Line data Source code
1 : /* Copyright (C) 2004 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : #include "pari.h"
16 : #include "paripriv.h"
17 :
18 : /* Not so fast arithmetic with polynomials with small coefficients. */
19 :
20 : static GEN
21 898594009 : get_Flx_red(GEN T, GEN *B)
22 : {
23 898594009 : if (typ(T)!=t_VEC) { *B=NULL; return T; }
24 631677 : *B = gel(T,1); return gel(T,2);
25 : }
26 :
27 : /***********************************************************************/
28 : /** Flx **/
29 : /***********************************************************************/
30 : /* Flx objects are defined as follows:
31 : * Let l an ulong. An Flx is a t_VECSMALL:
32 : * x[0] = codeword
33 : * x[1] = evalvarn(variable number) (signe is not stored).
34 : * x[2] = a_0 x[3] = a_1, etc. with 0 <= a_i < l
35 : *
36 : * signe(x) is not valid. Use degpol(x)>0 instead. */
37 : /***********************************************************************/
38 : /** Conversion from Flx **/
39 : /***********************************************************************/
40 :
41 : GEN
42 36154752 : Flx_to_ZX(GEN z)
43 : {
44 36154752 : long i, l = lg(z);
45 36154752 : GEN x = cgetg(l,t_POL);
46 238732716 : for (i=2; i<l; i++) gel(x,i) = utoi(z[i]);
47 36132493 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
48 : }
49 :
50 : GEN
51 71332 : Flx_to_FlxX(GEN z, long sv)
52 : {
53 71332 : long i, l = lg(z);
54 71332 : GEN x = cgetg(l,t_POL);
55 277929 : for (i=2; i<l; i++) gel(x,i) = Fl_to_Flx(z[i], sv);
56 71331 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
57 : }
58 :
59 : /* same as Flx_to_ZX, in place */
60 : GEN
61 36346523 : Flx_to_ZX_inplace(GEN z)
62 : {
63 36346523 : long i, l = lg(z);
64 226966135 : for (i=2; i<l; i++) gel(z,i) = utoi(z[i]);
65 36305216 : settyp(z, t_POL); z[1]=evalsigne(l-2!=0)|z[1]; return z;
66 : }
67 :
68 : /*Flx_to_Flv=zx_to_zv*/
69 : GEN
70 63076675 : Flx_to_Flv(GEN x, long N)
71 : {
72 63076675 : GEN z = cgetg(N+1,t_VECSMALL);
73 63065934 : long i, l = lg(x)-1;
74 63065934 : x++;
75 692408830 : for (i=1; i<l ; i++) z[i]=x[i];
76 323648382 : for ( ; i<=N; i++) z[i]=0;
77 63065934 : return z;
78 : }
79 :
80 : /*Flv_to_Flx=zv_to_zx*/
81 : GEN
82 26786257 : Flv_to_Flx(GEN x, long sv)
83 : {
84 26786257 : long i, l=lg(x)+1;
85 26786257 : GEN z = cgetg(l,t_VECSMALL); z[1]=sv;
86 26776907 : x--;
87 291296366 : for (i=2; i<l ; i++) z[i]=x[i];
88 26776907 : return Flx_renormalize(z,l);
89 : }
90 :
91 : /*Flm_to_FlxV=zm_to_zxV*/
92 : GEN
93 2268 : Flm_to_FlxV(GEN x, long sv)
94 6153 : { pari_APPLY_type(t_VEC, Flv_to_Flx(gel(x,i), sv)) }
95 :
96 : /*FlxC_to_ZXC=zxC_to_ZXC*/
97 : GEN
98 108837 : FlxC_to_ZXC(GEN x)
99 561563 : { pari_APPLY_type(t_COL, Flx_to_ZX(gel(x,i))) }
100 :
101 : /*FlxC_to_ZXC=zxV_to_ZXV*/
102 : GEN
103 593773 : FlxV_to_ZXV(GEN x)
104 2406710 : { pari_APPLY_type(t_VEC, Flx_to_ZX(gel(x,i))) }
105 :
106 : void
107 2785923 : FlxV_to_ZXV_inplace(GEN v)
108 : {
109 : long i;
110 7442938 : for(i=1;i<lg(v);i++) gel(v,i)= Flx_to_ZX(gel(v,i));
111 2785885 : }
112 :
113 : /*FlxM_to_ZXM=zxM_to_ZXM*/
114 : GEN
115 4570 : FlxM_to_ZXM(GEN x)
116 15760 : { pari_APPLY_same(FlxC_to_ZXC(gel(x,i))) }
117 :
118 : GEN
119 393971 : FlxV_to_FlxX(GEN x, long v)
120 : {
121 393971 : long i, l = lg(x)+1;
122 393971 : GEN z = cgetg(l,t_POL); z[1] = evalvarn(v);
123 393971 : x--;
124 4960313 : for (i=2; i<l ; i++) gel(z,i) = gel(x,i);
125 393971 : return FlxX_renormalize(z,l);
126 : }
127 :
128 : GEN
129 0 : FlxM_to_FlxXV(GEN x, long v)
130 0 : { pari_APPLY_type(t_COL, FlxV_to_FlxX(gel(x,i), v)) }
131 :
132 : GEN
133 0 : FlxM_Flx_add_shallow(GEN x, GEN y, ulong p)
134 : {
135 0 : long l = lg(x), i, j;
136 0 : GEN z = cgetg(l,t_MAT);
137 :
138 0 : if (l==1) return z;
139 0 : if (l != lgcols(x)) pari_err_OP( "+", x, y);
140 0 : for (i=1; i<l; i++)
141 : {
142 0 : GEN zi = cgetg(l,t_COL), xi = gel(x,i);
143 0 : gel(z,i) = zi;
144 0 : for (j=1; j<l; j++) gel(zi,j) = gel(xi,j);
145 0 : gel(zi,i) = Flx_add(gel(zi,i), y, p);
146 : }
147 0 : return z;
148 : }
149 :
150 : /***********************************************************************/
151 : /** Conversion to Flx **/
152 : /***********************************************************************/
153 : /* Take an integer and return a scalar polynomial mod p, with evalvarn=vs */
154 : GEN
155 16805828 : Fl_to_Flx(ulong x, long sv) { return x? mkvecsmall2(sv, x): pol0_Flx(sv); }
156 :
157 : /* a X^d */
158 : GEN
159 907774 : monomial_Flx(ulong a, long d, long vs)
160 : {
161 : GEN P;
162 907774 : if (a==0) return pol0_Flx(vs);
163 907774 : P = const_vecsmall(d+2, 0);
164 907786 : P[1] = vs; P[d+2] = a; return P;
165 : }
166 :
167 : GEN
168 2658208 : Z_to_Flx(GEN x, ulong p, long sv)
169 : {
170 2658208 : long u = umodiu(x,p);
171 2658312 : return u? mkvecsmall2(sv, u): pol0_Flx(sv);
172 : }
173 :
174 : /* return x[0 .. dx] mod p as t_VECSMALL. Assume x a t_POL*/
175 : GEN
176 158116351 : ZX_to_Flx(GEN x, ulong p)
177 : {
178 158116351 : long i, lx = lg(x);
179 158116351 : GEN a = cgetg(lx, t_VECSMALL);
180 157938940 : a[1]=((ulong)x[1])&VARNBITS;
181 1066827311 : for (i=2; i<lx; i++) a[i] = umodiu(gel(x,i), p);
182 158102247 : return Flx_renormalize(a,lx);
183 : }
184 :
185 : /* return x[0 .. dx] mod p as t_VECSMALL. Assume x a t_POL*/
186 : GEN
187 6045910 : zx_to_Flx(GEN x, ulong p)
188 : {
189 6045910 : long i, lx = lg(x);
190 6045910 : GEN a = cgetg(lx, t_VECSMALL);
191 6038135 : a[1] = x[1];
192 18553259 : for (i=2; i<lx; i++) uel(a,i) = umodsu(x[i], p);
193 6036942 : return Flx_renormalize(a,lx);
194 : }
195 :
196 : ulong
197 53651853 : Rg_to_Fl(GEN x, ulong p)
198 : {
199 53651853 : switch(typ(x))
200 : {
201 27962006 : case t_INT: return umodiu(x, p);
202 330703 : case t_FRAC: {
203 330703 : ulong z = umodiu(gel(x,1), p);
204 330705 : if (!z) return 0;
205 326526 : return Fl_div(z, umodiu(gel(x,2), p), p);
206 : }
207 205961 : case t_PADIC: return padic_to_Fl(x, p);
208 25153209 : case t_INTMOD: {
209 25153209 : GEN q = gel(x,1), a = gel(x,2);
210 25153209 : if (absequaliu(q, p)) return itou(a);
211 0 : if (!dvdiu(q,p)) pari_err_MODULUS("Rg_to_Fl", q, utoipos(p));
212 0 : return umodiu(a, p);
213 : }
214 0 : default: pari_err_TYPE("Rg_to_Fl",x);
215 : return 0; /* LCOV_EXCL_LINE */
216 : }
217 : }
218 :
219 : ulong
220 1664653 : Rg_to_F2(GEN x)
221 : {
222 1664653 : switch(typ(x))
223 : {
224 261615 : case t_INT: return mpodd(x);
225 0 : case t_FRAC:
226 0 : if (!mpodd(gel(x,2))) (void)Fl_inv(0,2); /* error */
227 0 : return mpodd(gel(x,1));
228 0 : case t_PADIC:
229 0 : if (!absequaliu(gel(x,2),2)) pari_err_OP("",x, mkintmodu(1,2));
230 0 : if (valp(x) < 0) (void)Fl_inv(0,2);
231 0 : return valp(x) & 1;
232 1403038 : case t_INTMOD: {
233 1403038 : GEN q = gel(x,1), a = gel(x,2);
234 1403038 : if (mpodd(q)) pari_err_MODULUS("Rg_to_F2", q, gen_2);
235 1403038 : return mpodd(a);
236 : }
237 0 : default: pari_err_TYPE("Rg_to_F2",x);
238 : return 0; /* LCOV_EXCL_LINE */
239 : }
240 : }
241 :
242 : GEN
243 2825977 : RgX_to_Flx(GEN x, ulong p)
244 : {
245 2825977 : long i, lx = lg(x);
246 2825977 : GEN a = cgetg(lx, t_VECSMALL);
247 2825976 : a[1]=((ulong)x[1])&VARNBITS;
248 22237150 : for (i=2; i<lx; i++) a[i] = Rg_to_Fl(gel(x,i), p);
249 2825974 : return Flx_renormalize(a,lx);
250 : }
251 :
252 : GEN
253 7 : RgXV_to_FlxV(GEN x, ulong p)
254 175 : { pari_APPLY_type(t_VEC, RgX_to_Flx(gel(x,i), p)) }
255 :
256 : /* If x is a POLMOD, assume modulus is a multiple of T. */
257 : GEN
258 3447333 : Rg_to_Flxq(GEN x, GEN T, ulong p)
259 : {
260 3447333 : long ta, tx = typ(x), v = get_Flx_var(T);
261 : ulong pi;
262 : GEN a, b;
263 3447331 : if (is_const_t(tx))
264 : {
265 3196839 : if (tx == t_FFELT) return FF_to_Flxq(x);
266 2465831 : return Fl_to_Flx(Rg_to_Fl(x, p), v);
267 : }
268 250498 : switch(tx)
269 : {
270 8576 : case t_POLMOD:
271 8576 : b = gel(x,1);
272 8576 : a = gel(x,2); ta = typ(a);
273 8576 : if (is_const_t(ta)) return Fl_to_Flx(Rg_to_Fl(a, p), v);
274 8422 : b = RgX_to_Flx(b, p); if (b[1] != v) break;
275 8422 : a = RgX_to_Flx(a, p); if (Flx_equal(b,T)) return a;
276 0 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
277 0 : if (lgpol(Flx_rem_pre(b,T,p,pi))==0) return Flx_rem_pre(a, T, p, pi);
278 0 : break;
279 241922 : case t_POL:
280 241922 : x = RgX_to_Flx(x,p);
281 241922 : if (x[1] != v) break;
282 241922 : return Flx_rem(x, T, p);
283 0 : case t_RFRAC:
284 0 : a = Rg_to_Flxq(gel(x,1), T,p);
285 0 : b = Rg_to_Flxq(gel(x,2), T,p);
286 0 : return Flxq_div(a,b, T,p);
287 : }
288 0 : pari_err_TYPE("Rg_to_Flxq",x);
289 : return NULL; /* LCOV_EXCL_LINE */
290 : }
291 :
292 : /***********************************************************************/
293 : /** Basic operation on Flx **/
294 : /***********************************************************************/
295 : /* = zx_renormalize. Similar to normalizepol, in place */
296 : GEN
297 2010450723 : Flx_renormalize(GEN /*in place*/ x, long lx)
298 : {
299 : long i;
300 2267424906 : for (i = lx-1; i>1; i--)
301 2174605203 : if (x[i]) break;
302 2010450723 : stackdummy((pari_sp)(x + lg(x)), (pari_sp)(x + i+1));
303 2009917167 : setlg(x, i+1); return x;
304 : }
305 :
306 : GEN
307 1890615 : Flx_red(GEN z, ulong p)
308 : {
309 1890615 : long i, l = lg(z);
310 1890615 : GEN x = cgetg(l, t_VECSMALL);
311 1890502 : x[1] = z[1];
312 35227205 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
313 1890502 : return Flx_renormalize(x,l);
314 : }
315 :
316 : int
317 26254902 : Flx_equal(GEN V, GEN W)
318 : {
319 26254902 : long l = lg(V);
320 26254902 : if (lg(W) != l) return 0;
321 27281529 : while (--l > 1) /* do not compare variables, V[1] */
322 26025322 : if (V[l] != W[l]) return 0;
323 1256207 : return 1;
324 : }
325 :
326 : GEN
327 2646132 : random_Flx(long d1, long vs, ulong p)
328 : {
329 2646132 : long i, d = d1+2;
330 2646132 : GEN y = cgetg(d,t_VECSMALL); y[1] = vs;
331 18322933 : for (i=2; i<d; i++) y[i] = random_Fl(p);
332 2646394 : return Flx_renormalize(y,d);
333 : }
334 :
335 : static GEN
336 7181202 : Flx_addspec(GEN x, GEN y, ulong p, long lx, long ly)
337 : {
338 : long i,lz;
339 : GEN z;
340 :
341 7181202 : if (ly>lx) swapspec(x,y, lx,ly);
342 7181202 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
343 107770597 : for (i=0; i<ly; i++) z[i+2] = Fl_add(x[i], y[i], p);
344 91187365 : for ( ; i<lx; i++) z[i+2] = x[i];
345 7181202 : z[1] = 0; return Flx_renormalize(z, lz);
346 : }
347 :
348 : GEN
349 60465193 : Flx_add(GEN x, GEN y, ulong p)
350 : {
351 : long i,lz;
352 : GEN z;
353 60465193 : long lx=lg(x);
354 60465193 : long ly=lg(y);
355 60465193 : if (ly>lx) swapspec(x,y, lx,ly);
356 60465193 : lz = lx; z = cgetg(lz, t_VECSMALL); z[1]=x[1];
357 559541035 : for (i=2; i<ly; i++) z[i] = Fl_add(x[i], y[i], p);
358 128333432 : for ( ; i<lx; i++) z[i] = x[i];
359 60420428 : return Flx_renormalize(z, lz);
360 : }
361 :
362 : GEN
363 9647459 : Flx_Fl_add(GEN y, ulong x, ulong p)
364 : {
365 : GEN z;
366 : long lz, i;
367 9647459 : if (!lgpol(y))
368 228264 : return Fl_to_Flx(x,y[1]);
369 9420386 : lz=lg(y);
370 9420386 : z=cgetg(lz,t_VECSMALL);
371 9419269 : z[1]=y[1];
372 9419269 : z[2] = Fl_add(y[2],x,p);
373 45985135 : for(i=3;i<lz;i++)
374 36565819 : z[i] = y[i];
375 9419316 : if (lz==3) z = Flx_renormalize(z,lz);
376 9419445 : return z;
377 : }
378 :
379 : static GEN
380 898985 : Flx_subspec(GEN x, GEN y, ulong p, long lx, long ly)
381 : {
382 : long i,lz;
383 : GEN z;
384 :
385 898985 : if (ly <= lx)
386 : {
387 898997 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
388 54363738 : for (i=0; i<ly; i++) z[i+2] = Fl_sub(x[i],y[i],p);
389 1443767 : for ( ; i<lx; i++) z[i+2] = x[i];
390 : }
391 : else
392 : {
393 0 : lz = ly+2; z = cgetg(lz, t_VECSMALL);
394 0 : for (i=0; i<lx; i++) z[i+2] = Fl_sub(x[i],y[i],p);
395 0 : for ( ; i<ly; i++) z[i+2] = Fl_neg(y[i],p);
396 : }
397 897809 : z[1] = 0; return Flx_renormalize(z, lz);
398 : }
399 :
400 : GEN
401 137806360 : Flx_sub(GEN x, GEN y, ulong p)
402 : {
403 137806360 : long i,lz,lx = lg(x), ly = lg(y);
404 : GEN z;
405 :
406 137806360 : if (ly <= lx)
407 : {
408 86599881 : lz = lx; z = cgetg(lz, t_VECSMALL);
409 455380476 : for (i=2; i<ly; i++) z[i] = Fl_sub(x[i],y[i],p);
410 173014213 : for ( ; i<lx; i++) z[i] = x[i];
411 : }
412 : else
413 : {
414 51206479 : lz = ly; z = cgetg(lz, t_VECSMALL);
415 267448340 : for (i=2; i<lx; i++) z[i] = Fl_sub(x[i],y[i],p);
416 237434054 : for ( ; i<ly; i++) z[i] = y[i]? (long)(p - y[i]): y[i];
417 : }
418 137747169 : z[1]=x[1]; return Flx_renormalize(z, lz);
419 : }
420 :
421 : GEN
422 43562 : Flx_Fl_sub(GEN y, ulong x, ulong p)
423 : {
424 : GEN z;
425 43562 : long lz = lg(y), i;
426 43562 : if (lz==2)
427 513 : return Fl_to_Flx(Fl_neg(x, p),y[1]);
428 43049 : z = cgetg(lz, t_VECSMALL);
429 43048 : z[1] = y[1];
430 43048 : z[2] = Fl_sub(uel(y,2), x, p);
431 281952 : for(i=3; i<lz; i++)
432 238903 : z[i] = y[i];
433 43049 : if (lz==3) z = Flx_renormalize(z,lz);
434 43049 : return z;
435 : }
436 :
437 : static GEN
438 3431777 : Flx_negspec(GEN x, ulong p, long l)
439 : {
440 : long i;
441 3431777 : GEN z = cgetg(l+2, t_VECSMALL) + 2;
442 21251158 : for (i=0; i<l; i++) z[i] = Fl_neg(x[i], p);
443 3431666 : return z-2;
444 : }
445 :
446 : GEN
447 3431833 : Flx_neg(GEN x, ulong p)
448 : {
449 3431833 : GEN z = Flx_negspec(x+2, p, lgpol(x));
450 3432138 : z[1] = x[1];
451 3432138 : return z;
452 : }
453 :
454 : GEN
455 1760375 : Flx_neg_inplace(GEN x, ulong p)
456 : {
457 1760375 : long i, l = lg(x);
458 52943181 : for (i=2; i<l; i++)
459 51182806 : if (x[i]) x[i] = p - x[i];
460 1760375 : return x;
461 : }
462 :
463 : GEN
464 2433686 : Flx_double(GEN y, ulong p)
465 : {
466 : long i, l;
467 2433686 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
468 20624785 : for(i=2; i<l; i++) z[i] = Fl_double(y[i], p);
469 2433686 : return Flx_renormalize(z, l);
470 : }
471 : GEN
472 1040382 : Flx_triple(GEN y, ulong p)
473 : {
474 : long i, l;
475 1040382 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
476 8294688 : for(i=2; i<l; i++) z[i] = Fl_triple(y[i], p);
477 1040382 : return Flx_renormalize(z, l);
478 : }
479 :
480 : GEN
481 18120247 : Flx_Fl_mul_pre(GEN y, ulong x, ulong p, ulong pi)
482 : {
483 : GEN z;
484 : long i, l;
485 18120247 : if (!x) return pol0_Flx(y[1]);
486 17432002 : z = cgetg_copy(y, &l); z[1] = y[1];
487 17431014 : if (pi==0)
488 : {
489 15478106 : if (HIGHWORD(x | p))
490 0 : for(i=2; i<l; i++) z[i] = Fl_mul(uel(y,i), x, p);
491 : else
492 91656234 : for(i=2; i<l; i++) z[i] = (uel(y,i) * x) % p;
493 : } else
494 14398930 : for(i=2; i<l; i++) z[i] = Fl_mul_pre(uel(y,i), x, p, pi);
495 17432106 : return Flx_renormalize(z, l);
496 : }
497 :
498 : GEN
499 6827455 : Flx_Fl_mul(GEN x, ulong y, ulong p)
500 6827455 : { return Flx_Fl_mul_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
501 :
502 : GEN
503 0 : Flx_convol(GEN x, GEN y, ulong p)
504 : {
505 0 : long lx = lg(x), ly = lg(y), i;
506 : GEN z;
507 0 : if (lx < ly) swapspec(x,y, lx,ly);
508 0 : z = cgetg(ly,t_VECSMALL); z[1] = x[1];
509 0 : for (i=2; i<ly; i++) uel(z,i) = Fl_mul(uel(x,i),uel(y,i), p);
510 0 : return Flx_renormalize(z, ly);
511 : }
512 :
513 : GEN
514 11864906 : Flx_Fl_mul_to_monic(GEN y, ulong x, ulong p)
515 : {
516 : GEN z;
517 : long i, l;
518 11864906 : z = cgetg_copy(y, &l); z[1] = y[1];
519 11860459 : if (HIGHWORD(x | p))
520 5353159 : for(i=2; i<l-1; i++) z[i] = Fl_mul(y[i], x, p);
521 : else
522 26605308 : for(i=2; i<l-1; i++) z[i] = (y[i] * x) % p;
523 11860450 : z[l-1] = 1; return z;
524 : }
525 :
526 : /* Return a*x^n if n>=0 and a\x^(-n) if n<0 */
527 : GEN
528 26185220 : Flx_shift(GEN a, long n)
529 : {
530 26185220 : long i, l = lg(a);
531 : GEN b;
532 26185220 : if (l==2 || !n) return Flx_copy(a);
533 25842215 : if (l+n<=2) return pol0_Flx(a[1]);
534 25627641 : b = cgetg(l+n, t_VECSMALL);
535 25624718 : b[1] = a[1];
536 25624718 : if (n < 0)
537 71548957 : for (i=2-n; i<l; i++) b[i+n] = a[i];
538 : else
539 : {
540 49821378 : for (i=0; i<n; i++) b[2+i] = 0;
541 145005847 : for (i=2; i<l; i++) b[i+n] = a[i];
542 : }
543 25624718 : return b;
544 : }
545 :
546 : GEN
547 61051521 : Flx_normalize(GEN z, ulong p)
548 : {
549 61051521 : long l = lg(z)-1;
550 61051521 : ulong p1 = z[l]; /* leading term */
551 61051521 : if (p1 == 1) return z;
552 11839762 : return Flx_Fl_mul_to_monic(z, Fl_inv(p1,p), p);
553 : }
554 :
555 : /* return (x * X^d) + y. Assume d > 0, shallow if x == 0*/
556 : static GEN
557 3689827 : Flx_addshift(GEN x, GEN y, ulong p, long d)
558 : {
559 3689827 : GEN xd,yd,zd = (GEN)avma;
560 3689827 : long a,lz,ny = lgpol(y), nx = lgpol(x);
561 3689827 : long vs = x[1];
562 3689827 : if (nx == 0) return y;
563 3687975 : x += 2; y += 2; a = ny-d;
564 3687975 : if (a <= 0)
565 : {
566 85237 : lz = (a>nx)? ny+2: nx+d+2;
567 85237 : (void)new_chunk(lz); xd = x+nx; yd = y+ny;
568 1735135 : while (xd > x) *--zd = *--xd;
569 85237 : x = zd + a;
570 166914 : while (zd > x) *--zd = 0;
571 : }
572 : else
573 : {
574 3602738 : xd = new_chunk(d); yd = y+d;
575 3602738 : x = Flx_addspec(x,yd,p, nx,a);
576 3602738 : lz = (a>nx)? ny+2: lg(x)+d;
577 134239541 : x += 2; while (xd > x) *--zd = *--xd;
578 : }
579 60972427 : while (yd > y) *--zd = *--yd;
580 3687975 : *--zd = vs;
581 3687975 : *--zd = evaltyp(t_VECSMALL) | evallg(lz); return zd;
582 : }
583 :
584 : /* shift polynomial + gerepile */
585 : /* Do not set evalvarn*/
586 : static GEN
587 612442541 : Flx_shiftip(pari_sp av, GEN x, long v)
588 : {
589 612442541 : long i, lx = lg(x), ly;
590 : GEN y;
591 612442541 : if (!v || lx==2) return gerepileuptoleaf(av, x);
592 171379772 : ly = lx + v; /* result length */
593 171379772 : (void)new_chunk(ly); /* check that result fits */
594 171263911 : x += lx; y = (GEN)av;
595 1224073940 : for (i = 2; i<lx; i++) *--y = *--x;
596 695961181 : for (i = 0; i< v; i++) *--y = 0;
597 171263911 : y -= 2; y[0] = evaltyp(t_VECSMALL) | evallg(ly);
598 171670446 : return gc_const((pari_sp)y, y);
599 : }
600 :
601 : static long
602 2182074412 : get_Fl_threshold(ulong p, long mul, long mul2)
603 : {
604 2182074412 : return SMALL_ULONG(p) ? mul: mul2;
605 : }
606 :
607 : #define BITS_IN_QUARTULONG (BITS_IN_HALFULONG >> 1)
608 : #define QUARTMASK ((1UL<<BITS_IN_QUARTULONG)-1UL)
609 : #define LLQUARTWORD(x) ((x) & QUARTMASK)
610 : #define HLQUARTWORD(x) (((x) >> BITS_IN_QUARTULONG) & QUARTMASK)
611 : #define LHQUARTWORD(x) (((x) >> (2*BITS_IN_QUARTULONG)) & QUARTMASK)
612 : #define HHQUARTWORD(x) (((x) >> (3*BITS_IN_QUARTULONG)) & QUARTMASK)
613 : INLINE long
614 8049672 : maxbitcoeffpol(ulong p, long n)
615 : {
616 8049672 : GEN z = muliu(sqru(p - 1), n);
617 8044550 : long b = expi(z) + 1;
618 : /* only do expensive bit-packing if it saves at least 1 limb */
619 8047285 : if (b <= BITS_IN_QUARTULONG)
620 : {
621 937962 : if (nbits2nlong(n*b) == (n + 3)>>2)
622 107358 : b = BITS_IN_QUARTULONG;
623 : }
624 7109323 : else if (b <= BITS_IN_HALFULONG)
625 : {
626 1551683 : if (nbits2nlong(n*b) == (n + 1)>>1)
627 5590 : b = BITS_IN_HALFULONG;
628 : }
629 : else
630 : {
631 5557640 : long l = lgefint(z) - 2;
632 5557640 : if (nbits2nlong(n*b) == n*l)
633 306535 : b = l*BITS_IN_LONG;
634 : }
635 8047105 : return b;
636 : }
637 :
638 : INLINE ulong
639 3380851018 : Flx_mullimb_ok(GEN x, GEN y, ulong p, long a, long b)
640 : { /* Assume OK_ULONG*/
641 3380851018 : ulong p1 = 0;
642 : long i;
643 16129254524 : for (i=a; i<b; i++)
644 12748403506 : if (y[i])
645 : {
646 10690824402 : p1 += y[i] * x[-i];
647 10690824402 : if (p1 & HIGHBIT) p1 %= p;
648 : }
649 3380851018 : return p1 % p;
650 : }
651 :
652 : INLINE ulong
653 1124741969 : Flx_mullimb(GEN x, GEN y, ulong p, ulong pi, long a, long b)
654 : {
655 1124741969 : ulong p1 = 0;
656 : long i;
657 3586766244 : for (i=a; i<b; i++)
658 2461381478 : if (y[i])
659 2419502933 : p1 = Fl_addmul_pre(p1, y[i], x[-i], p, pi);
660 1125384766 : return p1;
661 : }
662 :
663 : /* assume nx >= ny > 0 */
664 : static GEN
665 328690906 : Flx_mulspec_basecase(GEN x, GEN y, ulong p, ulong pi, long nx, long ny)
666 : {
667 : long i,lz,nz;
668 : GEN z;
669 :
670 328690906 : lz = nx+ny+1; nz = lz-2;
671 328690906 : z = cgetg(lz, t_VECSMALL) + 2; /* x:y:z [i] = term of degree i */
672 328279508 : if (!pi)
673 : {
674 1140564027 : for (i=0; i<ny; i++)z[i] = Flx_mullimb_ok(x+i,y,p,0,i+1);
675 732522479 : for ( ; i<nx; i++) z[i] = Flx_mullimb_ok(x+i,y,p,0,ny);
676 889183036 : for ( ; i<nz; i++) z[i] = Flx_mullimb_ok(x+i,y,p,i-nx+1,ny);
677 : }
678 : else
679 : {
680 275261611 : for (i=0; i<ny; i++)z[i] = Flx_mullimb(x+i,y,p,pi,0,i+1);
681 194490371 : for ( ; i<nx; i++) z[i] = Flx_mullimb(x+i,y,p,pi,0,ny);
682 199938691 : for ( ; i<nz; i++) z[i] = Flx_mullimb(x+i,y,p,pi,i-nx+1,ny);
683 : }
684 328285704 : z -= 2; return Flx_renormalize(z, lz);
685 : }
686 :
687 : static GEN
688 12329 : int_to_Flx(GEN z, ulong p)
689 : {
690 12329 : long i, l = lgefint(z);
691 12329 : GEN x = cgetg(l, t_VECSMALL);
692 1061037 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
693 12311 : return Flx_renormalize(x, l);
694 : }
695 :
696 : INLINE GEN
697 10051 : Flx_mulspec_mulii(GEN a, GEN b, ulong p, long na, long nb)
698 : {
699 10051 : GEN z=muliispec(a,b,na,nb);
700 10064 : return int_to_Flx(z,p);
701 : }
702 :
703 : static GEN
704 504129 : Flx_to_int_halfspec(GEN a, long na)
705 : {
706 : long j;
707 504129 : long n = (na+1)>>1UL;
708 504129 : GEN V = cgetipos(2+n);
709 : GEN w;
710 1429417 : for (w = int_LSW(V), j=0; j+1<na; j+=2, w=int_nextW(w))
711 925288 : *w = a[j]|(a[j+1]<<BITS_IN_HALFULONG);
712 504129 : if (j<na)
713 338170 : *w = a[j];
714 504129 : return V;
715 : }
716 :
717 : static GEN
718 591539 : int_to_Flx_half(GEN z, ulong p)
719 : {
720 : long i;
721 591539 : long lx = (lgefint(z)-2)*2+2;
722 591539 : GEN w, x = cgetg(lx, t_VECSMALL);
723 2016184 : for (w = int_LSW(z), i=2; i<lx; i+=2, w=int_nextW(w))
724 : {
725 1424645 : x[i] = LOWWORD((ulong)*w)%p;
726 1424645 : x[i+1] = HIGHWORD((ulong)*w)%p;
727 : }
728 591539 : return Flx_renormalize(x, lx);
729 : }
730 :
731 : static GEN
732 5454 : Flx_mulspec_halfmulii(GEN a, GEN b, ulong p, long na, long nb)
733 : {
734 5454 : GEN A = Flx_to_int_halfspec(a,na);
735 5454 : GEN B = Flx_to_int_halfspec(b,nb);
736 5454 : GEN z = mulii(A,B);
737 5454 : return int_to_Flx_half(z,p);
738 : }
739 :
740 : static GEN
741 204499 : Flx_to_int_quartspec(GEN a, long na)
742 : {
743 : long j;
744 204499 : long n = (na+3)>>2UL;
745 204499 : GEN V = cgetipos(2+n);
746 : GEN w;
747 4377834 : for (w = int_LSW(V), j=0; j+3<na; j+=4, w=int_nextW(w))
748 4173334 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG))|(a[j+3]<<(3*BITS_IN_QUARTULONG));
749 204500 : switch (na-j)
750 : {
751 116429 : case 3:
752 116429 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG));
753 116429 : break;
754 34455 : case 2:
755 34455 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG);
756 34455 : break;
757 27219 : case 1:
758 27219 : *w = a[j];
759 27219 : break;
760 26398 : case 0:
761 26398 : break;
762 : }
763 204500 : return V;
764 : }
765 :
766 : static GEN
767 107361 : int_to_Flx_quart(GEN z, ulong p)
768 : {
769 : long i;
770 107361 : long lx = (lgefint(z)-2)*4+2;
771 107361 : GEN w, x = cgetg(lx, t_VECSMALL);
772 4873993 : for (w = int_LSW(z), i=2; i<lx; i+=4, w=int_nextW(w))
773 : {
774 4766632 : x[i] = LLQUARTWORD((ulong)*w)%p;
775 4766632 : x[i+1] = HLQUARTWORD((ulong)*w)%p;
776 4766632 : x[i+2] = LHQUARTWORD((ulong)*w)%p;
777 4766632 : x[i+3] = HHQUARTWORD((ulong)*w)%p;
778 : }
779 107361 : return Flx_renormalize(x, lx);
780 : }
781 :
782 : static GEN
783 97138 : Flx_mulspec_quartmulii(GEN a, GEN b, ulong p, long na, long nb)
784 : {
785 97138 : GEN A = Flx_to_int_quartspec(a,na);
786 97141 : GEN B = Flx_to_int_quartspec(b,nb);
787 97141 : GEN z = mulii(A,B);
788 97140 : return int_to_Flx_quart(z,p);
789 : }
790 :
791 : /*Eval x in 2^(k*BIL) in linear time, k==2 or 3*/
792 : static GEN
793 580326 : Flx_eval2BILspec(GEN x, long k, long l)
794 : {
795 580326 : long i, lz = k*l, ki;
796 580326 : GEN pz = cgetipos(2+lz);
797 16327156 : for (i=0; i < lz; i++)
798 15746830 : *int_W(pz,i) = 0UL;
799 8453741 : for (i=0, ki=0; i<l; i++, ki+=k)
800 7873415 : *int_W(pz,ki) = x[i];
801 580326 : return int_normalize(pz,0);
802 : }
803 :
804 : static GEN
805 297149 : Z_mod2BIL_Flx_2(GEN x, long d, ulong p)
806 : {
807 297149 : long i, offset, lm = lgefint(x)-2, l = d+3;
808 297149 : ulong pi = get_Fl_red(p);
809 297149 : GEN pol = cgetg(l, t_VECSMALL);
810 297149 : pol[1] = 0;
811 7990008 : for (i=0, offset=0; offset+1 < lm; i++, offset += 2)
812 7692859 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
813 297149 : if (offset < lm)
814 224989 : pol[i+2] = (*int_W(x,offset)) % p;
815 297149 : return Flx_renormalize(pol,l);
816 : }
817 :
818 : static GEN
819 0 : Z_mod2BIL_Flx_3(GEN x, long d, ulong p)
820 : {
821 0 : long i, offset, lm = lgefint(x)-2, l = d+3;
822 0 : ulong pi = get_Fl_red(p);
823 0 : GEN pol = cgetg(l, t_VECSMALL);
824 0 : pol[1] = 0;
825 0 : for (i=0, offset=0; offset+2 < lm; i++, offset += 3)
826 0 : pol[i+2] = remlll_pre(*int_W(x,offset+2), *int_W(x,offset+1),
827 0 : *int_W(x,offset), p, pi);
828 0 : if (offset+1 < lm)
829 0 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
830 0 : else if (offset < lm)
831 0 : pol[i+2] = (*int_W(x,offset)) % p;
832 0 : return Flx_renormalize(pol,l);
833 : }
834 :
835 : static GEN
836 294219 : Z_mod2BIL_Flx(GEN x, long bs, long d, ulong p)
837 : {
838 294219 : return bs==2 ? Z_mod2BIL_Flx_2(x, d, p): Z_mod2BIL_Flx_3(x, d, p);
839 : }
840 :
841 : static GEN
842 282718 : Flx_mulspec_mulii_inflate(GEN x, GEN y, long N, ulong p, long nx, long ny)
843 : {
844 282718 : pari_sp av = avma;
845 282718 : GEN z = mulii(Flx_eval2BILspec(x,N,nx), Flx_eval2BILspec(y,N,ny));
846 282718 : return gerepileupto(av, Z_mod2BIL_Flx(z, N, nx+ny-2, p));
847 : }
848 :
849 : static GEN
850 20175719 : kron_pack_Flx_spec_bits(GEN x, long b, long l) {
851 : GEN y;
852 : long i;
853 20175719 : if (l == 0)
854 3426611 : return gen_0;
855 16749108 : y = cgetg(l + 1, t_VECSMALL);
856 811821349 : for(i = 1; i <= l; i++)
857 795080990 : y[i] = x[l - i];
858 16740359 : return nv_fromdigits_2k(y, b);
859 : }
860 :
861 : /* assume b < BITS_IN_LONG */
862 : static GEN
863 5699707 : kron_unpack_Flx_bits_narrow(GEN z, long b, ulong p) {
864 5699707 : GEN v = binary_2k_nv(z, b), x;
865 5699701 : long i, l = lg(v) + 1;
866 5699701 : x = cgetg(l, t_VECSMALL);
867 626482977 : for (i = 2; i < l; i++)
868 620782947 : x[i] = v[l - i] % p;
869 5700030 : return Flx_renormalize(x, l);
870 : }
871 :
872 : static GEN
873 5189331 : kron_unpack_Flx_bits_wide(GEN z, long b, ulong p, ulong pi) {
874 5189331 : GEN v = binary_2k(z, b), x, y;
875 5188598 : long i, l = lg(v) + 1, ly;
876 5188598 : x = cgetg(l, t_VECSMALL);
877 227671506 : for (i = 2; i < l; i++) {
878 222482292 : y = gel(v, l - i);
879 222482292 : ly = lgefint(y);
880 222482292 : switch (ly) {
881 6278081 : case 2: x[i] = 0; break;
882 28872738 : case 3: x[i] = *int_W_lg(y, 0, ly) % p; break;
883 171475242 : case 4: x[i] = remll_pre(*int_W_lg(y, 1, ly), *int_W_lg(y, 0, ly), p, pi); break;
884 31712625 : case 5: x[i] = remlll_pre(*int_W_lg(y, 2, ly), *int_W_lg(y, 1, ly),
885 15856231 : *int_W_lg(y, 0, ly), p, pi); break;
886 0 : default: x[i] = umodiu(gel(v, l - i), p);
887 : }
888 : }
889 5189214 : return Flx_renormalize(x, l);
890 : }
891 :
892 : static GEN
893 6938291 : Flx_mulspec_Kronecker(GEN A, GEN B, long b, ulong p, long lA, long lB)
894 : {
895 : GEN C, D;
896 6938291 : pari_sp av = avma;
897 6938291 : A = kron_pack_Flx_spec_bits(A, b, lA);
898 6942662 : B = kron_pack_Flx_spec_bits(B, b, lB);
899 6942735 : C = gerepileuptoint(av, mulii(A, B));
900 6940634 : if (b < BITS_IN_LONG)
901 2126486 : D = kron_unpack_Flx_bits_narrow(C, b, p);
902 : else
903 : {
904 4814148 : ulong pi = get_Fl_red(p);
905 4813430 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
906 : }
907 6939428 : return D;
908 : }
909 :
910 : static GEN
911 689075 : Flx_sqrspec_Kronecker(GEN A, long b, ulong p, long lA)
912 : {
913 : GEN C, D;
914 689075 : A = kron_pack_Flx_spec_bits(A, b, lA);
915 689151 : C = sqri(A);
916 689182 : if (b < BITS_IN_LONG)
917 480811 : D = kron_unpack_Flx_bits_narrow(C, b, p);
918 : else
919 : {
920 208371 : ulong pi = get_Fl_red(p);
921 208368 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
922 : }
923 689148 : return D;
924 : }
925 :
926 : /* fast product (Karatsuba) of polynomials a,b. These are not real GENs, a+2,
927 : * b+2 were sent instead. na, nb = number of terms of a, b.
928 : * Only c, c0, c1, c2 are genuine GEN.
929 : */
930 : static GEN
931 365608206 : Flx_mulspec(GEN a, GEN b, ulong p, ulong pi, long na, long nb)
932 : {
933 : GEN a0,c,c0;
934 365608206 : long n0, n0a, i, v = 0;
935 : pari_sp av;
936 :
937 470120912 : while (na && !a[0]) { a++; na--; v++; }
938 551263757 : while (nb && !b[0]) { b++; nb--; v++; }
939 365608206 : if (na < nb) swapspec(a,b, na,nb);
940 365608206 : if (!nb) return pol0_Flx(0);
941 :
942 337413974 : av = avma;
943 337413974 : if (nb >= get_Fl_threshold(p, Flx_MUL_MULII_LIMIT, Flx_MUL2_MULII_LIMIT))
944 : {
945 7336455 : long m = maxbitcoeffpol(p,nb);
946 7333075 : switch (m)
947 : {
948 97138 : case BITS_IN_QUARTULONG:
949 97138 : return Flx_shiftip(av,Flx_mulspec_quartmulii(a,b,p,na,nb), v);
950 5454 : case BITS_IN_HALFULONG:
951 5454 : return Flx_shiftip(av,Flx_mulspec_halfmulii(a,b,p,na,nb), v);
952 10053 : case BITS_IN_LONG:
953 10053 : return Flx_shiftip(av,Flx_mulspec_mulii(a,b,p,na,nb), v);
954 282718 : case 2*BITS_IN_LONG:
955 282718 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,2,p,na,nb), v);
956 0 : case 3*BITS_IN_LONG:
957 0 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,3,p,na,nb), v);
958 6937712 : default:
959 6937712 : return Flx_shiftip(av,Flx_mulspec_Kronecker(a,b,m,p,na,nb), v);
960 : }
961 : }
962 330418615 : if (nb < get_Fl_threshold(p, Flx_MUL_KARATSUBA_LIMIT, Flx_MUL2_KARATSUBA_LIMIT))
963 328661873 : return Flx_shiftip(av,Flx_mulspec_basecase(a,b,p,pi,na,nb), v);
964 1814005 : i=(na>>1); n0=na-i; na=i;
965 1814005 : a0=a+n0; n0a=n0;
966 2591242 : while (n0a && !a[n0a-1]) n0a--;
967 :
968 1814005 : if (nb > n0)
969 : {
970 : GEN b0,c1,c2;
971 : long n0b;
972 :
973 1760375 : nb -= n0; b0 = b+n0; n0b = n0;
974 2846603 : while (n0b && !b[n0b-1]) n0b--;
975 1760375 : c = Flx_mulspec(a,b,p,pi,n0a,n0b);
976 1760375 : c0 = Flx_mulspec(a0,b0,p,pi,na,nb);
977 :
978 1760375 : c2 = Flx_addspec(a0,a,p,na,n0a);
979 1760375 : c1 = Flx_addspec(b0,b,p,nb,n0b);
980 :
981 1760375 : c1 = Flx_mul_pre(c1,c2,p,pi);
982 1760375 : c2 = Flx_add(c0,c,p);
983 :
984 1760375 : c2 = Flx_neg_inplace(c2,p);
985 1760375 : c2 = Flx_add(c1,c2,p);
986 1760375 : c0 = Flx_addshift(c0,c2 ,p, n0);
987 : }
988 : else
989 : {
990 53630 : c = Flx_mulspec(a,b,p,pi,n0a,nb);
991 53630 : c0 = Flx_mulspec(a0,b,p,pi,na,nb);
992 : }
993 1814005 : c0 = Flx_addshift(c0,c,p,n0);
994 1814005 : return Flx_shiftip(av,c0, v);
995 : }
996 :
997 : GEN
998 360061622 : Flx_mul_pre(GEN x, GEN y, ulong p, ulong pi)
999 : {
1000 360061622 : GEN z = Flx_mulspec(x+2,y+2,p, pi, lgpol(x),lgpol(y));
1001 360261308 : z[1] = x[1]; return z;
1002 : }
1003 : GEN
1004 27923676 : Flx_mul(GEN x, GEN y, ulong p)
1005 27923676 : { return Flx_mul_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1006 :
1007 : static GEN
1008 276307492 : Flx_sqrspec_basecase(GEN x, ulong p, ulong pi, long nx)
1009 : {
1010 : long i, lz, nz;
1011 : ulong p1;
1012 : GEN z;
1013 :
1014 276307492 : if (!nx) return pol0_Flx(0);
1015 276307492 : lz = (nx << 1) + 1, nz = lz-2;
1016 276307492 : z = cgetg(lz, t_VECSMALL) + 2;
1017 275537487 : if (!pi)
1018 : {
1019 212507830 : z[0] = x[0]*x[0]%p;
1020 914691287 : for (i=1; i<nx; i++)
1021 : {
1022 702286402 : p1 = Flx_mullimb_ok(x+i,x,p,0, (i+1)>>1);
1023 702183457 : p1 <<= 1;
1024 702183457 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1025 702183457 : z[i] = p1 % p;
1026 : }
1027 919007639 : for ( ; i<nz; i++)
1028 : {
1029 706020786 : p1 = Flx_mullimb_ok(x+i,x,p,i-nx+1, (i+1)>>1);
1030 706602754 : p1 <<= 1;
1031 706602754 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1032 706602754 : z[i] = p1 % p;
1033 : }
1034 : }
1035 : else
1036 : {
1037 63029657 : z[0] = Fl_sqr_pre(x[0], p, pi);
1038 409808852 : for (i=1; i<nx; i++)
1039 : {
1040 346901607 : p1 = Flx_mullimb(x+i,x,p,pi,0, (i+1)>>1);
1041 347085642 : p1 = Fl_add(p1, p1, p);
1042 346613595 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1043 346825847 : z[i] = p1;
1044 : }
1045 409668416 : for ( ; i<nz; i++)
1046 : {
1047 346765828 : p1 = Flx_mullimb(x+i,x,p,pi,i-nx+1, (i+1)>>1);
1048 347494534 : p1 = Fl_add(p1, p1, p);
1049 347097476 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1050 346761171 : z[i] = p1;
1051 : }
1052 : }
1053 275889441 : z -= 2; return Flx_renormalize(z, lz);
1054 : }
1055 :
1056 : static GEN
1057 2263 : Flx_sqrspec_sqri(GEN a, ulong p, long na)
1058 : {
1059 2263 : GEN z=sqrispec(a,na);
1060 2265 : return int_to_Flx(z,p);
1061 : }
1062 :
1063 : static GEN
1064 136 : Flx_sqrspec_halfsqri(GEN a, ulong p, long na)
1065 : {
1066 136 : GEN z = sqri(Flx_to_int_halfspec(a,na));
1067 136 : return int_to_Flx_half(z,p);
1068 : }
1069 :
1070 : static GEN
1071 10221 : Flx_sqrspec_quartsqri(GEN a, ulong p, long na)
1072 : {
1073 10221 : GEN z = sqri(Flx_to_int_quartspec(a,na));
1074 10221 : return int_to_Flx_quart(z,p);
1075 : }
1076 :
1077 : static GEN
1078 11501 : Flx_sqrspec_sqri_inflate(GEN x, long N, ulong p, long nx)
1079 : {
1080 11501 : pari_sp av = avma;
1081 11501 : GEN z = sqri(Flx_eval2BILspec(x,N,nx));
1082 11501 : return gerepileupto(av, Z_mod2BIL_Flx(z, N, (nx-1)*2, p));
1083 : }
1084 :
1085 : static GEN
1086 276689597 : Flx_sqrspec(GEN a, ulong p, ulong pi, long na)
1087 : {
1088 : GEN a0, c, c0;
1089 276689597 : long n0, n0a, i, v = 0, m;
1090 : pari_sp av;
1091 :
1092 395850464 : while (na && !a[0]) { a++; na--; v += 2; }
1093 276689597 : if (!na) return pol0_Flx(0);
1094 :
1095 276453265 : av = avma;
1096 276453265 : if (na >= get_Fl_threshold(p, Flx_SQR_SQRI_LIMIT, Flx_SQR2_SQRI_LIMIT))
1097 : {
1098 713205 : m = maxbitcoeffpol(p,na);
1099 713188 : switch(m)
1100 : {
1101 10221 : case BITS_IN_QUARTULONG:
1102 10221 : return Flx_shiftip(av, Flx_sqrspec_quartsqri(a,p,na), v);
1103 136 : case BITS_IN_HALFULONG:
1104 136 : return Flx_shiftip(av, Flx_sqrspec_halfsqri(a,p,na), v);
1105 2263 : case BITS_IN_LONG:
1106 2263 : return Flx_shiftip(av, Flx_sqrspec_sqri(a,p,na), v);
1107 11501 : case 2*BITS_IN_LONG:
1108 11501 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,2,p,na), v);
1109 0 : case 3*BITS_IN_LONG:
1110 0 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,3,p,na), v);
1111 689067 : default:
1112 689067 : return Flx_shiftip(av, Flx_sqrspec_Kronecker(a,m,p,na), v);
1113 : }
1114 : }
1115 275937517 : if (na < get_Fl_threshold(p, Flx_SQR_KARATSUBA_LIMIT, Flx_SQR2_KARATSUBA_LIMIT))
1116 275880992 : return Flx_shiftip(av, Flx_sqrspec_basecase(a,p,pi,na), v);
1117 57733 : i=(na>>1); n0=na-i; na=i;
1118 57733 : a0=a+n0; n0a=n0;
1119 72524 : while (n0a && !a[n0a-1]) n0a--;
1120 :
1121 57733 : c = Flx_sqrspec(a,p,pi,n0a);
1122 57733 : c0= Flx_sqrspec(a0,p,pi,na);
1123 57733 : if (p == 2) n0 *= 2;
1124 : else
1125 : {
1126 57714 : GEN c1, t = Flx_addspec(a0,a,p,na,n0a);
1127 57714 : t = Flx_sqr_pre(t,p,pi);
1128 57714 : c1= Flx_add(c0,c, p);
1129 57714 : c1= Flx_sub(t, c1, p);
1130 57714 : c0 = Flx_addshift(c0,c1,p,n0);
1131 : }
1132 57733 : c0 = Flx_addshift(c0,c,p,n0);
1133 57733 : return Flx_shiftip(av,c0,v);
1134 : }
1135 :
1136 : GEN
1137 276407923 : Flx_sqr_pre(GEN x, ulong p, ulong pi)
1138 : {
1139 276407923 : GEN z = Flx_sqrspec(x+2,p, pi, lgpol(x));
1140 277375253 : z[1] = x[1]; return z;
1141 : }
1142 : GEN
1143 356154 : Flx_sqr(GEN x, ulong p)
1144 356154 : { return Flx_sqr_pre(x, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1145 :
1146 : GEN
1147 8461 : Flx_powu_pre(GEN x, ulong n, ulong p, ulong pi)
1148 : {
1149 8461 : GEN y = pol1_Flx(x[1]), z;
1150 : ulong m;
1151 8453 : if (n == 0) return y;
1152 8453 : m = n; z = x;
1153 : for (;;)
1154 : {
1155 32607 : if (m&1UL) y = Flx_mul_pre(y,z, p, pi);
1156 32622 : m >>= 1; if (!m) return y;
1157 24168 : z = Flx_sqr_pre(z, p, pi);
1158 : }
1159 : }
1160 : GEN
1161 0 : Flx_powu(GEN x, ulong n, ulong p)
1162 : {
1163 0 : if (n == 0) return pol1_Flx(x[1]);
1164 0 : return Flx_powu_pre(x, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p));
1165 : }
1166 :
1167 : GEN
1168 14237 : Flx_halve(GEN y, ulong p)
1169 : {
1170 : GEN z;
1171 : long i, l;
1172 14237 : z = cgetg_copy(y, &l); z[1] = y[1];
1173 59382 : for(i=2; i<l; i++) uel(z,i) = Fl_halve(uel(y,i), p);
1174 14237 : return z;
1175 : }
1176 :
1177 : static GEN
1178 7098532 : Flx_recipspec(GEN x, long l, long n)
1179 : {
1180 : long i;
1181 7098532 : GEN z=cgetg(n+2,t_VECSMALL)+2;
1182 116592032 : for(i=0; i<l; i++)
1183 109494920 : z[n-i-1] = x[i];
1184 15520306 : for( ; i<n; i++)
1185 8423194 : z[n-i-1] = 0;
1186 7097112 : return Flx_renormalize(z-2,n+2);
1187 : }
1188 :
1189 : GEN
1190 0 : Flx_recip(GEN x)
1191 : {
1192 0 : GEN z=Flx_recipspec(x+2,lgpol(x),lgpol(x));
1193 0 : z[1]=x[1];
1194 0 : return z;
1195 : }
1196 :
1197 : /* Return h^degpol(P) P(x / h) */
1198 : GEN
1199 1117 : Flx_rescale(GEN P, ulong h, ulong p)
1200 : {
1201 1117 : long i, l = lg(P);
1202 1117 : GEN Q = cgetg(l,t_VECSMALL);
1203 1117 : ulong hi = h;
1204 1117 : Q[l-1] = P[l-1];
1205 12538 : for (i=l-2; i>=2; i--)
1206 : {
1207 12538 : Q[i] = Fl_mul(P[i], hi, p);
1208 12538 : if (i == 2) break;
1209 11421 : hi = Fl_mul(hi,h, p);
1210 : }
1211 1117 : Q[1] = P[1]; return Q;
1212 : }
1213 :
1214 : /* x/polrecip(P)+O(x^n); allow pi = 0 */
1215 : static GEN
1216 134095 : Flx_invBarrett_basecase(GEN T, ulong p, ulong pi)
1217 : {
1218 134095 : long i, l=lg(T)-1, lr=l-1, k;
1219 134095 : GEN r=cgetg(lr,t_VECSMALL); r[1] = T[1];
1220 134095 : r[2] = 1;
1221 134095 : if (!pi)
1222 764068 : for (i=3;i<lr;i++)
1223 : {
1224 757076 : ulong u = uel(T, l-i+2);
1225 45377003 : for (k=3; k<i; k++)
1226 44619927 : { u += uel(T,l-i+k) * uel(r, k); if (u & HIGHBIT) u %= p; }
1227 757076 : r[i] = Fl_neg(u % p, p);
1228 : }
1229 : else
1230 2107913 : for (i=3;i<lr;i++)
1231 : {
1232 1980810 : ulong u = Fl_neg(uel(T,l-i+2), p);
1233 59503885 : for (k=3; k<i; k++)
1234 : {
1235 57523075 : ulong t = Fl_neg(uel(T,l-i+k), p);
1236 57523076 : u = Fl_addmul_pre(u, t, uel(r,k), p, pi);
1237 : }
1238 1980810 : r[i] = u;
1239 : }
1240 134095 : return Flx_renormalize(r,lr);
1241 : }
1242 :
1243 : /* Return new lgpol */
1244 : static long
1245 2134046 : Flx_lgrenormalizespec(GEN x, long lx)
1246 : {
1247 : long i;
1248 7415949 : for (i = lx-1; i>=0; i--)
1249 7415132 : if (x[i]) break;
1250 2134046 : return i+1;
1251 : }
1252 : /* allow pi = 0 */
1253 : static GEN
1254 23029 : Flx_invBarrett_Newton(GEN T, ulong p, ulong pi)
1255 : {
1256 23029 : long nold, lx, lz, lq, l = degpol(T), lQ;
1257 23029 : GEN q, y, z, x = zero_zv(l+1) + 2;
1258 23029 : ulong mask = quadratic_prec_mask(l-2); /* assume l > 2 */
1259 : pari_sp av;
1260 :
1261 23029 : y = T+2;
1262 23029 : q = Flx_recipspec(y,l+1,l+1); lQ = lgpol(q); q+=2;
1263 23029 : av = avma;
1264 : /* We work on _spec_ Flx's, all the l[xzq12] below are lgpol's */
1265 :
1266 : /* initialize */
1267 23029 : x[0] = Fl_inv(q[0], p);
1268 23028 : if (lQ>1 && q[1])
1269 5108 : {
1270 5108 : ulong u = q[1];
1271 5108 : if (x[0] != 1) u = Fl_mul(u, Fl_sqr(x[0],p), p);
1272 5108 : x[1] = p - u; lx = 2;
1273 : }
1274 : else
1275 17920 : lx = 1;
1276 23028 : nold = 1;
1277 158177 : for (; mask > 1; set_avma(av))
1278 : { /* set x -= x(x*q - 1) + O(t^(nnew + 1)), knowing x*q = 1 + O(t^(nold+1)) */
1279 135159 : long i, lnew, nnew = nold << 1;
1280 :
1281 135159 : if (mask & 1) nnew--;
1282 135159 : mask >>= 1;
1283 :
1284 135159 : lnew = nnew + 1;
1285 135159 : lq = Flx_lgrenormalizespec(q, minss(lQ, lnew));
1286 135161 : z = Flx_mulspec(x, q, p, pi, lx, lq); /* FIXME: high product */
1287 135146 : lz = lgpol(z); if (lz > lnew) lz = lnew;
1288 135150 : z += 2;
1289 : /* subtract 1 [=>first nold words are 0]: renormalize so that z(0) != 0 */
1290 290175 : for (i = nold; i < lz; i++) if (z[i]) break;
1291 135150 : nold = nnew;
1292 135150 : if (i >= lz) continue; /* z-1 = 0(t^(nnew + 1)) */
1293 :
1294 : /* z + i represents (x*q - 1) / t^i */
1295 100742 : lz = Flx_lgrenormalizespec (z+i, lz-i);
1296 100742 : z = Flx_mulspec(x, z+i, p, pi, lx, lz); /* FIXME: low product */
1297 100741 : lz = lgpol(z); z += 2;
1298 100740 : if (lz > lnew-i) lz = Flx_lgrenormalizespec(z, lnew-i);
1299 :
1300 100740 : lx = lz+ i;
1301 100740 : y = x + i; /* x -= z * t^i, in place */
1302 915339 : for (i = 0; i < lz; i++) y[i] = Fl_neg(z[i], p);
1303 : }
1304 23028 : x -= 2; setlg(x, lx + 2); x[1] = T[1];
1305 23028 : return x;
1306 : }
1307 :
1308 : /* allow pi = 0 */
1309 : static GEN
1310 158514 : Flx_invBarrett_pre(GEN T, ulong p, ulong pi)
1311 : {
1312 158514 : pari_sp ltop = avma;
1313 158514 : long l = lgpol(T);
1314 : GEN r;
1315 158514 : if (l < 3) return pol0_Flx(T[1]);
1316 157124 : if (l < get_Fl_threshold(p, Flx_INVBARRETT_LIMIT, Flx_INVBARRETT2_LIMIT))
1317 : {
1318 134095 : ulong c = T[l+1];
1319 134095 : if (c != 1)
1320 : {
1321 98118 : ulong ci = Fl_inv(c,p);
1322 98118 : T = Flx_Fl_mul_pre(T, ci, p, pi);
1323 98118 : r = Flx_invBarrett_basecase(T, p, pi);
1324 98118 : r = Flx_Fl_mul_pre(r, ci, p, pi);
1325 : }
1326 : else
1327 35977 : r = Flx_invBarrett_basecase(T, p, pi);
1328 : }
1329 : else
1330 23029 : r = Flx_invBarrett_Newton(T, p, pi);
1331 157123 : return gerepileuptoleaf(ltop, r);
1332 : }
1333 : GEN
1334 0 : Flx_invBarrett(GEN T, ulong p)
1335 0 : { return Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1336 :
1337 : /* allow pi = 0 */
1338 : GEN
1339 96959436 : Flx_get_red_pre(GEN T, ulong p, ulong pi)
1340 : {
1341 96959436 : if (typ(T)!=t_VECSMALL
1342 96927767 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1343 : Flx_BARRETT2_LIMIT))
1344 96939324 : return T;
1345 7611 : retmkvec2(Flx_invBarrett_pre(T, p, pi),T);
1346 : }
1347 : GEN
1348 13736550 : Flx_get_red(GEN T, ulong p)
1349 : {
1350 13736550 : if (typ(T)!=t_VECSMALL
1351 13736588 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1352 : Flx_BARRETT2_LIMIT))
1353 13730146 : return T;
1354 5194 : retmkvec2(Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)),T);
1355 : }
1356 :
1357 : /* separate from Flx_divrem for maximal speed. */
1358 : static GEN
1359 711751959 : Flx_rem_basecase(GEN x, GEN y, ulong p, ulong pi)
1360 : {
1361 : pari_sp av;
1362 : GEN z, c;
1363 : long dx,dy,dy1,dz,i,j;
1364 : ulong p1,inv;
1365 711751959 : long vs=x[1];
1366 :
1367 711751959 : dy = degpol(y); if (!dy) return pol0_Flx(x[1]);
1368 680526437 : dx = degpol(x);
1369 680543480 : dz = dx-dy; if (dz < 0) return Flx_copy(x);
1370 680543480 : x += 2; y += 2;
1371 680543480 : inv = y[dy];
1372 680543480 : if (inv != 1UL) inv = Fl_inv(inv,p);
1373 822607984 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1374 :
1375 682183656 : c = cgetg(dy+3, t_VECSMALL); c[1]=vs; c += 2; av=avma;
1376 680463348 : z = cgetg(dz+3, t_VECSMALL); z[1]=vs; z += 2;
1377 :
1378 678849699 : if (!pi)
1379 : {
1380 472471340 : z[dz] = (inv*x[dx]) % p;
1381 1796582301 : for (i=dx-1; i>=dy; --i)
1382 : {
1383 1324110961 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1384 10555803395 : for (j=i-dy1; j<=i && j<=dz; j++)
1385 : {
1386 9231692434 : p1 += z[j]*y[i-j];
1387 9231692434 : if (p1 & HIGHBIT) p1 %= p;
1388 : }
1389 1324110961 : p1 %= p;
1390 1324110961 : z[i-dy] = p1? ((p - p1)*inv) % p: 0;
1391 : }
1392 3256276709 : for (i=0; i<dy; i++)
1393 : {
1394 2784080369 : p1 = z[0]*y[i];
1395 14519267794 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1396 : {
1397 11735187425 : p1 += z[j]*y[i-j];
1398 11735187425 : if (p1 & HIGHBIT) p1 %= p;
1399 : }
1400 2784747769 : c[i] = Fl_sub(x[i], p1%p, p);
1401 : }
1402 : }
1403 : else
1404 : {
1405 206378359 : z[dz] = Fl_mul_pre(inv, x[dx], p, pi);
1406 714090359 : for (i=dx-1; i>=dy; --i)
1407 : {
1408 503885139 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1409 2236225093 : for (j=i-dy1; j<=i && j<=dz; j++)
1410 1728197803 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1411 508027290 : z[i-dy] = p1? Fl_mul_pre(p - p1, inv, p, pi): 0;
1412 : }
1413 1523359415 : for (i=0; i<dy; i++)
1414 : {
1415 1313374455 : p1 = Fl_mul_pre(z[0],y[i],p,pi);
1416 3904722852 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1417 2582259008 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1418 1303573331 : c[i] = Fl_sub(x[i], p1, p);
1419 : }
1420 : }
1421 847304258 : i = dy-1; while (i>=0 && !c[i]) i--;
1422 682181300 : set_avma(av); return Flx_renormalize(c-2, i+3);
1423 : }
1424 :
1425 : /* as FpX_divrem but working only on ulong types.
1426 : * if relevant, *pr is the last object on stack */
1427 : static GEN
1428 62220057 : Flx_divrem_basecase(GEN x, GEN y, ulong p, ulong pi, GEN *pr)
1429 : {
1430 : GEN z,q,c;
1431 : long dx,dy,dy1,dz,i,j;
1432 : ulong p1,inv;
1433 62220057 : long sv=x[1];
1434 :
1435 62220057 : dy = degpol(y);
1436 62218587 : if (dy<0) pari_err_INV("Flx_divrem",y);
1437 62218707 : if (pr == ONLY_REM) return Flx_rem_basecase(x, y, p, pi);
1438 62218309 : if (!dy)
1439 : {
1440 7506842 : if (pr && pr != ONLY_DIVIDES) *pr = pol0_Flx(sv);
1441 7506709 : if (y[2] == 1UL) return Flx_copy(x);
1442 5316109 : return Flx_Fl_mul_pre(x, Fl_inv(y[2], p), p, pi);
1443 : }
1444 54711467 : dx = degpol(x);
1445 54716782 : dz = dx-dy;
1446 54716782 : if (dz < 0)
1447 : {
1448 1017414 : q = pol0_Flx(sv);
1449 1017400 : if (pr && pr != ONLY_DIVIDES) *pr = Flx_copy(x);
1450 1017400 : return q;
1451 : }
1452 53699368 : x += 2;
1453 53699368 : y += 2;
1454 53699368 : z = cgetg(dz + 3, t_VECSMALL); z[1] = sv; z += 2;
1455 53687614 : inv = uel(y, dy);
1456 53687614 : if (inv != 1UL) inv = Fl_inv(inv,p);
1457 79053751 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1458 :
1459 53692752 : if (SMALL_ULONG(p))
1460 : {
1461 51825846 : z[dz] = (inv*x[dx]) % p;
1462 131777393 : for (i=dx-1; i>=dy; --i)
1463 : {
1464 79951547 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1465 258139161 : for (j=i-dy1; j<=i && j<=dz; j++)
1466 : {
1467 178187614 : p1 += z[j]*y[i-j];
1468 178187614 : if (p1 & HIGHBIT) p1 %= p;
1469 : }
1470 79951547 : p1 %= p;
1471 79951547 : z[i-dy] = p1? (long) ((p - p1)*inv) % p: 0;
1472 : }
1473 : }
1474 : else
1475 : {
1476 1866906 : z[dz] = Fl_mul(inv, x[dx], p);
1477 9215983 : for (i=dx-1; i>=dy; --i)
1478 : { /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1479 7348110 : p1 = p - uel(x,i);
1480 26307344 : for (j=i-dy1; j<=i && j<=dz; j++)
1481 18959238 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1482 7348106 : z[i-dy] = p1? Fl_mul(p - p1, inv, p): 0;
1483 : }
1484 : }
1485 53693719 : q = Flx_renormalize(z-2, dz+3);
1486 53698153 : if (!pr) return q;
1487 :
1488 27193080 : c = cgetg(dy + 3, t_VECSMALL); c[1] = sv; c += 2;
1489 27196947 : if (SMALL_ULONG(p))
1490 : {
1491 233561811 : for (i=0; i<dy; i++)
1492 : {
1493 207998784 : p1 = (ulong)z[0]*y[i];
1494 487436110 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1495 : {
1496 279437326 : p1 += (ulong)z[j]*y[i-j];
1497 279437326 : if (p1 & HIGHBIT) p1 %= p;
1498 : }
1499 207998741 : c[i] = Fl_sub(x[i], p1%p, p);
1500 : }
1501 : }
1502 : else
1503 : {
1504 16088327 : for (i=0; i<dy; i++)
1505 : {
1506 14454943 : p1 = Fl_mul(z[0],y[i],p);
1507 50348775 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1508 35893833 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1509 14454943 : c[i] = Fl_sub(x[i], p1, p);
1510 : }
1511 : }
1512 36590268 : i=dy-1; while (i>=0 && !c[i]) i--;
1513 27196411 : c = Flx_renormalize(c-2, i+3);
1514 27197641 : if (pr == ONLY_DIVIDES)
1515 513 : { if (lg(c) != 2) return NULL; }
1516 : else
1517 27197128 : *pr = c;
1518 27197501 : return q;
1519 : }
1520 :
1521 : /* Compute x mod T where 2 <= degpol(T) <= l+1 <= 2*(degpol(T)-1)
1522 : * and mg is the Barrett inverse of T. */
1523 : static GEN
1524 906605 : Flx_divrem_Barrettspec(GEN x, long l, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1525 : {
1526 : GEN q, r;
1527 906605 : long lt = degpol(T); /*We discard the leading term*/
1528 : long ld, lm, lT, lmg;
1529 906541 : ld = l-lt;
1530 906541 : lm = minss(ld, lgpol(mg));
1531 906794 : lT = Flx_lgrenormalizespec(T+2,lt);
1532 906838 : lmg = Flx_lgrenormalizespec(mg+2,lm);
1533 906759 : q = Flx_recipspec(x+lt,ld,ld); /* q = rec(x) lz<=ld*/
1534 906246 : q = Flx_mulspec(q+2,mg+2,p,pi,lgpol(q),lmg); /* q = rec(x) * mg lz<=ld+lm*/
1535 906806 : q = Flx_recipspec(q+2,minss(ld,lgpol(q)),ld);/* q = rec (rec(x) * mg) lz<=ld*/
1536 906136 : if (!pr) return q;
1537 898448 : r = Flx_mulspec(q+2,T+2,p,pi,lgpol(q),lT); /* r = q*pol lz<=ld+lt*/
1538 899142 : r = Flx_subspec(x,r+2,p,lt,minss(lt,lgpol(r)));/* r = x - q*pol lz<=lt */
1539 898787 : if (pr == ONLY_REM) return r;
1540 425063 : *pr = r; return q;
1541 : }
1542 :
1543 : static GEN
1544 608880 : Flx_divrem_Barrett(GEN x, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1545 : {
1546 608880 : GEN q = NULL, r = Flx_copy(x);
1547 608917 : long l = lgpol(x), lt = degpol(T), lm = 2*lt-1, v = T[1];
1548 : long i;
1549 608905 : if (l <= lt)
1550 : {
1551 0 : if (pr == ONLY_REM) return Flx_copy(x);
1552 0 : if (pr == ONLY_DIVIDES) return lgpol(x)? NULL: pol0_Flx(v);
1553 0 : if (pr) *pr = Flx_copy(x);
1554 0 : return pol0_Flx(v);
1555 : }
1556 608905 : if (lt <= 1)
1557 1390 : return Flx_divrem_basecase(x,T,p,pi,pr);
1558 607515 : if (pr != ONLY_REM && l>lm)
1559 28889 : { q = zero_zv(l-lt+1); q[1] = T[1]; }
1560 908022 : while (l>lm)
1561 : {
1562 300710 : GEN zr, zq = Flx_divrem_Barrettspec(r+2+l-lm,lm,mg,T,p,pi,&zr);
1563 300567 : long lz = lgpol(zr);
1564 300507 : if (pr != ONLY_REM)
1565 : {
1566 57929 : long lq = lgpol(zq);
1567 872174 : for(i=0; i<lq; i++) q[2+l-lm+i] = zq[2+i];
1568 : }
1569 4369923 : for(i=0; i<lz; i++) r[2+l-lm+i] = zr[2+i];
1570 300507 : l = l-lm+lz;
1571 : }
1572 607312 : if (pr == ONLY_REM)
1573 : {
1574 473778 : if (l > lt)
1575 473736 : r = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi,ONLY_REM);
1576 : else
1577 42 : r = Flx_renormalize(r, l+2);
1578 473761 : r[1] = v; return r;
1579 : }
1580 133534 : if (l > lt)
1581 : {
1582 132159 : GEN zq = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi, pr ? &r: NULL);
1583 132159 : if (!q) q = zq;
1584 : else
1585 : {
1586 27315 : long lq = lgpol(zq);
1587 158603 : for(i=0; i<lq; i++) q[2+i] = zq[2+i];
1588 : }
1589 : }
1590 1375 : else if (pr)
1591 1535 : r = Flx_renormalize(r, l+2);
1592 133534 : q[1] = v; q = Flx_renormalize(q, lg(q));
1593 133733 : if (pr == ONLY_DIVIDES) return lgpol(r)? NULL: q;
1594 133733 : if (pr) { r[1] = v; *pr = r; }
1595 133733 : return q;
1596 : }
1597 :
1598 : /* allow pi = 0 (SMALL_ULONG) */
1599 : GEN
1600 79609953 : Flx_divrem_pre(GEN x, GEN T, ulong p, ulong pi, GEN *pr)
1601 : {
1602 : GEN B, y;
1603 : long dy, dx, d;
1604 79609953 : if (pr==ONLY_REM) return Flx_rem_pre(x, T, p, pi);
1605 62343834 : y = get_Flx_red(T, &B);
1606 62363268 : dy = degpol(y); dx = degpol(x); d = dx-dy;
1607 62354087 : if (!B && d+3 < get_Fl_threshold(p, Flx_DIVREM_BARRETT_LIMIT,Flx_DIVREM2_BARRETT_LIMIT))
1608 62218051 : return Flx_divrem_basecase(x,y,p,pi,pr);
1609 : else
1610 : {
1611 134725 : pari_sp av = avma;
1612 134725 : GEN mg = B? B: Flx_invBarrett_pre(y, p, pi);
1613 134725 : GEN q1 = Flx_divrem_Barrett(x,mg,y,p,pi,pr);
1614 134725 : if (!q1) return gc_NULL(av);
1615 134725 : if (!pr || pr==ONLY_DIVIDES) return gerepileuptoleaf(av, q1);
1616 126437 : return gc_all(av, 2, &q1, pr);
1617 : }
1618 : }
1619 : GEN
1620 29984078 : Flx_divrem(GEN x, GEN T, ulong p, GEN *pr)
1621 29984078 : { return Flx_divrem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p), pr); }
1622 :
1623 : GEN
1624 834797948 : Flx_rem_pre(GEN x, GEN T, ulong p, ulong pi)
1625 : {
1626 834797948 : GEN B, y = get_Flx_red(T, &B);
1627 834769777 : long d = degpol(x) - degpol(y);
1628 834621649 : if (d < 0) return Flx_copy(x);
1629 712725871 : if (!B && d+3 < get_Fl_threshold(p, Flx_REM_BARRETT_LIMIT,Flx_REM2_BARRETT_LIMIT))
1630 711850966 : return Flx_rem_basecase(x,y,p, pi);
1631 : else
1632 : {
1633 474160 : pari_sp av=avma;
1634 474160 : GEN mg = B ? B: Flx_invBarrett_pre(y, p, pi);
1635 474159 : GEN r = Flx_divrem_Barrett(x, mg, y, p, pi, ONLY_REM);
1636 474157 : return gerepileuptoleaf(av, r);
1637 : }
1638 : }
1639 : GEN
1640 41002083 : Flx_rem(GEN x, GEN T, ulong p)
1641 41002083 : { return Flx_rem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1642 :
1643 : /* reduce T mod (X^n - 1, p). Shallow function */
1644 : GEN
1645 5089228 : Flx_mod_Xnm1(GEN T, ulong n, ulong p)
1646 : {
1647 5089228 : long i, j, L = lg(T), l = n+2;
1648 : GEN S;
1649 5089228 : if (L <= l || n & ~LGBITS) return T;
1650 3361 : S = cgetg(l, t_VECSMALL);
1651 3361 : S[1] = T[1];
1652 13657 : for (i = 2; i < l; i++) S[i] = T[i];
1653 9198 : for (j = 2; i < L; i++) {
1654 5837 : S[j] = Fl_add(S[j], T[i], p);
1655 5837 : if (++j == l) j = 2;
1656 : }
1657 3361 : return Flx_renormalize(S, l);
1658 : }
1659 : /* reduce T mod (X^n + 1, p). Shallow function */
1660 : GEN
1661 29712 : Flx_mod_Xn1(GEN T, ulong n, ulong p)
1662 : {
1663 29712 : long i, j, L = lg(T), l = n+2;
1664 : GEN S;
1665 29712 : if (L <= l || n & ~LGBITS) return T;
1666 2593 : S = cgetg(l, t_VECSMALL);
1667 2593 : S[1] = T[1];
1668 10991 : for (i = 2; i < l; i++) S[i] = T[i];
1669 6752 : for (j = 2; i < L; i++) {
1670 4159 : S[j] = Fl_sub(S[j], T[i], p);
1671 4159 : if (++j == l) j = 2;
1672 : }
1673 2593 : return Flx_renormalize(S, l);
1674 : }
1675 :
1676 : struct _Flxq {
1677 : GEN aut, T;
1678 : ulong p, pi;
1679 : };
1680 : /* allow pi = 0 */
1681 : static void
1682 69532687 : set_Flxq_pre(struct _Flxq *D, GEN T, ulong p, ulong pi)
1683 : {
1684 69532687 : D->p = p;
1685 69532687 : D->pi = pi;
1686 69532687 : D->T = Flx_get_red_pre(T, p, pi);
1687 69541364 : }
1688 : static void
1689 73018 : set_Flxq(struct _Flxq *D, GEN T, ulong p)
1690 73018 : { set_Flxq_pre(D, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1691 :
1692 : static GEN
1693 0 : _Flx_divrem(void * E, GEN x, GEN y, GEN *r)
1694 : {
1695 0 : struct _Flxq *D = (struct _Flxq*) E;
1696 0 : return Flx_divrem_pre(x, y, D->p, D->pi, r);
1697 : }
1698 : static GEN
1699 578248 : _Flx_add(void * E, GEN x, GEN y) {
1700 578248 : struct _Flxq *D = (struct _Flxq*) E;
1701 578248 : return Flx_add(x, y, D->p);
1702 : }
1703 : static GEN
1704 10449105 : _Flx_mul(void *E, GEN x, GEN y) {
1705 10449105 : struct _Flxq *D = (struct _Flxq*) E;
1706 10449105 : return Flx_mul_pre(x, y, D->p, D->pi);
1707 : }
1708 : static GEN
1709 0 : _Flx_sqr(void *E, GEN x) {
1710 0 : struct _Flxq *D = (struct _Flxq*) E;
1711 0 : return Flx_sqr_pre(x, D->p, D->pi);
1712 : }
1713 :
1714 : static struct bb_ring Flx_ring = { _Flx_add,_Flx_mul,_Flx_sqr };
1715 :
1716 : GEN
1717 0 : Flx_digits(GEN x, GEN T, ulong p)
1718 : {
1719 : struct _Flxq D;
1720 0 : long d = degpol(T), n = (lgpol(x)+d-1)/d;
1721 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1722 0 : return gen_digits(x,T,n,(void *)&D, &Flx_ring, _Flx_divrem);
1723 : }
1724 :
1725 : GEN
1726 0 : FlxV_Flx_fromdigits(GEN x, GEN T, ulong p)
1727 : {
1728 : struct _Flxq D;
1729 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1730 0 : return gen_fromdigits(x,T,(void *)&D, &Flx_ring);
1731 : }
1732 :
1733 : long
1734 3930025 : Flx_val(GEN x)
1735 : {
1736 3930025 : long i, l=lg(x);
1737 3930025 : if (l==2) return LONG_MAX;
1738 3938911 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1739 3930025 : return i-2;
1740 : }
1741 : long
1742 26232721 : Flx_valrem(GEN x, GEN *Z)
1743 : {
1744 26232721 : long v, i, l=lg(x);
1745 : GEN y;
1746 26232721 : if (l==2) { *Z = Flx_copy(x); return LONG_MAX; }
1747 28398809 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1748 26232721 : v = i-2;
1749 26232721 : if (v == 0) { *Z = x; return 0; }
1750 1013580 : l -= v;
1751 1013580 : y = cgetg(l, t_VECSMALL); y[1] = x[1];
1752 2607170 : for (i=2; i<l; i++) y[i] = x[i+v];
1753 1016534 : *Z = y; return v;
1754 : }
1755 :
1756 : GEN
1757 18507035 : Flx_deriv(GEN z, ulong p)
1758 : {
1759 18507035 : long i,l = lg(z)-1;
1760 : GEN x;
1761 18507035 : if (l < 2) l = 2;
1762 18507035 : x = cgetg(l, t_VECSMALL); x[1] = z[1]; z++;
1763 18500327 : if (HIGHWORD(l | p))
1764 53710071 : for (i=2; i<l; i++) x[i] = Fl_mul((ulong)i-1, z[i], p);
1765 : else
1766 75341929 : for (i=2; i<l; i++) x[i] = ((i-1) * z[i]) % p;
1767 18501697 : return Flx_renormalize(x,l);
1768 : }
1769 :
1770 : static GEN
1771 422649 : Flx_integXn(GEN x, long n, ulong p)
1772 : {
1773 422649 : long i, lx = lg(x);
1774 : GEN y;
1775 422649 : if (lx == 2) return Flx_copy(x);
1776 412836 : y = cgetg(lx, t_VECSMALL); y[1] = x[1];
1777 2096382 : for (i=2; i<lx; i++)
1778 : {
1779 1682925 : ulong xi = uel(x,i);
1780 1682925 : if (xi == 0)
1781 13345 : uel(y,i) = 0;
1782 : else
1783 : {
1784 1669580 : ulong j = n+i-1;
1785 1669580 : ulong d = ugcd(j, xi);
1786 1669476 : if (d==1)
1787 1018125 : uel(y,i) = Fl_div(xi, j, p);
1788 : else
1789 651351 : uel(y,i) = Fl_div(xi/d, j/d, p);
1790 : }
1791 : }
1792 413457 : return Flx_renormalize(y, lx);;
1793 : }
1794 :
1795 : GEN
1796 0 : Flx_integ(GEN x, ulong p)
1797 : {
1798 0 : long i, lx = lg(x);
1799 : GEN y;
1800 0 : if (lx == 2) return Flx_copy(x);
1801 0 : y = cgetg(lx+1, t_VECSMALL); y[1] = x[1];
1802 0 : uel(y,2) = 0;
1803 0 : for (i=3; i<=lx; i++)
1804 0 : uel(y,i) = uel(x,i-1) ? Fl_div(uel(x,i-1), (i-2)%p, p): 0UL;
1805 0 : return Flx_renormalize(y, lx+1);;
1806 : }
1807 :
1808 : /* assume p prime */
1809 : GEN
1810 13447 : Flx_diff1(GEN P, ulong p)
1811 : {
1812 13447 : return Flx_sub(Flx_translate1(P, p), P, p);
1813 : }
1814 :
1815 : GEN
1816 343977 : Flx_deflate(GEN x0, long d)
1817 : {
1818 : GEN z, y, x;
1819 343977 : long i,id, dy, dx = degpol(x0);
1820 343977 : if (d == 1 || dx <= 0) return Flx_copy(x0);
1821 280835 : dy = dx/d;
1822 280835 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1823 280830 : z = y + 2;
1824 280830 : x = x0+ 2;
1825 929205 : for (i=id=0; i<=dy; i++,id+=d) z[i] = x[id];
1826 280830 : return y;
1827 : }
1828 :
1829 : GEN
1830 164369 : Flx_inflate(GEN x0, long d)
1831 : {
1832 164369 : long i, id, dy, dx = degpol(x0);
1833 164363 : GEN x = x0 + 2, z, y;
1834 164363 : if (dx <= 0) return Flx_copy(x0);
1835 163231 : dy = dx*d;
1836 163231 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1837 163205 : z = y + 2;
1838 9386175 : for (i=0; i<=dy; i++) z[i] = 0;
1839 4576213 : for (i=id=0; i<=dx; i++,id+=d) z[id] = x[i];
1840 163205 : return y;
1841 : }
1842 :
1843 : /* write p(X) = a_0(X^k) + X*a_1(X^k) + ... + X^(k-1)*a_{k-1}(X^k) */
1844 : GEN
1845 149549 : Flx_splitting(GEN p, long k)
1846 : {
1847 149549 : long n = degpol(p), v = p[1], m, i, j, l;
1848 : GEN r;
1849 :
1850 149548 : m = n/k;
1851 149548 : r = cgetg(k+1,t_VEC);
1852 685916 : for(i=1; i<=k; i++)
1853 : {
1854 536379 : gel(r,i) = cgetg(m+3, t_VECSMALL);
1855 536372 : mael(r,i,1) = v;
1856 : }
1857 4741959 : for (j=1, i=0, l=2; i<=n; i++)
1858 : {
1859 4592422 : mael(r,j,l) = p[2+i];
1860 4592422 : if (j==k) { j=1; l++; } else j++;
1861 : }
1862 685929 : for(i=1; i<=k; i++)
1863 536407 : gel(r,i) = Flx_renormalize(gel(r,i),i<j?l+1:l);
1864 149522 : return r;
1865 : }
1866 :
1867 : /* ux + vy */
1868 : static GEN
1869 431286 : Flx_addmulmul(GEN u, GEN v, GEN x, GEN y, ulong p, ulong pi)
1870 431286 : { return Flx_add(Flx_mul_pre(u,x, p,pi), Flx_mul_pre(v,y, p,pi), p); }
1871 :
1872 : static GEN
1873 25344 : FlxM_Flx_mul2(GEN M, GEN x, GEN y, ulong p, ulong pi)
1874 : {
1875 25344 : GEN res = cgetg(3, t_COL);
1876 25344 : gel(res, 1) = Flx_addmulmul(gcoeff(M,1,1), gcoeff(M,1,2), x, y, p, pi);
1877 25344 : gel(res, 2) = Flx_addmulmul(gcoeff(M,2,1), gcoeff(M,2,2), x, y, p, pi);
1878 25344 : return res;
1879 : }
1880 :
1881 : #if 0
1882 : static GEN
1883 : FlxM_mul2_old(GEN M, GEN N, ulong p)
1884 : {
1885 : GEN res = cgetg(3, t_MAT);
1886 : gel(res, 1) = FlxM_Flx_mul2(M,gcoeff(N,1,1),gcoeff(N,2,1),p);
1887 : gel(res, 2) = FlxM_Flx_mul2(M,gcoeff(N,1,2),gcoeff(N,2,2),p);
1888 : return res;
1889 : }
1890 : #endif
1891 : /* A,B are 2x2 matrices, Flx entries. Return A x B using Strassen 7M formula */
1892 : static GEN
1893 6813 : FlxM_mul2(GEN A, GEN B, ulong p, ulong pi)
1894 : {
1895 6813 : GEN A11=gcoeff(A,1,1),A12=gcoeff(A,1,2), B11=gcoeff(B,1,1),B12=gcoeff(B,1,2);
1896 6813 : GEN A21=gcoeff(A,2,1),A22=gcoeff(A,2,2), B21=gcoeff(B,2,1),B22=gcoeff(B,2,2);
1897 6813 : GEN M1 = Flx_mul_pre(Flx_add(A11,A22, p), Flx_add(B11,B22, p), p, pi);
1898 6813 : GEN M2 = Flx_mul_pre(Flx_add(A21,A22, p), B11, p, pi);
1899 6813 : GEN M3 = Flx_mul_pre(A11, Flx_sub(B12,B22, p), p, pi);
1900 6813 : GEN M4 = Flx_mul_pre(A22, Flx_sub(B21,B11, p), p, pi);
1901 6813 : GEN M5 = Flx_mul_pre(Flx_add(A11,A12, p), B22, p, pi);
1902 6813 : GEN M6 = Flx_mul_pre(Flx_sub(A21,A11, p), Flx_add(B11,B12, p), p, pi);
1903 6813 : GEN M7 = Flx_mul_pre(Flx_sub(A12,A22, p), Flx_add(B21,B22, p), p, pi);
1904 6813 : GEN T1 = Flx_add(M1,M4, p), T2 = Flx_sub(M7,M5, p);
1905 6813 : GEN T3 = Flx_sub(M1,M2, p), T4 = Flx_add(M3,M6, p);
1906 6813 : retmkmat22(Flx_add(T1,T2, p), Flx_add(M3,M5, p),
1907 : Flx_add(M2,M4, p), Flx_add(T3,T4, p));
1908 : }
1909 :
1910 : /* Return [0,1;1,-q]*M */
1911 : static GEN
1912 6641 : Flx_FlxM_qmul(GEN q, GEN M, ulong p, ulong pi)
1913 : {
1914 6641 : GEN u = Flx_mul_pre(gcoeff(M,2,1), q, p, pi);
1915 6641 : GEN v = Flx_mul_pre(gcoeff(M,2,2), q, p, pi);
1916 6641 : retmkmat22(gcoeff(M,2,1), gcoeff(M,2,2),
1917 : Flx_sub(gcoeff(M,1,1), u, p), Flx_sub(gcoeff(M,1,2), v, p));
1918 : }
1919 :
1920 : static GEN
1921 895 : matid2_FlxM(long v)
1922 895 : { retmkmat22(pol1_Flx(v),pol0_Flx(v),pol0_Flx(v),pol1_Flx(v)); }
1923 :
1924 : static GEN
1925 13 : matJ2_FlxM(long v)
1926 13 : { retmkmat22(pol0_Flx(v),pol1_Flx(v),pol1_Flx(v),pol0_Flx(v)); }
1927 :
1928 : struct Flx_res
1929 : {
1930 : ulong res, lc;
1931 : long deg0, deg1, off;
1932 : };
1933 :
1934 : INLINE void
1935 9405 : Flx_halfres_update_pre(long da, long db, long dr, ulong p, ulong pi, struct Flx_res *res)
1936 : {
1937 9405 : if (dr >= 0)
1938 : {
1939 9405 : if (res->lc != 1)
1940 : {
1941 7596 : if (pi)
1942 : {
1943 3127 : res->lc = Fl_powu_pre(res->lc, da - dr, p, pi);
1944 3127 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1945 : } else
1946 : {
1947 4469 : res->lc = Fl_powu(res->lc, da - dr, p);
1948 4469 : res->res = Fl_mul(res->res, res->lc, p);
1949 : }
1950 : }
1951 9405 : if (both_odd(da + res->off, db + res->off))
1952 63 : res->res = Fl_neg(res->res, p);
1953 : } else
1954 : {
1955 0 : if (db == 0)
1956 : {
1957 0 : if (res->lc != 1)
1958 : {
1959 0 : if (pi)
1960 : {
1961 0 : res->lc = Fl_powu_pre(res->lc, da, p, pi);
1962 0 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1963 : } else
1964 : {
1965 0 : res->lc = Fl_powu(res->lc, da, p);
1966 0 : res->res = Fl_mul(res->res, res->lc, p);
1967 : }
1968 : }
1969 : } else
1970 0 : res->res = 0;
1971 : }
1972 9405 : }
1973 :
1974 : static GEN
1975 1115575 : Flx_halfres_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *pa, GEN *pb, struct Flx_res *res)
1976 : {
1977 1115575 : pari_sp av = avma;
1978 : GEN u, u1, v, v1, M;
1979 1115575 : long vx = a[1], n = lgpol(a)>>1;
1980 1115568 : u1 = v = pol0_Flx(vx);
1981 1115560 : u = v1 = pol1_Flx(vx);
1982 6993308 : while (lgpol(b)>n)
1983 : {
1984 : GEN r, q;
1985 5877748 : q = Flx_divrem_pre(a,b,p,pi, &r);
1986 5877841 : if (res)
1987 : {
1988 8362 : long da = degpol(a), db=degpol(b), dr = degpol(r);
1989 8362 : res->lc = b[db+2];
1990 8362 : if (dr >= n)
1991 7133 : Flx_halfres_update_pre(da, db, dr, p, pi, res);
1992 : else
1993 : {
1994 1229 : res->deg0 = da;
1995 1229 : res->deg1 = db;
1996 : }
1997 : }
1998 5877841 : a = b; b = r; swap(u,u1); swap(v,v1);
1999 5877841 : u1 = Flx_sub(u1, Flx_mul(u, q, p), p);
2000 5877748 : v1 = Flx_sub(v1, Flx_mul(v, q, p), p);
2001 5877760 : if (gc_needed(av,2))
2002 : {
2003 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_halfgcd (d = %ld)",degpol(b));
2004 0 : gerepileall(av,6, &a,&b,&u1,&v1,&u,&v);
2005 : }
2006 : }
2007 1115380 : M = mkmat22(u,v,u1,v1); *pa = a; *pb = b;
2008 1115544 : return gc_all(av,3, &M, pa, pb);
2009 : }
2010 :
2011 : static GEN Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res);
2012 :
2013 : static GEN
2014 19578 : Flx_halfres_split(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2015 : {
2016 19578 : pari_sp av = avma;
2017 : GEN R, S, T, V1, V2;
2018 : GEN x1, y1, r, q;
2019 19578 : long l = lgpol(x), n = l>>1, k;
2020 19578 : if (lgpol(y) <= n)
2021 855 : { *a = Flx_copy(x); *b = Flx_copy(y); return matid2_FlxM(x[1]); }
2022 18723 : if (res)
2023 : {
2024 3263 : res->lc = Flx_lead(y);
2025 3263 : res->deg0 -= n;
2026 3263 : res->deg1 -= n;
2027 3263 : res->off += n;
2028 : }
2029 18723 : R = Flx_halfres_i(Flx_shift(x,-n),Flx_shift(y,-n),p,pi,a,b,res);
2030 18723 : if (res)
2031 : {
2032 3263 : res->off -= n;
2033 3263 : res->deg0 += n;
2034 3263 : res->deg1 += n;
2035 : }
2036 18723 : V1 = FlxM_Flx_mul2(R, Flxn_red(x,n), Flxn_red(y,n), p, pi);
2037 18723 : x1 = Flx_add(Flx_shift(*a,n), gel(V1,1), p);
2038 18722 : y1 = Flx_add(Flx_shift(*b,n), gel(V1,2), p);
2039 18722 : if (lgpol(y1) <= n)
2040 12101 : { *a = x1; *b = y1; return gc_all(av, 3, &R, a, b); }
2041 6621 : k = 2*n-degpol(y1);
2042 6621 : q = Flx_divrem_pre(x1, y1, p, pi, &r);
2043 6621 : if (res)
2044 : {
2045 1043 : long dx1 = degpol(x1), dy1 = degpol(y1), dr = degpol(r);
2046 1043 : if (dy1 < degpol(y))
2047 185 : Flx_halfres_update_pre(res->deg0, res->deg1, dy1, p, pi, res);
2048 1043 : res->lc = uel(y1, dy1+2);
2049 1043 : res->deg0 = dx1;
2050 1043 : res->deg1 = dy1;
2051 1043 : if (dr >= n)
2052 : {
2053 1043 : Flx_halfres_update_pre(dx1, dy1, dr, p, pi, res);
2054 1043 : res->deg0 = dy1;
2055 1043 : res->deg1 = dr;
2056 : }
2057 1043 : res->deg0 -= k;
2058 1043 : res->deg1 -= k;
2059 1043 : res->off += k;
2060 : }
2061 6621 : S = Flx_halfres_i(Flx_shift(y1,-k), Flx_shift(r,-k), p, pi, a, b, res);
2062 6621 : if (res)
2063 : {
2064 1043 : res->deg0 += k;
2065 1043 : res->deg1 += k;
2066 1043 : res->off -= k;
2067 : }
2068 6621 : T = FlxM_mul2(S, Flx_FlxM_qmul(q, R, p,pi), p, pi);
2069 6621 : V2 = FlxM_Flx_mul2(S, Flxn_red(y1,k), Flxn_red(r,k), p, pi);
2070 6621 : *a = Flx_add(Flx_shift(*a,k), gel(V2,1), p);
2071 6621 : *b = Flx_add(Flx_shift(*b,k), gel(V2,2), p);
2072 6621 : return gc_all(av, 3, &T, a, b);
2073 : }
2074 :
2075 : static GEN
2076 1135152 : Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2077 : {
2078 1135152 : if (lgpol(x) < get_Fl_threshold(p, Flx_HALFGCD_LIMIT, Flx_HALFGCD2_LIMIT))
2079 1115572 : return Flx_halfres_basecase(x, y, p, pi, a, b, res);
2080 19578 : return Flx_halfres_split(x, y, p, pi, a, b, res);
2081 : }
2082 :
2083 : static GEN
2084 1108763 : Flx_halfgcd_all_i(GEN x, GEN y, ulong p, ulong pi, GEN *pa, GEN *pb)
2085 : {
2086 : GEN a, b, R;
2087 1108763 : R = Flx_halfres_i(x, y, p, pi, &a, &b, NULL);
2088 1108781 : if (pa) *pa = a;
2089 1108781 : if (pb) *pb = b;
2090 1108781 : return R;
2091 : }
2092 :
2093 : /* Return M in GL_2(Fl[X]) such that:
2094 : if [a',b']~=M*[a,b]~ then degpol(a')>= (lgpol(a)>>1) >degpol(b')
2095 : */
2096 :
2097 : GEN
2098 1108768 : Flx_halfgcd_all_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b)
2099 : {
2100 : pari_sp av;
2101 : GEN R, q ,r;
2102 1108768 : long lx = lgpol(x), ly = lgpol(y);
2103 1108762 : if (!lx)
2104 : {
2105 0 : if (a) *a = Flx_copy(y);
2106 0 : if (b) *b = Flx_copy(x);
2107 0 : return matJ2_FlxM(x[1]);
2108 : }
2109 1108762 : if (ly < lx) return Flx_halfgcd_all_i(x, y, p, pi, a, b);
2110 10919 : av = avma;
2111 10919 : q = Flx_divrem(y,x,p,&r);
2112 10919 : R = Flx_halfgcd_all_i(x, r, p, pi, a, b);
2113 10919 : gcoeff(R,1,1) = Flx_sub(gcoeff(R,1,1), Flx_mul_pre(q,gcoeff(R,1,2), p,pi), p);
2114 10919 : gcoeff(R,2,1) = Flx_sub(gcoeff(R,2,1), Flx_mul_pre(q,gcoeff(R,2,2), p,pi), p);
2115 10919 : return !a && b ? gc_all(av, 2, &R, b): gc_all(av, 1+!!a+!!b, &R, a, b);
2116 : }
2117 :
2118 : GEN
2119 126 : Flx_halfgcd_all(GEN x, GEN y, ulong p, GEN *a, GEN *b)
2120 126 : { return Flx_halfgcd_all_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), a, b); }
2121 :
2122 : GEN
2123 841473 : Flx_halfgcd_pre(GEN x, GEN y, ulong p, ulong pi)
2124 841473 : { return Flx_halfgcd_all_pre(x, y, p, pi, NULL, NULL); }
2125 :
2126 : GEN
2127 0 : Flx_halfgcd(GEN x, GEN y, ulong p)
2128 0 : { return Flx_halfgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2129 :
2130 : /*Do not garbage collect*/
2131 : static GEN
2132 76752870 : Flx_gcd_basecase(GEN a, GEN b, ulong p, ulong pi)
2133 : {
2134 76752870 : pari_sp av = avma;
2135 76752870 : ulong iter = 0;
2136 76752870 : if (lg(b) > lg(a)) swap(a, b);
2137 266764929 : while (lgpol(b))
2138 : {
2139 189507715 : GEN c = Flx_rem_pre(a,b,p,pi);
2140 190012059 : iter++; a = b; b = c;
2141 190012059 : if (gc_needed(av,2))
2142 : {
2143 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (d = %ld)",degpol(c));
2144 0 : gerepileall(av,2, &a,&b);
2145 : }
2146 : }
2147 76720377 : return iter < 2 ? Flx_copy(a) : a;
2148 : }
2149 :
2150 : GEN
2151 78367262 : Flx_gcd_pre(GEN x, GEN y, ulong p, ulong pi)
2152 : {
2153 78367262 : pari_sp av = avma;
2154 : long lim;
2155 78367262 : if (!lgpol(x)) return Flx_copy(y);
2156 76760761 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2157 76761504 : while (lgpol(y) >= lim)
2158 : {
2159 150 : if (lgpol(y)<=(lgpol(x)>>1))
2160 : {
2161 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2162 0 : x = y; y = r;
2163 : }
2164 150 : (void) Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2165 150 : if (gc_needed(av,2))
2166 : {
2167 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (y = %ld)",degpol(y));
2168 0 : gerepileall(av,2,&x,&y);
2169 : }
2170 : }
2171 76744825 : return gerepileuptoleaf(av, Flx_gcd_basecase(x,y,p,pi));
2172 : }
2173 : GEN
2174 27656578 : Flx_gcd(GEN x, GEN y, ulong p)
2175 27656578 : { return Flx_gcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2176 :
2177 : int
2178 6285216 : Flx_is_squarefree(GEN z, ulong p)
2179 : {
2180 6285216 : pari_sp av = avma;
2181 6285216 : GEN d = Flx_gcd(z, Flx_deriv(z,p) , p);
2182 6285200 : return gc_bool(av, degpol(d) == 0);
2183 : }
2184 :
2185 : static long
2186 139528 : Flx_is_smooth_squarefree(GEN f, long r, ulong p, ulong pi)
2187 : {
2188 139528 : pari_sp av = avma;
2189 : long i;
2190 139528 : GEN sx = polx_Flx(f[1]), a = sx;
2191 593417 : for(i=1;;i++)
2192 : {
2193 593417 : if (degpol(f)<=r) return gc_long(av,1);
2194 566940 : a = Flxq_powu_pre(Flx_rem_pre(a,f,p,pi), p, f, p, pi);
2195 571501 : if (Flx_equal(a, sx)) return gc_long(av,1);
2196 566950 : if (i==r) return gc_long(av,0);
2197 455305 : f = Flx_div_pre(f, Flx_gcd_pre(Flx_sub(a,sx,p),f,p,pi),p,pi);
2198 : }
2199 : }
2200 :
2201 : static long
2202 9134 : Flx_is_l_pow(GEN x, ulong p)
2203 : {
2204 9134 : ulong i, lx = lgpol(x);
2205 18369 : for (i=1; i<lx; i++)
2206 16413 : if (x[i+2] && i%p) return 0;
2207 1956 : return 1;
2208 : }
2209 :
2210 : int
2211 139547 : Flx_is_smooth_pre(GEN g, long r, ulong p, ulong pi)
2212 : {
2213 : while (1)
2214 9137 : {
2215 139547 : GEN f = Flx_gcd_pre(g, Flx_deriv(g, p), p, pi);
2216 139377 : if (!Flx_is_smooth_squarefree(Flx_div_pre(g, f, p, pi), r, p, pi))
2217 111619 : return 0;
2218 28016 : if (degpol(f)==0) return 1;
2219 9117 : g = Flx_is_l_pow(f,p) ? Flx_deflate(f, p): f;
2220 : }
2221 : }
2222 : int
2223 74256 : Flx_is_smooth(GEN g, long r, ulong p)
2224 74256 : { return Flx_is_smooth_pre(g, r, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2225 :
2226 : static GEN
2227 6618869 : Flx_extgcd_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2228 : {
2229 6618869 : pari_sp av=avma;
2230 : GEN u,v,u1,v1;
2231 6618869 : long vx = a[1];
2232 6618869 : v = pol0_Flx(vx); v1 = pol1_Flx(vx);
2233 6618624 : if (ptu) { u = pol1_Flx(vx); u1 = pol0_Flx(vx); }
2234 29277686 : while (lgpol(b))
2235 : {
2236 22657273 : GEN r, q = Flx_divrem_pre(a,b,p,pi, &r);
2237 22659180 : a = b; b = r;
2238 22659180 : if (ptu)
2239 : {
2240 2455902 : swap(u,u1);
2241 2455902 : u1 = Flx_sub(u1, Flx_mul_pre(u, q, p, pi), p);
2242 : }
2243 22659166 : swap(v,v1);
2244 22659166 : v1 = Flx_sub(v1, Flx_mul_pre(v, q, p, pi), p);
2245 22659093 : if (gc_needed(av,2))
2246 : {
2247 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_extgcd (d = %ld)",degpol(a));
2248 0 : gerepileall(av,ptu ? 6: 4, &a,&b,&v,&v1,&u,&u1);
2249 : }
2250 : }
2251 6618689 : if (ptu) *ptu = u;
2252 6618689 : *ptv = v;
2253 6618689 : return a;
2254 : }
2255 :
2256 : static GEN
2257 151992 : Flx_extgcd_halfgcd(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2258 : {
2259 : GEN u, v;
2260 151992 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2261 151992 : GEN V = cgetg(expu(lgpol(y))+2,t_VEC);
2262 151992 : long i, n = 0, vs = x[1];
2263 414963 : while (lgpol(y) >= lim)
2264 : {
2265 262970 : if (lgpol(y)<=(lgpol(x)>>1))
2266 : {
2267 26 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2268 26 : x = y; y = r;
2269 26 : gel(V,++n) = mkmat22(pol0_Flx(vs),pol1_Flx(vs),pol1_Flx(vs),Flx_neg(q,p));
2270 : } else
2271 262943 : gel(V,++n) = Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2272 : }
2273 151992 : y = Flx_extgcd_basecase(x,y,p,pi,&u,&v);
2274 262970 : for (i = n; i>1; i--)
2275 : {
2276 110979 : GEN R = gel(V,i);
2277 110979 : GEN u1 = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2278 110979 : GEN v1 = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2279 110979 : u = u1; v = v1;
2280 : }
2281 : {
2282 151991 : GEN R = gel(V,1);
2283 151991 : if (ptu)
2284 6543 : *ptu = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2285 151991 : *ptv = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2286 : }
2287 151992 : return y;
2288 : }
2289 :
2290 : /* x and y in Z[X], return lift(gcd(x mod p, y mod p)). Set u and v st
2291 : * ux + vy = gcd (mod p) */
2292 : GEN
2293 6618865 : Flx_extgcd_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2294 : {
2295 6618865 : pari_sp av = avma;
2296 : GEN d;
2297 6618865 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2298 6618887 : if (lgpol(y) >= lim)
2299 151992 : d = Flx_extgcd_halfgcd(x, y, p, pi, ptu, ptv);
2300 : else
2301 6466878 : d = Flx_extgcd_basecase(x, y, p, pi, ptu, ptv);
2302 6618690 : return gc_all(av, ptu?3:2, &d, ptv, ptu);
2303 : }
2304 : GEN
2305 841314 : Flx_extgcd(GEN x, GEN y, ulong p, GEN *ptu, GEN *ptv)
2306 841314 : { return Flx_extgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptu, ptv); }
2307 :
2308 : static GEN
2309 1044 : Flx_halfres_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, ulong *r)
2310 : {
2311 : struct Flx_res res;
2312 : GEN R;
2313 : long dB;
2314 :
2315 1044 : res.res = *r;
2316 1044 : res.lc = Flx_lead(y);
2317 1044 : res.deg0 = degpol(x);
2318 1044 : res.deg1 = degpol(y);
2319 1044 : res.off = 0;
2320 1044 : R = Flx_halfres_i(x, y, p, pi, a, b, &res);
2321 1044 : dB = degpol(*b);
2322 1044 : if (dB < degpol(y))
2323 1044 : Flx_halfres_update_pre(res.deg0, res.deg1, dB, p, pi, &res);
2324 1044 : *r = res.res;
2325 1044 : return R;
2326 : }
2327 :
2328 : static ulong
2329 7077535 : Flx_resultant_basecase_pre(GEN a, GEN b, ulong p, ulong pi)
2330 : {
2331 : pari_sp av;
2332 : long da,db,dc;
2333 7077535 : ulong lb, res = 1UL;
2334 : GEN c;
2335 :
2336 7077535 : da = degpol(a);
2337 7077414 : db = degpol(b);
2338 7077403 : if (db > da)
2339 : {
2340 0 : swapspec(a,b, da,db);
2341 0 : if (both_odd(da,db)) res = p-res;
2342 : }
2343 7077403 : else if (!da) return 1; /* = res * a[2] ^ db, since 0 <= db <= da = 0 */
2344 7077403 : av = avma;
2345 54695193 : while (db)
2346 : {
2347 47633101 : lb = b[db+2];
2348 47633101 : c = Flx_rem_pre(a,b, p,pi);
2349 47450760 : a = b; b = c; dc = degpol(c);
2350 47445674 : if (dc < 0) return gc_long(av,0);
2351 :
2352 47433109 : if (both_odd(da,db)) res = p - res;
2353 47437642 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, da - dc, p, pi), p);
2354 47617502 : if (gc_needed(av,2))
2355 : {
2356 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant (da = %ld)",da);
2357 0 : gerepileall(av,2, &a,&b);
2358 : }
2359 47617790 : da = db; /* = degpol(a) */
2360 47617790 : db = dc; /* = degpol(b) */
2361 : }
2362 7062092 : return gc_ulong(av, Fl_mul(res, Fl_powu_pre(b[2], da, p, pi), p));
2363 : }
2364 :
2365 : ulong
2366 7079406 : Flx_resultant_pre(GEN x, GEN y, ulong p, ulong pi)
2367 : {
2368 7079406 : pari_sp av = avma;
2369 : long lim;
2370 7079406 : ulong res = 1;
2371 7079406 : long dx = degpol(x), dy = degpol(y);
2372 7079114 : if (dx < 0 || dy < 0) return 0;
2373 7077672 : if (dx < dy)
2374 : {
2375 1064542 : swap(x,y);
2376 1064542 : if (both_odd(dx, dy))
2377 1906 : res = Fl_neg(res, p);
2378 : }
2379 7077673 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2380 7078530 : while (lgpol(y) >= lim)
2381 : {
2382 852 : if (lgpol(y)<=(lgpol(x)>>1))
2383 : {
2384 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2385 0 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2386 0 : ulong ly = y[dy+2];
2387 0 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2388 0 : if (both_odd(dx, dy))
2389 0 : res = Fl_neg(res, p);
2390 0 : x = y; y = r;
2391 : }
2392 852 : (void) Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2393 852 : if (gc_needed(av,2))
2394 : {
2395 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_res (y = %ld)",degpol(y));
2396 0 : gerepileall(av,2,&x,&y);
2397 : }
2398 : }
2399 7077570 : return gc_ulong(av, Fl_mul(res, Flx_resultant_basecase_pre(x, y, p, pi), p));
2400 : }
2401 :
2402 : ulong
2403 4810897 : Flx_resultant(GEN a, GEN b, ulong p)
2404 4810897 : { return Flx_resultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2405 :
2406 : /* If resultant is 0, *ptU and *ptV are not set */
2407 : static ulong
2408 53 : Flx_extresultant_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2409 : {
2410 53 : GEN z,q,u,v, x = a, y = b;
2411 53 : ulong lb, res = 1UL;
2412 53 : pari_sp av = avma;
2413 : long dx, dy, dz;
2414 53 : long vs = a[1];
2415 :
2416 53 : u = pol0_Flx(vs);
2417 53 : v = pol1_Flx(vs); /* v = 1 */
2418 53 : dx = degpol(x);
2419 53 : dy = degpol(y);
2420 764 : while (dy)
2421 : { /* b u = x (a), b v = y (a) */
2422 711 : lb = y[dy+2];
2423 711 : q = Flx_divrem_pre(x,y, p, pi, &z);
2424 711 : x = y; y = z; /* (x,y) = (y, x - q y) */
2425 711 : dz = degpol(z); if (dz < 0) return gc_ulong(av,0);
2426 711 : z = Flx_sub(u, Flx_mul_pre(q,v, p, pi), p);
2427 711 : u = v; v = z; /* (u,v) = (v, u - q v) */
2428 :
2429 711 : if (both_odd(dx,dy)) res = p - res;
2430 711 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, dx-dz, p, pi), p);
2431 711 : dx = dy; /* = degpol(x) */
2432 711 : dy = dz; /* = degpol(y) */
2433 : }
2434 53 : res = Fl_mul(res, Fl_powu_pre(y[2], dx, p, pi), p);
2435 53 : lb = Fl_mul(res, Fl_inv(y[2],p), p);
2436 53 : v = gerepileuptoleaf(av, Flx_Fl_mul_pre(v, lb, p, pi));
2437 53 : av = avma;
2438 53 : u = Flx_sub(Fl_to_Flx(res,vs), Flx_mul_pre(b,v,p,pi), p);
2439 53 : u = gerepileuptoleaf(av, Flx_div_pre(u,a,p,pi)); /* = (res - b v) / a */
2440 53 : *ptU = u;
2441 53 : *ptV = v; return res;
2442 : }
2443 :
2444 : ulong
2445 53 : Flx_extresultant_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2446 : {
2447 53 : pari_sp av=avma;
2448 : GEN u, v, R;
2449 53 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2450 53 : ulong res = 1, res1;
2451 53 : long dx = degpol(x), dy = degpol(y);
2452 53 : if (dy > dx)
2453 : {
2454 13 : swap(x,y); lswap(dx,dy);
2455 13 : if (both_odd(dx,dy)) res = p-res;
2456 13 : R = matJ2_FlxM(x[1]);
2457 40 : } else R = matid2_FlxM(x[1]);
2458 53 : if (dy < 0) return 0;
2459 245 : while (lgpol(y) >= lim)
2460 : {
2461 : GEN M;
2462 192 : if (lgpol(y)<=(lgpol(x)>>1))
2463 : {
2464 20 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2465 20 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2466 20 : ulong ly = y[dy+2];
2467 20 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2468 20 : if (both_odd(dx, dy))
2469 0 : res = Fl_neg(res, p);
2470 20 : x = y; y = r;
2471 20 : R = Flx_FlxM_qmul(q, R, p,pi);
2472 : }
2473 192 : M = Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2474 192 : if (!res) return gc_ulong(av, 0);
2475 192 : R = FlxM_mul2(M, R, p, pi);
2476 192 : gerepileall(av,3,&x,&y,&R);
2477 : }
2478 53 : res1 = Flx_extresultant_basecase(x,y,p,pi,&u,&v);
2479 53 : if (!res1) return gc_ulong(av, 0);
2480 53 : *ptU = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi), res, p, pi);
2481 53 : *ptV = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi), res, p, pi);
2482 53 : gerepileall(av, 2, ptU, ptV);
2483 53 : return Fl_mul(res1,res,p);
2484 : }
2485 :
2486 : ulong
2487 53 : Flx_extresultant(GEN a, GEN b, ulong p, GEN *ptU, GEN *ptV)
2488 53 : { return Flx_extresultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptU, ptV); }
2489 :
2490 : /* allow pi = 0 (SMALL_ULONG) */
2491 : ulong
2492 43754013 : Flx_eval_powers_pre(GEN x, GEN y, ulong p, ulong pi)
2493 : {
2494 43754013 : ulong l0, l1, h0, h1, v1, i = 1, lx = lg(x)-1;
2495 :
2496 43754013 : if (lx == 1) return 0;
2497 41002671 : x++;
2498 41002671 : if (pi)
2499 : {
2500 : LOCAL_OVERFLOW;
2501 : LOCAL_HIREMAINDER;
2502 40938496 : l1 = mulll(uel(x,i), uel(y,i)); h1 = hiremainder; v1 = 0;
2503 97851941 : while (++i < lx)
2504 : {
2505 56913445 : l0 = mulll(uel(x,i), uel(y,i)); h0 = hiremainder;
2506 56913445 : l1 = addll(l0, l1); h1 = addllx(h0, h1); v1 += overflow;
2507 : }
2508 81118 : return v1? remlll_pre(v1, h1, l1, p, pi)
2509 41019614 : : remll_pre(h1, l1, p, pi);
2510 : }
2511 : else
2512 : {
2513 64175 : l1 = x[i] * y[i];
2514 30924799 : while (++i < lx) { l1 += x[i] * y[i]; if (l1 & HIGHBIT) l1 %= p; }
2515 64175 : return l1 % p;
2516 : }
2517 : }
2518 :
2519 : /* allow pi = 0 (SMALL_ULONG) */
2520 : ulong
2521 44828307 : Flx_eval_pre(GEN x, ulong y, ulong p, ulong pi)
2522 : {
2523 44828307 : long i, n = degpol(x);
2524 : ulong t;
2525 44820224 : if (n <= 0) return n? 0: x[2];
2526 29638837 : if (n > 15)
2527 : {
2528 180373 : pari_sp av = avma;
2529 180373 : GEN v = Fl_powers_pre(y, n, p, pi);
2530 180361 : return gc_ulong(av, Flx_eval_powers_pre(x, v, p, pi));
2531 : }
2532 29458464 : i = n+2; t = x[i];
2533 29458464 : if (pi)
2534 : {
2535 116373441 : for (i--; i>=2; i--) t = Fl_addmul_pre(uel(x, i), t, y, p, pi);
2536 28316818 : return t;
2537 : }
2538 2671817 : for (i--; i>=2; i--) t = (t * y + x[i]) % p;
2539 1119179 : return t %= p;
2540 : }
2541 : ulong
2542 20377362 : Flx_eval(GEN x, ulong y, ulong p)
2543 20377362 : { return Flx_eval_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2544 :
2545 : ulong
2546 3255 : Flv_prod_pre(GEN x, ulong p, ulong pi)
2547 : {
2548 3255 : pari_sp ltop = avma;
2549 : GEN v;
2550 3255 : long i,k,lx = lg(x);
2551 3255 : if (lx == 1) return 1UL;
2552 3255 : if (lx == 2) return uel(x,1);
2553 3003 : v = cgetg(1+(lx << 1), t_VECSMALL);
2554 3003 : k = 1;
2555 28593 : for (i=1; i<lx-1; i+=2)
2556 25590 : uel(v,k++) = Fl_mul_pre(uel(x,i), uel(x,i+1), p, pi);
2557 3003 : if (i < lx) uel(v,k++) = uel(x,i);
2558 13529 : while (k > 2)
2559 : {
2560 10526 : lx = k; k = 1;
2561 36116 : for (i=1; i<lx-1; i+=2)
2562 25590 : uel(v,k++) = Fl_mul_pre(uel(v,i), uel(v,i+1), p, pi);
2563 10526 : if (i < lx) uel(v,k++) = uel(v,i);
2564 : }
2565 3003 : return gc_ulong(ltop, uel(v,1));
2566 : }
2567 :
2568 : ulong
2569 0 : Flv_prod(GEN v, ulong p)
2570 : {
2571 0 : return Flv_prod_pre(v, p, get_Fl_red(p));
2572 : }
2573 :
2574 : GEN
2575 0 : FlxV_prod(GEN V, ulong p)
2576 : {
2577 : struct _Flxq D;
2578 0 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2579 0 : return gen_product(V, (void *)&D, &_Flx_mul);
2580 : }
2581 :
2582 : /* compute prod (x - a[i]) */
2583 : GEN
2584 725691 : Flv_roots_to_pol(GEN a, ulong p, long vs)
2585 : {
2586 : struct _Flxq D;
2587 725691 : long i,k,lx = lg(a);
2588 : GEN p1;
2589 725691 : if (lx == 1) return pol1_Flx(vs);
2590 725691 : p1 = cgetg(lx, t_VEC);
2591 11829656 : for (k=1,i=1; i<lx-1; i+=2)
2592 11103195 : gel(p1,k++) = mkvecsmall4(vs, Fl_mul(a[i], a[i+1], p),
2593 11104305 : Fl_neg(Fl_add(a[i],a[i+1],p),p), 1);
2594 725351 : if (i < lx)
2595 58112 : gel(p1,k++) = mkvecsmall3(vs, Fl_neg(a[i],p), 1);
2596 725355 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2597 725355 : setlg(p1, k); return gen_product(p1, (void *)&D, _Flx_mul);
2598 : }
2599 :
2600 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for "large" p */
2601 : INLINE void
2602 18700311 : Flv_inv_pre_indir(GEN w, GEN v, ulong p, ulong pi)
2603 : {
2604 18700311 : pari_sp av = avma;
2605 18700311 : long n = lg(w), i;
2606 : ulong u;
2607 : GEN c;
2608 :
2609 18700311 : if (n == 1) return;
2610 18700311 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2611 77092081 : for (i = 2; i < n; ++i) c[i] = Fl_mul_pre(w[i], c[i-1], p, pi);
2612 18853037 : i = n-1; u = Fl_inv(c[i], p);
2613 77474897 : for ( ; i > 1; --i)
2614 : {
2615 58620237 : ulong t = Fl_mul_pre(u, c[i-1], p, pi);
2616 58608096 : u = Fl_mul_pre(u, w[i], p, pi); v[i] = t;
2617 : }
2618 18854660 : v[1] = u; set_avma(av);
2619 : }
2620 :
2621 : void
2622 18251116 : Flv_inv_pre_inplace(GEN v, ulong p, ulong pi) { Flv_inv_pre_indir(v,v, p, pi); }
2623 :
2624 : GEN
2625 10849 : Flv_inv_pre(GEN w, ulong p, ulong pi)
2626 10849 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_pre_indir(w, v, p, pi); return v; }
2627 :
2628 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for SMALL_ULONG p */
2629 : INLINE void
2630 48737 : Flv_inv_indir(GEN w, GEN v, ulong p)
2631 : {
2632 48737 : pari_sp av = avma;
2633 48737 : long n = lg(w), i;
2634 : ulong u;
2635 : GEN c;
2636 :
2637 48737 : if (n == 1) return;
2638 48737 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2639 1709480 : for (i = 2; i < n; ++i) c[i] = Fl_mul(w[i], c[i-1], p);
2640 48754 : i = n-1; u = Fl_inv(c[i], p);
2641 1709559 : for ( ; i > 1; --i)
2642 : {
2643 1660817 : ulong t = Fl_mul(u, c[i-1], p);
2644 1660816 : u = Fl_mul(u, w[i], p); v[i] = t;
2645 : }
2646 48742 : v[1] = u; set_avma(av);
2647 : }
2648 : static void
2649 443269 : Flv_inv_i(GEN v, GEN w, ulong p)
2650 : {
2651 443269 : if (SMALL_ULONG(p)) Flv_inv_indir(w, v, p);
2652 394531 : else Flv_inv_pre_indir(w, v, p, get_Fl_red(p));
2653 443278 : }
2654 : void
2655 12017 : Flv_inv_inplace(GEN v, ulong p) { Flv_inv_i(v, v, p); }
2656 : GEN
2657 431257 : Flv_inv(GEN w, ulong p)
2658 431257 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_i(v, w, p); return v; }
2659 :
2660 : GEN
2661 32897706 : Flx_div_by_X_x(GEN a, ulong x, ulong p, ulong *rem)
2662 : {
2663 32897706 : long l = lg(a), i;
2664 : GEN a0, z0, z;
2665 32897706 : if (l <= 3)
2666 : {
2667 0 : if (rem) *rem = l == 2? 0: a[2];
2668 0 : return zero_Flx(a[1]);
2669 : }
2670 32897706 : z = cgetg(l-1,t_VECSMALL); z[1] = a[1];
2671 32747530 : a0 = a + l-1;
2672 32747530 : z0 = z + l-2; *z0 = *a0--;
2673 32747530 : if (SMALL_ULONG(p))
2674 : {
2675 79409344 : for (i=l-3; i>1; i--) /* z[i] = (a[i+1] + x*z[i+1]) % p */
2676 : {
2677 58862005 : ulong t = (*a0-- + x * *z0--) % p;
2678 58862005 : *z0 = (long)t;
2679 : }
2680 20547339 : if (rem) *rem = (*a0 + x * *z0) % p;
2681 : }
2682 : else
2683 : {
2684 48205703 : for (i=l-3; i>1; i--)
2685 : {
2686 35954137 : ulong t = Fl_add((ulong)*a0--, Fl_mul(x, *z0--, p), p);
2687 36005512 : *z0 = (long)t;
2688 : }
2689 12251566 : if (rem) *rem = Fl_add((ulong)*a0, Fl_mul(x, *z0, p), p);
2690 : }
2691 32829049 : return z;
2692 : }
2693 :
2694 : /* xa, ya = t_VECSMALL */
2695 : static GEN
2696 432462 : Flv_producttree(GEN xa, GEN s, ulong p, ulong pi, long vs)
2697 : {
2698 432462 : long n = lg(xa)-1;
2699 432462 : long m = n==1 ? 1: expu(n-1)+1;
2700 432459 : long i, j, k, ls = lg(s);
2701 432459 : GEN T = cgetg(m+1, t_VEC);
2702 432446 : GEN t = cgetg(ls, t_VEC);
2703 4572370 : for (j=1, k=1; j<ls; k+=s[j++])
2704 4139750 : gel(t, j) = s[j] == 1 ?
2705 4139916 : mkvecsmall3(vs, Fl_neg(xa[k], p), 1):
2706 1304182 : mkvecsmall4(vs, Fl_mul(xa[k], xa[k+1], p),
2707 1304177 : Fl_neg(Fl_add(xa[k],xa[k+1],p),p), 1);
2708 432454 : gel(T,1) = t;
2709 1395495 : for (i=2; i<=m; i++)
2710 : {
2711 963077 : GEN u = gel(T, i-1);
2712 963077 : long n = lg(u)-1;
2713 963077 : GEN t = cgetg(((n+1)>>1)+1, t_VEC);
2714 4669976 : for (j=1, k=1; k<n; j++, k+=2)
2715 3706935 : gel(t, j) = Flx_mul_pre(gel(u, k), gel(u, k+1), p, pi);
2716 963041 : gel(T, i) = t;
2717 : }
2718 432418 : return T;
2719 : }
2720 :
2721 : static GEN
2722 472757 : Flx_Flv_multieval_tree(GEN P, GEN xa, GEN T, ulong p, ulong pi)
2723 : {
2724 : long i,j,k;
2725 472757 : long m = lg(T)-1;
2726 472757 : GEN R = cgetg(lg(xa), t_VECSMALL);
2727 472751 : GEN Tp = cgetg(m+1, t_VEC), t;
2728 472747 : gel(Tp, m) = mkvec(P);
2729 1621159 : for (i=m-1; i>=1; i--)
2730 : {
2731 1148408 : GEN u = gel(T, i), v = gel(Tp, i+1);
2732 1148408 : long n = lg(u)-1;
2733 1148408 : t = cgetg(n+1, t_VEC);
2734 5885156 : for (j=1, k=1; k<n; j++, k+=2)
2735 : {
2736 4736765 : gel(t, k) = Flx_rem_pre(gel(v, j), gel(u, k), p, pi);
2737 4736746 : gel(t, k+1) = Flx_rem_pre(gel(v, j), gel(u, k+1), p, pi);
2738 : }
2739 1148391 : gel(Tp, i) = t;
2740 : }
2741 : {
2742 472751 : GEN u = gel(T, i+1), v = gel(Tp, i+1);
2743 472751 : long n = lg(u)-1;
2744 5685205 : for (j=1, k=1; j<=n; j++)
2745 : {
2746 5212399 : long c, d = degpol(gel(u,j));
2747 11978256 : for (c=1; c<=d; c++, k++) R[k] = Flx_eval_pre(gel(v, j), xa[k], p, pi);
2748 : }
2749 472806 : return gc_const((pari_sp)R, R);
2750 : }
2751 : }
2752 :
2753 : static GEN
2754 1184446 : FlvV_polint_tree(GEN T, GEN R, GEN s, GEN xa, GEN ya, ulong p, ulong pi, long vs)
2755 : {
2756 1184446 : pari_sp av = avma;
2757 1184446 : long m = lg(T)-1;
2758 1184446 : long i, j, k, ls = lg(s);
2759 1184446 : GEN Tp = cgetg(m+1, t_VEC);
2760 1183919 : GEN t = cgetg(ls, t_VEC);
2761 21569943 : for (j=1, k=1; j<ls; k+=s[j++])
2762 20386192 : if (s[j]==2)
2763 : {
2764 6708750 : ulong a = Fl_mul(ya[k], R[k], p);
2765 6710079 : ulong b = Fl_mul(ya[k+1], R[k+1], p);
2766 6719959 : gel(t, j) = mkvecsmall3(vs, Fl_neg(Fl_add(Fl_mul(xa[k], b, p ),
2767 6716746 : Fl_mul(xa[k+1], a, p), p), p), Fl_add(a, b, p));
2768 6710708 : gel(t, j) = Flx_renormalize(gel(t, j), 4);
2769 : }
2770 : else
2771 13677442 : gel(t, j) = Fl_to_Flx(Fl_mul(ya[k], R[k], p), vs);
2772 1183751 : gel(Tp, 1) = t;
2773 5386717 : for (i=2; i<=m; i++)
2774 : {
2775 4202956 : GEN u = gel(T, i-1);
2776 4202956 : GEN t = cgetg(lg(gel(T,i)), t_VEC);
2777 4200006 : GEN v = gel(Tp, i-1);
2778 4200006 : long n = lg(v)-1;
2779 23358653 : for (j=1, k=1; k<n; j++, k+=2)
2780 19174262 : gel(t, j) = Flx_add(Flx_mul_pre(gel(u, k), gel(v, k+1), p, pi),
2781 19155687 : Flx_mul_pre(gel(u, k+1), gel(v, k), p, pi), p);
2782 4202966 : gel(Tp, i) = t;
2783 : }
2784 1183761 : return gerepileuptoleaf(av, gmael(Tp,m,1));
2785 : }
2786 :
2787 : GEN
2788 0 : Flx_Flv_multieval(GEN P, GEN xa, ulong p)
2789 : {
2790 0 : pari_sp av = avma;
2791 0 : GEN s = producttree_scheme(lg(xa)-1);
2792 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2793 0 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2794 0 : return gerepileuptoleaf(av, Flx_Flv_multieval_tree(P, xa, T, p, pi));
2795 : }
2796 :
2797 : static GEN
2798 2471 : FlxV_Flv_multieval_tree(GEN x, GEN xa, GEN T, ulong p, ulong pi)
2799 45248 : { pari_APPLY_same(Flx_Flv_multieval_tree(gel(x,i), xa, T, p, pi)) }
2800 :
2801 : GEN
2802 2471 : FlxV_Flv_multieval(GEN P, GEN xa, ulong p)
2803 : {
2804 2471 : pari_sp av = avma;
2805 2471 : GEN s = producttree_scheme(lg(xa)-1);
2806 2471 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2807 2471 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2808 2471 : return gerepileupto(av, FlxV_Flv_multieval_tree(P, xa, T, p, pi));
2809 : }
2810 :
2811 : GEN
2812 177136 : Flv_polint(GEN xa, GEN ya, ulong p, long vs)
2813 : {
2814 177136 : pari_sp av = avma;
2815 177136 : GEN s = producttree_scheme(lg(xa)-1);
2816 177140 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2817 177140 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2818 177142 : long m = lg(T)-1;
2819 177142 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2820 177140 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2821 177142 : return gerepileuptoleaf(av, FlvV_polint_tree(T, R, s, xa, ya, p, pi, vs));
2822 : }
2823 :
2824 : GEN
2825 99864 : Flv_Flm_polint(GEN xa, GEN ya, ulong p, long vs)
2826 : {
2827 99864 : pari_sp av = avma;
2828 99864 : GEN s = producttree_scheme(lg(xa)-1);
2829 99864 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2830 99864 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2831 99850 : long i, m = lg(T)-1, l = lg(ya)-1;
2832 99850 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2833 99850 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2834 99860 : GEN M = cgetg(l+1, t_VEC);
2835 1106972 : for (i=1; i<=l; i++)
2836 1007118 : gel(M,i) = FlvV_polint_tree(T, R, s, xa, gel(ya,i), p, pi, vs);
2837 99854 : return gerepileupto(av, M);
2838 : }
2839 :
2840 : GEN
2841 152987 : Flv_invVandermonde(GEN L, ulong den, ulong p)
2842 : {
2843 152987 : pari_sp av = avma;
2844 152987 : long i, n = lg(L);
2845 : GEN M, R;
2846 152987 : GEN s = producttree_scheme(n-1);
2847 152987 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2848 152987 : GEN tree = Flv_producttree(L, s, p, pi, 0);
2849 152987 : long m = lg(tree)-1;
2850 152987 : GEN T = gmael(tree, m, 1);
2851 152987 : R = Flv_inv(Flx_Flv_multieval_tree(Flx_deriv(T, p), L, tree, p, pi), p);
2852 152987 : if (den!=1) R = Flv_Fl_mul(R, den, p);
2853 152987 : M = cgetg(n, t_MAT);
2854 600481 : for (i = 1; i < n; i++)
2855 : {
2856 447494 : GEN P = Flx_Fl_mul(Flx_div_by_X_x(T, uel(L,i), p, NULL), uel(R,i), p);
2857 447494 : gel(M,i) = Flx_to_Flv(P, n-1);
2858 : }
2859 152987 : return gerepilecopy(av, M);
2860 : }
2861 :
2862 : /***********************************************************************/
2863 : /** Flxq **/
2864 : /***********************************************************************/
2865 : /* Flxq objects are Flx modulo another Flx called q. */
2866 :
2867 : /* Product of y and x in Z/pZ[X]/(T), as t_VECSMALL. */
2868 : GEN
2869 186579162 : Flxq_mul_pre(GEN x,GEN y,GEN T,ulong p,ulong pi)
2870 186579162 : { return Flx_rem_pre(Flx_mul_pre(x,y,p,pi),T,p,pi); }
2871 : GEN
2872 13149316 : Flxq_mul(GEN x,GEN y,GEN T,ulong p)
2873 13149316 : { return Flxq_mul_pre(x,y,T,p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2874 :
2875 : GEN
2876 275366581 : Flxq_sqr_pre(GEN x,GEN T,ulong p,ulong pi)
2877 275366581 : { return Flx_rem_pre(Flx_sqr_pre(x, p,pi), T, p,pi); }
2878 : /* Square of y in Z/pZ[X]/(T), as t_VECSMALL. */
2879 : GEN
2880 2725270 : Flxq_sqr(GEN x,GEN T,ulong p)
2881 2725270 : { return Flxq_sqr_pre(x,T,p,SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2882 :
2883 : static GEN
2884 1767494 : _Flxq_red(void *E, GEN x)
2885 1767494 : { struct _Flxq *s = (struct _Flxq *)E;
2886 1767494 : return Flx_rem_pre(x, s->T, s->p, s->pi); }
2887 : #if 0
2888 : static GEN
2889 : _Flx_sub(void *E, GEN x, GEN y)
2890 : { struct _Flxq *s = (struct _Flxq *)E;
2891 : return Flx_sub(x,y,s->p); }
2892 : #endif
2893 : static GEN
2894 268273446 : _Flxq_sqr(void *data, GEN x)
2895 : {
2896 268273446 : struct _Flxq *D = (struct _Flxq*)data;
2897 268273446 : return Flxq_sqr_pre(x, D->T, D->p, D->pi);
2898 : }
2899 : static GEN
2900 145262702 : _Flxq_mul(void *data, GEN x, GEN y)
2901 : {
2902 145262702 : struct _Flxq *D = (struct _Flxq*)data;
2903 145262702 : return Flxq_mul_pre(x,y, D->T, D->p, D->pi);
2904 : }
2905 : static GEN
2906 22497338 : _Flxq_one(void *data)
2907 : {
2908 22497338 : struct _Flxq *D = (struct _Flxq*)data;
2909 22497338 : return pol1_Flx(get_Flx_var(D->T));
2910 : }
2911 :
2912 : static GEN
2913 22157215 : _Flxq_powu_i(struct _Flxq *D, GEN x, ulong n)
2914 22157215 : { return gen_powu_i(x, n, (void*)D, &_Flxq_sqr, &_Flxq_mul); }
2915 : static GEN
2916 68 : _Flxq_powu(struct _Flxq *D, GEN x, ulong n)
2917 68 : { pari_sp av = avma; return gerepileuptoleaf(av, _Flxq_powu_i(D, x, n)); }
2918 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2919 : GEN
2920 22887388 : Flxq_powu_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
2921 : {
2922 : pari_sp av;
2923 : struct _Flxq D;
2924 22887388 : switch(n)
2925 : {
2926 0 : case 0: return pol1_Flx(get_Flx_var(T));
2927 141620 : case 1: return Flx_copy(x);
2928 593475 : case 2: return Flxq_sqr_pre(x, T, p, pi);
2929 : }
2930 22152293 : av = avma; set_Flxq_pre(&D, T, p, pi);
2931 22157943 : return gerepileuptoleaf(av, _Flxq_powu_i(&D, x, n));
2932 : }
2933 : GEN
2934 486591 : Flxq_powu(GEN x, ulong n, GEN T, ulong p)
2935 486591 : { return Flxq_powu_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2936 :
2937 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2938 : GEN
2939 24697026 : Flxq_pow_pre(GEN x, GEN n, GEN T, ulong p, ulong pi)
2940 : {
2941 24697026 : pari_sp av = avma;
2942 : struct _Flxq D;
2943 : GEN y;
2944 24697026 : long s = signe(n);
2945 24697026 : if (!s) return pol1_Flx(get_Flx_var(T));
2946 24461311 : if (s < 0) x = Flxq_inv_pre(x,T,p,pi);
2947 24461302 : if (is_pm1(n)) return s < 0 ? x : Flx_copy(x);
2948 23285400 : set_Flxq_pre(&D, T, p, pi);
2949 23285439 : y = gen_pow_i(x, n, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2950 23285297 : return gerepileuptoleaf(av, y);
2951 : }
2952 : GEN
2953 871176 : Flxq_pow(GEN x, GEN n, GEN T, ulong p)
2954 871176 : { return Flxq_pow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2955 :
2956 : GEN
2957 28 : Flxq_pow_init_pre(GEN x, GEN n, long k, GEN T, ulong p, ulong pi)
2958 : {
2959 28 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2960 28 : return gen_pow_init(x, n, k, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2961 : }
2962 : GEN
2963 0 : Flxq_pow_init(GEN x, GEN n, long k, GEN T, ulong p)
2964 0 : { return Flxq_pow_init_pre(x, n, k, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2965 :
2966 : GEN
2967 4400 : Flxq_pow_table_pre(GEN R, GEN n, GEN T, ulong p, ulong pi)
2968 : {
2969 4400 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2970 4400 : return gen_pow_table(R, n, (void*)&D, &_Flxq_one, &_Flxq_mul);
2971 : }
2972 : GEN
2973 0 : Flxq_pow_table(GEN R, GEN n, GEN T, ulong p)
2974 0 : { return Flxq_pow_table_pre(R, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2975 :
2976 : /* Inverse of x in Z/lZ[X]/(T) or NULL if inverse doesn't exist
2977 : * not stack clean. */
2978 : GEN
2979 5777580 : Flxq_invsafe_pre(GEN x, GEN T, ulong p, ulong pi)
2980 : {
2981 5777580 : GEN V, z = Flx_extgcd_pre(get_Flx_mod(T), x, p, pi, NULL, &V);
2982 : ulong iz;
2983 5777634 : if (degpol(z)) return NULL;
2984 5776954 : iz = Fl_inv(uel(z,2), p);
2985 5776950 : return Flx_Fl_mul_pre(V, iz, p, pi);
2986 : }
2987 : GEN
2988 540473 : Flxq_invsafe(GEN x, GEN T, ulong p)
2989 540473 : { return Flxq_invsafe_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2990 :
2991 : GEN
2992 4793365 : Flxq_inv_pre(GEN x, GEN T, ulong p, ulong pi)
2993 : {
2994 4793365 : pari_sp av=avma;
2995 4793365 : GEN U = Flxq_invsafe_pre(x, T, p, pi);
2996 4793326 : if (!U) pari_err_INV("Flxq_inv",Flx_to_ZX(x));
2997 4793298 : return gerepileuptoleaf(av, U);
2998 : }
2999 : GEN
3000 345215 : Flxq_inv(GEN x, GEN T, ulong p)
3001 345215 : { return Flxq_inv_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3002 :
3003 : GEN
3004 2399862 : Flxq_div_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
3005 : {
3006 2399862 : pari_sp av = avma;
3007 2399862 : return gerepileuptoleaf(av, Flxq_mul_pre(x,Flxq_inv_pre(y,T,p,pi),T,p,pi));
3008 : }
3009 : GEN
3010 225649 : Flxq_div(GEN x, GEN y, GEN T, ulong p)
3011 225649 : { return Flxq_div_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3012 :
3013 : GEN
3014 22499843 : Flxq_powers_pre(GEN x, long l, GEN T, ulong p, ulong pi)
3015 : {
3016 22499843 : int use_sqr = 2*degpol(x) >= get_Flx_degree(T);
3017 22490393 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3018 22489892 : return gen_powers(x, l, use_sqr, (void*)&D, &_Flxq_sqr, &_Flxq_mul, &_Flxq_one);
3019 : }
3020 : GEN
3021 232165 : Flxq_powers(GEN x, long l, GEN T, ulong p)
3022 232165 : { return Flxq_powers_pre(x, l, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3023 :
3024 : GEN
3025 167577 : Flxq_matrix_pow_pre(GEN y, long n, long m, GEN P, ulong l, ulong li)
3026 167577 : { return FlxV_to_Flm(Flxq_powers_pre(y,m-1,P,l,li),n); }
3027 : GEN
3028 399 : Flxq_matrix_pow(GEN y, long n, long m, GEN P, ulong l)
3029 399 : { return Flxq_matrix_pow_pre(y, n, m, P, l, SMALL_ULONG(l)? 0: get_Fl_red(l)); }
3030 :
3031 : GEN
3032 12625746 : Flx_Frobenius_pre(GEN T, ulong p, ulong pi)
3033 12625746 : { return Flxq_powu_pre(polx_Flx(get_Flx_var(T)), p, T, p, pi); }
3034 : GEN
3035 86550 : Flx_Frobenius(GEN T, ulong p)
3036 86550 : { return Flx_Frobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3037 :
3038 : GEN
3039 85034 : Flx_matFrobenius_pre(GEN T, ulong p, ulong pi)
3040 : {
3041 85034 : long n = get_Flx_degree(T);
3042 85034 : return Flxq_matrix_pow_pre(Flx_Frobenius_pre(T, p, pi), n, n, T, p, pi);
3043 : }
3044 : GEN
3045 0 : Flx_matFrobenius(GEN T, ulong p)
3046 0 : { return Flx_matFrobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3047 :
3048 : static GEN
3049 12956621 : Flx_blocks_Flm(GEN P, long n, long m)
3050 : {
3051 12956621 : GEN z = cgetg(m+1,t_MAT);
3052 12954705 : long i,j, k=2, l = lg(P);
3053 38379054 : for(i=1; i<=m; i++)
3054 : {
3055 25428048 : GEN zi = cgetg(n+1,t_VECSMALL);
3056 25424349 : gel(z,i) = zi;
3057 113011280 : for(j=1; j<=n; j++)
3058 87586931 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3059 : }
3060 12951006 : return z;
3061 : }
3062 :
3063 : GEN
3064 516592 : Flx_blocks(GEN P, long n, long m)
3065 : {
3066 516592 : GEN z = cgetg(m+1,t_VEC);
3067 516397 : long i,j, k=2, l = lg(P);
3068 1547503 : for(i=1; i<=m; i++)
3069 : {
3070 1031192 : GEN zi = cgetg(n+2,t_VECSMALL);
3071 1030555 : zi[1] = P[1];
3072 1030555 : gel(z,i) = zi;
3073 6464781 : for(j=2; j<n+2; j++)
3074 5434226 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3075 1030555 : zi = Flx_renormalize(zi, n+2);
3076 : }
3077 516311 : return z;
3078 : }
3079 :
3080 : static GEN
3081 12956295 : FlxV_to_Flm_lg(GEN x, long m, long n)
3082 : {
3083 : long i;
3084 12956295 : GEN y = cgetg(n+1, t_MAT);
3085 58956172 : for (i=1; i<=n; i++) gel(y,i) = Flx_to_Flv(gel(x,i), m);
3086 12953713 : return y;
3087 : }
3088 :
3089 : /* allow pi = 0 (SMALL_ULONG) */
3090 : GEN
3091 13153783 : Flx_FlxqV_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3092 : {
3093 13153783 : pari_sp btop, av = avma;
3094 13153783 : long sv = get_Flx_var(T), m = get_Flx_degree(T);
3095 13153486 : long i, l = lg(x)-1, lQ = lgpol(Q), n, d;
3096 : GEN A, B, C, S, g;
3097 13154540 : if (lQ == 0) return pol0_Flx(sv);
3098 12957626 : if (lQ <= l)
3099 : {
3100 5779269 : n = l;
3101 5779269 : d = 1;
3102 : }
3103 : else
3104 : {
3105 7178357 : n = l-1;
3106 7178357 : d = (lQ+n-1)/n;
3107 : }
3108 12957626 : A = FlxV_to_Flm_lg(x, m, n);
3109 12955901 : B = Flx_blocks_Flm(Q, n, d);
3110 12953732 : C = gerepileupto(av, Flm_mul(A, B, p));
3111 12958474 : g = gel(x, l);
3112 12958474 : if (pi && SMALL_ULONG(p)) pi = 0;
3113 12958474 : T = Flx_get_red_pre(T, p, pi);
3114 12957387 : btop = avma;
3115 12957387 : S = Flv_to_Flx(gel(C, d), sv);
3116 25436753 : for (i = d-1; i>0; i--)
3117 : {
3118 12480915 : S = Flx_add(Flxq_mul_pre(S, g, T, p, pi), Flv_to_Flx(gel(C,i), sv), p);
3119 12480481 : if (gc_needed(btop,1))
3120 0 : S = gerepileuptoleaf(btop, S);
3121 : }
3122 12955838 : return gerepileuptoleaf(av, S);
3123 : }
3124 : GEN
3125 5089 : Flx_FlxqV_eval(GEN Q, GEN x, GEN T, ulong p)
3126 5089 : { return Flx_FlxqV_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3127 :
3128 : /* allow pi = 0 (SMALL_ULONG) */
3129 : GEN
3130 2622311 : Flx_Flxq_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3131 : {
3132 2622311 : pari_sp av = avma;
3133 : GEN z, V;
3134 2622311 : long d = degpol(Q), rtd;
3135 2622302 : if (d < 0) return pol0_Flx(get_Flx_var(T));
3136 2622211 : rtd = (long) sqrt((double)d);
3137 2622211 : T = Flx_get_red_pre(T, p, pi);
3138 2622232 : V = Flxq_powers_pre(x, rtd, T, p, pi);
3139 2622275 : z = Flx_FlxqV_eval_pre(Q, V, T, p, pi);
3140 2622268 : return gerepileupto(av, z);
3141 : }
3142 : GEN
3143 786193 : Flx_Flxq_eval(GEN Q, GEN x, GEN T, ulong p)
3144 786193 : { return Flx_Flxq_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3145 :
3146 : /* allow pi = 0 (SMALL_ULONG) */
3147 : GEN
3148 0 : FlxC_FlxqV_eval_pre(GEN x, GEN v, GEN T, ulong p, ulong pi)
3149 0 : { pari_APPLY_type(t_COL, Flx_FlxqV_eval_pre(gel(x,i), v, T, p, pi)) }
3150 : GEN
3151 0 : FlxC_FlxqV_eval(GEN x, GEN v, GEN T, ulong p)
3152 0 : { return FlxC_FlxqV_eval_pre(x, v, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3153 :
3154 : /* allow pi = 0 (SMALL_ULONG) */
3155 : GEN
3156 0 : FlxC_Flxq_eval_pre(GEN x, GEN F, GEN T, ulong p, ulong pi)
3157 : {
3158 0 : long d = brent_kung_optpow(get_Flx_degree(T)-1,lg(x)-1,1);
3159 0 : GEN Fp = Flxq_powers_pre(F, d, T, p, pi);
3160 0 : return FlxC_FlxqV_eval_pre(x, Fp, T, p, pi);
3161 : }
3162 : GEN
3163 0 : FlxC_Flxq_eval(GEN x, GEN F, GEN T, ulong p)
3164 0 : { return FlxC_Flxq_eval_pre(x, F, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3165 :
3166 : #if 0
3167 : static struct bb_algebra Flxq_algebra = { _Flxq_red, _Flx_add, _Flx_sub,
3168 : _Flxq_mul, _Flxq_sqr, _Flxq_one, _Flxq_zero};
3169 : #endif
3170 :
3171 : static GEN
3172 386267 : Flxq_autpow_sqr(void *E, GEN x)
3173 : {
3174 386267 : struct _Flxq *D = (struct _Flxq*)E;
3175 386267 : return Flx_Flxq_eval_pre(x, x, D->T, D->p, D->pi);
3176 : }
3177 : static GEN
3178 20714 : Flxq_autpow_msqr(void *E, GEN x)
3179 : {
3180 20714 : struct _Flxq *D = (struct _Flxq*)E;
3181 20714 : return Flx_FlxqV_eval_pre(Flxq_autpow_sqr(E, x), D->aut, D->T, D->p, D->pi);
3182 : }
3183 :
3184 : GEN
3185 304446 : Flxq_autpow_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3186 : {
3187 304446 : pari_sp av = avma;
3188 : struct _Flxq D;
3189 : long d;
3190 304446 : if (n==0) return Flx_rem_pre(polx_Flx(x[1]), T, p, pi);
3191 304439 : if (n==1) return Flx_rem_pre(x, T, p, pi);
3192 303956 : set_Flxq_pre(&D, T, p, pi);
3193 303956 : d = brent_kung_optpow(get_Flx_degree(T), hammingl(n)-1, 1);
3194 303956 : D.aut = Flxq_powers_pre(x, d, T, p, D.pi);
3195 303956 : x = gen_powu_fold_i(x,n,(void*)&D,Flxq_autpow_sqr,Flxq_autpow_msqr);
3196 303956 : return gerepilecopy(av, x);
3197 : }
3198 : GEN
3199 7 : Flxq_autpow(GEN x, ulong n, GEN T, ulong p)
3200 7 : { return Flxq_autpow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3201 :
3202 : GEN
3203 1733 : Flxq_autpowers(GEN x, ulong l, GEN T, ulong p)
3204 : {
3205 1733 : long d, vT = get_Flx_var(T), dT = get_Flx_degree(T);
3206 : ulong i, pi;
3207 1733 : pari_sp av = avma;
3208 1733 : GEN xp, V = cgetg(l+2,t_VEC);
3209 1733 : gel(V,1) = polx_Flx(vT); if (l==0) return V;
3210 1733 : gel(V,2) = gcopy(x); if (l==1) return V;
3211 1733 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3212 1733 : T = Flx_get_red_pre(T, p, pi);
3213 1733 : d = brent_kung_optpow(dT-1, l-1, 1);
3214 1733 : xp = Flxq_powers_pre(x, d, T, p, pi);
3215 7376 : for(i = 3; i < l+2; i++)
3216 5643 : gel(V,i) = Flx_FlxqV_eval_pre(gel(V,i-1), xp, T, p, pi);
3217 1733 : return gerepilecopy(av, V);
3218 : }
3219 :
3220 : static GEN
3221 588680 : Flxq_autsum_mul(void *E, GEN x, GEN y)
3222 : {
3223 588680 : struct _Flxq *D = (struct _Flxq*)E;
3224 588680 : GEN T = D->T;
3225 588680 : ulong p = D->p, pi = D->pi;
3226 588680 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3227 588680 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3228 588680 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3229 588680 : GEN V2 = Flxq_powers_pre(phi2, d, T, p, pi);
3230 588680 : GEN phi3 = Flx_FlxqV_eval_pre(phi1, V2, T, p, pi);
3231 588680 : GEN aphi = Flx_FlxqV_eval_pre(a1, V2, T, p, pi);
3232 588680 : GEN a3 = Flxq_mul_pre(aphi, a2, T, p, pi);
3233 588680 : return mkvec2(phi3, a3);
3234 : }
3235 : static GEN
3236 379824 : Flxq_autsum_sqr(void *E, GEN x)
3237 379824 : { return Flxq_autsum_mul(E, x, x); }
3238 :
3239 : static GEN
3240 315643 : Flxq_autsum_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3241 : {
3242 315643 : pari_sp av = avma;
3243 315643 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3244 315643 : x = gen_powu_i(x,n,(void*)&D,Flxq_autsum_sqr,Flxq_autsum_mul);
3245 315643 : return gerepilecopy(av, x);
3246 : }
3247 : GEN
3248 0 : Flxq_autsum(GEN x, ulong n, GEN T, ulong p)
3249 0 : { return Flxq_autsum_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3250 :
3251 : static GEN
3252 759940 : Flxq_auttrace_mul(void *E, GEN x, GEN y)
3253 : {
3254 759940 : struct _Flxq *D = (struct _Flxq*)E;
3255 759940 : GEN T = D->T;
3256 759940 : ulong p = D->p, pi = D->pi;
3257 759940 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3258 759940 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3259 759940 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3260 759963 : GEN V1 = Flxq_powers_pre(phi1, d, T, p, pi);
3261 759903 : GEN phi3 = Flx_FlxqV_eval_pre(phi2, V1, T, p, pi);
3262 759945 : GEN aphi = Flx_FlxqV_eval_pre(a2, V1, T, p, pi);
3263 759938 : GEN a3 = Flx_add(a1, aphi, p);
3264 759919 : return mkvec2(phi3, a3);
3265 : }
3266 :
3267 : static GEN
3268 633097 : Flxq_auttrace_sqr(void *E, GEN x)
3269 633097 : { return Flxq_auttrace_mul(E, x, x); }
3270 :
3271 : GEN
3272 929094 : Flxq_auttrace_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3273 : {
3274 929094 : pari_sp av = avma;
3275 : struct _Flxq D;
3276 929094 : set_Flxq_pre(&D, T, p, pi);
3277 929097 : x = gen_powu_i(x,n,(void*)&D,Flxq_auttrace_sqr,Flxq_auttrace_mul);
3278 929076 : return gerepilecopy(av, x);
3279 : }
3280 : GEN
3281 0 : Flxq_auttrace(GEN x, ulong n, GEN T, ulong p)
3282 0 : { return Flxq_auttrace_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3283 :
3284 : static long
3285 849844 : bounded_order(ulong p, GEN b, long k)
3286 : {
3287 849844 : GEN a = modii(utoipos(p), b);
3288 : long i;
3289 2078556 : for(i = 1; i < k; i++)
3290 : {
3291 1612911 : if (equali1(a)) return i;
3292 1228714 : a = modii(muliu(a,p),b);
3293 : }
3294 465645 : return 0;
3295 : }
3296 :
3297 : /*
3298 : n = (p^d-a)\b
3299 : b = bb*p^vb
3300 : p^k = 1 [bb]
3301 : d = m*k+r+vb
3302 : u = (p^k-1)/bb;
3303 : v = (p^(r+vb)-a)/b;
3304 : w = (p^(m*k)-1)/(p^k-1)
3305 : n = p^r*w*u+v
3306 : w*u = p^vb*(p^(m*k)-1)/b
3307 : n = p^(r+vb)*(p^(m*k)-1)/b+(p^(r+vb)-a)/b
3308 : */
3309 :
3310 : static GEN
3311 23991296 : Flxq_pow_Frobenius(GEN x, GEN n, GEN aut, GEN T, ulong p, ulong pi)
3312 : {
3313 23991296 : pari_sp av=avma;
3314 23991296 : long d = get_Flx_degree(T);
3315 23991306 : GEN an = absi_shallow(n), z, q;
3316 23991296 : if (abscmpiu(an,p)<0 || cmpis(an,d)<=0) return Flxq_pow_pre(x, n, T, p, pi);
3317 850248 : q = powuu(p, d);
3318 850243 : if (dvdii(q, n))
3319 : {
3320 371 : long vn = logint(an, utoipos(p));
3321 371 : GEN autvn = vn==1 ? aut: Flxq_autpow_pre(aut,vn,T,p,pi);
3322 371 : z = Flx_Flxq_eval_pre(x,autvn,T,p,pi);
3323 : } else
3324 : {
3325 849874 : GEN b = diviiround(q, an), a = subii(q, mulii(an,b));
3326 : GEN bb, u, v, autk;
3327 849876 : long vb = Z_lvalrem(b,p,&bb);
3328 849877 : long m, r, k = is_pm1(bb)? 1: bounded_order(p,bb,d);
3329 849875 : if (!k || d-vb < k) return Flxq_pow_pre(x,n, T,p,pi);
3330 384223 : m = (d-vb)/k; r = (d-vb)%k;
3331 384223 : u = diviiexact(subiu(powuu(p,k),1),bb);
3332 384223 : v = diviiexact(subii(powuu(p,r+vb),a),b);
3333 384223 : autk = k==1 ? aut: Flxq_autpow_pre(aut,k,T,p,pi);
3334 384223 : if (r)
3335 : {
3336 94534 : GEN autr = r==1 ? aut: Flxq_autpow_pre(aut,r,T,p,pi);
3337 94534 : z = Flx_Flxq_eval_pre(x,autr,T,p,pi);
3338 289689 : } else z = x;
3339 384223 : if (m > 1) z = gel(Flxq_autsum_pre(mkvec2(autk, z), m, T, p, pi), 2);
3340 384223 : if (!is_pm1(u)) z = Flxq_pow_pre(z, u, T, p, pi);
3341 384223 : if (signe(v)) z = Flxq_mul_pre(z, Flxq_pow_pre(x, v, T, p, pi), T, p, pi);
3342 : }
3343 384594 : return gerepileupto(av,signe(n)>0 ? z : Flxq_inv_pre(z,T,p,pi));
3344 : }
3345 :
3346 : static GEN
3347 23983810 : _Flxq_pow(void *data, GEN x, GEN n)
3348 : {
3349 23983810 : struct _Flxq *D = (struct _Flxq*)data;
3350 23983810 : return Flxq_pow_Frobenius(x, n, D->aut, D->T, D->p, D->pi);
3351 : }
3352 :
3353 : static GEN
3354 329127 : _Flxq_rand(void *data)
3355 : {
3356 329127 : pari_sp av=avma;
3357 329127 : struct _Flxq *D = (struct _Flxq*)data;
3358 : GEN z;
3359 : do
3360 : {
3361 330445 : set_avma(av);
3362 330445 : z = random_Flx(get_Flx_degree(D->T),get_Flx_var(D->T),D->p);
3363 330446 : } while (lgpol(z)==0);
3364 329127 : return z;
3365 : }
3366 :
3367 : /* discrete log in FpXQ for a in Fp^*, g in FpXQ^* of order ord */
3368 : static GEN
3369 35754 : Fl_Flxq_log(ulong a, GEN g, GEN o, GEN T, ulong p)
3370 : {
3371 35754 : pari_sp av = avma;
3372 : GEN q,n_q,ord,ordp, op;
3373 :
3374 35754 : if (a == 1UL) return gen_0;
3375 : /* p > 2 */
3376 :
3377 35754 : ordp = utoi(p - 1);
3378 35754 : ord = get_arith_Z(o);
3379 35754 : if (!ord) ord = T? subiu(powuu(p, get_FpX_degree(T)), 1): ordp;
3380 35754 : if (a == p - 1) /* -1 */
3381 7718 : return gerepileuptoint(av, shifti(ord,-1));
3382 28036 : ordp = gcdii(ordp, ord);
3383 28036 : op = typ(o)==t_MAT ? famat_Z_gcd(o, ordp) : ordp;
3384 :
3385 28036 : q = NULL;
3386 28036 : if (T)
3387 : { /* we want < g > = Fp^* */
3388 28036 : if (!equalii(ord,ordp)) {
3389 11899 : q = diviiexact(ord,ordp);
3390 11899 : g = Flxq_pow(g,q,T,p);
3391 : }
3392 : }
3393 28036 : n_q = Fp_log(utoi(a), utoipos(uel(g,2)), op, utoipos(p));
3394 28036 : if (lg(n_q)==1) return gerepileuptoleaf(av, n_q);
3395 28036 : if (q) n_q = mulii(q, n_q);
3396 28036 : return gerepileuptoint(av, n_q);
3397 : }
3398 :
3399 : static GEN
3400 691242 : Flxq_easylog(void* E, GEN a, GEN g, GEN ord)
3401 : {
3402 691242 : struct _Flxq *f = (struct _Flxq *)E;
3403 691242 : GEN T = f->T;
3404 691242 : ulong p = f->p;
3405 691242 : long d = get_Flx_degree(T);
3406 691242 : if (Flx_equal1(a)) return gen_0;
3407 531136 : if (Flx_equal(a,g)) return gen_1;
3408 174113 : if (!degpol(a))
3409 35754 : return Fl_Flxq_log(uel(a,2), g, ord, T, p);
3410 138359 : if (typ(ord)!=t_INT || d <= 4 || d == 6 || abscmpiu(ord,1UL<<27)<0)
3411 138331 : return NULL;
3412 28 : return Flxq_log_index(a, g, ord, T, p);
3413 : }
3414 :
3415 : static const struct bb_group Flxq_star={_Flxq_mul,_Flxq_pow,_Flxq_rand,hash_GEN,Flx_equal,Flx_equal1,Flxq_easylog};
3416 :
3417 : const struct bb_group *
3418 427700 : get_Flxq_star(void **E, GEN T, ulong p)
3419 : {
3420 427700 : struct _Flxq *e = (struct _Flxq *) stack_malloc(sizeof(struct _Flxq));
3421 427700 : e->T = T; e->p = p; e->pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3422 427700 : e->aut = Flx_Frobenius_pre(T, p, e->pi);
3423 427700 : *E = (void*)e; return &Flxq_star;
3424 : }
3425 :
3426 : GEN
3427 95205 : Flxq_order(GEN a, GEN ord, GEN T, ulong p)
3428 : {
3429 : void *E;
3430 95205 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3431 95205 : return gen_order(a,ord,E,S);
3432 : }
3433 :
3434 : GEN
3435 164532 : Flxq_log(GEN a, GEN g, GEN ord, GEN T, ulong p)
3436 : {
3437 : void *E;
3438 164532 : pari_sp av = avma;
3439 164532 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3440 164532 : GEN v = get_arith_ZZM(ord), F = gmael(v,2,1);
3441 164532 : if (Flxq_log_use_index(veclast(F), T, p))
3442 24311 : v = mkvec2(gel(v, 1), ZM_famat_limit(gel(v, 2), int2n(27)));
3443 164532 : return gerepileuptoleaf(av, gen_PH_log(a, g, v, E, S));
3444 : }
3445 :
3446 : GEN
3447 170917 : Flxq_sqrtn(GEN a, GEN n, GEN T, ulong p, GEN *zeta)
3448 : {
3449 170917 : if (!lgpol(a))
3450 : {
3451 2954 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
3452 2947 : if (zeta)
3453 0 : *zeta=pol1_Flx(get_Flx_var(T));
3454 2947 : return pol0_Flx(get_Flx_var(T));
3455 : }
3456 : else
3457 : {
3458 : void *E;
3459 167963 : pari_sp av = avma;
3460 167963 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3461 167963 : GEN o = subiu(powuu(p,get_Flx_degree(T)), 1);
3462 167963 : GEN s = gen_Shanks_sqrtn(a,n,o,zeta,E,S);
3463 167962 : if (!s) return gc_NULL(av);
3464 167864 : return gc_all(av, zeta?2:1, &s, zeta);
3465 : }
3466 : }
3467 :
3468 : GEN
3469 218670 : Flxq_sqrt_pre(GEN z, GEN T, ulong p, ulong pi)
3470 : {
3471 218670 : pari_sp av = avma;
3472 218670 : if (p==2)
3473 : {
3474 0 : GEN r = F2xq_sqrt(Flx_to_F2x(z), Flx_to_F2x(get_Flx_mod(T)));
3475 0 : return gerepileupto(av, F2x_to_Flx(r));
3476 : }
3477 218670 : if (get_Flx_degree(T)==2)
3478 : {
3479 67856 : GEN P = get_Flx_mod(T), s;
3480 67856 : ulong c = uel(P,2), b = uel(P,3), a = uel(P,4);
3481 67856 : ulong y = degpol(z)<1 ? 0: uel(z,3);
3482 67856 : if (a==1 && b==0)
3483 14890 : {
3484 15670 : ulong x = degpol(z)<1 ? Flx_constant(z): uel(z,2);
3485 15670 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), Fl_neg(c, p), p, pi);
3486 15670 : if (!r) return gc_NULL(av);
3487 14890 : s = mkvecsmall3(P[1], uel(r,1), uel(r,2));
3488 : }
3489 : else
3490 : {
3491 52186 : ulong b2 = Fl_halve(b, p), t = Fl_div(b2, a, p);
3492 52186 : ulong D = Fl_sub(Fl_sqr(b2, p), Fl_mul(a, c, p), p);
3493 52186 : ulong x = degpol(z)<1 ? Flx_constant(z): Fl_sub(uel(z,2), Fl_mul(uel(z,3), t, p), p);
3494 52186 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), D, p, pi);
3495 52186 : if (!r) return gc_NULL(av);
3496 49792 : s = mkvecsmall3(P[1], Fl_add(uel(r,1), Fl_mul(uel(r,2),t,p), p), uel(r,2));
3497 : }
3498 64682 : return gerepileuptoleaf(av, Flx_renormalize(s, 4));
3499 : }
3500 : else
3501 150814 : return Flxq_sqrtn(z, gen_2, T, p, NULL);
3502 : }
3503 :
3504 : GEN
3505 218670 : Flxq_sqrt(GEN a, GEN T, ulong p)
3506 218670 : { return Flxq_sqrt_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3507 :
3508 : /* assume T irreducible mod p */
3509 : int
3510 380306 : Flxq_issquare(GEN x, GEN T, ulong p)
3511 : {
3512 380306 : if (lgpol(x) == 0 || p == 2) return 1;
3513 377107 : return krouu(Flxq_norm(x,T,p), p) == 1;
3514 : }
3515 :
3516 : /* assume T irreducible mod p */
3517 : int
3518 0 : Flxq_is2npower(GEN x, long n, GEN T, ulong p)
3519 : {
3520 : pari_sp av;
3521 : GEN m;
3522 0 : if (n==1) return Flxq_issquare(x, T, p);
3523 0 : if (lgpol(x) == 0 || p == 2) return 1;
3524 0 : av = avma;
3525 0 : m = shifti(subiu(powuu(p, get_Flx_degree(T)), 1), -n);
3526 0 : return gc_bool(av, Flx_equal1(Flxq_pow(x, m, T, p)));
3527 : }
3528 :
3529 : GEN
3530 113589 : Flxq_lroot_fast_pre(GEN a, GEN sqx, GEN T, long p, ulong pi)
3531 : {
3532 113589 : pari_sp av=avma;
3533 113589 : GEN A = Flx_splitting(a,p);
3534 113589 : return gerepileuptoleaf(av, FlxqV_dotproduct_pre(A,sqx,T,p,pi));
3535 : }
3536 : GEN
3537 0 : Flxq_lroot_fast(GEN a, GEN sqx, GEN T, long p)
3538 0 : { return Flxq_lroot_fast_pre(a, sqx, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3539 :
3540 : GEN
3541 25053 : Flxq_lroot_pre(GEN a, GEN T, long p, ulong pi)
3542 : {
3543 25053 : pari_sp av=avma;
3544 25053 : long n = get_Flx_degree(T), d = degpol(a);
3545 : GEN sqx, V;
3546 25053 : if (n==1) return leafcopy(a);
3547 25053 : if (n==2) return Flxq_powu_pre(a, p, T, p, pi);
3548 25053 : sqx = Flxq_autpow_pre(Flx_Frobenius_pre(T, p, pi), n-1, T, p, pi);
3549 25053 : if (d==1 && a[2]==0 && a[3]==1) return gerepileuptoleaf(av, sqx);
3550 0 : if (d>=p)
3551 : {
3552 0 : V = Flxq_powers_pre(sqx,p-1,T,p,pi);
3553 0 : return gerepileuptoleaf(av, Flxq_lroot_fast_pre(a,V,T,p,pi));
3554 : } else
3555 0 : return gerepileuptoleaf(av, Flx_Flxq_eval_pre(a,sqx,T,p,pi));
3556 : }
3557 : GEN
3558 0 : Flxq_lroot(GEN a, GEN T, long p)
3559 0 : { return Flxq_lroot_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3560 :
3561 : ulong
3562 422462 : Flxq_norm(GEN x, GEN TB, ulong p)
3563 : {
3564 422462 : GEN T = get_Flx_mod(TB);
3565 422462 : ulong y = Flx_resultant(T, x, p), L = Flx_lead(T);
3566 422462 : if (L==1 || lgpol(x)==0) return y;
3567 0 : return Fl_div(y, Fl_powu(L, (ulong)degpol(x), p), p);
3568 : }
3569 :
3570 : ulong
3571 3219 : Flxq_trace(GEN x, GEN TB, ulong p)
3572 : {
3573 3219 : pari_sp av = avma;
3574 : ulong t;
3575 3219 : GEN T = get_Flx_mod(TB);
3576 3219 : long n = degpol(T)-1;
3577 3219 : GEN z = Flxq_mul(x, Flx_deriv(T, p), TB, p);
3578 3219 : t = degpol(z)<n ? 0 : Fl_div(z[2+n],T[3+n],p);
3579 3219 : return gc_ulong(av, t);
3580 : }
3581 :
3582 : /*x must be reduced*/
3583 : GEN
3584 3632 : Flxq_charpoly(GEN x, GEN TB, ulong p)
3585 : {
3586 3632 : pari_sp ltop=avma;
3587 3632 : GEN T = get_Flx_mod(TB);
3588 3632 : long vs = evalvarn(fetch_var());
3589 3632 : GEN xm1 = deg1pol_shallow(pol1_Flx(x[1]),Flx_neg(x,p),vs);
3590 3632 : GEN r = Flx_FlxY_resultant(T, xm1, p);
3591 3632 : r[1] = x[1];
3592 3632 : (void)delete_var(); return gerepileupto(ltop, r);
3593 : }
3594 :
3595 : /* Computing minimal polynomial : */
3596 : /* cf Shoup 'Efficient Computation of Minimal Polynomials */
3597 : /* in Algebraic Extensions of Finite Fields' */
3598 :
3599 : /* Let v a linear form, return the linear form z->v(tau*z)
3600 : that is, v*(M_tau) */
3601 :
3602 : static GEN
3603 1682935 : Flxq_transmul_init(GEN tau, GEN T, ulong p, ulong pi)
3604 : {
3605 : GEN bht;
3606 1682935 : GEN h, Tp = get_Flx_red(T, &h);
3607 1682932 : long n = degpol(Tp), vT = Tp[1];
3608 1682923 : GEN ft = Flx_recipspec(Tp+2, n+1, n+1);
3609 1682915 : GEN bt = Flx_recipspec(tau+2, lgpol(tau), n);
3610 1682901 : ft[1] = vT; bt[1] = vT;
3611 1682901 : if (h)
3612 2688 : bht = Flxn_mul_pre(bt, h, n-1, p, pi);
3613 : else
3614 : {
3615 1680213 : GEN bh = Flx_div_pre(Flx_shift(tau, n-1), T, p, pi);
3616 1680198 : bht = Flx_recipspec(bh+2, lgpol(bh), n-1);
3617 1680217 : bht[1] = vT;
3618 : }
3619 1682905 : return mkvec3(bt, bht, ft);
3620 : }
3621 :
3622 : static GEN
3623 4064720 : Flxq_transmul(GEN tau, GEN a, long n, ulong p, ulong pi)
3624 : {
3625 4064720 : pari_sp ltop = avma;
3626 : GEN t1, t2, t3, vec;
3627 4064720 : GEN bt = gel(tau, 1), bht = gel(tau, 2), ft = gel(tau, 3);
3628 4064720 : if (lgpol(a)==0) return pol0_Flx(a[1]);
3629 4034555 : t2 = Flx_shift(Flx_mul_pre(bt, a, p, pi),1-n);
3630 4034307 : if (lgpol(bht)==0) return gerepileuptoleaf(ltop, t2);
3631 3046956 : t1 = Flx_shift(Flx_mul_pre(ft, a, p, pi),-n);
3632 3046980 : t3 = Flxn_mul_pre(t1, bht, n-1, p, pi);
3633 3046946 : vec = Flx_sub(t2, Flx_shift(t3, 1), p);
3634 3046970 : return gerepileuptoleaf(ltop, vec);
3635 : }
3636 :
3637 : GEN
3638 780377 : Flxq_minpoly_pre(GEN x, GEN T, ulong p, ulong pi)
3639 : {
3640 780377 : pari_sp ltop = avma;
3641 780377 : long vT = get_Flx_var(T), n = get_Flx_degree(T);
3642 : GEN v_x;
3643 780369 : GEN g = pol1_Flx(vT), tau = pol1_Flx(vT);
3644 780347 : T = Flx_get_red_pre(T, p, pi);
3645 780345 : v_x = Flxq_powers_pre(x, usqrt(2*n), T, p, pi);
3646 1621810 : while (lgpol(tau) != 0)
3647 : {
3648 : long i, j, m, k1;
3649 : GEN M, v, tr, g_prime, c;
3650 841458 : if (degpol(g) == n) { tau = pol1_Flx(vT); g = pol1_Flx(vT); }
3651 841459 : v = random_Flx(n, vT, p);
3652 841479 : tr = Flxq_transmul_init(tau, T, p, pi);
3653 841460 : v = Flxq_transmul(tr, v, n, p, pi);
3654 841473 : m = 2*(n-degpol(g));
3655 841471 : k1 = usqrt(m);
3656 841473 : tr = Flxq_transmul_init(gel(v_x,k1+1), T, p, pi);
3657 841463 : c = cgetg(m+2,t_VECSMALL);
3658 841404 : c[1] = vT;
3659 4064635 : for (i=0; i<m; i+=k1)
3660 : {
3661 3223166 : long mj = minss(m-i, k1);
3662 12599605 : for (j=0; j<mj; j++)
3663 9376054 : uel(c,m+1-(i+j)) = Flx_dotproduct_pre(v, gel(v_x,j+1), p, pi);
3664 3223551 : v = Flxq_transmul(tr, v, n, p, pi);
3665 : }
3666 841469 : c = Flx_renormalize(c, m+2);
3667 : /* now c contains <v,x^i> , i = 0..m-1 */
3668 841465 : M = Flx_halfgcd_pre(monomial_Flx(1, m, vT), c, p, pi);
3669 841484 : g_prime = gmael(M, 2, 2);
3670 841484 : if (degpol(g_prime) < 1) continue;
3671 829875 : g = Flx_mul_pre(g, g_prime, p, pi);
3672 829856 : tau = Flxq_mul_pre(tau, Flx_FlxqV_eval_pre(g_prime, v_x, T,p,pi), T,p,pi);
3673 : }
3674 780325 : g = Flx_normalize(g,p);
3675 780357 : return gerepileuptoleaf(ltop,g);
3676 : }
3677 : GEN
3678 44381 : Flxq_minpoly(GEN x, GEN T, ulong p)
3679 44381 : { return Flxq_minpoly_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3680 :
3681 : GEN
3682 20 : Flxq_conjvec(GEN x, GEN T, ulong p)
3683 : {
3684 20 : long i, l = 1+get_Flx_degree(T);
3685 20 : GEN z = cgetg(l,t_COL);
3686 20 : struct _Flxq D; set_Flxq(&D, T, p);
3687 20 : gel(z,1) = Flx_copy(x);
3688 88 : for (i=2; i<l; i++) gel(z,i) = _Flxq_powu(&D, gel(z,i-1), p);
3689 20 : return z;
3690 : }
3691 :
3692 : GEN
3693 7166 : gener_Flxq(GEN T, ulong p, GEN *po)
3694 : {
3695 7166 : long i, j, vT = get_Flx_var(T), f = get_Flx_degree(T);
3696 : ulong p_1, pi;
3697 : GEN g, L, L2, o, q, F;
3698 : pari_sp av0, av;
3699 :
3700 7166 : if (f == 1) {
3701 : GEN fa;
3702 28 : o = utoipos(p-1);
3703 28 : fa = Z_factor(o);
3704 28 : L = gel(fa,1);
3705 28 : L = vecslice(L, 2, lg(L)-1); /* remove 2 for efficiency */
3706 28 : g = Fl_to_Flx(pgener_Fl_local(p, vec_to_vecsmall(L)), vT);
3707 28 : if (po) *po = mkvec2(o, fa);
3708 28 : return g;
3709 : }
3710 :
3711 7138 : av0 = avma; p_1 = p - 1;
3712 7138 : q = diviuexact(subiu(powuu(p,f), 1), p_1);
3713 :
3714 7138 : L = cgetg(1, t_VECSMALL);
3715 7138 : if (p > 3)
3716 : {
3717 2343 : ulong t = p_1 >> vals(p_1);
3718 2343 : GEN P = gel(factoru(t), 1);
3719 2343 : L = cgetg_copy(P, &i);
3720 3745 : while (--i) L[i] = p_1 / P[i];
3721 : }
3722 7138 : o = factor_pn_1(utoipos(p),f);
3723 7138 : L2 = leafcopy( gel(o, 1) );
3724 19100 : for (i = j = 1; i < lg(L2); i++)
3725 : {
3726 11962 : if (umodui(p_1, gel(L2,i)) == 0) continue;
3727 6460 : gel(L2,j++) = diviiexact(q, gel(L2,i));
3728 : }
3729 7138 : setlg(L2, j); pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3730 7138 : F = Flx_Frobenius_pre(T, p, pi);
3731 17615 : for (av = avma;; set_avma(av))
3732 10477 : {
3733 : GEN tt;
3734 17615 : g = random_Flx(f, vT, p);
3735 17615 : if (degpol(g) < 1) continue;
3736 11948 : if (p == 2) tt = g;
3737 : else
3738 : {
3739 8749 : ulong t = Flxq_norm(g, T, p);
3740 8749 : if (t == 1 || !is_gener_Fl(t, p, p_1, L)) continue;
3741 4794 : tt = Flxq_powu_pre(g, p_1>>1, T, p, pi);
3742 : }
3743 14573 : for (i = 1; i < j; i++)
3744 : {
3745 7435 : GEN a = Flxq_pow_Frobenius(tt, gel(L2,i), F, T, p, pi);
3746 7435 : if (!degpol(a) && uel(a,2) == p_1) break;
3747 : }
3748 7993 : if (i == j) break;
3749 : }
3750 7138 : if (!po)
3751 : {
3752 187 : set_avma((pari_sp)g);
3753 187 : g = gerepileuptoleaf(av0, g);
3754 : }
3755 : else {
3756 6951 : *po = mkvec2(subiu(powuu(p,f), 1), o);
3757 6951 : gerepileall(av0, 2, &g, po);
3758 : }
3759 7138 : return g;
3760 : }
3761 :
3762 : static GEN
3763 553283 : _Flxq_neg(void *E, GEN x)
3764 553283 : { struct _Flxq *s = (struct _Flxq *)E;
3765 553283 : return Flx_neg(x,s->p); }
3766 :
3767 : static GEN
3768 1571963 : _Flxq_rmul(void *E, GEN x, GEN y)
3769 1571963 : { struct _Flxq *s = (struct _Flxq *)E;
3770 1571963 : return Flx_mul_pre(x,y,s->p,s->pi); }
3771 :
3772 : static GEN
3773 18987 : _Flxq_inv(void *E, GEN x)
3774 18987 : { struct _Flxq *s = (struct _Flxq *)E;
3775 18987 : return Flxq_inv(x,s->T,s->p); }
3776 :
3777 : static int
3778 165019 : _Flxq_equal0(GEN x) { return lgpol(x)==0; }
3779 :
3780 : static GEN
3781 25164 : _Flxq_s(void *E, long x)
3782 25164 : { struct _Flxq *s = (struct _Flxq *)E;
3783 25164 : ulong u = x<0 ? s->p+x: (ulong)x;
3784 25164 : return Fl_to_Flx(u, get_Flx_var(s->T));
3785 : }
3786 :
3787 : static const struct bb_field Flxq_field={_Flxq_red,_Flx_add,_Flxq_rmul,_Flxq_neg,
3788 : _Flxq_inv,_Flxq_equal0,_Flxq_s};
3789 :
3790 72998 : const struct bb_field *get_Flxq_field(void **E, GEN T, ulong p)
3791 : {
3792 72998 : GEN z = new_chunk(sizeof(struct _Flxq));
3793 72998 : set_Flxq((struct _Flxq *)z, T, p); *E = (void*)z; return &Flxq_field;
3794 : }
3795 :
3796 : /***********************************************************************/
3797 : /** Flxn **/
3798 : /***********************************************************************/
3799 :
3800 : GEN
3801 54316 : Flx_invLaplace(GEN x, ulong p)
3802 : {
3803 54316 : long i, d = degpol(x);
3804 : ulong t;
3805 : GEN y;
3806 54309 : if (d <= 1) return Flx_copy(x);
3807 54309 : t = Fl_inv(factorial_Fl(d, p), p);
3808 54368 : y = cgetg(d+3, t_VECSMALL);
3809 54306 : y[1] = x[1];
3810 1328382 : for (i=d; i>=2; i--)
3811 : {
3812 1274005 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
3813 1273952 : t = Fl_mul(t, i, p);
3814 : }
3815 54377 : uel(y,3) = uel(x,3);
3816 54377 : uel(y,2) = uel(x,2);
3817 54377 : return y;
3818 : }
3819 :
3820 : GEN
3821 27314 : Flx_Laplace(GEN x, ulong p)
3822 : {
3823 27314 : long i, d = degpol(x);
3824 27312 : ulong t = 1;
3825 : GEN y;
3826 27312 : if (d <= 1) return Flx_copy(x);
3827 27312 : y = cgetg(d+3, t_VECSMALL);
3828 27300 : y[1] = x[1];
3829 27300 : uel(y,2) = uel(x,2);
3830 27300 : uel(y,3) = uel(x,3);
3831 757411 : for (i=2; i<=d; i++)
3832 : {
3833 730062 : t = Fl_mul(t, i%p, p);
3834 730102 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
3835 : }
3836 27349 : return y;
3837 : }
3838 :
3839 : GEN
3840 6220152 : Flxn_red(GEN a, long n)
3841 : {
3842 6220152 : long i, L, l = lg(a);
3843 : GEN b;
3844 6220152 : if (l == 2 || !n) return zero_Flx(a[1]);
3845 5830251 : L = n+2; if (L > l) L = l;
3846 5830251 : b = cgetg(L, t_VECSMALL); b[1] = a[1];
3847 58572143 : for (i=2; i<L; i++) b[i] = a[i];
3848 5827065 : return Flx_renormalize(b,L);
3849 : }
3850 :
3851 : GEN
3852 5050866 : Flxn_mul_pre(GEN a, GEN b, long n, ulong p, ulong pi)
3853 5050866 : { return Flxn_red(Flx_mul_pre(a, b, p, pi), n); }
3854 : GEN
3855 75203 : Flxn_mul(GEN a, GEN b, long n, ulong p)
3856 75203 : { return Flxn_mul_pre(a, b, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3857 :
3858 : GEN
3859 0 : Flxn_sqr_pre(GEN a, long n, ulong p, ulong pi)
3860 0 : { return Flxn_red(Flx_sqr_pre(a, p, pi), n); }
3861 : GEN
3862 0 : Flxn_sqr(GEN a, long n, ulong p)
3863 0 : { return Flxn_sqr_pre(a, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3864 :
3865 : /* (f*g) \/ x^n */
3866 : static GEN
3867 938247 : Flx_mulhigh_i(GEN f, GEN g, long n, ulong p, ulong pi)
3868 938247 : { return Flx_shift(Flx_mul_pre(f, g, p, pi),-n); }
3869 :
3870 : static GEN
3871 516570 : Flxn_mulhigh(GEN f, GEN g, long n2, long n, ulong p, ulong pi)
3872 : {
3873 516570 : GEN F = Flx_blocks(f, n2, 2), fl = gel(F,1), fh = gel(F,2);
3874 516405 : return Flx_add(Flx_mulhigh_i(fl, g, n2, p, pi),
3875 : Flxn_mul_pre(fh, g, n - n2, p, pi), p);
3876 : }
3877 :
3878 : /* g==NULL -> assume g==1 */
3879 : GEN
3880 55169 : Flxn_div_pre(GEN g, GEN f, long e, ulong p, ulong pi)
3881 : {
3882 55169 : pari_sp av = avma, av2;
3883 : ulong mask;
3884 : GEN W;
3885 55169 : long n = 1;
3886 55169 : if (lg(f) <= 2) pari_err_INV("Flxn_inv",f);
3887 55169 : W = Fl_to_Flx(Fl_inv(uel(f,2),p), f[1]);
3888 55139 : mask = quadratic_prec_mask(e);
3889 55157 : av2 = avma;
3890 258806 : for (;mask>1;)
3891 : {
3892 : GEN u, fr;
3893 203642 : long n2 = n;
3894 203642 : n<<=1; if (mask & 1) n--;
3895 203642 : mask >>= 1;
3896 203642 : fr = Flxn_red(f, n);
3897 203537 : if (mask>1 || !g)
3898 : {
3899 149460 : u = Flxn_mul_pre(W, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
3900 149703 : W = Flx_sub(W, Flx_shift(u, n2), p);
3901 : } else
3902 : {
3903 54077 : GEN y = Flxn_mul_pre(g, W, n, p, pi), yt = Flxn_red(y, n-n2);
3904 54082 : u = Flxn_mul_pre(yt, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
3905 54060 : W = Flx_sub(y, Flx_shift(u, n2), p);
3906 : }
3907 203577 : if (gc_needed(av2,2))
3908 : {
3909 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxn_div, e = %ld", n);
3910 0 : W = gerepileupto(av2, W);
3911 : }
3912 : }
3913 55164 : return gerepileupto(av, W);
3914 : }
3915 : GEN
3916 55127 : Flxn_div(GEN g, GEN f, long e, ulong p)
3917 55127 : { return Flxn_div_pre(g, f, e, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3918 :
3919 : GEN
3920 1039 : Flxn_inv(GEN f, long e, ulong p)
3921 1039 : { return Flxn_div(NULL, f, e, p); }
3922 :
3923 : GEN
3924 109381 : Flxn_expint(GEN h, long e, ulong p)
3925 : {
3926 109381 : pari_sp av = avma, av2;
3927 109381 : long v = h[1], n=1;
3928 109381 : GEN f = pol1_Flx(v), g = pol1_Flx(v);
3929 109347 : ulong mask = quadratic_prec_mask(e), pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3930 109363 : av2 = avma;
3931 422726 : for (;mask>1;)
3932 : {
3933 : GEN u, w;
3934 422660 : long n2 = n;
3935 422660 : n<<=1; if (mask & 1) n--;
3936 422660 : mask >>= 1;
3937 422660 : u = Flxn_mul_pre(g, Flx_mulhigh_i(f, Flxn_red(h, n2-1), n2-1, p,pi), n-n2, p,pi);
3938 422629 : u = Flx_add(u, Flx_shift(Flxn_red(h, n-1), 1-n2), p);
3939 422655 : w = Flxn_mul_pre(f, Flx_integXn(u, n2-1, p), n-n2, p, pi);
3940 422612 : f = Flx_add(f, Flx_shift(w, n2), p);
3941 422750 : if (mask<=1) break;
3942 313374 : u = Flxn_mul_pre(g, Flxn_mulhigh(f, g, n2, n, p, pi), n-n2, p, pi);
3943 313342 : g = Flx_sub(g, Flx_shift(u, n2), p);
3944 313363 : if (gc_needed(av2,2))
3945 : {
3946 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flxn_exp, e = %ld", n);
3947 0 : gerepileall(av2, 2, &f, &g);
3948 : }
3949 : }
3950 109442 : return gerepileupto(av, f);
3951 : }
3952 :
3953 : GEN
3954 0 : Flxn_exp(GEN h, long e, ulong p)
3955 : {
3956 0 : if (degpol(h)<1 || uel(h,2)!=0)
3957 0 : pari_err_DOMAIN("Flxn_exp","valuation", "<", gen_1, h);
3958 0 : return Flxn_expint(Flx_deriv(h, p), e, p);
3959 : }
3960 :
3961 : INLINE GEN
3962 217206 : Flxn_recip(GEN x, long n)
3963 : {
3964 217206 : GEN z=Flx_recipspec(x+2,lgpol(x),n);
3965 217067 : z[1]=x[1];
3966 217067 : return z;
3967 : }
3968 :
3969 : GEN
3970 54063 : Flx_Newton(GEN P, long n, ulong p)
3971 : {
3972 54063 : pari_sp av = avma;
3973 54063 : long d = degpol(P);
3974 54051 : GEN dP = Flxn_recip(Flx_deriv(P, p), d);
3975 53982 : GEN Q = Flxn_div(dP, Flxn_recip(P, d+1), n, p);
3976 54018 : return gerepileuptoleaf(av, Q);
3977 : }
3978 :
3979 : GEN
3980 109380 : Flx_fromNewton(GEN P, ulong p)
3981 : {
3982 109380 : pari_sp av = avma;
3983 109380 : ulong n = Flx_constant(P)+1;
3984 109378 : GEN z = Flx_neg(Flx_shift(P, -1), p);
3985 109383 : GEN Q = Flxn_recip(Flxn_expint(z, n, p), n);
3986 109368 : return gerepileuptoleaf(av, Q);
3987 : }
3988 :
3989 : static void
3990 12514 : init_invlaplace(long d, ulong p, GEN *pt_P, GEN *pt_V)
3991 : {
3992 : long i;
3993 : ulong e;
3994 12514 : GEN P = cgetg(d+1, t_VECSMALL);
3995 12514 : GEN V = cgetg(d+1, t_VECSMALL);
3996 1396581 : for (i=1, e=1; i<=d; i++, e++)
3997 : {
3998 1384067 : if (e==p)
3999 : {
4000 459153 : e = 0;
4001 459153 : V[i] = u_lvalrem(i, p, &uel(P,i));
4002 : } else
4003 : {
4004 924914 : V[i] = 0; uel(P,i) = i;
4005 : }
4006 : }
4007 12514 : *pt_P = P; *pt_V = V;
4008 12514 : }
4009 :
4010 : /* return p^val * FpX_invLaplace(1+x+...x^(n-1), q), with q a power of p and
4011 : * val large enough to compensate for the power of p in the factorials */
4012 :
4013 : static GEN
4014 497 : ZpX_invLaplace_init(long n, GEN q, ulong p, long v, long sv)
4015 : {
4016 497 : pari_sp av = avma;
4017 497 : long i, d = n-1, w;
4018 : GEN y, W, E, t;
4019 497 : init_invlaplace(d, p, &E, &W);
4020 497 : t = Fp_inv(FpV_prod(Flv_to_ZV(E), q), q);
4021 497 : w = zv_sum(W);
4022 497 : if (v > w) t = Fp_mul(t, powuu(p, v-w), q);
4023 497 : y = cgetg(d+3,t_POL);
4024 497 : y[1] = evalsigne(1) | sv;
4025 28882 : for (i=d; i>=1; i--)
4026 : {
4027 28385 : gel(y,i+2) = t;
4028 28385 : t = Fp_mulu(t, uel(E,i), q);
4029 28385 : if (uel(W,i)) t = Fp_mul(t, powuu(p, uel(W,i)), q);
4030 : }
4031 497 : gel(y,2) = t;
4032 497 : return gerepilecopy(av, ZX_renormalize(y, d+3));
4033 : }
4034 :
4035 : GEN
4036 27520 : Flx_composedsum(GEN P, GEN Q, ulong p)
4037 : {
4038 27520 : pari_sp av = avma;
4039 27520 : long n = 1 + degpol(P)*degpol(Q);
4040 27514 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4041 27509 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4042 : GEN R;
4043 27527 : if (p >= (ulong)n)
4044 : {
4045 27030 : GEN Pl = Flx_invLaplace(Flx_Newton(P,n,p), p);
4046 27036 : GEN Ql = Flx_invLaplace(Flx_Newton(Q,n,p), p);
4047 27039 : GEN L = Flx_Laplace(Flxn_mul(Pl, Ql, n, p), p);
4048 27045 : R = Flx_fromNewton(L, p);
4049 : } else
4050 : {
4051 497 : long v = factorial_lval(n-1, p);
4052 497 : long w = 1 + ulogint(n-1, p);
4053 497 : GEN pv = powuu(p, v);
4054 497 : GEN qf = powuu(p, w), q = mulii(pv, qf), q2 = mulii(q, pv);
4055 497 : GEN iL = ZpX_invLaplace_init(n, q, p, v, P[1]);
4056 497 : GEN Pl = FpX_convol(iL, FpX_Newton(Flx_to_ZX(P), n, qf), q);
4057 497 : GEN Ql = FpX_convol(iL, FpX_Newton(Flx_to_ZX(Q), n, qf), q);
4058 497 : GEN Ln = ZX_Z_divexact(FpXn_mul(Pl, Ql, n, q2), pv);
4059 497 : GEN L = ZX_Z_divexact(FpX_Laplace(Ln, q), pv);
4060 497 : R = ZX_to_Flx(FpX_fromNewton(L, qf), p);
4061 : }
4062 27521 : return gerepileuptoleaf(av, Flx_Fl_mul(R, lead, p));
4063 : }
4064 :
4065 : static GEN
4066 3826 : _Flx_composedsum(void *E, GEN a, GEN b)
4067 3826 : { return Flx_composedsum(a, b, (ulong)E); }
4068 :
4069 : GEN
4070 28895 : FlxV_composedsum(GEN V, ulong p)
4071 28895 : { return gen_product(V, (void *)p, &_Flx_composedsum); }
4072 :
4073 : GEN
4074 0 : Flx_composedprod(GEN P, GEN Q, ulong p)
4075 : {
4076 0 : pari_sp av = avma;
4077 0 : long n = 1+ degpol(P)*degpol(Q);
4078 0 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4079 0 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4080 : GEN R;
4081 0 : if (p >= (ulong)n)
4082 : {
4083 0 : GEN L = Flx_convol(Flx_Newton(P,n,p), Flx_Newton(Q,n,p), p);
4084 0 : R = Flx_fromNewton(L, p);
4085 : } else
4086 : {
4087 0 : long w = 1 + ulogint(n, p);
4088 0 : GEN qf = powuu(p, w);
4089 0 : GEN Pl = FpX_convol(FpX_Newton(Flx_to_ZX(P), n, qf), FpX_Newton(Flx_to_ZX(Q), n, qf), qf);
4090 0 : R = ZX_to_Flx(FpX_fromNewton(Pl, qf), p);
4091 : }
4092 0 : return gerepileuptoleaf(av, Flx_Fl_mul(R, lead, p));
4093 :
4094 : }
4095 :
4096 : /* (x+1)^n mod p; assume 2 <= n < 2p prime */
4097 : static GEN
4098 0 : Fl_Xp1_powu(ulong n, ulong p, long v)
4099 : {
4100 0 : ulong k, d = (n + 1) >> 1;
4101 0 : GEN C, V = identity_zv(d);
4102 :
4103 0 : Flv_inv_inplace(V, p); /* could restrict to odd integers in [3,d] */
4104 0 : C = cgetg(n+3, t_VECSMALL);
4105 0 : C[1] = v;
4106 0 : uel(C,2) = 1UL;
4107 0 : uel(C,3) = n%p;
4108 0 : uel(C,4) = Fl_mul(odd(n)? n: n-1, n >> 1, p);
4109 : /* binom(n,k) = binom(n,k-1) * (n-k+1) / k */
4110 0 : if (SMALL_ULONG(p))
4111 0 : for (k = 3; k <= d; k++)
4112 0 : uel(C,k+2) = Fl_mul(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p);
4113 : else
4114 : {
4115 0 : ulong pi = get_Fl_red(p);
4116 0 : for (k = 3; k <= d; k++)
4117 0 : uel(C,k+2) = Fl_mul_pre(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p, pi);
4118 : }
4119 0 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4120 0 : return C; /* normalized */
4121 : }
4122 :
4123 : /* p arbitrary */
4124 : GEN
4125 28194 : Flx_translate1_basecase(GEN P, ulong p)
4126 : {
4127 28194 : GEN R = Flx_copy(P);
4128 28194 : long i, k, n = degpol(P);
4129 654732 : for (i = 1; i <= n; i++)
4130 14846488 : for (k = n-i; k < n; k++) uel(R,k+2) = Fl_add(uel(R,k+2), uel(R,k+3), p);
4131 28194 : return R;
4132 : }
4133 :
4134 : static int
4135 41359 : translate_basecase(long n, ulong p)
4136 : {
4137 : #ifdef LONG_IS_64BIT
4138 36066 : if (p <= 19) return n < 40;
4139 29910 : if (p < 1UL<<30) return n < 58;
4140 0 : if (p < 1UL<<59) return n < 100;
4141 0 : if (p < 1UL<<62) return n < 120;
4142 0 : if (p < 1UL<<63) return n < 240;
4143 0 : return n < 250;
4144 : #else
4145 5293 : if (p <= 13) return n < 18;
4146 4136 : if (p <= 17) return n < 22;
4147 4078 : if (p <= 29) return n < 39;
4148 3886 : if (p <= 67) return n < 69;
4149 3667 : if (p < 1UL<< 15) return n < 80;
4150 2047 : if (p < 1UL<< 16) return n < 100;
4151 0 : if (p < 1UL<< 28) return n < 300;
4152 0 : return n < 650;
4153 : #endif
4154 : }
4155 : /* assume p prime */
4156 : GEN
4157 16100 : Flx_translate1(GEN P, ulong p)
4158 : {
4159 16100 : long d, n = degpol(P);
4160 : GEN R, Q, S;
4161 16100 : if (translate_basecase(n, p)) return Flx_translate1_basecase(P, p);
4162 : /* n > 0 */
4163 1148 : d = n >> 1;
4164 1148 : if ((ulong)n < p)
4165 : {
4166 0 : R = Flx_translate1(Flxn_red(P, d), p);
4167 0 : Q = Flx_translate1(Flx_shift(P, -d), p);
4168 0 : S = Fl_Xp1_powu(d, p, P[1]);
4169 0 : return Flx_add(Flx_mul(Q, S, p), R, p);
4170 : }
4171 : else
4172 : {
4173 : ulong q;
4174 1148 : if ((ulong)d > p) (void)ulogintall(d, p, &q); else q = p;
4175 1148 : R = Flx_translate1(Flxn_red(P, q), p);
4176 1148 : Q = Flx_translate1(Flx_shift(P, -q), p);
4177 1148 : S = Flx_add(Flx_shift(Q, q), Q, p);
4178 1148 : return Flx_add(S, R, p); /* P(x+1) = Q(x+1) (x^q+1) + R(x+1) */
4179 : }
4180 : }
4181 :
4182 : static GEN
4183 12017 : zl_Xp1_powu(ulong n, ulong p, ulong q, long e, long vs)
4184 : {
4185 12017 : ulong k, d = n >> 1, c, v = 0;
4186 12017 : GEN C, V, W, U = upowers(p, e-1);
4187 12017 : init_invlaplace(d, p, &V, &W);
4188 12017 : Flv_inv_inplace(V, q);
4189 12017 : C = cgetg(n+3, t_VECSMALL);
4190 12017 : C[1] = vs;
4191 12017 : uel(C,2) = 1UL;
4192 12017 : uel(C,3) = n%q;
4193 12017 : v = u_lvalrem(n, p, &c);
4194 1355682 : for (k = 2; k <= d; k++)
4195 : {
4196 : ulong w;
4197 1343665 : v += u_lvalrem(n-k+1, p, &w) - W[k];
4198 1343665 : c = Fl_mul(Fl_mul(w%q, c, q), uel(V,k), q);
4199 1343665 : uel(C,2+k) = v >= (ulong)e ? 0: v==0 ? c : Fl_mul(c, uel(U, v+1), q);
4200 : }
4201 1374521 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4202 12017 : return C; /* normalized */
4203 : }
4204 :
4205 : GEN
4206 25259 : zlx_translate1(GEN P, ulong p, long e)
4207 : {
4208 25259 : ulong d, q = upowuu(p,e), n = degpol(P);
4209 : GEN R, Q, S;
4210 25259 : if (translate_basecase(n, q)) return Flx_translate1_basecase(P, q);
4211 : /* n > 0 */
4212 12017 : d = n >> 1;
4213 12017 : R = zlx_translate1(Flxn_red(P, d), p, e);
4214 12017 : Q = zlx_translate1(Flx_shift(P, -d), p, e);
4215 12017 : S = zl_Xp1_powu(d, p, q, e, P[1]);
4216 12017 : return Flx_add(Flx_mul(Q, S, q), R, q);
4217 : }
4218 :
4219 : /***********************************************************************/
4220 : /** Fl2 **/
4221 : /***********************************************************************/
4222 : /* Fl2 objects are Flv of length 2 [a,b] representing a+bsqrt(D) for
4223 : * a nonsquare D. */
4224 :
4225 : INLINE GEN
4226 7141299 : mkF2(ulong a, ulong b) { return mkvecsmall2(a,b); }
4227 :
4228 : /* allow pi = 0 */
4229 : GEN
4230 1902935 : Fl2_mul_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4231 : {
4232 : ulong xaya, xbyb, Db2, mid, z1, z2;
4233 1902935 : ulong x1 = x[1], x2 = x[2], y1 = y[1], y2 = y[2];
4234 1902935 : if (pi)
4235 : {
4236 1903172 : xaya = Fl_mul_pre(x1,y1,p,pi);
4237 1903501 : if (x2==0 && y2==0) return mkF2(xaya,0);
4238 1835438 : if (x2==0) return mkF2(xaya,Fl_mul_pre(x1,y2,p,pi));
4239 1811073 : if (y2==0) return mkF2(xaya,Fl_mul_pre(x2,y1,p,pi));
4240 1810771 : xbyb = Fl_mul_pre(x2,y2,p,pi);
4241 1810723 : mid = Fl_mul_pre(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p,pi);
4242 1810799 : Db2 = Fl_mul_pre(D, xbyb, p,pi);
4243 : }
4244 0 : else if (p & HIGHMASK)
4245 : {
4246 0 : xaya = Fl_mul(x1,y1,p);
4247 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4248 0 : if (x2==0) return mkF2(xaya,Fl_mul(x1,y2,p));
4249 0 : if (y2==0) return mkF2(xaya,Fl_mul(x2,y1,p));
4250 0 : xbyb = Fl_mul(x2,y2,p);
4251 0 : mid = Fl_mul(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p);
4252 0 : Db2 = Fl_mul(D, xbyb, p);
4253 : }
4254 : else
4255 : {
4256 0 : xaya = (x1 * y1) % p;
4257 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4258 0 : if (x2==0) return mkF2(xaya, (x1 * y2) % p);
4259 0 : if (y2==0) return mkF2(xaya, (x2 * y1) % p);
4260 0 : xbyb = (x2 * y2) % p;
4261 0 : mid = (Fl_add(x1,x2,p) * Fl_add(y1,y2,p)) % p;
4262 0 : Db2 = (D * xbyb) % p;
4263 : }
4264 1810768 : z1 = Fl_add(xaya,Db2,p);
4265 1810702 : z2 = Fl_sub(mid,Fl_add(xaya,xbyb,p),p);
4266 1810697 : return mkF2(z1,z2);
4267 : }
4268 :
4269 : /* allow pi = 0 */
4270 : GEN
4271 4788378 : Fl2_sqr_pre(GEN x, ulong D, ulong p, ulong pi)
4272 : {
4273 4788378 : ulong a = x[1], b = x[2];
4274 : ulong a2, Db2, ab;
4275 4788378 : if (pi)
4276 : {
4277 4788866 : a2 = Fl_sqr_pre(a,p,pi);
4278 4791104 : if (b==0) return mkF2(a2,0);
4279 4582332 : Db2= Fl_mul_pre(D, Fl_sqr_pre(b,p,pi), p,pi);
4280 4582328 : ab = Fl_mul_pre(a,b,p,pi);
4281 : }
4282 0 : else if (p & HIGHMASK)
4283 : {
4284 0 : a2 = Fl_sqr(a,p);
4285 0 : if (b==0) return mkF2(a2,0);
4286 0 : Db2= Fl_mul(D, Fl_sqr(b,p), p);
4287 0 : ab = Fl_mul(a,b,p);
4288 : }
4289 : else
4290 : {
4291 0 : a2 = (a * a) % p;
4292 0 : if (b==0) return mkF2(a2,0);
4293 0 : Db2= (D * ((b * b) % p)) % p;
4294 0 : ab = (a * b) % p;
4295 : }
4296 4581892 : return mkF2(Fl_add(a2,Db2,p), Fl_double(ab,p));
4297 : }
4298 :
4299 : /* allow pi = 0 */
4300 : ulong
4301 125532 : Fl2_norm_pre(GEN x, ulong D, ulong p, ulong pi)
4302 : {
4303 125532 : ulong a = x[1], b = x[2], a2;
4304 125532 : if (pi)
4305 : {
4306 71747 : a2 = Fl_sqr_pre(a,p,pi);
4307 71747 : return b? Fl_sub(a2, Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p): a2;
4308 : }
4309 53785 : else if (p & HIGHMASK)
4310 : {
4311 0 : a2 = Fl_sqr(a,p);
4312 0 : return b? Fl_sub(a2, Fl_mul(D, Fl_sqr(b, p), p), p): a2;
4313 : }
4314 : else
4315 : {
4316 53785 : a2 = (a * a) % p;
4317 53785 : return b? Fl_sub(a2, (D * ((b * b) % p)) % p, p): a2;
4318 : }
4319 : }
4320 :
4321 : /* allow pi = 0 */
4322 : GEN
4323 191891 : Fl2_inv_pre(GEN x, ulong D, ulong p, ulong pi)
4324 : {
4325 191891 : ulong a = x[1], b = x[2], n, ni;
4326 191891 : if (b == 0) return mkF2(Fl_inv(a,p), 0);
4327 161453 : b = Fl_neg(b, p);
4328 161455 : if (pi)
4329 : {
4330 161455 : n = Fl_sub(Fl_sqr_pre(a, p,pi),
4331 : Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p);
4332 161453 : ni = Fl_inv(n,p);
4333 161453 : return mkF2(Fl_mul_pre(a, ni, p,pi), Fl_mul_pre(b, ni, p,pi));
4334 : }
4335 0 : else if (p & HIGHMASK)
4336 : {
4337 0 : n = Fl_sub(Fl_sqr(a, p), Fl_mul(D, Fl_sqr(b, p), p), p);
4338 0 : ni = Fl_inv(n,p);
4339 0 : return mkF2(Fl_mul(a, ni, p), Fl_mul(b, ni, p));
4340 : }
4341 : else
4342 : {
4343 0 : n = Fl_sub((a * a) % p, (D * ((b * b) % p)) % p, p);
4344 0 : ni = Fl_inv(n,p);
4345 0 : return mkF2((a * ni) % p, (b * ni) % p);
4346 : }
4347 : }
4348 :
4349 : int
4350 436752 : Fl2_equal1(GEN x) { return x[1]==1 && x[2]==0; }
4351 :
4352 : struct _Fl2 {
4353 : ulong p, pi, D;
4354 : };
4355 :
4356 : static GEN
4357 4787823 : _Fl2_sqr(void *data, GEN x)
4358 : {
4359 4787823 : struct _Fl2 *D = (struct _Fl2*)data;
4360 4787823 : return Fl2_sqr_pre(x, D->D, D->p, D->pi);
4361 : }
4362 : static GEN
4363 1874706 : _Fl2_mul(void *data, GEN x, GEN y)
4364 : {
4365 1874706 : struct _Fl2 *D = (struct _Fl2*)data;
4366 1874706 : return Fl2_mul_pre(x,y, D->D, D->p, D->pi);
4367 : }
4368 :
4369 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL; allow pi = 0 */
4370 : GEN
4371 652819 : Fl2_pow_pre(GEN x, GEN n, ulong D, ulong p, ulong pi)
4372 : {
4373 652819 : pari_sp av = avma;
4374 : struct _Fl2 d;
4375 : GEN y;
4376 652819 : long s = signe(n);
4377 652819 : if (!s) return mkF2(1,0);
4378 578674 : if (s < 0)
4379 191891 : x = Fl2_inv_pre(x,D,p,pi);
4380 578670 : if (is_pm1(n)) return s < 0 ? x : zv_copy(x);
4381 426035 : d.p = p; d.pi = pi; d.D=D;
4382 426035 : y = gen_pow_i(x, n, (void*)&d, &_Fl2_sqr, &_Fl2_mul);
4383 426045 : return gerepileuptoleaf(av, y);
4384 : }
4385 :
4386 : static GEN
4387 652803 : _Fl2_pow(void *data, GEN x, GEN n)
4388 : {
4389 652803 : struct _Fl2 *D = (struct _Fl2*)data;
4390 652803 : return Fl2_pow_pre(x, n, D->D, D->p, D->pi);
4391 : }
4392 :
4393 : static GEN
4394 109951 : _Fl2_rand(void *data)
4395 : {
4396 109951 : struct _Fl2 *D = (struct _Fl2*)data;
4397 109951 : ulong a = random_Fl(D->p), b=random_Fl(D->p-1)+1;
4398 109950 : return mkF2(a,b);
4399 : }
4400 :
4401 : GEN
4402 67856 : Fl2_sqrt_pre(GEN z, ulong D, ulong p, ulong pi)
4403 : {
4404 67856 : ulong a = uel(z,1), b = uel(z,2), as2, u, v, s;
4405 67856 : ulong y = Fl_2gener_pre_i(D, p, pi);
4406 67856 : if (b == 0)
4407 19504 : return krouu(a, p)==1 ? mkF2(Fl_sqrt_pre_i(a, y, p, pi), 0)
4408 19504 : : mkF2(0, Fl_sqrt_pre_i(Fl_div(a, D, p), y, p, pi));
4409 54615 : s = Fl_sqrt_pre_i(Fl2_norm_pre(z, D, p, pi), y, p, pi);
4410 54615 : if (s==~0UL) return NULL;
4411 51441 : as2 = Fl_halve(Fl_add(a, s, p), p);
4412 51441 : if (krouu(as2, p)==-1) as2 = Fl_sub(as2, s, p);
4413 51441 : u = Fl_sqrt_pre_i(as2, y, p, pi);
4414 51441 : v = Fl_div(b, Fl_double(u, p), p);
4415 51441 : return mkF2(u,v);
4416 : }
4417 :
4418 : static const struct bb_group Fl2_star={_Fl2_mul, _Fl2_pow, _Fl2_rand,
4419 : hash_GEN, zv_equal, Fl2_equal1, NULL};
4420 :
4421 : /* allow pi = 0 */
4422 : GEN
4423 74146 : Fl2_sqrtn_pre(GEN a, GEN n, ulong D, ulong p, ulong pi, GEN *zeta)
4424 : {
4425 : struct _Fl2 E;
4426 : GEN o;
4427 74146 : if (a[1]==0 && a[2]==0)
4428 : {
4429 0 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
4430 0 : if (zeta) *zeta=mkF2(1,0);
4431 0 : return zv_copy(a);
4432 : }
4433 74146 : E.p=p; E.pi = pi; E.D = D;
4434 74146 : o = subiu(powuu(p,2), 1);
4435 74144 : return gen_Shanks_sqrtn(a,n,o,zeta,(void*)&E,&Fl2_star);
4436 : }
4437 :
4438 : /* allow pi = 0 */
4439 : GEN
4440 10402 : Flx_Fl2_eval_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4441 : {
4442 : GEN p1;
4443 10402 : long i = lg(x)-1;
4444 10402 : if (i <= 2)
4445 2065 : return mkF2(i == 2? x[2]: 0, 0);
4446 8337 : p1 = mkF2(x[i], 0);
4447 36456 : for (i--; i>=2; i--)
4448 : {
4449 28119 : p1 = Fl2_mul_pre(p1, y, D, p, pi);
4450 28119 : uel(p1,1) = Fl_add(uel(p1,1), uel(x,i), p);
4451 : }
4452 8337 : return p1;
4453 : }
4454 :
4455 : /***********************************************************************/
4456 : /** FlxV **/
4457 : /***********************************************************************/
4458 : /* FlxV are t_VEC with Flx coefficients. */
4459 :
4460 : GEN
4461 34482 : FlxV_Flc_mul(GEN V, GEN W, ulong p)
4462 : {
4463 34482 : pari_sp ltop=avma;
4464 : long i;
4465 34482 : GEN z = Flx_Fl_mul(gel(V,1),W[1],p);
4466 257068 : for(i=2;i<lg(V);i++)
4467 222586 : z=Flx_add(z,Flx_Fl_mul(gel(V,i),W[i],p),p);
4468 34482 : return gerepileuptoleaf(ltop,z);
4469 : }
4470 :
4471 : GEN
4472 0 : ZXV_to_FlxV(GEN x, ulong p)
4473 0 : { pari_APPLY_type(t_VEC, ZX_to_Flx(gel(x,i), p)) }
4474 :
4475 : GEN
4476 3738927 : ZXT_to_FlxT(GEN x, ulong p)
4477 : {
4478 3738927 : if (typ(x) == t_POL)
4479 3678522 : return ZX_to_Flx(x, p);
4480 : else
4481 198073 : pari_APPLY_type(t_VEC, ZXT_to_FlxT(gel(x,i), p))
4482 : }
4483 :
4484 : GEN
4485 168779 : FlxV_to_Flm(GEN x, long n)
4486 916934 : { pari_APPLY_type(t_MAT, Flx_to_Flv(gel(x,i), n)) }
4487 :
4488 : GEN
4489 0 : FlxV_red(GEN x, ulong p)
4490 0 : { pari_APPLY_type(t_VEC, Flx_red(gel(x,i), p)) }
4491 :
4492 : GEN
4493 313913 : FlxT_red(GEN x, ulong p)
4494 : {
4495 313913 : if (typ(x) == t_VECSMALL)
4496 211164 : return Flx_red(x, p);
4497 : else
4498 344632 : pari_APPLY_type(t_VEC, FlxT_red(gel(x,i), p))
4499 : }
4500 :
4501 : GEN
4502 113589 : FlxqV_dotproduct_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
4503 : {
4504 113589 : long i, lx = lg(x);
4505 : pari_sp av;
4506 : GEN c;
4507 113589 : if (lx == 1) return pol0_Flx(get_Flx_var(T));
4508 113589 : av = avma; c = Flx_mul_pre(gel(x,1),gel(y,1), p, pi);
4509 464499 : for (i=2; i<lx; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4510 113589 : return gerepileuptoleaf(av, Flx_rem_pre(c,T,p,pi));
4511 : }
4512 : GEN
4513 0 : FlxqV_dotproduct(GEN x, GEN y, GEN T, ulong p)
4514 0 : { return FlxqV_dotproduct_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4515 :
4516 : GEN
4517 2060 : FlxqX_dotproduct(GEN x, GEN y, GEN T, ulong p)
4518 : {
4519 2060 : long i, l = minss(lg(x), lg(y));
4520 : ulong pi;
4521 : pari_sp av;
4522 : GEN c;
4523 2060 : if (l == 2) return pol0_Flx(get_Flx_var(T));
4524 1977 : av = avma; pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4525 1977 : c = Flx_mul_pre(gel(x,2),gel(y,2), p, pi);
4526 6329 : for (i=3; i<l; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4527 1977 : return gerepileuptoleaf(av, Flx_rem_pre(c,T,p,pi));
4528 : }
4529 :
4530 : /* allow pi = 0 */
4531 : GEN
4532 250777 : FlxC_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4533 : {
4534 250777 : long i, l = lg(z);
4535 250777 : GEN y = cgetg(l, t_VECSMALL);
4536 12737219 : for (i=1; i<l; i++) uel(y,i) = Flx_eval_powers_pre(gel(z,i), x, p, pi);
4537 250888 : return y;
4538 : }
4539 :
4540 : /***********************************************************************/
4541 : /** FlxM **/
4542 : /***********************************************************************/
4543 : /* allow pi = 0 */
4544 : GEN
4545 19404 : FlxM_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4546 : {
4547 19404 : long i, l = lg(z);
4548 19404 : GEN y = cgetg(l, t_MAT);
4549 270181 : for (i=1; i<l; i++) gel(y,i) = FlxC_eval_powers_pre(gel(z,i), x, p, pi);
4550 19404 : return y;
4551 : }
4552 :
4553 : GEN
4554 0 : zero_FlxC(long n, long sv)
4555 : {
4556 0 : GEN x = cgetg(n + 1, t_COL), z = zero_Flx(sv);
4557 : long i;
4558 0 : for (i = 1; i <= n; i++) gel(x, i) = z;
4559 0 : return x;
4560 : }
4561 :
4562 : GEN
4563 0 : FlxC_neg(GEN x, ulong p)
4564 0 : { pari_APPLY_type(t_COL, Flx_neg(gel(x, i), p)) }
4565 :
4566 : GEN
4567 0 : FlxC_sub(GEN x, GEN y, ulong p)
4568 0 : { pari_APPLY_type(t_COL, Flx_sub(gel(x, i), gel(y, i), p)) }
4569 :
4570 : GEN
4571 0 : zero_FlxM(long r, long c, long sv)
4572 : {
4573 0 : GEN x = cgetg(c + 1, t_MAT), z = zero_FlxC(r, sv);
4574 : long j;
4575 0 : for (j = 1; j <= c; j++) gel(x, j) = z;
4576 0 : return x;
4577 : }
4578 :
4579 : GEN
4580 0 : FlxM_neg(GEN x, ulong p)
4581 0 : { pari_APPLY_same(FlxC_neg(gel(x, i), p)) }
4582 :
4583 : GEN
4584 0 : FlxM_sub(GEN x, GEN y, ulong p)
4585 0 : { pari_APPLY_same(FlxC_sub(gel(x, i), gel(y,i), p)) }
4586 :
4587 : GEN
4588 0 : FlxqC_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4589 0 : { pari_APPLY_type(t_COL, Flxq_mul(gel(x, i), y, T, p)) }
4590 :
4591 : GEN
4592 0 : FlxqM_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4593 0 : { pari_APPLY_same(FlxqC_Flxq_mul(gel(x, i), y, T, p)) }
4594 :
4595 : static GEN
4596 57458 : FlxM_pack_ZM(GEN M, GEN (*pack)(GEN, long)) {
4597 : long i, j, l, lc;
4598 57458 : GEN N = cgetg_copy(M, &l), x;
4599 57458 : if (l == 1)
4600 0 : return N;
4601 57458 : lc = lgcols(M);
4602 276559 : for (j = 1; j < l; j++) {
4603 219101 : gel(N, j) = cgetg(lc, t_COL);
4604 1127224 : for (i = 1; i < lc; i++) {
4605 908123 : x = gcoeff(M, i, j);
4606 908123 : gcoeff(N, i, j) = pack(x + 2, lgpol(x));
4607 : }
4608 : }
4609 57458 : return N;
4610 : }
4611 :
4612 : static GEN
4613 850912 : kron_pack_Flx_spec_half(GEN x, long l) {
4614 850912 : if (l == 0) return gen_0;
4615 493085 : return Flx_to_int_halfspec(x, l);
4616 : }
4617 :
4618 : static GEN
4619 53822 : kron_pack_Flx_spec(GEN x, long l) {
4620 : long i;
4621 : GEN w, y;
4622 53822 : if (l == 0)
4623 10075 : return gen_0;
4624 43747 : y = cgetipos(l + 2);
4625 159446 : for (i = 0, w = int_LSW(y); i < l; i++, w = int_nextW(w))
4626 115699 : *w = x[i];
4627 43747 : return y;
4628 : }
4629 :
4630 : static GEN
4631 3389 : kron_pack_Flx_spec_2(GEN x, long l) { return Flx_eval2BILspec(x, 2, l); }
4632 :
4633 : static GEN
4634 0 : kron_pack_Flx_spec_3(GEN x, long l) { return Flx_eval2BILspec(x, 3, l); }
4635 :
4636 : static GEN
4637 42982 : kron_unpack_Flx(GEN z, ulong p)
4638 : {
4639 42982 : long i, l = lgefint(z);
4640 42982 : GEN x = cgetg(l, t_VECSMALL), w;
4641 202068 : for (w = int_LSW(z), i = 2; i < l; w = int_nextW(w), i++)
4642 159086 : x[i] = ((ulong) *w) % p;
4643 42982 : return Flx_renormalize(x, l);
4644 : }
4645 :
4646 : static GEN
4647 2930 : kron_unpack_Flx_2(GEN x, ulong p) {
4648 2930 : long d = (lgefint(x)-1)/2 - 1;
4649 2930 : return Z_mod2BIL_Flx_2(x, d, p);
4650 : }
4651 :
4652 : static GEN
4653 0 : kron_unpack_Flx_3(GEN x, ulong p) {
4654 0 : long d = lgefint(x)/3 - 1;
4655 0 : return Z_mod2BIL_Flx_3(x, d, p);
4656 : }
4657 :
4658 : static GEN
4659 118719 : FlxM_pack_ZM_bits(GEN M, long b)
4660 : {
4661 : long i, j, l, lc;
4662 118719 : GEN N = cgetg_copy(M, &l), x;
4663 118719 : if (l == 1)
4664 0 : return N;
4665 118719 : lc = lgcols(M);
4666 491688 : for (j = 1; j < l; j++) {
4667 372969 : gel(N, j) = cgetg(lc, t_COL);
4668 5983078 : for (i = 1; i < lc; i++) {
4669 5610109 : x = gcoeff(M, i, j);
4670 5610109 : gcoeff(N, i, j) = kron_pack_Flx_spec_bits(x + 2, b, lgpol(x));
4671 : }
4672 : }
4673 118719 : return N;
4674 : }
4675 :
4676 : static GEN
4677 28729 : ZM_unpack_FlxqM(GEN M, GEN T, ulong p, ulong pi, GEN (*unpack)(GEN, ulong))
4678 : {
4679 28729 : long i, j, l, lc, sv = get_Flx_var(T);
4680 28729 : GEN N = cgetg_copy(M, &l), x;
4681 28729 : if (l == 1)
4682 0 : return N;
4683 28729 : lc = lgcols(M);
4684 162662 : for (j = 1; j < l; j++) {
4685 133933 : gel(N, j) = cgetg(lc, t_COL);
4686 765794 : for (i = 1; i < lc; i++) {
4687 631861 : x = unpack(gcoeff(M, i, j), p);
4688 631861 : x[1] = sv;
4689 631861 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4690 : }
4691 : }
4692 28729 : return N;
4693 : }
4694 :
4695 : static GEN
4696 59400 : ZM_unpack_FlxqM_bits(GEN M, long b, GEN T, ulong p, ulong pi)
4697 : {
4698 59400 : long i, j, l, lc, sv = get_Flx_var(T);
4699 59400 : GEN N = cgetg_copy(M, &l), x;
4700 59400 : if (l == 1)
4701 0 : return N;
4702 59400 : lc = lgcols(M);
4703 59400 : if (b < BITS_IN_LONG) {
4704 200902 : for (j = 1; j < l; j++) {
4705 143161 : gel(N, j) = cgetg(lc, t_COL);
4706 3235607 : for (i = 1; i < lc; i++) {
4707 3092446 : x = kron_unpack_Flx_bits_narrow(gcoeff(M, i, j), b, p);
4708 3092446 : x[1] = sv;
4709 3092446 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4710 : }
4711 : }
4712 : } else {
4713 1659 : ulong pi = get_Fl_red(p);
4714 9796 : for (j = 1; j < l; j++) {
4715 8137 : gel(N, j) = cgetg(lc, t_COL);
4716 175265 : for (i = 1; i < lc; i++) {
4717 167128 : x = kron_unpack_Flx_bits_wide(gcoeff(M, i, j), b, p, pi);
4718 167128 : x[1] = sv;
4719 167128 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4720 : }
4721 : }
4722 : }
4723 59400 : return N;
4724 : }
4725 :
4726 : GEN
4727 88129 : FlxqM_mul_Kronecker(GEN A, GEN B, GEN T, ulong p)
4728 : {
4729 88129 : pari_sp av = avma;
4730 88129 : long b, d = get_Flx_degree(T), n = lg(A) - 1;
4731 : GEN C, D, z;
4732 : GEN (*pack)(GEN, long), (*unpack)(GEN, ulong);
4733 88129 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4734 88129 : int is_sqr = A==B;
4735 :
4736 88129 : z = muliu(muliu(sqru(p - 1), d), n);
4737 88129 : b = expi(z) + 1;
4738 : /* only do expensive bit-packing if it saves at least 1 limb */
4739 88129 : if (b <= BITS_IN_HALFULONG)
4740 83726 : { if (nbits2nlong(d*b) == (d + 1)/2) b = BITS_IN_HALFULONG; }
4741 : else
4742 : {
4743 4403 : long l = lgefint(z) - 2;
4744 4403 : if (nbits2nlong(d*b) == d*l) b = l*BITS_IN_LONG;
4745 : }
4746 88129 : set_avma(av);
4747 :
4748 88129 : switch (b) {
4749 27668 : case BITS_IN_HALFULONG:
4750 27668 : pack = kron_pack_Flx_spec_half;
4751 27668 : unpack = int_to_Flx_half;
4752 27668 : break;
4753 1012 : case BITS_IN_LONG:
4754 1012 : pack = kron_pack_Flx_spec;
4755 1012 : unpack = kron_unpack_Flx;
4756 1012 : break;
4757 49 : case 2*BITS_IN_LONG:
4758 49 : pack = kron_pack_Flx_spec_2;
4759 49 : unpack = kron_unpack_Flx_2;
4760 49 : break;
4761 0 : case 3*BITS_IN_LONG:
4762 0 : pack = kron_pack_Flx_spec_3;
4763 0 : unpack = kron_unpack_Flx_3;
4764 0 : break;
4765 59400 : default:
4766 59400 : A = FlxM_pack_ZM_bits(A, b);
4767 59400 : B = is_sqr? A: FlxM_pack_ZM_bits(B, b);
4768 59400 : C = ZM_mul(A, B);
4769 59400 : D = ZM_unpack_FlxqM_bits(C, b, T, p, pi);
4770 59400 : return gerepilecopy(av, D);
4771 : }
4772 28729 : A = FlxM_pack_ZM(A, pack);
4773 28729 : B = is_sqr? A: FlxM_pack_ZM(B, pack);
4774 28729 : C = ZM_mul(A, B);
4775 28729 : D = ZM_unpack_FlxqM(C, T, p, pi, unpack);
4776 28729 : return gerepilecopy(av, D);
4777 : }
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