-i), B(m-i)); |
---|
653 | if (U) swap((*U)(m+dep-i), (*U)(m-i)); |
---|
654 | } |
---|
655 | } |
---|
656 | |
---|
657 | |
---|
658 | // clean-up |
---|
659 | |
---|
660 | for (i = 1; i <= m+dep; i++) { |
---|
661 | delete [] B1[i]; |
---|
662 | } |
---|
663 | |
---|
664 | delete [] B1; |
---|
665 | |
---|
666 | for (i = 1; i <= m+dep; i++) { |
---|
667 | delete [] mu[i]; |
---|
668 | } |
---|
669 | |
---|
670 | delete [] mu; |
---|
671 | |
---|
672 | for (i = 1; i <= m+dep; i++) { |
---|
673 | delete [] aux[i]; |
---|
674 | } |
---|
675 | |
---|
676 | delete [] aux; |
---|
677 | |
---|
678 | return m; |
---|
679 | } |
---|
680 | |
---|
681 | |
---|
682 | |
---|
683 | long G_LLL_XD(mat_ZZ& B, double delta, long deep, |
---|
684 | LLLCheckFct check, long verb) |
---|
685 | { |
---|
686 | verbose = verb; |
---|
687 | NumSwaps = 0; |
---|
688 | |
---|
689 | if (delta < 0.50 || delta >= 1) Error("G_LLL_XD: bad delta"); |
---|
690 | if (deep < 0) Error("G_LLL_XD: bad deep"); |
---|
691 | return G_LLL_XD(B, 0, to_xdouble(delta), deep, check); |
---|
692 | } |
---|
693 | |
---|
694 | long G_LLL_XD(mat_ZZ& B, mat_ZZ& U, double delta, long deep, |
---|
695 | LLLCheckFct check, long verb) |
---|
696 | { |
---|
697 | verbose = verb; |
---|
698 | NumSwaps = 0; |
---|
699 | |
---|
700 | if (delta < 0.50 || delta >= 1) Error("G_LLL_XD: bad delta"); |
---|
701 | if (deep < 0) Error("G_LLL_XD: bad deep"); |
---|
702 | return G_LLL_XD(B, &U, to_xdouble(delta), deep, check); |
---|
703 | } |
---|
704 | |
---|
705 | |
---|
706 | |
---|
707 | static vec_xdouble G_BKZConstant; |
---|
708 | |
---|
709 | static |
---|
710 | void ComputeG_BKZConstant(long beta, long p) |
---|
711 | { |
---|
712 | const double c_PI = 3.14159265358979323846264338328; |
---|
713 | const double LogPI = 1.14472988584940017414342735135; |
---|
714 | |
---|
715 | G_BKZConstant.SetLength(beta-1); |
---|
716 | |
---|
717 | vec_double Log; |
---|
718 | Log.SetLength(beta); |
---|
719 | |
---|
720 | |
---|
721 | long i, j, k; |
---|
722 | double x, y; |
---|
723 | |
---|
724 | for (j = 1; j <= beta; j++) |
---|
725 | Log(j) = log(double(j)); |
---|
726 | |
---|
727 | for (i = 1; i <= beta-1; i++) { |
---|
728 | // First, we compute x = gamma(i/2)^{2/i} |
---|
729 | |
---|
730 | k = i/2; |
---|
731 | |
---|
732 | if ((i & 1) == 0) { // i even |
---|
733 | x = 0; |
---|
734 | for (j = 1; j <= k; j++) |
---|
735 | x = x + Log(j); |
---|
736 | |
---|
737 | x = x * (1/double(k)); |
---|
738 | |
---|
739 | x = exp(x); |
---|
740 | } |
---|
741 | else { // i odd |
---|
742 | x = 0; |
---|
743 | for (j = k + 2; j <= 2*k + 2; j++) |
---|
744 | x = x + Log(j); |
---|
745 | |
---|
746 | x = 0.5*LogPI + x - 2*(k+1)*Log(2); |
---|
747 | |
---|
748 | x = x * (2.0/double(i)); |
---|
749 | |
---|
750 | x = exp(x); |
---|
751 | } |
---|
752 | |
---|
753 | // Second, we compute y = 2^{2*p/i} |
---|
754 | |
---|
755 | y = -(2*p/double(i))*Log(2); |
---|
756 | y = exp(y); |
---|
757 | |
---|
758 | G_BKZConstant(i) = x*y/c_PI; |
---|
759 | } |
---|
760 | } |
---|
761 | |
---|
762 | static vec_xdouble G_BKZThresh; |
---|
763 | |
---|
764 | static |
---|
765 | void ComputeG_BKZThresh(xdouble *c, long beta) |
---|
766 | { |
---|
767 | G_BKZThresh.SetLength(beta-1); |
---|
768 | |
---|
769 | long i; |
---|
770 | double x; |
---|
771 | |
---|
772 | x = 0; |
---|
773 | |
---|
774 | for (i = 1; i <= beta-1; i++) { |
---|
775 | x += log(c[i-1]); |
---|
776 | G_BKZThresh(i) = xexp(x/double(i))*G_BKZConstant(i); |
---|
777 | } |
---|
778 | } |
---|
779 | |
---|
780 | |
---|
781 | static |
---|
782 | long G_BKZ_XD(mat_ZZ& BB, mat_ZZ* UU, xdouble delta, |
---|
783 | long beta, long prune, LLLCheckFct check) |
---|
784 | { |
---|
785 | long m = BB.NumRows(); |
---|
786 | long n = BB.NumCols(); |
---|
787 | long m_orig = m; |
---|
788 | |
---|
789 | long i, j; |
---|
790 | ZZ MU; |
---|
791 | |
---|
792 | xdouble t1; |
---|
793 | ZZ T1; |
---|
794 | xdouble *tp; |
---|
795 | |
---|
796 | init_red_fudge(); |
---|
797 | |
---|
798 | mat_ZZ B; |
---|
799 | B = BB; |
---|
800 | |
---|
801 | B.SetDims(m+1, n); |
---|
802 | |
---|
803 | |
---|
804 | xdouble **B1; // approximates B |
---|
805 | |
---|
806 | typedef xdouble *xdoubleptr; |
---|
807 | |
---|
808 | B1 = NTL_NEW_OP xdoubleptr[m+2]; |
---|
809 | if (!B1) Error("G_BKZ_XD: out of memory"); |
---|
810 | |
---|
811 | for (i = 1; i <= m+1; i++) { |
---|
812 | B1[i] = NTL_NEW_OP xdouble[n+1]; |
---|
813 | if (!B1[i]) Error("G_BKZ_XD: out of memory"); |
---|
814 | } |
---|
815 | |
---|
816 | xdouble **mu; |
---|
817 | mu = NTL_NEW_OP xdoubleptr[m+2]; |
---|
818 | if (!mu) Error("G_BKZ_XD: out of memory"); |
---|
819 | |
---|
820 | for (i = 1; i <= m+1; i++) { |
---|
821 | mu[i] = NTL_NEW_OP xdouble[n+2]; |
---|
822 | if (!mu[i]) Error("G_BKZ_XD: out of memory"); |
---|
823 | } |
---|
824 | |
---|
825 | xdouble **aux; |
---|
826 | aux = NTL_NEW_OP xdoubleptr[m+2]; |
---|
827 | if (!aux) Error("G_BKZ_XD: out of memory"); |
---|
828 | |
---|
829 | for (i = 1; i <= m+1; i++) { |
---|
830 | aux[i] = NTL_NEW_OP xdouble[n+1]; |
---|
831 | if (!aux[i]) Error("G_BKZ_XD: out of memory"); |
---|
832 | } |
---|
833 | |
---|
834 | xdouble *c; // squared lengths of Gramm-Schmidt basis vectors |
---|
835 | |
---|
836 | c = NTL_NEW_OP xdouble[m+2]; |
---|
837 | if (!c) Error("G_BKZ_XD: out of memory"); |
---|
838 | |
---|
839 | xdouble cbar; |
---|
840 | |
---|
841 | xdouble *ctilda; |
---|
842 | ctilda = NTL_NEW_OP xdouble[m+2]; |
---|
843 | if (!ctilda) Error("G_BKZ_XD: out of memory"); |
---|
844 | |
---|
845 | xdouble *vvec; |
---|
846 | vvec = NTL_NEW_OP xdouble[m+2]; |
---|
847 | if (!vvec) Error("G_BKZ_XD: out of memory"); |
---|
848 | |
---|
849 | xdouble *yvec; |
---|
850 | yvec = NTL_NEW_OP xdouble[m+2]; |
---|
851 | if (!yvec) Error("G_BKZ_XD: out of memory"); |
---|
852 | |
---|
853 | xdouble *uvec; |
---|
854 | uvec = NTL_NEW_OP xdouble[m+2]; |
---|
855 | if (!uvec) Error("G_BKZ_XD: out of memory"); |
---|
856 | |
---|
857 | xdouble *utildavec; |
---|
858 | utildavec = NTL_NEW_OP xdouble[m+2]; |
---|
859 | if (!utildavec) Error("G_BKZ_XD: out of memory"); |
---|
860 | |
---|
861 | |
---|
862 | long *Deltavec; |
---|
863 | Deltavec = NTL_NEW_OP long[m+2]; |
---|
864 | if (!Deltavec) Error("G_BKZ_XD: out of memory"); |
---|
865 | |
---|
866 | long *deltavec; |
---|
867 | deltavec = NTL_NEW_OP long[m+2]; |
---|
868 | if (!deltavec) Error("G_BKZ_XD: out of memory"); |
---|
869 | |
---|
870 | mat_ZZ Ulocal; |
---|
871 | mat_ZZ *U; |
---|
872 | |
---|
873 | if (UU) { |
---|
874 | Ulocal.SetDims(m+1, m); |
---|
875 | for (i = 1; i <= m; i++) |
---|
876 | conv(Ulocal(i, i), 1); |
---|
877 | U = &Ulocal; |
---|
878 | } |
---|
879 | else |
---|
880 | U = 0; |
---|
881 | |
---|
882 | long quit; |
---|
883 | long new_m; |
---|
884 | long z, jj, kk; |
---|
885 | long s, t; |
---|
886 | long h; |
---|
887 | xdouble eta; |
---|
888 | |
---|
889 | |
---|
890 | for (i = 1; i <=m; i++) |
---|
891 | for (j = 1; j <= n; j++) |
---|
892 | conv(B1[i][j], B(i, j)); |
---|
893 | |
---|
894 | // cerr << "\n"; |
---|
895 | // cerr << "first G_LLL\n"; |
---|
896 | |
---|
897 | GivensCache_XD cache(m, n); |
---|
898 | |
---|
899 | m = ll_G_LLL_XD(B, U, delta, 0, check, B1, mu, aux, m, 1, quit, cache); |
---|
900 | |
---|
901 | |
---|
902 | double tt; |
---|
903 | |
---|
904 | double enum_time = 0; |
---|
905 | unsigned long NumIterations = 0; |
---|
906 | unsigned long NumTrivial = 0; |
---|
907 | unsigned long NumNonTrivial = 0; |
---|
908 | unsigned long NumNoOps = 0; |
---|
909 | |
---|
910 | long verb = verbose; |
---|
911 | |
---|
912 | verbose = 0; |
---|
913 | |
---|
914 | |
---|
915 | |
---|
916 | if (m < m_orig) { |
---|
917 | for (i = m_orig+1; i >= m+2; i--) { |
---|
918 | // swap i, i-1 |
---|
919 | |
---|
920 | swap(B(i), B(i-1)); |
---|
921 | if (U) swap((*U)(i), (*U)(i-1)); |
---|
922 | } |
---|
923 | } |
---|
924 | |
---|
925 | long clean = 1; |
---|
926 | |
---|
927 | if (!quit && m > 1) { |
---|
928 | // cerr << "continuing\n"; |
---|
929 | if (beta > m) beta = m; |
---|
930 | |
---|
931 | if (prune > 0) |
---|
932 | ComputeG_BKZConstant(beta, prune); |
---|
933 | |
---|
934 | z = 0; |
---|
935 | jj = 0; |
---|
936 | |
---|
937 | while (z < m-1) { |
---|
938 | jj++; |
---|
939 | kk = min(jj+beta-1, m); |
---|
940 | |
---|
941 | if (jj == m) { |
---|
942 | jj = 1; |
---|
943 | kk = beta; |
---|
944 | clean = 1; |
---|
945 | } |
---|
946 | |
---|
947 | // ENUM |
---|
948 | |
---|
949 | double tt1; |
---|
950 | |
---|
951 | for (i = jj; i <= kk; i++) |
---|
952 | c[i] = mu[i][i]*mu[i][i]; |
---|
953 | |
---|
954 | if (prune > 0) |
---|
955 | ComputeG_BKZThresh(&c[jj], kk-jj+1); |
---|
956 | |
---|
957 | cbar = c[jj]; |
---|
958 | utildavec[jj] = uvec[jj] = 1; |
---|
959 | |
---|
960 | yvec[jj] = vvec[jj] = 0; |
---|
961 | Deltavec[jj] = 0; |
---|
962 | |
---|
963 | |
---|
964 | s = t = jj; |
---|
965 | deltavec[jj] = 1; |
---|
966 | |
---|
967 | for (i = jj+1; i <= kk+1; i++) { |
---|
968 | ctilda[i] = uvec[i] = utildavec[i] = yvec[i] = 0; |
---|
969 | Deltavec[i] = 0; |
---|
970 | vvec[i] = 0; |
---|
971 | deltavec[i] = 1; |
---|
972 | } |
---|
973 | |
---|
974 | long enum_cnt = 0; |
---|
975 | |
---|
976 | while (t <= kk) { |
---|
977 | |
---|
978 | ctilda[t] = ctilda[t+1] + |
---|
979 | (yvec[t]+utildavec[t])*(yvec[t]+utildavec[t])*c[t]; |
---|
980 | |
---|
981 | if (prune > 0 && t > jj) { |
---|
982 | eta = G_BKZThresh(t-jj); |
---|
983 | } |
---|
984 | else |
---|
985 | eta = 0; |
---|
986 | |
---|
987 | if (ctilda[t] < cbar - eta) { |
---|
988 | if (t > jj) { |
---|
989 | t--; |
---|
990 | t1 = 0; |
---|
991 | for (i = t+1; i <= s; i++) { |
---|
992 | t1 += utildavec[i]*mu[i][t]; |
---|
993 | } |
---|
994 | |
---|
995 | |
---|
996 | yvec[t] = t1; |
---|
997 | t1 = -t1; |
---|
998 | if (t1 >= 0) |
---|
999 | t1 = ceil(t1-0.5); |
---|
1000 | else |
---|
1001 | t1 = floor(t1+0.5); |
---|
1002 | |
---|
1003 | utildavec[t] = vvec[t] = t1; |
---|
1004 | Deltavec[t] = 0; |
---|
1005 | if (utildavec[t] > -yvec[t]) |
---|
1006 | deltavec[t] = -1; |
---|
1007 | else |
---|
1008 | deltavec[t] = 1; |
---|
1009 | } |
---|
1010 | else { |
---|
1011 | cbar = ctilda[jj]; |
---|
1012 | for (i = jj; i <= kk; i++) { |
---|
1013 | uvec[i] = utildavec[i]; |
---|
1014 | } |
---|
1015 | } |
---|
1016 | } |
---|
1017 | else { |
---|
1018 | t++; |
---|
1019 | s = max(s, t); |
---|
1020 | if (t < s) Deltavec[t] = -Deltavec[t]; |
---|
1021 | if (Deltavec[t]*deltavec[t] >= 0) Deltavec[t] += deltavec[t]; |
---|
1022 | utildavec[t] = vvec[t] + Deltavec[t]; |
---|
1023 | } |
---|
1024 | } |
---|
1025 | |
---|
1026 | NumIterations++; |
---|
1027 | |
---|
1028 | h = min(kk+1, m); |
---|
1029 | |
---|
1030 | if ((delta-8*red_fudge)*c[jj] > cbar) { |
---|
1031 | |
---|
1032 | clean = 0; |
---|
1033 | |
---|
1034 | // we treat the case that the new vector is b_s (jj < s <= kk) |
---|
1035 | // as a special case that appears to occur most of the time. |
---|
1036 | |
---|
1037 | s = 0; |
---|
1038 | for (i = jj+1; i <= kk; i++) { |
---|
1039 | if (uvec[i] != 0) { |
---|
1040 | if (s == 0) |
---|
1041 | s = i; |
---|
1042 | else |
---|
1043 | s = -1; |
---|
1044 | } |
---|
1045 | } |
---|
1046 | |
---|
1047 | if (s == 0) Error("G_BKZ_XD: internal error"); |
---|
1048 | |
---|
1049 | if (s > 0) { |
---|
1050 | // special case |
---|
1051 | |
---|
1052 | NumTrivial++; |
---|
1053 | |
---|
1054 | for (i = s; i > jj; i--) { |
---|
1055 | // swap i, i-1 |
---|
1056 | swap(B(i-1), B(i)); |
---|
1057 | if (U) swap((*U)(i-1), (*U)(i)); |
---|
1058 | tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp; |
---|
1059 | } |
---|
1060 | |
---|
1061 | // cerr << "special case\n"; |
---|
1062 | new_m = ll_G_LLL_XD(B, U, delta, 0, check, |
---|
1063 | B1, mu, aux, h, jj, quit, cache); |
---|
1064 | if (new_m != h) Error("G_BKZ_XD: internal error"); |
---|
1065 | if (quit) break; |
---|
1066 | } |
---|
1067 | else { |
---|
1068 | // the general case |
---|
1069 | |
---|
1070 | NumNonTrivial++; |
---|
1071 | |
---|
1072 | for (i = 1; i <= n; i++) conv(B(m+1, i), 0); |
---|
1073 | |
---|
1074 | if (U) { |
---|
1075 | for (i = 1; i <= m_orig; i++) |
---|
1076 | conv((*U)(m+1, i), 0); |
---|
1077 | } |
---|
1078 | |
---|
1079 | for (i = jj; i <= kk; i++) { |
---|
1080 | if (uvec[i] == 0) continue; |
---|
1081 | conv(MU, uvec[i]); |
---|
1082 | RowTransform2(B(m+1), B(i), MU); |
---|
1083 | if (U) RowTransform2((*U)(m+1), (*U)(i), MU); |
---|
1084 | } |
---|
1085 | |
---|
1086 | for (i = m+1; i >= jj+1; i--) { |
---|
1087 | // swap i, i-1 |
---|
1088 | swap(B(i-1), B(i)); |
---|
1089 | if (U) swap((*U)(i-1), (*U)(i)); |
---|
1090 | tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp; |
---|
1091 | } |
---|
1092 | |
---|
1093 | for (i = 1; i <= n; i++) |
---|
1094 | conv(B1[jj][i], B(jj, i)); |
---|
1095 | |
---|
1096 | if (IsZero(B(jj))) Error("G_BKZ_XD: internal error"); |
---|
1097 | |
---|
1098 | // remove linear dependencies |
---|
1099 | |
---|
1100 | // cerr << "general case\n"; |
---|
1101 | new_m = ll_G_LLL_XD(B, U, delta, 0, 0, B1, mu, aux, |
---|
1102 | kk+1, jj, quit, cache); |
---|
1103 | |
---|
1104 | |
---|
1105 | if (new_m != kk) Error("G_BKZ_XD: internal error"); |
---|
1106 | |
---|
1107 | // remove zero vector |
---|
1108 | |
---|
1109 | for (i = kk+2; i <= m+1; i++) { |
---|
1110 | // swap i, i-1 |
---|
1111 | swap(B(i-1), B(i)); |
---|
1112 | if (U) swap((*U)(i-1), (*U)(i)); |
---|
1113 | tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp; |
---|
1114 | } |
---|
1115 | |
---|
1116 | quit = 0; |
---|
1117 | if (check) { |
---|
1118 | for (i = 1; i <= kk; i++) |
---|
1119 | if ((*check)(B(i))) { |
---|
1120 | quit = 1; |
---|
1121 | break; |
---|
1122 | } |
---|
1123 | } |
---|
1124 | |
---|
1125 | if (quit) break; |
---|
1126 | |
---|
1127 | if (h > kk) { |
---|
1128 | // extend reduced basis |
---|
1129 | |
---|
1130 | new_m = ll_G_LLL_XD(B, U, delta, 0, check, |
---|
1131 | B1, mu, aux, h, h, quit, cache); |
---|
1132 | |
---|
1133 | |
---|
1134 | if (new_m != h) Error("G_BKZ_XD: internal error"); |
---|
1135 | if (quit) break; |
---|
1136 | } |
---|
1137 | } |
---|
1138 | |
---|
1139 | z = 0; |
---|
1140 | } |
---|
1141 | else { |
---|
1142 | // G_LLL_XD |
---|
1143 | // cerr << "progress\n"; |
---|
1144 | |
---|
1145 | NumNoOps++; |
---|
1146 | |
---|
1147 | if (!clean) { |
---|
1148 | new_m = ll_G_LLL_XD(B, U, delta, 0, check, B1, mu, aux, |
---|
1149 | h, h, quit, cache); |
---|
1150 | if (new_m != h) Error("G_BKZ_XD: internal error"); |
---|
1151 | if (quit) break; |
---|
1152 | } |
---|
1153 | |
---|
1154 | z++; |
---|
1155 | } |
---|
1156 | } |
---|
1157 | } |
---|
1158 | |
---|
1159 | // clean up |
---|
1160 | |
---|
1161 | if (m_orig > m) { |
---|
1162 | // for consistency, we move zero vectors to the front |
---|
1163 | |
---|
1164 | for (i = m+1; i <= m_orig; i++) { |
---|
1165 | swap(B(i), B(i+1)); |
---|
1166 | if (U) swap((*U)(i), (*U)(i+1)); |
---|
1167 | } |
---|
1168 | |
---|
1169 | for (i = 0; i < m; i++) { |
---|
1170 | swap(B(m_orig-i), B(m-i)); |
---|
1171 | if (U) swap((*U)(m_orig-i), (*U)(m-i)); |
---|
1172 | } |
---|
1173 | } |
---|
1174 | |
---|
1175 | B.SetDims(m_orig, n); |
---|
1176 | BB = B; |
---|
1177 | |
---|
1178 | if (U) { |
---|
1179 | U->SetDims(m_orig, m_orig); |
---|
1180 | *UU = *U; |
---|
1181 | } |
---|
1182 | |
---|
1183 | for (i = 1; i <= m_orig+1; i++) { |
---|
1184 | delete [] B1[i]; |
---|
1185 | } |
---|
1186 | |
---|
1187 | delete [] B1; |
---|
1188 | |
---|
1189 | for (i = 1; i <= m_orig+1; i++) { |
---|
1190 | delete [] mu[i]; |
---|
1191 | } |
---|
1192 | |
---|
1193 | delete [] mu; |
---|
1194 | |
---|
1195 | for (i = 1; i <= m_orig+1; i++) { |
---|
1196 | delete [] aux[i]; |
---|
1197 | } |
---|
1198 | |
---|
1199 | delete [] aux; |
---|
1200 | |
---|
1201 | |
---|
1202 | delete [] c; |
---|
1203 | delete [] ctilda; |
---|
1204 | delete [] vvec; |
---|
1205 | delete [] yvec; |
---|
1206 | delete [] uvec; |
---|
1207 | delete [] utildavec; |
---|
1208 | delete [] Deltavec; |
---|
1209 | delete [] deltavec; |
---|
1210 | |
---|
1211 | return m; |
---|
1212 | } |
---|
1213 | |
---|
1214 | long G_BKZ_XD(mat_ZZ& BB, mat_ZZ& UU, double delta, |
---|
1215 | long beta, long prune, LLLCheckFct check, long verb) |
---|
1216 | { |
---|
1217 | verbose = verb; |
---|
1218 | NumSwaps = 0; |
---|
1219 | |
---|
1220 | if (delta < 0.50 || delta >= 1) Error("G_BKZ_XD: bad delta"); |
---|
1221 | if (beta < 2) Error("G_BKZ_XD: bad block size"); |
---|
1222 | |
---|
1223 | return G_BKZ_XD(BB, &UU, to_xdouble(delta), beta, prune, check); |
---|
1224 | } |
---|
1225 | |
---|
1226 | long G_BKZ_XD(mat_ZZ& BB, double delta, |
---|
1227 | long beta, long prune, LLLCheckFct check, long verb) |
---|
1228 | { |
---|
1229 | verbose = verb; |
---|
1230 | NumSwaps = 0; |
---|
1231 | |
---|
1232 | if (delta < 0.50 || delta >= 1) Error("G_BKZ_XD: bad delta"); |
---|
1233 | if (beta < 2) Error("G_BKZ_XD: bad block size"); |
---|
1234 | |
---|
1235 | return G_BKZ_XD(BB, 0, to_xdouble(delta), beta, prune, check); |
---|
1236 | } |
---|
1237 | |
---|
1238 | NTL_END_IMPL |
---|