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 986970553 : get_Flx_red(GEN T, GEN *B)
22 : {
23 986970553 : if (typ(T)!=t_VEC) { *B=NULL; return T; }
24 669619 : *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 37308781 : Flx_to_ZX(GEN z)
43 : {
44 37308781 : long i, l = lg(z);
45 37308781 : GEN x = cgetg(l,t_POL);
46 243299071 : for (i=2; i<l; i++) gel(x,i) = utoi(z[i]);
47 37295342 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
48 : }
49 :
50 : GEN
51 71629 : Flx_to_FlxX(GEN z, long sv)
52 : {
53 71629 : long i, l = lg(z);
54 71629 : GEN x = cgetg(l,t_POL);
55 279716 : for (i=2; i<l; i++) gel(x,i) = Fl_to_Flx(z[i], sv);
56 71629 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
57 : }
58 :
59 : /* same as Flx_to_ZX, in place */
60 : GEN
61 36555337 : Flx_to_ZX_inplace(GEN z)
62 : {
63 36555337 : long i, l = lg(z);
64 227873687 : for (i=2; i<l; i++) gel(z,i) = utoi(z[i]);
65 36547567 : 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 66627555 : Flx_to_Flv(GEN x, long N)
71 : {
72 66627555 : GEN z = cgetg(N+1,t_VECSMALL);
73 66622003 : long i, l = lg(x)-1;
74 66622003 : x++;
75 708064923 : for (i=1; i<l ; i++) z[i]=x[i];
76 330150015 : for ( ; i<=N; i++) z[i]=0;
77 66622003 : return z;
78 : }
79 :
80 : /*Flv_to_Flx=zv_to_zx*/
81 : GEN
82 25482901 : Flv_to_Flx(GEN x, long sv)
83 : {
84 25482901 : long i, l=lg(x)+1;
85 25482901 : GEN z = cgetg(l,t_VECSMALL); z[1]=sv;
86 25478913 : x--;
87 279646604 : for (i=2; i<l ; i++) z[i]=x[i];
88 25478913 : return Flx_renormalize(z,l);
89 : }
90 :
91 : /*Flm_to_FlxV=zm_to_zxV*/
92 : GEN
93 2772 : Flm_to_FlxV(GEN x, long sv)
94 7455 : { pari_APPLY_type(t_VEC, Flv_to_Flx(gel(x,i), sv)) }
95 :
96 : /*FlxC_to_ZXC=zxC_to_ZXC*/
97 : GEN
98 104060 : FlxC_to_ZXC(GEN x)
99 527425 : { pari_APPLY_type(t_COL, Flx_to_ZX(gel(x,i))) }
100 :
101 : /*FlxC_to_ZXC=zxV_to_ZXV*/
102 : GEN
103 606888 : FlxV_to_ZXV(GEN x)
104 2454611 : { pari_APPLY_type(t_VEC, Flx_to_ZX(gel(x,i))) }
105 :
106 : void
107 3028161 : FlxV_to_ZXV_inplace(GEN v)
108 : {
109 : long i;
110 8036736 : for(i=1;i<lg(v);i++) gel(v,i)= Flx_to_ZX(gel(v,i));
111 3028070 : }
112 :
113 : /*FlxM_to_ZXM=zxM_to_ZXM*/
114 : GEN
115 2485 : FlxM_to_ZXM(GEN x)
116 8351 : { pari_APPLY_same(FlxC_to_ZXC(gel(x,i))) }
117 :
118 : GEN
119 397977 : FlxV_to_FlxX(GEN x, long v)
120 : {
121 397977 : long i, l = lg(x)+1;
122 397977 : GEN z = cgetg(l,t_POL); z[1] = evalvarn(v);
123 397977 : x--;
124 4999132 : for (i=2; i<l ; i++) gel(z,i) = gel(x,i);
125 397977 : 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 20367064 : Fl_to_Flx(ulong x, long sv) { return x? mkvecsmall2(sv, x): pol0_Flx(sv); }
156 :
157 : /* a X^d */
158 : GEN
159 942760 : monomial_Flx(ulong a, long d, long vs)
160 : {
161 : GEN P;
162 942760 : if (a==0) return pol0_Flx(vs);
163 942760 : P = const_vecsmall(d+2, 0);
164 942763 : P[1] = vs; P[d+2] = a; return P;
165 : }
166 :
167 : GEN
168 2631516 : Z_to_Flx(GEN x, ulong p, long sv)
169 : {
170 2631516 : long u = umodiu(x,p);
171 2631511 : 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 169914915 : ZX_to_Flx(GEN x, ulong p)
177 : {
178 169914915 : long i, lx = lg(x);
179 169914915 : GEN a = cgetg(lx, t_VECSMALL);
180 169864429 : a[1]=((ulong)x[1])&VARNBITS;
181 1121643320 : for (i=2; i<lx; i++) a[i] = umodiu(gel(x,i), p);
182 169874079 : 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 6486495 : zx_to_Flx(GEN x, ulong p)
188 : {
189 6486495 : long i, lx = lg(x);
190 6486495 : GEN a = cgetg(lx, t_VECSMALL);
191 6481662 : a[1] = x[1];
192 19947338 : for (i=2; i<lx; i++) uel(a,i) = umodsu(x[i], p);
193 6481288 : return Flx_renormalize(a,lx);
194 : }
195 :
196 : ulong
197 73221877 : Rg_to_Fl(GEN x, ulong p)
198 : {
199 73221877 : switch(typ(x))
200 : {
201 48462823 : case t_INT: return umodiu(x, p);
202 456040 : case t_FRAC: {
203 456040 : ulong z = umodiu(gel(x,1), p);
204 456040 : if (!z) return 0;
205 446302 : return Fl_div(z, umodiu(gel(x,2), p), p);
206 : }
207 205942 : case t_PADIC: return padic_to_Fl(x, p);
208 24097083 : case t_INTMOD: {
209 24097083 : GEN q = gel(x,1), a = gel(x,2);
210 24097083 : 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 1706863 : Rg_to_F2(GEN x)
221 : {
222 1706863 : switch(typ(x))
223 : {
224 274054 : 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(padic_p(x),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 1432809 : case t_INTMOD: {
233 1432809 : GEN q = gel(x,1), a = gel(x,2);
234 1432809 : if (mpodd(q)) pari_err_MODULUS("Rg_to_F2", q, gen_2);
235 1432809 : 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 2250314 : RgX_to_Flx(GEN x, ulong p)
244 : {
245 2250314 : long i, lx = lg(x);
246 2250314 : GEN a = cgetg(lx, t_VECSMALL);
247 2250314 : a[1]=((ulong)x[1])&VARNBITS;
248 20058964 : for (i=2; i<lx; i++) a[i] = Rg_to_Fl(gel(x,i), p);
249 2250305 : 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 3598613 : Rg_to_Flxq(GEN x, GEN T, ulong p)
259 : {
260 3598613 : long ta, tx = typ(x), v = get_Flx_var(T);
261 : ulong pi;
262 : GEN a, b;
263 3598614 : if (is_const_t(tx))
264 : {
265 3338987 : if (tx == t_FFELT) return FF_to_Flxq(x);
266 2607979 : return Fl_to_Flx(Rg_to_Fl(x, p), v);
267 : }
268 259625 : 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 251049 : case t_POL:
280 251049 : x = RgX_to_Flx(x,p);
281 251049 : if (x[1] != v) break;
282 251049 : 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 2137874493 : Flx_renormalize(GEN /*in place*/ x, long lx)
298 : {
299 : long i;
300 2388386676 : for (i = lx-1; i>1; i--)
301 2292649531 : if (x[i]) break;
302 2137874493 : stackdummy((pari_sp)(x + lg(x)), (pari_sp)(x + i+1));
303 2137647115 : setlg(x, i+1); return x;
304 : }
305 :
306 : GEN
307 1882912 : Flx_red(GEN z, ulong p)
308 : {
309 1882912 : long i, l = lg(z);
310 1882912 : GEN x = cgetg(l, t_VECSMALL);
311 1882768 : x[1] = z[1];
312 33644095 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
313 1882768 : return Flx_renormalize(x,l);
314 : }
315 :
316 : int
317 28944446 : Flx_equal(GEN V, GEN W)
318 : {
319 28944446 : long l = lg(V);
320 28944446 : if (lg(W) != l) return 0;
321 29944505 : while (--l > 1) /* do not compare variables, V[1] */
322 28840465 : if (V[l] != W[l]) return 0;
323 1104040 : return 1;
324 : }
325 :
326 : GEN
327 2646651 : random_Flx(long d1, long vs, ulong p)
328 : {
329 2646651 : long i, d = d1+2;
330 2646651 : GEN y = cgetg(d,t_VECSMALL); y[1] = vs;
331 18210236 : for (i=2; i<d; i++) y[i] = random_Fl(p);
332 2646734 : return Flx_renormalize(y,d);
333 : }
334 :
335 : static GEN
336 7169811 : Flx_addspec(GEN x, GEN y, ulong p, long lx, long ly)
337 : {
338 : long i,lz;
339 : GEN z;
340 :
341 7169811 : if (ly>lx) swapspec(x,y, lx,ly);
342 7169811 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
343 106257124 : for (i=0; i<ly; i++) z[i+2] = Fl_add(x[i], y[i], p);
344 90014993 : for ( ; i<lx; i++) z[i+2] = x[i];
345 7169811 : z[1] = 0; return Flx_renormalize(z, lz);
346 : }
347 :
348 : GEN
349 63389275 : Flx_add(GEN x, GEN y, ulong p)
350 : {
351 : long i,lz;
352 : GEN z;
353 63389275 : long lx=lg(x);
354 63389275 : long ly=lg(y);
355 63389275 : if (ly>lx) swapspec(x,y, lx,ly);
356 63389275 : lz = lx; z = cgetg(lz, t_VECSMALL); z[1]=x[1];
357 581463324 : for (i=2; i<ly; i++) z[i] = Fl_add(x[i], y[i], p);
358 129097268 : for ( ; i<lx; i++) z[i] = x[i];
359 63356689 : return Flx_renormalize(z, lz);
360 : }
361 :
362 : GEN
363 10145298 : Flx_Fl_add(GEN y, ulong x, ulong p)
364 : {
365 : GEN z;
366 : long lz, i;
367 10145298 : if (!lgpol(y))
368 229937 : return Fl_to_Flx(x,y[1]);
369 9916530 : lz=lg(y);
370 9916530 : z=cgetg(lz,t_VECSMALL);
371 9916185 : z[1]=y[1];
372 9916185 : z[2] = Fl_add(y[2],x,p);
373 48047406 : for(i=3;i<lz;i++)
374 38131852 : z[i] = y[i];
375 9915554 : if (lz==3) z = Flx_renormalize(z,lz);
376 9915506 : return z;
377 : }
378 :
379 : static GEN
380 898265 : Flx_subspec(GEN x, GEN y, ulong p, long lx, long ly)
381 : {
382 : long i,lz;
383 : GEN z;
384 :
385 898265 : if (ly <= lx)
386 : {
387 898275 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
388 53986329 : for (i=0; i<ly; i++) z[i+2] = Fl_sub(x[i],y[i],p);
389 1448798 : 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 897788 : z[1] = 0; return Flx_renormalize(z, lz);
398 : }
399 :
400 : GEN
401 138653385 : Flx_sub(GEN x, GEN y, ulong p)
402 : {
403 138653385 : long i,lz,lx = lg(x), ly = lg(y);
404 : GEN z;
405 :
406 138653385 : if (ly <= lx)
407 : {
408 88471101 : lz = lx; z = cgetg(lz, t_VECSMALL);
409 458068823 : for (i=2; i<ly; i++) z[i] = Fl_sub(x[i],y[i],p);
410 176557693 : for ( ; i<lx; i++) z[i] = x[i];
411 : }
412 : else
413 : {
414 50182284 : lz = ly; z = cgetg(lz, t_VECSMALL);
415 262337524 : for (i=2; i<lx; i++) z[i] = Fl_sub(x[i],y[i],p);
416 232669800 : for ( ; i<ly; i++) z[i] = y[i]? (long)(p - y[i]): y[i];
417 : }
418 138594409 : z[1]=x[1]; return Flx_renormalize(z, lz);
419 : }
420 :
421 : GEN
422 151843 : Flx_Fl_sub(GEN y, ulong x, ulong p)
423 : {
424 : GEN z;
425 151843 : long lz = lg(y), i;
426 151843 : if (lz==2)
427 513 : return Fl_to_Flx(Fl_neg(x, p),y[1]);
428 151330 : z = cgetg(lz, t_VECSMALL);
429 151330 : z[1] = y[1];
430 151330 : z[2] = Fl_sub(uel(y,2), x, p);
431 753900 : for(i=3; i<lz; i++)
432 602570 : z[i] = y[i];
433 151330 : if (lz==3) z = Flx_renormalize(z,lz);
434 151330 : return z;
435 : }
436 :
437 : static GEN
438 3265091 : Flx_negspec(GEN x, ulong p, long l)
439 : {
440 : long i;
441 3265091 : GEN z = cgetg(l+2, t_VECSMALL) + 2;
442 20993816 : for (i=0; i<l; i++) z[i] = Fl_neg(x[i], p);
443 3265061 : return z-2;
444 : }
445 :
446 : GEN
447 3265093 : Flx_neg(GEN x, ulong p)
448 : {
449 3265093 : GEN z = Flx_negspec(x+2, p, lgpol(x));
450 3265142 : z[1] = x[1];
451 3265142 : return z;
452 : }
453 :
454 : GEN
455 1756902 : Flx_neg_inplace(GEN x, ulong p)
456 : {
457 1756902 : long i, l = lg(x);
458 52171730 : for (i=2; i<l; i++)
459 50414828 : if (x[i]) x[i] = p - x[i];
460 1756902 : return x;
461 : }
462 :
463 : GEN
464 2445177 : Flx_double(GEN y, ulong p)
465 : {
466 : long i, l;
467 2445177 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
468 20335150 : for(i=2; i<l; i++) z[i] = Fl_double(y[i], p);
469 2445177 : return Flx_renormalize(z, l);
470 : }
471 : GEN
472 1049887 : Flx_triple(GEN y, ulong p)
473 : {
474 : long i, l;
475 1049887 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
476 8278717 : for(i=2; i<l; i++) z[i] = Fl_triple(y[i], p);
477 1049887 : return Flx_renormalize(z, l);
478 : }
479 :
480 : GEN
481 18289575 : Flx_Fl_mul_pre(GEN y, ulong x, ulong p, ulong pi)
482 : {
483 : GEN z;
484 : long i, l;
485 18289575 : if (!x) return pol0_Flx(y[1]);
486 17513818 : z = cgetg_copy(y, &l); z[1] = y[1];
487 17513680 : if (pi==0)
488 : {
489 15306266 : if (HIGHWORD(x | p))
490 0 : for(i=2; i<l; i++) z[i] = Fl_mul(uel(y,i), x, p);
491 : else
492 92195563 : for(i=2; i<l; i++) z[i] = (uel(y,i) * x) % p;
493 : } else
494 18073608 : for(i=2; i<l; i++) z[i] = Fl_mul_pre(uel(y,i), x, p, pi);
495 17514725 : return Flx_renormalize(z, l);
496 : }
497 :
498 : GEN
499 7326089 : Flx_Fl_mul(GEN x, ulong y, ulong p)
500 7326089 : { 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 12139273 : Flx_Fl_mul_to_monic(GEN y, ulong x, ulong p)
515 : {
516 : GEN z;
517 : long i, l;
518 12139273 : z = cgetg_copy(y, &l); z[1] = y[1];
519 12135763 : if (HIGHWORD(x | p))
520 5445115 : for(i=2; i<l-1; i++) z[i] = Fl_mul(y[i], x, p);
521 : else
522 27281156 : for(i=2; i<l-1; i++) z[i] = (y[i] * x) % p;
523 12135752 : 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 27390889 : Flx_shift(GEN a, long n)
529 : {
530 27390889 : long i, l = lg(a);
531 : GEN b;
532 27390889 : if (l==2 || !n) return Flx_copy(a);
533 27046021 : if (l+n<=2) return pol0_Flx(a[1]);
534 26830859 : b = cgetg(l+n, t_VECSMALL);
535 26828673 : b[1] = a[1];
536 26828673 : if (n < 0)
537 73752149 : for (i=2-n; i<l; i++) b[i+n] = a[i];
538 : else
539 : {
540 52389707 : for (i=0; i<n; i++) b[2+i] = 0;
541 150040822 : for (i=2; i<l; i++) b[i+n] = a[i];
542 : }
543 26828673 : return b;
544 : }
545 :
546 : GEN
547 64210448 : Flx_normalize(GEN z, ulong p)
548 : {
549 64210448 : long l = lg(z)-1;
550 64210448 : ulong p1 = z[l]; /* leading term */
551 64210448 : if (p1 == 1) return z;
552 12116698 : 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 3684621 : Flx_addshift(GEN x, GEN y, ulong p, long d)
558 : {
559 3684621 : GEN xd,yd,zd = (GEN)avma;
560 3684621 : long a,lz,ny = lgpol(y), nx = lgpol(x);
561 3684621 : long vs = x[1];
562 3684621 : if (nx == 0) return y;
563 3682769 : x += 2; y += 2; a = ny-d;
564 3682769 : if (a <= 0)
565 : {
566 85096 : lz = (a>nx)? ny+2: nx+d+2;
567 85096 : (void)new_chunk(lz); xd = x+nx; yd = y+ny;
568 1732234 : while (xd > x) *--zd = *--xd;
569 85096 : x = zd + a;
570 164576 : while (zd > x) *--zd = 0;
571 : }
572 : else
573 : {
574 3597673 : xd = new_chunk(d); yd = y+d;
575 3597673 : x = Flx_addspec(x,yd,p, nx,a);
576 3597673 : lz = (a>nx)? ny+2: lg(x)+d;
577 132343793 : x += 2; while (xd > x) *--zd = *--xd;
578 : }
579 60217683 : while (yd > y) *--zd = *--yd;
580 3682769 : *--zd = vs;
581 3682769 : *--zd = evaltyp(t_VECSMALL) | evallg(lz); return zd;
582 : }
583 :
584 : /* shift polynomial + GC; do not set evalvarn*/
585 : static GEN
586 636473315 : Flx_shiftip(pari_sp av, GEN x, long v)
587 : {
588 636473315 : long i, lx = lg(x), ly;
589 : GEN y;
590 636473315 : if (!v || lx==2) return gc_leaf(av, x);
591 175909786 : ly = lx + v; /* result length */
592 175909786 : (void)new_chunk(ly); /* check that result fits */
593 175834511 : x += lx; y = (GEN)av;
594 1242807240 : for (i = 2; i<lx; i++) *--y = *--x;
595 708408191 : for (i = 0; i< v; i++) *--y = 0;
596 175834511 : y -= 2; y[0] = evaltyp(t_VECSMALL) | evallg(ly);
597 175991158 : return gc_const((pari_sp)y, y);
598 : }
599 :
600 : static long
601 2330370782 : get_Fl_threshold(ulong p, long mul, long mul2)
602 : {
603 2330370782 : return SMALL_ULONG(p) ? mul: mul2;
604 : }
605 :
606 : #define BITS_IN_QUARTULONG (BITS_IN_HALFULONG >> 1)
607 : #define QUARTMASK ((1UL<<BITS_IN_QUARTULONG)-1UL)
608 : #define LLQUARTWORD(x) ((x) & QUARTMASK)
609 : #define HLQUARTWORD(x) (((x) >> BITS_IN_QUARTULONG) & QUARTMASK)
610 : #define LHQUARTWORD(x) (((x) >> (2*BITS_IN_QUARTULONG)) & QUARTMASK)
611 : #define HHQUARTWORD(x) (((x) >> (3*BITS_IN_QUARTULONG)) & QUARTMASK)
612 : INLINE long
613 8383286 : maxbitcoeffpol(ulong p, long n)
614 : {
615 8383286 : GEN z = muliu(sqru(p - 1), n);
616 8380808 : long b = expi(z) + 1;
617 : /* only do expensive bit-packing if it saves at least 1 limb */
618 8381618 : if (b <= BITS_IN_QUARTULONG)
619 : {
620 875410 : if (nbits2nlong(n*b) == (n + 3)>>2)
621 107384 : b = BITS_IN_QUARTULONG;
622 : }
623 7506208 : else if (b <= BITS_IN_HALFULONG)
624 : {
625 1553330 : if (nbits2nlong(n*b) == (n + 1)>>1)
626 5805 : b = BITS_IN_HALFULONG;
627 : }
628 : else
629 : {
630 5952878 : long l = lgefint(z) - 2;
631 5952878 : if (nbits2nlong(n*b) == n*l)
632 308071 : b = l*BITS_IN_LONG;
633 : }
634 8381365 : return b;
635 : }
636 :
637 : INLINE ulong
638 3439048178 : Flx_mullimb_ok(GEN x, GEN y, ulong p, long a, long b)
639 : { /* Assume OK_ULONG*/
640 3439048178 : ulong p1 = 0;
641 : long i;
642 16218354673 : for (i=a; i<b; i++)
643 12779306495 : if (y[i])
644 : {
645 10752299909 : p1 += y[i] * x[-i];
646 10752299909 : if (p1 & HIGHBIT) p1 %= p;
647 : }
648 3439048178 : return p1 % p;
649 : }
650 :
651 : INLINE ulong
652 1172337689 : Flx_mullimb(GEN x, GEN y, ulong p, ulong pi, long a, long b)
653 : {
654 1172337689 : ulong p1 = 0;
655 : long i;
656 3716448311 : for (i=a; i<b; i++)
657 2542841596 : if (y[i])
658 2503136566 : p1 = Fl_addmul_pre(p1, y[i], x[-i], p, pi);
659 1173606715 : return p1;
660 : }
661 :
662 : /* assume nx >= ny > 0 */
663 : static GEN
664 345426784 : Flx_mulspec_basecase(GEN x, GEN y, ulong p, ulong pi, long nx, long ny)
665 : {
666 : long i,lz,nz;
667 : GEN z;
668 :
669 345426784 : lz = nx+ny+1; nz = lz-2;
670 345426784 : z = cgetg(lz, t_VECSMALL) + 2; /* x:y:z [i] = term of degree i */
671 345247326 : if (!pi)
672 : {
673 1161925531 : for (i=0; i<ny; i++)z[i] = Flx_mullimb_ok(x+i,y,p,0,i+1);
674 739509716 : for ( ; i<nx; i++) z[i] = Flx_mullimb_ok(x+i,y,p,0,ny);
675 905643861 : for ( ; i<nz; i++) z[i] = Flx_mullimb_ok(x+i,y,p,i-nx+1,ny);
676 : }
677 : else
678 : {
679 312239462 : for (i=0; i<ny; i++)z[i] = Flx_mullimb(x+i,y,p,pi,0,i+1);
680 213411115 : for ( ; i<nx; i++) z[i] = Flx_mullimb(x+i,y,p,pi,0,ny);
681 224873341 : for ( ; i<nz; i++) z[i] = Flx_mullimb(x+i,y,p,pi,i-nx+1,ny);
682 : }
683 345133234 : z -= 2; return Flx_renormalize(z, lz);
684 : }
685 :
686 : static GEN
687 12412 : int_to_Flx(GEN z, ulong p)
688 : {
689 12412 : long i, l = lgefint(z);
690 12412 : GEN x = cgetg(l, t_VECSMALL);
691 1068124 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
692 12409 : return Flx_renormalize(x, l);
693 : }
694 :
695 : INLINE GEN
696 10145 : Flx_mulspec_mulii(GEN a, GEN b, ulong p, long na, long nb)
697 : {
698 10145 : GEN z=muliispec(a,b,na,nb);
699 10147 : return int_to_Flx(z,p);
700 : }
701 :
702 : static GEN
703 468814 : Flx_to_int_halfspec(GEN a, long na)
704 : {
705 : long j;
706 468814 : long n = (na+1)>>1UL;
707 468814 : GEN V = cgetipos(2+n);
708 : GEN w;
709 1381543 : for (w = int_LSW(V), j=0; j+1<na; j+=2, w=int_nextW(w))
710 912730 : *w = a[j]|(a[j+1]<<BITS_IN_HALFULONG);
711 468813 : if (j<na)
712 319655 : *w = a[j];
713 468813 : return V;
714 : }
715 :
716 : static GEN
717 507265 : int_to_Flx_half(GEN z, ulong p)
718 : {
719 : long i;
720 507265 : long lx = (lgefint(z)-2)*2+2;
721 507265 : GEN w, x = cgetg(lx, t_VECSMALL);
722 1919061 : for (w = int_LSW(z), i=2; i<lx; i+=2, w=int_nextW(w))
723 : {
724 1411797 : x[i] = LOWWORD((ulong)*w)%p;
725 1411797 : x[i+1] = HIGHWORD((ulong)*w)%p;
726 : }
727 507264 : return Flx_renormalize(x, lx);
728 : }
729 :
730 : static GEN
731 5481 : Flx_mulspec_halfmulii(GEN a, GEN b, ulong p, long na, long nb)
732 : {
733 5481 : GEN A = Flx_to_int_halfspec(a,na);
734 5481 : GEN B = Flx_to_int_halfspec(b,nb);
735 5481 : GEN z = mulii(A,B);
736 5481 : return int_to_Flx_half(z,p);
737 : }
738 :
739 : static GEN
740 204549 : Flx_to_int_quartspec(GEN a, long na)
741 : {
742 : long j;
743 204549 : long n = (na+3)>>2UL;
744 204549 : GEN V = cgetipos(2+n);
745 : GEN w;
746 4378074 : for (w = int_LSW(V), j=0; j+3<na; j+=4, w=int_nextW(w))
747 4173523 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG))|(a[j+3]<<(3*BITS_IN_QUARTULONG));
748 204551 : switch (na-j)
749 : {
750 116239 : case 3:
751 116239 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG));
752 116239 : break;
753 34469 : case 2:
754 34469 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG);
755 34469 : break;
756 27346 : case 1:
757 27346 : *w = a[j];
758 27346 : break;
759 26497 : case 0:
760 26497 : break;
761 : }
762 204551 : return V;
763 : }
764 :
765 : static GEN
766 107385 : int_to_Flx_quart(GEN z, ulong p)
767 : {
768 : long i;
769 107385 : long lx = (lgefint(z)-2)*4+2;
770 107385 : GEN w, x = cgetg(lx, t_VECSMALL);
771 4874013 : for (w = int_LSW(z), i=2; i<lx; i+=4, w=int_nextW(w))
772 : {
773 4766628 : x[i] = LLQUARTWORD((ulong)*w)%p;
774 4766628 : x[i+1] = HLQUARTWORD((ulong)*w)%p;
775 4766628 : x[i+2] = LHQUARTWORD((ulong)*w)%p;
776 4766628 : x[i+3] = HHQUARTWORD((ulong)*w)%p;
777 : }
778 107385 : return Flx_renormalize(x, lx);
779 : }
780 :
781 : static GEN
782 97166 : Flx_mulspec_quartmulii(GEN a, GEN b, ulong p, long na, long nb)
783 : {
784 97166 : GEN A = Flx_to_int_quartspec(a,na);
785 97166 : GEN B = Flx_to_int_quartspec(b,nb);
786 97166 : GEN z = mulii(A,B);
787 97167 : return int_to_Flx_quart(z,p);
788 : }
789 :
790 : /*Eval x in 2^(k*BIL) in linear time, k==2 or 3*/
791 : static GEN
792 583219 : Flx_eval2BILspec(GEN x, long k, long l)
793 : {
794 583219 : long i, lz = k*l, ki;
795 583219 : GEN pz = cgetipos(2+lz);
796 16393255 : for (i=0; i < lz; i++)
797 15810036 : *int_W(pz,i) = 0UL;
798 8488237 : for (i=0, ki=0; i<l; i++, ki+=k)
799 7905018 : *int_W(pz,ki) = x[i];
800 583219 : return int_normalize(pz,0);
801 : }
802 :
803 : static GEN
804 298594 : Z_mod2BIL_Flx_2(GEN x, long d, ulong p)
805 : {
806 298594 : long i, offset, lm = lgefint(x)-2, l = d+3;
807 298594 : ulong pi = get_Fl_red(p);
808 298594 : GEN pol = cgetg(l, t_VECSMALL);
809 298594 : pol[1] = 0;
810 8020806 : for (i=0, offset=0; offset+1 < lm; i++, offset += 2)
811 7722212 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
812 298594 : if (offset < lm)
813 225746 : pol[i+2] = (*int_W(x,offset)) % p;
814 298594 : return Flx_renormalize(pol,l);
815 : }
816 :
817 : static GEN
818 0 : Z_mod2BIL_Flx_3(GEN x, long d, ulong p)
819 : {
820 0 : long i, offset, lm = lgefint(x)-2, l = d+3;
821 0 : ulong pi = get_Fl_red(p);
822 0 : GEN pol = cgetg(l, t_VECSMALL);
823 0 : pol[1] = 0;
824 0 : for (i=0, offset=0; offset+2 < lm; i++, offset += 3)
825 0 : pol[i+2] = remlll_pre(*int_W(x,offset+2), *int_W(x,offset+1),
826 0 : *int_W(x,offset), p, pi);
827 0 : if (offset+1 < lm)
828 0 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
829 0 : else if (offset < lm)
830 0 : pol[i+2] = (*int_W(x,offset)) % p;
831 0 : return Flx_renormalize(pol,l);
832 : }
833 :
834 : static GEN
835 295664 : Z_mod2BIL_Flx(GEN x, long bs, long d, ulong p)
836 : {
837 295664 : return bs==2 ? Z_mod2BIL_Flx_2(x, d, p): Z_mod2BIL_Flx_3(x, d, p);
838 : }
839 :
840 : static GEN
841 284166 : Flx_mulspec_mulii_inflate(GEN x, GEN y, long N, ulong p, long nx, long ny)
842 : {
843 284166 : pari_sp av = avma;
844 284166 : GEN z = mulii(Flx_eval2BILspec(x,N,nx), Flx_eval2BILspec(y,N,ny));
845 284166 : return gc_upto(av, Z_mod2BIL_Flx(z, N, nx+ny-2, p));
846 : }
847 :
848 : static GEN
849 20821484 : kron_pack_Flx_spec_bits(GEN x, long b, long l) {
850 : GEN y;
851 : long i;
852 20821484 : if (l == 0)
853 3429494 : return gen_0;
854 17391990 : y = cgetg(l + 1, t_VECSMALL);
855 817719088 : for(i = 1; i <= l; i++)
856 800330547 : y[i] = x[l - i];
857 17388541 : return nv_fromdigits_2k(y, b);
858 : }
859 :
860 : /* assume b < BITS_IN_LONG */
861 : static GEN
862 5658279 : kron_unpack_Flx_bits_narrow(GEN z, long b, ulong p) {
863 5658279 : GEN v = binary_2k_nv(z, b), x;
864 5658280 : long i, l = lg(v) + 1;
865 5658280 : x = cgetg(l, t_VECSMALL);
866 624769402 : for (i = 2; i < l; i++)
867 619110985 : x[i] = v[l - i] % p;
868 5658417 : return Flx_renormalize(x, l);
869 : }
870 :
871 : static GEN
872 5583988 : kron_unpack_Flx_bits_wide(GEN z, long b, ulong p, ulong pi) {
873 5583988 : GEN v = binary_2k(z, b), x, y;
874 5583189 : long i, l = lg(v) + 1, ly;
875 5583189 : x = cgetg(l, t_VECSMALL);
876 234606606 : for (i = 2; i < l; i++) {
877 229025653 : y = gel(v, l - i);
878 229025653 : ly = lgefint(y);
879 229025653 : switch (ly) {
880 6276268 : case 2: x[i] = 0; break;
881 29471675 : case 3: x[i] = *int_W_lg(y, 0, ly) % p; break;
882 177366298 : case 4: x[i] = remll_pre(*int_W_lg(y, 1, ly), *int_W_lg(y, 0, ly), p, pi); break;
883 31822085 : case 5: x[i] = remlll_pre(*int_W_lg(y, 2, ly), *int_W_lg(y, 1, ly),
884 15911412 : *int_W_lg(y, 0, ly), p, pi); break;
885 0 : default: x[i] = umodiu(gel(v, l - i), p);
886 : }
887 : }
888 5580953 : return Flx_renormalize(x, l);
889 : }
890 :
891 : static GEN
892 7268789 : Flx_mulspec_Kronecker(GEN A, GEN B, long b, ulong p, long lA, long lB)
893 : {
894 : GEN C, D;
895 7268789 : pari_sp av = avma;
896 7268789 : A = kron_pack_Flx_spec_bits(A, b, lA);
897 7274734 : B = kron_pack_Flx_spec_bits(B, b, lB);
898 7274855 : C = gc_INT(av, mulii(A, B));
899 7273721 : if (b < BITS_IN_LONG)
900 2069900 : D = kron_unpack_Flx_bits_narrow(C, b, p);
901 : else
902 : {
903 5203821 : ulong pi = get_Fl_red(p);
904 5203450 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
905 : }
906 7270903 : return D;
907 : }
908 :
909 : static GEN
910 690285 : Flx_sqrspec_Kronecker(GEN A, long b, ulong p, long lA)
911 : {
912 : GEN C, D;
913 690285 : A = kron_pack_Flx_spec_bits(A, b, lA);
914 690370 : C = sqri(A);
915 690383 : if (b < BITS_IN_LONG)
916 476924 : D = kron_unpack_Flx_bits_narrow(C, b, p);
917 : else
918 : {
919 213459 : ulong pi = get_Fl_red(p);
920 213459 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
921 : }
922 690354 : return D;
923 : }
924 :
925 : /* fast product (Karatsuba) of polynomials a,b. These are not real GENs, a+2,
926 : * b+2 were sent instead. na, nb = number of terms of a, b.
927 : * Only c, c0, c1, c2 are genuine GEN.
928 : */
929 : static GEN
930 382946565 : Flx_mulspec(GEN a, GEN b, ulong p, ulong pi, long na, long nb)
931 : {
932 : GEN a0,c,c0;
933 382946565 : long n0, n0a, i, v = 0;
934 : pari_sp av;
935 :
936 487826516 : while (na && !a[0]) { a++; na--; v++; }
937 569272055 : while (nb && !b[0]) { b++; nb--; v++; }
938 382946565 : if (na < nb) swapspec(a,b, na,nb);
939 382946565 : if (!nb) return pol0_Flx(0);
940 :
941 354593504 : av = avma;
942 354593504 : if (nb >= get_Fl_threshold(p, Flx_MUL_MULII_LIMIT, Flx_MUL2_MULII_LIMIT))
943 : {
944 7668970 : long m = maxbitcoeffpol(p,nb);
945 7665399 : switch (m)
946 : {
947 97166 : case BITS_IN_QUARTULONG:
948 97166 : return Flx_shiftip(av,Flx_mulspec_quartmulii(a,b,p,na,nb), v);
949 5481 : case BITS_IN_HALFULONG:
950 5481 : return Flx_shiftip(av,Flx_mulspec_halfmulii(a,b,p,na,nb), v);
951 10145 : case BITS_IN_LONG:
952 10145 : return Flx_shiftip(av,Flx_mulspec_mulii(a,b,p,na,nb), v);
953 284166 : case 2*BITS_IN_LONG:
954 284166 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,2,p,na,nb), v);
955 0 : case 3*BITS_IN_LONG:
956 0 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,3,p,na,nb), v);
957 7268441 : default:
958 7268441 : return Flx_shiftip(av,Flx_mulspec_Kronecker(a,b,m,p,na,nb), v);
959 : }
960 : }
961 347154449 : if (nb < get_Fl_threshold(p, Flx_MUL_KARATSUBA_LIMIT, Flx_MUL2_KARATSUBA_LIMIT))
962 345361576 : return Flx_shiftip(av,Flx_mulspec_basecase(a,b,p,pi,na,nb), v);
963 1811032 : i=(na>>1); n0=na-i; na=i;
964 1811032 : a0=a+n0; n0a=n0;
965 2579372 : while (n0a && !a[n0a-1]) n0a--;
966 :
967 1811032 : if (nb > n0)
968 : {
969 : GEN b0,c1,c2;
970 : long n0b;
971 :
972 1756902 : nb -= n0; b0 = b+n0; n0b = n0;
973 2837703 : while (n0b && !b[n0b-1]) n0b--;
974 1756902 : c = Flx_mulspec(a,b,p,pi,n0a,n0b);
975 1756902 : c0 = Flx_mulspec(a0,b0,p,pi,na,nb);
976 :
977 1756902 : c2 = Flx_addspec(a0,a,p,na,n0a);
978 1756902 : c1 = Flx_addspec(b0,b,p,nb,n0b);
979 :
980 1756902 : c1 = Flx_mul_pre(c1,c2,p,pi);
981 1756902 : c2 = Flx_add(c0,c,p);
982 :
983 1756902 : c2 = Flx_neg_inplace(c2,p);
984 1756902 : c2 = Flx_add(c1,c2,p);
985 1756902 : c0 = Flx_addshift(c0,c2 ,p, n0);
986 : }
987 : else
988 : {
989 54130 : c = Flx_mulspec(a,b,p,pi,n0a,nb);
990 54130 : c0 = Flx_mulspec(a0,b,p,pi,na,nb);
991 : }
992 1811032 : c0 = Flx_addshift(c0,c,p,n0);
993 1811032 : return Flx_shiftip(av,c0, v);
994 : }
995 :
996 : GEN
997 377418772 : Flx_mul_pre(GEN x, GEN y, ulong p, ulong pi)
998 : {
999 377418772 : GEN z = Flx_mulspec(x+2,y+2,p, pi, lgpol(x),lgpol(y));
1000 377463959 : z[1] = x[1]; return z;
1001 : }
1002 : GEN
1003 27778543 : Flx_mul(GEN x, GEN y, ulong p)
1004 27778543 : { return Flx_mul_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1005 :
1006 : static GEN
1007 282289883 : Flx_sqrspec_basecase(GEN x, ulong p, ulong pi, long nx)
1008 : {
1009 : long i, lz, nz;
1010 : ulong p1;
1011 : GEN z;
1012 :
1013 282289883 : if (!nx) return pol0_Flx(0);
1014 282289883 : lz = (nx << 1) + 1, nz = lz-2;
1015 282289883 : z = cgetg(lz, t_VECSMALL) + 2;
1016 281855252 : if (!pi)
1017 : {
1018 217106246 : z[0] = x[0]*x[0]%p;
1019 931620432 : for (i=1; i<nx; i++)
1020 : {
1021 714517821 : p1 = Flx_mullimb_ok(x+i,x,p,0, (i+1)>>1);
1022 714514186 : p1 <<= 1;
1023 714514186 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1024 714514186 : z[i] = p1 % p;
1025 : }
1026 935947981 : for ( ; i<nz; i++)
1027 : {
1028 718332658 : p1 = Flx_mullimb_ok(x+i,x,p,i-nx+1, (i+1)>>1);
1029 718845370 : p1 <<= 1;
1030 718845370 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1031 718845370 : z[i] = p1 % p;
1032 : }
1033 : }
1034 : else
1035 : {
1036 64749006 : z[0] = Fl_sqr_pre(x[0], p, pi);
1037 413183844 : for (i=1; i<nx; i++)
1038 : {
1039 348403048 : p1 = Flx_mullimb(x+i,x,p,pi,0, (i+1)>>1);
1040 348956543 : p1 = Fl_add(p1, p1, p);
1041 348452250 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1042 348332064 : z[i] = p1;
1043 : }
1044 413435158 : for ( ; i<nz; i++)
1045 : {
1046 348614951 : p1 = Flx_mullimb(x+i,x,p,pi,i-nx+1, (i+1)>>1);
1047 349381622 : p1 = Fl_add(p1, p1, p);
1048 348939311 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1049 348654362 : z[i] = p1;
1050 : }
1051 : }
1052 282435530 : z -= 2; return Flx_renormalize(z, lz);
1053 : }
1054 :
1055 : static GEN
1056 2262 : Flx_sqrspec_sqri(GEN a, ulong p, long na)
1057 : {
1058 2262 : GEN z=sqrispec(a,na);
1059 2265 : return int_to_Flx(z,p);
1060 : }
1061 :
1062 : static GEN
1063 324 : Flx_sqrspec_halfsqri(GEN a, ulong p, long na)
1064 : {
1065 324 : GEN z = sqri(Flx_to_int_halfspec(a,na));
1066 325 : return int_to_Flx_half(z,p);
1067 : }
1068 :
1069 : static GEN
1070 10218 : Flx_sqrspec_quartsqri(GEN a, ulong p, long na)
1071 : {
1072 10218 : GEN z = sqri(Flx_to_int_quartspec(a,na));
1073 10218 : return int_to_Flx_quart(z,p);
1074 : }
1075 :
1076 : static GEN
1077 11498 : Flx_sqrspec_sqri_inflate(GEN x, long N, ulong p, long nx)
1078 : {
1079 11498 : pari_sp av = avma;
1080 11498 : GEN z = sqri(Flx_eval2BILspec(x,N,nx));
1081 11498 : return gc_upto(av, Z_mod2BIL_Flx(z, N, (nx-1)*2, p));
1082 : }
1083 :
1084 : static GEN
1085 283077987 : Flx_sqrspec(GEN a, ulong p, ulong pi, long na)
1086 : {
1087 : GEN a0, c, c0;
1088 283077987 : long n0, n0a, i, v = 0, m;
1089 : pari_sp av;
1090 :
1091 405605339 : while (na && !a[0]) { a++; na--; v += 2; }
1092 283077987 : if (!na) return pol0_Flx(0);
1093 :
1094 282832123 : av = avma;
1095 282832123 : if (na >= get_Fl_threshold(p, Flx_SQR_SQRI_LIMIT, Flx_SQR2_SQRI_LIMIT))
1096 : {
1097 714551 : m = maxbitcoeffpol(p,na);
1098 714583 : switch(m)
1099 : {
1100 10218 : case BITS_IN_QUARTULONG:
1101 10218 : return Flx_shiftip(av, Flx_sqrspec_quartsqri(a,p,na), v);
1102 324 : case BITS_IN_HALFULONG:
1103 324 : return Flx_shiftip(av, Flx_sqrspec_halfsqri(a,p,na), v);
1104 2262 : case BITS_IN_LONG:
1105 2262 : return Flx_shiftip(av, Flx_sqrspec_sqri(a,p,na), v);
1106 11498 : case 2*BITS_IN_LONG:
1107 11498 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,2,p,na), v);
1108 0 : case 3*BITS_IN_LONG:
1109 0 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,3,p,na), v);
1110 690281 : default:
1111 690281 : return Flx_shiftip(av, Flx_sqrspec_Kronecker(a,m,p,na), v);
1112 : }
1113 : }
1114 282117850 : if (na < get_Fl_threshold(p, Flx_SQR_KARATSUBA_LIMIT, Flx_SQR2_KARATSUBA_LIMIT))
1115 281985706 : return Flx_shiftip(av, Flx_sqrspec_basecase(a,p,pi,na), v);
1116 58353 : i=(na>>1); n0=na-i; na=i;
1117 58353 : a0=a+n0; n0a=n0;
1118 73095 : while (n0a && !a[n0a-1]) n0a--;
1119 :
1120 58353 : c = Flx_sqrspec(a,p,pi,n0a);
1121 58353 : c0= Flx_sqrspec(a0,p,pi,na);
1122 58353 : if (p == 2) n0 *= 2;
1123 : else
1124 : {
1125 58334 : GEN c1, t = Flx_addspec(a0,a,p,na,n0a);
1126 58334 : t = Flx_sqr_pre(t,p,pi);
1127 58334 : c1= Flx_add(c0,c, p);
1128 58334 : c1= Flx_sub(t, c1, p);
1129 58334 : c0 = Flx_addshift(c0,c1,p,n0);
1130 : }
1131 58353 : c0 = Flx_addshift(c0,c,p,n0);
1132 58353 : return Flx_shiftip(av,c0,v);
1133 : }
1134 :
1135 : GEN
1136 283004198 : Flx_sqr_pre(GEN x, ulong p, ulong pi)
1137 : {
1138 283004198 : GEN z = Flx_sqrspec(x+2,p, pi, lgpol(x));
1139 283575578 : z[1] = x[1]; return z;
1140 : }
1141 : GEN
1142 354675 : Flx_sqr(GEN x, ulong p)
1143 354675 : { return Flx_sqr_pre(x, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1144 :
1145 : GEN
1146 7921 : Flx_powu_pre(GEN x, ulong n, ulong p, ulong pi)
1147 : {
1148 7921 : GEN y = pol1_Flx(x[1]), z;
1149 : ulong m;
1150 7919 : if (n == 0) return y;
1151 7919 : m = n; z = x;
1152 : for (;;)
1153 : {
1154 30550 : if (m&1UL) y = Flx_mul_pre(y,z, p, pi);
1155 30556 : m >>= 1; if (!m) return y;
1156 22640 : z = Flx_sqr_pre(z, p, pi);
1157 : }
1158 : }
1159 : GEN
1160 0 : Flx_powu(GEN x, ulong n, ulong p)
1161 : {
1162 0 : if (n == 0) return pol1_Flx(x[1]);
1163 0 : return Flx_powu_pre(x, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p));
1164 : }
1165 :
1166 : GEN
1167 14047 : Flx_halve(GEN y, ulong p)
1168 : {
1169 : GEN z;
1170 : long i, l;
1171 14047 : z = cgetg_copy(y, &l); z[1] = y[1];
1172 58609 : for(i=2; i<l; i++) uel(z,i) = Fl_halve(uel(y,i), p);
1173 14047 : return z;
1174 : }
1175 :
1176 : static GEN
1177 7294683 : Flx_recipspec(GEN x, long l, long n)
1178 : {
1179 : long i;
1180 7294683 : GEN z=cgetg(n+2,t_VECSMALL)+2;
1181 116521612 : for(i=0; i<l; i++)
1182 109228141 : z[n-i-1] = x[i];
1183 15943093 : for( ; i<n; i++)
1184 8649622 : z[n-i-1] = 0;
1185 7293471 : return Flx_renormalize(z-2,n+2);
1186 : }
1187 :
1188 : GEN
1189 0 : Flx_recip(GEN x)
1190 : {
1191 0 : GEN z=Flx_recipspec(x+2,lgpol(x),lgpol(x));
1192 0 : z[1]=x[1];
1193 0 : return z;
1194 : }
1195 :
1196 : /* Return P(x * h) */
1197 : GEN
1198 0 : Flx_unscale(GEN P, ulong h, ulong p)
1199 : {
1200 : long i, l;
1201 0 : ulong hi = 1UL;
1202 0 : GEN Q = cgetg_copy(P, &l);
1203 0 : Q[1] = P[1];
1204 0 : if (l == 2) return Q;
1205 0 : uel(Q,2) = uel(P,2);
1206 0 : for (i=3; i<l; i++)
1207 : {
1208 0 : hi = Fl_mul(hi, h ,p);
1209 0 : uel(Q,i) = Fl_mul(uel(P,i), hi, p);
1210 : }
1211 0 : return Q;
1212 : }
1213 : /* Return h^degpol(P) P(x / h) */
1214 : GEN
1215 1117 : Flx_rescale(GEN P, ulong h, ulong p)
1216 : {
1217 1117 : long i, l = lg(P);
1218 1117 : GEN Q = cgetg(l,t_VECSMALL);
1219 1117 : ulong hi = h;
1220 1117 : Q[l-1] = P[l-1];
1221 12538 : for (i=l-2; i>=2; i--)
1222 : {
1223 12538 : Q[i] = Fl_mul(P[i], hi, p);
1224 12538 : if (i == 2) break;
1225 11421 : hi = Fl_mul(hi,h, p);
1226 : }
1227 1117 : Q[1] = P[1]; return Q;
1228 : }
1229 :
1230 : /* x/polrecip(P)+O(x^n); allow pi = 0 */
1231 : static GEN
1232 134328 : Flx_invBarrett_basecase(GEN T, ulong p, ulong pi)
1233 : {
1234 134328 : long i, l=lg(T)-1, lr=l-1, k;
1235 134328 : GEN r=cgetg(lr,t_VECSMALL); r[1] = T[1];
1236 134328 : r[2] = 1;
1237 134328 : if (!pi)
1238 768031 : for (i=3;i<lr;i++)
1239 : {
1240 760982 : ulong u = uel(T, l-i+2);
1241 45653640 : for (k=3; k<i; k++)
1242 44892658 : { u += uel(T,l-i+k) * uel(r, k); if (u & HIGHBIT) u %= p; }
1243 760982 : r[i] = Fl_neg(u % p, p);
1244 : }
1245 : else
1246 2110281 : for (i=3;i<lr;i++)
1247 : {
1248 1983005 : ulong u = Fl_neg(uel(T,l-i+2), p);
1249 59531981 : for (k=3; k<i; k++)
1250 : {
1251 57548979 : ulong t = Fl_neg(uel(T,l-i+k), p);
1252 57548979 : u = Fl_addmul_pre(u, t, uel(r,k), p, pi);
1253 : }
1254 1983002 : r[i] = u;
1255 : }
1256 134325 : return Flx_renormalize(r,lr);
1257 : }
1258 :
1259 : /* Return new lgpol */
1260 : static long
1261 2133831 : Flx_lgrenormalizespec(GEN x, long lx)
1262 : {
1263 : long i;
1264 7479161 : for (i = lx-1; i>=0; i--)
1265 7478326 : if (x[i]) break;
1266 2133831 : return i+1;
1267 : }
1268 : /* allow pi = 0 */
1269 : static GEN
1270 23161 : Flx_invBarrett_Newton(GEN T, ulong p, ulong pi)
1271 : {
1272 23161 : long nold, lx, lz, lq, l = degpol(T), lQ;
1273 23161 : GEN q, y, z, x = zero_zv(l+1) + 2;
1274 23161 : ulong mask = quadratic_prec_mask(l-2); /* assume l > 2 */
1275 : pari_sp av;
1276 :
1277 23161 : y = T+2;
1278 23161 : q = Flx_recipspec(y,l+1,l+1); lQ = lgpol(q); q+=2;
1279 23160 : av = avma;
1280 : /* We work on _spec_ Flx's, all the l[xzq12] below are lgpol's */
1281 :
1282 : /* initialize */
1283 23160 : x[0] = Fl_inv(q[0], p);
1284 23160 : if (lQ>1 && q[1])
1285 5109 : {
1286 5109 : ulong u = q[1];
1287 5109 : if (x[0] != 1) u = Fl_mul(u, Fl_sqr(x[0],p), p);
1288 5109 : x[1] = p - u; lx = 2;
1289 : }
1290 : else
1291 18051 : lx = 1;
1292 23160 : nold = 1;
1293 159201 : for (; mask > 1; set_avma(av))
1294 : { /* set x -= x(x*q - 1) + O(t^(nnew + 1)), knowing x*q = 1 + O(t^(nold+1)) */
1295 136051 : long i, lnew, nnew = nold << 1;
1296 :
1297 136051 : if (mask & 1) nnew--;
1298 136051 : mask >>= 1;
1299 :
1300 136051 : lnew = nnew + 1;
1301 136051 : lq = Flx_lgrenormalizespec(q, minss(lQ, lnew));
1302 136049 : z = Flx_mulspec(x, q, p, pi, lx, lq); /* FIXME: high product */
1303 136035 : lz = lgpol(z); if (lz > lnew) lz = lnew;
1304 136037 : z += 2;
1305 : /* subtract 1 [=>first nold words are 0]: renormalize so that z(0) != 0 */
1306 298856 : for (i = nold; i < lz; i++) if (z[i]) break;
1307 136037 : nold = nnew;
1308 136037 : if (i >= lz) continue; /* z-1 = 0(t^(nnew + 1)) */
1309 :
1310 : /* z + i represents (x*q - 1) / t^i */
1311 100987 : lz = Flx_lgrenormalizespec (z+i, lz-i);
1312 100987 : z = Flx_mulspec(x, z+i, p, pi, lx, lz); /* FIXME: low product */
1313 100989 : lz = lgpol(z); z += 2;
1314 100989 : if (lz > lnew-i) lz = Flx_lgrenormalizespec(z, lnew-i);
1315 :
1316 100989 : lx = lz+ i;
1317 100989 : y = x + i; /* x -= z * t^i, in place */
1318 999750 : for (i = 0; i < lz; i++) y[i] = Fl_neg(z[i], p);
1319 : }
1320 23161 : x -= 2; setlg(x, lx + 2); x[1] = T[1];
1321 23161 : return x;
1322 : }
1323 :
1324 : /* allow pi = 0 */
1325 : static GEN
1326 158789 : Flx_invBarrett_pre(GEN T, ulong p, ulong pi)
1327 : {
1328 158789 : pari_sp ltop = avma;
1329 158789 : long l = lgpol(T);
1330 : GEN r;
1331 158789 : if (l < 3) return pol0_Flx(T[1]);
1332 157489 : if (l < get_Fl_threshold(p, Flx_INVBARRETT_LIMIT, Flx_INVBARRETT2_LIMIT))
1333 : {
1334 134328 : ulong c = T[l+1];
1335 134328 : if (c != 1)
1336 : {
1337 98118 : ulong ci = Fl_inv(c,p);
1338 98118 : T = Flx_Fl_mul_pre(T, ci, p, pi);
1339 98118 : r = Flx_invBarrett_basecase(T, p, pi);
1340 98117 : r = Flx_Fl_mul_pre(r, ci, p, pi);
1341 : }
1342 : else
1343 36210 : r = Flx_invBarrett_basecase(T, p, pi);
1344 : }
1345 : else
1346 23161 : r = Flx_invBarrett_Newton(T, p, pi);
1347 157488 : return gc_leaf(ltop, r);
1348 : }
1349 : GEN
1350 0 : Flx_invBarrett(GEN T, ulong p)
1351 0 : { return Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1352 :
1353 : /* allow pi = 0 */
1354 : GEN
1355 99358324 : Flx_get_red_pre(GEN T, ulong p, ulong pi)
1356 : {
1357 99358324 : if (typ(T)!=t_VECSMALL
1358 99321926 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1359 : Flx_BARRETT2_LIMIT))
1360 99337055 : return T;
1361 7616 : retmkvec2(Flx_invBarrett_pre(T, p, pi),T);
1362 : }
1363 : GEN
1364 14490736 : Flx_get_red(GEN T, ulong p)
1365 : {
1366 14490736 : if (typ(T)!=t_VECSMALL
1367 14490638 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1368 : Flx_BARRETT2_LIMIT))
1369 14484947 : return T;
1370 5194 : retmkvec2(Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)),T);
1371 : }
1372 :
1373 : /* separate from Flx_divrem for maximal speed. */
1374 : static GEN
1375 798246995 : Flx_rem_basecase(GEN x, GEN y, ulong p, ulong pi)
1376 : {
1377 : pari_sp av;
1378 : GEN z, c;
1379 : long dx,dy,dy1,dz,i,j;
1380 : ulong p1,inv;
1381 798246995 : long vs=x[1];
1382 :
1383 798246995 : dy = degpol(y); if (!dy) return pol0_Flx(x[1]);
1384 761675600 : dx = degpol(x);
1385 761613054 : dz = dx-dy; if (dz < 0) return Flx_copy(x);
1386 761613054 : x += 2; y += 2;
1387 761613054 : inv = y[dy];
1388 761613054 : if (inv != 1UL) inv = Fl_inv(inv,p);
1389 912458046 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1390 :
1391 763162244 : c = cgetg(dy+3, t_VECSMALL); c[1]=vs; c += 2; av=avma;
1392 761888661 : z = cgetg(dz+3, t_VECSMALL); z[1]=vs; z += 2;
1393 :
1394 760449213 : if (!pi)
1395 : {
1396 488568515 : z[dz] = (inv*x[dx]) % p;
1397 1833138421 : for (i=dx-1; i>=dy; --i)
1398 : {
1399 1344569906 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1400 10555541726 : for (j=i-dy1; j<=i && j<=dz; j++)
1401 : {
1402 9210971820 : p1 += z[j]*y[i-j];
1403 9210971820 : if (p1 & HIGHBIT) p1 %= p;
1404 : }
1405 1344569906 : p1 %= p;
1406 1344569906 : z[i-dy] = p1? ((p - p1)*inv) % p: 0;
1407 : }
1408 3327201860 : for (i=0; i<dy; i++)
1409 : {
1410 2838946510 : p1 = z[0]*y[i];
1411 14584478115 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1412 : {
1413 11745531605 : p1 += z[j]*y[i-j];
1414 11745531605 : if (p1 & HIGHBIT) p1 %= p;
1415 : }
1416 2839009022 : c[i] = Fl_sub(x[i], p1%p, p);
1417 : }
1418 : }
1419 : else
1420 : {
1421 271880698 : z[dz] = Fl_mul_pre(inv, x[dx], p, pi);
1422 838079759 : for (i=dx-1; i>=dy; --i)
1423 : {
1424 566023605 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1425 2373935179 : for (j=i-dy1; j<=i && j<=dz; j++)
1426 1807636921 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1427 566298258 : z[i-dy] = p1? Fl_mul_pre(p - p1, inv, p, pi): 0;
1428 : }
1429 2012345028 : for (i=0; i<dy; i++)
1430 : {
1431 1741294872 : p1 = Fl_mul_pre(z[0],y[i],p,pi);
1432 4703864017 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1433 2953442014 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1434 1727527077 : c[i] = Fl_sub(x[i], p1, p);
1435 : }
1436 : }
1437 927716065 : i = dy-1; while (i>=0 && !c[i]) i--;
1438 759305506 : set_avma(av); return Flx_renormalize(c-2, i+3);
1439 : }
1440 :
1441 : /* as FpX_divrem but working only on ulong types.
1442 : * if relevant, *pr is the last object on stack */
1443 : static GEN
1444 62270297 : Flx_divrem_basecase(GEN x, GEN y, ulong p, ulong pi, GEN *pr)
1445 : {
1446 : GEN z,q,c;
1447 : long dx,dy,dy1,dz,i,j;
1448 : ulong p1,inv;
1449 62270297 : long sv=x[1];
1450 :
1451 62270297 : dy = degpol(y);
1452 62267703 : if (dy<0) pari_err_INV("Flx_divrem",y);
1453 62267898 : if (pr == ONLY_REM) return Flx_rem_basecase(x, y, p, pi);
1454 62267500 : if (!dy)
1455 : {
1456 7154571 : if (pr && pr != ONLY_DIVIDES) *pr = pol0_Flx(sv);
1457 7154541 : if (y[2] == 1UL) return Flx_copy(x);
1458 5146162 : return Flx_Fl_mul_pre(x, Fl_inv(y[2], p), p, pi);
1459 : }
1460 55112929 : dx = degpol(x);
1461 55116105 : dz = dx-dy;
1462 55116105 : if (dz < 0)
1463 : {
1464 1057446 : q = pol0_Flx(sv);
1465 1057449 : if (pr && pr != ONLY_DIVIDES) *pr = Flx_copy(x);
1466 1057448 : return q;
1467 : }
1468 54058659 : x += 2;
1469 54058659 : y += 2;
1470 54058659 : z = cgetg(dz + 3, t_VECSMALL); z[1] = sv; z += 2;
1471 54057941 : inv = uel(y, dy);
1472 54057941 : if (inv != 1UL) inv = Fl_inv(inv,p);
1473 79414081 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1474 :
1475 54061265 : if (SMALL_ULONG(p))
1476 : {
1477 52179081 : z[dz] = (inv*x[dx]) % p;
1478 132391522 : for (i=dx-1; i>=dy; --i)
1479 : {
1480 80212441 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1481 259338122 : for (j=i-dy1; j<=i && j<=dz; j++)
1482 : {
1483 179125681 : p1 += z[j]*y[i-j];
1484 179125681 : if (p1 & HIGHBIT) p1 %= p;
1485 : }
1486 80212441 : p1 %= p;
1487 80212441 : z[i-dy] = p1? (long) ((p - p1)*inv) % p: 0;
1488 : }
1489 : }
1490 : else
1491 : {
1492 1882184 : z[dz] = Fl_mul(inv, x[dx], p);
1493 9261076 : for (i=dx-1; i>=dy; --i)
1494 : { /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1495 7378865 : p1 = p - uel(x,i);
1496 26402831 : for (j=i-dy1; j<=i && j<=dz; j++)
1497 19023972 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1498 7378859 : z[i-dy] = p1? Fl_mul(p - p1, inv, p): 0;
1499 : }
1500 : }
1501 54061292 : q = Flx_renormalize(z-2, dz+3);
1502 54060318 : if (!pr) return q;
1503 :
1504 26570173 : c = cgetg(dy + 3, t_VECSMALL); c[1] = sv; c += 2;
1505 26572278 : if (SMALL_ULONG(p))
1506 : {
1507 229369544 : for (i=0; i<dy; i++)
1508 : {
1509 204437729 : p1 = (ulong)z[0]*y[i];
1510 479702467 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1511 : {
1512 275264738 : p1 += (ulong)z[j]*y[i-j];
1513 275264738 : if (p1 & HIGHBIT) p1 %= p;
1514 : }
1515 204437157 : c[i] = Fl_sub(x[i], p1%p, p);
1516 : }
1517 : }
1518 : else
1519 : {
1520 16047961 : for (i=0; i<dy; i++)
1521 : {
1522 14408537 : p1 = Fl_mul(z[0],y[i],p);
1523 50253544 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1524 35845006 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1525 14408541 : c[i] = Fl_sub(x[i], p1, p);
1526 : }
1527 : }
1528 35797299 : i=dy-1; while (i>=0 && !c[i]) i--;
1529 26571239 : c = Flx_renormalize(c-2, i+3);
1530 26572451 : if (pr == ONLY_DIVIDES)
1531 453 : { if (lg(c) != 2) return NULL; }
1532 : else
1533 26571998 : *pr = c;
1534 26572304 : return q;
1535 : }
1536 :
1537 : /* Compute x mod T where 2 <= degpol(T) <= l+1 <= 2*(degpol(T)-1)
1538 : * and mg is the Barrett inverse of T. */
1539 : static GEN
1540 905749 : Flx_divrem_Barrettspec(GEN x, long l, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1541 : {
1542 : GEN q, r;
1543 905749 : long lt = degpol(T); /*We discard the leading term*/
1544 : long ld, lm, lT, lmg;
1545 905725 : ld = l-lt;
1546 905725 : lm = minss(ld, lgpol(mg));
1547 906007 : lT = Flx_lgrenormalizespec(T+2,lt);
1548 906143 : lmg = Flx_lgrenormalizespec(mg+2,lm);
1549 905952 : q = Flx_recipspec(x+lt,ld,ld); /* q = rec(x) lz<=ld*/
1550 905548 : q = Flx_mulspec(q+2,mg+2,p,pi,lgpol(q),lmg); /* q = rec(x) * mg lz<=ld+lm*/
1551 906118 : q = Flx_recipspec(q+2,minss(ld,lgpol(q)),ld);/* q = rec (rec(x) * mg) lz<=ld*/
1552 905608 : if (!pr) return q;
1553 897902 : r = Flx_mulspec(q+2,T+2,p,pi,lgpol(q),lT); /* r = q*pol lz<=ld+lt*/
1554 898400 : r = Flx_subspec(x,r+2,p,lt,minss(lt,lgpol(r)));/* r = x - q*pol lz<=lt */
1555 898105 : if (pr == ONLY_REM) return r;
1556 428241 : *pr = r; return q;
1557 : }
1558 :
1559 : static GEN
1560 605029 : Flx_divrem_Barrett(GEN x, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1561 : {
1562 605029 : GEN q = NULL, r = Flx_copy(x);
1563 605058 : long l = lgpol(x), lt = degpol(T), lm = 2*lt-1, v = T[1];
1564 : long i;
1565 605060 : if (l <= lt)
1566 : {
1567 0 : if (pr == ONLY_REM) return Flx_copy(x);
1568 0 : if (pr == ONLY_DIVIDES) return lgpol(x)? NULL: pol0_Flx(v);
1569 0 : if (pr) *pr = Flx_copy(x);
1570 0 : return pol0_Flx(v);
1571 : }
1572 605060 : if (lt <= 1)
1573 1300 : return Flx_divrem_basecase(x,T,p,pi,pr);
1574 603760 : if (pr != ONLY_REM && l>lm)
1575 28972 : { q = zero_zv(l-lt+1); q[1] = T[1]; }
1576 907364 : while (l>lm)
1577 : {
1578 303645 : GEN zr, zq = Flx_divrem_Barrettspec(r+2+l-lm,lm,mg,T,p,pi,&zr);
1579 303658 : long lz = lgpol(zr);
1580 303604 : if (pr != ONLY_REM)
1581 : {
1582 58215 : long lq = lgpol(zq);
1583 884279 : for(i=0; i<lq; i++) q[2+l-lm+i] = zq[2+i];
1584 : }
1585 4410969 : for(i=0; i<lz; i++) r[2+l-lm+i] = zr[2+i];
1586 303604 : l = l-lm+lz;
1587 : }
1588 603719 : if (pr == ONLY_REM)
1589 : {
1590 469911 : if (l > lt)
1591 469869 : r = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi,ONLY_REM);
1592 : else
1593 42 : r = Flx_renormalize(r, l+2);
1594 469907 : r[1] = v; return r;
1595 : }
1596 133808 : if (l > lt)
1597 : {
1598 132268 : GEN zq = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi, pr ? &r: NULL);
1599 132268 : if (!q) q = zq;
1600 : else
1601 : {
1602 27392 : long lq = lgpol(zq);
1603 160012 : for(i=0; i<lq; i++) q[2+i] = zq[2+i];
1604 : }
1605 : }
1606 1540 : else if (pr)
1607 1541 : r = Flx_renormalize(r, l+2);
1608 133808 : q[1] = v; q = Flx_renormalize(q, lg(q));
1609 133848 : if (pr == ONLY_DIVIDES) return lgpol(r)? NULL: q;
1610 133848 : if (pr) { r[1] = v; *pr = r; }
1611 133848 : return q;
1612 : }
1613 :
1614 : /* allow pi = 0 (SMALL_ULONG) */
1615 : GEN
1616 79710721 : Flx_divrem_pre(GEN x, GEN T, ulong p, ulong pi, GEN *pr)
1617 : {
1618 : GEN B, y;
1619 : long dy, dx, d;
1620 79710721 : if (pr==ONLY_REM) return Flx_rem_pre(x, T, p, pi);
1621 62394315 : y = get_Flx_red(T, &B);
1622 62406188 : dy = degpol(y); dx = degpol(x); d = dx-dy;
1623 62401757 : if (!B && d+3 < get_Fl_threshold(p, Flx_DIVREM_BARRETT_LIMIT,Flx_DIVREM2_BARRETT_LIMIT))
1624 62267488 : return Flx_divrem_basecase(x,y,p,pi,pr);
1625 : else
1626 : {
1627 134750 : pari_sp av = avma;
1628 134750 : GEN mg = B? B: Flx_invBarrett_pre(y, p, pi);
1629 134750 : GEN q1 = Flx_divrem_Barrett(x,mg,y,p,pi,pr);
1630 134750 : if (!q1) return gc_NULL(av);
1631 134750 : if (!pr || pr==ONLY_DIVIDES) return gc_leaf(av, q1);
1632 126450 : return gc_all(av, 2, &q1, pr);
1633 : }
1634 : }
1635 : GEN
1636 30447809 : Flx_divrem(GEN x, GEN T, ulong p, GEN *pr)
1637 30447809 : { return Flx_divrem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p), pr); }
1638 :
1639 : GEN
1640 922980927 : Flx_rem_pre(GEN x, GEN T, ulong p, ulong pi)
1641 : {
1642 922980927 : GEN B, y = get_Flx_red(T, &B);
1643 922881907 : long d = degpol(x) - degpol(y);
1644 922666425 : if (d < 0) return Flx_copy(x);
1645 799102565 : if (!B && d+3 < get_Fl_threshold(p, Flx_REM_BARRETT_LIMIT,Flx_REM2_BARRETT_LIMIT))
1646 798380334 : return Flx_rem_basecase(x,y,p, pi);
1647 : else
1648 : {
1649 470279 : pari_sp av=avma;
1650 470279 : GEN mg = B ? B: Flx_invBarrett_pre(y, p, pi);
1651 470278 : GEN r = Flx_divrem_Barrett(x, mg, y, p, pi, ONLY_REM);
1652 470302 : return gc_leaf(av, r);
1653 : }
1654 : }
1655 : GEN
1656 42213601 : Flx_rem(GEN x, GEN T, ulong p)
1657 42213601 : { return Flx_rem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1658 :
1659 : /* reduce T mod (X^n - 1, p). Shallow function */
1660 : GEN
1661 5177170 : Flx_mod_Xnm1(GEN T, ulong n, ulong p)
1662 : {
1663 5177170 : long i, j, L = lg(T), l = n+2;
1664 : GEN S;
1665 5177170 : if (L <= l || n & ~LGBITS) return T;
1666 3529 : S = cgetg(l, t_VECSMALL);
1667 3529 : S[1] = T[1];
1668 15097 : for (i = 2; i < l; i++) S[i] = T[i];
1669 9755 : for (j = 2; i < L; i++) {
1670 6226 : S[j] = Fl_add(S[j], T[i], p);
1671 6226 : if (++j == l) j = 2;
1672 : }
1673 3529 : return Flx_renormalize(S, l);
1674 : }
1675 : /* reduce T mod (X^n + 1, p). Shallow function */
1676 : GEN
1677 31764 : Flx_mod_Xn1(GEN T, ulong n, ulong p)
1678 : {
1679 31764 : long i, j, L = lg(T), l = n+2, s = -1;
1680 : GEN S;
1681 31764 : if (L <= l || n & ~LGBITS) return T;
1682 2740 : S = cgetg(l, t_VECSMALL);
1683 2740 : S[1] = T[1];
1684 12032 : for (i = 2; i < l; i++) S[i] = T[i];
1685 7267 : for (j = 2; i < L; i++) {
1686 4527 : S[j] = s==-1 ? Fl_sub(S[j], T[i], p): Fl_add(S[j], T[i], p);
1687 4527 : if (++j == l) { j = 2; s = -s; }
1688 : }
1689 2740 : return Flx_renormalize(S, l);
1690 : }
1691 :
1692 : struct _Flxq {
1693 : GEN aut, T;
1694 : ulong p, pi;
1695 : };
1696 : /* allow pi = 0 */
1697 : static void
1698 71616653 : set_Flxq_pre(struct _Flxq *D, GEN T, ulong p, ulong pi)
1699 : {
1700 71616653 : D->p = p;
1701 71616653 : D->pi = pi;
1702 71616653 : D->T = Flx_get_red_pre(T, p, pi);
1703 71610531 : }
1704 : static void
1705 68922 : set_Flxq(struct _Flxq *D, GEN T, ulong p)
1706 68922 : { set_Flxq_pre(D, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1707 :
1708 : static GEN
1709 0 : _Flx_divrem(void * E, GEN x, GEN y, GEN *r)
1710 : {
1711 0 : struct _Flxq *D = (struct _Flxq*) E;
1712 0 : return Flx_divrem_pre(x, y, D->p, D->pi, r);
1713 : }
1714 : static GEN
1715 389834 : _Flx_add(void * E, GEN x, GEN y) {
1716 389834 : struct _Flxq *D = (struct _Flxq*) E;
1717 389834 : return Flx_add(x, y, D->p);
1718 : }
1719 : static GEN
1720 11022915 : _Flx_mul(void *E, GEN x, GEN y) {
1721 11022915 : struct _Flxq *D = (struct _Flxq*) E;
1722 11022915 : return Flx_mul_pre(x, y, D->p, D->pi);
1723 : }
1724 : static GEN
1725 0 : _Flx_sqr(void *E, GEN x) {
1726 0 : struct _Flxq *D = (struct _Flxq*) E;
1727 0 : return Flx_sqr_pre(x, D->p, D->pi);
1728 : }
1729 :
1730 : static struct bb_ring Flx_ring = { _Flx_add,_Flx_mul,_Flx_sqr };
1731 :
1732 : GEN
1733 0 : Flx_digits(GEN x, GEN T, ulong p)
1734 : {
1735 : struct _Flxq D;
1736 0 : long d = degpol(T), n = (lgpol(x)+d-1)/d;
1737 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1738 0 : return gen_digits(x,T,n,(void *)&D, &Flx_ring, _Flx_divrem);
1739 : }
1740 :
1741 : GEN
1742 0 : FlxV_Flx_fromdigits(GEN x, GEN T, ulong p)
1743 : {
1744 : struct _Flxq D;
1745 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1746 0 : return gen_fromdigits(x,T,(void *)&D, &Flx_ring);
1747 : }
1748 :
1749 : long
1750 4471824 : Flx_val(GEN x)
1751 : {
1752 4471824 : long i, l=lg(x);
1753 4471824 : if (l==2) return LONG_MAX;
1754 4481010 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1755 4471824 : return i-2;
1756 : }
1757 : long
1758 27592951 : Flx_valrem(GEN x, GEN *Z)
1759 : {
1760 27592951 : long v, i, l=lg(x);
1761 : GEN y;
1762 27592951 : if (l==2) { *Z = Flx_copy(x); return LONG_MAX; }
1763 29771342 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1764 27592951 : v = i-2;
1765 27592951 : if (v == 0) { *Z = x; return 0; }
1766 1021401 : l -= v;
1767 1021401 : y = cgetg(l, t_VECSMALL); y[1] = x[1];
1768 2629562 : for (i=2; i<l; i++) y[i] = x[i+v];
1769 1024242 : *Z = y; return v;
1770 : }
1771 :
1772 : GEN
1773 21861388 : Flx_deriv(GEN z, ulong p)
1774 : {
1775 21861388 : long i,l = lg(z)-1;
1776 : GEN x;
1777 21861388 : if (l < 2) l = 2;
1778 21861388 : x = cgetg(l, t_VECSMALL); x[1] = z[1]; z++;
1779 21859580 : if (HIGHWORD(l | p))
1780 57499186 : for (i=2; i<l; i++) x[i] = Fl_mul((ulong)i-1, z[i], p);
1781 : else
1782 87931573 : for (i=2; i<l; i++) x[i] = ((i-1) * z[i]) % p;
1783 21859649 : return Flx_renormalize(x,l);
1784 : }
1785 :
1786 : static GEN
1787 422841 : Flx_integXn(GEN x, long n, ulong p)
1788 : {
1789 422841 : long i, lx = lg(x);
1790 : GEN y;
1791 422841 : if (lx == 2) return Flx_copy(x);
1792 413029 : y = cgetg(lx, t_VECSMALL); y[1] = x[1];
1793 2097011 : for (i=2; i<lx; i++)
1794 : {
1795 1683436 : ulong xi = uel(x,i);
1796 1683436 : if (xi == 0)
1797 13345 : uel(y,i) = 0;
1798 : else
1799 : {
1800 1670091 : ulong j = n+i-1;
1801 1670091 : ulong d = ugcd(j, xi);
1802 1670090 : if (d==1)
1803 1018468 : uel(y,i) = Fl_div(xi, j, p);
1804 : else
1805 651622 : uel(y,i) = Fl_div(xi/d, j/d, p);
1806 : }
1807 : }
1808 413575 : return Flx_renormalize(y, lx);;
1809 : }
1810 :
1811 : GEN
1812 0 : Flx_integ(GEN x, ulong p)
1813 : {
1814 0 : long i, lx = lg(x);
1815 : GEN y;
1816 0 : if (lx == 2) return Flx_copy(x);
1817 0 : y = cgetg(lx+1, t_VECSMALL); y[1] = x[1];
1818 0 : uel(y,2) = 0;
1819 0 : for (i=3; i<=lx; i++)
1820 0 : uel(y,i) = uel(x,i-1) ? Fl_div(uel(x,i-1), (i-2)%p, p): 0UL;
1821 0 : return Flx_renormalize(y, lx+1);;
1822 : }
1823 :
1824 : /* assume p prime */
1825 : GEN
1826 13482 : Flx_diff1(GEN P, ulong p)
1827 : {
1828 13482 : return Flx_sub(Flx_translate1(P, p), P, p);
1829 : }
1830 :
1831 : GEN
1832 420980 : Flx_deflate(GEN x0, long d)
1833 : {
1834 : GEN z, y, x;
1835 420980 : long i,id, dy, dx = degpol(x0);
1836 420980 : if (d == 1 || dx <= 0) return Flx_copy(x0);
1837 357467 : dy = dx/d;
1838 357467 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1839 357466 : z = y + 2;
1840 357466 : x = x0+ 2;
1841 1162498 : for (i=id=0; i<=dy; i++,id+=d) z[i] = x[id];
1842 357466 : return y;
1843 : }
1844 :
1845 : GEN
1846 160638 : Flx_inflate(GEN x0, long d)
1847 : {
1848 160638 : long i, id, dy, dx = degpol(x0);
1849 160635 : GEN x = x0 + 2, z, y;
1850 160635 : if (dx <= 0) return Flx_copy(x0);
1851 159573 : dy = dx*d;
1852 159573 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1853 159559 : z = y + 2;
1854 8879980 : for (i=0; i<=dy; i++) z[i] = 0;
1855 4321521 : for (i=id=0; i<=dx; i++,id+=d) z[id] = x[i];
1856 159559 : return y;
1857 : }
1858 :
1859 : /* write p(X) = a_0(X^k) + X*a_1(X^k) + ... + X^(k-1)*a_{k-1}(X^k) */
1860 : GEN
1861 147608 : Flx_splitting(GEN p, long k)
1862 : {
1863 147608 : long n = degpol(p), v = p[1], m, i, j, l;
1864 : GEN r;
1865 :
1866 147607 : m = n/k;
1867 147607 : r = cgetg(k+1,t_VEC);
1868 680120 : for(i=1; i<=k; i++)
1869 : {
1870 532520 : gel(r,i) = cgetg(m+3, t_VECSMALL);
1871 532511 : mael(r,i,1) = v;
1872 : }
1873 4494191 : for (j=1, i=0, l=2; i<=n; i++)
1874 : {
1875 4346591 : mael(r,j,l) = p[2+i];
1876 4346591 : if (j==k) { j=1; l++; } else j++;
1877 : }
1878 680116 : for(i=1; i<=k; i++)
1879 532532 : gel(r,i) = Flx_renormalize(gel(r,i),i<j?l+1:l);
1880 147584 : return r;
1881 : }
1882 :
1883 : /* ux + vy */
1884 : static GEN
1885 416641 : Flx_addmulmul(GEN u, GEN v, GEN x, GEN y, ulong p, ulong pi)
1886 416641 : { return Flx_add(Flx_mul_pre(u,x, p,pi), Flx_mul_pre(v,y, p,pi), p); }
1887 :
1888 : static GEN
1889 26000 : FlxM_Flx_mul2(GEN M, GEN x, GEN y, ulong p, ulong pi)
1890 : {
1891 26000 : GEN res = cgetg(3, t_COL);
1892 26000 : gel(res, 1) = Flx_addmulmul(gcoeff(M,1,1), gcoeff(M,1,2), x, y, p, pi);
1893 26000 : gel(res, 2) = Flx_addmulmul(gcoeff(M,2,1), gcoeff(M,2,2), x, y, p, pi);
1894 26000 : return res;
1895 : }
1896 :
1897 : #if 0
1898 : static GEN
1899 : FlxM_mul2_old(GEN M, GEN N, ulong p)
1900 : {
1901 : GEN res = cgetg(3, t_MAT);
1902 : gel(res, 1) = FlxM_Flx_mul2(M,gcoeff(N,1,1),gcoeff(N,2,1),p);
1903 : gel(res, 2) = FlxM_Flx_mul2(M,gcoeff(N,1,2),gcoeff(N,2,2),p);
1904 : return res;
1905 : }
1906 : #endif
1907 : /* A,B are 2x2 matrices, Flx entries. Return A x B using Strassen 7M formula */
1908 : static GEN
1909 7099 : FlxM_mul2(GEN A, GEN B, ulong p, ulong pi)
1910 : {
1911 7099 : GEN A11=gcoeff(A,1,1),A12=gcoeff(A,1,2), B11=gcoeff(B,1,1),B12=gcoeff(B,1,2);
1912 7099 : GEN A21=gcoeff(A,2,1),A22=gcoeff(A,2,2), B21=gcoeff(B,2,1),B22=gcoeff(B,2,2);
1913 7099 : GEN M1 = Flx_mul_pre(Flx_add(A11,A22, p), Flx_add(B11,B22, p), p, pi);
1914 7099 : GEN M2 = Flx_mul_pre(Flx_add(A21,A22, p), B11, p, pi);
1915 7099 : GEN M3 = Flx_mul_pre(A11, Flx_sub(B12,B22, p), p, pi);
1916 7099 : GEN M4 = Flx_mul_pre(A22, Flx_sub(B21,B11, p), p, pi);
1917 7099 : GEN M5 = Flx_mul_pre(Flx_add(A11,A12, p), B22, p, pi);
1918 7099 : GEN M6 = Flx_mul_pre(Flx_sub(A21,A11, p), Flx_add(B11,B12, p), p, pi);
1919 7099 : GEN M7 = Flx_mul_pre(Flx_sub(A12,A22, p), Flx_add(B21,B22, p), p, pi);
1920 7099 : GEN T1 = Flx_add(M1,M4, p), T2 = Flx_sub(M7,M5, p);
1921 7099 : GEN T3 = Flx_sub(M1,M2, p), T4 = Flx_add(M3,M6, p);
1922 7099 : retmkmat22(Flx_add(T1,T2, p), Flx_add(M3,M5, p),
1923 : Flx_add(M2,M4, p), Flx_add(T3,T4, p));
1924 : }
1925 :
1926 : /* Return [0,1;1,-q]*M */
1927 : static GEN
1928 6927 : Flx_FlxM_qmul(GEN q, GEN M, ulong p, ulong pi)
1929 : {
1930 6927 : GEN u = Flx_mul_pre(gcoeff(M,2,1), q, p, pi);
1931 6927 : GEN v = Flx_mul_pre(gcoeff(M,2,2), q, p, pi);
1932 6927 : retmkmat22(gcoeff(M,2,1), gcoeff(M,2,2),
1933 : Flx_sub(gcoeff(M,1,1), u, p), Flx_sub(gcoeff(M,1,2), v, p));
1934 : }
1935 :
1936 : static GEN
1937 911 : matid2_FlxM(long v)
1938 911 : { retmkmat22(pol1_Flx(v),pol0_Flx(v),pol0_Flx(v),pol1_Flx(v)); }
1939 :
1940 : static GEN
1941 13 : matJ2_FlxM(long v)
1942 13 : { retmkmat22(pol0_Flx(v),pol1_Flx(v),pol1_Flx(v),pol0_Flx(v)); }
1943 :
1944 : struct Flx_res
1945 : {
1946 : ulong res, lc;
1947 : long deg0, deg1, off;
1948 : };
1949 :
1950 : INLINE void
1951 9405 : Flx_halfres_update_pre(long da, long db, long dr, ulong p, ulong pi, struct Flx_res *res)
1952 : {
1953 9405 : if (dr >= 0)
1954 : {
1955 9405 : if (res->lc != 1)
1956 : {
1957 7596 : if (pi)
1958 : {
1959 3127 : res->lc = Fl_powu_pre(res->lc, da - dr, p, pi);
1960 3127 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1961 : } else
1962 : {
1963 4469 : res->lc = Fl_powu(res->lc, da - dr, p);
1964 4469 : res->res = Fl_mul(res->res, res->lc, p);
1965 : }
1966 : }
1967 9405 : if (both_odd(da + res->off, db + res->off))
1968 63 : res->res = Fl_neg(res->res, p);
1969 : } else
1970 : {
1971 0 : if (db == 0)
1972 : {
1973 0 : if (res->lc != 1)
1974 : {
1975 0 : if (pi)
1976 : {
1977 0 : res->lc = Fl_powu_pre(res->lc, da, p, pi);
1978 0 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1979 : } else
1980 : {
1981 0 : res->lc = Fl_powu(res->lc, da, p);
1982 0 : res->res = Fl_mul(res->res, res->lc, p);
1983 : }
1984 : }
1985 : } else
1986 0 : res->res = 0;
1987 : }
1988 9405 : }
1989 :
1990 : static GEN
1991 1134994 : Flx_halfres_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *pa, GEN *pb, struct Flx_res *res)
1992 : {
1993 1134994 : pari_sp av = avma;
1994 : GEN u, u1, v, v1, M;
1995 1134994 : long vx = a[1], n = lgpol(a)>>1;
1996 1134991 : u1 = v = pol0_Flx(vx);
1997 1134983 : u = v1 = pol1_Flx(vx);
1998 6941497 : while (lgpol(b)>n)
1999 : {
2000 : GEN r, q;
2001 5806534 : q = Flx_divrem_pre(a,b,p,pi, &r);
2002 5806627 : if (res)
2003 : {
2004 8362 : long da = degpol(a), db=degpol(b), dr = degpol(r);
2005 8362 : res->lc = b[db+2];
2006 8362 : if (dr >= n)
2007 7133 : Flx_halfres_update_pre(da, db, dr, p, pi, res);
2008 : else
2009 : {
2010 1229 : res->deg0 = da;
2011 1229 : res->deg1 = db;
2012 : }
2013 : }
2014 5806627 : a = b; b = r; swap(u,u1); swap(v,v1);
2015 5806627 : u1 = Flx_sub(u1, Flx_mul(u, q, p), p);
2016 5806493 : v1 = Flx_sub(v1, Flx_mul(v, q, p), p);
2017 5806518 : if (gc_needed(av,2))
2018 : {
2019 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_halfgcd (d = %ld)",degpol(b));
2020 0 : (void)gc_all(av,6, &a,&b,&u1,&v1,&u,&v);
2021 : }
2022 : }
2023 1134825 : M = mkmat22(u,v,u1,v1); *pa = a; *pb = b;
2024 1134969 : return gc_all(av,3, &M, pa, pb);
2025 : }
2026 :
2027 : static GEN Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res);
2028 :
2029 : static GEN
2030 19964 : Flx_halfres_split(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2031 : {
2032 19964 : pari_sp av = avma;
2033 : GEN R, S, T, V1, V2;
2034 : GEN x1, y1, r, q;
2035 19964 : long l = lgpol(x), n = l>>1, k;
2036 19964 : if (lgpol(y) <= n)
2037 871 : { *a = Flx_copy(x); *b = Flx_copy(y); return matid2_FlxM(x[1]); }
2038 19093 : if (res)
2039 : {
2040 3263 : res->lc = Flx_lead(y);
2041 3263 : res->deg0 -= n;
2042 3263 : res->deg1 -= n;
2043 3263 : res->off += n;
2044 : }
2045 19093 : R = Flx_halfres_i(Flx_shift(x,-n),Flx_shift(y,-n),p,pi,a,b,res);
2046 19093 : if (res)
2047 : {
2048 3263 : res->off -= n;
2049 3263 : res->deg0 += n;
2050 3263 : res->deg1 += n;
2051 : }
2052 19093 : V1 = FlxM_Flx_mul2(R, Flxn_red(x,n), Flxn_red(y,n), p, pi);
2053 19093 : x1 = Flx_add(Flx_shift(*a,n), gel(V1,1), p);
2054 19093 : y1 = Flx_add(Flx_shift(*b,n), gel(V1,2), p);
2055 19091 : if (lgpol(y1) <= n)
2056 12185 : { *a = x1; *b = y1; return gc_all(av, 3, &R, a, b); }
2057 6907 : k = 2*n-degpol(y1);
2058 6907 : q = Flx_divrem_pre(x1, y1, p, pi, &r);
2059 6907 : if (res)
2060 : {
2061 1043 : long dx1 = degpol(x1), dy1 = degpol(y1), dr = degpol(r);
2062 1043 : if (dy1 < degpol(y))
2063 185 : Flx_halfres_update_pre(res->deg0, res->deg1, dy1, p, pi, res);
2064 1043 : res->lc = uel(y1, dy1+2);
2065 1043 : res->deg0 = dx1;
2066 1043 : res->deg1 = dy1;
2067 1043 : if (dr >= n)
2068 : {
2069 1043 : Flx_halfres_update_pre(dx1, dy1, dr, p, pi, res);
2070 1043 : res->deg0 = dy1;
2071 1043 : res->deg1 = dr;
2072 : }
2073 1043 : res->deg0 -= k;
2074 1043 : res->deg1 -= k;
2075 1043 : res->off += k;
2076 : }
2077 6907 : S = Flx_halfres_i(Flx_shift(y1,-k), Flx_shift(r,-k), p, pi, a, b, res);
2078 6907 : if (res)
2079 : {
2080 1043 : res->deg0 += k;
2081 1043 : res->deg1 += k;
2082 1043 : res->off -= k;
2083 : }
2084 6907 : T = FlxM_mul2(S, Flx_FlxM_qmul(q, R, p,pi), p, pi);
2085 6907 : V2 = FlxM_Flx_mul2(S, Flxn_red(y1,k), Flxn_red(r,k), p, pi);
2086 6907 : *a = Flx_add(Flx_shift(*a,k), gel(V2,1), p);
2087 6907 : *b = Flx_add(Flx_shift(*b,k), gel(V2,2), p);
2088 6907 : return gc_all(av, 3, &T, a, b);
2089 : }
2090 :
2091 : static GEN
2092 1154960 : Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2093 : {
2094 1154960 : if (lgpol(x) < get_Fl_threshold(p, Flx_HALFGCD_LIMIT, Flx_HALFGCD2_LIMIT))
2095 1134994 : return Flx_halfres_basecase(x, y, p, pi, a, b, res);
2096 19964 : return Flx_halfres_split(x, y, p, pi, a, b, res);
2097 : }
2098 :
2099 : static GEN
2100 1127916 : Flx_halfgcd_all_i(GEN x, GEN y, ulong p, ulong pi, GEN *pa, GEN *pb)
2101 : {
2102 : GEN a, b, R;
2103 1127916 : R = Flx_halfres_i(x, y, p, pi, &a, &b, NULL);
2104 1127925 : if (pa) *pa = a;
2105 1127925 : if (pb) *pb = b;
2106 1127925 : return R;
2107 : }
2108 :
2109 : /* Return M in GL_2(Fl[X]) such that:
2110 : if [a',b']~=M*[a,b]~ then degpol(a')>= (lgpol(a)>>1) >degpol(b')
2111 : */
2112 :
2113 : GEN
2114 1127915 : Flx_halfgcd_all_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b)
2115 : {
2116 : pari_sp av;
2117 : GEN R, q ,r;
2118 1127915 : long lx = lgpol(x), ly = lgpol(y);
2119 1127915 : if (!lx)
2120 : {
2121 0 : if (a) *a = Flx_copy(y);
2122 0 : if (b) *b = Flx_copy(x);
2123 0 : return matJ2_FlxM(x[1]);
2124 : }
2125 1127915 : if (ly < lx) return Flx_halfgcd_all_i(x, y, p, pi, a, b);
2126 7918 : av = avma;
2127 7918 : q = Flx_divrem(y,x,p,&r);
2128 7918 : R = Flx_halfgcd_all_i(x, r, p, pi, a, b);
2129 7918 : gcoeff(R,1,1) = Flx_sub(gcoeff(R,1,1), Flx_mul_pre(q,gcoeff(R,1,2), p,pi), p);
2130 7918 : gcoeff(R,2,1) = Flx_sub(gcoeff(R,2,1), Flx_mul_pre(q,gcoeff(R,2,2), p,pi), p);
2131 7918 : return !a && b ? gc_all(av, 2, &R, b): gc_all(av, 1+!!a+!!b, &R, a, b);
2132 : }
2133 :
2134 : GEN
2135 154 : Flx_halfgcd_all(GEN x, GEN y, ulong p, GEN *a, GEN *b)
2136 154 : { return Flx_halfgcd_all_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), a, b); }
2137 :
2138 : GEN
2139 874332 : Flx_halfgcd_pre(GEN x, GEN y, ulong p, ulong pi)
2140 874332 : { return Flx_halfgcd_all_pre(x, y, p, pi, NULL, NULL); }
2141 :
2142 : GEN
2143 0 : Flx_halfgcd(GEN x, GEN y, ulong p)
2144 0 : { return Flx_halfgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2145 :
2146 : /*Do not garbage collect*/
2147 : static GEN
2148 85103371 : Flx_gcd_basecase(GEN a, GEN b, ulong p, ulong pi)
2149 : {
2150 85103371 : pari_sp av = avma;
2151 85103371 : ulong iter = 0;
2152 85103371 : if (lg(b) > lg(a)) swap(a, b);
2153 293003589 : while (lgpol(b))
2154 : {
2155 207504600 : GEN c = Flx_rem_pre(a,b,p,pi);
2156 207900218 : iter++; a = b; b = c;
2157 207900218 : if (gc_needed(av,2))
2158 : {
2159 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (d = %ld)",degpol(c));
2160 0 : (void)gc_all(av,2, &a,&b);
2161 : }
2162 : }
2163 85056712 : return iter < 2 ? Flx_copy(a) : a;
2164 : }
2165 :
2166 : GEN
2167 86766419 : Flx_gcd_pre(GEN x, GEN y, ulong p, ulong pi)
2168 : {
2169 86766419 : pari_sp av = avma;
2170 : long lim;
2171 86766419 : if (!lgpol(x)) return Flx_copy(y);
2172 85106178 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2173 85111367 : while (lgpol(y) >= lim)
2174 : {
2175 229 : if (lgpol(y)<=(lgpol(x)>>1))
2176 : {
2177 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2178 0 : x = y; y = r;
2179 : }
2180 229 : (void) Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2181 229 : if (gc_needed(av,2))
2182 : {
2183 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (y = %ld)",degpol(y));
2184 0 : (void)gc_all(av,2,&x,&y);
2185 : }
2186 : }
2187 85099337 : return gc_leaf(av, Flx_gcd_basecase(x,y,p,pi));
2188 : }
2189 : GEN
2190 34068052 : Flx_gcd(GEN x, GEN y, ulong p)
2191 34068052 : { return Flx_gcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2192 :
2193 : int
2194 9183417 : Flx_is_squarefree(GEN z, ulong p)
2195 : {
2196 9183417 : pari_sp av = avma;
2197 9183417 : GEN d = Flx_gcd(z, Flx_deriv(z,p) , p);
2198 9183222 : return gc_bool(av, degpol(d) == 0);
2199 : }
2200 :
2201 : static long
2202 126637 : Flx_is_smooth_squarefree(GEN f, long r, ulong p, ulong pi)
2203 : {
2204 126637 : pari_sp av = avma;
2205 : long i;
2206 126637 : GEN sx = polx_Flx(f[1]), a = sx;
2207 534081 : for(i=1;;i++)
2208 : {
2209 534081 : if (degpol(f)<=r) return gc_long(av,1);
2210 512315 : a = Flxq_powu_pre(Flx_rem_pre(a,f,p,pi), p, f, p, pi);
2211 512312 : if (Flx_equal(a, sx)) return gc_long(av,1);
2212 508599 : if (i==r) return gc_long(av,0);
2213 407133 : f = Flx_div_pre(f, Flx_gcd_pre(Flx_sub(a,sx,p),f,p,pi),p,pi);
2214 : }
2215 : }
2216 :
2217 : static long
2218 8177 : Flx_is_l_pow(GEN x, ulong p)
2219 : {
2220 8177 : ulong i, lx = lgpol(x);
2221 16357 : for (i=1; i<lx; i++)
2222 14670 : if (x[i+2] && i%p) return 0;
2223 1687 : return 1;
2224 : }
2225 :
2226 : int
2227 126583 : Flx_is_smooth_pre(GEN g, long r, ulong p, ulong pi)
2228 : {
2229 : while (1)
2230 8177 : {
2231 126583 : GEN f = Flx_gcd_pre(g, Flx_deriv(g, p), p, pi);
2232 126461 : if (!Flx_is_smooth_squarefree(Flx_div_pre(g, f, p, pi), r, p, pi))
2233 101466 : return 0;
2234 25171 : if (degpol(f)==0) return 1;
2235 8169 : g = Flx_is_l_pow(f,p) ? Flx_deflate(f, p): f;
2236 : }
2237 : }
2238 : int
2239 74256 : Flx_is_smooth(GEN g, long r, ulong p)
2240 74256 : { return Flx_is_smooth_pre(g, r, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2241 :
2242 : static GEN
2243 6286150 : Flx_extgcd_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2244 : {
2245 6286150 : pari_sp av=avma;
2246 : GEN u,v,u1,v1;
2247 6286150 : long vx = a[1];
2248 6286150 : v = pol0_Flx(vx); v1 = pol1_Flx(vx);
2249 6285951 : if (ptu) { u = pol1_Flx(vx); u1 = pol0_Flx(vx); }
2250 28064431 : while (lgpol(b))
2251 : {
2252 21777266 : GEN r, q = Flx_divrem_pre(a,b,p,pi, &r);
2253 21778979 : a = b; b = r;
2254 21778979 : if (ptu)
2255 : {
2256 2430900 : swap(u,u1);
2257 2430900 : u1 = Flx_sub(u1, Flx_mul_pre(u, q, p, pi), p);
2258 : }
2259 21778969 : swap(v,v1);
2260 21778969 : v1 = Flx_sub(v1, Flx_mul_pre(v, q, p, pi), p);
2261 21778453 : if (gc_needed(av,2))
2262 : {
2263 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_extgcd (d = %ld)",degpol(a));
2264 0 : (void)gc_all(av,ptu ? 6: 4, &a,&b,&v,&v1,&u,&u1);
2265 : }
2266 : }
2267 6286018 : if (ptu) *ptu = u;
2268 6286018 : *ptv = v;
2269 6286018 : return a;
2270 : }
2271 :
2272 : static GEN
2273 146590 : Flx_extgcd_halfgcd(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2274 : {
2275 : GEN u, v;
2276 146590 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2277 146590 : GEN V = cgetg(expu(lgpol(y))+2,t_VEC);
2278 146590 : long i, n = 0, vs = x[1];
2279 398867 : while (lgpol(y) >= lim)
2280 : {
2281 252276 : if (lgpol(y)<=(lgpol(x)>>1))
2282 : {
2283 26 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2284 26 : x = y; y = r;
2285 26 : gel(V,++n) = mkmat22(pol0_Flx(vs),pol1_Flx(vs),pol1_Flx(vs),Flx_neg(q,p));
2286 : } else
2287 252250 : gel(V,++n) = Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2288 : }
2289 146591 : y = Flx_extgcd_basecase(x,y,p,pi,&u,&v);
2290 252276 : for (i = n; i>1; i--)
2291 : {
2292 105686 : GEN R = gel(V,i);
2293 105686 : GEN u1 = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2294 105686 : GEN v1 = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2295 105686 : u = u1; v = v1;
2296 : }
2297 : {
2298 146590 : GEN R = gel(V,1);
2299 146590 : if (ptu)
2300 6574 : *ptu = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2301 146590 : *ptv = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2302 : }
2303 146590 : return y;
2304 : }
2305 :
2306 : /* x and y in Z[X], return lift(gcd(x mod p, y mod p)). Set u and v st
2307 : * ux + vy = gcd (mod p) */
2308 : GEN
2309 6286141 : Flx_extgcd_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2310 : {
2311 6286141 : pari_sp av = avma;
2312 : GEN d;
2313 6286141 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2314 6286142 : if (lgpol(y) >= lim)
2315 146590 : d = Flx_extgcd_halfgcd(x, y, p, pi, ptu, ptv);
2316 : else
2317 6139540 : d = Flx_extgcd_basecase(x, y, p, pi, ptu, ptv);
2318 6286016 : return gc_all(av, ptu?3:2, &d, ptv, ptu);
2319 : }
2320 : GEN
2321 857188 : Flx_extgcd(GEN x, GEN y, ulong p, GEN *ptu, GEN *ptv)
2322 857188 : { return Flx_extgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptu, ptv); }
2323 :
2324 : static GEN
2325 1044 : Flx_halfres_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, ulong *r)
2326 : {
2327 : struct Flx_res res;
2328 : GEN R;
2329 : long dB;
2330 :
2331 1044 : res.res = *r;
2332 1044 : res.lc = Flx_lead(y);
2333 1044 : res.deg0 = degpol(x);
2334 1044 : res.deg1 = degpol(y);
2335 1044 : res.off = 0;
2336 1044 : R = Flx_halfres_i(x, y, p, pi, a, b, &res);
2337 1044 : dB = degpol(*b);
2338 1044 : if (dB < degpol(y))
2339 1044 : Flx_halfres_update_pre(res.deg0, res.deg1, dB, p, pi, &res);
2340 1044 : *r = res.res;
2341 1044 : return R;
2342 : }
2343 :
2344 : static ulong
2345 10272820 : Flx_resultant_basecase_pre(GEN a, GEN b, ulong p, ulong pi)
2346 : {
2347 : pari_sp av;
2348 : long da,db,dc;
2349 10272820 : ulong lb, res = 1UL;
2350 : GEN c;
2351 :
2352 10272820 : da = degpol(a);
2353 10272716 : db = degpol(b);
2354 10272805 : if (db > da)
2355 : {
2356 0 : swapspec(a,b, da,db);
2357 0 : if (both_odd(da,db)) res = p-res;
2358 : }
2359 10272805 : else if (!da) return 1; /* = res * a[2] ^ db, since 0 <= db <= da = 0 */
2360 10272805 : av = avma;
2361 107312693 : while (db)
2362 : {
2363 97062970 : lb = b[db+2];
2364 97062970 : c = Flx_rem_pre(a,b, p,pi);
2365 96721304 : a = b; b = c; dc = degpol(c);
2366 96694723 : if (dc < 0) return gc_long(av,0);
2367 :
2368 96689245 : if (both_odd(da,db)) res = p - res;
2369 96672318 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, da - dc, p, pi), p);
2370 97049742 : if (gc_needed(av,2))
2371 : {
2372 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant (da = %ld)",da);
2373 0 : (void)gc_all(av,2, &a,&b);
2374 : }
2375 97039888 : da = db; /* = degpol(a) */
2376 97039888 : db = dc; /* = degpol(b) */
2377 : }
2378 10249723 : return gc_ulong(av, Fl_mul(res, Fl_powu_pre(b[2], da, p, pi), p));
2379 : }
2380 :
2381 : ulong
2382 10274703 : Flx_resultant_pre(GEN x, GEN y, ulong p, ulong pi)
2383 : {
2384 10274703 : pari_sp av = avma;
2385 : long lim;
2386 10274703 : ulong res = 1;
2387 10274703 : long dx = degpol(x), dy = degpol(y);
2388 10274338 : if (dx < 0 || dy < 0) return 0;
2389 10272896 : if (dx < dy)
2390 : {
2391 1065350 : swap(x,y);
2392 1065350 : if (both_odd(dx, dy))
2393 1906 : res = Fl_neg(res, p);
2394 : }
2395 10272897 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2396 10273789 : while (lgpol(y) >= lim)
2397 : {
2398 852 : if (lgpol(y)<=(lgpol(x)>>1))
2399 : {
2400 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2401 0 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2402 0 : ulong ly = y[dy+2];
2403 0 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2404 0 : if (both_odd(dx, dy))
2405 0 : res = Fl_neg(res, p);
2406 0 : x = y; y = r;
2407 : }
2408 852 : (void) Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2409 852 : if (gc_needed(av,2))
2410 : {
2411 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_res (y = %ld)",degpol(y));
2412 0 : (void)gc_all(av,2,&x,&y);
2413 : }
2414 : }
2415 10272834 : return gc_ulong(av, Fl_mul(res, Flx_resultant_basecase_pre(x, y, p, pi), p));
2416 : }
2417 :
2418 : ulong
2419 4734200 : Flx_resultant(GEN a, GEN b, ulong p)
2420 4734200 : { return Flx_resultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2421 :
2422 : /* If resultant is 0, *ptU and *ptV are not set */
2423 : static ulong
2424 53 : Flx_extresultant_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2425 : {
2426 53 : GEN z,q,u,v, x = a, y = b;
2427 53 : ulong lb, res = 1UL;
2428 53 : pari_sp av = avma;
2429 : long dx, dy, dz;
2430 53 : long vs = a[1];
2431 :
2432 53 : u = pol0_Flx(vs);
2433 53 : v = pol1_Flx(vs); /* v = 1 */
2434 53 : dx = degpol(x);
2435 53 : dy = degpol(y);
2436 764 : while (dy)
2437 : { /* b u = x (a), b v = y (a) */
2438 711 : lb = y[dy+2];
2439 711 : q = Flx_divrem_pre(x,y, p, pi, &z);
2440 711 : x = y; y = z; /* (x,y) = (y, x - q y) */
2441 711 : dz = degpol(z); if (dz < 0) return gc_ulong(av,0);
2442 711 : z = Flx_sub(u, Flx_mul_pre(q,v, p, pi), p);
2443 711 : u = v; v = z; /* (u,v) = (v, u - q v) */
2444 :
2445 711 : if (both_odd(dx,dy)) res = p - res;
2446 711 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, dx-dz, p, pi), p);
2447 711 : dx = dy; /* = degpol(x) */
2448 711 : dy = dz; /* = degpol(y) */
2449 : }
2450 53 : res = Fl_mul(res, Fl_powu_pre(y[2], dx, p, pi), p);
2451 53 : lb = Fl_mul(res, Fl_inv(y[2],p), p);
2452 53 : v = gc_leaf(av, Flx_Fl_mul_pre(v, lb, p, pi));
2453 53 : av = avma;
2454 53 : u = Flx_sub(Fl_to_Flx(res,vs), Flx_mul_pre(b,v,p,pi), p);
2455 53 : u = gc_leaf(av, Flx_div_pre(u,a,p,pi)); /* = (res - b v) / a */
2456 53 : *ptU = u;
2457 53 : *ptV = v; return res;
2458 : }
2459 :
2460 : ulong
2461 53 : Flx_extresultant_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2462 : {
2463 53 : pari_sp av=avma;
2464 : GEN u, v, R;
2465 53 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2466 53 : ulong res = 1, res1;
2467 53 : long dx = degpol(x), dy = degpol(y);
2468 53 : if (dy > dx)
2469 : {
2470 13 : swap(x,y); lswap(dx,dy);
2471 13 : if (both_odd(dx,dy)) res = p-res;
2472 13 : R = matJ2_FlxM(x[1]);
2473 40 : } else R = matid2_FlxM(x[1]);
2474 53 : if (dy < 0) return 0;
2475 245 : while (lgpol(y) >= lim)
2476 : {
2477 : GEN M;
2478 192 : if (lgpol(y)<=(lgpol(x)>>1))
2479 : {
2480 20 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2481 20 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2482 20 : ulong ly = y[dy+2];
2483 20 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2484 20 : if (both_odd(dx, dy))
2485 0 : res = Fl_neg(res, p);
2486 20 : x = y; y = r;
2487 20 : R = Flx_FlxM_qmul(q, R, p,pi);
2488 : }
2489 192 : M = Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2490 192 : if (!res) return gc_ulong(av, 0);
2491 192 : R = FlxM_mul2(M, R, p, pi);
2492 192 : (void)gc_all(av,3,&x,&y,&R);
2493 : }
2494 53 : res1 = Flx_extresultant_basecase(x,y,p,pi,&u,&v);
2495 53 : if (!res1) return gc_ulong(av, 0);
2496 53 : *ptU = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi), res, p, pi);
2497 53 : *ptV = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi), res, p, pi);
2498 53 : (void)gc_all(av, 2, ptU, ptV);
2499 53 : return Fl_mul(res1,res,p);
2500 : }
2501 :
2502 : ulong
2503 53 : Flx_extresultant(GEN a, GEN b, ulong p, GEN *ptU, GEN *ptV)
2504 53 : { return Flx_extresultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptU, ptV); }
2505 :
2506 : /* allow pi = 0 (SMALL_ULONG) */
2507 : ulong
2508 48396037 : Flx_eval_powers_pre(GEN x, GEN y, ulong p, ulong pi)
2509 : {
2510 48396037 : ulong l0, l1, h0, h1, v1, i = 1, lx = lg(x)-1;
2511 :
2512 48396037 : if (lx == 1) return 0;
2513 45355170 : x++;
2514 45355170 : if (pi)
2515 : {
2516 : LOCAL_OVERFLOW;
2517 : LOCAL_HIREMAINDER;
2518 45292360 : l1 = mulll(uel(x,i), uel(y,i)); h1 = hiremainder; v1 = 0;
2519 114816219 : while (++i < lx)
2520 : {
2521 69523859 : l0 = mulll(uel(x,i), uel(y,i)); h0 = hiremainder;
2522 69523859 : l1 = addll(l0, l1); h1 = addllx(h0, h1); v1 += overflow;
2523 : }
2524 81325 : return v1? remlll_pre(v1, h1, l1, p, pi)
2525 45373685 : : remll_pre(h1, l1, p, pi);
2526 : }
2527 : else
2528 : {
2529 62810 : l1 = x[i] * y[i];
2530 30926389 : while (++i < lx) { l1 += x[i] * y[i]; if (l1 & HIGHBIT) l1 %= p; }
2531 62810 : return l1 % p;
2532 : }
2533 : }
2534 :
2535 : /* allow pi = 0 (SMALL_ULONG) */
2536 : ulong
2537 100786341 : Flx_eval_pre(GEN x, ulong y, ulong p, ulong pi)
2538 : {
2539 100786341 : long i, n = degpol(x);
2540 : ulong t;
2541 100779070 : if (n <= 0) return n? 0: x[2];
2542 32997973 : if (n > 15)
2543 : {
2544 180167 : pari_sp av = avma;
2545 180167 : GEN v = Fl_powers_pre(y, n, p, pi);
2546 180171 : return gc_ulong(av, Flx_eval_powers_pre(x, v, p, pi));
2547 : }
2548 32817806 : i = n+2; t = x[i];
2549 32817806 : if (pi)
2550 : {
2551 123191311 : for (i--; i>=2; i--) t = Fl_addmul_pre(uel(x, i), t, y, p, pi);
2552 31675173 : return t;
2553 : }
2554 2769674 : for (i--; i>=2; i--) t = (t * y + x[i]) % p;
2555 1142952 : return t %= p;
2556 : }
2557 : ulong
2558 20420064 : Flx_eval(GEN x, ulong y, ulong p)
2559 20420064 : { return Flx_eval_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2560 :
2561 : ulong
2562 3304 : Flv_prod_pre(GEN x, ulong p, ulong pi)
2563 : {
2564 3304 : pari_sp ltop = avma;
2565 : GEN v;
2566 3304 : long i,k,lx = lg(x);
2567 3304 : if (lx == 1) return 1UL;
2568 3304 : if (lx == 2) return uel(x,1);
2569 2863 : v = cgetg(1+(lx << 1), t_VECSMALL);
2570 2863 : k = 1;
2571 26955 : for (i=1; i<lx-1; i+=2)
2572 24092 : uel(v,k++) = Fl_mul_pre(uel(x,i), uel(x,i+1), p, pi);
2573 2863 : if (i < lx) uel(v,k++) = uel(x,i);
2574 12836 : while (k > 2)
2575 : {
2576 9973 : lx = k; k = 1;
2577 34065 : for (i=1; i<lx-1; i+=2)
2578 24092 : uel(v,k++) = Fl_mul_pre(uel(v,i), uel(v,i+1), p, pi);
2579 9973 : if (i < lx) uel(v,k++) = uel(v,i);
2580 : }
2581 2863 : return gc_ulong(ltop, uel(v,1));
2582 : }
2583 :
2584 : ulong
2585 0 : Flv_prod(GEN v, ulong p)
2586 : {
2587 0 : return Flv_prod_pre(v, p, get_Fl_red(p));
2588 : }
2589 :
2590 : GEN
2591 0 : FlxV_prod(GEN V, ulong p)
2592 : {
2593 : struct _Flxq D;
2594 0 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2595 0 : return gen_product(V, (void *)&D, &_Flx_mul);
2596 : }
2597 :
2598 : /* compute prod (x - a[i]) */
2599 : GEN
2600 797666 : Flv_roots_to_pol(GEN a, ulong p, long vs)
2601 : {
2602 : struct _Flxq D;
2603 797666 : long i,k,lx = lg(a);
2604 : GEN p1;
2605 797666 : if (lx == 1) return pol1_Flx(vs);
2606 797666 : p1 = cgetg(lx, t_VEC);
2607 12561536 : for (k=1,i=1; i<lx-1; i+=2)
2608 11761797 : gel(p1,k++) = mkvecsmall4(vs, Fl_mul(a[i], a[i+1], p),
2609 11764177 : Fl_neg(Fl_add(a[i],a[i+1],p),p), 1);
2610 797359 : if (i < lx)
2611 67572 : gel(p1,k++) = mkvecsmall3(vs, Fl_neg(a[i],p), 1);
2612 797356 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2613 797354 : setlg(p1, k); return gen_product(p1, (void *)&D, _Flx_mul);
2614 : }
2615 :
2616 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for "large" p */
2617 : INLINE void
2618 20265776 : Flv_inv_pre_indir(GEN w, GEN v, ulong p, ulong pi)
2619 : {
2620 20265776 : pari_sp av = avma;
2621 20265776 : long n = lg(w), i;
2622 : ulong u;
2623 : GEN c;
2624 :
2625 20265776 : if (n == 1) return;
2626 20265776 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2627 87267301 : for (i = 2; i < n; ++i) c[i] = Fl_mul_pre(w[i], c[i-1], p, pi);
2628 20434811 : i = n-1; u = Fl_inv(c[i], p);
2629 87637817 : for ( ; i > 1; --i)
2630 : {
2631 67173262 : ulong t = Fl_mul_pre(u, c[i-1], p, pi);
2632 67117158 : u = Fl_mul_pre(u, w[i], p, pi); v[i] = t;
2633 : }
2634 20464555 : v[1] = u; set_avma(av);
2635 : }
2636 :
2637 : void
2638 19660189 : Flv_inv_pre_inplace(GEN v, ulong p, ulong pi) { Flv_inv_pre_indir(v,v, p, pi); }
2639 :
2640 : GEN
2641 10038 : Flv_inv_pre(GEN w, ulong p, ulong pi)
2642 10038 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_pre_indir(w, v, p, pi); return v; }
2643 :
2644 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for SMALL_ULONG p */
2645 : INLINE void
2646 51381 : Flv_inv_indir(GEN w, GEN v, ulong p)
2647 : {
2648 51381 : pari_sp av = avma;
2649 51381 : long n = lg(w), i;
2650 : ulong u;
2651 : GEN c;
2652 :
2653 51381 : if (n == 1) return;
2654 51381 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2655 1757261 : for (i = 2; i < n; ++i) c[i] = Fl_mul(w[i], c[i-1], p);
2656 51393 : i = n-1; u = Fl_inv(c[i], p);
2657 1757314 : for ( ; i > 1; --i)
2658 : {
2659 1705931 : ulong t = Fl_mul(u, c[i-1], p);
2660 1705927 : u = Fl_mul(u, w[i], p); v[i] = t;
2661 : }
2662 51383 : v[1] = u; set_avma(av);
2663 : }
2664 : static void
2665 638820 : Flv_inv_i(GEN v, GEN w, ulong p)
2666 : {
2667 638820 : if (SMALL_ULONG(p)) Flv_inv_indir(w, v, p);
2668 587439 : else Flv_inv_pre_indir(w, v, p, get_Fl_red(p));
2669 638826 : }
2670 : void
2671 12017 : Flv_inv_inplace(GEN v, ulong p) { Flv_inv_i(v, v, p); }
2672 : GEN
2673 626804 : Flv_inv(GEN w, ulong p)
2674 626804 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_i(v, w, p); return v; }
2675 :
2676 : GEN
2677 34735859 : Flx_div_by_X_x(GEN a, ulong x, ulong p, ulong *rem)
2678 : {
2679 34735859 : long l = lg(a), i;
2680 : GEN a0, z0, z;
2681 34735859 : if (l <= 3)
2682 : {
2683 0 : if (rem) *rem = l == 2? 0: a[2];
2684 0 : return zero_Flx(a[1]);
2685 : }
2686 34735859 : z = cgetg(l-1,t_VECSMALL); z[1] = a[1];
2687 34594468 : a0 = a + l-1;
2688 34594468 : z0 = z + l-2; *z0 = *a0--;
2689 34594468 : if (SMALL_ULONG(p))
2690 : {
2691 84414609 : for (i=l-3; i>1; i--) /* z[i] = (a[i+1] + x*z[i+1]) % p */
2692 : {
2693 62258167 : ulong t = (*a0-- + x * *z0--) % p;
2694 62258167 : *z0 = (long)t;
2695 : }
2696 22156442 : if (rem) *rem = (*a0 + x * *z0) % p;
2697 : }
2698 : else
2699 : {
2700 48871838 : for (i=l-3; i>1; i--)
2701 : {
2702 36421256 : ulong t = Fl_add((ulong)*a0--, Fl_mul(x, *z0--, p), p);
2703 36433812 : *z0 = (long)t;
2704 : }
2705 12450582 : if (rem) *rem = Fl_add((ulong)*a0, Fl_mul(x, *z0, p), p);
2706 : }
2707 34613167 : return z;
2708 : }
2709 :
2710 : /* xa, ya = t_VECSMALL */
2711 : static GEN
2712 628013 : Flv_producttree(GEN xa, GEN s, ulong p, ulong pi, long vs)
2713 : {
2714 628013 : long n = lg(xa)-1;
2715 628013 : long m = n==1 ? 1: expu(n-1)+1;
2716 628013 : long i, j, k, ls = lg(s);
2717 628013 : GEN T = cgetg(m+1, t_VEC);
2718 628007 : GEN t = cgetg(ls, t_VEC);
2719 7878661 : for (j=1, k=1; j<ls; k+=s[j++])
2720 7250500 : gel(t, j) = s[j] == 1 ?
2721 7250653 : mkvecsmall3(vs, Fl_neg(xa[k], p), 1):
2722 1529922 : mkvecsmall4(vs, Fl_mul(xa[k], xa[k+1], p),
2723 1529928 : Fl_neg(Fl_add(xa[k],xa[k+1],p),p), 1);
2724 628008 : gel(T,1) = t;
2725 2370781 : for (i=2; i<=m; i++)
2726 : {
2727 1742812 : GEN u = gel(T, i-1);
2728 1742812 : long n = lg(u)-1;
2729 1742812 : GEN t = cgetg(((n+1)>>1)+1, t_VEC);
2730 8364922 : for (j=1, k=1; k<n; j++, k+=2)
2731 6622149 : gel(t, j) = Flx_mul_pre(gel(u, k), gel(u, k+1), p, pi);
2732 1742773 : gel(T, i) = t;
2733 : }
2734 627969 : return T;
2735 : }
2736 :
2737 : static GEN
2738 668313 : Flx_Flv_multieval_tree(GEN P, GEN xa, GEN T, ulong p, ulong pi)
2739 : {
2740 : long i,j,k;
2741 668313 : long m = lg(T)-1;
2742 668313 : GEN R = cgetg(lg(xa), t_VECSMALL);
2743 668310 : GEN Tp = cgetg(m+1, t_VEC), t;
2744 668306 : gel(Tp, m) = mkvec(P);
2745 2596463 : for (i=m-1; i>=1; i--)
2746 : {
2747 1928159 : GEN u = gel(T, i), v = gel(Tp, i+1);
2748 1928159 : long n = lg(u)-1;
2749 1928159 : t = cgetg(n+1, t_VEC);
2750 9581274 : for (j=1, k=1; k<n; j++, k+=2)
2751 : {
2752 7653151 : gel(t, k) = Flx_rem_pre(gel(v, j), gel(u, k), p, pi);
2753 7653078 : gel(t, k+1) = Flx_rem_pre(gel(v, j), gel(u, k+1), p, pi);
2754 : }
2755 1928123 : gel(Tp, i) = t;
2756 : }
2757 : {
2758 668304 : GEN u = gel(T, i+1), v = gel(Tp, i+1);
2759 668304 : long n = lg(u)-1;
2760 8991628 : for (j=1, k=1; j<=n; j++)
2761 : {
2762 8323282 : long c, d = degpol(gel(u,j));
2763 18425688 : for (c=1; c<=d; c++, k++) R[k] = Flx_eval_pre(gel(v, j), xa[k], p, pi);
2764 : }
2765 668346 : return gc_const((pari_sp)R, R);
2766 : }
2767 : }
2768 :
2769 : static GEN
2770 1424460 : FlvV_polint_tree(GEN T, GEN R, GEN s, GEN xa, GEN ya, ulong p, ulong pi, long vs)
2771 : {
2772 1424460 : pari_sp av = avma;
2773 1424460 : long m = lg(T)-1;
2774 1424460 : long i, j, k, ls = lg(s);
2775 1424460 : GEN Tp = cgetg(m+1, t_VEC);
2776 1424016 : GEN t = cgetg(ls, t_VEC);
2777 25734428 : for (j=1, k=1; j<ls; k+=s[j++])
2778 24310492 : if (s[j]==2)
2779 : {
2780 7204185 : ulong a = Fl_mul(ya[k], R[k], p);
2781 7203539 : ulong b = Fl_mul(ya[k+1], R[k+1], p);
2782 7209724 : gel(t, j) = mkvecsmall3(vs, Fl_neg(Fl_add(Fl_mul(xa[k], b, p ),
2783 7203410 : Fl_mul(xa[k+1], a, p), p), p), Fl_add(a, b, p));
2784 7208121 : gel(t, j) = Flx_renormalize(gel(t, j), 4);
2785 : }
2786 : else
2787 17106307 : gel(t, j) = Fl_to_Flx(Fl_mul(ya[k], R[k], p), vs);
2788 1423936 : gel(Tp, 1) = t;
2789 6598409 : for (i=2; i<=m; i++)
2790 : {
2791 5174579 : GEN u = gel(T, i-1);
2792 5174579 : GEN t = cgetg(lg(gel(T,i)), t_VEC);
2793 5172145 : GEN v = gel(Tp, i-1);
2794 5172145 : long n = lg(v)-1;
2795 27979008 : for (j=1, k=1; k<n; j++, k+=2)
2796 22807676 : gel(t, j) = Flx_add(Flx_mul_pre(gel(u, k), gel(v, k+1), p, pi),
2797 22804535 : Flx_mul_pre(gel(u, k+1), gel(v, k), p, pi), p);
2798 5174473 : gel(Tp, i) = t;
2799 : }
2800 1423830 : return gc_leaf(av, gmael(Tp,m,1));
2801 : }
2802 :
2803 : GEN
2804 0 : Flx_Flv_multieval(GEN P, GEN xa, ulong p)
2805 : {
2806 0 : pari_sp av = avma;
2807 0 : GEN s = producttree_scheme(lg(xa)-1);
2808 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2809 0 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2810 0 : return gc_leaf(av, Flx_Flv_multieval_tree(P, xa, T, p, pi));
2811 : }
2812 :
2813 : static GEN
2814 2471 : FlxV_Flv_multieval_tree(GEN x, GEN xa, GEN T, ulong p, ulong pi)
2815 45248 : { pari_APPLY_same(Flx_Flv_multieval_tree(gel(x,i), xa, T, p, pi)) }
2816 :
2817 : GEN
2818 2471 : FlxV_Flv_multieval(GEN P, GEN xa, ulong p)
2819 : {
2820 2471 : pari_sp av = avma;
2821 2471 : GEN s = producttree_scheme(lg(xa)-1);
2822 2471 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2823 2471 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2824 2471 : return gc_upto(av, FlxV_Flv_multieval_tree(P, xa, T, p, pi));
2825 : }
2826 :
2827 : GEN
2828 368511 : Flv_polint(GEN xa, GEN ya, ulong p, long vs)
2829 : {
2830 368511 : pari_sp av = avma;
2831 368511 : GEN s = producttree_scheme(lg(xa)-1);
2832 368521 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2833 368521 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2834 368524 : long m = lg(T)-1;
2835 368524 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2836 368519 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2837 368523 : return gc_leaf(av, FlvV_polint_tree(T, R, s, xa, ya, p, pi, vs));
2838 : }
2839 :
2840 : GEN
2841 103577 : Flv_Flm_polint(GEN xa, GEN ya, ulong p, long vs)
2842 : {
2843 103577 : pari_sp av = avma;
2844 103577 : GEN s = producttree_scheme(lg(xa)-1);
2845 103577 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2846 103577 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2847 103571 : long i, m = lg(T)-1, l = lg(ya)-1;
2848 103571 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2849 103572 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2850 103569 : GEN M = cgetg(l+1, t_VEC);
2851 1159288 : for (i=1; i<=l; i++)
2852 1055719 : gel(M,i) = FlvV_polint_tree(T, R, s, xa, gel(ya,i), p, pi, vs);
2853 103569 : return gc_upto(av, M);
2854 : }
2855 :
2856 : GEN
2857 153445 : Flv_invVandermonde(GEN L, ulong den, ulong p)
2858 : {
2859 153445 : pari_sp av = avma;
2860 153445 : long i, n = lg(L);
2861 : GEN M, R;
2862 153445 : GEN s = producttree_scheme(n-1);
2863 153445 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2864 153445 : GEN tree = Flv_producttree(L, s, p, pi, 0);
2865 153445 : long m = lg(tree)-1;
2866 153445 : GEN T = gmael(tree, m, 1);
2867 153445 : R = Flv_inv(Flx_Flv_multieval_tree(Flx_deriv(T, p), L, tree, p, pi), p);
2868 153445 : if (den!=1) R = Flv_Fl_mul(R, den, p);
2869 153445 : M = cgetg(n, t_MAT);
2870 603585 : for (i = 1; i < n; i++)
2871 : {
2872 450140 : GEN P = Flx_Fl_mul(Flx_div_by_X_x(T, uel(L,i), p, NULL), uel(R,i), p);
2873 450140 : gel(M,i) = Flx_to_Flv(P, n-1);
2874 : }
2875 153445 : return gc_GEN(av, M);
2876 : }
2877 :
2878 : /***********************************************************************/
2879 : /** Flxq **/
2880 : /***********************************************************************/
2881 : /* Flxq objects are Flx modulo another Flx called q. */
2882 :
2883 : /* Product of y and x in Z/pZ[X]/(T), as t_VECSMALL. */
2884 : GEN
2885 193788737 : Flxq_mul_pre(GEN x,GEN y,GEN T,ulong p,ulong pi)
2886 193788737 : { return Flx_rem_pre(Flx_mul_pre(x,y,p,pi),T,p,pi); }
2887 : GEN
2888 13191092 : Flxq_mul(GEN x,GEN y,GEN T,ulong p)
2889 13191092 : { return Flxq_mul_pre(x,y,T,p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2890 :
2891 : GEN
2892 282009361 : Flxq_sqr_pre(GEN x,GEN T,ulong p,ulong pi)
2893 282009361 : { return Flx_rem_pre(Flx_sqr_pre(x, p,pi), T, p,pi); }
2894 : /* Square of y in Z/pZ[X]/(T), as t_VECSMALL. */
2895 : GEN
2896 2759489 : Flxq_sqr(GEN x,GEN T,ulong p)
2897 2759489 : { return Flxq_sqr_pre(x,T,p,SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2898 :
2899 : static GEN
2900 1550928 : _Flxq_red(void *E, GEN x)
2901 1550928 : { struct _Flxq *s = (struct _Flxq *)E;
2902 1550928 : return Flx_rem_pre(x, s->T, s->p, s->pi); }
2903 : #if 0
2904 : static GEN
2905 : _Flx_sub(void *E, GEN x, GEN y)
2906 : { struct _Flxq *s = (struct _Flxq *)E;
2907 : return Flx_sub(x,y,s->p); }
2908 : #endif
2909 : static GEN
2910 274002867 : _Flxq_sqr(void *data, GEN x)
2911 : {
2912 274002867 : struct _Flxq *D = (struct _Flxq*)data;
2913 274002867 : return Flxq_sqr_pre(x, D->T, D->p, D->pi);
2914 : }
2915 : static GEN
2916 152578554 : _Flxq_mul(void *data, GEN x, GEN y)
2917 : {
2918 152578554 : struct _Flxq *D = (struct _Flxq*)data;
2919 152578554 : return Flxq_mul_pre(x,y, D->T, D->p, D->pi);
2920 : }
2921 : static GEN
2922 22417236 : _Flxq_one(void *data)
2923 : {
2924 22417236 : struct _Flxq *D = (struct _Flxq*)data;
2925 22417236 : return pol1_Flx(get_Flx_var(D->T));
2926 : }
2927 :
2928 : static GEN
2929 23215649 : _Flxq_powu_i(struct _Flxq *D, GEN x, ulong n)
2930 23215649 : { return gen_powu_i(x, n, (void*)D, &_Flxq_sqr, &_Flxq_mul); }
2931 : static GEN
2932 68 : _Flxq_powu(struct _Flxq *D, GEN x, ulong n)
2933 68 : { pari_sp av = avma; return gc_leaf(av, _Flxq_powu_i(D, x, n)); }
2934 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2935 : GEN
2936 24472061 : Flxq_powu_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
2937 : {
2938 : pari_sp av;
2939 : struct _Flxq D;
2940 24472061 : switch(n)
2941 : {
2942 0 : case 0: return pol1_Flx(get_Flx_var(T));
2943 279942 : case 1: return Flx_copy(x);
2944 974890 : case 2: return Flxq_sqr_pre(x, T, p, pi);
2945 : }
2946 23217229 : av = avma; set_Flxq_pre(&D, T, p, pi);
2947 23216331 : return gc_leaf(av, _Flxq_powu_i(&D, x, n));
2948 : }
2949 : GEN
2950 488426 : Flxq_powu(GEN x, ulong n, GEN T, ulong p)
2951 488426 : { return Flxq_powu_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2952 :
2953 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2954 : GEN
2955 25426834 : Flxq_pow_pre(GEN x, GEN n, GEN T, ulong p, ulong pi)
2956 : {
2957 25426834 : pari_sp av = avma;
2958 : struct _Flxq D;
2959 : GEN y;
2960 25426834 : long s = signe(n);
2961 25426834 : if (!s) return pol1_Flx(get_Flx_var(T));
2962 25350204 : if (s < 0) x = Flxq_inv_pre(x,T,p,pi);
2963 25350204 : if (is_pm1(n)) return s < 0 ? x : Flx_copy(x);
2964 24830496 : set_Flxq_pre(&D, T, p, pi);
2965 24830494 : y = gen_pow_i(x, n, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2966 24830422 : return gc_leaf(av, y);
2967 : }
2968 : GEN
2969 932569 : Flxq_pow(GEN x, GEN n, GEN T, ulong p)
2970 932569 : { return Flxq_pow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2971 :
2972 : GEN
2973 28 : Flxq_pow_init_pre(GEN x, GEN n, long k, GEN T, ulong p, ulong pi)
2974 : {
2975 28 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2976 28 : return gen_pow_init(x, n, k, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2977 : }
2978 : GEN
2979 0 : Flxq_pow_init(GEN x, GEN n, long k, GEN T, ulong p)
2980 0 : { return Flxq_pow_init_pre(x, n, k, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2981 :
2982 : GEN
2983 4391 : Flxq_pow_table_pre(GEN R, GEN n, GEN T, ulong p, ulong pi)
2984 : {
2985 4391 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2986 4391 : return gen_pow_table(R, n, (void*)&D, &_Flxq_one, &_Flxq_mul);
2987 : }
2988 : GEN
2989 0 : Flxq_pow_table(GEN R, GEN n, GEN T, ulong p)
2990 0 : { return Flxq_pow_table_pre(R, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2991 :
2992 : /* Inverse of x in Z/lZ[X]/(T) or NULL if inverse doesn't exist
2993 : * not stack clean. */
2994 : GEN
2995 5428980 : Flxq_invsafe_pre(GEN x, GEN T, ulong p, ulong pi)
2996 : {
2997 5428980 : GEN V, z = Flx_extgcd_pre(get_Flx_mod(T), x, p, pi, NULL, &V);
2998 : ulong iz;
2999 5429122 : if (degpol(z)) return NULL;
3000 5428455 : iz = Fl_inv(uel(z,2), p);
3001 5428444 : return Flx_Fl_mul_pre(V, iz, p, pi);
3002 : }
3003 : GEN
3004 670606 : Flxq_invsafe(GEN x, GEN T, ulong p)
3005 670606 : { return Flxq_invsafe_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3006 :
3007 : GEN
3008 4284256 : Flxq_inv_pre(GEN x, GEN T, ulong p, ulong pi)
3009 : {
3010 4284256 : pari_sp av=avma;
3011 4284256 : GEN U = Flxq_invsafe_pre(x, T, p, pi);
3012 4284249 : if (!U) pari_err_INV("Flxq_inv",Flx_to_ZX(x));
3013 4284242 : return gc_leaf(av, U);
3014 : }
3015 : GEN
3016 335768 : Flxq_inv(GEN x, GEN T, ulong p)
3017 335768 : { return Flxq_inv_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3018 :
3019 : GEN
3020 2417764 : Flxq_div_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
3021 : {
3022 2417764 : pari_sp av = avma;
3023 2417764 : return gc_leaf(av, Flxq_mul_pre(x,Flxq_inv_pre(y,T,p,pi),T,p,pi));
3024 : }
3025 : GEN
3026 237864 : Flxq_div(GEN x, GEN y, GEN T, ulong p)
3027 237864 : { return Flxq_div_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3028 :
3029 : GEN
3030 22420364 : Flxq_powers_pre(GEN x, long l, GEN T, ulong p, ulong pi)
3031 : {
3032 22420364 : int use_sqr = 2*degpol(x) >= get_Flx_degree(T);
3033 22416748 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3034 22414455 : return gen_powers(x, l, use_sqr, (void*)&D, &_Flxq_sqr, &_Flxq_mul, &_Flxq_one);
3035 : }
3036 : GEN
3037 232068 : Flxq_powers(GEN x, long l, GEN T, ulong p)
3038 232068 : { return Flxq_powers_pre(x, l, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3039 :
3040 : GEN
3041 170668 : Flxq_matrix_pow_pre(GEN y, long n, long m, GEN P, ulong l, ulong li)
3042 170668 : { return FlxV_to_Flm(Flxq_powers_pre(y,m-1,P,l,li),n); }
3043 : GEN
3044 399 : Flxq_matrix_pow(GEN y, long n, long m, GEN P, ulong l)
3045 399 : { return Flxq_matrix_pow_pre(y, n, m, P, l, SMALL_ULONG(l)? 0: get_Fl_red(l)); }
3046 :
3047 : GEN
3048 13803875 : Flx_Frobenius_pre(GEN T, ulong p, ulong pi)
3049 13803875 : { return Flxq_powu_pre(polx_Flx(get_Flx_var(T)), p, T, p, pi); }
3050 : GEN
3051 86486 : Flx_Frobenius(GEN T, ulong p)
3052 86486 : { return Flx_Frobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3053 :
3054 : GEN
3055 86568 : Flx_matFrobenius_pre(GEN T, ulong p, ulong pi)
3056 : {
3057 86568 : long n = get_Flx_degree(T);
3058 86568 : return Flxq_matrix_pow_pre(Flx_Frobenius_pre(T, p, pi), n, n, T, p, pi);
3059 : }
3060 : GEN
3061 0 : Flx_matFrobenius(GEN T, ulong p)
3062 0 : { return Flx_matFrobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3063 :
3064 : static GEN
3065 12968040 : Flx_blocks_Flm(GEN P, long n, long m)
3066 : {
3067 12968040 : GEN z = cgetg(m+1,t_MAT);
3068 12967844 : long i,j, k=2, l = lg(P);
3069 37085313 : for(i=1; i<=m; i++)
3070 : {
3071 24121110 : GEN zi = cgetg(n+1,t_VECSMALL);
3072 24117469 : gel(z,i) = zi;
3073 111710177 : for(j=1; j<=n; j++)
3074 87592708 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3075 : }
3076 12964203 : return z;
3077 : }
3078 :
3079 : GEN
3080 517122 : Flx_blocks(GEN P, long n, long m)
3081 : {
3082 517122 : GEN z = cgetg(m+1,t_VEC);
3083 516893 : long i,j, k=2, l = lg(P);
3084 1549252 : for(i=1; i<=m; i++)
3085 : {
3086 1032463 : GEN zi = cgetg(n+2,t_VECSMALL);
3087 1031815 : zi[1] = P[1];
3088 1031815 : gel(z,i) = zi;
3089 6471950 : for(j=2; j<n+2; j++)
3090 5440135 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3091 1031815 : zi = Flx_renormalize(zi, n+2);
3092 : }
3093 516789 : return z;
3094 : }
3095 :
3096 : static GEN
3097 12968779 : FlxV_to_Flm_lg(GEN x, long m, long n)
3098 : {
3099 : long i;
3100 12968779 : GEN y = cgetg(n+1, t_MAT);
3101 61473155 : for (i=1; i<=n; i++) gel(y,i) = Flx_to_Flv(gel(x,i), m);
3102 12966087 : return y;
3103 : }
3104 :
3105 : /* allow pi = 0 (SMALL_ULONG) */
3106 : GEN
3107 13167577 : Flx_FlxqV_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3108 : {
3109 13167577 : pari_sp btop, av = avma;
3110 13167577 : long sv = get_Flx_var(T), m = get_Flx_degree(T);
3111 13167770 : long i, l = lg(x)-1, lQ = lgpol(Q), n, d;
3112 : GEN A, B, C, S, g;
3113 13168560 : if (lQ == 0) return pol0_Flx(sv);
3114 12969639 : if (lQ <= l)
3115 : {
3116 6459185 : n = l;
3117 6459185 : d = 1;
3118 : }
3119 : else
3120 : {
3121 6510454 : n = l-1;
3122 6510454 : d = (lQ+n-1)/n;
3123 : }
3124 12969639 : A = FlxV_to_Flm_lg(x, m, n);
3125 12967973 : B = Flx_blocks_Flm(Q, n, d);
3126 12966933 : C = gc_upto(av, Flm_mul(A, B, p));
3127 12970089 : g = gel(x, l);
3128 12970089 : if (pi && SMALL_ULONG(p)) pi = 0;
3129 12970089 : T = Flx_get_red_pre(T, p, pi);
3130 12969674 : btop = avma;
3131 12969674 : S = Flv_to_Flx(gel(C, d), sv);
3132 24125634 : for (i = d-1; i>0; i--)
3133 : {
3134 11156955 : S = Flx_add(Flxq_mul_pre(S, g, T, p, pi), Flv_to_Flx(gel(C,i), sv), p);
3135 11155879 : if (gc_needed(btop,1))
3136 0 : S = gc_leaf(btop, S);
3137 : }
3138 12968679 : return gc_leaf(av, S);
3139 : }
3140 : GEN
3141 5082 : Flx_FlxqV_eval(GEN Q, GEN x, GEN T, ulong p)
3142 5082 : { return Flx_FlxqV_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3143 :
3144 : /* allow pi = 0 (SMALL_ULONG) */
3145 : GEN
3146 2444054 : Flx_Flxq_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3147 : {
3148 2444054 : pari_sp av = avma;
3149 : GEN z, V;
3150 2444054 : long d = degpol(Q), rtd;
3151 2444053 : if (d < 0) return pol0_Flx(get_Flx_var(T));
3152 2443962 : rtd = (long) sqrt((double)d);
3153 2443962 : T = Flx_get_red_pre(T, p, pi);
3154 2443974 : V = Flxq_powers_pre(x, rtd, T, p, pi);
3155 2444007 : z = Flx_FlxqV_eval_pre(Q, V, T, p, pi);
3156 2444007 : return gc_upto(av, z);
3157 : }
3158 : GEN
3159 790612 : Flx_Flxq_eval(GEN Q, GEN x, GEN T, ulong p)
3160 790612 : { return Flx_Flxq_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3161 :
3162 : /* allow pi = 0 (SMALL_ULONG) */
3163 : GEN
3164 0 : FlxC_FlxqV_eval_pre(GEN x, GEN v, GEN T, ulong p, ulong pi)
3165 0 : { pari_APPLY_type(t_COL, Flx_FlxqV_eval_pre(gel(x,i), v, T, p, pi)) }
3166 : GEN
3167 0 : FlxC_FlxqV_eval(GEN x, GEN v, GEN T, ulong p)
3168 0 : { return FlxC_FlxqV_eval_pre(x, v, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3169 :
3170 : /* allow pi = 0 (SMALL_ULONG) */
3171 : GEN
3172 0 : FlxC_Flxq_eval_pre(GEN x, GEN F, GEN T, ulong p, ulong pi)
3173 : {
3174 0 : long d = brent_kung_optpow(get_Flx_degree(T)-1,lg(x)-1,1);
3175 0 : GEN Fp = Flxq_powers_pre(F, d, T, p, pi);
3176 0 : return FlxC_FlxqV_eval_pre(x, Fp, T, p, pi);
3177 : }
3178 : GEN
3179 0 : FlxC_Flxq_eval(GEN x, GEN F, GEN T, ulong p)
3180 0 : { return FlxC_Flxq_eval_pre(x, F, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3181 :
3182 : #if 0
3183 : static struct bb_algebra Flxq_algebra = { _Flxq_red, _Flx_add, _Flx_sub,
3184 : _Flxq_mul, _Flxq_sqr, _Flxq_one, _Flxq_zero};
3185 : #endif
3186 :
3187 : static GEN
3188 46252 : Flxq_autpow_sqr(void *E, GEN x)
3189 : {
3190 46252 : struct _Flxq *D = (struct _Flxq*)E;
3191 46252 : return Flx_Flxq_eval_pre(x, x, D->T, D->p, D->pi);
3192 : }
3193 : static GEN
3194 20696 : Flxq_autpow_msqr(void *E, GEN x)
3195 : {
3196 20696 : struct _Flxq *D = (struct _Flxq*)E;
3197 20696 : return Flx_FlxqV_eval_pre(Flxq_autpow_sqr(E, x), D->aut, D->T, D->p, D->pi);
3198 : }
3199 :
3200 : GEN
3201 67493 : Flxq_autpow_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3202 : {
3203 67493 : pari_sp av = avma;
3204 : struct _Flxq D;
3205 : long d;
3206 67493 : if (n==0) return Flx_rem_pre(polx_Flx(x[1]), T, p, pi);
3207 67486 : if (n==1) return Flx_rem_pre(x, T, p, pi);
3208 31380 : set_Flxq_pre(&D, T, p, pi);
3209 31380 : d = brent_kung_optpow(get_Flx_degree(T), hammingu(n)-1, 1);
3210 31380 : D.aut = Flxq_powers_pre(x, d, T, p, D.pi);
3211 31380 : x = gen_powu_fold_i(x,n,(void*)&D,Flxq_autpow_sqr,Flxq_autpow_msqr);
3212 31380 : return gc_GEN(av, x);
3213 : }
3214 : GEN
3215 7 : Flxq_autpow(GEN x, ulong n, GEN T, ulong p)
3216 7 : { return Flxq_autpow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3217 :
3218 : GEN
3219 1667 : Flxq_autpowers(GEN x, ulong l, GEN T, ulong p)
3220 : {
3221 1667 : long d, vT = get_Flx_var(T), dT = get_Flx_degree(T);
3222 : ulong i, pi;
3223 1667 : pari_sp av = avma;
3224 1667 : GEN xp, V = cgetg(l+2,t_VEC);
3225 1667 : gel(V,1) = polx_Flx(vT); if (l==0) return V;
3226 1667 : gel(V,2) = gcopy(x); if (l==1) return V;
3227 1667 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3228 1667 : T = Flx_get_red_pre(T, p, pi);
3229 1667 : d = brent_kung_optpow(dT-1, l-1, 1);
3230 1667 : xp = Flxq_powers_pre(x, d, T, p, pi);
3231 6998 : for(i = 3; i < l+2; i++)
3232 5331 : gel(V,i) = Flx_FlxqV_eval_pre(gel(V,i-1), xp, T, p, pi);
3233 1667 : return gc_GEN(av, V);
3234 : }
3235 :
3236 : static GEN
3237 112472 : Flxq_autsum_mul(void *E, GEN x, GEN y)
3238 : {
3239 112472 : struct _Flxq *D = (struct _Flxq*)E;
3240 112472 : GEN T = D->T;
3241 112472 : ulong p = D->p, pi = D->pi;
3242 112472 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3243 112472 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3244 112472 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3245 112472 : GEN V2 = Flxq_powers_pre(phi2, d, T, p, pi);
3246 112472 : GEN phi3 = Flx_FlxqV_eval_pre(phi1, V2, T, p, pi);
3247 112472 : GEN aphi = Flx_FlxqV_eval_pre(a1, V2, T, p, pi);
3248 112472 : GEN a3 = Flxq_mul_pre(aphi, a2, T, p, pi);
3249 112472 : return mkvec2(phi3, a3);
3250 : }
3251 : static GEN
3252 105110 : Flxq_autsum_sqr(void *E, GEN x)
3253 105110 : { return Flxq_autsum_mul(E, x, x); }
3254 :
3255 : static GEN
3256 98768 : Flxq_autsum_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3257 : {
3258 98768 : pari_sp av = avma;
3259 98768 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3260 98768 : x = gen_powu_i(x,n,(void*)&D,Flxq_autsum_sqr,Flxq_autsum_mul);
3261 98768 : return gc_GEN(av, x);
3262 : }
3263 : GEN
3264 0 : Flxq_autsum(GEN x, ulong n, GEN T, ulong p)
3265 0 : { return Flxq_autsum_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3266 :
3267 : static GEN
3268 782833 : Flxq_auttrace_mul(void *E, GEN x, GEN y)
3269 : {
3270 782833 : struct _Flxq *D = (struct _Flxq*)E;
3271 782833 : GEN T = D->T;
3272 782833 : ulong p = D->p, pi = D->pi;
3273 782833 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3274 782833 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3275 782833 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3276 782853 : GEN V1 = Flxq_powers_pre(phi1, d, T, p, pi);
3277 782798 : GEN phi3 = Flx_FlxqV_eval_pre(phi2, V1, T, p, pi);
3278 782794 : GEN aphi = Flx_FlxqV_eval_pre(a2, V1, T, p, pi);
3279 782803 : GEN a3 = Flx_add(a1, aphi, p);
3280 782804 : return mkvec2(phi3, a3);
3281 : }
3282 :
3283 : static GEN
3284 655317 : Flxq_auttrace_sqr(void *E, GEN x)
3285 655317 : { return Flxq_auttrace_mul(E, x, x); }
3286 :
3287 : GEN
3288 962207 : Flxq_auttrace_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3289 : {
3290 962207 : pari_sp av = avma;
3291 : struct _Flxq D;
3292 962207 : set_Flxq_pre(&D, T, p, pi);
3293 962210 : x = gen_powu_i(x,n,(void*)&D,Flxq_auttrace_sqr,Flxq_auttrace_mul);
3294 962166 : return gc_GEN(av, x);
3295 : }
3296 : GEN
3297 0 : Flxq_auttrace(GEN x, ulong n, GEN T, ulong p)
3298 0 : { return Flxq_auttrace_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3299 :
3300 : static long
3301 394046 : bounded_order(ulong p, GEN b, long k)
3302 : {
3303 394046 : GEN a = modii(utoipos(p), b);
3304 : long i;
3305 809790 : for(i = 1; i < k; i++)
3306 : {
3307 515152 : if (equali1(a)) return i;
3308 415745 : a = modii(muliu(a,p),b);
3309 : }
3310 294638 : return 0;
3311 : }
3312 :
3313 : /* n = (p^d-a)\b
3314 : * b = bb*p^vb
3315 : * p^k = 1 [bb]
3316 : * d = m*k+r+vb
3317 : * u = (p^k-1)/bb;
3318 : * v = (p^(r+vb)-a)/b;
3319 : * w = (p^(m*k)-1)/(p^k-1)
3320 : * n = p^r*w*u+v
3321 : * w*u = p^vb*(p^(m*k)-1)/b
3322 : * n = p^(r+vb)*(p^(m*k)-1)/b+(p^(r+vb)-a)/b */
3323 : static GEN
3324 24385336 : Flxq_pow_Frobenius(GEN x, GEN n, GEN aut, GEN T, ulong p, ulong pi)
3325 : {
3326 24385336 : pari_sp av=avma;
3327 24385336 : long d = get_Flx_degree(T);
3328 24385336 : GEN an = absi_shallow(n), z, q;
3329 24385337 : if (abscmpiu(an,p)<0 || cmpis(an,d)<=0) return Flxq_pow_pre(x, n, T, p, pi);
3330 394411 : q = powuu(p, d);
3331 394410 : if (dvdii(q, n))
3332 : {
3333 317 : long vn = logint(an, utoipos(p));
3334 317 : GEN autvn = vn==1 ? aut: Flxq_autpow_pre(aut,vn,T,p,pi);
3335 317 : z = Flx_Flxq_eval_pre(x,autvn,T,p,pi);
3336 : } else
3337 : {
3338 394094 : GEN b = diviiround(q, an), a = subii(q, mulii(an,b));
3339 : GEN bb, u, v, autk;
3340 394094 : long vb = Z_lvalrem(b,p,&bb);
3341 394094 : long m, r, k = is_pm1(bb)? 1: bounded_order(p,bb,d);
3342 394093 : if (!k || d-vb < k) return Flxq_pow_pre(x,n, T,p,pi);
3343 99448 : m = (d-vb)/k; r = (d-vb)%k;
3344 99448 : u = diviiexact(subiu(powuu(p,k),1),bb);
3345 99448 : v = diviiexact(subii(powuu(p,r+vb),a),b);
3346 99448 : autk = k==1 ? aut: Flxq_autpow_pre(aut,k,T,p,pi);
3347 99448 : if (r)
3348 : {
3349 487 : GEN autr = r==1 ? aut: Flxq_autpow_pre(aut,r,T,p,pi);
3350 487 : z = Flx_Flxq_eval_pre(x,autr,T,p,pi);
3351 98961 : } else z = x;
3352 99448 : if (m > 1) z = gel(Flxq_autsum_pre(mkvec2(autk, z), m, T, p, pi), 2);
3353 99448 : if (!is_pm1(u)) z = Flxq_pow_pre(z, u, T, p, pi);
3354 99448 : if (signe(v)) z = Flxq_mul_pre(z, Flxq_pow_pre(x, v, T, p, pi), T, p, pi);
3355 : }
3356 99765 : return gc_upto(av,signe(n)>0 ? z : Flxq_inv_pre(z,T,p,pi));
3357 : }
3358 :
3359 : static GEN
3360 24377929 : _Flxq_pow(void *data, GEN x, GEN n)
3361 : {
3362 24377929 : struct _Flxq *D = (struct _Flxq*)data;
3363 24377929 : return Flxq_pow_Frobenius(x, n, D->aut, D->T, D->p, D->pi);
3364 : }
3365 :
3366 : static GEN
3367 5583 : _Flxq_rand(void *data)
3368 : {
3369 5583 : pari_sp av=avma;
3370 5583 : struct _Flxq *D = (struct _Flxq*)data;
3371 : GEN z;
3372 : do
3373 : {
3374 5584 : set_avma(av);
3375 5584 : z = random_Flx(get_Flx_degree(D->T),get_Flx_var(D->T),D->p);
3376 5584 : } while (lgpol(z)==0);
3377 5583 : return z;
3378 : }
3379 :
3380 : /* discrete log in FpXQ for a in Fp^*, g in FpXQ^* of order ord */
3381 : static GEN
3382 35539 : Fl_Flxq_log(ulong a, GEN g, GEN o, GEN T, ulong p)
3383 : {
3384 35539 : pari_sp av = avma;
3385 : GEN q,n_q,ord,ordp, op;
3386 :
3387 35539 : if (a == 1UL) return gen_0;
3388 : /* p > 2 */
3389 :
3390 35539 : ordp = utoi(p - 1);
3391 35539 : ord = get_arith_Z(o);
3392 35539 : if (!ord) ord = T? subiu(powuu(p, get_FpX_degree(T)), 1): ordp;
3393 35539 : if (a == p - 1) /* -1 */
3394 7739 : return gc_INT(av, shifti(ord,-1));
3395 27800 : ordp = gcdii(ordp, ord);
3396 27800 : op = typ(o)==t_MAT ? famat_Z_gcd(o, ordp) : ordp;
3397 :
3398 27800 : q = NULL;
3399 27800 : if (T)
3400 : { /* we want < g > = Fp^* */
3401 27800 : if (!equalii(ord,ordp)) {
3402 11906 : q = diviiexact(ord,ordp);
3403 11906 : g = Flxq_pow(g,q,T,p);
3404 : }
3405 : }
3406 27800 : n_q = Fp_log(utoi(a), utoipos(uel(g,2)), op, utoipos(p));
3407 27800 : if (lg(n_q)==1) return gc_leaf(av, n_q);
3408 27800 : if (q) n_q = mulii(q, n_q);
3409 27800 : return gc_INT(av, n_q);
3410 : }
3411 :
3412 : static GEN
3413 519205 : Flxq_easylog(void* E, GEN a, GEN g, GEN ord)
3414 : {
3415 519205 : struct _Flxq *f = (struct _Flxq *)E;
3416 519205 : GEN T = f->T;
3417 519205 : ulong p = f->p;
3418 519205 : long d = get_Flx_degree(T);
3419 519205 : if (Flx_equal1(a)) return gen_0;
3420 359485 : if (Flx_equal(a,g)) return gen_1;
3421 174421 : if (!degpol(a))
3422 35539 : return Fl_Flxq_log(uel(a,2), g, ord, T, p);
3423 138882 : if (typ(ord)!=t_INT || d <= 4 || d == 6 || abscmpiu(ord,1UL<<27)<0)
3424 138854 : return NULL;
3425 28 : return Flxq_log_index(a, g, ord, T, p);
3426 : }
3427 :
3428 : static const struct bb_group Flxq_star={_Flxq_mul,_Flxq_pow,_Flxq_rand,hash_GEN,Flx_equal,Flx_equal1,Flxq_easylog};
3429 :
3430 : const struct bb_group *
3431 280807 : get_Flxq_star(void **E, GEN T, ulong p)
3432 : {
3433 280807 : struct _Flxq *e = (struct _Flxq *) stack_malloc(sizeof(struct _Flxq));
3434 280807 : e->T = T; e->p = p; e->pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3435 280807 : e->aut = Flx_Frobenius_pre(T, p, e->pi);
3436 280807 : *E = (void*)e; return &Flxq_star;
3437 : }
3438 :
3439 : GEN
3440 97328 : Flxq_order(GEN a, GEN ord, GEN T, ulong p)
3441 : {
3442 : void *E;
3443 97328 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3444 97328 : return gen_order(a,ord,E,S);
3445 : }
3446 :
3447 : GEN
3448 164217 : Flxq_log(GEN a, GEN g, GEN ord, GEN T, ulong p)
3449 : {
3450 : void *E;
3451 164217 : pari_sp av = avma;
3452 164217 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3453 164217 : GEN v = get_arith_ZZM(ord), F = gmael(v,2,1);
3454 164217 : if (lg(F) > 1 && Flxq_log_use_index(veclast(F), T, p))
3455 24290 : v = mkvec2(gel(v, 1), ZM_famat_limit(gel(v, 2), int2n(27)));
3456 164217 : return gc_leaf(av, gen_PH_log(a, g, v, E, S));
3457 : }
3458 :
3459 : static GEN
3460 292682 : Flxq_sumautsum_sqr(void *E, GEN xzd)
3461 : {
3462 292682 : struct _Flxq *D = (struct _Flxq*)E;
3463 292682 : pari_sp av = avma;
3464 : GEN xi, zeta, delta, xi2, zeta2, delta2, temp, xipow;
3465 292682 : GEN T = D->T;
3466 292682 : ulong d, p = D-> p, pi = D->pi;
3467 292682 : xi = gel(xzd, 1); zeta = gel(xzd, 2); delta = gel(xzd, 3);
3468 :
3469 292682 : d = brent_kung_optpow(get_Flx_degree(T)-1,3,1);
3470 292682 : xipow = Flxq_powers_pre(xi, d, T, p, pi);
3471 :
3472 292682 : xi2 = Flx_FlxqV_eval_pre(xi, xipow, T, p, pi);
3473 292682 : zeta2 = Flxq_mul_pre(zeta, Flx_FlxqV_eval_pre(zeta, xipow, T, p, pi), T, p, pi);
3474 292682 : temp = Flxq_mul_pre(zeta, Flx_FlxqV_eval_pre(delta, xipow, T, p, pi), T, p, pi);
3475 292682 : delta2 = Flx_add(delta, temp, p);
3476 292682 : return gc_GEN(av, mkvec3(xi2, zeta2, delta2));
3477 : }
3478 :
3479 : static GEN
3480 40558 : Flxq_sumautsum_msqr(void *E, GEN xzd)
3481 : {
3482 40558 : struct _Flxq *D = (struct _Flxq*)E;
3483 40558 : pari_sp av = avma;
3484 : GEN xii, zetai, deltai, xzd2;
3485 40558 : GEN T = D->T, xi0pow = gel(D->aut, 1), zeta0 = gel(D->aut, 2);
3486 40558 : ulong p = D-> p, pi = D->pi;
3487 40558 : xzd2 = Flxq_sumautsum_sqr(E, xzd);
3488 40558 : xii = Flx_FlxqV_eval_pre(gel(xzd2, 1), xi0pow, T, p, pi);
3489 40558 : zetai = Flxq_mul_pre(zeta0, Flx_FlxqV_eval_pre(gel(xzd2, 2), xi0pow, T, p, pi), T, p, pi);
3490 40558 : deltai = Flx_add(gel(xzd2, 3), zetai, p);
3491 :
3492 40558 : return gc_GEN(av, mkvec3(xii, zetai, deltai));
3493 : }
3494 :
3495 : /*returns a + a^(1+s) + a^(1+s+2s) + ... + a^(1+s+...+is)
3496 : where ax = [a,s] with s an automorphism */
3497 : static GEN
3498 208704 : Flxq_sumautsum_pre(GEN ax, long i, GEN T, ulong p, ulong pi) {
3499 208704 : pari_sp av = avma;
3500 : GEN a, xi, zeta, vec, res;
3501 : struct _Flxq D;
3502 : ulong d;
3503 208704 : D.T = Flx_get_red(T, p); D.p = p; D.pi = pi;
3504 208704 : a = gel(ax, 1); xi = gel(ax,2);
3505 208704 : d = brent_kung_optpow(get_Flx_degree(T)-1,2*(hammingu(i)-1),1);
3506 208704 : zeta = Flx_Flxq_eval_pre(a, xi, T, p, pi);
3507 208704 : D.aut = mkvec2(Flxq_powers_pre(xi, d, T, p, pi), zeta);
3508 :
3509 208704 : vec = gen_powu_fold(mkvec3(xi, zeta, zeta), i, (void *)&D, Flxq_sumautsum_sqr, Flxq_sumautsum_msqr);
3510 208704 : res = Flxq_mul_pre(a, Flx_add(pol1_Flx(get_Flx_var(T)), gel(vec, 3), p), T, p, pi);
3511 :
3512 208704 : return gc_GEN(av, res);
3513 : }
3514 :
3515 : /*algorithm from
3516 : Doliskani, J., & Schost, E. (2014).
3517 : Taking roots over high extensions of finite fields*/
3518 : static GEN
3519 35707 : Flxq_sqrtl_spec_pre(GEN z, GEN n, GEN T, ulong p, ulong pi, GEN *zetan)
3520 : {
3521 35707 : pari_sp av = avma;
3522 : GEN psn, c, b, new_z, beta, x, y, w, ax, g, zeta;
3523 35707 : long s, l, v = get_Flx_var(T), d = get_Flx_degree(T);
3524 : ulong zeta2, beta2;
3525 35707 : s = itos(Fp_order(utoi(p), stoi(d), n));
3526 35707 : if(s >= d || d % s != 0)
3527 0 : pari_err(e_MISC, "expected p's order mod n to divide the degree of T");
3528 35707 : l = d/s;
3529 35707 : if (!lgpol(z)) return pol0_Flx(get_Flx_var(T));
3530 35707 : T = Flx_get_red(T, p);
3531 35707 : ax = mkvec2(NULL, Flxq_autpow_pre(Flx_Frobenius_pre(T,p,pi), s, T, p,pi));
3532 35707 : psn = diviiexact(subiu(powuu(p, s), 1), n);
3533 : do {
3534 39689 : do c = random_Flx(d, v, p); while (!lgpol(c));
3535 39186 : new_z = Flxq_mul_pre(z, Flxq_pow_pre(c, n, T, p,pi), T, p,pi);
3536 39184 : gel(ax,1) = Flxq_pow_pre(new_z, psn, T, p,pi);
3537 :
3538 : /*If l == 2, b has to be 1 + a^((p^s-1)/n)*/
3539 39184 : if(l == 2) y = gel(ax, 1);
3540 1235 : else y = Flxq_sumautsum_pre(ax, l-2, T, p, pi);
3541 39184 : b = Flx_Fl_add(y, 1, p);
3542 39185 : } while (!lgpol(b));
3543 :
3544 35707 : x = Flxq_mul_pre(new_z, Flxq_pow_pre(b, n, T, p,pi), T, p,pi);
3545 35705 : if(s == 1) {
3546 35621 : if (degpol(x) > 0) return gc_NULL(av);
3547 35579 : beta2 = Fl_sqrtn(Flx_constant(x), umodiu(n, p), p, &zeta2);
3548 35581 : if (beta2==~0UL) return gc_NULL(av);
3549 35581 : if(zetan) *zetan = monomial_Flx(zeta2, 0, get_Flx_var(T));
3550 35581 : w = Flx_Fl_mul(Flxq_inv_pre(Flxq_mul_pre(b, c, T, p,pi), T, p,pi), beta2, p);
3551 35580 : (void)gc_all(av, zetan? 2: 1, &w, zetan);
3552 35581 : return w;
3553 : }
3554 84 : g = Flxq_minpoly(x, T, p);
3555 84 : if (degpol(g) != s) return gc_NULL(av);
3556 77 : beta = Flxq_sqrtn(polx_Flx(get_Flx_var(T)), n, g, p, &zeta);
3557 77 : if (!beta) return gc_NULL(av);
3558 :
3559 77 : if(zetan) *zetan = Flx_Flxq_eval(zeta, x, T, p);
3560 77 : beta = Flx_Flxq_eval(beta, x, T, p);
3561 77 : w = Flxq_mul_pre(Flxq_inv_pre(Flxq_mul_pre(b, c, T, p,pi), T, p,pi), beta, T, p,pi);
3562 77 : (void)gc_all(av, zetan? 2: 1, &w, zetan);
3563 77 : return w;
3564 : }
3565 :
3566 : static GEN
3567 19262 : Flxq_sqrtn_spec_pre(GEN a, GEN n, GEN T, ulong p, ulong pi, GEN q, GEN *zetan)
3568 : {
3569 19262 : pari_sp ltop = avma;
3570 : GEN z, m, u1, u2;
3571 : int is_1;
3572 19262 : if (is_pm1(n))
3573 : {
3574 847 : if (zetan) *zetan = pol1_Flx(get_Flx_var(T));
3575 847 : return signe(n) < 0? Flxq_inv_pre(a, T, p,pi): gcopy(a);
3576 : }
3577 18415 : is_1 = gequal1(a);
3578 18415 : if (is_1 && !zetan) return gcopy(a);
3579 18415 : z = pol1_Flx(get_Flx_var(T));
3580 18415 : m = bezout(n,q,&u1,&u2);
3581 18415 : if (!is_pm1(m))
3582 : {
3583 18415 : GEN F = Z_factor(m);
3584 18415 : long i, j, j2 = 0; /* -Wall */
3585 : GEN y, l;
3586 18415 : pari_sp av1 = avma;
3587 36907 : for (i = nbrows(F); i; i--)
3588 : {
3589 18541 : l = gcoeff(F,i,1);
3590 18541 : j = itos(gcoeff(F,i,2));
3591 18541 : if(zetan) {
3592 188 : a = Flxq_sqrtl_spec_pre(a,l,T,p,pi,&y);
3593 237 : if (!a) return gc_NULL(ltop);
3594 188 : j--;
3595 188 : j2 = j;
3596 : }
3597 18541 : if (!is_1 && j > 0) {
3598 : do
3599 : {
3600 35309 : a = Flxq_sqrtl_spec_pre(a,l,T,p,pi,NULL);
3601 35309 : if (!a) return gc_NULL(ltop);
3602 35260 : } while (--j);
3603 : }
3604 : /*This is below finding a's root,
3605 : so we don't spend time doing this, if a is not n-th root*/
3606 18492 : if(zetan) {
3607 391 : for(; j2>0; j2--) y = Flxq_sqrtl_spec_pre(y, l, T, p,pi,NULL);
3608 181 : z = Flxq_mul_pre(z, y, T, p,pi);
3609 : }
3610 18492 : if (gc_needed(ltop,1))
3611 : { /* n can have lots of prime factors*/
3612 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxq_sqrtn_spec");
3613 0 : (void)gc_all(av1, zetan? 2: 1, &a, &z);
3614 : }
3615 : }
3616 : }
3617 :
3618 18366 : if (!equalii(m, n))
3619 119 : a = Flxq_pow_pre(a,modii(u1,q), T, p,pi);
3620 18366 : if (zetan)
3621 : {
3622 181 : *zetan = z;
3623 181 : (void)gc_all(ltop,2,&a,zetan);
3624 : }
3625 : else /* is_1 is 0: a was modified above -> gc_upto valid */
3626 18185 : a = gc_upto(ltop, a);
3627 18366 : return a;
3628 : }
3629 :
3630 : GEN
3631 20456 : Flxq_sqrtn(GEN a, GEN n, GEN T, ulong p, GEN *zeta)
3632 : {
3633 20456 : if (!lgpol(a))
3634 : {
3635 7 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
3636 0 : if (zeta)
3637 0 : *zeta=pol1_Flx(get_Flx_var(T));
3638 0 : return pol0_Flx(get_Flx_var(T));
3639 : }
3640 20449 : else if(p == 2) {
3641 1187 : pari_sp av = avma;
3642 : GEN z;
3643 1187 : z = F2xq_sqrtn(Flx_to_F2x(a), n, Flx_to_F2x(get_FpX_mod(T)), zeta);
3644 1187 : if (!z) return NULL;
3645 1187 : z = F2x_to_Flx(z);
3646 1187 : if (!zeta) return gc_leaf(av, z);
3647 0 : *zeta=F2x_to_Flx(*zeta);
3648 0 : return gc_all(av, 2, &z,zeta);
3649 : }
3650 : else
3651 : {
3652 : void *E;
3653 19262 : pari_sp av = avma;
3654 19262 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3655 19262 : GEN o = subiu(powuu(p,get_Flx_degree(T)), 1);
3656 : GEN m, u1, u2, l, zeta2, F, n2, z;
3657 19262 : long i, s, pi, d = get_Flx_degree(T);
3658 19262 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3659 19262 : m = bezout(n,o,&u1,&u2);
3660 19262 : F = Z_factor(m);
3661 41492 : for (i = nbrows(F); i; i--)
3662 : {
3663 22230 : l = gcoeff(F,i,1);
3664 22230 : s = itos(Fp_order(utoi(p), subiu(l, 1), l));
3665 : /*Flxq_sqrtn_spec only works if d > s and s | d
3666 : for those factors of m we use Flxq_sqrtn_spec
3667 : for the other factor we stay with gen_Shanks_sqrtn*/
3668 22230 : if(d <= s || d % s != 0) {
3669 3689 : gcoeff(F,i,2) = gen_0;
3670 : }
3671 18541 : else gcoeff(F,i,2) = stoi(Z_pval(n,l));
3672 : }
3673 19262 : F = factorback(F);
3674 19262 : z = Flxq_sqrtn_spec_pre(a,F,T, p,pi,o,zeta);
3675 19262 : if(!z) return gc_NULL(av);
3676 19213 : n2 = diviiexact(n, F);
3677 19213 : if(!gequal1(n2)) {
3678 3934 : if(zeta) zeta2 = gcopy(*zeta);
3679 3934 : z = gen_Shanks_sqrtn(z, n2, o, zeta, E, S);
3680 3934 : if (!z) return gc_NULL(av);
3681 3934 : if(zeta) *zeta = Flxq_mul_pre(*zeta, zeta2, T, p,pi);
3682 : }
3683 19213 : return gc_all(av, zeta?2:1, &z, zeta);
3684 : }
3685 : }
3686 :
3687 : GEN
3688 230484 : Flxq_sqrt_pre(GEN z, GEN T, ulong p, ulong pi)
3689 : {
3690 230484 : pari_sp av = avma;
3691 : long d;
3692 230484 : if (p==2)
3693 : {
3694 0 : GEN r = F2xq_sqrt(Flx_to_F2x(z), Flx_to_F2x(get_Flx_mod(T)));
3695 0 : return gc_upto(av, F2x_to_Flx(r));
3696 : }
3697 230484 : d = get_Flx_degree(T);
3698 230484 : if (d==2)
3699 : {
3700 65765 : GEN P = get_Flx_mod(T), s;
3701 65765 : ulong c = uel(P,2), b = uel(P,3), a = uel(P,4);
3702 65765 : ulong y = degpol(z)<1 ? 0: uel(z,3);
3703 65765 : if (a==1 && b==0)
3704 15226 : {
3705 16006 : ulong x = degpol(z)<1 ? Flx_constant(z): uel(z,2);
3706 16006 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), Fl_neg(c, p), p, pi);
3707 16006 : if (!r) return gc_NULL(av);
3708 15226 : s = mkvecsmall3(P[1], uel(r,1), uel(r,2));
3709 : }
3710 : else
3711 : {
3712 49759 : ulong b2 = Fl_halve(b, p), t = Fl_div(b2, a, p);
3713 49759 : ulong D = Fl_sub(Fl_sqr(b2, p), Fl_mul(a, c, p), p);
3714 49759 : ulong x = degpol(z)<1 ? Flx_constant(z): Fl_sub(uel(z,2), Fl_mul(uel(z,3), t, p), p);
3715 49759 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), D, p, pi);
3716 49759 : if (!r) return gc_NULL(av);
3717 47365 : s = mkvecsmall3(P[1], Fl_add(uel(r,1), Fl_mul(uel(r,2),t,p), p), uel(r,2));
3718 : }
3719 62591 : return gc_leaf(av, Flx_renormalize(s, 4));
3720 : }
3721 164719 : if (lgpol(z)<=1 && odd(d))
3722 : {
3723 11710 : pari_sp av = avma;
3724 11710 : ulong s = Fl_sqrt(Flx_constant(z), p);
3725 11710 : if (s==~0UL) return gc_NULL(av);
3726 11696 : return gc_GEN(av, Fl_to_Flx(s, get_Flx_var(T)));
3727 : } else
3728 : {
3729 : GEN c, b, new_z, x, y, w, ax;
3730 : ulong p2, beta;
3731 153009 : long v = get_Flx_var(T);
3732 153009 : if (!lgpol(z)) return pol0_Flx(v);
3733 152344 : T = Flx_get_red_pre(T, p, pi);
3734 152344 : ax = mkvec2(NULL, Flx_Frobenius_pre(T, p, pi));
3735 152344 : p2 = p >> 1; /* (p-1) / 2 */
3736 : do {
3737 208141 : do c = random_Flx(d, v, p); while (!lgpol(c));
3738 :
3739 207469 : new_z = Flxq_mul_pre(z, Flxq_sqr_pre(c, T, p, pi), T, p, pi);
3740 207469 : gel(ax, 1) = Flxq_powu_pre(new_z, p2, T, p, pi);
3741 207469 : y = Flxq_sumautsum_pre(ax, d-2, T, p, pi); /* d > 2 */
3742 207469 : b = Flx_Fl_add(y, 1UL, p);
3743 207469 : } while (!lgpol(b));
3744 :
3745 152344 : x = Flxq_mul_pre(new_z, Flxq_sqr_pre(b, T, p, pi), T, p, pi);
3746 152344 : if (degpol(x) > 0) return gc_NULL(av);
3747 145302 : beta = Fl_sqrt_pre(Flx_constant(x), p, pi);
3748 145302 : if (beta==~0UL) return gc_NULL(av);
3749 145302 : w = Flx_Fl_mul(Flxq_inv_pre(Flxq_mul_pre(b, c, T,p,pi), T,p,pi), beta, p);
3750 145302 : return gc_GEN(av, w);
3751 : }
3752 : }
3753 :
3754 : GEN
3755 230484 : Flxq_sqrt(GEN a, GEN T, ulong p)
3756 230484 : { return Flxq_sqrt_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3757 :
3758 : /* assume T irreducible mod p */
3759 : int
3760 404492 : Flxq_issquare(GEN x, GEN T, ulong p)
3761 : {
3762 404492 : if (lgpol(x) == 0 || p == 2) return 1;
3763 397989 : return krouu(Flxq_norm(x,T,p), p) == 1;
3764 : }
3765 :
3766 : /* assume T irreducible mod p */
3767 : int
3768 0 : Flxq_is2npower(GEN x, long n, GEN T, ulong p)
3769 : {
3770 : pari_sp av;
3771 : GEN m;
3772 0 : if (n==1) return Flxq_issquare(x, T, p);
3773 0 : if (lgpol(x) == 0 || p == 2) return 1;
3774 0 : av = avma;
3775 0 : m = shifti(subiu(powuu(p, get_Flx_degree(T)), 1), -n);
3776 0 : return gc_bool(av, Flx_equal1(Flxq_pow(x, m, T, p)));
3777 : }
3778 :
3779 : GEN
3780 113589 : Flxq_lroot_fast_pre(GEN a, GEN sqx, GEN T, long p, ulong pi)
3781 : {
3782 113589 : pari_sp av=avma;
3783 113589 : GEN A = Flx_splitting(a,p);
3784 113589 : return gc_leaf(av, FlxqV_dotproduct_pre(A,sqx,T,p,pi));
3785 : }
3786 : GEN
3787 0 : Flxq_lroot_fast(GEN a, GEN sqx, GEN T, long p)
3788 0 : { return Flxq_lroot_fast_pre(a, sqx, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3789 :
3790 : GEN
3791 25053 : Flxq_lroot_pre(GEN a, GEN T, long p, ulong pi)
3792 : {
3793 25053 : pari_sp av=avma;
3794 25053 : long n = get_Flx_degree(T), d = degpol(a);
3795 : GEN sqx, V;
3796 25053 : if (n==1) return leafcopy(a);
3797 25053 : if (n==2) return Flxq_powu_pre(a, p, T, p, pi);
3798 25053 : sqx = Flxq_autpow_pre(Flx_Frobenius_pre(T, p, pi), n-1, T, p, pi);
3799 25053 : if (d==1 && a[2]==0 && a[3]==1) return gc_leaf(av, sqx);
3800 0 : if (d>=p)
3801 : {
3802 0 : V = Flxq_powers_pre(sqx,p-1,T,p,pi);
3803 0 : return gc_leaf(av, Flxq_lroot_fast_pre(a,V,T,p,pi));
3804 : } else
3805 0 : return gc_leaf(av, Flx_Flxq_eval_pre(a,sqx,T,p,pi));
3806 : }
3807 : GEN
3808 0 : Flxq_lroot(GEN a, GEN T, long p)
3809 0 : { return Flxq_lroot_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3810 :
3811 : ulong
3812 443326 : Flxq_norm(GEN x, GEN TB, ulong p)
3813 : {
3814 443326 : GEN T = get_Flx_mod(TB);
3815 443326 : ulong y = Flx_resultant(T, x, p), L = Flx_lead(T);
3816 443326 : if (L==1 || lgpol(x)==0) return y;
3817 0 : return Fl_div(y, Fl_powu(L, (ulong)degpol(x), p), p);
3818 : }
3819 :
3820 : ulong
3821 4696 : Flxq_trace(GEN x, GEN TB, ulong p)
3822 : {
3823 4696 : pari_sp av = avma;
3824 : ulong t;
3825 4696 : GEN T = get_Flx_mod(TB);
3826 4696 : long n = degpol(T)-1;
3827 4696 : GEN z = Flxq_mul(x, Flx_deriv(T, p), TB, p);
3828 4696 : t = degpol(z)<n ? 0 : Fl_div(z[2+n],T[3+n],p);
3829 4696 : return gc_ulong(av, t);
3830 : }
3831 :
3832 : /*x must be reduced*/
3833 : GEN
3834 3624 : Flxq_charpoly(GEN x, GEN TB, ulong p)
3835 : {
3836 3624 : pari_sp ltop=avma;
3837 3624 : GEN T = get_Flx_mod(TB);
3838 3624 : long vs = evalvarn(fetch_var());
3839 3624 : GEN xm1 = deg1pol_shallow(pol1_Flx(x[1]),Flx_neg(x,p),vs);
3840 3624 : GEN r = Flx_FlxY_resultant(T, xm1, p);
3841 3624 : r[1] = x[1];
3842 3624 : (void)delete_var(); return gc_upto(ltop, r);
3843 : }
3844 :
3845 : /* Computing minimal polynomial : */
3846 : /* cf Shoup 'Efficient Computation of Minimal Polynomials */
3847 : /* in Algebraic Extensions of Finite Fields' */
3848 :
3849 : /* Let v a linear form, return the linear form z->v(tau*z)
3850 : that is, v*(M_tau) */
3851 :
3852 : static GEN
3853 1748638 : Flxq_transmul_init(GEN tau, GEN T, ulong p, ulong pi)
3854 : {
3855 : GEN bht;
3856 1748638 : GEN h, Tp = get_Flx_red(T, &h);
3857 1748631 : long n = degpol(Tp), vT = Tp[1];
3858 1748620 : GEN ft = Flx_recipspec(Tp+2, n+1, n+1);
3859 1748617 : GEN bt = Flx_recipspec(tau+2, lgpol(tau), n);
3860 1748627 : ft[1] = vT; bt[1] = vT;
3861 1748627 : if (h)
3862 2688 : bht = Flxn_mul_pre(bt, h, n-1, p, pi);
3863 : else
3864 : {
3865 1745939 : GEN bh = Flx_div_pre(Flx_shift(tau, n-1), T, p, pi);
3866 1745936 : bht = Flx_recipspec(bh+2, lgpol(bh), n-1);
3867 1745921 : bht[1] = vT;
3868 : }
3869 1748609 : return mkvec3(bt, bht, ft);
3870 : }
3871 :
3872 : static GEN
3873 4218010 : Flxq_transmul(GEN tau, GEN a, long n, ulong p, ulong pi)
3874 : {
3875 4218010 : pari_sp ltop = avma;
3876 : GEN t1, t2, t3, vec;
3877 4218010 : GEN bt = gel(tau, 1), bht = gel(tau, 2), ft = gel(tau, 3);
3878 4218010 : if (lgpol(a)==0) return pol0_Flx(a[1]);
3879 4186541 : t2 = Flx_shift(Flx_mul_pre(bt, a, p, pi),1-n);
3880 4186241 : if (lgpol(bht)==0) return gc_leaf(ltop, t2);
3881 3157763 : t1 = Flx_shift(Flx_mul_pre(ft, a, p, pi),-n);
3882 3157734 : t3 = Flxn_mul_pre(t1, bht, n-1, p, pi);
3883 3157792 : vec = Flx_sub(t2, Flx_shift(t3, 1), p);
3884 3157810 : return gc_leaf(ltop, vec);
3885 : }
3886 :
3887 : GEN
3888 810674 : Flxq_minpoly_pre(GEN x, GEN T, ulong p, ulong pi)
3889 : {
3890 810674 : pari_sp ltop = avma;
3891 810674 : long vT = get_Flx_var(T), n = get_Flx_degree(T);
3892 : GEN v_x;
3893 810666 : GEN g = pol1_Flx(vT), tau = pol1_Flx(vT);
3894 810632 : T = Flx_get_red_pre(T, p, pi);
3895 810633 : v_x = Flxq_powers_pre(x, usqrt(2*n), T, p, pi);
3896 1684966 : while (lgpol(tau) != 0)
3897 : {
3898 : long i, j, m, k1;
3899 : GEN M, v, tr, g_prime, c;
3900 874308 : if (degpol(g) == n) { tau = pol1_Flx(vT); g = pol1_Flx(vT); }
3901 874308 : v = random_Flx(n, vT, p);
3902 874325 : tr = Flxq_transmul_init(tau, T, p, pi);
3903 874307 : v = Flxq_transmul(tr, v, n, p, pi);
3904 874325 : m = 2*(n-degpol(g));
3905 874327 : k1 = usqrt(m);
3906 874331 : tr = Flxq_transmul_init(gel(v_x,k1+1), T, p, pi);
3907 874305 : c = cgetg(m+2,t_VECSMALL);
3908 874308 : c[1] = vT;
3909 4217833 : for (i=0; i<m; i+=k1)
3910 : {
3911 3343503 : long mj = minss(m-i, k1);
3912 12980148 : for (j=0; j<mj; j++)
3913 9636326 : uel(c,m+1-(i+j)) = Flx_dotproduct_pre(v, gel(v_x,j+1), p, pi);
3914 3343822 : v = Flxq_transmul(tr, v, n, p, pi);
3915 : }
3916 874330 : c = Flx_renormalize(c, m+2);
3917 : /* now c contains <v,x^i> , i = 0..m-1 */
3918 874330 : M = Flx_halfgcd_pre(monomial_Flx(1, m, vT), c, p, pi);
3919 874340 : g_prime = gmael(M, 2, 2);
3920 874340 : if (degpol(g_prime) < 1) continue;
3921 861659 : g = Flx_mul_pre(g, g_prime, p, pi);
3922 861645 : tau = Flxq_mul_pre(tau, Flx_FlxqV_eval_pre(g_prime, v_x, T,p,pi), T,p,pi);
3923 : }
3924 810630 : g = Flx_normalize(g,p);
3925 810668 : return gc_leaf(ltop,g);
3926 : }
3927 : GEN
3928 44915 : Flxq_minpoly(GEN x, GEN T, ulong p)
3929 44915 : { return Flxq_minpoly_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3930 :
3931 : GEN
3932 20 : Flxq_conjvec(GEN x, GEN T, ulong p)
3933 : {
3934 20 : long i, l = 1+get_Flx_degree(T);
3935 20 : GEN z = cgetg(l,t_COL);
3936 20 : struct _Flxq D; set_Flxq(&D, T, p);
3937 20 : gel(z,1) = Flx_copy(x);
3938 88 : for (i=2; i<l; i++) gel(z,i) = _Flxq_powu(&D, gel(z,i-1), p);
3939 20 : return z;
3940 : }
3941 :
3942 : GEN
3943 7201 : gener_Flxq(GEN T, ulong p, GEN *po)
3944 : {
3945 7201 : long i, j, vT = get_Flx_var(T), f = get_Flx_degree(T);
3946 : ulong p_1, pi;
3947 : GEN g, L, L2, o, q, F;
3948 : pari_sp av0, av;
3949 :
3950 7201 : if (f == 1) {
3951 : GEN fa;
3952 28 : o = utoipos(p-1);
3953 28 : fa = Z_factor(o);
3954 28 : L = gel(fa,1);
3955 28 : L = vecslice(L, 2, lg(L)-1); /* remove 2 for efficiency */
3956 28 : g = Fl_to_Flx(pgener_Fl_local(p, vec_to_vecsmall(L)), vT);
3957 28 : if (po) *po = mkvec2(o, fa);
3958 28 : return g;
3959 : }
3960 :
3961 7173 : av0 = avma; p_1 = p - 1;
3962 7173 : q = diviuexact(subiu(powuu(p,f), 1), p_1);
3963 :
3964 7173 : L = cgetg(1, t_VECSMALL);
3965 7173 : if (p > 3)
3966 : {
3967 2371 : ulong t = p_1 >> vals(p_1);
3968 2371 : GEN P = gel(factoru(t), 1);
3969 2371 : L = cgetg_copy(P, &i);
3970 3787 : while (--i) L[i] = p_1 / P[i];
3971 : }
3972 7173 : o = factor_pn_1(utoipos(p),f);
3973 7173 : L2 = leafcopy( gel(o, 1) );
3974 19212 : for (i = j = 1; i < lg(L2); i++)
3975 : {
3976 12039 : if (umodui(p_1, gel(L2,i)) == 0) continue;
3977 6488 : gel(L2,j++) = diviiexact(q, gel(L2,i));
3978 : }
3979 7173 : setlg(L2, j); pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3980 7173 : F = Flx_Frobenius_pre(T, p, pi);
3981 17708 : for (av = avma;; set_avma(av))
3982 10535 : {
3983 : GEN tt;
3984 17708 : g = random_Flx(f, vT, p);
3985 17708 : if (degpol(g) < 1) continue;
3986 12112 : if (p == 2) tt = g;
3987 : else
3988 : {
3989 8913 : ulong t = Flxq_norm(g, T, p);
3990 8913 : if (t == 1 || !is_gener_Fl(t, p, p_1, L)) continue;
3991 4771 : tt = Flxq_powu_pre(g, p_1>>1, T, p, pi);
3992 : }
3993 14580 : for (i = 1; i < j; i++)
3994 : {
3995 7407 : GEN a = Flxq_pow_Frobenius(tt, gel(L2,i), F, T, p, pi);
3996 7407 : if (!degpol(a) && uel(a,2) == p_1) break;
3997 : }
3998 7970 : if (i == j) break;
3999 : }
4000 7173 : if (!po)
4001 : {
4002 187 : set_avma((pari_sp)g);
4003 187 : g = gc_leaf(av0, g);
4004 : }
4005 : else {
4006 6986 : *po = mkvec2(subiu(powuu(p,f), 1), o);
4007 6986 : (void)gc_all(av0, 2, &g, po);
4008 : }
4009 7173 : return g;
4010 : }
4011 :
4012 : static GEN
4013 366572 : _Flxq_neg(void *E, GEN x)
4014 366572 : { struct _Flxq *s = (struct _Flxq *)E;
4015 366572 : return Flx_neg(x,s->p); }
4016 :
4017 : static GEN
4018 1461838 : _Flxq_rmul(void *E, GEN x, GEN y)
4019 1461838 : { struct _Flxq *s = (struct _Flxq *)E;
4020 1461838 : return Flx_mul_pre(x,y,s->p,s->pi); }
4021 :
4022 : static GEN
4023 9460 : _Flxq_inv(void *E, GEN x)
4024 9460 : { struct _Flxq *s = (struct _Flxq *)E;
4025 9460 : return Flxq_inv(x,s->T,s->p); }
4026 :
4027 : static int
4028 69139 : _Flxq_equal0(GEN x) { return lgpol(x)==0; }
4029 :
4030 : static GEN
4031 6567 : _Flxq_s(void *E, long x)
4032 6567 : { struct _Flxq *s = (struct _Flxq *)E;
4033 6567 : ulong u = x<0 ? s->p+x: (ulong)x;
4034 6567 : return Fl_to_Flx(u, get_Flx_var(s->T));
4035 : }
4036 :
4037 : static const struct bb_field Flxq_field={_Flxq_red,_Flx_add,_Flxq_rmul,_Flxq_neg,
4038 : _Flxq_inv,_Flxq_equal0,_Flxq_s};
4039 :
4040 68902 : const struct bb_field *get_Flxq_field(void **E, GEN T, ulong p)
4041 : {
4042 68902 : GEN z = new_chunk(sizeof(struct _Flxq));
4043 68902 : set_Flxq((struct _Flxq *)z, T, p); *E = (void*)z; return &Flxq_field;
4044 : }
4045 :
4046 : /***********************************************************************/
4047 : /** Flxn **/
4048 : /***********************************************************************/
4049 :
4050 : GEN
4051 54420 : Flx_invLaplace(GEN x, ulong p)
4052 : {
4053 54420 : long i, d = degpol(x);
4054 : ulong t;
4055 : GEN y;
4056 54418 : if (d <= 1) return Flx_copy(x);
4057 54418 : t = Fl_inv(factorial_Fl(d, p), p);
4058 54468 : y = cgetg(d+3, t_VECSMALL);
4059 54424 : y[1] = x[1];
4060 1330602 : for (i=d; i>=2; i--)
4061 : {
4062 1276162 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
4063 1276149 : t = Fl_mul(t, i, p);
4064 : }
4065 54440 : uel(y,3) = uel(x,3);
4066 54440 : uel(y,2) = uel(x,2);
4067 54440 : return y;
4068 : }
4069 :
4070 : GEN
4071 27373 : Flx_Laplace(GEN x, ulong p)
4072 : {
4073 27373 : long i, d = degpol(x);
4074 27373 : ulong t = 1;
4075 : GEN y;
4076 27373 : if (d <= 1) return Flx_copy(x);
4077 27373 : y = cgetg(d+3, t_VECSMALL);
4078 27365 : y[1] = x[1];
4079 27365 : uel(y,2) = uel(x,2);
4080 27365 : uel(y,3) = uel(x,3);
4081 759618 : for (i=2; i<=d; i++)
4082 : {
4083 732234 : t = Fl_mul(t, i%p, p);
4084 732243 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
4085 : }
4086 27384 : return y;
4087 : }
4088 :
4089 : GEN
4090 6335311 : Flxn_red(GEN a, long n)
4091 : {
4092 6335311 : long i, L, l = lg(a);
4093 : GEN b;
4094 6335311 : if (l == 2 || !n) return zero_Flx(a[1]);
4095 5944203 : L = n+2; if (L > l) L = l;
4096 5944203 : b = cgetg(L, t_VECSMALL); b[1] = a[1];
4097 59751404 : for (i=2; i<L; i++) b[i] = a[i];
4098 5942278 : return Flx_renormalize(b,L);
4099 : }
4100 :
4101 : GEN
4102 5164239 : Flxn_mul_pre(GEN a, GEN b, long n, ulong p, ulong pi)
4103 5164239 : { return Flxn_red(Flx_mul_pre(a, b, p, pi), n); }
4104 : GEN
4105 75867 : Flxn_mul(GEN a, GEN b, long n, ulong p)
4106 75867 : { return Flxn_mul_pre(a, b, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4107 :
4108 : GEN
4109 0 : Flxn_sqr_pre(GEN a, long n, ulong p, ulong pi)
4110 0 : { return Flxn_red(Flx_sqr_pre(a, p, pi), n); }
4111 : GEN
4112 0 : Flxn_sqr(GEN a, long n, ulong p)
4113 0 : { return Flxn_sqr_pre(a, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4114 :
4115 : /* (f*g) \/ x^n */
4116 : static GEN
4117 938994 : Flx_mulhigh_i(GEN f, GEN g, long n, ulong p, ulong pi)
4118 938994 : { return Flx_shift(Flx_mul_pre(f, g, p, pi),-n); }
4119 :
4120 : static GEN
4121 517064 : Flxn_mulhigh(GEN f, GEN g, long n2, long n, ulong p, ulong pi)
4122 : {
4123 517064 : GEN F = Flx_blocks(f, n2, 2), fl = gel(F,1), fh = gel(F,2);
4124 516900 : return Flx_add(Flx_mulhigh_i(fl, g, n2, p, pi),
4125 : Flxn_mul_pre(fh, g, n - n2, p, pi), p);
4126 : }
4127 :
4128 : /* g==NULL -> assume g==1 */
4129 : GEN
4130 55240 : Flxn_div_pre(GEN g, GEN f, long e, ulong p, ulong pi)
4131 : {
4132 55240 : pari_sp av = avma, av2;
4133 : ulong mask;
4134 : GEN W;
4135 55240 : long n = 1;
4136 55240 : if (lg(f) <= 2) pari_err_INV("Flxn_inv",f);
4137 55240 : W = Fl_to_Flx(Fl_inv(uel(f,2),p), f[1]);
4138 55262 : mask = quadratic_prec_mask(e);
4139 55257 : av2 = avma;
4140 259270 : for (;mask>1;)
4141 : {
4142 : GEN u, fr;
4143 204006 : long n2 = n;
4144 204006 : n<<=1; if (mask & 1) n--;
4145 204006 : mask >>= 1;
4146 204006 : fr = Flxn_red(f, n);
4147 203920 : if (mask>1 || !g)
4148 : {
4149 149745 : u = Flxn_mul_pre(W, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
4150 149991 : W = Flx_sub(W, Flx_shift(u, n2), p);
4151 : } else
4152 : {
4153 54175 : GEN y = Flxn_mul_pre(g, W, n, p, pi), yt = Flxn_red(y, n-n2);
4154 54165 : u = Flxn_mul_pre(yt, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
4155 54182 : W = Flx_sub(y, Flx_shift(u, n2), p);
4156 : }
4157 203954 : if (gc_needed(av2,2))
4158 : {
4159 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxn_div, e = %ld", n);
4160 0 : W = gc_upto(av2, W);
4161 : }
4162 : }
4163 55264 : return gc_upto(av, W);
4164 : }
4165 : GEN
4166 55228 : Flxn_div(GEN g, GEN f, long e, ulong p)
4167 55228 : { return Flxn_div_pre(g, f, e, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4168 :
4169 : GEN
4170 1030 : Flxn_inv(GEN f, long e, ulong p)
4171 1030 : { return Flxn_div(NULL, f, e, p); }
4172 :
4173 : GEN
4174 109431 : Flxn_expint(GEN h, long e, ulong p)
4175 : {
4176 109431 : pari_sp av = avma, av2;
4177 109431 : long v = h[1], n=1;
4178 109431 : GEN f = pol1_Flx(v), g = pol1_Flx(v);
4179 109412 : ulong mask = quadratic_prec_mask(e), pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4180 109419 : av2 = avma;
4181 422917 : for (;mask>1;)
4182 : {
4183 : GEN u, w;
4184 422868 : long n2 = n;
4185 422868 : n<<=1; if (mask & 1) n--;
4186 422868 : mask >>= 1;
4187 422868 : u = Flxn_mul_pre(g, Flx_mulhigh_i(f, Flxn_red(h, n2-1), n2-1, p,pi), n-n2, p,pi);
4188 422826 : u = Flx_add(u, Flx_shift(Flxn_red(h, n-1), 1-n2), p);
4189 422835 : w = Flxn_mul_pre(f, Flx_integXn(u, n2-1, p), n-n2, p, pi);
4190 422833 : f = Flx_add(f, Flx_shift(w, n2), p);
4191 422923 : if (mask<=1) break;
4192 313505 : u = Flxn_mul_pre(g, Flxn_mulhigh(f, g, n2, n, p, pi), n-n2, p, pi);
4193 313497 : g = Flx_sub(g, Flx_shift(u, n2), p);
4194 313498 : if (gc_needed(av2,2))
4195 : {
4196 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flxn_exp, e = %ld", n);
4197 0 : (void)gc_all(av2, 2, &f, &g);
4198 : }
4199 : }
4200 109467 : return gc_upto(av, f);
4201 : }
4202 :
4203 : GEN
4204 0 : Flxn_exp(GEN h, long e, ulong p)
4205 : {
4206 0 : if (degpol(h)<1 || uel(h,2)!=0)
4207 0 : pari_err_DOMAIN("Flxn_exp","valuation", "<", gen_1, h);
4208 0 : return Flxn_expint(Flx_deriv(h, p), e, p);
4209 : }
4210 :
4211 : INLINE GEN
4212 217513 : Flxn_recip(GEN x, long n)
4213 : {
4214 217513 : GEN z=Flx_recipspec(x+2,lgpol(x),n);
4215 217421 : z[1]=x[1];
4216 217421 : return z;
4217 : }
4218 :
4219 : GEN
4220 54175 : Flx_Newton(GEN P, long n, ulong p)
4221 : {
4222 54175 : pari_sp av = avma;
4223 54175 : long d = degpol(P);
4224 54171 : GEN dP = Flxn_recip(Flx_deriv(P, p), d);
4225 54104 : GEN Q = Flxn_div(dP, Flxn_recip(P, d+1), n, p);
4226 54123 : return gc_leaf(av, Q);
4227 : }
4228 :
4229 : GEN
4230 109433 : Flx_fromNewton(GEN P, ulong p)
4231 : {
4232 109433 : pari_sp av = avma;
4233 109433 : ulong n = Flx_constant(P)+1;
4234 109433 : GEN z = Flx_neg(Flx_shift(P, -1), p);
4235 109431 : GEN Q = Flxn_recip(Flxn_expint(z, n, p), n);
4236 109416 : return gc_leaf(av, Q);
4237 : }
4238 :
4239 : static void
4240 12514 : init_invlaplace(long d, ulong p, GEN *pt_P, GEN *pt_V)
4241 : {
4242 : long i;
4243 : ulong e;
4244 12514 : GEN P = cgetg(d+1, t_VECSMALL);
4245 12514 : GEN V = cgetg(d+1, t_VECSMALL);
4246 1396581 : for (i=1, e=1; i<=d; i++, e++)
4247 : {
4248 1384067 : if (e==p)
4249 : {
4250 459153 : e = 0;
4251 459153 : V[i] = u_lvalrem(i, p, &uel(P,i));
4252 : } else
4253 : {
4254 924914 : V[i] = 0; uel(P,i) = i;
4255 : }
4256 : }
4257 12514 : *pt_P = P; *pt_V = V;
4258 12514 : }
4259 :
4260 : /* return p^val * FpX_invLaplace(1+x+...x^(n-1), q), with q a power of p and
4261 : * val large enough to compensate for the power of p in the factorials */
4262 :
4263 : static GEN
4264 497 : ZpX_invLaplace_init(long n, GEN q, ulong p, long v, long sv)
4265 : {
4266 497 : pari_sp av = avma;
4267 497 : long i, d = n-1, w;
4268 : GEN y, W, E, t;
4269 497 : init_invlaplace(d, p, &E, &W);
4270 497 : t = Fp_inv(FpV_prod(Flv_to_ZV(E), q), q);
4271 497 : w = zv_sum(W);
4272 497 : if (v > w) t = Fp_mul(t, powuu(p, v-w), q);
4273 497 : y = cgetg(d+3,t_POL);
4274 497 : y[1] = evalsigne(1) | sv;
4275 28882 : for (i=d; i>=1; i--)
4276 : {
4277 28385 : gel(y,i+2) = t;
4278 28385 : t = Fp_mulu(t, uel(E,i), q);
4279 28385 : if (uel(W,i)) t = Fp_mul(t, powuu(p, uel(W,i)), q);
4280 : }
4281 497 : gel(y,2) = t;
4282 497 : return gc_GEN(av, ZX_renormalize(y, d+3));
4283 : }
4284 :
4285 : GEN
4286 27589 : Flx_composedsum(GEN P, GEN Q, ulong p)
4287 : {
4288 27589 : pari_sp av = avma;
4289 27589 : long n = 1 + degpol(P)*degpol(Q);
4290 27586 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4291 27587 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4292 : GEN R;
4293 27590 : if (p >= (ulong)n)
4294 : {
4295 27093 : GEN Pl = Flx_invLaplace(Flx_Newton(P,n,p), p);
4296 27087 : GEN Ql = Flx_invLaplace(Flx_Newton(Q,n,p), p);
4297 27078 : GEN L = Flx_Laplace(Flxn_mul(Pl, Ql, n, p), p);
4298 27089 : R = Flx_fromNewton(L, p);
4299 : } else
4300 : {
4301 497 : long v = factorial_lval(n-1, p);
4302 497 : long w = 1 + ulogint(n-1, p);
4303 497 : GEN pv = powuu(p, v);
4304 497 : GEN qf = powuu(p, w), q = mulii(pv, qf), q2 = mulii(q, pv);
4305 497 : GEN iL = ZpX_invLaplace_init(n, q, p, v, P[1]);
4306 497 : GEN Pl = FpX_convol(iL, FpX_Newton(Flx_to_ZX(P), n, qf), q);
4307 497 : GEN Ql = FpX_convol(iL, FpX_Newton(Flx_to_ZX(Q), n, qf), q);
4308 497 : GEN Ln = ZX_Z_divexact(FpXn_mul(Pl, Ql, n, q2), pv);
4309 497 : GEN L = ZX_Z_divexact(FpX_Laplace(Ln, q), pv);
4310 497 : R = ZX_to_Flx(FpX_fromNewton(L, qf), p);
4311 : }
4312 27576 : return gc_leaf(av, Flx_Fl_mul(R, lead, p));
4313 : }
4314 :
4315 : static GEN
4316 3910 : _Flx_composedsum(void *E, GEN a, GEN b)
4317 3910 : { return Flx_composedsum(a, b, (ulong)E); }
4318 :
4319 : GEN
4320 28994 : FlxV_composedsum(GEN V, ulong p)
4321 28994 : { return gen_product(V, (void *)p, &_Flx_composedsum); }
4322 :
4323 : GEN
4324 0 : Flx_composedprod(GEN P, GEN Q, ulong p)
4325 : {
4326 0 : pari_sp av = avma;
4327 0 : long n = 1+ degpol(P)*degpol(Q);
4328 0 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4329 0 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4330 : GEN R;
4331 0 : if (p >= (ulong)n)
4332 : {
4333 0 : GEN L = Flx_convol(Flx_Newton(P,n,p), Flx_Newton(Q,n,p), p);
4334 0 : R = Flx_fromNewton(L, p);
4335 : } else
4336 : {
4337 0 : long w = 1 + ulogint(n, p);
4338 0 : GEN qf = powuu(p, w);
4339 0 : GEN Pl = FpX_convol(FpX_Newton(Flx_to_ZX(P), n, qf), FpX_Newton(Flx_to_ZX(Q), n, qf), qf);
4340 0 : R = ZX_to_Flx(FpX_fromNewton(Pl, qf), p);
4341 : }
4342 0 : return gc_leaf(av, Flx_Fl_mul(R, lead, p));
4343 :
4344 : }
4345 :
4346 : /* (x+1)^n mod p; assume 2 <= n < 2p prime */
4347 : static GEN
4348 0 : Fl_Xp1_powu(ulong n, ulong p, long v)
4349 : {
4350 0 : ulong k, d = (n + 1) >> 1;
4351 0 : GEN C, V = identity_zv(d);
4352 :
4353 0 : Flv_inv_inplace(V, p); /* could restrict to odd integers in [3,d] */
4354 0 : C = cgetg(n+3, t_VECSMALL);
4355 0 : C[1] = v;
4356 0 : uel(C,2) = 1UL;
4357 0 : uel(C,3) = n%p;
4358 0 : uel(C,4) = Fl_mul(odd(n)? n: n-1, n >> 1, p);
4359 : /* binom(n,k) = binom(n,k-1) * (n-k+1) / k */
4360 0 : if (SMALL_ULONG(p))
4361 0 : for (k = 3; k <= d; k++)
4362 0 : uel(C,k+2) = Fl_mul(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p);
4363 : else
4364 : {
4365 0 : ulong pi = get_Fl_red(p);
4366 0 : for (k = 3; k <= d; k++)
4367 0 : uel(C,k+2) = Fl_mul_pre(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p, pi);
4368 : }
4369 0 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4370 0 : return C; /* normalized */
4371 : }
4372 :
4373 : /* p arbitrary */
4374 : GEN
4375 28236 : Flx_translate1_basecase(GEN P, ulong p)
4376 : {
4377 28236 : GEN R = Flx_copy(P);
4378 28236 : long i, k, n = degpol(P);
4379 654893 : for (i = 1; i <= n; i++)
4380 14846873 : for (k = n-i; k < n; k++) uel(R,k+2) = Fl_add(uel(R,k+2), uel(R,k+3), p);
4381 28236 : return R;
4382 : }
4383 :
4384 : static int
4385 41401 : translate_basecase(long n, ulong p)
4386 : {
4387 : #ifdef LONG_IS_64BIT
4388 36102 : if (p <= 19) return n < 40;
4389 29910 : if (p < 1UL<<30) return n < 58;
4390 0 : if (p < 1UL<<59) return n < 100;
4391 0 : if (p < 1UL<<62) return n < 120;
4392 0 : if (p < 1UL<<63) return n < 240;
4393 0 : return n < 250;
4394 : #else
4395 5299 : if (p <= 13) return n < 18;
4396 4136 : if (p <= 17) return n < 22;
4397 4078 : if (p <= 29) return n < 39;
4398 3886 : if (p <= 67) return n < 69;
4399 3667 : if (p < 1UL<< 15) return n < 80;
4400 2047 : if (p < 1UL<< 16) return n < 100;
4401 0 : if (p < 1UL<< 28) return n < 300;
4402 0 : return n < 650;
4403 : #endif
4404 : }
4405 : /* assume p prime */
4406 : GEN
4407 16142 : Flx_translate1(GEN P, ulong p)
4408 : {
4409 16142 : long d, n = degpol(P);
4410 : GEN R, Q, S;
4411 16142 : if (translate_basecase(n, p)) return Flx_translate1_basecase(P, p);
4412 : /* n > 0 */
4413 1148 : d = n >> 1;
4414 1148 : if ((ulong)n < p)
4415 : {
4416 0 : R = Flx_translate1(Flxn_red(P, d), p);
4417 0 : Q = Flx_translate1(Flx_shift(P, -d), p);
4418 0 : S = Fl_Xp1_powu(d, p, P[1]);
4419 0 : return Flx_add(Flx_mul(Q, S, p), R, p);
4420 : }
4421 : else
4422 : {
4423 : ulong q;
4424 1148 : if ((ulong)d > p) (void)ulogintall(d, p, &q); else q = p;
4425 1148 : R = Flx_translate1(Flxn_red(P, q), p);
4426 1148 : Q = Flx_translate1(Flx_shift(P, -q), p);
4427 1148 : S = Flx_add(Flx_shift(Q, q), Q, p);
4428 1148 : return Flx_add(S, R, p); /* P(x+1) = Q(x+1) (x^q+1) + R(x+1) */
4429 : }
4430 : }
4431 :
4432 : GEN
4433 0 : Flx_Fl_translate(GEN P, ulong c, ulong p)
4434 : {
4435 0 : pari_sp av = avma;
4436 : GEN Q;
4437 0 : if (c==0) return Flx_copy(P);
4438 0 : if (c==1) return Flx_translate1(P, p);
4439 0 : Q = Flx_unscale(Flx_translate1(Flx_unscale(P, c, p), p), Fl_inv(c, p), p);
4440 0 : return gc_leaf(av, Q);
4441 : }
4442 :
4443 : static GEN
4444 12017 : zl_Xp1_powu(ulong n, ulong p, ulong q, long e, long vs)
4445 : {
4446 12017 : ulong k, d = n >> 1, c, v = 0;
4447 12017 : GEN C, V, W, U = upowers(p, e-1);
4448 12017 : init_invlaplace(d, p, &V, &W);
4449 12017 : Flv_inv_inplace(V, q);
4450 12017 : C = cgetg(n+3, t_VECSMALL);
4451 12017 : C[1] = vs;
4452 12017 : uel(C,2) = 1UL;
4453 12017 : uel(C,3) = n%q;
4454 12017 : v = u_lvalrem(n, p, &c);
4455 1355682 : for (k = 2; k <= d; k++)
4456 : {
4457 : ulong w;
4458 1343665 : v += u_lvalrem(n-k+1, p, &w) - W[k];
4459 1343665 : c = Fl_mul(Fl_mul(w%q, c, q), uel(V,k), q);
4460 1343665 : uel(C,2+k) = v >= (ulong)e ? 0: v==0 ? c : Fl_mul(c, uel(U, v+1), q);
4461 : }
4462 1374521 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4463 12017 : return C; /* normalized */
4464 : }
4465 :
4466 : GEN
4467 25259 : zlx_translate1(GEN P, ulong p, long e)
4468 : {
4469 25259 : ulong d, q = upowuu(p,e), n = degpol(P);
4470 : GEN R, Q, S;
4471 25259 : if (translate_basecase(n, q)) return Flx_translate1_basecase(P, q);
4472 : /* n > 0 */
4473 12017 : d = n >> 1;
4474 12017 : R = zlx_translate1(Flxn_red(P, d), p, e);
4475 12017 : Q = zlx_translate1(Flx_shift(P, -d), p, e);
4476 12017 : S = zl_Xp1_powu(d, p, q, e, P[1]);
4477 12017 : return Flx_add(Flx_mul(Q, S, q), R, q);
4478 : }
4479 :
4480 : /***********************************************************************/
4481 : /** Fl2 **/
4482 : /***********************************************************************/
4483 : /* Fl2 objects are Flv of length 2 [a,b] representing a+bsqrt(D) for
4484 : * a nonsquare D. */
4485 :
4486 : INLINE GEN
4487 7601242 : mkF2(ulong a, ulong b) { return mkvecsmall2(a,b); }
4488 :
4489 : /* allow pi = 0 */
4490 : GEN
4491 2023785 : Fl2_mul_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4492 : {
4493 : ulong xaya, xbyb, Db2, mid, z1, z2;
4494 2023785 : ulong x1 = x[1], x2 = x[2], y1 = y[1], y2 = y[2];
4495 2023785 : if (pi)
4496 : {
4497 2023813 : xaya = Fl_mul_pre(x1,y1,p,pi);
4498 2024549 : if (x2==0 && y2==0) return mkF2(xaya,0);
4499 1950403 : if (x2==0) return mkF2(xaya,Fl_mul_pre(x1,y2,p,pi));
4500 1925198 : if (y2==0) return mkF2(xaya,Fl_mul_pre(x2,y1,p,pi));
4501 1924958 : xbyb = Fl_mul_pre(x2,y2,p,pi);
4502 1924810 : mid = Fl_mul_pre(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p,pi);
4503 1924977 : Db2 = Fl_mul_pre(D, xbyb, p,pi);
4504 : }
4505 0 : else if (p & HIGHMASK)
4506 : {
4507 0 : xaya = Fl_mul(x1,y1,p);
4508 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4509 0 : if (x2==0) return mkF2(xaya,Fl_mul(x1,y2,p));
4510 0 : if (y2==0) return mkF2(xaya,Fl_mul(x2,y1,p));
4511 0 : xbyb = Fl_mul(x2,y2,p);
4512 0 : mid = Fl_mul(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p);
4513 0 : Db2 = Fl_mul(D, xbyb, p);
4514 : }
4515 : else
4516 : {
4517 0 : xaya = (x1 * y1) % p;
4518 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4519 0 : if (x2==0) return mkF2(xaya, (x1 * y2) % p);
4520 0 : if (y2==0) return mkF2(xaya, (x2 * y1) % p);
4521 0 : xbyb = (x2 * y2) % p;
4522 0 : mid = (Fl_add(x1,x2,p) * Fl_add(y1,y2,p)) % p;
4523 0 : Db2 = (D * xbyb) % p;
4524 : }
4525 1924918 : z1 = Fl_add(xaya,Db2,p);
4526 1924873 : z2 = Fl_sub(mid,Fl_add(xaya,xbyb,p),p);
4527 1924760 : return mkF2(z1,z2);
4528 : }
4529 :
4530 : /* allow pi = 0 */
4531 : GEN
4532 5104246 : Fl2_sqr_pre(GEN x, ulong D, ulong p, ulong pi)
4533 : {
4534 5104246 : ulong a = x[1], b = x[2];
4535 : ulong a2, Db2, ab;
4536 5104246 : if (pi)
4537 : {
4538 5104280 : a2 = Fl_sqr_pre(a,p,pi);
4539 5108068 : if (b==0) return mkF2(a2,0);
4540 4876114 : Db2= Fl_mul_pre(D, Fl_sqr_pre(b,p,pi), p,pi);
4541 4876349 : ab = Fl_mul_pre(a,b,p,pi);
4542 : }
4543 0 : else if (p & HIGHMASK)
4544 : {
4545 0 : a2 = Fl_sqr(a,p);
4546 0 : if (b==0) return mkF2(a2,0);
4547 0 : Db2= Fl_mul(D, Fl_sqr(b,p), p);
4548 0 : ab = Fl_mul(a,b,p);
4549 : }
4550 : else
4551 : {
4552 0 : a2 = (a * a) % p;
4553 0 : if (b==0) return mkF2(a2,0);
4554 0 : Db2= (D * ((b * b) % p)) % p;
4555 0 : ab = (a * b) % p;
4556 : }
4557 4876498 : return mkF2(Fl_add(a2,Db2,p), Fl_double(ab,p));
4558 : }
4559 :
4560 : /* allow pi = 0 */
4561 : ulong
4562 127093 : Fl2_norm_pre(GEN x, ulong D, ulong p, ulong pi)
4563 : {
4564 127093 : ulong a = x[1], b = x[2], a2;
4565 127093 : if (pi)
4566 : {
4567 75214 : a2 = Fl_sqr_pre(a,p,pi);
4568 75214 : return b? Fl_sub(a2, Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p): a2;
4569 : }
4570 51879 : else if (p & HIGHMASK)
4571 : {
4572 0 : a2 = Fl_sqr(a,p);
4573 0 : return b? Fl_sub(a2, Fl_mul(D, Fl_sqr(b, p), p), p): a2;
4574 : }
4575 : else
4576 : {
4577 51879 : a2 = (a * a) % p;
4578 51879 : return b? Fl_sub(a2, (D * ((b * b) % p)) % p, p): a2;
4579 : }
4580 : }
4581 :
4582 : /* allow pi = 0 */
4583 : GEN
4584 202501 : Fl2_inv_pre(GEN x, ulong D, ulong p, ulong pi)
4585 : {
4586 202501 : ulong a = x[1], b = x[2], n, ni;
4587 202501 : if (b == 0) return mkF2(Fl_inv(a,p), 0);
4588 169377 : b = Fl_neg(b, p);
4589 169377 : if (pi)
4590 : {
4591 169377 : n = Fl_sub(Fl_sqr_pre(a, p,pi),
4592 : Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p);
4593 169376 : ni = Fl_inv(n,p);
4594 169377 : return mkF2(Fl_mul_pre(a, ni, p,pi), Fl_mul_pre(b, ni, p,pi));
4595 : }
4596 0 : else if (p & HIGHMASK)
4597 : {
4598 0 : n = Fl_sub(Fl_sqr(a, p), Fl_mul(D, Fl_sqr(b, p), p), p);
4599 0 : ni = Fl_inv(n,p);
4600 0 : return mkF2(Fl_mul(a, ni, p), Fl_mul(b, ni, p));
4601 : }
4602 : else
4603 : {
4604 0 : n = Fl_sub((a * a) % p, (D * ((b * b) % p)) % p, p);
4605 0 : ni = Fl_inv(n,p);
4606 0 : return mkF2((a * ni) % p, (b * ni) % p);
4607 : }
4608 : }
4609 :
4610 : int
4611 463829 : Fl2_equal1(GEN x) { return x[1]==1 && x[2]==0; }
4612 :
4613 : struct _Fl2 {
4614 : ulong p, pi, D;
4615 : };
4616 :
4617 : static GEN
4618 5104188 : _Fl2_sqr(void *data, GEN x)
4619 : {
4620 5104188 : struct _Fl2 *D = (struct _Fl2*)data;
4621 5104188 : return Fl2_sqr_pre(x, D->D, D->p, D->pi);
4622 : }
4623 : static GEN
4624 1995357 : _Fl2_mul(void *data, GEN x, GEN y)
4625 : {
4626 1995357 : struct _Fl2 *D = (struct _Fl2*)data;
4627 1995357 : return Fl2_mul_pre(x,y, D->D, D->p, D->pi);
4628 : }
4629 :
4630 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL; allow pi = 0 */
4631 : GEN
4632 691969 : Fl2_pow_pre(GEN x, GEN n, ulong D, ulong p, ulong pi)
4633 : {
4634 691969 : pari_sp av = avma;
4635 : struct _Fl2 d;
4636 : GEN y;
4637 691969 : long s = signe(n);
4638 691969 : if (!s) return mkF2(1,0);
4639 613282 : if (s < 0)
4640 202501 : x = Fl2_inv_pre(x,D,p,pi);
4641 613274 : if (is_pm1(n)) return s < 0 ? x : zv_copy(x);
4642 452912 : d.p = p; d.pi = pi; d.D=D;
4643 452912 : y = gen_pow_i(x, n, (void*)&d, &_Fl2_sqr, &_Fl2_mul);
4644 452926 : return gc_leaf(av, y);
4645 : }
4646 :
4647 : static GEN
4648 691957 : _Fl2_pow(void *data, GEN x, GEN n)
4649 : {
4650 691957 : struct _Fl2 *D = (struct _Fl2*)data;
4651 691957 : return Fl2_pow_pre(x, n, D->D, D->p, D->pi);
4652 : }
4653 :
4654 : static GEN
4655 118755 : _Fl2_rand(void *data)
4656 : {
4657 118755 : struct _Fl2 *D = (struct _Fl2*)data;
4658 118755 : ulong a = random_Fl(D->p), b=random_Fl(D->p-1)+1;
4659 118755 : return mkF2(a,b);
4660 : }
4661 :
4662 : GEN
4663 65765 : Fl2_sqrt_pre(GEN z, ulong D, ulong p, ulong pi)
4664 : {
4665 65765 : ulong a = uel(z,1), b = uel(z,2), as2, u, v, s;
4666 65765 : ulong y = Fl_2gener_pre_i(D, p, pi);
4667 65765 : if (b == 0)
4668 18930 : return krouu(a, p)==1 ? mkF2(Fl_sqrt_pre_i(a, y, p, pi), 0)
4669 18930 : : mkF2(0, Fl_sqrt_pre_i(Fl_div(a, D, p), y, p, pi));
4670 52709 : s = Fl_sqrt_pre_i(Fl2_norm_pre(z, D, p, pi), y, p, pi);
4671 52709 : if (s==~0UL) return NULL;
4672 49535 : as2 = Fl_halve(Fl_add(a, s, p), p);
4673 49535 : if (krouu(as2, p)==-1) as2 = Fl_sub(as2, s, p);
4674 49535 : u = Fl_sqrt_pre_i(as2, y, p, pi);
4675 49535 : v = Fl_div(b, Fl_double(u, p), p);
4676 49535 : return mkF2(u,v);
4677 : }
4678 :
4679 : static const struct bb_group Fl2_star={_Fl2_mul, _Fl2_pow, _Fl2_rand,
4680 : hash_GEN, zv_equal, Fl2_equal1, NULL};
4681 :
4682 : /* allow pi = 0 */
4683 : GEN
4684 78685 : Fl2_sqrtn_pre(GEN a, GEN n, ulong D, ulong p, ulong pi, GEN *zeta)
4685 : {
4686 : struct _Fl2 E;
4687 : GEN o;
4688 78685 : if (a[1]==0 && a[2]==0)
4689 : {
4690 0 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
4691 0 : if (zeta) *zeta=mkF2(1,0);
4692 0 : return zv_copy(a);
4693 : }
4694 78685 : E.p=p; E.pi = pi; E.D = D;
4695 78685 : o = subiu(powuu(p,2), 1);
4696 78686 : return gen_Shanks_sqrtn(a,n,o,zeta,(void*)&E,&Fl2_star);
4697 : }
4698 :
4699 : /* allow pi = 0 */
4700 : GEN
4701 10528 : Flx_Fl2_eval_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4702 : {
4703 : GEN p1;
4704 10528 : long i = lg(x)-1;
4705 10528 : if (i <= 2)
4706 2086 : return mkF2(i == 2? x[2]: 0, 0);
4707 8442 : p1 = mkF2(x[i], 0);
4708 36876 : for (i--; i>=2; i--)
4709 : {
4710 28434 : p1 = Fl2_mul_pre(p1, y, D, p, pi);
4711 28434 : uel(p1,1) = Fl_add(uel(p1,1), uel(x,i), p);
4712 : }
4713 8442 : return p1;
4714 : }
4715 :
4716 : /***********************************************************************/
4717 : /** FlxV **/
4718 : /***********************************************************************/
4719 : /* FlxV are t_VEC with Flx coefficients. */
4720 :
4721 : GEN
4722 34482 : FlxV_Flc_mul(GEN V, GEN W, ulong p)
4723 : {
4724 34482 : pari_sp ltop=avma;
4725 : long i;
4726 34482 : GEN z = Flx_Fl_mul(gel(V,1),W[1],p);
4727 257068 : for(i=2;i<lg(V);i++)
4728 222586 : z=Flx_add(z,Flx_Fl_mul(gel(V,i),W[i],p),p);
4729 34482 : return gc_leaf(ltop,z);
4730 : }
4731 :
4732 : GEN
4733 0 : ZXV_to_FlxV(GEN x, ulong p)
4734 0 : { pari_APPLY_type(t_VEC, ZX_to_Flx(gel(x,i), p)) }
4735 :
4736 : GEN
4737 3798250 : ZXT_to_FlxT(GEN x, ulong p)
4738 : {
4739 3798250 : if (typ(x) == t_POL)
4740 3739360 : return ZX_to_Flx(x, p);
4741 : else
4742 193299 : pari_APPLY_type(t_VEC, ZXT_to_FlxT(gel(x,i), p))
4743 : }
4744 :
4745 : GEN
4746 171886 : FlxV_to_Flm(GEN x, long n)
4747 927616 : { pari_APPLY_type(t_MAT, Flx_to_Flv(gel(x,i), n)) }
4748 :
4749 : GEN
4750 0 : FlxV_red(GEN x, ulong p)
4751 0 : { pari_APPLY_type(t_VEC, Flx_red(gel(x,i), p)) }
4752 :
4753 : GEN
4754 296647 : FlxT_red(GEN x, ulong p)
4755 : {
4756 296647 : if (typ(x) == t_VECSMALL)
4757 199573 : return Flx_red(x, p);
4758 : else
4759 325468 : pari_APPLY_type(t_VEC, FlxT_red(gel(x,i), p))
4760 : }
4761 :
4762 : GEN
4763 113589 : FlxqV_dotproduct_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
4764 : {
4765 113589 : long i, lx = lg(x);
4766 : pari_sp av;
4767 : GEN c;
4768 113589 : if (lx == 1) return pol0_Flx(get_Flx_var(T));
4769 113589 : av = avma; c = Flx_mul_pre(gel(x,1),gel(y,1), p, pi);
4770 464499 : for (i=2; i<lx; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4771 113589 : return gc_leaf(av, Flx_rem_pre(c,T,p,pi));
4772 : }
4773 : GEN
4774 0 : FlxqV_dotproduct(GEN x, GEN y, GEN T, ulong p)
4775 0 : { return FlxqV_dotproduct_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4776 :
4777 : GEN
4778 1918 : FlxqX_dotproduct(GEN x, GEN y, GEN T, ulong p)
4779 : {
4780 1918 : long i, l = minss(lg(x), lg(y));
4781 : ulong pi;
4782 : pari_sp av;
4783 : GEN c;
4784 1918 : if (l == 2) return pol0_Flx(get_Flx_var(T));
4785 1905 : av = avma; pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4786 1905 : c = Flx_mul_pre(gel(x,2),gel(y,2), p, pi);
4787 6202 : for (i=3; i<l; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4788 1905 : return gc_leaf(av, Flx_rem_pre(c,T,p,pi));
4789 : }
4790 :
4791 : /* allow pi = 0 */
4792 : GEN
4793 325155 : FlxC_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4794 : {
4795 325155 : long i, l = lg(z);
4796 325155 : GEN y = cgetg(l, t_VECSMALL);
4797 15065416 : for (i=1; i<l; i++) uel(y,i) = Flx_eval_powers_pre(gel(z,i), x, p, pi);
4798 325250 : return y;
4799 : }
4800 :
4801 : /***********************************************************************/
4802 : /** FlxM **/
4803 : /***********************************************************************/
4804 : /* allow pi = 0 */
4805 : GEN
4806 22302 : FlxM_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4807 : {
4808 22302 : long i, l = lg(z);
4809 22302 : GEN y = cgetg(l, t_MAT);
4810 347456 : for (i=1; i<l; i++) gel(y,i) = FlxC_eval_powers_pre(gel(z,i), x, p, pi);
4811 22300 : return y;
4812 : }
4813 :
4814 : GEN
4815 0 : zero_FlxC(long n, long sv)
4816 : {
4817 0 : GEN x = cgetg(n + 1, t_COL), z = zero_Flx(sv);
4818 : long i;
4819 0 : for (i = 1; i <= n; i++) gel(x, i) = z;
4820 0 : return x;
4821 : }
4822 :
4823 : GEN
4824 0 : FlxC_neg(GEN x, ulong p)
4825 0 : { pari_APPLY_type(t_COL, Flx_neg(gel(x, i), p)) }
4826 :
4827 : GEN
4828 0 : FlxC_sub(GEN x, GEN y, ulong p)
4829 0 : { pari_APPLY_type(t_COL, Flx_sub(gel(x, i), gel(y, i), p)) }
4830 :
4831 : GEN
4832 0 : zero_FlxM(long r, long c, long sv)
4833 : {
4834 0 : GEN x = cgetg(c + 1, t_MAT), z = zero_FlxC(r, sv);
4835 : long j;
4836 0 : for (j = 1; j <= c; j++) gel(x, j) = z;
4837 0 : return x;
4838 : }
4839 :
4840 : GEN
4841 0 : zero_FlxM_copy(long r, long c, long sv)
4842 : {
4843 0 : GEN x = cgetg(c + 1, t_MAT);
4844 : long j;
4845 0 : for (j = 1; j <= c; j++) gel(x, j) = zero_FlxC(r, sv);
4846 0 : return x;
4847 : }
4848 :
4849 : GEN
4850 0 : FlxM_neg(GEN x, ulong p)
4851 0 : { pari_APPLY_same(FlxC_neg(gel(x, i), p)) }
4852 :
4853 : GEN
4854 0 : FlxM_sub(GEN x, GEN y, ulong p)
4855 0 : { pari_APPLY_same(FlxC_sub(gel(x, i), gel(y,i), p)) }
4856 :
4857 : GEN
4858 0 : FlxC_Fl_translate(GEN x, ulong c, ulong p)
4859 0 : { pari_APPLY_type(t_COL, Flx_Fl_translate(gel(x,i), c, p)) }
4860 :
4861 : GEN
4862 0 : FlxM_Fl_translate(GEN x, ulong c, ulong p)
4863 0 : { pari_APPLY_same(FlxC_Fl_translate(gel(x,i), c, p)) }
4864 :
4865 : GEN
4866 234845 : FlxqC_red_pre(GEN x, GEN T, ulong p, ulong pi)
4867 4060693 : { pari_APPLY_type(t_COL, Flx_rem_pre(gel(x,i), T, p, pi)) }
4868 :
4869 : GEN
4870 81581 : FlxqM_red_pre(GEN x, GEN T, ulong p, ulong pi)
4871 316426 : { pari_APPLY_same(FlxqC_red_pre(gel(x,i), T, p, pi)) }
4872 :
4873 : GEN
4874 0 : FlxqC_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4875 0 : { pari_APPLY_type(t_COL, Flxq_mul(gel(x, i), y, T, p)) }
4876 :
4877 : GEN
4878 0 : FlxqM_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4879 0 : { pari_APPLY_same(FlxqC_Flxq_mul(gel(x, i), y, T, p)) }
4880 :
4881 : static GEN
4882 46835 : FlxM_pack_ZM(GEN M, GEN (*pack)(GEN, long)) {
4883 : long i, j, l, lc;
4884 46835 : GEN N = cgetg_copy(M, &l), x;
4885 46835 : if (l == 1)
4886 0 : return N;
4887 46835 : lc = lgcols(M);
4888 205007 : for (j = 1; j < l; j++) {
4889 158172 : gel(N, j) = cgetg(lc, t_COL);
4890 902833 : for (i = 1; i < lc; i++) {
4891 744661 : x = gcoeff(M, i, j);
4892 744661 : gcoeff(N, i, j) = pack(x + 2, lgpol(x));
4893 : }
4894 : }
4895 46835 : return N;
4896 : }
4897 :
4898 : static GEN
4899 688104 : kron_pack_Flx_spec_half(GEN x, long l) {
4900 688104 : if (l == 0) return gen_0;
4901 457528 : return Flx_to_int_halfspec(x, l);
4902 : }
4903 :
4904 : static GEN
4905 53168 : kron_pack_Flx_spec(GEN x, long l) {
4906 : long i;
4907 : GEN w, y;
4908 53168 : if (l == 0)
4909 9964 : return gen_0;
4910 43204 : y = cgetipos(l + 2);
4911 157864 : for (i = 0, w = int_LSW(y); i < l; i++, w = int_nextW(w))
4912 114660 : *w = x[i];
4913 43204 : return y;
4914 : }
4915 :
4916 : static GEN
4917 3389 : kron_pack_Flx_spec_2(GEN x, long l) { return Flx_eval2BILspec(x, 2, l); }
4918 :
4919 : static GEN
4920 0 : kron_pack_Flx_spec_3(GEN x, long l) { return Flx_eval2BILspec(x, 3, l); }
4921 :
4922 : static GEN
4923 42785 : kron_unpack_Flx(GEN z, ulong p)
4924 : {
4925 42785 : long i, l = lgefint(z);
4926 42785 : GEN x = cgetg(l, t_VECSMALL), w;
4927 201296 : for (w = int_LSW(z), i = 2; i < l; w = int_nextW(w), i++)
4928 158511 : x[i] = ((ulong) *w) % p;
4929 42785 : return Flx_renormalize(x, l);
4930 : }
4931 :
4932 : static GEN
4933 2930 : kron_unpack_Flx_2(GEN x, ulong p) {
4934 2930 : long d = (lgefint(x)-1)/2 - 1;
4935 2930 : return Z_mod2BIL_Flx_2(x, d, p);
4936 : }
4937 :
4938 : static GEN
4939 0 : kron_unpack_Flx_3(GEN x, ulong p) {
4940 0 : long d = lgefint(x)/3 - 1;
4941 0 : return Z_mod2BIL_Flx_3(x, d, p);
4942 : }
4943 :
4944 : static GEN
4945 116239 : FlxM_pack_ZM_bits(GEN M, long b)
4946 : {
4947 : long i, j, l, lc;
4948 116239 : GEN N = cgetg_copy(M, &l), x;
4949 116239 : if (l == 1)
4950 0 : return N;
4951 116239 : lc = lgcols(M);
4952 479672 : for (j = 1; j < l; j++) {
4953 363433 : gel(N, j) = cgetg(lc, t_COL);
4954 5955086 : for (i = 1; i < lc; i++) {
4955 5591653 : x = gcoeff(M, i, j);
4956 5591653 : gcoeff(N, i, j) = kron_pack_Flx_spec_bits(x + 2, b, lgpol(x));
4957 : }
4958 : }
4959 116239 : return N;
4960 : }
4961 :
4962 : static GEN
4963 23421 : ZM_unpack_FlxM(GEN M, ulong p, ulong sv, GEN (*unpack)(GEN, ulong))
4964 : {
4965 : long i, j, l, lc;
4966 23421 : GEN N = cgetg_copy(M, &l), x;
4967 23421 : if (l == 1)
4968 0 : return N;
4969 23421 : lc = lgcols(M);
4970 111236 : for (j = 1; j < l; j++) {
4971 87815 : gel(N, j) = cgetg(lc, t_COL);
4972 634989 : for (i = 1; i < lc; i++) {
4973 547174 : x = unpack(gcoeff(M, i, j), p);
4974 547174 : x[1] = sv;
4975 547174 : gcoeff(N, i, j) = x;
4976 : }
4977 : }
4978 23421 : return N;
4979 : }
4980 :
4981 : static GEN
4982 58160 : ZM_unpack_FlxM_bits(GEN M, long b, ulong p, ulong pi, long sv)
4983 : {
4984 : long i, j, l, lc;
4985 58160 : GEN N = cgetg_copy(M, &l), x;
4986 58160 : if (l == 1)
4987 0 : return N;
4988 58160 : lc = lgcols(M);
4989 58160 : if (b < BITS_IN_LONG) {
4990 195346 : for (j = 1; j < l; j++) {
4991 138869 : gel(N, j) = cgetg(lc, t_COL);
4992 3250343 : for (i = 1; i < lc; i++) {
4993 3111474 : x = kron_unpack_Flx_bits_narrow(gcoeff(M, i, j), b, p);
4994 3111474 : x[1] = sv;
4995 3111474 : gcoeff(N, i, j) = x;
4996 : }
4997 : }
4998 : } else {
4999 1683 : if (!pi) pi = get_Fl_red(p); /* unset if !SMALL_ULONG(p) */
5000 9844 : for (j = 1; j < l; j++) {
5001 8161 : gel(N, j) = cgetg(lc, t_COL);
5002 175361 : for (i = 1; i < lc; i++) {
5003 167200 : x = kron_unpack_Flx_bits_wide(gcoeff(M, i, j), b, p, pi);
5004 167200 : x[1] = sv;
5005 167200 : gcoeff(N, i, j) = x;
5006 : }
5007 : }
5008 : }
5009 58160 : return N;
5010 : }
5011 :
5012 : static GEN
5013 81581 : FlxM_mul_Kronecker_i(GEN A, GEN B, ulong p, ulong pi, long d, long sv)
5014 : {
5015 81581 : long b, n = lg(A) - 1;
5016 : GEN C, z;
5017 : GEN (*pack)(GEN, long), (*unpack)(GEN, ulong);
5018 81581 : int is_sqr = A==B;
5019 :
5020 81581 : z = muliu(muliu(sqru(p - 1), d), n);
5021 81581 : b = expi(z) + 1;
5022 : /* only do expensive bit-packing if it saves at least 1 limb */
5023 81581 : if (b <= BITS_IN_HALFULONG)
5024 77198 : { if (nbits2nlong(d*b) == (d + 1)/2) b = BITS_IN_HALFULONG; }
5025 : else
5026 : {
5027 4383 : long l = lgefint(z) - 2;
5028 4383 : if (nbits2nlong(d*b) == d*l) b = l*BITS_IN_LONG;
5029 : }
5030 :
5031 81581 : switch (b) {
5032 22410 : case BITS_IN_HALFULONG:
5033 22410 : pack = kron_pack_Flx_spec_half;
5034 22410 : unpack = int_to_Flx_half;
5035 22410 : break;
5036 962 : case BITS_IN_LONG:
5037 962 : pack = kron_pack_Flx_spec;
5038 962 : unpack = kron_unpack_Flx;
5039 962 : break;
5040 49 : case 2*BITS_IN_LONG:
5041 49 : pack = kron_pack_Flx_spec_2;
5042 49 : unpack = kron_unpack_Flx_2;
5043 49 : break;
5044 0 : case 3*BITS_IN_LONG:
5045 0 : pack = kron_pack_Flx_spec_3;
5046 0 : unpack = kron_unpack_Flx_3;
5047 0 : break;
5048 58160 : default:
5049 58160 : A = FlxM_pack_ZM_bits(A, b);
5050 58160 : B = is_sqr? A: FlxM_pack_ZM_bits(B, b);
5051 58160 : C = ZM_mul(A, B);
5052 58160 : return ZM_unpack_FlxM_bits(C, b, p, pi, sv);
5053 : }
5054 23421 : A = FlxM_pack_ZM(A, pack);
5055 23421 : B = is_sqr? A: FlxM_pack_ZM(B, pack);
5056 23421 : C = ZM_mul(A, B);
5057 23421 : return ZM_unpack_FlxM(C, p, sv, unpack);
5058 : }
5059 :
5060 : GEN
5061 81581 : FlxqM_mul_Kronecker(GEN A, GEN B, GEN T, ulong p)
5062 : {
5063 81581 : pari_sp av = avma;
5064 81581 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
5065 81581 : long sv = get_Flx_var(T), d = get_Flx_degree(T);
5066 81581 : GEN C = FlxM_mul_Kronecker_i(A, B, p, pi, d, sv);
5067 81581 : C = FlxqM_red_pre(C, T, p, pi);
5068 81581 : return gc_upto(av, C);
5069 : }
5070 :
5071 : /* assume m > 1 */
5072 : static long
5073 0 : FlxV_max_degree_i(GEN x, long m)
5074 : {
5075 0 : long i, l = degpol(gel(x,1));
5076 0 : for (i = 2; i < m; i++) l = maxss(l, degpol(gel(x,i)));
5077 0 : return l;
5078 : }
5079 :
5080 : /* assume n > 1 and m > 1 */
5081 : static long
5082 0 : FlxM_max_degree_i(GEN x, long n, long m)
5083 : {
5084 0 : long j, l = FlxV_max_degree_i(gel(x,1), m);
5085 0 : for (j = 2; j < n; j++) l = maxss(l, FlxV_max_degree_i(gel(x,j), m));
5086 0 : return l;
5087 : }
5088 :
5089 : static long
5090 0 : FlxM_max_degree(GEN x)
5091 : {
5092 0 : long n = lg(x), m;
5093 0 : if (n == 1) return -1;
5094 0 : m = lgcols(x); return m == 1? -1: FlxM_max_degree_i(x, n, m);
5095 : }
5096 :
5097 : GEN
5098 0 : FlxM_mul(GEN x, GEN y, ulong p)
5099 : {
5100 0 : pari_sp av = avma;
5101 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
5102 : long sv, d;
5103 0 : if (lg(x) == 1) return cgetg(1,t_MAT);
5104 0 : if (lg(gel(x,1))==1) return FlxqM_mul(x, y, NULL, p);
5105 0 : sv = mael3(x,1,1,1);
5106 0 : d = maxss(FlxM_max_degree(x), FlxM_max_degree(y));
5107 0 : return gc_GEN(av, FlxM_mul_Kronecker_i(x, y, p, pi, d+1, sv));
5108 : }
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