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 986781924 : get_Flx_red(GEN T, GEN *B)
22 : {
23 986781924 : if (typ(T)!=t_VEC) { *B=NULL; return T; }
24 656398 : *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 38195985 : Flx_to_ZX(GEN z)
43 : {
44 38195985 : long i, l = lg(z);
45 38195985 : GEN x = cgetg(l,t_POL);
46 247453815 : for (i=2; i<l; i++) gel(x,i) = utoi(z[i]);
47 38182985 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
48 : }
49 :
50 : GEN
51 71328 : Flx_to_FlxX(GEN z, long sv)
52 : {
53 71328 : long i, l = lg(z);
54 71328 : GEN x = cgetg(l,t_POL);
55 278513 : for (i=2; i<l; i++) gel(x,i) = Fl_to_Flx(z[i], sv);
56 71328 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
57 : }
58 :
59 : /* same as Flx_to_ZX, in place */
60 : GEN
61 36394664 : Flx_to_ZX_inplace(GEN z)
62 : {
63 36394664 : long i, l = lg(z);
64 227106013 : for (i=2; i<l; i++) gel(z,i) = utoi(z[i]);
65 36389339 : 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 66312699 : Flx_to_Flv(GEN x, long N)
71 : {
72 66312699 : GEN z = cgetg(N+1,t_VECSMALL);
73 66305660 : long i, l = lg(x)-1;
74 66305660 : x++;
75 705878433 : for (i=1; i<l ; i++) z[i]=x[i];
76 328591243 : for ( ; i<=N; i++) z[i]=0;
77 66305660 : return z;
78 : }
79 :
80 : /*Flv_to_Flx=zv_to_zx*/
81 : GEN
82 25369116 : Flv_to_Flx(GEN x, long sv)
83 : {
84 25369116 : long i, l=lg(x)+1;
85 25369116 : GEN z = cgetg(l,t_VECSMALL); z[1]=sv;
86 25364904 : x--;
87 278546593 : for (i=2; i<l ; i++) z[i]=x[i];
88 25364904 : 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 102363 : FlxC_to_ZXC(GEN x)
99 502097 : { pari_APPLY_type(t_COL, Flx_to_ZX(gel(x,i))) }
100 :
101 : /*FlxC_to_ZXC=zxV_to_ZXV*/
102 : GEN
103 606386 : FlxV_to_ZXV(GEN x)
104 2451992 : { pari_APPLY_type(t_VEC, Flx_to_ZX(gel(x,i))) }
105 :
106 : void
107 3021306 : FlxV_to_ZXV_inplace(GEN v)
108 : {
109 : long i;
110 8014331 : for(i=1;i<lg(v);i++) gel(v,i)= Flx_to_ZX(gel(v,i));
111 3021216 : }
112 :
113 : /*FlxM_to_ZXM=zxM_to_ZXM*/
114 : GEN
115 1894 : FlxM_to_ZXM(GEN x)
116 5978 : { pari_APPLY_same(FlxC_to_ZXC(gel(x,i))) }
117 :
118 : GEN
119 394876 : FlxV_to_FlxX(GEN x, long v)
120 : {
121 394876 : long i, l = lg(x)+1;
122 394876 : GEN z = cgetg(l,t_POL); z[1] = evalvarn(v);
123 394876 : x--;
124 4961479 : for (i=2; i<l ; i++) gel(z,i) = gel(x,i);
125 394876 : 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 21193327 : Fl_to_Flx(ulong x, long sv) { return x? mkvecsmall2(sv, x): pol0_Flx(sv); }
156 :
157 : /* a X^d */
158 : GEN
159 939458 : monomial_Flx(ulong a, long d, long vs)
160 : {
161 : GEN P;
162 939458 : if (a==0) return pol0_Flx(vs);
163 939458 : P = const_vecsmall(d+2, 0);
164 939464 : P[1] = vs; P[d+2] = a; return P;
165 : }
166 :
167 : GEN
168 7503212 : Z_to_Flx(GEN x, ulong p, long sv)
169 : {
170 7503212 : long u = umodiu(x,p);
171 7503211 : 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 170360852 : ZX_to_Flx(GEN x, ulong p)
177 : {
178 170360852 : long i, lx = lg(x);
179 170360852 : GEN a = cgetg(lx, t_VECSMALL);
180 170313217 : a[1]=((ulong)x[1])&VARNBITS;
181 1121488663 : for (i=2; i<lx; i++) a[i] = umodiu(gel(x,i), p);
182 170329325 : 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 6539138 : zx_to_Flx(GEN x, ulong p)
188 : {
189 6539138 : long i, lx = lg(x);
190 6539138 : GEN a = cgetg(lx, t_VECSMALL);
191 6533600 : a[1] = x[1];
192 20124492 : for (i=2; i<lx; i++) uel(a,i) = umodsu(x[i], p);
193 6533965 : return Flx_renormalize(a,lx);
194 : }
195 :
196 : ulong
197 63876875 : Rg_to_Fl(GEN x, ulong p)
198 : {
199 63876875 : switch(typ(x))
200 : {
201 39938272 : case t_INT: return umodiu(x, p);
202 455746 : case t_FRAC: {
203 455746 : ulong z = umodiu(gel(x,1), p);
204 455746 : if (!z) return 0;
205 446319 : return Fl_div(z, umodiu(gel(x,2), p), p);
206 : }
207 205955 : case t_PADIC: return padic_to_Fl(x, p);
208 23276914 : case t_INTMOD: {
209 23276914 : GEN q = gel(x,1), a = gel(x,2);
210 23276914 : 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 1704463 : Rg_to_F2(GEN x)
221 : {
222 1704463 : switch(typ(x))
223 : {
224 277534 : 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 1426929 : case t_INTMOD: {
233 1426929 : GEN q = gel(x,1), a = gel(x,2);
234 1426929 : if (mpodd(q)) pari_err_MODULUS("Rg_to_F2", q, gen_2);
235 1426929 : 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 2162743 : RgX_to_Flx(GEN x, ulong p)
244 : {
245 2162743 : long i, lx = lg(x);
246 2162743 : GEN a = cgetg(lx, t_VECSMALL);
247 2162743 : a[1]=((ulong)x[1])&VARNBITS;
248 19607959 : for (i=2; i<lx; i++) a[i] = Rg_to_Fl(gel(x,i), p);
249 2162743 : 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 3589621 : Rg_to_Flxq(GEN x, GEN T, ulong p)
259 : {
260 3589621 : long ta, tx = typ(x), v = get_Flx_var(T);
261 : ulong pi;
262 : GEN a, b;
263 3589621 : if (is_const_t(tx))
264 : {
265 3329989 : if (tx == t_FFELT) return FF_to_Flxq(x);
266 2598981 : return Fl_to_Flx(Rg_to_Fl(x, p), v);
267 : }
268 259631 : 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 251055 : case t_POL:
280 251055 : x = RgX_to_Flx(x,p);
281 251055 : if (x[1] != v) break;
282 251055 : 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 2147635892 : Flx_renormalize(GEN /*in place*/ x, long lx)
298 : {
299 : long i;
300 2396254781 : for (i = lx-1; i>1; i--)
301 2300747454 : if (x[i]) break;
302 2147635892 : stackdummy((pari_sp)(x + lg(x)), (pari_sp)(x + i+1));
303 2146396166 : setlg(x, i+1); return x;
304 : }
305 :
306 : GEN
307 1879111 : Flx_red(GEN z, ulong p)
308 : {
309 1879111 : long i, l = lg(z);
310 1879111 : GEN x = cgetg(l, t_VECSMALL);
311 1878971 : x[1] = z[1];
312 33411629 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
313 1878971 : return Flx_renormalize(x,l);
314 : }
315 :
316 : int
317 26928717 : Flx_equal(GEN V, GEN W)
318 : {
319 26928717 : long l = lg(V);
320 26928717 : if (lg(W) != l) return 0;
321 27927946 : while (--l > 1) /* do not compare variables, V[1] */
322 26826787 : if (V[l] != W[l]) return 0;
323 1101159 : return 1;
324 : }
325 :
326 : GEN
327 2640265 : random_Flx(long d1, long vs, ulong p)
328 : {
329 2640265 : long i, d = d1+2;
330 2640265 : GEN y = cgetg(d,t_VECSMALL); y[1] = vs;
331 18167650 : for (i=2; i<d; i++) y[i] = random_Fl(p);
332 2640416 : return Flx_renormalize(y,d);
333 : }
334 :
335 : static GEN
336 7250247 : Flx_addspec(GEN x, GEN y, ulong p, long lx, long ly)
337 : {
338 : long i,lz;
339 : GEN z;
340 :
341 7250247 : if (ly>lx) swapspec(x,y, lx,ly);
342 7250247 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
343 106607358 : for (i=0; i<ly; i++) z[i+2] = Fl_add(x[i], y[i], p);
344 90323907 : for ( ; i<lx; i++) z[i+2] = x[i];
345 7250247 : z[1] = 0; return Flx_renormalize(z, lz);
346 : }
347 :
348 : GEN
349 64364318 : Flx_add(GEN x, GEN y, ulong p)
350 : {
351 : long i,lz;
352 : GEN z;
353 64364318 : long lx=lg(x);
354 64364318 : long ly=lg(y);
355 64364318 : if (ly>lx) swapspec(x,y, lx,ly);
356 64364318 : lz = lx; z = cgetg(lz, t_VECSMALL); z[1]=x[1];
357 583586967 : for (i=2; i<ly; i++) z[i] = Fl_add(x[i], y[i], p);
358 128568187 : for ( ; i<lx; i++) z[i] = x[i];
359 64339227 : return Flx_renormalize(z, lz);
360 : }
361 :
362 : GEN
363 10006254 : Flx_Fl_add(GEN y, ulong x, ulong p)
364 : {
365 : GEN z;
366 : long lz, i;
367 10006254 : if (!lgpol(y))
368 229285 : return Fl_to_Flx(x,y[1]);
369 9778083 : lz=lg(y);
370 9778083 : z=cgetg(lz,t_VECSMALL);
371 9777348 : z[1]=y[1];
372 9777348 : z[2] = Fl_add(y[2],x,p);
373 47664069 : for(i=3;i<lz;i++)
374 37887029 : z[i] = y[i];
375 9777040 : if (lz==3) z = Flx_renormalize(z,lz);
376 9776939 : return z;
377 : }
378 :
379 : static GEN
380 882139 : Flx_subspec(GEN x, GEN y, ulong p, long lx, long ly)
381 : {
382 : long i,lz;
383 : GEN z;
384 :
385 882139 : if (ly <= lx)
386 : {
387 882158 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
388 52429712 : for (i=0; i<ly; i++) z[i+2] = Fl_sub(x[i],y[i],p);
389 1432832 : 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 881886 : z[1] = 0; return Flx_renormalize(z, lz);
398 : }
399 :
400 : GEN
401 138086354 : Flx_sub(GEN x, GEN y, ulong p)
402 : {
403 138086354 : long i,lz,lx = lg(x), ly = lg(y);
404 : GEN z;
405 :
406 138086354 : if (ly <= lx)
407 : {
408 88077907 : lz = lx; z = cgetg(lz, t_VECSMALL);
409 455780914 : for (i=2; i<ly; i++) z[i] = Fl_sub(x[i],y[i],p);
410 175504545 : for ( ; i<lx; i++) z[i] = x[i];
411 : }
412 : else
413 : {
414 50008447 : lz = ly; z = cgetg(lz, t_VECSMALL);
415 261254876 : for (i=2; i<lx; i++) z[i] = Fl_sub(x[i],y[i],p);
416 231917319 : for ( ; i<ly; i++) z[i] = y[i]? (long)(p - y[i]): y[i];
417 : }
418 138075992 : z[1]=x[1]; return Flx_renormalize(z, lz);
419 : }
420 :
421 : GEN
422 151664 : Flx_Fl_sub(GEN y, ulong x, ulong p)
423 : {
424 : GEN z;
425 151664 : long lz = lg(y), i;
426 151664 : if (lz==2)
427 513 : return Fl_to_Flx(Fl_neg(x, p),y[1]);
428 151151 : z = cgetg(lz, t_VECSMALL);
429 151151 : z[1] = y[1];
430 151151 : z[2] = Fl_sub(uel(y,2), x, p);
431 752666 : for(i=3; i<lz; i++)
432 601515 : z[i] = y[i];
433 151151 : if (lz==3) z = Flx_renormalize(z,lz);
434 151151 : return z;
435 : }
436 :
437 : static GEN
438 2926432 : Flx_negspec(GEN x, ulong p, long l)
439 : {
440 : long i;
441 2926432 : GEN z = cgetg(l+2, t_VECSMALL) + 2;
442 17832584 : for (i=0; i<l; i++) z[i] = Fl_neg(x[i], p);
443 2926373 : return z-2;
444 : }
445 :
446 : GEN
447 2926437 : Flx_neg(GEN x, ulong p)
448 : {
449 2926437 : GEN z = Flx_negspec(x+2, p, lgpol(x));
450 2926542 : z[1] = x[1];
451 2926542 : return z;
452 : }
453 :
454 : GEN
455 1777029 : Flx_neg_inplace(GEN x, ulong p)
456 : {
457 1777029 : long i, l = lg(x);
458 52326165 : for (i=2; i<l; i++)
459 50549136 : if (x[i]) x[i] = p - x[i];
460 1777029 : 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 18134170 : Flx_Fl_mul_pre(GEN y, ulong x, ulong p, ulong pi)
482 : {
483 : GEN z;
484 : long i, l;
485 18134170 : if (!x) return pol0_Flx(y[1]);
486 17400655 : z = cgetg_copy(y, &l); z[1] = y[1];
487 17400504 : if (pi==0)
488 : {
489 15190897 : if (HIGHWORD(x | p))
490 0 : for(i=2; i<l; i++) z[i] = Fl_mul(uel(y,i), x, p);
491 : else
492 89677738 : for(i=2; i<l; i++) z[i] = (uel(y,i) * x) % p;
493 : } else
494 18083918 : for(i=2; i<l; i++) z[i] = Fl_mul_pre(uel(y,i), x, p, pi);
495 17400892 : return Flx_renormalize(z, l);
496 : }
497 :
498 : GEN
499 7201719 : Flx_Fl_mul(GEN x, ulong y, ulong p)
500 7201719 : { 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 12007730 : Flx_Fl_mul_to_monic(GEN y, ulong x, ulong p)
515 : {
516 : GEN z;
517 : long i, l;
518 12007730 : z = cgetg_copy(y, &l); z[1] = y[1];
519 12004101 : if (HIGHWORD(x | p))
520 5385345 : for(i=2; i<l-1; i++) z[i] = Fl_mul(y[i], x, p);
521 : else
522 27026587 : for(i=2; i<l-1; i++) z[i] = (y[i] * x) % p;
523 12004097 : 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 27256933 : Flx_shift(GEN a, long n)
529 : {
530 27256933 : long i, l = lg(a);
531 : GEN b;
532 27256933 : if (l==2 || !n) return Flx_copy(a);
533 26912202 : if (l+n<=2) return pol0_Flx(a[1]);
534 26696947 : b = cgetg(l+n, t_VECSMALL);
535 26695168 : b[1] = a[1];
536 26695168 : if (n < 0)
537 73556698 : for (i=2-n; i<l; i++) b[i+n] = a[i];
538 : else
539 : {
540 52147945 : for (i=0; i<n; i++) b[2+i] = 0;
541 148547648 : for (i=2; i<l; i++) b[i+n] = a[i];
542 : }
543 26695168 : return b;
544 : }
545 :
546 : GEN
547 63840720 : Flx_normalize(GEN z, ulong p)
548 : {
549 63840720 : long l = lg(z)-1;
550 63840720 : ulong p1 = z[l]; /* leading term */
551 63840720 : if (p1 == 1) return z;
552 11983370 : 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 3724819 : Flx_addshift(GEN x, GEN y, ulong p, long d)
558 : {
559 3724819 : GEN xd,yd,zd = (GEN)avma;
560 3724819 : long a,lz,ny = lgpol(y), nx = lgpol(x);
561 3724819 : long vs = x[1];
562 3724819 : if (nx == 0) return y;
563 3722976 : x += 2; y += 2; a = ny-d;
564 3722976 : if (a <= 0)
565 : {
566 85099 : lz = (a>nx)? ny+2: nx+d+2;
567 85099 : (void)new_chunk(lz); xd = x+nx; yd = y+ny;
568 1732405 : while (xd > x) *--zd = *--xd;
569 85099 : x = zd + a;
570 164579 : while (zd > x) *--zd = 0;
571 : }
572 : else
573 : {
574 3637877 : xd = new_chunk(d); yd = y+d;
575 3637877 : x = Flx_addspec(x,yd,p, nx,a);
576 3637877 : lz = (a>nx)? ny+2: lg(x)+d;
577 132727937 : x += 2; while (xd > x) *--zd = *--xd;
578 : }
579 60411380 : while (yd > y) *--zd = *--yd;
580 3722976 : *--zd = vs;
581 3722976 : *--zd = evaltyp(t_VECSMALL) | evallg(lz); return zd;
582 : }
583 :
584 : /* shift polynomial + GC; do not set evalvarn*/
585 : static GEN
586 631535656 : Flx_shiftip(pari_sp av, GEN x, long v)
587 : {
588 631535656 : long i, lx = lg(x), ly;
589 : GEN y;
590 631535656 : if (!v || lx==2) return gc_leaf(av, x);
591 176445291 : ly = lx + v; /* result length */
592 176445291 : (void)new_chunk(ly); /* check that result fits */
593 176341575 : x += lx; y = (GEN)av;
594 1244636786 : for (i = 2; i<lx; i++) *--y = *--x;
595 707734172 : for (i = 0; i< v; i++) *--y = 0;
596 176341575 : y -= 2; y[0] = evaltyp(t_VECSMALL) | evallg(ly);
597 176458013 : return gc_const((pari_sp)y, y);
598 : }
599 :
600 : static long
601 2322278137 : get_Fl_threshold(ulong p, long mul, long mul2)
602 : {
603 2322278137 : 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 8358695 : maxbitcoeffpol(ulong p, long n)
614 : {
615 8358695 : GEN z = muliu(sqru(p - 1), n);
616 8356444 : long b = expi(z) + 1;
617 : /* only do expensive bit-packing if it saves at least 1 limb */
618 8357205 : if (b <= BITS_IN_QUARTULONG)
619 : {
620 873005 : if (nbits2nlong(n*b) == (n + 3)>>2)
621 107039 : b = BITS_IN_QUARTULONG;
622 : }
623 7484200 : else if (b <= BITS_IN_HALFULONG)
624 : {
625 1548618 : if (nbits2nlong(n*b) == (n + 1)>>1)
626 5809 : b = BITS_IN_HALFULONG;
627 : }
628 : else
629 : {
630 5935582 : long l = lgefint(z) - 2;
631 5935582 : if (nbits2nlong(n*b) == n*l)
632 305784 : b = l*BITS_IN_LONG;
633 : }
634 8357076 : return b;
635 : }
636 :
637 : INLINE ulong
638 3367439324 : Flx_mullimb_ok(GEN x, GEN y, ulong p, long a, long b)
639 : { /* Assume OK_ULONG*/
640 3367439324 : ulong p1 = 0;
641 : long i;
642 15960425276 : for (i=a; i<b; i++)
643 12592985952 : if (y[i])
644 : {
645 10599040532 : p1 += y[i] * x[-i];
646 10599040532 : if (p1 & HIGHBIT) p1 %= p;
647 : }
648 3367439324 : return p1 % p;
649 : }
650 :
651 : INLINE ulong
652 1189521705 : Flx_mullimb(GEN x, GEN y, ulong p, ulong pi, long a, long b)
653 : {
654 1189521705 : ulong p1 = 0;
655 : long i;
656 3763871466 : for (i=a; i<b; i++)
657 2574860420 : if (y[i])
658 2533253818 : p1 = Fl_addmul_pre(p1, y[i], x[-i], p, pi);
659 1189011046 : return p1;
660 : }
661 :
662 : /* assume nx >= ny > 0 */
663 : static GEN
664 344480500 : 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 344480500 : lz = nx+ny+1; nz = lz-2;
670 344480500 : z = cgetg(lz, t_VECSMALL) + 2; /* x:y:z [i] = term of degree i */
671 344262074 : if (!pi)
672 : {
673 1132871538 : for (i=0; i<ny; i++)z[i] = Flx_mullimb_ok(x+i,y,p,0,i+1);
674 733202221 : for ( ; i<nx; i++) z[i] = Flx_mullimb_ok(x+i,y,p,0,ny);
675 881181623 : for ( ; i<nz; i++) z[i] = Flx_mullimb_ok(x+i,y,p,i-nx+1,ny);
676 : }
677 : else
678 : {
679 322797734 : for (i=0; i<ny; i++)z[i] = Flx_mullimb(x+i,y,p,pi,0,i+1);
680 222227419 : for ( ; i<nx; i++) z[i] = Flx_mullimb(x+i,y,p,pi,0,ny);
681 231593959 : for ( ; i<nz; i++) z[i] = Flx_mullimb(x+i,y,p,pi,i-nx+1,ny);
682 : }
683 344033194 : z -= 2; return Flx_renormalize(z, lz);
684 : }
685 :
686 : static GEN
687 12215 : int_to_Flx(GEN z, ulong p)
688 : {
689 12215 : long i, l = lgefint(z);
690 12215 : GEN x = cgetg(l, t_VECSMALL);
691 1047253 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
692 12212 : return Flx_renormalize(x, l);
693 : }
694 :
695 : INLINE GEN
696 10146 : Flx_mulspec_mulii(GEN a, GEN b, ulong p, long na, long nb)
697 : {
698 10146 : GEN z=muliispec(a,b,na,nb);
699 10147 : return int_to_Flx(z,p);
700 : }
701 :
702 : static GEN
703 330361 : Flx_to_int_halfspec(GEN a, long na)
704 : {
705 : long j;
706 330361 : long n = (na+1)>>1UL;
707 330361 : GEN V = cgetipos(2+n);
708 : GEN w;
709 1126179 : for (w = int_LSW(V), j=0; j+1<na; j+=2, w=int_nextW(w))
710 795818 : *w = a[j]|(a[j+1]<<BITS_IN_HALFULONG);
711 330361 : if (j<na)
712 206638 : *w = a[j];
713 330361 : return V;
714 : }
715 :
716 : static GEN
717 335184 : int_to_Flx_half(GEN z, ulong p)
718 : {
719 : long i;
720 335184 : long lx = (lgefint(z)-2)*2+2;
721 335184 : GEN w, x = cgetg(lx, t_VECSMALL);
722 1520014 : for (w = int_LSW(z), i=2; i<lx; i+=2, w=int_nextW(w))
723 : {
724 1184830 : x[i] = LOWWORD((ulong)*w)%p;
725 1184830 : x[i+1] = HIGHWORD((ulong)*w)%p;
726 : }
727 335184 : return Flx_renormalize(x, lx);
728 : }
729 :
730 : static GEN
731 5484 : Flx_mulspec_halfmulii(GEN a, GEN b, ulong p, long na, long nb)
732 : {
733 5484 : GEN A = Flx_to_int_halfspec(a,na);
734 5484 : GEN B = Flx_to_int_halfspec(b,nb);
735 5484 : GEN z = mulii(A,B);
736 5484 : return int_to_Flx_half(z,p);
737 : }
738 :
739 : static GEN
740 203882 : Flx_to_int_quartspec(GEN a, long na)
741 : {
742 : long j;
743 203882 : long n = (na+3)>>2UL;
744 203882 : GEN V = cgetipos(2+n);
745 : GEN w;
746 4364633 : for (w = int_LSW(V), j=0; j+3<na; j+=4, w=int_nextW(w))
747 4160751 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG))|(a[j+3]<<(3*BITS_IN_QUARTULONG));
748 203882 : switch (na-j)
749 : {
750 115846 : case 3:
751 115846 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG));
752 115846 : break;
753 34292 : case 2:
754 34292 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG);
755 34292 : break;
756 27334 : case 1:
757 27334 : *w = a[j];
758 27334 : break;
759 26411 : case 0:
760 26411 : break;
761 : }
762 203882 : return V;
763 : }
764 :
765 : static GEN
766 107040 : int_to_Flx_quart(GEN z, ulong p)
767 : {
768 : long i;
769 107040 : long lx = (lgefint(z)-2)*4+2;
770 107040 : GEN w, x = cgetg(lx, t_VECSMALL);
771 4860014 : for (w = int_LSW(z), i=2; i<lx; i+=4, w=int_nextW(w))
772 : {
773 4752974 : x[i] = LLQUARTWORD((ulong)*w)%p;
774 4752974 : x[i+1] = HLQUARTWORD((ulong)*w)%p;
775 4752974 : x[i+2] = LHQUARTWORD((ulong)*w)%p;
776 4752974 : x[i+3] = HHQUARTWORD((ulong)*w)%p;
777 : }
778 107040 : return Flx_renormalize(x, lx);
779 : }
780 :
781 : static GEN
782 96842 : Flx_mulspec_quartmulii(GEN a, GEN b, ulong p, long na, long nb)
783 : {
784 96842 : GEN A = Flx_to_int_quartspec(a,na);
785 96843 : GEN B = Flx_to_int_quartspec(b,nb);
786 96843 : GEN z = mulii(A,B);
787 96843 : 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 579005 : Flx_eval2BILspec(GEN x, long k, long l)
793 : {
794 579005 : long i, lz = k*l, ki;
795 579005 : GEN pz = cgetipos(2+lz);
796 16208751 : for (i=0; i < lz; i++)
797 15629746 : *int_W(pz,i) = 0UL;
798 8393878 : for (i=0, ki=0; i<l; i++, ki+=k)
799 7814873 : *int_W(pz,ki) = x[i];
800 579005 : return int_normalize(pz,0);
801 : }
802 :
803 : static GEN
804 296499 : Z_mod2BIL_Flx_2(GEN x, long d, ulong p)
805 : {
806 296499 : long i, offset, lm = lgefint(x)-2, l = d+3;
807 296499 : ulong pi = get_Fl_red(p);
808 296499 : GEN pol = cgetg(l, t_VECSMALL);
809 296499 : pol[1] = 0;
810 7934330 : for (i=0, offset=0; offset+1 < lm; i++, offset += 2)
811 7637831 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
812 296499 : if (offset < lm)
813 223628 : pol[i+2] = (*int_W(x,offset)) % p;
814 296499 : 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 293569 : Z_mod2BIL_Flx(GEN x, long bs, long d, ulong p)
836 : {
837 293569 : return bs==2 ? Z_mod2BIL_Flx_2(x, d, p): Z_mod2BIL_Flx_3(x, d, p);
838 : }
839 :
840 : static GEN
841 282047 : Flx_mulspec_mulii_inflate(GEN x, GEN y, long N, ulong p, long nx, long ny)
842 : {
843 282047 : pari_sp av = avma;
844 282047 : GEN z = mulii(Flx_eval2BILspec(x,N,nx), Flx_eval2BILspec(y,N,ny));
845 282047 : return gc_upto(av, Z_mod2BIL_Flx(z, N, nx+ny-2, p));
846 : }
847 :
848 : static GEN
849 20685718 : kron_pack_Flx_spec_bits(GEN x, long b, long l) {
850 : GEN y;
851 : long i;
852 20685718 : if (l == 0)
853 3417050 : return gen_0;
854 17268668 : y = cgetg(l + 1, t_VECSMALL);
855 806293460 : for(i = 1; i <= l; i++)
856 789029133 : y[i] = x[l - i];
857 17264327 : return nv_fromdigits_2k(y, b);
858 : }
859 :
860 : /* assume b < BITS_IN_LONG */
861 : static GEN
862 5549547 : kron_unpack_Flx_bits_narrow(GEN z, long b, ulong p) {
863 5549547 : GEN v = binary_2k_nv(z, b), x;
864 5549610 : long i, l = lg(v) + 1;
865 5549610 : x = cgetg(l, t_VECSMALL);
866 614516376 : for (i = 2; i < l; i++)
867 608966655 : x[i] = v[l - i] % p;
868 5549721 : return Flx_renormalize(x, l);
869 : }
870 :
871 : static GEN
872 5595676 : kron_unpack_Flx_bits_wide(GEN z, long b, ulong p, ulong pi) {
873 5595676 : GEN v = binary_2k(z, b), x, y;
874 5596431 : long i, l = lg(v) + 1, ly;
875 5596431 : x = cgetg(l, t_VECSMALL);
876 232259495 : for (i = 2; i < l; i++) {
877 226667152 : y = gel(v, l - i);
878 226667152 : ly = lgefint(y);
879 226667152 : switch (ly) {
880 6277676 : case 2: x[i] = 0; break;
881 29562034 : case 3: x[i] = *int_W_lg(y, 0, ly) % p; break;
882 174837056 : case 4: x[i] = remll_pre(*int_W_lg(y, 1, ly), *int_W_lg(y, 0, ly), p, pi); break;
883 31981499 : case 5: x[i] = remlll_pre(*int_W_lg(y, 2, ly), *int_W_lg(y, 1, ly),
884 15990386 : *int_W_lg(y, 0, ly), p, pi); break;
885 0 : default: x[i] = umodiu(gel(v, l - i), p);
886 : }
887 : }
888 5592343 : return Flx_renormalize(x, l);
889 : }
890 :
891 : static GEN
892 7247033 : Flx_mulspec_Kronecker(GEN A, GEN B, long b, ulong p, long lA, long lB)
893 : {
894 : GEN C, D;
895 7247033 : pari_sp av = avma;
896 7247033 : A = kron_pack_Flx_spec_bits(A, b, lA);
897 7252271 : B = kron_pack_Flx_spec_bits(B, b, lB);
898 7252290 : C = gc_INT(av, mulii(A, B));
899 7250415 : if (b < BITS_IN_LONG)
900 2031850 : D = kron_unpack_Flx_bits_narrow(C, b, p);
901 : else
902 : {
903 5218565 : ulong pi = get_Fl_red(p);
904 5217634 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
905 : }
906 7248050 : return D;
907 : }
908 :
909 : static GEN
910 690499 : Flx_sqrspec_Kronecker(GEN A, long b, ulong p, long lA)
911 : {
912 : GEN C, D;
913 690499 : A = kron_pack_Flx_spec_bits(A, b, lA);
914 690587 : C = sqri(A);
915 690608 : if (b < BITS_IN_LONG)
916 477136 : D = kron_unpack_Flx_bits_narrow(C, b, p);
917 : else
918 : {
919 213472 : ulong pi = get_Fl_red(p);
920 213468 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
921 : }
922 690583 : 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 381724616 : Flx_mulspec(GEN a, GEN b, ulong p, ulong pi, long na, long nb)
931 : {
932 : GEN a0,c,c0;
933 381724616 : long n0, n0a, i, v = 0;
934 : pari_sp av;
935 :
936 486917226 : while (na && !a[0]) { a++; na--; v++; }
937 567981860 : while (nb && !b[0]) { b++; nb--; v++; }
938 381724616 : if (na < nb) swapspec(a,b, na,nb);
939 381724616 : if (!nb) return pol0_Flx(0);
940 :
941 353491848 : av = avma;
942 353491848 : if (nb >= get_Fl_threshold(p, Flx_MUL_MULII_LIMIT, Flx_MUL2_MULII_LIMIT))
943 : {
944 7644267 : long m = maxbitcoeffpol(p,nb);
945 7641306 : switch (m)
946 : {
947 96842 : case BITS_IN_QUARTULONG:
948 96842 : return Flx_shiftip(av,Flx_mulspec_quartmulii(a,b,p,na,nb), v);
949 5484 : case BITS_IN_HALFULONG:
950 5484 : return Flx_shiftip(av,Flx_mulspec_halfmulii(a,b,p,na,nb), v);
951 10146 : case BITS_IN_LONG:
952 10146 : return Flx_shiftip(av,Flx_mulspec_mulii(a,b,p,na,nb), v);
953 282047 : case 2*BITS_IN_LONG:
954 282047 : 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 7246787 : default:
958 7246787 : return Flx_shiftip(av,Flx_mulspec_Kronecker(a,b,m,p,na,nb), v);
959 : }
960 : }
961 346220331 : if (nb < get_Fl_threshold(p, Flx_MUL_KARATSUBA_LIMIT, Flx_MUL2_KARATSUBA_LIMIT))
962 344407426 : return Flx_shiftip(av,Flx_mulspec_basecase(a,b,p,pi,na,nb), v);
963 1831147 : i=(na>>1); n0=na-i; na=i;
964 1831147 : a0=a+n0; n0a=n0;
965 2598960 : while (n0a && !a[n0a-1]) n0a--;
966 :
967 1831147 : if (nb > n0)
968 : {
969 : GEN b0,c1,c2;
970 : long n0b;
971 :
972 1777029 : nb -= n0; b0 = b+n0; n0b = n0;
973 2857236 : while (n0b && !b[n0b-1]) n0b--;
974 1777029 : c = Flx_mulspec(a,b,p,pi,n0a,n0b);
975 1777029 : c0 = Flx_mulspec(a0,b0,p,pi,na,nb);
976 :
977 1777029 : c2 = Flx_addspec(a0,a,p,na,n0a);
978 1777029 : c1 = Flx_addspec(b0,b,p,nb,n0b);
979 :
980 1777029 : c1 = Flx_mul_pre(c1,c2,p,pi);
981 1777029 : c2 = Flx_add(c0,c,p);
982 :
983 1777029 : c2 = Flx_neg_inplace(c2,p);
984 1777029 : c2 = Flx_add(c1,c2,p);
985 1777029 : c0 = Flx_addshift(c0,c2 ,p, n0);
986 : }
987 : else
988 : {
989 54118 : c = Flx_mulspec(a,b,p,pi,n0a,nb);
990 54118 : c0 = Flx_mulspec(a0,b,p,pi,na,nb);
991 : }
992 1831147 : c0 = Flx_addshift(c0,c,p,n0);
993 1831147 : return Flx_shiftip(av,c0, v);
994 : }
995 :
996 : GEN
997 376151906 : Flx_mul_pre(GEN x, GEN y, ulong p, ulong pi)
998 : {
999 376151906 : GEN z = Flx_mulspec(x+2,y+2,p, pi, lgpol(x),lgpol(y));
1000 376336997 : z[1] = x[1]; return z;
1001 : }
1002 : GEN
1003 27729962 : Flx_mul(GEN x, GEN y, ulong p)
1004 27729962 : { return Flx_mul_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1005 :
1006 : static GEN
1007 278821700 : 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 278821700 : if (!nx) return pol0_Flx(0);
1014 278821700 : lz = (nx << 1) + 1, nz = lz-2;
1015 278821700 : z = cgetg(lz, t_VECSMALL) + 2;
1016 278245777 : if (!pi)
1017 : {
1018 213776127 : z[0] = x[0]*x[0]%p;
1019 915588252 : for (i=1; i<nx; i++)
1020 : {
1021 702167199 : p1 = Flx_mullimb_ok(x+i,x,p,0, (i+1)>>1);
1022 701812125 : p1 <<= 1;
1023 701812125 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1024 701812125 : z[i] = p1 % p;
1025 : }
1026 919815149 : for ( ; i<nz; i++)
1027 : {
1028 705512813 : p1 = Flx_mullimb_ok(x+i,x,p,i-nx+1, (i+1)>>1);
1029 706394096 : p1 <<= 1;
1030 706394096 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1031 706394096 : z[i] = p1 % p;
1032 : }
1033 : }
1034 : else
1035 : {
1036 64469650 : z[0] = Fl_sqr_pre(x[0], p, pi);
1037 413698410 : for (i=1; i<nx; i++)
1038 : {
1039 349387813 : p1 = Flx_mullimb(x+i,x,p,pi,0, (i+1)>>1);
1040 349521299 : p1 = Fl_add(p1, p1, p);
1041 349128825 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1042 349249064 : z[i] = p1;
1043 : }
1044 413578801 : for ( ; i<nz; i++)
1045 : {
1046 349196763 : p1 = Flx_mullimb(x+i,x,p,pi,i-nx+1, (i+1)>>1);
1047 349956719 : p1 = Fl_add(p1, p1, p);
1048 349616607 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1049 349268204 : z[i] = p1;
1050 : }
1051 : }
1052 278684374 : z -= 2; return Flx_renormalize(z, lz);
1053 : }
1054 :
1055 : static GEN
1056 2069 : Flx_sqrspec_sqri(GEN a, ulong p, long na)
1057 : {
1058 2069 : GEN z=sqrispec(a,na);
1059 2068 : return int_to_Flx(z,p);
1060 : }
1061 :
1062 : static GEN
1063 325 : Flx_sqrspec_halfsqri(GEN a, ulong p, long na)
1064 : {
1065 325 : GEN z = sqri(Flx_to_int_halfspec(a,na));
1066 325 : return int_to_Flx_half(z,p);
1067 : }
1068 :
1069 : static GEN
1070 10197 : Flx_sqrspec_quartsqri(GEN a, ulong p, long na)
1071 : {
1072 10197 : GEN z = sqri(Flx_to_int_quartspec(a,na));
1073 10197 : return int_to_Flx_quart(z,p);
1074 : }
1075 :
1076 : static GEN
1077 11522 : Flx_sqrspec_sqri_inflate(GEN x, long N, ulong p, long nx)
1078 : {
1079 11522 : pari_sp av = avma;
1080 11522 : GEN z = sqri(Flx_eval2BILspec(x,N,nx));
1081 11522 : return gc_upto(av, Z_mod2BIL_Flx(z, N, (nx-1)*2, p));
1082 : }
1083 :
1084 : static GEN
1085 278552768 : Flx_sqrspec(GEN a, ulong p, ulong pi, long na)
1086 : {
1087 : GEN a0, c, c0;
1088 278552768 : long n0, n0a, i, v = 0, m;
1089 : pari_sp av;
1090 :
1091 400441096 : while (na && !a[0]) { a++; na--; v += 2; }
1092 278552768 : if (!na) return pol0_Flx(0);
1093 :
1094 278307908 : av = avma;
1095 278307908 : if (na >= get_Fl_threshold(p, Flx_SQR_SQRI_LIMIT, Flx_SQR2_SQRI_LIMIT))
1096 : {
1097 714597 : m = maxbitcoeffpol(p,na);
1098 714608 : switch(m)
1099 : {
1100 10197 : case BITS_IN_QUARTULONG:
1101 10197 : return Flx_shiftip(av, Flx_sqrspec_quartsqri(a,p,na), v);
1102 325 : case BITS_IN_HALFULONG:
1103 325 : return Flx_shiftip(av, Flx_sqrspec_halfsqri(a,p,na), v);
1104 2069 : case BITS_IN_LONG:
1105 2069 : return Flx_shiftip(av, Flx_sqrspec_sqri(a,p,na), v);
1106 11522 : case 2*BITS_IN_LONG:
1107 11522 : 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 690495 : default:
1111 690495 : return Flx_shiftip(av, Flx_sqrspec_Kronecker(a,m,p,na), v);
1112 : }
1113 : }
1114 278237411 : if (na < get_Fl_threshold(p, Flx_SQR_KARATSUBA_LIMIT, Flx_SQR2_KARATSUBA_LIMIT))
1115 278260497 : return Flx_shiftip(av, Flx_sqrspec_basecase(a,p,pi,na), v);
1116 58331 : i=(na>>1); n0=na-i; na=i;
1117 58331 : a0=a+n0; n0a=n0;
1118 73073 : while (n0a && !a[n0a-1]) n0a--;
1119 :
1120 58331 : c = Flx_sqrspec(a,p,pi,n0a);
1121 58331 : c0= Flx_sqrspec(a0,p,pi,na);
1122 58331 : if (p == 2) n0 *= 2;
1123 : else
1124 : {
1125 58312 : GEN c1, t = Flx_addspec(a0,a,p,na,n0a);
1126 58312 : t = Flx_sqr_pre(t,p,pi);
1127 58312 : c1= Flx_add(c0,c, p);
1128 58312 : c1= Flx_sub(t, c1, p);
1129 58312 : c0 = Flx_addshift(c0,c1,p,n0);
1130 : }
1131 58331 : c0 = Flx_addshift(c0,c,p,n0);
1132 58331 : return Flx_shiftip(av,c0,v);
1133 : }
1134 :
1135 : GEN
1136 278259859 : Flx_sqr_pre(GEN x, ulong p, ulong pi)
1137 : {
1138 278259859 : GEN z = Flx_sqrspec(x+2,p, pi, lgpol(x));
1139 279839154 : z[1] = x[1]; return z;
1140 : }
1141 : GEN
1142 354916 : Flx_sqr(GEN x, ulong p)
1143 354916 : { return Flx_sqr_pre(x, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1144 :
1145 : GEN
1146 3915 : Flx_powu_pre(GEN x, ulong n, ulong p, ulong pi)
1147 : {
1148 3915 : GEN y = pol1_Flx(x[1]), z;
1149 : ulong m;
1150 3915 : if (n == 0) return y;
1151 3915 : m = n; z = x;
1152 : for (;;)
1153 : {
1154 12693 : if (m&1UL) y = Flx_mul_pre(y,z, p, pi);
1155 12695 : m >>= 1; if (!m) return y;
1156 8780 : 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 7243996 : Flx_recipspec(GEN x, long l, long n)
1178 : {
1179 : long i;
1180 7243996 : GEN z=cgetg(n+2,t_VECSMALL)+2;
1181 113345448 : for(i=0; i<l; i++)
1182 106103010 : z[n-i-1] = x[i];
1183 15858859 : for( ; i<n; i++)
1184 8616421 : z[n-i-1] = 0;
1185 7242438 : 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 134277 : Flx_invBarrett_basecase(GEN T, ulong p, ulong pi)
1233 : {
1234 134277 : long i, l=lg(T)-1, lr=l-1, k;
1235 134277 : GEN r=cgetg(lr,t_VECSMALL); r[1] = T[1];
1236 134277 : r[2] = 1;
1237 134277 : 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 2110533 : for (i=3;i<lr;i++)
1247 : {
1248 1983306 : ulong u = Fl_neg(uel(T,l-i+2), p);
1249 59548138 : for (k=3; k<i; k++)
1250 : {
1251 57564833 : ulong t = Fl_neg(uel(T,l-i+k), p);
1252 57564833 : u = Fl_addmul_pre(u, t, uel(r,k), p, pi);
1253 : }
1254 1983305 : r[i] = u;
1255 : }
1256 134276 : return Flx_renormalize(r,lr);
1257 : }
1258 :
1259 : /* Return new lgpol */
1260 : static long
1261 2101993 : Flx_lgrenormalizespec(GEN x, long lx)
1262 : {
1263 : long i;
1264 7450338 : for (i = lx-1; i>=0; i--)
1265 7449560 : if (x[i]) break;
1266 2101993 : return i+1;
1267 : }
1268 : /* allow pi = 0 */
1269 : static GEN
1270 23160 : Flx_invBarrett_Newton(GEN T, ulong p, ulong pi)
1271 : {
1272 23160 : long nold, lx, lz, lq, l = degpol(T), lQ;
1273 23160 : 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 23161 : av = avma;
1280 : /* We work on _spec_ Flx's, all the l[xzq12] below are lgpol's */
1281 :
1282 : /* initialize */
1283 23161 : x[0] = Fl_inv(q[0], p);
1284 23161 : 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 18052 : lx = 1;
1292 23161 : 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 136049 : long i, lnew, nnew = nold << 1;
1296 :
1297 136049 : if (mask & 1) nnew--;
1298 136049 : mask >>= 1;
1299 :
1300 136049 : lnew = nnew + 1;
1301 136049 : lq = Flx_lgrenormalizespec(q, minss(lQ, lnew));
1302 136056 : z = Flx_mulspec(x, q, p, pi, lx, lq); /* FIXME: high product */
1303 136041 : lz = lgpol(z); if (lz > lnew) lz = lnew;
1304 136040 : z += 2;
1305 : /* subtract 1 [=>first nold words are 0]: renormalize so that z(0) != 0 */
1306 298863 : for (i = nold; i < lz; i++) if (z[i]) break;
1307 136040 : nold = nnew;
1308 136040 : if (i >= lz) continue; /* z-1 = 0(t^(nnew + 1)) */
1309 :
1310 : /* z + i represents (x*q - 1) / t^i */
1311 100989 : lz = Flx_lgrenormalizespec (z+i, lz-i);
1312 100991 : z = Flx_mulspec(x, z+i, p, pi, lx, lz); /* FIXME: low product */
1313 100991 : lz = lgpol(z); z += 2;
1314 100991 : if (lz > lnew-i) lz = Flx_lgrenormalizespec(z, lnew-i);
1315 :
1316 100990 : lx = lz+ i;
1317 100990 : y = x + i; /* x -= z * t^i, in place */
1318 999771 : for (i = 0; i < lz; i++) y[i] = Fl_neg(z[i], p);
1319 : }
1320 23160 : x -= 2; setlg(x, lx + 2); x[1] = T[1];
1321 23160 : return x;
1322 : }
1323 :
1324 : /* allow pi = 0 */
1325 : static GEN
1326 158713 : Flx_invBarrett_pre(GEN T, ulong p, ulong pi)
1327 : {
1328 158713 : pari_sp ltop = avma;
1329 158713 : long l = lgpol(T);
1330 : GEN r;
1331 158713 : if (l < 3) return pol0_Flx(T[1]);
1332 157437 : if (l < get_Fl_threshold(p, Flx_INVBARRETT_LIMIT, Flx_INVBARRETT2_LIMIT))
1333 : {
1334 134277 : ulong c = T[l+1];
1335 134277 : 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 98118 : r = Flx_Fl_mul_pre(r, ci, p, pi);
1341 : }
1342 : else
1343 36159 : r = Flx_invBarrett_basecase(T, p, pi);
1344 : }
1345 : else
1346 23160 : r = Flx_invBarrett_Newton(T, p, pi);
1347 157437 : 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 97006009 : Flx_get_red_pre(GEN T, ulong p, ulong pi)
1356 : {
1357 97006009 : if (typ(T)!=t_VECSMALL
1358 96969605 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1359 : Flx_BARRETT2_LIMIT))
1360 96990318 : return T;
1361 7626 : retmkvec2(Flx_invBarrett_pre(T, p, pi),T);
1362 : }
1363 : GEN
1364 14443274 : Flx_get_red(GEN T, ulong p)
1365 : {
1366 14443274 : if (typ(T)!=t_VECSMALL
1367 14443179 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1368 : Flx_BARRETT2_LIMIT))
1369 14437462 : 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 799810212 : 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 799810212 : long vs=x[1];
1382 :
1383 799810212 : dy = degpol(y); if (!dy) return pol0_Flx(x[1]);
1384 763467877 : dx = degpol(x);
1385 763264031 : dz = dx-dy; if (dz < 0) return Flx_copy(x);
1386 763264031 : x += 2; y += 2;
1387 763264031 : inv = y[dy];
1388 763264031 : if (inv != 1UL) inv = Fl_inv(inv,p);
1389 916074047 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1390 :
1391 765136746 : c = cgetg(dy+3, t_VECSMALL); c[1]=vs; c += 2; av=avma;
1392 763530903 : z = cgetg(dz+3, t_VECSMALL); z[1]=vs; z += 2;
1393 :
1394 761576930 : if (!pi)
1395 : {
1396 479835160 : z[dz] = (inv*x[dx]) % p;
1397 1802145527 : for (i=dx-1; i>=dy; --i)
1398 : {
1399 1322310367 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1400 10448866680 : for (j=i-dy1; j<=i && j<=dz; j++)
1401 : {
1402 9126556313 : p1 += z[j]*y[i-j];
1403 9126556313 : if (p1 & HIGHBIT) p1 %= p;
1404 : }
1405 1322310367 : p1 %= p;
1406 1322310367 : z[i-dy] = p1? ((p - p1)*inv) % p: 0;
1407 : }
1408 3277253962 : for (i=0; i<dy; i++)
1409 : {
1410 2798742502 : p1 = z[0]*y[i];
1411 14414377826 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1412 : {
1413 11615635324 : p1 += z[j]*y[i-j];
1414 11615635324 : if (p1 & HIGHBIT) p1 %= p;
1415 : }
1416 2798890393 : c[i] = Fl_sub(x[i], p1%p, p);
1417 : }
1418 : }
1419 : else
1420 : {
1421 281741770 : z[dz] = Fl_mul_pre(inv, x[dx], p, pi);
1422 862002951 : for (i=dx-1; i>=dy; --i)
1423 : {
1424 578475707 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1425 2402974158 : for (j=i-dy1; j<=i && j<=dz; j++)
1426 1822019217 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1427 580954941 : z[i-dy] = p1? Fl_mul_pre(p - p1, inv, p, pi): 0;
1428 : }
1429 2041606999 : for (i=0; i<dy; i++)
1430 : {
1431 1758302900 : p1 = Fl_mul_pre(z[0],y[i],p,pi);
1432 4735880796 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1433 2966428332 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1434 1744416960 : c[i] = Fl_sub(x[i], p1, p);
1435 : }
1436 : }
1437 930832409 : i = dy-1; while (i>=0 && !c[i]) i--;
1438 761815559 : 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 62086754 : 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 62086754 : long sv=x[1];
1450 :
1451 62086754 : dy = degpol(y);
1452 62084812 : if (dy<0) pari_err_INV("Flx_divrem",y);
1453 62085009 : if (pr == ONLY_REM) return Flx_rem_basecase(x, y, p, pi);
1454 62084611 : if (!dy)
1455 : {
1456 7142814 : if (pr && pr != ONLY_DIVIDES) *pr = pol0_Flx(sv);
1457 7142771 : if (y[2] == 1UL) return Flx_copy(x);
1458 5129975 : return Flx_Fl_mul_pre(x, Fl_inv(y[2], p), p, pi);
1459 : }
1460 54941797 : dx = degpol(x);
1461 54944688 : dz = dx-dy;
1462 54944688 : if (dz < 0)
1463 : {
1464 1054257 : q = pol0_Flx(sv);
1465 1054255 : if (pr && pr != ONLY_DIVIDES) *pr = Flx_copy(x);
1466 1054255 : return q;
1467 : }
1468 53890431 : x += 2;
1469 53890431 : y += 2;
1470 53890431 : z = cgetg(dz + 3, t_VECSMALL); z[1] = sv; z += 2;
1471 53888633 : inv = uel(y, dy);
1472 53888633 : if (inv != 1UL) inv = Fl_inv(inv,p);
1473 79192530 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1474 :
1475 53892095 : if (SMALL_ULONG(p))
1476 : {
1477 52011006 : z[dz] = (inv*x[dx]) % p;
1478 131943093 : for (i=dx-1; i>=dy; --i)
1479 : {
1480 79932087 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1481 258444771 : for (j=i-dy1; j<=i && j<=dz; j++)
1482 : {
1483 178512684 : p1 += z[j]*y[i-j];
1484 178512684 : if (p1 & HIGHBIT) p1 %= p;
1485 : }
1486 79932087 : p1 %= p;
1487 79932087 : z[i-dy] = p1? (long) ((p - p1)*inv) % p: 0;
1488 : }
1489 : }
1490 : else
1491 : {
1492 1881089 : z[dz] = Fl_mul(inv, x[dx], p);
1493 9254659 : for (i=dx-1; i>=dy; --i)
1494 : { /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1495 7373246 : p1 = p - uel(x,i);
1496 26383310 : for (j=i-dy1; j<=i && j<=dz; j++)
1497 19010068 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1498 7373242 : z[i-dy] = p1? Fl_mul(p - p1, inv, p): 0;
1499 : }
1500 : }
1501 53892419 : q = Flx_renormalize(z-2, dz+3);
1502 53891250 : if (!pr) return q;
1503 :
1504 26496893 : c = cgetg(dy + 3, t_VECSMALL); c[1] = sv; c += 2;
1505 26498999 : if (SMALL_ULONG(p))
1506 : {
1507 228283135 : for (i=0; i<dy; i++)
1508 : {
1509 203426888 : p1 = (ulong)z[0]*y[i];
1510 477745303 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1511 : {
1512 274318415 : p1 += (ulong)z[j]*y[i-j];
1513 274318415 : if (p1 & HIGHBIT) p1 %= p;
1514 : }
1515 203426647 : c[i] = Fl_sub(x[i], p1%p, p);
1516 : }
1517 : }
1518 : else
1519 : {
1520 15958161 : for (i=0; i<dy; i++)
1521 : {
1522 14316257 : p1 = Fl_mul(z[0],y[i],p);
1523 50063595 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1524 35747337 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1525 14316257 : c[i] = Fl_sub(x[i], p1, p);
1526 : }
1527 : }
1528 35707784 : i=dy-1; while (i>=0 && !c[i]) i--;
1529 26498151 : c = Flx_renormalize(c-2, i+3);
1530 26498766 : if (pr == ONLY_DIVIDES)
1531 462 : { if (lg(c) != 2) return NULL; }
1532 : else
1533 26498304 : *pr = c;
1534 26498619 : 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 889783 : Flx_divrem_Barrettspec(GEN x, long l, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1541 : {
1542 : GEN q, r;
1543 889783 : long lt = degpol(T); /*We discard the leading term*/
1544 : long ld, lm, lT, lmg;
1545 889736 : ld = l-lt;
1546 889736 : lm = minss(ld, lgpol(mg));
1547 890026 : lT = Flx_lgrenormalizespec(T+2,lt);
1548 890176 : lmg = Flx_lgrenormalizespec(mg+2,lm);
1549 890095 : q = Flx_recipspec(x+lt,ld,ld); /* q = rec(x) lz<=ld*/
1550 889230 : q = Flx_mulspec(q+2,mg+2,p,pi,lgpol(q),lmg); /* q = rec(x) * mg lz<=ld+lm*/
1551 890069 : q = Flx_recipspec(q+2,minss(ld,lgpol(q)),ld);/* q = rec (rec(x) * mg) lz<=ld*/
1552 889083 : if (!pr) return q;
1553 881389 : r = Flx_mulspec(q+2,T+2,p,pi,lgpol(q),lT); /* r = q*pol lz<=ld+lt*/
1554 882288 : r = Flx_subspec(x,r+2,p,lt,minss(lt,lgpol(r)));/* r = x - q*pol lz<=lt */
1555 882184 : if (pr == ONLY_REM) return r;
1556 428106 : *pr = r; return q;
1557 : }
1558 :
1559 : static GEN
1560 589174 : Flx_divrem_Barrett(GEN x, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1561 : {
1562 589174 : GEN q = NULL, r = Flx_copy(x);
1563 589197 : long l = lgpol(x), lt = degpol(T), lm = 2*lt-1, v = T[1];
1564 : long i;
1565 589197 : 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 589197 : if (lt <= 1)
1573 1276 : return Flx_divrem_basecase(x,T,p,pi,pr);
1574 587921 : if (pr != ONLY_REM && l>lm)
1575 28910 : { q = zero_zv(l-lt+1); q[1] = T[1]; }
1576 891403 : while (l>lm)
1577 : {
1578 303537 : GEN zr, zq = Flx_divrem_Barrettspec(r+2+l-lm,lm,mg,T,p,pi,&zr);
1579 303549 : long lz = lgpol(zr);
1580 303482 : if (pr != ONLY_REM)
1581 : {
1582 57950 : long lq = lgpol(zq);
1583 883186 : for(i=0; i<lq; i++) q[2+l-lm+i] = zq[2+i];
1584 : }
1585 4412043 : for(i=0; i<lz; i++) r[2+l-lm+i] = zr[2+i];
1586 303482 : l = l-lm+lz;
1587 : }
1588 587866 : if (pr == ONLY_REM)
1589 : {
1590 454134 : if (l > lt)
1591 454092 : r = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi,ONLY_REM);
1592 : else
1593 42 : r = Flx_renormalize(r, l+2);
1594 454119 : r[1] = v; return r;
1595 : }
1596 133732 : if (l > lt)
1597 : {
1598 132206 : GEN zq = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi, pr ? &r: NULL);
1599 132206 : if (!q) q = zq;
1600 : else
1601 : {
1602 27330 : long lq = lgpol(zq);
1603 159759 : for(i=0; i<lq; i++) q[2+i] = zq[2+i];
1604 : }
1605 : }
1606 1526 : else if (pr)
1607 1541 : r = Flx_renormalize(r, l+2);
1608 133732 : q[1] = v; q = Flx_renormalize(q, lg(q));
1609 133786 : if (pr == ONLY_DIVIDES) return lgpol(r)? NULL: q;
1610 133786 : if (pr) { r[1] = v; *pr = r; }
1611 133786 : return q;
1612 : }
1613 :
1614 : /* allow pi = 0 (SMALL_ULONG) */
1615 : GEN
1616 79442965 : Flx_divrem_pre(GEN x, GEN T, ulong p, ulong pi, GEN *pr)
1617 : {
1618 : GEN B, y;
1619 : long dy, dx, d;
1620 79442965 : if (pr==ONLY_REM) return Flx_rem_pre(x, T, p, pi);
1621 62211318 : y = get_Flx_red(T, &B);
1622 62223159 : dy = degpol(y); dx = degpol(x); d = dx-dy;
1623 62218932 : if (!B && d+3 < get_Fl_threshold(p, Flx_DIVREM_BARRETT_LIMIT,Flx_DIVREM2_BARRETT_LIMIT))
1624 62084490 : return Flx_divrem_basecase(x,y,p,pi,pr);
1625 : else
1626 : {
1627 134664 : pari_sp av = avma;
1628 134664 : GEN mg = B? B: Flx_invBarrett_pre(y, p, pi);
1629 134664 : GEN q1 = Flx_divrem_Barrett(x,mg,y,p,pi,pr);
1630 134664 : if (!q1) return gc_NULL(av);
1631 134664 : if (!pr || pr==ONLY_DIVIDES) return gc_leaf(av, q1);
1632 126400 : return gc_all(av, 2, &q1, pr);
1633 : }
1634 : }
1635 : GEN
1636 30324120 : Flx_divrem(GEN x, GEN T, ulong p, GEN *pr)
1637 30324120 : { return Flx_divrem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p), pr); }
1638 :
1639 : GEN
1640 923012026 : Flx_rem_pre(GEN x, GEN T, ulong p, ulong pi)
1641 : {
1642 923012026 : GEN B, y = get_Flx_red(T, &B);
1643 922839152 : long d = degpol(x) - degpol(y);
1644 922584298 : if (d < 0) return Flx_copy(x);
1645 800185449 : if (!B && d+3 < get_Fl_threshold(p, Flx_REM_BARRETT_LIMIT,Flx_REM2_BARRETT_LIMIT))
1646 799582541 : return Flx_rem_basecase(x,y,p, pi);
1647 : else
1648 : {
1649 454511 : pari_sp av=avma;
1650 454511 : GEN mg = B ? B: Flx_invBarrett_pre(y, p, pi);
1651 454511 : GEN r = Flx_divrem_Barrett(x, mg, y, p, pi, ONLY_REM);
1652 454518 : return gc_leaf(av, r);
1653 : }
1654 : }
1655 : GEN
1656 42095028 : Flx_rem(GEN x, GEN T, ulong p)
1657 42095028 : { 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 5057363 : Flx_mod_Xnm1(GEN T, ulong n, ulong p)
1662 : {
1663 5057363 : long i, j, L = lg(T), l = n+2;
1664 : GEN S;
1665 5057363 : if (L <= l || n & ~LGBITS) return T;
1666 3534 : S = cgetg(l, t_VECSMALL);
1667 3534 : S[1] = T[1];
1668 15112 : for (i = 2; i < l; i++) S[i] = T[i];
1669 9770 : for (j = 2; i < L; i++) {
1670 6236 : S[j] = Fl_add(S[j], T[i], p);
1671 6236 : if (++j == l) j = 2;
1672 : }
1673 3534 : return Flx_renormalize(S, l);
1674 : }
1675 : /* reduce T mod (X^n + 1, p). Shallow function */
1676 : GEN
1677 31654 : Flx_mod_Xn1(GEN T, ulong n, ulong p)
1678 : {
1679 31654 : long i, j, L = lg(T), l = n+2, s = -1;
1680 : GEN S;
1681 31654 : if (L <= l || n & ~LGBITS) return T;
1682 2745 : S = cgetg(l, t_VECSMALL);
1683 2745 : S[1] = T[1];
1684 12047 : for (i = 2; i < l; i++) S[i] = T[i];
1685 7282 : for (j = 2; i < L; i++) {
1686 4537 : S[j] = s==-1 ? Fl_sub(S[j], T[i], p): Fl_add(S[j], T[i], p);
1687 4537 : if (++j == l) { j = 2; s = -s; }
1688 : }
1689 2745 : 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 69384069 : set_Flxq_pre(struct _Flxq *D, GEN T, ulong p, ulong pi)
1699 : {
1700 69384069 : D->p = p;
1701 69384069 : D->pi = pi;
1702 69384069 : D->T = Flx_get_red_pre(T, p, pi);
1703 69380315 : }
1704 : static void
1705 66764 : set_Flxq(struct _Flxq *D, GEN T, ulong p)
1706 66764 : { set_Flxq_pre(D, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1707 :
1708 : struct _Flx {
1709 : long v;
1710 : ulong p, pi;
1711 : };
1712 :
1713 : static GEN
1714 0 : _Flx_divrem(void * E, GEN x, GEN y, GEN *r)
1715 : {
1716 0 : struct _Flx *D = (struct _Flx*) E;
1717 0 : return Flx_divrem_pre(x, y, D->p, D->pi, r);
1718 : }
1719 : static GEN
1720 38600 : _Flx_add(void * E, GEN x, GEN y) {
1721 38600 : struct _Flx *D = (struct _Flx*) E;
1722 38600 : return Flx_add(x, y, D->p);
1723 : }
1724 : static GEN
1725 11010673 : _Flx_mul(void *E, GEN x, GEN y) {
1726 11010673 : struct _Flx *D = (struct _Flx*) E;
1727 11010673 : return Flx_mul_pre(x, y, D->p, D->pi);
1728 : }
1729 : static GEN
1730 0 : _Flx_sqr(void *E, GEN x) {
1731 0 : struct _Flx *D = (struct _Flx*) E;
1732 0 : return Flx_sqr_pre(x, D->p, D->pi);
1733 : }
1734 :
1735 : static struct bb_ring Flx_ring = { _Flx_add,_Flx_mul,_Flx_sqr };
1736 :
1737 : GEN
1738 0 : Flx_digits(GEN x, GEN T, ulong p)
1739 : {
1740 : struct _Flx D;
1741 0 : long d = degpol(T), n = (lgpol(x)+d-1)/d;
1742 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1743 0 : return gen_digits(x,T,n,(void *)&D, &Flx_ring, _Flx_divrem);
1744 : }
1745 :
1746 : GEN
1747 0 : FlxV_Flx_fromdigits(GEN x, GEN T, ulong p)
1748 : {
1749 : struct _Flx D;
1750 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1751 0 : return gen_fromdigits(x,T,(void *)&D, &Flx_ring);
1752 : }
1753 :
1754 : long
1755 4493533 : Flx_val(GEN x)
1756 : {
1757 4493533 : long i, l=lg(x);
1758 4493533 : if (l==2) return LONG_MAX;
1759 4502627 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1760 4493533 : return i-2;
1761 : }
1762 : long
1763 27449196 : Flx_valrem(GEN x, GEN *Z)
1764 : {
1765 27449196 : long v, i, l=lg(x);
1766 : GEN y;
1767 27449196 : if (l==2) { *Z = Flx_copy(x); return LONG_MAX; }
1768 29625428 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1769 27449196 : v = i-2;
1770 27449196 : if (v == 0) { *Z = x; return 0; }
1771 1021121 : l -= v;
1772 1021121 : y = cgetg(l, t_VECSMALL); y[1] = x[1];
1773 2626573 : for (i=2; i<l; i++) y[i] = x[i+v];
1774 1023423 : *Z = y; return v;
1775 : }
1776 :
1777 : GEN
1778 22942262 : Flx_deriv(GEN z, ulong p)
1779 : {
1780 22942262 : long i,l = lg(z)-1;
1781 : GEN x;
1782 22942262 : if (l < 2) l = 2;
1783 22942262 : x = cgetg(l, t_VECSMALL); x[1] = z[1]; z++;
1784 22940830 : if (HIGHWORD(l | p))
1785 62866188 : for (i=2; i<l; i++) x[i] = Fl_mul((ulong)i-1, z[i], p);
1786 : else
1787 87690361 : for (i=2; i<l; i++) x[i] = ((i-1) * z[i]) % p;
1788 22940474 : return Flx_renormalize(x,l);
1789 : }
1790 :
1791 : static GEN
1792 422795 : Flx_integXn(GEN x, long n, ulong p)
1793 : {
1794 422795 : long i, lx = lg(x);
1795 : GEN y;
1796 422795 : if (lx == 2) return Flx_copy(x);
1797 412986 : y = cgetg(lx, t_VECSMALL); y[1] = x[1];
1798 2096870 : for (i=2; i<lx; i++)
1799 : {
1800 1683547 : ulong xi = uel(x,i);
1801 1683547 : if (xi == 0)
1802 13345 : uel(y,i) = 0;
1803 : else
1804 : {
1805 1670202 : ulong j = n+i-1;
1806 1670202 : ulong d = ugcd(j, xi);
1807 1670127 : if (d==1)
1808 1018415 : uel(y,i) = Fl_div(xi, j, p);
1809 : else
1810 651712 : uel(y,i) = Fl_div(xi/d, j/d, p);
1811 : }
1812 : }
1813 413323 : return Flx_renormalize(y, lx);;
1814 : }
1815 :
1816 : GEN
1817 0 : Flx_integ(GEN x, ulong p)
1818 : {
1819 0 : long i, lx = lg(x);
1820 : GEN y;
1821 0 : if (lx == 2) return Flx_copy(x);
1822 0 : y = cgetg(lx+1, t_VECSMALL); y[1] = x[1];
1823 0 : uel(y,2) = 0;
1824 0 : for (i=3; i<=lx; i++)
1825 0 : uel(y,i) = uel(x,i-1) ? Fl_div(uel(x,i-1), (i-2)%p, p): 0UL;
1826 0 : return Flx_renormalize(y, lx+1);;
1827 : }
1828 :
1829 : /* assume p prime */
1830 : GEN
1831 14448 : Flx_diff1(GEN P, ulong p)
1832 : {
1833 14448 : return Flx_sub(Flx_translate1(P, p), P, p);
1834 : }
1835 :
1836 : GEN
1837 419602 : Flx_deflate(GEN x0, long d)
1838 : {
1839 : GEN z, y, x;
1840 419602 : long i,id, dy, dx = degpol(x0);
1841 419602 : if (d == 1 || dx <= 0) return Flx_copy(x0);
1842 356130 : dy = dx/d;
1843 356130 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1844 356128 : z = y + 2;
1845 356128 : x = x0+ 2;
1846 1155436 : for (i=id=0; i<=dy; i++,id+=d) z[i] = x[id];
1847 356128 : return y;
1848 : }
1849 :
1850 : GEN
1851 160265 : Flx_inflate(GEN x0, long d)
1852 : {
1853 160265 : long i, id, dy, dx = degpol(x0);
1854 160262 : GEN x = x0 + 2, z, y;
1855 160262 : if (dx <= 0) return Flx_copy(x0);
1856 159205 : dy = dx*d;
1857 159205 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1858 159206 : z = y + 2;
1859 8807429 : for (i=0; i<=dy; i++) z[i] = 0;
1860 4284336 : for (i=id=0; i<=dx; i++,id+=d) z[id] = x[i];
1861 159206 : return y;
1862 : }
1863 :
1864 : /* write p(X) = a_0(X^k) + X*a_1(X^k) + ... + X^(k-1)*a_{k-1}(X^k) */
1865 : GEN
1866 147326 : Flx_splitting(GEN p, long k)
1867 : {
1868 147326 : long n = degpol(p), v = p[1], m, i, j, l;
1869 : GEN r;
1870 :
1871 147325 : m = n/k;
1872 147325 : r = cgetg(k+1,t_VEC);
1873 679280 : for(i=1; i<=k; i++)
1874 : {
1875 531961 : gel(r,i) = cgetg(m+3, t_VECSMALL);
1876 531955 : mael(r,i,1) = v;
1877 : }
1878 4458438 : for (j=1, i=0, l=2; i<=n; i++)
1879 : {
1880 4311119 : mael(r,j,l) = p[2+i];
1881 4311119 : if (j==k) { j=1; l++; } else j++;
1882 : }
1883 679280 : for(i=1; i<=k; i++)
1884 531970 : gel(r,i) = Flx_renormalize(gel(r,i),i<j?l+1:l);
1885 147310 : return r;
1886 : }
1887 :
1888 : /* ux + vy */
1889 : static GEN
1890 415784 : Flx_addmulmul(GEN u, GEN v, GEN x, GEN y, ulong p, ulong pi)
1891 415784 : { return Flx_add(Flx_mul_pre(u,x, p,pi), Flx_mul_pre(v,y, p,pi), p); }
1892 :
1893 : static GEN
1894 26000 : FlxM_Flx_mul2(GEN M, GEN x, GEN y, ulong p, ulong pi)
1895 : {
1896 26000 : GEN res = cgetg(3, t_COL);
1897 25999 : gel(res, 1) = Flx_addmulmul(gcoeff(M,1,1), gcoeff(M,1,2), x, y, p, pi);
1898 25999 : gel(res, 2) = Flx_addmulmul(gcoeff(M,2,1), gcoeff(M,2,2), x, y, p, pi);
1899 26000 : return res;
1900 : }
1901 :
1902 : #if 0
1903 : static GEN
1904 : FlxM_mul2_old(GEN M, GEN N, ulong p)
1905 : {
1906 : GEN res = cgetg(3, t_MAT);
1907 : gel(res, 1) = FlxM_Flx_mul2(M,gcoeff(N,1,1),gcoeff(N,2,1),p);
1908 : gel(res, 2) = FlxM_Flx_mul2(M,gcoeff(N,1,2),gcoeff(N,2,2),p);
1909 : return res;
1910 : }
1911 : #endif
1912 : /* A,B are 2x2 matrices, Flx entries. Return A x B using Strassen 7M formula */
1913 : static GEN
1914 7099 : FlxM_mul2(GEN A, GEN B, ulong p, ulong pi)
1915 : {
1916 7099 : GEN A11=gcoeff(A,1,1),A12=gcoeff(A,1,2), B11=gcoeff(B,1,1),B12=gcoeff(B,1,2);
1917 7099 : GEN A21=gcoeff(A,2,1),A22=gcoeff(A,2,2), B21=gcoeff(B,2,1),B22=gcoeff(B,2,2);
1918 7099 : GEN M1 = Flx_mul_pre(Flx_add(A11,A22, p), Flx_add(B11,B22, p), p, pi);
1919 7099 : GEN M2 = Flx_mul_pre(Flx_add(A21,A22, p), B11, p, pi);
1920 7099 : GEN M3 = Flx_mul_pre(A11, Flx_sub(B12,B22, p), p, pi);
1921 7099 : GEN M4 = Flx_mul_pre(A22, Flx_sub(B21,B11, p), p, pi);
1922 7099 : GEN M5 = Flx_mul_pre(Flx_add(A11,A12, p), B22, p, pi);
1923 7099 : GEN M6 = Flx_mul_pre(Flx_sub(A21,A11, p), Flx_add(B11,B12, p), p, pi);
1924 7099 : GEN M7 = Flx_mul_pre(Flx_sub(A12,A22, p), Flx_add(B21,B22, p), p, pi);
1925 7099 : GEN T1 = Flx_add(M1,M4, p), T2 = Flx_sub(M7,M5, p);
1926 7099 : GEN T3 = Flx_sub(M1,M2, p), T4 = Flx_add(M3,M6, p);
1927 7099 : retmkmat22(Flx_add(T1,T2, p), Flx_add(M3,M5, p),
1928 : Flx_add(M2,M4, p), Flx_add(T3,T4, p));
1929 : }
1930 :
1931 : /* Return [0,1;1,-q]*M */
1932 : static GEN
1933 6927 : Flx_FlxM_qmul(GEN q, GEN M, ulong p, ulong pi)
1934 : {
1935 6927 : GEN u = Flx_mul_pre(gcoeff(M,2,1), q, p, pi);
1936 6927 : GEN v = Flx_mul_pre(gcoeff(M,2,2), q, p, pi);
1937 6927 : retmkmat22(gcoeff(M,2,1), gcoeff(M,2,2),
1938 : Flx_sub(gcoeff(M,1,1), u, p), Flx_sub(gcoeff(M,1,2), v, p));
1939 : }
1940 :
1941 : static GEN
1942 911 : matid2_FlxM(long v)
1943 911 : { retmkmat22(pol1_Flx(v),pol0_Flx(v),pol0_Flx(v),pol1_Flx(v)); }
1944 :
1945 : static GEN
1946 13 : matJ2_FlxM(long v)
1947 13 : { retmkmat22(pol0_Flx(v),pol1_Flx(v),pol1_Flx(v),pol0_Flx(v)); }
1948 :
1949 : struct Flx_res
1950 : {
1951 : ulong res, lc;
1952 : long deg0, deg1, off;
1953 : };
1954 :
1955 : INLINE void
1956 9405 : Flx_halfres_update_pre(long da, long db, long dr, ulong p, ulong pi, struct Flx_res *res)
1957 : {
1958 9405 : if (dr >= 0)
1959 : {
1960 9405 : if (res->lc != 1)
1961 : {
1962 7596 : if (pi)
1963 : {
1964 3127 : res->lc = Fl_powu_pre(res->lc, da - dr, p, pi);
1965 3127 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1966 : } else
1967 : {
1968 4469 : res->lc = Fl_powu(res->lc, da - dr, p);
1969 4469 : res->res = Fl_mul(res->res, res->lc, p);
1970 : }
1971 : }
1972 9405 : if (both_odd(da + res->off, db + res->off))
1973 63 : res->res = Fl_neg(res->res, p);
1974 : } else
1975 : {
1976 0 : if (db == 0)
1977 : {
1978 0 : if (res->lc != 1)
1979 : {
1980 0 : if (pi)
1981 : {
1982 0 : res->lc = Fl_powu_pre(res->lc, da, p, pi);
1983 0 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1984 : } else
1985 : {
1986 0 : res->lc = Fl_powu(res->lc, da, p);
1987 0 : res->res = Fl_mul(res->res, res->lc, p);
1988 : }
1989 : }
1990 : } else
1991 0 : res->res = 0;
1992 : }
1993 9405 : }
1994 :
1995 : static GEN
1996 1131486 : Flx_halfres_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *pa, GEN *pb, struct Flx_res *res)
1997 : {
1998 1131486 : pari_sp av = avma;
1999 : GEN u, u1, v, v1, M;
2000 1131486 : long vx = a[1], n = lgpol(a)>>1;
2001 1131499 : u1 = v = pol0_Flx(vx);
2002 1131497 : u = v1 = pol1_Flx(vx);
2003 6912090 : while (lgpol(b)>n)
2004 : {
2005 : GEN r, q;
2006 5780612 : q = Flx_divrem_pre(a,b,p,pi, &r);
2007 5780609 : if (res)
2008 : {
2009 8362 : long da = degpol(a), db=degpol(b), dr = degpol(r);
2010 8362 : res->lc = b[db+2];
2011 8362 : if (dr >= n)
2012 7133 : Flx_halfres_update_pre(da, db, dr, p, pi, res);
2013 : else
2014 : {
2015 1229 : res->deg0 = da;
2016 1229 : res->deg1 = db;
2017 : }
2018 : }
2019 5780609 : a = b; b = r; swap(u,u1); swap(v,v1);
2020 5780609 : u1 = Flx_sub(u1, Flx_mul(u, q, p), p);
2021 5780544 : v1 = Flx_sub(v1, Flx_mul(v, q, p), p);
2022 5780589 : if (gc_needed(av,2))
2023 : {
2024 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_halfgcd (d = %ld)",degpol(b));
2025 0 : (void)gc_all(av,6, &a,&b,&u1,&v1,&u,&v);
2026 : }
2027 : }
2028 1131339 : M = mkmat22(u,v,u1,v1); *pa = a; *pb = b;
2029 1131472 : return gc_all(av,3, &M, pa, pb);
2030 : }
2031 :
2032 : static GEN Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res);
2033 :
2034 : static GEN
2035 19964 : Flx_halfres_split(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2036 : {
2037 19964 : pari_sp av = avma;
2038 : GEN R, S, T, V1, V2;
2039 : GEN x1, y1, r, q;
2040 19964 : long l = lgpol(x), n = l>>1, k;
2041 19964 : if (lgpol(y) <= n)
2042 871 : { *a = Flx_copy(x); *b = Flx_copy(y); return matid2_FlxM(x[1]); }
2043 19093 : if (res)
2044 : {
2045 3263 : res->lc = Flx_lead(y);
2046 3263 : res->deg0 -= n;
2047 3263 : res->deg1 -= n;
2048 3263 : res->off += n;
2049 : }
2050 19093 : R = Flx_halfres_i(Flx_shift(x,-n),Flx_shift(y,-n),p,pi,a,b,res);
2051 19092 : if (res)
2052 : {
2053 3262 : res->off -= n;
2054 3262 : res->deg0 += n;
2055 3262 : res->deg1 += n;
2056 : }
2057 19092 : V1 = FlxM_Flx_mul2(R, Flxn_red(x,n), Flxn_red(y,n), p, pi);
2058 19093 : x1 = Flx_add(Flx_shift(*a,n), gel(V1,1), p);
2059 19093 : y1 = Flx_add(Flx_shift(*b,n), gel(V1,2), p);
2060 19092 : if (lgpol(y1) <= n)
2061 12185 : { *a = x1; *b = y1; return gc_all(av, 3, &R, a, b); }
2062 6907 : k = 2*n-degpol(y1);
2063 6907 : q = Flx_divrem_pre(x1, y1, p, pi, &r);
2064 6907 : if (res)
2065 : {
2066 1043 : long dx1 = degpol(x1), dy1 = degpol(y1), dr = degpol(r);
2067 1043 : if (dy1 < degpol(y))
2068 185 : Flx_halfres_update_pre(res->deg0, res->deg1, dy1, p, pi, res);
2069 1043 : res->lc = uel(y1, dy1+2);
2070 1043 : res->deg0 = dx1;
2071 1043 : res->deg1 = dy1;
2072 1043 : if (dr >= n)
2073 : {
2074 1043 : Flx_halfres_update_pre(dx1, dy1, dr, p, pi, res);
2075 1043 : res->deg0 = dy1;
2076 1043 : res->deg1 = dr;
2077 : }
2078 1043 : res->deg0 -= k;
2079 1043 : res->deg1 -= k;
2080 1043 : res->off += k;
2081 : }
2082 6907 : S = Flx_halfres_i(Flx_shift(y1,-k), Flx_shift(r,-k), p, pi, a, b, res);
2083 6907 : if (res)
2084 : {
2085 1043 : res->deg0 += k;
2086 1043 : res->deg1 += k;
2087 1043 : res->off -= k;
2088 : }
2089 6907 : T = FlxM_mul2(S, Flx_FlxM_qmul(q, R, p,pi), p, pi);
2090 6907 : V2 = FlxM_Flx_mul2(S, Flxn_red(y1,k), Flxn_red(r,k), p, pi);
2091 6907 : *a = Flx_add(Flx_shift(*a,k), gel(V2,1), p);
2092 6907 : *b = Flx_add(Flx_shift(*b,k), gel(V2,2), p);
2093 6907 : return gc_all(av, 3, &T, a, b);
2094 : }
2095 :
2096 : static GEN
2097 1151466 : Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2098 : {
2099 1151466 : if (lgpol(x) < get_Fl_threshold(p, Flx_HALFGCD_LIMIT, Flx_HALFGCD2_LIMIT))
2100 1131499 : return Flx_halfres_basecase(x, y, p, pi, a, b, res);
2101 19964 : return Flx_halfres_split(x, y, p, pi, a, b, res);
2102 : }
2103 :
2104 : static GEN
2105 1124423 : Flx_halfgcd_all_i(GEN x, GEN y, ulong p, ulong pi, GEN *pa, GEN *pb)
2106 : {
2107 : GEN a, b, R;
2108 1124423 : R = Flx_halfres_i(x, y, p, pi, &a, &b, NULL);
2109 1124428 : if (pa) *pa = a;
2110 1124428 : if (pb) *pb = b;
2111 1124428 : return R;
2112 : }
2113 :
2114 : /* Return M in GL_2(Fl[X]) such that:
2115 : if [a',b']~=M*[a,b]~ then degpol(a')>= (lgpol(a)>>1) >degpol(b')
2116 : */
2117 :
2118 : GEN
2119 1124426 : Flx_halfgcd_all_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b)
2120 : {
2121 : pari_sp av;
2122 : GEN R, q ,r;
2123 1124426 : long lx = lgpol(x), ly = lgpol(y);
2124 1124422 : if (!lx)
2125 : {
2126 0 : if (a) *a = Flx_copy(y);
2127 0 : if (b) *b = Flx_copy(x);
2128 0 : return matJ2_FlxM(x[1]);
2129 : }
2130 1124422 : if (ly < lx) return Flx_halfgcd_all_i(x, y, p, pi, a, b);
2131 8030 : av = avma;
2132 8030 : q = Flx_divrem(y,x,p,&r);
2133 8030 : R = Flx_halfgcd_all_i(x, r, p, pi, a, b);
2134 8030 : gcoeff(R,1,1) = Flx_sub(gcoeff(R,1,1), Flx_mul_pre(q,gcoeff(R,1,2), p,pi), p);
2135 8030 : gcoeff(R,2,1) = Flx_sub(gcoeff(R,2,1), Flx_mul_pre(q,gcoeff(R,2,2), p,pi), p);
2136 8030 : return !a && b ? gc_all(av, 2, &R, b): gc_all(av, 1+!!a+!!b, &R, a, b);
2137 : }
2138 :
2139 : GEN
2140 154 : Flx_halfgcd_all(GEN x, GEN y, ulong p, GEN *a, GEN *b)
2141 154 : { return Flx_halfgcd_all_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), a, b); }
2142 :
2143 : GEN
2144 871236 : Flx_halfgcd_pre(GEN x, GEN y, ulong p, ulong pi)
2145 871236 : { return Flx_halfgcd_all_pre(x, y, p, pi, NULL, NULL); }
2146 :
2147 : GEN
2148 0 : Flx_halfgcd(GEN x, GEN y, ulong p)
2149 0 : { return Flx_halfgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2150 :
2151 : /*Do not garbage collect*/
2152 : static GEN
2153 84948317 : Flx_gcd_basecase(GEN a, GEN b, ulong p, ulong pi)
2154 : {
2155 84948317 : pari_sp av = avma;
2156 84948317 : ulong iter = 0;
2157 84948317 : if (lg(b) > lg(a)) swap(a, b);
2158 292289784 : while (lgpol(b))
2159 : {
2160 206890705 : GEN c = Flx_rem_pre(a,b,p,pi);
2161 207341467 : iter++; a = b; b = c;
2162 207341467 : if (gc_needed(av,2))
2163 : {
2164 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (d = %ld)",degpol(c));
2165 0 : (void)gc_all(av,2, &a,&b);
2166 : }
2167 : }
2168 84890635 : return iter < 2 ? Flx_copy(a) : a;
2169 : }
2170 :
2171 : GEN
2172 86603873 : Flx_gcd_pre(GEN x, GEN y, ulong p, ulong pi)
2173 : {
2174 86603873 : pari_sp av = avma;
2175 : long lim;
2176 86603873 : if (!lgpol(x)) return Flx_copy(y);
2177 84947156 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2178 84954884 : while (lgpol(y) >= lim)
2179 : {
2180 229 : if (lgpol(y)<=(lgpol(x)>>1))
2181 : {
2182 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2183 0 : x = y; y = r;
2184 : }
2185 229 : (void) Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2186 229 : if (gc_needed(av,2))
2187 : {
2188 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (y = %ld)",degpol(y));
2189 0 : (void)gc_all(av,2,&x,&y);
2190 : }
2191 : }
2192 84944278 : return gc_leaf(av, Flx_gcd_basecase(x,y,p,pi));
2193 : }
2194 : GEN
2195 34224735 : Flx_gcd(GEN x, GEN y, ulong p)
2196 34224735 : { return Flx_gcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2197 :
2198 : int
2199 9165433 : Flx_is_squarefree(GEN z, ulong p)
2200 : {
2201 9165433 : pari_sp av = avma;
2202 9165433 : GEN d = Flx_gcd(z, Flx_deriv(z,p) , p);
2203 9165303 : return gc_bool(av, degpol(d) == 0);
2204 : }
2205 :
2206 : static long
2207 126540 : Flx_is_smooth_squarefree(GEN f, long r, ulong p, ulong pi)
2208 : {
2209 126540 : pari_sp av = avma;
2210 : long i;
2211 126540 : GEN sx = polx_Flx(f[1]), a = sx;
2212 533262 : for(i=1;;i++)
2213 : {
2214 533262 : if (degpol(f)<=r) return gc_long(av,1);
2215 511438 : a = Flxq_powu_pre(Flx_rem_pre(a,f,p,pi), p, f, p, pi);
2216 511795 : if (Flx_equal(a, sx)) return gc_long(av,1);
2217 508200 : if (i==r) return gc_long(av,0);
2218 406760 : f = Flx_div_pre(f, Flx_gcd_pre(Flx_sub(a,sx,p),f,p,pi),p,pi);
2219 : }
2220 : }
2221 :
2222 : static long
2223 8143 : Flx_is_l_pow(GEN x, ulong p)
2224 : {
2225 8143 : ulong i, lx = lgpol(x);
2226 16302 : for (i=1; i<lx; i++)
2227 14624 : if (x[i+2] && i%p) return 0;
2228 1678 : return 1;
2229 : }
2230 :
2231 : int
2232 126587 : Flx_is_smooth_pre(GEN g, long r, ulong p, ulong pi)
2233 : {
2234 : while (1)
2235 8144 : {
2236 126587 : GEN f = Flx_gcd_pre(g, Flx_deriv(g, p), p, pi);
2237 126355 : if (!Flx_is_smooth_squarefree(Flx_div_pre(g, f, p, pi), r, p, pi))
2238 101451 : return 0;
2239 25109 : if (degpol(f)==0) return 1;
2240 8134 : g = Flx_is_l_pow(f,p) ? Flx_deflate(f, p): f;
2241 : }
2242 : }
2243 : int
2244 74256 : Flx_is_smooth(GEN g, long r, ulong p)
2245 74256 : { return Flx_is_smooth_pre(g, r, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2246 :
2247 : static GEN
2248 6266105 : Flx_extgcd_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2249 : {
2250 6266105 : pari_sp av=avma;
2251 : GEN u,v,u1,v1;
2252 6266105 : long vx = a[1];
2253 6266105 : v = pol0_Flx(vx); v1 = pol1_Flx(vx);
2254 6265969 : if (ptu) { u = pol1_Flx(vx); u1 = pol0_Flx(vx); }
2255 27997870 : while (lgpol(b))
2256 : {
2257 21730876 : GEN r, q = Flx_divrem_pre(a,b,p,pi, &r);
2258 21732001 : a = b; b = r;
2259 21732001 : if (ptu)
2260 : {
2261 2421133 : swap(u,u1);
2262 2421133 : u1 = Flx_sub(u1, Flx_mul_pre(u, q, p, pi), p);
2263 : }
2264 21732015 : swap(v,v1);
2265 21732015 : v1 = Flx_sub(v1, Flx_mul_pre(v, q, p, pi), p);
2266 21731911 : if (gc_needed(av,2))
2267 : {
2268 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_extgcd (d = %ld)",degpol(a));
2269 0 : (void)gc_all(av,ptu ? 6: 4, &a,&b,&v,&v1,&u,&u1);
2270 : }
2271 : }
2272 6265871 : if (ptu) *ptu = u;
2273 6265871 : *ptv = v;
2274 6265871 : return a;
2275 : }
2276 :
2277 : static GEN
2278 146433 : Flx_extgcd_halfgcd(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2279 : {
2280 : GEN u, v;
2281 146433 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2282 146433 : GEN V = cgetg(expu(lgpol(y))+2,t_VEC);
2283 146433 : long i, n = 0, vs = x[1];
2284 398202 : while (lgpol(y) >= lim)
2285 : {
2286 251773 : if (lgpol(y)<=(lgpol(x)>>1))
2287 : {
2288 26 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2289 26 : x = y; y = r;
2290 26 : gel(V,++n) = mkmat22(pol0_Flx(vs),pol1_Flx(vs),pol1_Flx(vs),Flx_neg(q,p));
2291 : } else
2292 251745 : gel(V,++n) = Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2293 : }
2294 146427 : y = Flx_extgcd_basecase(x,y,p,pi,&u,&v);
2295 251765 : for (i = n; i>1; i--)
2296 : {
2297 105341 : GEN R = gel(V,i);
2298 105341 : GEN u1 = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2299 105341 : GEN v1 = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2300 105340 : u = u1; v = v1;
2301 : }
2302 : {
2303 146424 : GEN R = gel(V,1);
2304 146424 : if (ptu)
2305 6574 : *ptu = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2306 146424 : *ptv = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2307 : }
2308 146427 : return y;
2309 : }
2310 :
2311 : /* x and y in Z[X], return lift(gcd(x mod p, y mod p)). Set u and v st
2312 : * ux + vy = gcd (mod p) */
2313 : GEN
2314 6266113 : Flx_extgcd_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2315 : {
2316 6266113 : pari_sp av = avma;
2317 : GEN d;
2318 6266113 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2319 6266111 : if (lgpol(y) >= lim)
2320 146433 : d = Flx_extgcd_halfgcd(x, y, p, pi, ptu, ptv);
2321 : else
2322 6119669 : d = Flx_extgcd_basecase(x, y, p, pi, ptu, ptv);
2323 6265874 : return gc_all(av, ptu?3:2, &d, ptv, ptu);
2324 : }
2325 : GEN
2326 851872 : Flx_extgcd(GEN x, GEN y, ulong p, GEN *ptu, GEN *ptv)
2327 851872 : { return Flx_extgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptu, ptv); }
2328 :
2329 : static GEN
2330 1044 : Flx_halfres_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, ulong *r)
2331 : {
2332 : struct Flx_res res;
2333 : GEN R;
2334 : long dB;
2335 :
2336 1044 : res.res = *r;
2337 1044 : res.lc = Flx_lead(y);
2338 1044 : res.deg0 = degpol(x);
2339 1044 : res.deg1 = degpol(y);
2340 1044 : res.off = 0;
2341 1044 : R = Flx_halfres_i(x, y, p, pi, a, b, &res);
2342 1044 : dB = degpol(*b);
2343 1044 : if (dB < degpol(y))
2344 1044 : Flx_halfres_update_pre(res.deg0, res.deg1, dB, p, pi, &res);
2345 1044 : *r = res.res;
2346 1044 : return R;
2347 : }
2348 :
2349 : static ulong
2350 14610914 : Flx_resultant_basecase_pre(GEN a, GEN b, ulong p, ulong pi)
2351 : {
2352 : pari_sp av;
2353 : long da,db,dc;
2354 14610914 : ulong lb, res = 1UL;
2355 : GEN c;
2356 :
2357 14610914 : da = degpol(a);
2358 14610809 : db = degpol(b);
2359 14611063 : if (db > da)
2360 : {
2361 0 : swapspec(a,b, da,db);
2362 0 : if (both_odd(da,db)) res = p-res;
2363 : }
2364 14611063 : else if (!da) return 1; /* = res * a[2] ^ db, since 0 <= db <= da = 0 */
2365 14611063 : av = avma;
2366 119619151 : while (db)
2367 : {
2368 105029429 : lb = b[db+2];
2369 105029429 : c = Flx_rem_pre(a,b, p,pi);
2370 104681607 : a = b; b = c; dc = degpol(c);
2371 104658760 : if (dc < 0) return gc_long(av,0);
2372 :
2373 104651234 : if (both_odd(da,db)) res = p - res;
2374 104628543 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, da - dc, p, pi), p);
2375 105007738 : if (gc_needed(av,2))
2376 : {
2377 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant (da = %ld)",da);
2378 0 : (void)gc_all(av,2, &a,&b);
2379 : }
2380 105008088 : da = db; /* = degpol(a) */
2381 105008088 : db = dc; /* = degpol(b) */
2382 : }
2383 14589722 : return gc_ulong(av, Fl_mul(res, Fl_powu_pre(b[2], da, p, pi), p));
2384 : }
2385 :
2386 : ulong
2387 14612885 : Flx_resultant_pre(GEN x, GEN y, ulong p, ulong pi)
2388 : {
2389 14612885 : pari_sp av = avma;
2390 : long lim;
2391 14612885 : ulong res = 1;
2392 14612885 : long dx = degpol(x), dy = degpol(y);
2393 14612494 : if (dx < 0 || dy < 0) return 0;
2394 14611038 : if (dx < dy)
2395 : {
2396 1066131 : swap(x,y);
2397 1066131 : if (both_odd(dx, dy))
2398 1906 : res = Fl_neg(res, p);
2399 : }
2400 14611037 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2401 14611902 : while (lgpol(y) >= lim)
2402 : {
2403 852 : if (lgpol(y)<=(lgpol(x)>>1))
2404 : {
2405 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2406 0 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2407 0 : ulong ly = y[dy+2];
2408 0 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2409 0 : if (both_odd(dx, dy))
2410 0 : res = Fl_neg(res, p);
2411 0 : x = y; y = r;
2412 : }
2413 852 : (void) Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2414 852 : if (gc_needed(av,2))
2415 : {
2416 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_res (y = %ld)",degpol(y));
2417 0 : (void)gc_all(av,2,&x,&y);
2418 : }
2419 : }
2420 14610934 : return gc_ulong(av, Fl_mul(res, Flx_resultant_basecase_pre(x, y, p, pi), p));
2421 : }
2422 :
2423 : ulong
2424 4727391 : Flx_resultant(GEN a, GEN b, ulong p)
2425 4727391 : { return Flx_resultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2426 :
2427 : /* If resultant is 0, *ptU and *ptV are not set */
2428 : static ulong
2429 53 : Flx_extresultant_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2430 : {
2431 53 : GEN z,q,u,v, x = a, y = b;
2432 53 : ulong lb, res = 1UL;
2433 53 : pari_sp av = avma;
2434 : long dx, dy, dz;
2435 53 : long vs = a[1];
2436 :
2437 53 : u = pol0_Flx(vs);
2438 53 : v = pol1_Flx(vs); /* v = 1 */
2439 53 : dx = degpol(x);
2440 53 : dy = degpol(y);
2441 764 : while (dy)
2442 : { /* b u = x (a), b v = y (a) */
2443 711 : lb = y[dy+2];
2444 711 : q = Flx_divrem_pre(x,y, p, pi, &z);
2445 711 : x = y; y = z; /* (x,y) = (y, x - q y) */
2446 711 : dz = degpol(z); if (dz < 0) return gc_ulong(av,0);
2447 711 : z = Flx_sub(u, Flx_mul_pre(q,v, p, pi), p);
2448 711 : u = v; v = z; /* (u,v) = (v, u - q v) */
2449 :
2450 711 : if (both_odd(dx,dy)) res = p - res;
2451 711 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, dx-dz, p, pi), p);
2452 711 : dx = dy; /* = degpol(x) */
2453 711 : dy = dz; /* = degpol(y) */
2454 : }
2455 53 : res = Fl_mul(res, Fl_powu_pre(y[2], dx, p, pi), p);
2456 53 : lb = Fl_mul(res, Fl_inv(y[2],p), p);
2457 53 : v = gc_leaf(av, Flx_Fl_mul_pre(v, lb, p, pi));
2458 53 : av = avma;
2459 53 : u = Flx_sub(Fl_to_Flx(res,vs), Flx_mul_pre(b,v,p,pi), p);
2460 53 : u = gc_leaf(av, Flx_div_pre(u,a,p,pi)); /* = (res - b v) / a */
2461 53 : *ptU = u;
2462 53 : *ptV = v; return res;
2463 : }
2464 :
2465 : ulong
2466 53 : Flx_extresultant_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2467 : {
2468 53 : pari_sp av=avma;
2469 : GEN u, v, R;
2470 53 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2471 53 : ulong res = 1, res1;
2472 53 : long dx = degpol(x), dy = degpol(y);
2473 53 : if (dy > dx)
2474 : {
2475 13 : swap(x,y); lswap(dx,dy);
2476 13 : if (both_odd(dx,dy)) res = p-res;
2477 13 : R = matJ2_FlxM(x[1]);
2478 40 : } else R = matid2_FlxM(x[1]);
2479 53 : if (dy < 0) return 0;
2480 245 : while (lgpol(y) >= lim)
2481 : {
2482 : GEN M;
2483 192 : if (lgpol(y)<=(lgpol(x)>>1))
2484 : {
2485 20 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2486 20 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2487 20 : ulong ly = y[dy+2];
2488 20 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2489 20 : if (both_odd(dx, dy))
2490 0 : res = Fl_neg(res, p);
2491 20 : x = y; y = r;
2492 20 : R = Flx_FlxM_qmul(q, R, p,pi);
2493 : }
2494 192 : M = Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2495 192 : if (!res) return gc_ulong(av, 0);
2496 192 : R = FlxM_mul2(M, R, p, pi);
2497 192 : (void)gc_all(av,3,&x,&y,&R);
2498 : }
2499 53 : res1 = Flx_extresultant_basecase(x,y,p,pi,&u,&v);
2500 53 : if (!res1) return gc_ulong(av, 0);
2501 53 : *ptU = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi), res, p, pi);
2502 53 : *ptV = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi), res, p, pi);
2503 53 : (void)gc_all(av, 2, ptU, ptV);
2504 53 : return Fl_mul(res1,res,p);
2505 : }
2506 :
2507 : ulong
2508 53 : Flx_extresultant(GEN a, GEN b, ulong p, GEN *ptU, GEN *ptV)
2509 53 : { return Flx_extresultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptU, ptV); }
2510 :
2511 : /* allow pi = 0 (SMALL_ULONG) */
2512 : ulong
2513 48598429 : Flx_eval_powers_pre(GEN x, GEN y, ulong p, ulong pi)
2514 : {
2515 48598429 : ulong l0, l1, h0, h1, v1, i = 1, lx = lg(x)-1;
2516 :
2517 48598429 : if (lx == 1) return 0;
2518 45538922 : x++;
2519 45538922 : if (pi)
2520 : {
2521 : LOCAL_OVERFLOW;
2522 : LOCAL_HIREMAINDER;
2523 45476376 : l1 = mulll(uel(x,i), uel(y,i)); h1 = hiremainder; v1 = 0;
2524 115316303 : while (++i < lx)
2525 : {
2526 69839927 : l0 = mulll(uel(x,i), uel(y,i)); h0 = hiremainder;
2527 69839927 : l1 = addll(l0, l1); h1 = addllx(h0, h1); v1 += overflow;
2528 : }
2529 81325 : return v1? remlll_pre(v1, h1, l1, p, pi)
2530 45557701 : : remll_pre(h1, l1, p, pi);
2531 : }
2532 : else
2533 : {
2534 62546 : l1 = x[i] * y[i];
2535 30921034 : while (++i < lx) { l1 += x[i] * y[i]; if (l1 & HIGHBIT) l1 %= p; }
2536 62546 : return l1 % p;
2537 : }
2538 : }
2539 :
2540 : /* allow pi = 0 (SMALL_ULONG) */
2541 : ulong
2542 136287178 : Flx_eval_pre(GEN x, ulong y, ulong p, ulong pi)
2543 : {
2544 136287178 : long i, n = degpol(x);
2545 : ulong t;
2546 136285500 : if (n <= 0) return n? 0: x[2];
2547 39402411 : if (n > 15)
2548 : {
2549 180187 : pari_sp av = avma;
2550 180187 : GEN v = Fl_powers_pre(y, n, p, pi);
2551 180194 : return gc_ulong(av, Flx_eval_powers_pre(x, v, p, pi));
2552 : }
2553 39222224 : i = n+2; t = x[i];
2554 39222224 : if (pi)
2555 : {
2556 135936128 : for (i--; i>=2; i--) t = Fl_addmul_pre(uel(x, i), t, y, p, pi);
2557 38100350 : return t;
2558 : }
2559 2735561 : for (i--; i>=2; i--) t = (t * y + x[i]) % p;
2560 1132227 : return t %= p;
2561 : }
2562 : ulong
2563 20368803 : Flx_eval(GEN x, ulong y, ulong p)
2564 20368803 : { return Flx_eval_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2565 :
2566 : ulong
2567 3304 : Flv_prod_pre(GEN x, ulong p, ulong pi)
2568 : {
2569 3304 : pari_sp ltop = avma;
2570 : GEN v;
2571 3304 : long i,k,lx = lg(x);
2572 3304 : if (lx == 1) return 1UL;
2573 3304 : if (lx == 2) return uel(x,1);
2574 2863 : v = cgetg(1+(lx << 1), t_VECSMALL);
2575 2863 : k = 1;
2576 26955 : for (i=1; i<lx-1; i+=2)
2577 24092 : uel(v,k++) = Fl_mul_pre(uel(x,i), uel(x,i+1), p, pi);
2578 2863 : if (i < lx) uel(v,k++) = uel(x,i);
2579 12836 : while (k > 2)
2580 : {
2581 9973 : lx = k; k = 1;
2582 34065 : for (i=1; i<lx-1; i+=2)
2583 24092 : uel(v,k++) = Fl_mul_pre(uel(v,i), uel(v,i+1), p, pi);
2584 9973 : if (i < lx) uel(v,k++) = uel(v,i);
2585 : }
2586 2863 : return gc_ulong(ltop, uel(v,1));
2587 : }
2588 :
2589 : ulong
2590 0 : Flv_prod(GEN v, ulong p)
2591 : {
2592 0 : return Flv_prod_pre(v, p, get_Fl_red(p));
2593 : }
2594 :
2595 : GEN
2596 0 : FlxV_prod(GEN V, ulong p)
2597 : {
2598 : struct _Flx D;
2599 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2600 0 : return gen_product(V, (void *)&D, &_Flx_mul);
2601 : }
2602 :
2603 : static GEN
2604 0 : _Flx_pow(void* E, GEN x, GEN y)
2605 0 : { struct _Flx *D = (struct _Flx *)E; return Flx_powu(x, itou(y), D->p); }
2606 : static GEN
2607 0 : _Flx_one(void *E)
2608 0 : { struct _Flx *D = (struct _Flx *)E; return pol1_Flx(D->v); }
2609 :
2610 : GEN
2611 0 : FlxV_factorback(GEN f, GEN e, ulong p, long v)
2612 : {
2613 : struct _Flx D;
2614 0 : D.p = p; D.v = v;
2615 0 : D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2616 0 : return gen_factorback(f, e, (void *)&D, &_Flx_mul, &_Flx_pow, &_Flx_one);
2617 : }
2618 :
2619 : /* compute prod (x - a[i]) */
2620 : GEN
2621 795669 : Flv_roots_to_pol(GEN a, ulong p, long vs)
2622 : {
2623 : struct _Flx D;
2624 795669 : long i,k,lx = lg(a);
2625 : GEN p1;
2626 795669 : if (lx == 1) return pol1_Flx(vs);
2627 795669 : p1 = cgetg(lx, t_VEC);
2628 12542519 : for (k=1,i=1; i<lx-1; i+=2)
2629 11745302 : gel(p1,k++) = mkvecsmall4(vs, Fl_mul(a[i], a[i+1], p),
2630 11747135 : Fl_neg(Fl_add(a[i],a[i+1],p),p), 1);
2631 795384 : if (i < lx)
2632 64187 : gel(p1,k++) = mkvecsmall3(vs, Fl_neg(a[i],p), 1);
2633 795382 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2634 795378 : setlg(p1, k); return gen_product(p1, (void *)&D, _Flx_mul);
2635 : }
2636 :
2637 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for "large" p */
2638 : INLINE void
2639 21407521 : Flv_inv_pre_indir(GEN w, GEN v, ulong p, ulong pi)
2640 : {
2641 21407521 : pari_sp av = avma;
2642 21407521 : long n = lg(w), i;
2643 : ulong u;
2644 : GEN c;
2645 :
2646 21407521 : if (n == 1) return;
2647 21407521 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2648 90569000 : for (i = 2; i < n; ++i) c[i] = Fl_mul_pre(w[i], c[i-1], p, pi);
2649 21530809 : i = n-1; u = Fl_inv(c[i], p);
2650 90920039 : for ( ; i > 1; --i)
2651 : {
2652 69324072 : ulong t = Fl_mul_pre(u, c[i-1], p, pi);
2653 69253098 : u = Fl_mul_pre(u, w[i], p, pi); v[i] = t;
2654 : }
2655 21595967 : v[1] = u; set_avma(av);
2656 : }
2657 :
2658 : void
2659 19680828 : Flv_inv_pre_inplace(GEN v, ulong p, ulong pi) { Flv_inv_pre_indir(v,v, p, pi); }
2660 :
2661 : GEN
2662 10048 : Flv_inv_pre(GEN w, ulong p, ulong pi)
2663 10048 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_pre_indir(w, v, p, pi); return v; }
2664 :
2665 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for SMALL_ULONG p */
2666 : INLINE void
2667 51282 : Flv_inv_indir(GEN w, GEN v, ulong p)
2668 : {
2669 51282 : pari_sp av = avma;
2670 51282 : long n = lg(w), i;
2671 : ulong u;
2672 : GEN c;
2673 :
2674 51282 : if (n == 1) return;
2675 51282 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2676 1755462 : for (i = 2; i < n; ++i) c[i] = Fl_mul(w[i], c[i-1], p);
2677 51292 : i = n-1; u = Fl_inv(c[i], p);
2678 1755504 : for ( ; i > 1; --i)
2679 : {
2680 1704220 : ulong t = Fl_mul(u, c[i-1], p);
2681 1704213 : u = Fl_mul(u, w[i], p); v[i] = t;
2682 : }
2683 51284 : v[1] = u; set_avma(av);
2684 : }
2685 : static void
2686 1756262 : Flv_inv_i(GEN v, GEN w, ulong p)
2687 : {
2688 1756262 : if (SMALL_ULONG(p)) Flv_inv_indir(w, v, p);
2689 1704980 : else Flv_inv_pre_indir(w, v, p, get_Fl_red(p));
2690 1756264 : }
2691 : void
2692 12017 : Flv_inv_inplace(GEN v, ulong p) { Flv_inv_i(v, v, p); }
2693 : GEN
2694 1744250 : Flv_inv(GEN w, ulong p)
2695 1744250 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_i(v, w, p); return v; }
2696 :
2697 : GEN
2698 34745368 : Flx_div_by_X_x(GEN a, ulong x, ulong p, ulong *rem)
2699 : {
2700 34745368 : long l = lg(a), i;
2701 : GEN a0, z0, z;
2702 34745368 : if (l <= 3)
2703 : {
2704 0 : if (rem) *rem = l == 2? 0: a[2];
2705 0 : return zero_Flx(a[1]);
2706 : }
2707 34745368 : z = cgetg(l-1,t_VECSMALL); z[1] = a[1];
2708 34603101 : a0 = a + l-1;
2709 34603101 : z0 = z + l-2; *z0 = *a0--;
2710 34603101 : if (SMALL_ULONG(p))
2711 : {
2712 84389774 : for (i=l-3; i>1; i--) /* z[i] = (a[i+1] + x*z[i+1]) % p */
2713 : {
2714 62209188 : ulong t = (*a0-- + x * *z0--) % p;
2715 62209188 : *z0 = (long)t;
2716 : }
2717 22180586 : if (rem) *rem = (*a0 + x * *z0) % p;
2718 : }
2719 : else
2720 : {
2721 48834992 : for (i=l-3; i>1; i--)
2722 : {
2723 36389395 : ulong t = Fl_add((ulong)*a0--, Fl_mul(x, *z0--, p), p);
2724 36412477 : *z0 = (long)t;
2725 : }
2726 12445597 : if (rem) *rem = Fl_add((ulong)*a0, Fl_mul(x, *z0, p), p);
2727 : }
2728 34634247 : return z;
2729 : }
2730 :
2731 : /* xa, ya = t_VECSMALL */
2732 : static GEN
2733 1745438 : Flv_producttree(GEN xa, GEN s, ulong p, ulong pi, long vs)
2734 : {
2735 1745438 : long n = lg(xa)-1;
2736 1745438 : long m = n==1 ? 1: expu(n-1)+1;
2737 1745437 : long i, j, k, ls = lg(s);
2738 1745437 : GEN T = cgetg(m+1, t_VEC);
2739 1745432 : GEN t = cgetg(ls, t_VEC);
2740 11590013 : for (j=1, k=1; j<ls; k+=s[j++])
2741 9844455 : gel(t, j) = s[j] == 1 ?
2742 9844583 : mkvecsmall3(vs, Fl_neg(xa[k], p), 1):
2743 3275997 : mkvecsmall4(vs, Fl_mul(xa[k], xa[k+1], p),
2744 3276000 : Fl_neg(Fl_add(xa[k],xa[k+1],p),p), 1);
2745 1745430 : gel(T,1) = t;
2746 4766761 : for (i=2; i<=m; i++)
2747 : {
2748 3021387 : GEN u = gel(T, i-1);
2749 3021387 : long n = lg(u)-1;
2750 3021387 : GEN t = cgetg(((n+1)>>1)+1, t_VEC);
2751 11119491 : for (j=1, k=1; k<n; j++, k+=2)
2752 8098160 : gel(t, j) = Flx_mul_pre(gel(u, k), gel(u, k+1), p, pi);
2753 3021331 : gel(T, i) = t;
2754 : }
2755 1745374 : return T;
2756 : }
2757 :
2758 : static GEN
2759 1785566 : Flx_Flv_multieval_tree(GEN P, GEN xa, GEN T, ulong p, ulong pi)
2760 : {
2761 : long i,j,k;
2762 1785566 : long m = lg(T)-1;
2763 1785566 : GEN R = cgetg(lg(xa), t_VECSMALL);
2764 1785560 : GEN Tp = cgetg(m+1, t_VEC), t;
2765 1785554 : gel(Tp, m) = mkvec(P);
2766 4991566 : for (i=m-1; i>=1; i--)
2767 : {
2768 3206005 : GEN u = gel(T, i), v = gel(Tp, i+1);
2769 3206005 : long n = lg(u)-1;
2770 3206005 : t = cgetg(n+1, t_VEC);
2771 12331290 : for (j=1, k=1; k<n; j++, k+=2)
2772 : {
2773 9125310 : gel(t, k) = Flx_rem_pre(gel(v, j), gel(u, k), p, pi);
2774 9125361 : gel(t, k+1) = Flx_rem_pre(gel(v, j), gel(u, k+1), p, pi);
2775 : }
2776 3205980 : gel(Tp, i) = t;
2777 : }
2778 : {
2779 1785561 : GEN u = gel(T, i+1), v = gel(Tp, i+1);
2780 1785561 : long n = lg(u)-1;
2781 12699390 : for (j=1, k=1; j<=n; j++)
2782 : {
2783 10913799 : long c, d = degpol(gel(u,j));
2784 25352480 : for (c=1; c<=d; c++, k++) R[k] = Flx_eval_pre(gel(v, j), xa[k], p, pi);
2785 : }
2786 1785591 : return gc_const((pari_sp)R, R);
2787 : }
2788 : }
2789 :
2790 : static GEN
2791 2539798 : FlvV_polint_tree(GEN T, GEN R, GEN s, GEN xa, GEN ya, ulong p, ulong pi, long vs)
2792 : {
2793 2539798 : pari_sp av = avma;
2794 2539798 : long m = lg(T)-1;
2795 2539798 : long i, j, k, ls = lg(s);
2796 2539798 : GEN Tp = cgetg(m+1, t_VEC);
2797 2539429 : GEN t = cgetg(ls, t_VEC);
2798 29416901 : for (j=1, k=1; j<ls; k+=s[j++])
2799 26877629 : if (s[j]==2)
2800 : {
2801 8929627 : ulong a = Fl_mul(ya[k], R[k], p);
2802 8929119 : ulong b = Fl_mul(ya[k+1], R[k+1], p);
2803 8934575 : gel(t, j) = mkvecsmall3(vs, Fl_neg(Fl_add(Fl_mul(xa[k], b, p ),
2804 8929479 : Fl_mul(xa[k+1], a, p), p), p), Fl_add(a, b, p));
2805 8933452 : gel(t, j) = Flx_renormalize(gel(t, j), 4);
2806 : }
2807 : else
2808 17948002 : gel(t, j) = Fl_to_Flx(Fl_mul(ya[k], R[k], p), vs);
2809 2539272 : gel(Tp, 1) = t;
2810 8986322 : for (i=2; i<=m; i++)
2811 : {
2812 6447020 : GEN u = gel(T, i-1);
2813 6447020 : GEN t = cgetg(lg(gel(T,i)), t_VEC);
2814 6443896 : GEN v = gel(Tp, i-1);
2815 6443896 : long n = lg(v)-1;
2816 30732362 : for (j=1, k=1; k<n; j++, k+=2)
2817 24280451 : gel(t, j) = Flx_add(Flx_mul_pre(gel(u, k), gel(v, k+1), p, pi),
2818 24285312 : Flx_mul_pre(gel(u, k+1), gel(v, k), p, pi), p);
2819 6447050 : gel(Tp, i) = t;
2820 : }
2821 2539302 : return gc_leaf(av, gmael(Tp,m,1));
2822 : }
2823 :
2824 : GEN
2825 0 : Flx_Flv_multieval(GEN P, GEN xa, ulong p)
2826 : {
2827 0 : pari_sp av = avma;
2828 0 : GEN s = producttree_scheme(lg(xa)-1);
2829 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2830 0 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2831 0 : return gc_leaf(av, Flx_Flv_multieval_tree(P, xa, T, p, pi));
2832 : }
2833 :
2834 : static GEN
2835 2457 : FlxV_Flv_multieval_tree(GEN x, GEN xa, GEN T, ulong p, ulong pi)
2836 45045 : { pari_APPLY_same(Flx_Flv_multieval_tree(gel(x,i), xa, T, p, pi)) }
2837 :
2838 : GEN
2839 2457 : FlxV_Flv_multieval(GEN P, GEN xa, ulong p)
2840 : {
2841 2457 : pari_sp av = avma;
2842 2457 : GEN s = producttree_scheme(lg(xa)-1);
2843 2457 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2844 2457 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2845 2457 : return gc_upto(av, FlxV_Flv_multieval_tree(P, xa, T, p, pi));
2846 : }
2847 :
2848 : GEN
2849 1486274 : Flv_polint(GEN xa, GEN ya, ulong p, long vs)
2850 : {
2851 1486274 : pari_sp av = avma;
2852 1486274 : GEN s = producttree_scheme(lg(xa)-1);
2853 1486279 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2854 1486278 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2855 1486278 : long m = lg(T)-1;
2856 1486278 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2857 1486277 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2858 1486273 : return gc_leaf(av, FlvV_polint_tree(T, R, s, xa, ya, p, pi, vs));
2859 : }
2860 :
2861 : GEN
2862 103258 : Flv_Flm_polint(GEN xa, GEN ya, ulong p, long vs)
2863 : {
2864 103258 : pari_sp av = avma;
2865 103258 : GEN s = producttree_scheme(lg(xa)-1);
2866 103258 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2867 103258 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2868 103259 : long i, m = lg(T)-1, l = lg(ya)-1;
2869 103259 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2870 103259 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2871 103256 : GEN M = cgetg(l+1, t_VEC);
2872 1156653 : for (i=1; i<=l; i++)
2873 1053398 : gel(M,i) = FlvV_polint_tree(T, R, s, xa, gel(ya,i), p, pi, vs);
2874 103255 : return gc_upto(av, M);
2875 : }
2876 :
2877 : GEN
2878 153445 : Flv_invVandermonde(GEN L, ulong den, ulong p)
2879 : {
2880 153445 : pari_sp av = avma;
2881 153445 : long i, n = lg(L);
2882 : GEN M, R;
2883 153445 : GEN s = producttree_scheme(n-1);
2884 153445 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2885 153445 : GEN tree = Flv_producttree(L, s, p, pi, 0);
2886 153445 : long m = lg(tree)-1;
2887 153445 : GEN T = gmael(tree, m, 1);
2888 153445 : R = Flv_inv(Flx_Flv_multieval_tree(Flx_deriv(T, p), L, tree, p, pi), p);
2889 153445 : if (den!=1) R = Flv_Fl_mul(R, den, p);
2890 153445 : M = cgetg(n, t_MAT);
2891 603585 : for (i = 1; i < n; i++)
2892 : {
2893 450140 : GEN P = Flx_Fl_mul(Flx_div_by_X_x(T, uel(L,i), p, NULL), uel(R,i), p);
2894 450140 : gel(M,i) = Flx_to_Flv(P, n-1);
2895 : }
2896 153445 : return gc_GEN(av, M);
2897 : }
2898 :
2899 : /***********************************************************************/
2900 : /** Flxq **/
2901 : /***********************************************************************/
2902 : /* Flxq objects are Flx modulo another Flx called q. */
2903 :
2904 : /* Product of y and x in Z/pZ[X]/(T), as t_VECSMALL. */
2905 : GEN
2906 188655689 : Flxq_mul_pre(GEN x,GEN y,GEN T,ulong p,ulong pi)
2907 188655689 : { return Flx_rem_pre(Flx_mul_pre(x,y,p,pi),T,p,pi); }
2908 : GEN
2909 13164147 : Flxq_mul(GEN x,GEN y,GEN T,ulong p)
2910 13164147 : { return Flxq_mul_pre(x,y,T,p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2911 :
2912 : GEN
2913 277317823 : Flxq_sqr_pre(GEN x,GEN T,ulong p,ulong pi)
2914 277317823 : { return Flx_rem_pre(Flx_sqr_pre(x, p,pi), T, p,pi); }
2915 : /* Square of y in Z/pZ[X]/(T), as t_VECSMALL. */
2916 : GEN
2917 2763118 : Flxq_sqr(GEN x,GEN T,ulong p)
2918 2763118 : { return Flxq_sqr_pre(x,T,p,SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2919 :
2920 : static GEN
2921 1270700 : _Flxq_red(void *E, GEN x)
2922 1270700 : { struct _Flxq *s = (struct _Flxq *)E;
2923 1270700 : return Flx_rem_pre(x, s->T, s->p, s->pi); }
2924 : #if 0
2925 : static GEN
2926 : _Flxq_add(void *E, GEN x, GEN y)
2927 : { struct _Flxq *s = (struct _Flxq *)E;
2928 : return Flx_add(x,y,s->p); }
2929 : static GEN
2930 : _Flxq_sub(void *E, GEN x, GEN y)
2931 : { struct _Flxq *s = (struct _Flxq *)E;
2932 : return Flx_sub(x,y,s->p); }
2933 : #endif
2934 : static GEN
2935 269451224 : _Flxq_sqr(void *data, GEN x)
2936 : {
2937 269451224 : struct _Flxq *D = (struct _Flxq*)data;
2938 269451224 : return Flxq_sqr_pre(x, D->T, D->p, D->pi);
2939 : }
2940 : static GEN
2941 147625721 : _Flxq_mul(void *data, GEN x, GEN y)
2942 : {
2943 147625721 : struct _Flxq *D = (struct _Flxq*)data;
2944 147625721 : return Flxq_mul_pre(x,y, D->T, D->p, D->pi);
2945 : }
2946 : static GEN
2947 22345651 : _Flxq_one(void *data)
2948 : {
2949 22345651 : struct _Flxq *D = (struct _Flxq*)data;
2950 22345651 : return pol1_Flx(get_Flx_var(D->T));
2951 : }
2952 :
2953 : static GEN
2954 23042300 : _Flxq_powu_i(struct _Flxq *D, GEN x, ulong n)
2955 23042300 : { return gen_powu_i(x, n, (void*)D, &_Flxq_sqr, &_Flxq_mul); }
2956 : static GEN
2957 68 : _Flxq_powu(struct _Flxq *D, GEN x, ulong n)
2958 68 : { pari_sp av = avma; return gc_leaf(av, _Flxq_powu_i(D, x, n)); }
2959 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2960 : GEN
2961 24291916 : Flxq_powu_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
2962 : {
2963 : pari_sp av;
2964 : struct _Flxq D;
2965 24291916 : switch(n)
2966 : {
2967 0 : case 0: return pol1_Flx(get_Flx_var(T));
2968 277633 : case 1: return Flx_copy(x);
2969 972469 : case 2: return Flxq_sqr_pre(x, T, p, pi);
2970 : }
2971 23041814 : av = avma; set_Flxq_pre(&D, T, p, pi);
2972 23041709 : return gc_leaf(av, _Flxq_powu_i(&D, x, n));
2973 : }
2974 : GEN
2975 488198 : Flxq_powu(GEN x, ulong n, GEN T, ulong p)
2976 488198 : { return Flxq_powu_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2977 :
2978 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2979 : GEN
2980 23450613 : Flxq_pow_pre(GEN x, GEN n, GEN T, ulong p, ulong pi)
2981 : {
2982 23450613 : pari_sp av = avma;
2983 : struct _Flxq D;
2984 : GEN y;
2985 23450613 : long s = signe(n);
2986 23450613 : if (!s) return pol1_Flx(get_Flx_var(T));
2987 23372918 : if (s < 0) x = Flxq_inv_pre(x,T,p,pi);
2988 23372918 : if (is_pm1(n)) return s < 0 ? x : Flx_copy(x);
2989 22851675 : set_Flxq_pre(&D, T, p, pi);
2990 22851696 : y = gen_pow_i(x, n, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2991 22851629 : return gc_leaf(av, y);
2992 : }
2993 : GEN
2994 945679 : Flxq_pow(GEN x, GEN n, GEN T, ulong p)
2995 945679 : { return Flxq_pow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2996 :
2997 : GEN
2998 28 : Flxq_pow_init_pre(GEN x, GEN n, long k, GEN T, ulong p, ulong pi)
2999 : {
3000 28 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3001 28 : return gen_pow_init(x, n, k, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
3002 : }
3003 : GEN
3004 0 : Flxq_pow_init(GEN x, GEN n, long k, GEN T, ulong p)
3005 0 : { return Flxq_pow_init_pre(x, n, k, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3006 :
3007 : GEN
3008 4393 : Flxq_pow_table_pre(GEN R, GEN n, GEN T, ulong p, ulong pi)
3009 : {
3010 4393 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3011 4393 : return gen_pow_table(R, n, (void*)&D, &_Flxq_one, &_Flxq_mul);
3012 : }
3013 : GEN
3014 0 : Flxq_pow_table(GEN R, GEN n, GEN T, ulong p)
3015 0 : { return Flxq_pow_table_pre(R, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3016 :
3017 : /* Inverse of x in Z/lZ[X]/(T) or NULL if inverse doesn't exist
3018 : * not stack clean. */
3019 : GEN
3020 5414255 : Flxq_invsafe_pre(GEN x, GEN T, ulong p, ulong pi)
3021 : {
3022 5414255 : GEN V, z = Flx_extgcd_pre(get_Flx_mod(T), x, p, pi, NULL, &V);
3023 : ulong iz;
3024 5414232 : if (degpol(z)) return NULL;
3025 5413563 : iz = Fl_inv(uel(z,2), p);
3026 5413572 : return Flx_Fl_mul_pre(V, iz, p, pi);
3027 : }
3028 : GEN
3029 669320 : Flxq_invsafe(GEN x, GEN T, ulong p)
3030 669320 : { return Flxq_invsafe_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3031 :
3032 : GEN
3033 4273223 : Flxq_inv_pre(GEN x, GEN T, ulong p, ulong pi)
3034 : {
3035 4273223 : pari_sp av=avma;
3036 4273223 : GEN U = Flxq_invsafe_pre(x, T, p, pi);
3037 4273093 : if (!U) pari_err_INV("Flxq_inv",Flx_to_ZX(x));
3038 4273079 : return gc_leaf(av, U);
3039 : }
3040 : GEN
3041 327066 : Flxq_inv(GEN x, GEN T, ulong p)
3042 327066 : { return Flxq_inv_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3043 :
3044 : GEN
3045 2417764 : Flxq_div_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
3046 : {
3047 2417764 : pari_sp av = avma;
3048 2417764 : return gc_leaf(av, Flxq_mul_pre(x,Flxq_inv_pre(y,T,p,pi),T,p,pi));
3049 : }
3050 : GEN
3051 237864 : Flxq_div(GEN x, GEN y, GEN T, ulong p)
3052 237864 : { return Flxq_div_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3053 :
3054 : GEN
3055 22344438 : Flxq_powers_pre(GEN x, long l, GEN T, ulong p, ulong pi)
3056 : {
3057 22344438 : int use_sqr = 2*degpol(x) >= get_Flx_degree(T);
3058 22341806 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3059 22340488 : return gen_powers(x, l, use_sqr, (void*)&D, &_Flxq_sqr, &_Flxq_mul, &_Flxq_one);
3060 : }
3061 : GEN
3062 231725 : Flxq_powers(GEN x, long l, GEN T, ulong p)
3063 231725 : { return Flxq_powers_pre(x, l, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3064 :
3065 : GEN
3066 171061 : Flxq_matrix_pow_pre(GEN y, long n, long m, GEN P, ulong l, ulong li)
3067 171061 : { return FlxV_to_Flm(Flxq_powers_pre(y,m-1,P,l,li),n); }
3068 : GEN
3069 399 : Flxq_matrix_pow(GEN y, long n, long m, GEN P, ulong l)
3070 399 : { return Flxq_matrix_pow_pre(y, n, m, P, l, SMALL_ULONG(l)? 0: get_Fl_red(l)); }
3071 :
3072 : GEN
3073 13768973 : Flx_Frobenius_pre(GEN T, ulong p, ulong pi)
3074 13768973 : { return Flxq_powu_pre(polx_Flx(get_Flx_var(T)), p, T, p, pi); }
3075 : GEN
3076 86100 : Flx_Frobenius(GEN T, ulong p)
3077 86100 : { return Flx_Frobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3078 :
3079 : GEN
3080 86772 : Flx_matFrobenius_pre(GEN T, ulong p, ulong pi)
3081 : {
3082 86772 : long n = get_Flx_degree(T);
3083 86772 : return Flxq_matrix_pow_pre(Flx_Frobenius_pre(T, p, pi), n, n, T, p, pi);
3084 : }
3085 : GEN
3086 0 : Flx_matFrobenius(GEN T, ulong p)
3087 0 : { return Flx_matFrobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3088 :
3089 : static GEN
3090 12902741 : Flx_blocks_Flm(GEN P, long n, long m)
3091 : {
3092 12902741 : GEN z = cgetg(m+1,t_MAT);
3093 12902509 : long i,j, k=2, l = lg(P);
3094 36905728 : for(i=1; i<=m; i++)
3095 : {
3096 24006751 : GEN zi = cgetg(n+1,t_VECSMALL);
3097 24003219 : gel(z,i) = zi;
3098 111216346 : for(j=1; j<=n; j++)
3099 87213127 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3100 : }
3101 12898977 : return z;
3102 : }
3103 :
3104 : GEN
3105 517097 : Flx_blocks(GEN P, long n, long m)
3106 : {
3107 517097 : GEN z = cgetg(m+1,t_VEC);
3108 516852 : long i,j, k=2, l = lg(P);
3109 1549118 : for(i=1; i<=m; i++)
3110 : {
3111 1032520 : GEN zi = cgetg(n+2,t_VECSMALL);
3112 1031885 : zi[1] = P[1];
3113 1031885 : gel(z,i) = zi;
3114 6475023 : for(j=2; j<n+2; j++)
3115 5443138 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3116 1031885 : zi = Flx_renormalize(zi, n+2);
3117 : }
3118 516598 : return z;
3119 : }
3120 :
3121 : static GEN
3122 12903561 : FlxV_to_Flm_lg(GEN x, long m, long n)
3123 : {
3124 : long i;
3125 12903561 : GEN y = cgetg(n+1, t_MAT);
3126 61175260 : for (i=1; i<=n; i++) gel(y,i) = Flx_to_Flv(gel(x,i), m);
3127 12900663 : return y;
3128 : }
3129 :
3130 : /* allow pi = 0 (SMALL_ULONG) */
3131 : GEN
3132 13101804 : Flx_FlxqV_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3133 : {
3134 13101804 : pari_sp btop, av = avma;
3135 13101804 : long sv = get_Flx_var(T), m = get_Flx_degree(T);
3136 13102040 : long i, l = lg(x)-1, lQ = lgpol(Q), n, d;
3137 : GEN A, B, C, S, g;
3138 13102851 : if (lQ == 0) return pol0_Flx(sv);
3139 12904343 : if (lQ <= l)
3140 : {
3141 6423404 : n = l;
3142 6423404 : d = 1;
3143 : }
3144 : else
3145 : {
3146 6480939 : n = l-1;
3147 6480939 : d = (lQ+n-1)/n;
3148 : }
3149 12904343 : A = FlxV_to_Flm_lg(x, m, n);
3150 12902620 : B = Flx_blocks_Flm(Q, n, d);
3151 12901712 : C = gc_upto(av, Flm_mul(A, B, p));
3152 12904690 : g = gel(x, l);
3153 12904690 : if (pi && SMALL_ULONG(p)) pi = 0;
3154 12904690 : T = Flx_get_red_pre(T, p, pi);
3155 12904478 : btop = avma;
3156 12904478 : S = Flv_to_Flx(gel(C, d), sv);
3157 24011179 : for (i = d-1; i>0; i--)
3158 : {
3159 11107852 : S = Flx_add(Flxq_mul_pre(S, g, T, p, pi), Flv_to_Flx(gel(C,i), sv), p);
3160 11107613 : if (gc_needed(btop,1))
3161 0 : S = gc_leaf(btop, S);
3162 : }
3163 12903327 : return gc_leaf(av, S);
3164 : }
3165 : GEN
3166 5082 : Flx_FlxqV_eval(GEN Q, GEN x, GEN T, ulong p)
3167 5082 : { return Flx_FlxqV_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3168 :
3169 : /* allow pi = 0 (SMALL_ULONG) */
3170 : GEN
3171 2437495 : Flx_Flxq_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3172 : {
3173 2437495 : pari_sp av = avma;
3174 : GEN z, V;
3175 2437495 : long d = degpol(Q), rtd;
3176 2437502 : if (d < 0) return pol0_Flx(get_Flx_var(T));
3177 2437411 : rtd = (long) sqrt((double)d);
3178 2437411 : T = Flx_get_red_pre(T, p, pi);
3179 2437433 : V = Flxq_powers_pre(x, rtd, T, p, pi);
3180 2437492 : z = Flx_FlxqV_eval_pre(Q, V, T, p, pi);
3181 2437459 : return gc_upto(av, z);
3182 : }
3183 : GEN
3184 788990 : Flx_Flxq_eval(GEN Q, GEN x, GEN T, ulong p)
3185 788990 : { return Flx_Flxq_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3186 :
3187 : /* allow pi = 0 (SMALL_ULONG) */
3188 : GEN
3189 0 : FlxC_FlxqV_eval_pre(GEN x, GEN v, GEN T, ulong p, ulong pi)
3190 0 : { pari_APPLY_type(t_COL, Flx_FlxqV_eval_pre(gel(x,i), v, T, p, pi)) }
3191 : GEN
3192 0 : FlxC_FlxqV_eval(GEN x, GEN v, GEN T, ulong p)
3193 0 : { return FlxC_FlxqV_eval_pre(x, v, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3194 :
3195 : /* allow pi = 0 (SMALL_ULONG) */
3196 : GEN
3197 0 : FlxC_Flxq_eval_pre(GEN x, GEN F, GEN T, ulong p, ulong pi)
3198 : {
3199 0 : long d = brent_kung_optpow(get_Flx_degree(T)-1,lg(x)-1,1);
3200 0 : GEN Fp = Flxq_powers_pre(F, d, T, p, pi);
3201 0 : return FlxC_FlxqV_eval_pre(x, Fp, T, p, pi);
3202 : }
3203 : GEN
3204 0 : FlxC_Flxq_eval(GEN x, GEN F, GEN T, ulong p)
3205 0 : { return FlxC_Flxq_eval_pre(x, F, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3206 :
3207 : #if 0
3208 : static struct bb_algebra Flxq_algebra = { _Flxq_red, _Flxq_add, _Flxq_sub,
3209 : _Flxq_mul, _Flxq_sqr, _Flxq_one, _Flxq_zero};
3210 : #endif
3211 :
3212 : static GEN
3213 47324 : Flxq_autpow_sqr(void *E, GEN x)
3214 : {
3215 47324 : struct _Flxq *D = (struct _Flxq*)E;
3216 47324 : return Flx_Flxq_eval_pre(x, x, D->T, D->p, D->pi);
3217 : }
3218 : static GEN
3219 20698 : Flxq_autpow_msqr(void *E, GEN x)
3220 : {
3221 20698 : struct _Flxq *D = (struct _Flxq*)E;
3222 20698 : return Flx_FlxqV_eval_pre(Flxq_autpow_sqr(E, x), D->aut, D->T, D->p, D->pi);
3223 : }
3224 :
3225 : GEN
3226 69455 : Flxq_autpow_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3227 : {
3228 69455 : pari_sp av = avma;
3229 : struct _Flxq D;
3230 : long d;
3231 69455 : if (n==0) return Flx_rem_pre(polx_Flx(x[1]), T, p, pi);
3232 69448 : if (n==1) return Flx_rem_pre(x, T, p, pi);
3233 32448 : set_Flxq_pre(&D, T, p, pi);
3234 32448 : d = brent_kung_optpow(get_Flx_degree(T), hammingu(n)-1, 1);
3235 32448 : D.aut = Flxq_powers_pre(x, d, T, p, D.pi);
3236 32448 : x = gen_powu_fold_i(x,n,(void*)&D,Flxq_autpow_sqr,Flxq_autpow_msqr);
3237 32448 : return gc_GEN(av, x);
3238 : }
3239 : GEN
3240 7 : Flxq_autpow(GEN x, ulong n, GEN T, ulong p)
3241 7 : { return Flxq_autpow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3242 :
3243 : GEN
3244 1667 : Flxq_autpowers(GEN x, ulong l, GEN T, ulong p)
3245 : {
3246 1667 : long d, vT = get_Flx_var(T), dT = get_Flx_degree(T);
3247 : ulong i, pi;
3248 1667 : pari_sp av = avma;
3249 1667 : GEN xp, V = cgetg(l+2,t_VEC);
3250 1667 : gel(V,1) = polx_Flx(vT); if (l==0) return V;
3251 1667 : gel(V,2) = gcopy(x); if (l==1) return V;
3252 1667 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3253 1667 : T = Flx_get_red_pre(T, p, pi);
3254 1667 : d = brent_kung_optpow(dT-1, l-1, 1);
3255 1667 : xp = Flxq_powers_pre(x, d, T, p, pi);
3256 6998 : for(i = 3; i < l+2; i++)
3257 5331 : gel(V,i) = Flx_FlxqV_eval_pre(gel(V,i-1), xp, T, p, pi);
3258 1667 : return gc_GEN(av, V);
3259 : }
3260 :
3261 : static GEN
3262 112497 : Flxq_autsum_mul(void *E, GEN x, GEN y)
3263 : {
3264 112497 : struct _Flxq *D = (struct _Flxq*)E;
3265 112497 : GEN T = D->T;
3266 112497 : ulong p = D->p, pi = D->pi;
3267 112497 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3268 112497 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3269 112497 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3270 112497 : GEN V2 = Flxq_powers_pre(phi2, d, T, p, pi);
3271 112497 : GEN phi3 = Flx_FlxqV_eval_pre(phi1, V2, T, p, pi);
3272 112497 : GEN aphi = Flx_FlxqV_eval_pre(a1, V2, T, p, pi);
3273 112497 : GEN a3 = Flxq_mul_pre(aphi, a2, T, p, pi);
3274 112497 : return mkvec2(phi3, a3);
3275 : }
3276 : static GEN
3277 105124 : Flxq_autsum_sqr(void *E, GEN x)
3278 105124 : { return Flxq_autsum_mul(E, x, x); }
3279 :
3280 : static GEN
3281 98774 : Flxq_autsum_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3282 : {
3283 98774 : pari_sp av = avma;
3284 98774 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3285 98774 : x = gen_powu_i(x,n,(void*)&D,Flxq_autsum_sqr,Flxq_autsum_mul);
3286 98774 : return gc_GEN(av, x);
3287 : }
3288 : GEN
3289 0 : Flxq_autsum(GEN x, ulong n, GEN T, ulong p)
3290 0 : { return Flxq_autsum_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3291 :
3292 : static GEN
3293 777153 : Flxq_auttrace_mul(void *E, GEN x, GEN y)
3294 : {
3295 777153 : struct _Flxq *D = (struct _Flxq*)E;
3296 777153 : GEN T = D->T;
3297 777153 : ulong p = D->p, pi = D->pi;
3298 777153 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3299 777153 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3300 777153 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3301 777163 : GEN V1 = Flxq_powers_pre(phi1, d, T, p, pi);
3302 777110 : GEN phi3 = Flx_FlxqV_eval_pre(phi2, V1, T, p, pi);
3303 777128 : GEN aphi = Flx_FlxqV_eval_pre(a2, V1, T, p, pi);
3304 777149 : GEN a3 = Flx_add(a1, aphi, p);
3305 777158 : return mkvec2(phi3, a3);
3306 : }
3307 :
3308 : static GEN
3309 651513 : Flxq_auttrace_sqr(void *E, GEN x)
3310 651513 : { return Flxq_auttrace_mul(E, x, x); }
3311 :
3312 : GEN
3313 956619 : Flxq_auttrace_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3314 : {
3315 956619 : pari_sp av = avma;
3316 : struct _Flxq D;
3317 956619 : set_Flxq_pre(&D, T, p, pi);
3318 956620 : x = gen_powu_i(x,n,(void*)&D,Flxq_auttrace_sqr,Flxq_auttrace_mul);
3319 956596 : return gc_GEN(av, x);
3320 : }
3321 : GEN
3322 0 : Flxq_auttrace(GEN x, ulong n, GEN T, ulong p)
3323 0 : { return Flxq_auttrace_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3324 :
3325 : static long
3326 395379 : bounded_order(ulong p, GEN b, long k)
3327 : {
3328 395379 : GEN a = modii(utoipos(p), b);
3329 : long i;
3330 812479 : for(i = 1; i < k; i++)
3331 : {
3332 516506 : if (equali1(a)) return i;
3333 417100 : a = modii(muliu(a,p),b);
3334 : }
3335 295973 : return 0;
3336 : }
3337 :
3338 : /* n = (p^d-a)\b
3339 : * b = bb*p^vb
3340 : * p^k = 1 [bb]
3341 : * d = m*k+r+vb
3342 : * u = (p^k-1)/bb;
3343 : * v = (p^(r+vb)-a)/b;
3344 : * w = (p^(m*k)-1)/(p^k-1)
3345 : * n = p^r*w*u+v
3346 : * w*u = p^vb*(p^(m*k)-1)/b
3347 : * n = p^(r+vb)*(p^(m*k)-1)/b+(p^(r+vb)-a)/b */
3348 : static GEN
3349 22389784 : Flxq_pow_Frobenius(GEN x, GEN n, GEN aut, GEN T, ulong p, ulong pi)
3350 : {
3351 22389784 : pari_sp av=avma;
3352 22389784 : long d = get_Flx_degree(T);
3353 22389784 : GEN an = absi_shallow(n), z, q;
3354 22389784 : if (abscmpiu(an,p)<0 || cmpis(an,d)<=0) return Flxq_pow_pre(x, n, T, p, pi);
3355 395749 : q = powuu(p, d);
3356 395749 : if (dvdii(q, n))
3357 : {
3358 315 : long vn = logint(an, utoipos(p));
3359 315 : GEN autvn = vn==1 ? aut: Flxq_autpow_pre(aut,vn,T,p,pi);
3360 315 : z = Flx_Flxq_eval_pre(x,autvn,T,p,pi);
3361 : } else
3362 : {
3363 395434 : GEN b = diviiround(q, an), a = subii(q, mulii(an,b));
3364 : GEN bb, u, v, autk;
3365 395434 : long vb = Z_lvalrem(b,p,&bb);
3366 395434 : long m, r, k = is_pm1(bb)? 1: bounded_order(p,bb,d);
3367 395434 : if (!k || d-vb < k) return Flxq_pow_pre(x,n, T,p,pi);
3368 99454 : m = (d-vb)/k; r = (d-vb)%k;
3369 99454 : u = diviiexact(subiu(powuu(p,k),1),bb);
3370 99454 : v = diviiexact(subii(powuu(p,r+vb),a),b);
3371 99454 : autk = k==1 ? aut: Flxq_autpow_pre(aut,k,T,p,pi);
3372 99454 : if (r)
3373 : {
3374 487 : GEN autr = r==1 ? aut: Flxq_autpow_pre(aut,r,T,p,pi);
3375 487 : z = Flx_Flxq_eval_pre(x,autr,T,p,pi);
3376 98967 : } else z = x;
3377 99454 : if (m > 1) z = gel(Flxq_autsum_pre(mkvec2(autk, z), m, T, p, pi), 2);
3378 99454 : if (!is_pm1(u)) z = Flxq_pow_pre(z, u, T, p, pi);
3379 99454 : if (signe(v)) z = Flxq_mul_pre(z, Flxq_pow_pre(x, v, T, p, pi), T, p, pi);
3380 : }
3381 99769 : return gc_upto(av,signe(n)>0 ? z : Flxq_inv_pre(z,T,p,pi));
3382 : }
3383 :
3384 : static GEN
3385 22382372 : _Flxq_pow(void *data, GEN x, GEN n)
3386 : {
3387 22382372 : struct _Flxq *D = (struct _Flxq*)data;
3388 22382372 : return Flxq_pow_Frobenius(x, n, D->aut, D->T, D->p, D->pi);
3389 : }
3390 :
3391 : static GEN
3392 6567 : _Flxq_rand(void *data)
3393 : {
3394 6567 : pari_sp av=avma;
3395 6567 : struct _Flxq *D = (struct _Flxq*)data;
3396 : GEN z;
3397 : do
3398 : {
3399 6589 : set_avma(av);
3400 6589 : z = random_Flx(get_Flx_degree(D->T),get_Flx_var(D->T),D->p);
3401 6589 : } while (lgpol(z)==0);
3402 6567 : return z;
3403 : }
3404 :
3405 : /* discrete log in FpXQ for a in Fp^*, g in FpXQ^* of order ord */
3406 : static GEN
3407 35544 : Fl_Flxq_log(ulong a, GEN g, GEN o, GEN T, ulong p)
3408 : {
3409 35544 : pari_sp av = avma;
3410 : GEN q,n_q,ord,ordp, op;
3411 :
3412 35544 : if (a == 1UL) return gen_0;
3413 : /* p > 2 */
3414 :
3415 35544 : ordp = utoi(p - 1);
3416 35544 : ord = get_arith_Z(o);
3417 35544 : if (!ord) ord = T? subiu(powuu(p, get_FpX_degree(T)), 1): ordp;
3418 35544 : if (a == p - 1) /* -1 */
3419 7739 : return gc_INT(av, shifti(ord,-1));
3420 27805 : ordp = gcdii(ordp, ord);
3421 27805 : op = typ(o)==t_MAT ? famat_Z_gcd(o, ordp) : ordp;
3422 :
3423 27805 : q = NULL;
3424 27805 : if (T)
3425 : { /* we want < g > = Fp^* */
3426 27805 : if (!equalii(ord,ordp)) {
3427 11906 : q = diviiexact(ord,ordp);
3428 11906 : g = Flxq_pow(g,q,T,p);
3429 : }
3430 : }
3431 27805 : n_q = Fp_log(utoi(a), utoipos(uel(g,2)), op, utoipos(p));
3432 27805 : if (lg(n_q)==1) return gc_leaf(av, n_q);
3433 27805 : if (q) n_q = mulii(q, n_q);
3434 27805 : return gc_INT(av, n_q);
3435 : }
3436 :
3437 : static GEN
3438 519321 : Flxq_easylog(void* E, GEN a, GEN g, GEN ord)
3439 : {
3440 519321 : struct _Flxq *f = (struct _Flxq *)E;
3441 519321 : GEN T = f->T;
3442 519321 : ulong p = f->p;
3443 519321 : long d = get_Flx_degree(T);
3444 519321 : if (Flx_equal1(a)) return gen_0;
3445 359618 : if (Flx_equal(a,g)) return gen_1;
3446 174457 : if (!degpol(a))
3447 35544 : return Fl_Flxq_log(uel(a,2), g, ord, T, p);
3448 138913 : if (typ(ord)!=t_INT || d <= 4 || d == 6 || abscmpiu(ord,1UL<<27)<0)
3449 138885 : return NULL;
3450 28 : return Flxq_log_index(a, g, ord, T, p);
3451 : }
3452 :
3453 : static const struct bb_group Flxq_star={_Flxq_mul,_Flxq_pow,_Flxq_rand,hash_GEN,Flx_equal,Flx_equal1,Flxq_easylog};
3454 :
3455 : const struct bb_group *
3456 283438 : get_Flxq_star(void **E, GEN T, ulong p)
3457 : {
3458 283438 : struct _Flxq *e = (struct _Flxq *) stack_malloc(sizeof(struct _Flxq));
3459 283438 : e->T = T; e->p = p; e->pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3460 283438 : e->aut = Flx_Frobenius_pre(T, p, e->pi);
3461 283439 : *E = (void*)e; return &Flxq_star;
3462 : }
3463 :
3464 : GEN
3465 97328 : Flxq_order(GEN a, GEN ord, GEN T, ulong p)
3466 : {
3467 : void *E;
3468 97328 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3469 97328 : return gen_order(a,ord,E,S);
3470 : }
3471 :
3472 : GEN
3473 164209 : Flxq_log(GEN a, GEN g, GEN ord, GEN T, ulong p)
3474 : {
3475 : void *E;
3476 164209 : pari_sp av = avma;
3477 164209 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3478 164210 : GEN v = get_arith_ZZM(ord), F = gmael(v,2,1);
3479 164210 : if (lg(F) > 1 && Flxq_log_use_index(veclast(F), T, p))
3480 24283 : v = mkvec2(gel(v, 1), ZM_famat_limit(gel(v, 2), int2n(27)));
3481 164210 : return gc_leaf(av, gen_PH_log(a, g, v, E, S));
3482 : }
3483 :
3484 : static GEN
3485 295314 : Flxq_sumautsum_sqr(void *E, GEN xzd)
3486 : {
3487 295314 : struct _Flxq *D = (struct _Flxq*)E;
3488 295314 : pari_sp av = avma;
3489 : GEN xi, zeta, delta, xi2, zeta2, delta2, temp, xipow;
3490 295314 : GEN T = D->T;
3491 295314 : ulong d, p = D-> p, pi = D->pi;
3492 295314 : xi = gel(xzd, 1); zeta = gel(xzd, 2); delta = gel(xzd, 3);
3493 :
3494 295314 : d = brent_kung_optpow(get_Flx_degree(T)-1,3,1);
3495 295314 : xipow = Flxq_powers_pre(xi, d, T, p, pi);
3496 :
3497 295314 : xi2 = Flx_FlxqV_eval_pre(xi, xipow, T, p, pi);
3498 295314 : zeta2 = Flxq_mul_pre(zeta, Flx_FlxqV_eval_pre(zeta, xipow, T, p, pi), T, p, pi);
3499 295314 : temp = Flxq_mul_pre(zeta, Flx_FlxqV_eval_pre(delta, xipow, T, p, pi), T, p, pi);
3500 295314 : delta2 = Flx_add(delta, temp, p);
3501 295314 : return gc_GEN(av, mkvec3(xi2, zeta2, delta2));
3502 : }
3503 :
3504 : static GEN
3505 40915 : Flxq_sumautsum_msqr(void *E, GEN xzd)
3506 : {
3507 40915 : struct _Flxq *D = (struct _Flxq*)E;
3508 40915 : pari_sp av = avma;
3509 : GEN xii, zetai, deltai, xzd2;
3510 40915 : GEN T = D->T, xi0pow = gel(D->aut, 1), zeta0 = gel(D->aut, 2);
3511 40915 : ulong p = D-> p, pi = D->pi;
3512 40915 : xzd2 = Flxq_sumautsum_sqr(E, xzd);
3513 40915 : xii = Flx_FlxqV_eval_pre(gel(xzd2, 1), xi0pow, T, p, pi);
3514 40915 : zetai = Flxq_mul_pre(zeta0, Flx_FlxqV_eval_pre(gel(xzd2, 2), xi0pow, T, p, pi), T, p, pi);
3515 40915 : deltai = Flx_add(gel(xzd2, 3), zetai, p);
3516 :
3517 40915 : return gc_GEN(av, mkvec3(xii, zetai, deltai));
3518 : }
3519 :
3520 : /*returns a + a^(1+s) + a^(1+s+2s) + ... + a^(1+s+...+is)
3521 : where ax = [a,s] with s an automorphism */
3522 : static GEN
3523 210713 : Flxq_sumautsum_pre(GEN ax, long i, GEN T, ulong p, ulong pi) {
3524 210713 : pari_sp av = avma;
3525 : GEN a, xi, zeta, vec, res;
3526 : struct _Flxq D;
3527 : ulong d;
3528 210713 : D.T = Flx_get_red(T, p); D.p = p; D.pi = pi;
3529 210713 : a = gel(ax, 1); xi = gel(ax,2);
3530 210713 : d = brent_kung_optpow(get_Flx_degree(T)-1,2*(hammingu(i)-1),1);
3531 210713 : zeta = Flx_Flxq_eval_pre(a, xi, T, p, pi);
3532 210713 : D.aut = mkvec2(Flxq_powers_pre(xi, d, T, p, pi), zeta);
3533 :
3534 210713 : vec = gen_powu_fold(mkvec3(xi, zeta, zeta), i, (void *)&D, Flxq_sumautsum_sqr, Flxq_sumautsum_msqr);
3535 210713 : res = Flxq_mul_pre(a, Flx_add(pol1_Flx(get_Flx_var(T)), gel(vec, 3), p), T, p, pi);
3536 :
3537 210713 : return gc_GEN(av, res);
3538 : }
3539 :
3540 : /*algorithm from
3541 : Doliskani, J., & Schost, E. (2014).
3542 : Taking roots over high extensions of finite fields
3543 : https://arxiv.org/abs/1110.4350
3544 : */
3545 : static GEN
3546 37667 : Flxq_sqrtl_spec_pre(GEN z, GEN n, GEN T, ulong p, ulong pi, GEN *zetan)
3547 : {
3548 37667 : pari_sp av = avma;
3549 : GEN psn, c, b, new_z, beta, x, y, w, ax, g, zeta;
3550 37667 : long s, l, v = get_Flx_var(T), d = get_Flx_degree(T);
3551 : ulong zeta2, beta2;
3552 37666 : s = itos(Fp_order(utoi(p), stoi(d), n));
3553 37667 : if(s >= d || d % s != 0)
3554 0 : pari_err(e_MISC, "expected p's order mod n to divide the degree of T");
3555 37667 : l = d/s;
3556 37667 : if (!lgpol(z)) return pol0_Flx(get_Flx_var(T));
3557 37667 : T = Flx_get_red(T, p);
3558 37667 : ax = mkvec2(NULL, Flxq_autpow_pre(Flx_Frobenius_pre(T,p,pi), s, T, p,pi));
3559 37664 : psn = diviiexact(subiu(powuu(p, s), 1), n);
3560 : do {
3561 41668 : do c = random_Flx(d, v, p); while (!lgpol(c));
3562 41166 : new_z = Flxq_mul_pre(z, Flxq_pow_pre(c, n, T, p,pi), T, p,pi);
3563 41165 : gel(ax,1) = Flxq_pow_pre(new_z, psn, T, p,pi);
3564 :
3565 : /*If l == 2, b has to be 1 + a^((p^s-1)/n)*/
3566 41166 : if(l == 2) y = gel(ax, 1);
3567 3244 : else y = Flxq_sumautsum_pre(ax, l-2, T, p, pi);
3568 41166 : b = Flx_Fl_add(y, 1, p);
3569 41166 : } while (!lgpol(b));
3570 :
3571 37666 : x = Flxq_mul_pre(new_z, Flxq_pow_pre(b, n, T, p,pi), T, p,pi);
3572 37665 : if(s == 1) {
3573 36517 : if (degpol(x) > 0) return gc_NULL(av);
3574 36475 : beta2 = Fl_sqrtn(Flx_constant(x), umodiu(n, p), p, &zeta2);
3575 36475 : if (beta2==~0UL) return gc_NULL(av);
3576 36475 : if(zetan) *zetan = monomial_Flx(zeta2, 0, get_Flx_var(T));
3577 36475 : w = Flx_Fl_mul(Flxq_inv_pre(Flxq_mul_pre(b, c, T, p,pi), T, p,pi), beta2, p);
3578 36476 : (void)gc_all(av, zetan? 2: 1, &w, zetan);
3579 36477 : return w;
3580 : }
3581 1148 : g = Flxq_minpoly(x, T, p);
3582 1148 : if (degpol(g) > s) return gc_NULL(av);
3583 1148 : beta = Flxq_sqrtn(polx_Flx(get_Flx_var(T)), n, g, p, &zeta);
3584 1148 : if (!beta) return gc_NULL(av);
3585 :
3586 1148 : if(zetan) *zetan = Flx_Flxq_eval(zeta, x, T, p);
3587 1148 : beta = Flx_Flxq_eval(beta, x, T, p);
3588 1148 : w = Flxq_mul_pre(Flxq_inv_pre(Flxq_mul_pre(b, c, T, p,pi), T, p,pi), beta, T, p,pi);
3589 1148 : (void)gc_all(av, zetan? 2: 1, &w, zetan);
3590 1148 : return w;
3591 : }
3592 :
3593 : static GEN
3594 21901 : Flxq_sqrtn_spec_pre(GEN a, GEN n, GEN T, ulong p, ulong pi, GEN q, GEN *zetan)
3595 : {
3596 21901 : pari_sp ltop = avma;
3597 : GEN z, m, u1, u2;
3598 : int is_1;
3599 21901 : if (is_pm1(n))
3600 : {
3601 1918 : if (zetan) *zetan = pol1_Flx(get_Flx_var(T));
3602 1918 : return signe(n) < 0? Flxq_inv_pre(a, T, p,pi): gcopy(a);
3603 : }
3604 19983 : is_1 = gequal1(a);
3605 19983 : if (is_1 && !zetan) return gcopy(a);
3606 19983 : z = pol1_Flx(get_Flx_var(T));
3607 19983 : m = bezout(n,q,&u1,&u2);
3608 19983 : if (!is_pm1(m))
3609 : {
3610 19983 : GEN F = Z_factor(m);
3611 19983 : long i, j, j2 = 0; /* -Wall */
3612 : GEN y, l;
3613 19983 : pari_sp av1 = avma;
3614 40050 : for (i = nbrows(F); i; i--)
3615 : {
3616 20109 : l = gcoeff(F,i,1);
3617 20109 : j = itos(gcoeff(F,i,2));
3618 20109 : if(zetan) {
3619 104 : a = Flxq_sqrtl_spec_pre(a,l,T,p,pi,&y);
3620 146 : if (!a) return gc_NULL(ltop);
3621 104 : j--;
3622 104 : j2 = j;
3623 : }
3624 20109 : if (!is_1 && j > 0) {
3625 : do
3626 : {
3627 37451 : a = Flxq_sqrtl_spec_pre(a,l,T,p,pi,NULL);
3628 37451 : if (!a) return gc_NULL(ltop);
3629 37409 : } while (--j);
3630 : }
3631 : /*This is below finding a's root,
3632 : so we don't spend time doing this, if a is not n-th root*/
3633 20067 : if(zetan) {
3634 216 : for(; j2>0; j2--) y = Flxq_sqrtl_spec_pre(y, l, T, p,pi,NULL);
3635 104 : z = Flxq_mul_pre(z, y, T, p,pi);
3636 : }
3637 20067 : if (gc_needed(ltop,1))
3638 : { /* n can have lots of prime factors*/
3639 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxq_sqrtn_spec");
3640 0 : (void)gc_all(av1, zetan? 2: 1, &a, &z);
3641 : }
3642 : }
3643 : }
3644 :
3645 19941 : if (!equalii(m, n))
3646 434 : a = Flxq_pow_pre(a,modii(u1,q), T, p,pi);
3647 19941 : if (zetan)
3648 : {
3649 104 : *zetan = z;
3650 104 : (void)gc_all(ltop,2,&a,zetan);
3651 : }
3652 : else /* is_1 is 0: a was modified above -> gc_upto valid */
3653 19837 : a = gc_upto(ltop, a);
3654 19941 : return a;
3655 : }
3656 :
3657 : GEN
3658 23095 : Flxq_sqrtn(GEN a, GEN n, GEN T, ulong p, GEN *zeta)
3659 : {
3660 23095 : if (!lgpol(a))
3661 : {
3662 7 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
3663 0 : if (zeta)
3664 0 : *zeta=pol1_Flx(get_Flx_var(T));
3665 0 : return pol0_Flx(get_Flx_var(T));
3666 : }
3667 23088 : else if(p == 2) {
3668 1187 : pari_sp av = avma;
3669 : GEN z;
3670 1187 : z = F2xq_sqrtn(Flx_to_F2x(a), n, Flx_to_F2x(get_FpX_mod(T)), zeta);
3671 1187 : if (!z) return NULL;
3672 1187 : z = F2x_to_Flx(z);
3673 1187 : if (!zeta) return gc_leaf(av, z);
3674 0 : *zeta=F2x_to_Flx(*zeta);
3675 0 : return gc_all(av, 2, &z,zeta);
3676 : }
3677 : else
3678 : {
3679 : void *E;
3680 21901 : pari_sp av = avma;
3681 21901 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3682 21901 : GEN o = subiu(powuu(p,get_Flx_degree(T)), 1);
3683 : GEN m, u1, u2, l, zeta2, F, n2, z;
3684 21901 : long i, s, pi, d = get_Flx_degree(T);
3685 21901 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3686 21901 : m = bezout(n,o,&u1,&u2);
3687 21900 : F = Z_factor(m);
3688 46433 : for (i = nbrows(F); i; i--)
3689 : {
3690 24533 : l = gcoeff(F,i,1);
3691 24533 : s = itos(Fp_order(utoi(p), subiu(l, 1), l));
3692 : /*Flxq_sqrtn_spec only works if d > s and s | d
3693 : for those factors of m we use Flxq_sqrtn_spec
3694 : for the other factor we stay with gen_Shanks_sqrtn*/
3695 24532 : if(d <= s || d % s != 0) {
3696 4424 : gcoeff(F,i,2) = gen_0;
3697 : }
3698 20108 : else gcoeff(F,i,2) = stoi(Z_pval(n,l));
3699 : }
3700 21900 : F = factorback(F);
3701 21901 : z = Flxq_sqrtn_spec_pre(a,F,T, p,pi,o,zeta);
3702 21901 : if(!z) return gc_NULL(av);
3703 21859 : n2 = diviiexact(n, F);
3704 21859 : if(!gequal1(n2)) {
3705 5005 : if(zeta) zeta2 = gcopy(*zeta);
3706 5005 : z = gen_Shanks_sqrtn(z, n2, o, zeta, E, S);
3707 5005 : if (!z) return gc_NULL(av);
3708 5005 : if(zeta) *zeta = Flxq_mul_pre(*zeta, zeta2, T, p,pi);
3709 : }
3710 21859 : return gc_all(av, zeta?2:1, &z, zeta);
3711 : }
3712 : }
3713 :
3714 : GEN
3715 230484 : Flxq_sqrt_pre(GEN z, GEN T, ulong p, ulong pi)
3716 : {
3717 230484 : pari_sp av = avma;
3718 : long d;
3719 230484 : if (p==2)
3720 : {
3721 0 : GEN r = F2xq_sqrt(Flx_to_F2x(z), Flx_to_F2x(get_Flx_mod(T)));
3722 0 : return gc_upto(av, F2x_to_Flx(r));
3723 : }
3724 230484 : d = get_Flx_degree(T);
3725 230484 : if (d==2)
3726 : {
3727 65765 : GEN P = get_Flx_mod(T), s;
3728 65765 : ulong c = uel(P,2), b = uel(P,3), a = uel(P,4);
3729 65765 : ulong y = degpol(z)<1 ? 0: uel(z,3);
3730 65765 : if (a==1 && b==0)
3731 15226 : {
3732 16006 : ulong x = degpol(z)<1 ? Flx_constant(z): uel(z,2);
3733 16006 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), Fl_neg(c, p), p, pi);
3734 16006 : if (!r) return gc_NULL(av);
3735 15226 : s = mkvecsmall3(P[1], uel(r,1), uel(r,2));
3736 : }
3737 : else
3738 : {
3739 49759 : ulong b2 = Fl_halve(b, p), t = Fl_div(b2, a, p);
3740 49759 : ulong D = Fl_sub(Fl_sqr(b2, p), Fl_mul(a, c, p), p);
3741 49759 : ulong x = degpol(z)<1 ? Flx_constant(z): Fl_sub(uel(z,2), Fl_mul(uel(z,3), t, p), p);
3742 49759 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), D, p, pi);
3743 49759 : if (!r) return gc_NULL(av);
3744 47365 : s = mkvecsmall3(P[1], Fl_add(uel(r,1), Fl_mul(uel(r,2),t,p), p), uel(r,2));
3745 : }
3746 62591 : return gc_leaf(av, Flx_renormalize(s, 4));
3747 : }
3748 164719 : if (lgpol(z)<=1 && odd(d))
3749 : {
3750 11710 : pari_sp av = avma;
3751 11710 : ulong s = Fl_sqrt(Flx_constant(z), p);
3752 11710 : if (s==~0UL) return gc_NULL(av);
3753 11696 : return gc_GEN(av, Fl_to_Flx(s, get_Flx_var(T)));
3754 : } else
3755 : {
3756 : GEN c, b, new_z, x, y, w, ax;
3757 : ulong p2, beta;
3758 153009 : long v = get_Flx_var(T);
3759 153009 : if (!lgpol(z)) return pol0_Flx(v);
3760 152344 : T = Flx_get_red_pre(T, p, pi);
3761 152344 : ax = mkvec2(NULL, Flx_Frobenius_pre(T, p, pi));
3762 152344 : p2 = p >> 1; /* (p-1) / 2 */
3763 : do {
3764 208141 : do c = random_Flx(d, v, p); while (!lgpol(c));
3765 :
3766 207469 : new_z = Flxq_mul_pre(z, Flxq_sqr_pre(c, T, p, pi), T, p, pi);
3767 207469 : gel(ax, 1) = Flxq_powu_pre(new_z, p2, T, p, pi);
3768 207469 : y = Flxq_sumautsum_pre(ax, d-2, T, p, pi); /* d > 2 */
3769 207469 : b = Flx_Fl_add(y, 1UL, p);
3770 207469 : } while (!lgpol(b));
3771 :
3772 152344 : x = Flxq_mul_pre(new_z, Flxq_sqr_pre(b, T, p, pi), T, p, pi);
3773 152344 : if (degpol(x) > 0) return gc_NULL(av);
3774 145302 : beta = Fl_sqrt_pre(Flx_constant(x), p, pi);
3775 145302 : if (beta==~0UL) return gc_NULL(av);
3776 145302 : w = Flx_Fl_mul(Flxq_inv_pre(Flxq_mul_pre(b, c, T,p,pi), T,p,pi), beta, p);
3777 145302 : return gc_GEN(av, w);
3778 : }
3779 : }
3780 :
3781 : GEN
3782 230484 : Flxq_sqrt(GEN a, GEN T, ulong p)
3783 230484 : { return Flxq_sqrt_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3784 :
3785 : /* assume T irreducible mod p */
3786 : int
3787 404492 : Flxq_issquare(GEN x, GEN T, ulong p)
3788 : {
3789 404492 : if (lgpol(x) == 0 || p == 2) return 1;
3790 397989 : return krouu(Flxq_norm(x,T,p), p) == 1;
3791 : }
3792 :
3793 : /* assume T irreducible mod p */
3794 : int
3795 0 : Flxq_is2npower(GEN x, long n, GEN T, ulong p)
3796 : {
3797 : pari_sp av;
3798 : GEN m;
3799 0 : if (n==1) return Flxq_issquare(x, T, p);
3800 0 : if (lgpol(x) == 0 || p == 2) return 1;
3801 0 : av = avma;
3802 0 : m = shifti(subiu(powuu(p, get_Flx_degree(T)), 1), -n);
3803 0 : return gc_bool(av, Flx_equal1(Flxq_pow(x, m, T, p)));
3804 : }
3805 :
3806 : GEN
3807 113589 : Flxq_lroot_fast_pre(GEN a, GEN sqx, GEN T, long p, ulong pi)
3808 : {
3809 113589 : pari_sp av=avma;
3810 113589 : GEN A = Flx_splitting(a,p);
3811 113589 : return gc_leaf(av, FlxqV_dotproduct_pre(A,sqx,T,p,pi));
3812 : }
3813 : GEN
3814 0 : Flxq_lroot_fast(GEN a, GEN sqx, GEN T, long p)
3815 0 : { return Flxq_lroot_fast_pre(a, sqx, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3816 :
3817 : GEN
3818 25053 : Flxq_lroot_pre(GEN a, GEN T, long p, ulong pi)
3819 : {
3820 25053 : pari_sp av=avma;
3821 25053 : long n = get_Flx_degree(T), d = degpol(a);
3822 : GEN sqx, V;
3823 25053 : if (n==1) return leafcopy(a);
3824 25053 : if (n==2) return Flxq_powu_pre(a, p, T, p, pi);
3825 25053 : sqx = Flxq_autpow_pre(Flx_Frobenius_pre(T, p, pi), n-1, T, p, pi);
3826 25053 : if (d==1 && a[2]==0 && a[3]==1) return gc_leaf(av, sqx);
3827 0 : if (d>=p)
3828 : {
3829 0 : V = Flxq_powers_pre(sqx,p-1,T,p,pi);
3830 0 : return gc_leaf(av, Flxq_lroot_fast_pre(a,V,T,p,pi));
3831 : } else
3832 0 : return gc_leaf(av, Flx_Flxq_eval_pre(a,sqx,T,p,pi));
3833 : }
3834 : GEN
3835 0 : Flxq_lroot(GEN a, GEN T, long p)
3836 0 : { return Flxq_lroot_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3837 :
3838 : ulong
3839 443318 : Flxq_norm(GEN x, GEN TB, ulong p)
3840 : {
3841 443318 : GEN T = get_Flx_mod(TB);
3842 443318 : ulong y = Flx_resultant(T, x, p), L = Flx_lead(T);
3843 443318 : if (L==1 || lgpol(x)==0) return y;
3844 0 : return Fl_div(y, Fl_powu(L, (ulong)degpol(x), p), p);
3845 : }
3846 :
3847 : ulong
3848 4696 : Flxq_trace(GEN x, GEN TB, ulong p)
3849 : {
3850 4696 : pari_sp av = avma;
3851 : ulong t;
3852 4696 : GEN T = get_Flx_mod(TB);
3853 4696 : long n = degpol(T)-1;
3854 4696 : GEN z = Flxq_mul(x, Flx_deriv(T, p), TB, p);
3855 4696 : t = degpol(z)<n ? 0 : Fl_div(z[2+n],T[3+n],p);
3856 4696 : return gc_ulong(av, t);
3857 : }
3858 :
3859 : /*x must be reduced*/
3860 : GEN
3861 3624 : Flxq_charpoly(GEN x, GEN TB, ulong p)
3862 : {
3863 3624 : pari_sp ltop=avma;
3864 3624 : GEN T = get_Flx_mod(TB);
3865 3624 : long vs = evalvarn(fetch_var());
3866 3624 : GEN xm1 = deg1pol_shallow(pol1_Flx(x[1]),Flx_neg(x,p),vs);
3867 3624 : GEN r = Flx_FlxY_resultant(T, xm1, p);
3868 3624 : r[1] = x[1];
3869 3624 : (void)delete_var(); return gc_upto(ltop, r);
3870 : }
3871 :
3872 : /* Computing minimal polynomial : */
3873 : /* cf Shoup 'Efficient Computation of Minimal Polynomials */
3874 : /* in Algebraic Extensions of Finite Fields' */
3875 :
3876 : /* Let v a linear form, return the linear form z->v(tau*z)
3877 : that is, v*(M_tau) */
3878 :
3879 : static GEN
3880 1742459 : Flxq_transmul_init(GEN tau, GEN T, ulong p, ulong pi)
3881 : {
3882 : GEN bht;
3883 1742459 : GEN h, Tp = get_Flx_red(T, &h);
3884 1742460 : long n = degpol(Tp), vT = Tp[1];
3885 1742445 : GEN ft = Flx_recipspec(Tp+2, n+1, n+1);
3886 1742438 : GEN bt = Flx_recipspec(tau+2, lgpol(tau), n);
3887 1742421 : ft[1] = vT; bt[1] = vT;
3888 1742421 : if (h)
3889 2688 : bht = Flxn_mul_pre(bt, h, n-1, p, pi);
3890 : else
3891 : {
3892 1739733 : GEN bh = Flx_div_pre(Flx_shift(tau, n-1), T, p, pi);
3893 1739738 : bht = Flx_recipspec(bh+2, lgpol(bh), n-1);
3894 1739743 : bht[1] = vT;
3895 : }
3896 1742431 : return mkvec3(bt, bht, ft);
3897 : }
3898 :
3899 : static GEN
3900 4203149 : Flxq_transmul(GEN tau, GEN a, long n, ulong p, ulong pi)
3901 : {
3902 4203149 : pari_sp ltop = avma;
3903 : GEN t1, t2, t3, vec;
3904 4203149 : GEN bt = gel(tau, 1), bht = gel(tau, 2), ft = gel(tau, 3);
3905 4203149 : if (lgpol(a)==0) return pol0_Flx(a[1]);
3906 4172081 : t2 = Flx_shift(Flx_mul_pre(bt, a, p, pi),1-n);
3907 4171752 : if (lgpol(bht)==0) return gc_leaf(ltop, t2);
3908 3146379 : t1 = Flx_shift(Flx_mul_pre(ft, a, p, pi),-n);
3909 3146385 : t3 = Flxn_mul_pre(t1, bht, n-1, p, pi);
3910 3146401 : vec = Flx_sub(t2, Flx_shift(t3, 1), p);
3911 3146461 : return gc_leaf(ltop, vec);
3912 : }
3913 :
3914 : GEN
3915 807806 : Flxq_minpoly_pre(GEN x, GEN T, ulong p, ulong pi)
3916 : {
3917 807806 : pari_sp ltop = avma;
3918 807806 : long vT = get_Flx_var(T), n = get_Flx_degree(T);
3919 : GEN v_x;
3920 807805 : GEN g = pol1_Flx(vT), tau = pol1_Flx(vT);
3921 807775 : T = Flx_get_red_pre(T, p, pi);
3922 807774 : v_x = Flxq_powers_pre(x, usqrt(2*n), T, p, pi);
3923 1678993 : while (lgpol(tau) != 0)
3924 : {
3925 : long i, j, m, k1;
3926 : GEN M, v, tr, g_prime, c;
3927 871213 : if (degpol(g) == n) { tau = pol1_Flx(vT); g = pol1_Flx(vT); }
3928 871214 : v = random_Flx(n, vT, p);
3929 871234 : tr = Flxq_transmul_init(tau, T, p, pi);
3930 871218 : v = Flxq_transmul(tr, v, n, p, pi);
3931 871230 : m = 2*(n-degpol(g));
3932 871231 : k1 = usqrt(m);
3933 871232 : tr = Flxq_transmul_init(gel(v_x,k1+1), T, p, pi);
3934 871221 : c = cgetg(m+2,t_VECSMALL);
3935 871165 : c[1] = vT;
3936 4203024 : for (i=0; i<m; i+=k1)
3937 : {
3938 3331793 : long mj = minss(m-i, k1);
3939 12933095 : for (j=0; j<mj; j++)
3940 9601051 : uel(c,m+1-(i+j)) = Flx_dotproduct_pre(v, gel(v_x,j+1), p, pi);
3941 3332044 : v = Flxq_transmul(tr, v, n, p, pi);
3942 : }
3943 871231 : c = Flx_renormalize(c, m+2);
3944 : /* now c contains <v,x^i> , i = 0..m-1 */
3945 871230 : M = Flx_halfgcd_pre(monomial_Flx(1, m, vT), c, p, pi);
3946 871238 : g_prime = gmael(M, 2, 2);
3947 871238 : if (degpol(g_prime) < 1) continue;
3948 858607 : g = Flx_mul_pre(g, g_prime, p, pi);
3949 858596 : tau = Flxq_mul_pre(tau, Flx_FlxqV_eval_pre(g_prime, v_x, T,p,pi), T,p,pi);
3950 : }
3951 807756 : g = Flx_normalize(g,p);
3952 807797 : return gc_leaf(ltop,g);
3953 : }
3954 : GEN
3955 45776 : Flxq_minpoly(GEN x, GEN T, ulong p)
3956 45776 : { return Flxq_minpoly_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3957 :
3958 : GEN
3959 20 : Flxq_conjvec(GEN x, GEN T, ulong p)
3960 : {
3961 20 : long i, l = 1+get_Flx_degree(T);
3962 20 : GEN z = cgetg(l,t_COL);
3963 20 : struct _Flxq D; set_Flxq(&D, T, p);
3964 20 : gel(z,1) = Flx_copy(x);
3965 88 : for (i=2; i<l; i++) gel(z,i) = _Flxq_powu(&D, gel(z,i-1), p);
3966 20 : return z;
3967 : }
3968 :
3969 : GEN
3970 7194 : gener_Flxq(GEN T, ulong p, GEN *po)
3971 : {
3972 7194 : long i, j, vT = get_Flx_var(T), f = get_Flx_degree(T);
3973 : ulong p_1, pi;
3974 : GEN g, L, L2, o, q, F;
3975 : pari_sp av0, av;
3976 :
3977 7194 : if (f == 1) {
3978 : GEN fa;
3979 28 : o = utoipos(p-1);
3980 28 : fa = Z_factor(o);
3981 28 : L = gel(fa,1);
3982 28 : L = vecslice(L, 2, lg(L)-1); /* remove 2 for efficiency */
3983 28 : g = Fl_to_Flx(pgener_Fl_local(p, vec_to_vecsmall(L)), vT);
3984 28 : if (po) *po = mkvec2(o, fa);
3985 28 : return g;
3986 : }
3987 :
3988 7166 : av0 = avma; p_1 = p - 1;
3989 7166 : q = diviuexact(subiu(powuu(p,f), 1), p_1);
3990 :
3991 7166 : L = cgetg(1, t_VECSMALL);
3992 7166 : if (p > 3)
3993 : {
3994 2371 : ulong t = p_1 >> vals(p_1);
3995 2371 : GEN P = gel(factoru(t), 1);
3996 2371 : L = cgetg_copy(P, &i);
3997 3787 : while (--i) L[i] = p_1 / P[i];
3998 : }
3999 7166 : o = factor_pn_1(utoipos(p),f);
4000 7166 : L2 = leafcopy( gel(o, 1) );
4001 19198 : for (i = j = 1; i < lg(L2); i++)
4002 : {
4003 12032 : if (umodui(p_1, gel(L2,i)) == 0) continue;
4004 6488 : gel(L2,j++) = diviiexact(q, gel(L2,i));
4005 : }
4006 7166 : setlg(L2, j); pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4007 7166 : F = Flx_Frobenius_pre(T, p, pi);
4008 17686 : for (av = avma;; set_avma(av))
4009 10520 : {
4010 : GEN tt;
4011 17686 : g = random_Flx(f, vT, p);
4012 17686 : if (degpol(g) < 1) continue;
4013 12104 : if (p == 2) tt = g;
4014 : else
4015 : {
4016 8905 : ulong t = Flxq_norm(g, T, p);
4017 8905 : if (t == 1 || !is_gener_Fl(t, p, p_1, L)) continue;
4018 4765 : tt = Flxq_powu_pre(g, p_1>>1, T, p, pi);
4019 : }
4020 14578 : for (i = 1; i < j; i++)
4021 : {
4022 7412 : GEN a = Flxq_pow_Frobenius(tt, gel(L2,i), F, T, p, pi);
4023 7412 : if (!degpol(a) && uel(a,2) == p_1) break;
4024 : }
4025 7964 : if (i == j) break;
4026 : }
4027 7166 : if (!po)
4028 : {
4029 187 : set_avma((pari_sp)g);
4030 187 : g = gc_leaf(av0, g);
4031 : }
4032 : else {
4033 6979 : *po = mkvec2(subiu(powuu(p,f), 1), o);
4034 6979 : (void)gc_all(av0, 2, &g, po);
4035 : }
4036 7166 : return g;
4037 : }
4038 :
4039 : static GEN
4040 35607 : _Flxq_neg(void *E, GEN x)
4041 35607 : { struct _Flxq *s = (struct _Flxq *)E;
4042 35607 : return Flx_neg(x,s->p); }
4043 :
4044 : static GEN
4045 1260698 : _Flxq_rmul(void *E, GEN x, GEN y)
4046 1260698 : { struct _Flxq *s = (struct _Flxq *)E;
4047 1260698 : return Flx_mul_pre(x,y,s->p,s->pi); }
4048 :
4049 : static GEN
4050 1532 : _Flxq_inv(void *E, GEN x)
4051 1532 : { struct _Flxq *s = (struct _Flxq *)E;
4052 1532 : return Flxq_inv(x,s->T,s->p); }
4053 :
4054 : static int
4055 6540 : _Flxq_equal0(GEN x) { return lgpol(x)==0; }
4056 :
4057 : static GEN
4058 1393 : _Flxq_s(void *E, long x)
4059 1393 : { struct _Flxq *s = (struct _Flxq *)E;
4060 1393 : ulong u = x<0 ? s->p+x: (ulong)x;
4061 1393 : return Fl_to_Flx(u, get_Flx_var(s->T));
4062 : }
4063 :
4064 : static const struct bb_field Flxq_field={_Flxq_red,_Flx_add,_Flxq_rmul,_Flxq_neg,
4065 : _Flxq_inv,_Flxq_equal0,_Flxq_s};
4066 :
4067 66744 : const struct bb_field *get_Flxq_field(void **E, GEN T, ulong p)
4068 : {
4069 66744 : GEN z = new_chunk(sizeof(struct _Flxq));
4070 66744 : set_Flxq((struct _Flxq *)z, T, p); *E = (void*)z; return &Flxq_field;
4071 : }
4072 :
4073 : /***********************************************************************/
4074 : /** Flxn **/
4075 : /***********************************************************************/
4076 :
4077 : GEN
4078 54385 : Flx_invLaplace(GEN x, ulong p)
4079 : {
4080 54385 : long i, d = degpol(x);
4081 : ulong t;
4082 : GEN y;
4083 54384 : if (d <= 1) return Flx_copy(x);
4084 54384 : t = Fl_inv(factorial_Fl(d, p), p);
4085 54417 : y = cgetg(d+3, t_VECSMALL);
4086 54386 : y[1] = x[1];
4087 1332654 : for (i=d; i>=2; i--)
4088 : {
4089 1278250 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
4090 1278248 : t = Fl_mul(t, i, p);
4091 : }
4092 54404 : uel(y,3) = uel(x,3);
4093 54404 : uel(y,2) = uel(x,2);
4094 54404 : return y;
4095 : }
4096 :
4097 : GEN
4098 27346 : Flx_Laplace(GEN x, ulong p)
4099 : {
4100 27346 : long i, d = degpol(x);
4101 27345 : ulong t = 1;
4102 : GEN y;
4103 27345 : if (d <= 1) return Flx_copy(x);
4104 27345 : y = cgetg(d+3, t_VECSMALL);
4105 27334 : y[1] = x[1];
4106 27334 : uel(y,2) = uel(x,2);
4107 27334 : uel(y,3) = uel(x,3);
4108 761161 : for (i=2; i<=d; i++)
4109 : {
4110 733802 : t = Fl_mul(t, i%p, p);
4111 733811 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
4112 : }
4113 27359 : return y;
4114 : }
4115 :
4116 : GEN
4117 6324649 : Flxn_red(GEN a, long n)
4118 : {
4119 6324649 : long i, L, l = lg(a);
4120 : GEN b;
4121 6324649 : if (l == 2 || !n) return zero_Flx(a[1]);
4122 5933261 : L = n+2; if (L > l) L = l;
4123 5933261 : b = cgetg(L, t_VECSMALL); b[1] = a[1];
4124 59681175 : for (i=2; i<L; i++) b[i] = a[i];
4125 5931210 : return Flx_renormalize(b,L);
4126 : }
4127 :
4128 : GEN
4129 5152865 : Flxn_mul_pre(GEN a, GEN b, long n, ulong p, ulong pi)
4130 5152865 : { return Flxn_red(Flx_mul_pre(a, b, p, pi), n); }
4131 : GEN
4132 75845 : Flxn_mul(GEN a, GEN b, long n, ulong p)
4133 75845 : { return Flxn_mul_pre(a, b, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4134 :
4135 : GEN
4136 0 : Flxn_sqr_pre(GEN a, long n, ulong p, ulong pi)
4137 0 : { return Flxn_red(Flx_sqr_pre(a, p, pi), n); }
4138 : GEN
4139 0 : Flxn_sqr(GEN a, long n, ulong p)
4140 0 : { return Flxn_sqr_pre(a, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4141 :
4142 : /* (f*g) \/ x^n */
4143 : static GEN
4144 938974 : Flx_mulhigh_i(GEN f, GEN g, long n, ulong p, ulong pi)
4145 938974 : { return Flx_shift(Flx_mul_pre(f, g, p, pi),-n); }
4146 :
4147 : static GEN
4148 517019 : Flxn_mulhigh(GEN f, GEN g, long n2, long n, ulong p, ulong pi)
4149 : {
4150 517019 : GEN F = Flx_blocks(f, n2, 2), fl = gel(F,1), fh = gel(F,2);
4151 516758 : return Flx_add(Flx_mulhigh_i(fl, g, n2, p, pi),
4152 : Flxn_mul_pre(fh, g, n - n2, p, pi), p);
4153 : }
4154 :
4155 : /* g==NULL -> assume g==1 */
4156 : GEN
4157 55195 : Flxn_div_pre(GEN g, GEN f, long e, ulong p, ulong pi)
4158 : {
4159 55195 : pari_sp av = avma, av2;
4160 : ulong mask;
4161 : GEN W;
4162 55195 : long n = 1;
4163 55195 : if (lg(f) <= 2) pari_err_INV("Flxn_inv",f);
4164 55195 : W = Fl_to_Flx(Fl_inv(uel(f,2),p), f[1]);
4165 55206 : mask = quadratic_prec_mask(e);
4166 55208 : av2 = avma;
4167 259169 : for (;mask>1;)
4168 : {
4169 : GEN u, fr;
4170 203939 : long n2 = n;
4171 203939 : n<<=1; if (mask & 1) n--;
4172 203939 : mask >>= 1;
4173 203939 : fr = Flxn_red(f, n);
4174 203808 : if (mask>1 || !g)
4175 : {
4176 149667 : u = Flxn_mul_pre(W, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
4177 149952 : W = Flx_sub(W, Flx_shift(u, n2), p);
4178 : } else
4179 : {
4180 54141 : GEN y = Flxn_mul_pre(g, W, n, p, pi), yt = Flxn_red(y, n-n2);
4181 54112 : u = Flxn_mul_pre(yt, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
4182 54137 : W = Flx_sub(y, Flx_shift(u, n2), p);
4183 : }
4184 203971 : if (gc_needed(av2,2))
4185 : {
4186 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxn_div, e = %ld", n);
4187 0 : W = gc_upto(av2, W);
4188 : }
4189 : }
4190 55230 : return gc_upto(av, W);
4191 : }
4192 : GEN
4193 55176 : Flxn_div(GEN g, GEN f, long e, ulong p)
4194 55176 : { return Flxn_div_pre(g, f, e, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4195 :
4196 : GEN
4197 1037 : Flxn_inv(GEN f, long e, ulong p)
4198 1037 : { return Flxn_div(NULL, f, e, p); }
4199 :
4200 : GEN
4201 109397 : Flxn_expint(GEN h, long e, ulong p)
4202 : {
4203 109397 : pari_sp av = avma, av2;
4204 109397 : long v = h[1], n=1;
4205 109397 : GEN f = pol1_Flx(v), g = pol1_Flx(v);
4206 109366 : ulong mask = quadratic_prec_mask(e), pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4207 109374 : av2 = avma;
4208 422801 : for (;mask>1;)
4209 : {
4210 : GEN u, w;
4211 422756 : long n2 = n;
4212 422756 : n<<=1; if (mask & 1) n--;
4213 422756 : mask >>= 1;
4214 422756 : u = Flxn_mul_pre(g, Flx_mulhigh_i(f, Flxn_red(h, n2-1), n2-1, p,pi), n-n2, p,pi);
4215 422734 : u = Flx_add(u, Flx_shift(Flxn_red(h, n-1), 1-n2), p);
4216 422782 : w = Flxn_mul_pre(f, Flx_integXn(u, n2-1, p), n-n2, p, pi);
4217 422714 : f = Flx_add(f, Flx_shift(w, n2), p);
4218 422828 : if (mask<=1) break;
4219 313445 : u = Flxn_mul_pre(g, Flxn_mulhigh(f, g, n2, n, p, pi), n-n2, p, pi);
4220 313416 : g = Flx_sub(g, Flx_shift(u, n2), p);
4221 313427 : if (gc_needed(av2,2))
4222 : {
4223 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flxn_exp, e = %ld", n);
4224 0 : (void)gc_all(av2, 2, &f, &g);
4225 : }
4226 : }
4227 109428 : return gc_upto(av, f);
4228 : }
4229 :
4230 : GEN
4231 0 : Flxn_exp(GEN h, long e, ulong p)
4232 : {
4233 0 : if (degpol(h)<1 || uel(h,2)!=0)
4234 0 : pari_err_DOMAIN("Flxn_exp","valuation", "<", gen_1, h);
4235 0 : return Flxn_expint(Flx_deriv(h, p), e, p);
4236 : }
4237 :
4238 : INLINE GEN
4239 217365 : Flxn_recip(GEN x, long n)
4240 : {
4241 217365 : GEN z=Flx_recipspec(x+2,lgpol(x),n);
4242 217244 : z[1]=x[1];
4243 217244 : return z;
4244 : }
4245 :
4246 : GEN
4247 54101 : Flx_Newton(GEN P, long n, ulong p)
4248 : {
4249 54101 : pari_sp av = avma;
4250 54101 : long d = degpol(P);
4251 54091 : GEN dP = Flxn_recip(Flx_deriv(P, p), d);
4252 54022 : GEN Q = Flxn_div(dP, Flxn_recip(P, d+1), n, p);
4253 54089 : return gc_leaf(av, Q);
4254 : }
4255 :
4256 : GEN
4257 109401 : Flx_fromNewton(GEN P, ulong p)
4258 : {
4259 109401 : pari_sp av = avma;
4260 109401 : ulong n = Flx_constant(P)+1;
4261 109401 : GEN z = Flx_neg(Flx_shift(P, -1), p);
4262 109397 : GEN Q = Flxn_recip(Flxn_expint(z, n, p), n);
4263 109364 : return gc_leaf(av, Q);
4264 : }
4265 :
4266 : static void
4267 12514 : init_invlaplace(long d, ulong p, GEN *pt_P, GEN *pt_V)
4268 : {
4269 : long i;
4270 : ulong e;
4271 12514 : GEN P = cgetg(d+1, t_VECSMALL);
4272 12514 : GEN V = cgetg(d+1, t_VECSMALL);
4273 1396581 : for (i=1, e=1; i<=d; i++, e++)
4274 : {
4275 1384067 : if (e==p)
4276 : {
4277 459153 : e = 0;
4278 459153 : V[i] = u_lvalrem(i, p, &uel(P,i));
4279 : } else
4280 : {
4281 924914 : V[i] = 0; uel(P,i) = i;
4282 : }
4283 : }
4284 12514 : *pt_P = P; *pt_V = V;
4285 12514 : }
4286 :
4287 : /* return p^val * FpX_invLaplace(1+x+...x^(n-1), q), with q a power of p and
4288 : * val large enough to compensate for the power of p in the factorials */
4289 :
4290 : static GEN
4291 497 : ZpX_invLaplace_init(long n, GEN q, ulong p, long v, long sv)
4292 : {
4293 497 : pari_sp av = avma;
4294 497 : long i, d = n-1, w;
4295 : GEN y, W, E, t;
4296 497 : init_invlaplace(d, p, &E, &W);
4297 497 : t = Fp_inv(FpV_prod(Flv_to_ZV(E), q), q);
4298 497 : w = zv_sum(W);
4299 497 : if (v > w) t = Fp_mul(t, powuu(p, v-w), q);
4300 497 : y = cgetg(d+3,t_POL);
4301 497 : y[1] = evalsigne(1) | sv;
4302 28882 : for (i=d; i>=1; i--)
4303 : {
4304 28385 : gel(y,i+2) = t;
4305 28385 : t = Fp_mulu(t, uel(E,i), q);
4306 28385 : if (uel(W,i)) t = Fp_mul(t, powuu(p, uel(W,i)), q);
4307 : }
4308 497 : gel(y,2) = t;
4309 497 : return gc_GEN(av, ZX_renormalize(y, d+3));
4310 : }
4311 :
4312 : GEN
4313 27542 : Flx_composedsum(GEN P, GEN Q, ulong p)
4314 : {
4315 27542 : pari_sp av = avma;
4316 27542 : long n = 1 + degpol(P)*degpol(Q);
4317 27536 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4318 27537 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4319 : GEN R;
4320 27543 : if (p >= (ulong)n)
4321 : {
4322 27046 : GEN Pl = Flx_invLaplace(Flx_Newton(P,n,p), p);
4323 27058 : GEN Ql = Flx_invLaplace(Flx_Newton(Q,n,p), p);
4324 27055 : GEN L = Flx_Laplace(Flxn_mul(Pl, Ql, n, p), p);
4325 27057 : R = Flx_fromNewton(L, p);
4326 : } else
4327 : {
4328 497 : long v = factorial_lval(n-1, p);
4329 497 : long w = 1 + ulogint(n-1, p);
4330 497 : GEN pv = powuu(p, v);
4331 497 : GEN qf = powuu(p, w), q = mulii(pv, qf), q2 = mulii(q, pv);
4332 497 : GEN iL = ZpX_invLaplace_init(n, q, p, v, P[1]);
4333 497 : GEN Pl = FpX_convol(iL, FpX_Newton(Flx_to_ZX(P), n, qf), q);
4334 497 : GEN Ql = FpX_convol(iL, FpX_Newton(Flx_to_ZX(Q), n, qf), q);
4335 497 : GEN Ln = ZX_Z_divexact(FpXn_mul(Pl, Ql, n, q2), pv);
4336 497 : GEN L = ZX_Z_divexact(FpX_Laplace(Ln, q), pv);
4337 497 : R = ZX_to_Flx(FpX_fromNewton(L, qf), p);
4338 : }
4339 27530 : return gc_leaf(av, Flx_Fl_mul(R, lead, p));
4340 : }
4341 :
4342 : static GEN
4343 3882 : _Flx_composedsum(void *E, GEN a, GEN b)
4344 3882 : { return Flx_composedsum(a, b, (ulong)E); }
4345 :
4346 : GEN
4347 28961 : FlxV_composedsum(GEN V, ulong p)
4348 28961 : { return gen_product(V, (void *)p, &_Flx_composedsum); }
4349 :
4350 : GEN
4351 0 : Flx_composedprod(GEN P, GEN Q, ulong p)
4352 : {
4353 0 : pari_sp av = avma;
4354 0 : long n = 1+ degpol(P)*degpol(Q);
4355 0 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4356 0 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4357 : GEN R;
4358 0 : if (p >= (ulong)n)
4359 : {
4360 0 : GEN L = Flx_convol(Flx_Newton(P,n,p), Flx_Newton(Q,n,p), p);
4361 0 : R = Flx_fromNewton(L, p);
4362 : } else
4363 : {
4364 0 : long w = 1 + ulogint(n, p);
4365 0 : GEN qf = powuu(p, w);
4366 0 : GEN Pl = FpX_convol(FpX_Newton(Flx_to_ZX(P), n, qf), FpX_Newton(Flx_to_ZX(Q), n, qf), qf);
4367 0 : R = ZX_to_Flx(FpX_fromNewton(Pl, qf), p);
4368 : }
4369 0 : return gc_leaf(av, Flx_Fl_mul(R, lead, p));
4370 :
4371 : }
4372 :
4373 : /* (x+1)^n mod p; assume 2 <= n < 2p prime */
4374 : static GEN
4375 0 : Fl_Xp1_powu(ulong n, ulong p, long v)
4376 : {
4377 0 : ulong k, d = (n + 1) >> 1;
4378 0 : GEN C, V = identity_zv(d);
4379 :
4380 0 : Flv_inv_inplace(V, p); /* could restrict to odd integers in [3,d] */
4381 0 : C = cgetg(n+3, t_VECSMALL);
4382 0 : C[1] = v;
4383 0 : uel(C,2) = 1UL;
4384 0 : uel(C,3) = n%p;
4385 0 : uel(C,4) = Fl_mul(odd(n)? n: n-1, n >> 1, p);
4386 : /* binom(n,k) = binom(n,k-1) * (n-k+1) / k */
4387 0 : if (SMALL_ULONG(p))
4388 0 : for (k = 3; k <= d; k++)
4389 0 : uel(C,k+2) = Fl_mul(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p);
4390 : else
4391 : {
4392 0 : ulong pi = get_Fl_red(p);
4393 0 : for (k = 3; k <= d; k++)
4394 0 : uel(C,k+2) = Fl_mul_pre(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p, pi);
4395 : }
4396 0 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4397 0 : return C; /* normalized */
4398 : }
4399 :
4400 : /* p arbitrary */
4401 : GEN
4402 29202 : Flx_translate1_basecase(GEN P, ulong p)
4403 : {
4404 29202 : GEN R = Flx_copy(P);
4405 29202 : long i, k, n = degpol(P);
4406 659170 : for (i = 1; i <= n; i++)
4407 14859060 : for (k = n-i; k < n; k++) uel(R,k+2) = Fl_add(uel(R,k+2), uel(R,k+3), p);
4408 29202 : return R;
4409 : }
4410 :
4411 : static int
4412 42367 : translate_basecase(long n, ulong p)
4413 : {
4414 : #ifdef LONG_IS_64BIT
4415 36930 : if (p <= 19) return n < 40;
4416 30522 : if (p < 1UL<<30) return n < 58;
4417 0 : if (p < 1UL<<59) return n < 100;
4418 0 : if (p < 1UL<<62) return n < 120;
4419 0 : if (p < 1UL<<63) return n < 240;
4420 0 : return n < 250;
4421 : #else
4422 5437 : if (p <= 13) return n < 18;
4423 4250 : if (p <= 17) return n < 22;
4424 4186 : if (p <= 29) return n < 39;
4425 3976 : if (p <= 67) return n < 69;
4426 3703 : if (p < 1UL<< 15) return n < 80;
4427 2047 : if (p < 1UL<< 16) return n < 100;
4428 0 : if (p < 1UL<< 28) return n < 300;
4429 0 : return n < 650;
4430 : #endif
4431 : }
4432 : /* assume p prime */
4433 : GEN
4434 17108 : Flx_translate1(GEN P, ulong p)
4435 : {
4436 17108 : long d, n = degpol(P);
4437 : GEN R, Q, S;
4438 17108 : if (translate_basecase(n, p)) return Flx_translate1_basecase(P, p);
4439 : /* n > 0 */
4440 1148 : d = n >> 1;
4441 1148 : if ((ulong)n < p)
4442 : {
4443 0 : R = Flx_translate1(Flxn_red(P, d), p);
4444 0 : Q = Flx_translate1(Flx_shift(P, -d), p);
4445 0 : S = Fl_Xp1_powu(d, p, P[1]);
4446 0 : return Flx_add(Flx_mul(Q, S, p), R, p);
4447 : }
4448 : else
4449 : {
4450 : ulong q;
4451 1148 : if ((ulong)d > p) (void)ulogintall(d, p, &q); else q = p;
4452 1148 : R = Flx_translate1(Flxn_red(P, q), p);
4453 1148 : Q = Flx_translate1(Flx_shift(P, -q), p);
4454 1148 : S = Flx_add(Flx_shift(Q, q), Q, p);
4455 1148 : return Flx_add(S, R, p); /* P(x+1) = Q(x+1) (x^q+1) + R(x+1) */
4456 : }
4457 : }
4458 :
4459 : GEN
4460 0 : Flx_Fl_translate(GEN P, ulong c, ulong p)
4461 : {
4462 0 : pari_sp av = avma;
4463 : GEN Q;
4464 0 : if (c==0) return Flx_copy(P);
4465 0 : if (c==1) return Flx_translate1(P, p);
4466 0 : Q = Flx_unscale(Flx_translate1(Flx_unscale(P, c, p), p), Fl_inv(c, p), p);
4467 0 : return gc_leaf(av, Q);
4468 : }
4469 :
4470 : static GEN
4471 12017 : zl_Xp1_powu(ulong n, ulong p, ulong q, long e, long vs)
4472 : {
4473 12017 : ulong k, d = n >> 1, c, v = 0;
4474 12017 : GEN C, V, W, U = upowers(p, e-1);
4475 12017 : init_invlaplace(d, p, &V, &W);
4476 12017 : Flv_inv_inplace(V, q);
4477 12017 : C = cgetg(n+3, t_VECSMALL);
4478 12017 : C[1] = vs;
4479 12017 : uel(C,2) = 1UL;
4480 12017 : uel(C,3) = n%q;
4481 12017 : v = u_lvalrem(n, p, &c);
4482 1355682 : for (k = 2; k <= d; k++)
4483 : {
4484 : ulong w;
4485 1343665 : v += u_lvalrem(n-k+1, p, &w) - W[k];
4486 1343665 : c = Fl_mul(Fl_mul(w%q, c, q), uel(V,k), q);
4487 1343665 : uel(C,2+k) = v >= (ulong)e ? 0: v==0 ? c : Fl_mul(c, uel(U, v+1), q);
4488 : }
4489 1374521 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4490 12017 : return C; /* normalized */
4491 : }
4492 :
4493 : GEN
4494 25259 : zlx_translate1(GEN P, ulong p, long e)
4495 : {
4496 25259 : ulong d, q = upowuu(p,e), n = degpol(P);
4497 : GEN R, Q, S;
4498 25259 : if (translate_basecase(n, q)) return Flx_translate1_basecase(P, q);
4499 : /* n > 0 */
4500 12017 : d = n >> 1;
4501 12017 : R = zlx_translate1(Flxn_red(P, d), p, e);
4502 12017 : Q = zlx_translate1(Flx_shift(P, -d), p, e);
4503 12017 : S = zl_Xp1_powu(d, p, q, e, P[1]);
4504 12017 : return Flx_add(Flx_mul(Q, S, q), R, q);
4505 : }
4506 :
4507 : /***********************************************************************/
4508 : /** Fl2 **/
4509 : /***********************************************************************/
4510 : /* Fl2 objects are Flv of length 2 [a,b] representing a+bsqrt(D) for
4511 : * a nonsquare D. */
4512 :
4513 : INLINE GEN
4514 7558683 : mkF2(ulong a, ulong b) { return mkvecsmall2(a,b); }
4515 :
4516 : /* allow pi = 0 */
4517 : GEN
4518 2014572 : Fl2_mul_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4519 : {
4520 : ulong xaya, xbyb, Db2, mid, z1, z2;
4521 2014572 : ulong x1 = x[1], x2 = x[2], y1 = y[1], y2 = y[2];
4522 2014572 : if (pi)
4523 : {
4524 2014594 : xaya = Fl_mul_pre(x1,y1,p,pi);
4525 2014967 : if (x2==0 && y2==0) return mkF2(xaya,0);
4526 1938938 : if (x2==0) return mkF2(xaya,Fl_mul_pre(x1,y2,p,pi));
4527 1913206 : if (y2==0) return mkF2(xaya,Fl_mul_pre(x2,y1,p,pi));
4528 1912952 : xbyb = Fl_mul_pre(x2,y2,p,pi);
4529 1912816 : mid = Fl_mul_pre(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p,pi);
4530 1913014 : Db2 = Fl_mul_pre(D, xbyb, p,pi);
4531 : }
4532 0 : else if (p & HIGHMASK)
4533 : {
4534 0 : xaya = Fl_mul(x1,y1,p);
4535 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4536 0 : if (x2==0) return mkF2(xaya,Fl_mul(x1,y2,p));
4537 0 : if (y2==0) return mkF2(xaya,Fl_mul(x2,y1,p));
4538 0 : xbyb = Fl_mul(x2,y2,p);
4539 0 : mid = Fl_mul(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p);
4540 0 : Db2 = Fl_mul(D, xbyb, p);
4541 : }
4542 : else
4543 : {
4544 0 : xaya = (x1 * y1) % p;
4545 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4546 0 : if (x2==0) return mkF2(xaya, (x1 * y2) % p);
4547 0 : if (y2==0) return mkF2(xaya, (x2 * y1) % p);
4548 0 : xbyb = (x2 * y2) % p;
4549 0 : mid = (Fl_add(x1,x2,p) * Fl_add(y1,y2,p)) % p;
4550 0 : Db2 = (D * xbyb) % p;
4551 : }
4552 1912930 : z1 = Fl_add(xaya,Db2,p);
4553 1912928 : z2 = Fl_sub(mid,Fl_add(xaya,xbyb,p),p);
4554 1912935 : return mkF2(z1,z2);
4555 : }
4556 :
4557 : /* allow pi = 0 */
4558 : GEN
4559 5074028 : Fl2_sqr_pre(GEN x, ulong D, ulong p, ulong pi)
4560 : {
4561 5074028 : ulong a = x[1], b = x[2];
4562 : ulong a2, Db2, ab;
4563 5074028 : if (pi)
4564 : {
4565 5074105 : a2 = Fl_sqr_pre(a,p,pi);
4566 5075993 : if (b==0) return mkF2(a2,0);
4567 4839744 : Db2= Fl_mul_pre(D, Fl_sqr_pre(b,p,pi), p,pi);
4568 4839907 : ab = Fl_mul_pre(a,b,p,pi);
4569 : }
4570 0 : else if (p & HIGHMASK)
4571 : {
4572 0 : a2 = Fl_sqr(a,p);
4573 0 : if (b==0) return mkF2(a2,0);
4574 0 : Db2= Fl_mul(D, Fl_sqr(b,p), p);
4575 0 : ab = Fl_mul(a,b,p);
4576 : }
4577 : else
4578 : {
4579 0 : a2 = (a * a) % p;
4580 0 : if (b==0) return mkF2(a2,0);
4581 0 : Db2= (D * ((b * b) % p)) % p;
4582 0 : ab = (a * b) % p;
4583 : }
4584 4839676 : return mkF2(Fl_add(a2,Db2,p), Fl_double(ab,p));
4585 : }
4586 :
4587 : /* allow pi = 0 */
4588 : ulong
4589 128169 : Fl2_norm_pre(GEN x, ulong D, ulong p, ulong pi)
4590 : {
4591 128169 : ulong a = x[1], b = x[2], a2;
4592 128169 : if (pi)
4593 : {
4594 76290 : a2 = Fl_sqr_pre(a,p,pi);
4595 76290 : return b? Fl_sub(a2, Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p): a2;
4596 : }
4597 51879 : else if (p & HIGHMASK)
4598 : {
4599 0 : a2 = Fl_sqr(a,p);
4600 0 : return b? Fl_sub(a2, Fl_mul(D, Fl_sqr(b, p), p), p): a2;
4601 : }
4602 : else
4603 : {
4604 51879 : a2 = (a * a) % p;
4605 51879 : return b? Fl_sub(a2, (D * ((b * b) % p)) % p, p): a2;
4606 : }
4607 : }
4608 :
4609 : /* allow pi = 0 */
4610 : GEN
4611 202095 : Fl2_inv_pre(GEN x, ulong D, ulong p, ulong pi)
4612 : {
4613 202095 : ulong a = x[1], b = x[2], n, ni;
4614 202095 : if (b == 0) return mkF2(Fl_inv(a,p), 0);
4615 168193 : b = Fl_neg(b, p);
4616 168195 : if (pi)
4617 : {
4618 168195 : n = Fl_sub(Fl_sqr_pre(a, p,pi),
4619 : Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p);
4620 168194 : ni = Fl_inv(n,p);
4621 168197 : return mkF2(Fl_mul_pre(a, ni, p,pi), Fl_mul_pre(b, ni, p,pi));
4622 : }
4623 0 : else if (p & HIGHMASK)
4624 : {
4625 0 : n = Fl_sub(Fl_sqr(a, p), Fl_mul(D, Fl_sqr(b, p), p), p);
4626 0 : ni = Fl_inv(n,p);
4627 0 : return mkF2(Fl_mul(a, ni, p), Fl_mul(b, ni, p));
4628 : }
4629 : else
4630 : {
4631 0 : n = Fl_sub((a * a) % p, (D * ((b * b) % p)) % p, p);
4632 0 : ni = Fl_inv(n,p);
4633 0 : return mkF2((a * ni) % p, (b * ni) % p);
4634 : }
4635 : }
4636 :
4637 : int
4638 462956 : Fl2_equal1(GEN x) { return x[1]==1 && x[2]==0; }
4639 :
4640 : struct _Fl2 {
4641 : ulong p, pi, D;
4642 : };
4643 :
4644 : static GEN
4645 5074022 : _Fl2_sqr(void *data, GEN x)
4646 : {
4647 5074022 : struct _Fl2 *D = (struct _Fl2*)data;
4648 5074022 : return Fl2_sqr_pre(x, D->D, D->p, D->pi);
4649 : }
4650 : static GEN
4651 1985544 : _Fl2_mul(void *data, GEN x, GEN y)
4652 : {
4653 1985544 : struct _Fl2 *D = (struct _Fl2*)data;
4654 1985544 : return Fl2_mul_pre(x,y, D->D, D->p, D->pi);
4655 : }
4656 :
4657 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL; allow pi = 0 */
4658 : GEN
4659 690574 : Fl2_pow_pre(GEN x, GEN n, ulong D, ulong p, ulong pi)
4660 : {
4661 690574 : pari_sp av = avma;
4662 : struct _Fl2 d;
4663 : GEN y;
4664 690574 : long s = signe(n);
4665 690574 : if (!s) return mkF2(1,0);
4666 612182 : if (s < 0)
4667 202096 : x = Fl2_inv_pre(x,D,p,pi);
4668 612178 : if (is_pm1(n)) return s < 0 ? x : zv_copy(x);
4669 451891 : d.p = p; d.pi = pi; d.D=D;
4670 451891 : y = gen_pow_i(x, n, (void*)&d, &_Fl2_sqr, &_Fl2_mul);
4671 451938 : return gc_leaf(av, y);
4672 : }
4673 :
4674 : static GEN
4675 690561 : _Fl2_pow(void *data, GEN x, GEN n)
4676 : {
4677 690561 : struct _Fl2 *D = (struct _Fl2*)data;
4678 690561 : return Fl2_pow_pre(x, n, D->D, D->p, D->pi);
4679 : }
4680 :
4681 : static GEN
4682 118287 : _Fl2_rand(void *data)
4683 : {
4684 118287 : struct _Fl2 *D = (struct _Fl2*)data;
4685 118287 : ulong a = random_Fl(D->p), b=random_Fl(D->p-1)+1;
4686 118288 : return mkF2(a,b);
4687 : }
4688 :
4689 : GEN
4690 65765 : Fl2_sqrt_pre(GEN z, ulong D, ulong p, ulong pi)
4691 : {
4692 65765 : ulong a = uel(z,1), b = uel(z,2), as2, u, v, s;
4693 65765 : ulong y = Fl_2gener_pre_i(D, p, pi);
4694 65765 : if (b == 0)
4695 18930 : return krouu(a, p)==1 ? mkF2(Fl_sqrt_pre_i(a, y, p, pi), 0)
4696 18930 : : mkF2(0, Fl_sqrt_pre_i(Fl_div(a, D, p), y, p, pi));
4697 52709 : s = Fl_sqrt_pre_i(Fl2_norm_pre(z, D, p, pi), y, p, pi);
4698 52709 : if (s==~0UL) return NULL;
4699 49535 : as2 = Fl_halve(Fl_add(a, s, p), p);
4700 49535 : if (krouu(as2, p)==-1) as2 = Fl_sub(as2, s, p);
4701 49535 : u = Fl_sqrt_pre_i(as2, y, p, pi);
4702 49535 : v = Fl_div(b, Fl_double(u, p), p);
4703 49535 : return mkF2(u,v);
4704 : }
4705 :
4706 : static const struct bb_group Fl2_star={_Fl2_mul, _Fl2_pow, _Fl2_rand,
4707 : hash_GEN, zv_equal, Fl2_equal1, NULL};
4708 :
4709 : /* allow pi = 0 */
4710 : GEN
4711 78393 : Fl2_sqrtn_pre(GEN a, GEN n, ulong D, ulong p, ulong pi, GEN *zeta)
4712 : {
4713 : struct _Fl2 E;
4714 : GEN o;
4715 78393 : if (a[1]==0 && a[2]==0)
4716 : {
4717 0 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
4718 0 : if (zeta) *zeta=mkF2(1,0);
4719 0 : return zv_copy(a);
4720 : }
4721 78393 : E.p=p; E.pi = pi; E.D = D;
4722 78393 : o = subiu(powuu(p,2), 1);
4723 78392 : return gen_Shanks_sqrtn(a,n,o,zeta,(void*)&E,&Fl2_star);
4724 : }
4725 :
4726 : /* allow pi = 0 */
4727 : GEN
4728 10773 : Flx_Fl2_eval_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4729 : {
4730 : GEN p1;
4731 10773 : long i = lg(x)-1;
4732 10773 : if (i <= 2)
4733 2177 : return mkF2(i == 2? x[2]: 0, 0);
4734 8596 : p1 = mkF2(x[i], 0);
4735 37625 : for (i--; i>=2; i--)
4736 : {
4737 29029 : p1 = Fl2_mul_pre(p1, y, D, p, pi);
4738 29029 : uel(p1,1) = Fl_add(uel(p1,1), uel(x,i), p);
4739 : }
4740 8596 : return p1;
4741 : }
4742 :
4743 : /***********************************************************************/
4744 : /** FlxV **/
4745 : /***********************************************************************/
4746 : /* FlxV are t_VEC with Flx coefficients. */
4747 :
4748 : GEN
4749 34356 : FlxV_Flc_mul(GEN V, GEN W, ulong p)
4750 : {
4751 34356 : pari_sp ltop=avma;
4752 : long i;
4753 34356 : GEN z = Flx_Fl_mul(gel(V,1),W[1],p);
4754 255934 : for(i=2;i<lg(V);i++)
4755 221578 : z=Flx_add(z,Flx_Fl_mul(gel(V,i),W[i],p),p);
4756 34356 : return gc_leaf(ltop,z);
4757 : }
4758 :
4759 : GEN
4760 0 : ZXV_to_FlxV(GEN x, ulong p)
4761 0 : { pari_APPLY_type(t_VEC, ZX_to_Flx(gel(x,i), p)) }
4762 :
4763 : GEN
4764 3805468 : ZXT_to_FlxT(GEN x, ulong p)
4765 : {
4766 3805468 : if (typ(x) == t_POL)
4767 3746774 : return ZX_to_Flx(x, p);
4768 : else
4769 192685 : pari_APPLY_type(t_VEC, ZXT_to_FlxT(gel(x,i), p))
4770 : }
4771 :
4772 : GEN
4773 172280 : FlxV_to_Flm(GEN x, long n)
4774 928631 : { pari_APPLY_type(t_MAT, Flx_to_Flv(gel(x,i), n)) }
4775 :
4776 : GEN
4777 0 : FlxV_red(GEN x, ulong p)
4778 0 : { pari_APPLY_type(t_VEC, Flx_red(gel(x,i), p)) }
4779 :
4780 : GEN
4781 294167 : FlxT_red(GEN x, ulong p)
4782 : {
4783 294167 : if (typ(x) == t_VECSMALL)
4784 197930 : return Flx_red(x, p);
4785 : else
4786 322709 : pari_APPLY_type(t_VEC, FlxT_red(gel(x,i), p))
4787 : }
4788 :
4789 : GEN
4790 113589 : FlxqV_dotproduct_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
4791 : {
4792 113589 : long i, lx = lg(x);
4793 : pari_sp av;
4794 : GEN c;
4795 113589 : if (lx == 1) return pol0_Flx(get_Flx_var(T));
4796 113589 : av = avma; c = Flx_mul_pre(gel(x,1),gel(y,1), p, pi);
4797 464499 : for (i=2; i<lx; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4798 113589 : return gc_leaf(av, Flx_rem_pre(c,T,p,pi));
4799 : }
4800 : GEN
4801 0 : FlxqV_dotproduct(GEN x, GEN y, GEN T, ulong p)
4802 0 : { return FlxqV_dotproduct_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4803 :
4804 : GEN
4805 1526 : FlxqX_dotproduct(GEN x, GEN y, GEN T, ulong p)
4806 : {
4807 1526 : long i, l = minss(lg(x), lg(y));
4808 : ulong pi;
4809 : pari_sp av;
4810 : GEN c;
4811 1526 : if (l == 2) return pol0_Flx(get_Flx_var(T));
4812 1513 : av = avma; pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4813 1513 : c = Flx_mul_pre(gel(x,2),gel(y,2), p, pi);
4814 4494 : for (i=3; i<l; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4815 1513 : return gc_leaf(av, Flx_rem_pre(c,T,p,pi));
4816 : }
4817 :
4818 : /* allow pi = 0 */
4819 : GEN
4820 325154 : FlxC_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4821 : {
4822 325154 : long i, l = lg(z);
4823 325154 : GEN y = cgetg(l, t_VECSMALL);
4824 15065854 : for (i=1; i<l; i++) uel(y,i) = Flx_eval_powers_pre(gel(z,i), x, p, pi);
4825 325059 : return y;
4826 : }
4827 :
4828 : /***********************************************************************/
4829 : /** FlxM **/
4830 : /***********************************************************************/
4831 : /* allow pi = 0 */
4832 : GEN
4833 22302 : FlxM_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4834 : {
4835 22302 : long i, l = lg(z);
4836 22302 : GEN y = cgetg(l, t_MAT);
4837 347456 : for (i=1; i<l; i++) gel(y,i) = FlxC_eval_powers_pre(gel(z,i), x, p, pi);
4838 22303 : return y;
4839 : }
4840 :
4841 : GEN
4842 0 : zero_FlxC(long n, long sv)
4843 : {
4844 0 : GEN x = cgetg(n + 1, t_COL), z = zero_Flx(sv);
4845 : long i;
4846 0 : for (i = 1; i <= n; i++) gel(x, i) = z;
4847 0 : return x;
4848 : }
4849 :
4850 : GEN
4851 0 : FlxC_neg(GEN x, ulong p)
4852 0 : { pari_APPLY_type(t_COL, Flx_neg(gel(x, i), p)) }
4853 :
4854 : GEN
4855 0 : FlxC_sub(GEN x, GEN y, ulong p)
4856 0 : { pari_APPLY_type(t_COL, Flx_sub(gel(x, i), gel(y, i), p)) }
4857 :
4858 : GEN
4859 0 : zero_FlxM(long r, long c, long sv)
4860 : {
4861 0 : GEN x = cgetg(c + 1, t_MAT), z = zero_FlxC(r, sv);
4862 : long j;
4863 0 : for (j = 1; j <= c; j++) gel(x, j) = z;
4864 0 : return x;
4865 : }
4866 :
4867 : GEN
4868 0 : zero_FlxM_copy(long r, long c, long sv)
4869 : {
4870 0 : GEN x = cgetg(c + 1, t_MAT);
4871 : long j;
4872 0 : for (j = 1; j <= c; j++) gel(x, j) = zero_FlxC(r, sv);
4873 0 : return x;
4874 : }
4875 :
4876 : GEN
4877 0 : FlxM_neg(GEN x, ulong p)
4878 0 : { pari_APPLY_same(FlxC_neg(gel(x, i), p)) }
4879 :
4880 : GEN
4881 0 : FlxM_sub(GEN x, GEN y, ulong p)
4882 0 : { pari_APPLY_same(FlxC_sub(gel(x, i), gel(y,i), p)) }
4883 :
4884 : GEN
4885 0 : FlxC_Fl_translate(GEN x, ulong c, ulong p)
4886 0 : { pari_APPLY_type(t_COL, Flx_Fl_translate(gel(x,i), c, p)) }
4887 :
4888 : GEN
4889 0 : FlxM_Fl_translate(GEN x, ulong c, ulong p)
4890 0 : { pari_APPLY_same(FlxC_Fl_translate(gel(x,i), c, p)) }
4891 :
4892 : GEN
4893 170737 : FlxqC_red_pre(GEN x, GEN T, ulong p, ulong pi)
4894 3739496 : { pari_APPLY_type(t_COL, Flx_rem_pre(gel(x,i), T, p, pi)) }
4895 :
4896 : GEN
4897 74264 : FlxqM_red_pre(GEN x, GEN T, ulong p, ulong pi)
4898 245001 : { pari_APPLY_same(FlxqC_red_pre(gel(x,i), T, p, pi)) }
4899 :
4900 : GEN
4901 0 : FlxqC_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4902 0 : { pari_APPLY_type(t_COL, Flxq_mul(gel(x, i), y, T, p)) }
4903 :
4904 : GEN
4905 0 : FlxqM_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4906 0 : { pari_APPLY_same(FlxqC_Flxq_mul(gel(x, i), y, T, p)) }
4907 :
4908 : static GEN
4909 35257 : FlxM_pack_ZM(GEN M, GEN (*pack)(GEN, long)) {
4910 : long i, j, l, lc;
4911 35257 : GEN N = cgetg_copy(M, &l), x;
4912 35257 : if (l == 1)
4913 0 : return N;
4914 35257 : lc = lgcols(M);
4915 120938 : for (j = 1; j < l; j++) {
4916 85681 : gel(N, j) = cgetg(lc, t_COL);
4917 538944 : for (i = 1; i < lc; i++) {
4918 453263 : x = gcoeff(M, i, j);
4919 453263 : gcoeff(N, i, j) = pack(x + 2, lgpol(x));
4920 : }
4921 : }
4922 35257 : return N;
4923 : }
4924 :
4925 : static GEN
4926 414404 : kron_pack_Flx_spec_half(GEN x, long l) {
4927 414404 : if (l == 0) return gen_0;
4928 319068 : return Flx_to_int_halfspec(x, l);
4929 : }
4930 :
4931 : static GEN
4932 35470 : kron_pack_Flx_spec(GEN x, long l) {
4933 : long i;
4934 : GEN w, y;
4935 35470 : if (l == 0)
4936 6545 : return gen_0;
4937 28925 : y = cgetipos(l + 2);
4938 101157 : for (i = 0, w = int_LSW(y); i < l; i++, w = int_nextW(w))
4939 72232 : *w = x[i];
4940 28925 : return y;
4941 : }
4942 :
4943 : static GEN
4944 3389 : kron_pack_Flx_spec_2(GEN x, long l) { return Flx_eval2BILspec(x, 2, l); }
4945 :
4946 : static GEN
4947 0 : kron_pack_Flx_spec_3(GEN x, long l) { return Flx_eval2BILspec(x, 3, l); }
4948 :
4949 : static GEN
4950 31047 : kron_unpack_Flx(GEN z, ulong p)
4951 : {
4952 31047 : long i, l = lgefint(z);
4953 31047 : GEN x = cgetg(l, t_VECSMALL), w;
4954 144373 : for (w = int_LSW(z), i = 2; i < l; w = int_nextW(w), i++)
4955 113326 : x[i] = ((ulong) *w) % p;
4956 31047 : return Flx_renormalize(x, l);
4957 : }
4958 :
4959 : static GEN
4960 2930 : kron_unpack_Flx_2(GEN x, ulong p) {
4961 2930 : long d = (lgefint(x)-1)/2 - 1;
4962 2930 : return Z_mod2BIL_Flx_2(x, d, p);
4963 : }
4964 :
4965 : static GEN
4966 0 : kron_unpack_Flx_3(GEN x, ulong p) {
4967 0 : long d = lgefint(x)/3 - 1;
4968 0 : return Z_mod2BIL_Flx_3(x, d, p);
4969 : }
4970 :
4971 : static GEN
4972 113183 : FlxM_pack_ZM_bits(GEN M, long b)
4973 : {
4974 : long i, j, l, lc;
4975 113183 : GEN N = cgetg_copy(M, &l), x;
4976 113183 : if (l == 1)
4977 0 : return N;
4978 113183 : lc = lgcols(M);
4979 461587 : for (j = 1; j < l; j++) {
4980 348404 : gel(N, j) = cgetg(lc, t_COL);
4981 5848073 : for (i = 1; i < lc; i++) {
4982 5499669 : x = gcoeff(M, i, j);
4983 5499669 : gcoeff(N, i, j) = kron_pack_Flx_spec_bits(x + 2, b, lgpol(x));
4984 : }
4985 : }
4986 113183 : return N;
4987 : }
4988 :
4989 : static GEN
4990 17632 : ZM_unpack_FlxM(GEN M, ulong p, ulong sv, GEN (*unpack)(GEN, ulong))
4991 : {
4992 : long i, j, l, lc;
4993 17632 : GEN N = cgetg_copy(M, &l), x;
4994 17632 : if (l == 1)
4995 0 : return N;
4996 17632 : lc = lgcols(M);
4997 51323 : for (j = 1; j < l; j++) {
4998 33691 : gel(N, j) = cgetg(lc, t_COL);
4999 397043 : for (i = 1; i < lc; i++) {
5000 363352 : x = unpack(gcoeff(M, i, j), p);
5001 363352 : x[1] = sv;
5002 363352 : gcoeff(N, i, j) = x;
5003 : }
5004 : }
5005 17632 : return N;
5006 : }
5007 :
5008 : static GEN
5009 56632 : ZM_unpack_FlxM_bits(GEN M, long b, ulong p, ulong pi, long sv)
5010 : {
5011 : long i, j, l, lc;
5012 56632 : GEN N = cgetg_copy(M, &l), x;
5013 56632 : if (l == 1)
5014 0 : return N;
5015 56632 : lc = lgcols(M);
5016 56632 : if (b < BITS_IN_LONG) {
5017 184308 : for (j = 1; j < l; j++) {
5018 129314 : gel(N, j) = cgetg(lc, t_COL);
5019 3169907 : for (i = 1; i < lc; i++) {
5020 3040593 : x = kron_unpack_Flx_bits_narrow(gcoeff(M, i, j), b, p);
5021 3040593 : x[1] = sv;
5022 3040593 : gcoeff(N, i, j) = x;
5023 : }
5024 : }
5025 : } else {
5026 1638 : if (!pi) pi = get_Fl_red(p); /* unset if !SMALL_ULONG(p) */
5027 9370 : for (j = 1; j < l; j++) {
5028 7732 : gel(N, j) = cgetg(lc, t_COL);
5029 172546 : for (i = 1; i < lc; i++) {
5030 164814 : x = kron_unpack_Flx_bits_wide(gcoeff(M, i, j), b, p, pi);
5031 164814 : x[1] = sv;
5032 164814 : gcoeff(N, i, j) = x;
5033 : }
5034 : }
5035 : }
5036 56632 : return N;
5037 : }
5038 :
5039 : static GEN
5040 74264 : FlxM_mul_Kronecker_i(GEN A, GEN B, ulong p, ulong pi, long d, long sv)
5041 : {
5042 74264 : long b, n = lg(A) - 1;
5043 : GEN C, z;
5044 : GEN (*pack)(GEN, long), (*unpack)(GEN, ulong);
5045 74264 : int is_sqr = A==B;
5046 :
5047 74264 : z = muliu(muliu(sqru(p - 1), d), n);
5048 74264 : b = expi(z) + 1;
5049 : /* only do expensive bit-packing if it saves at least 1 limb */
5050 74264 : if (b <= BITS_IN_HALFULONG)
5051 70503 : { if (nbits2nlong(d*b) == (d + 1)/2) b = BITS_IN_HALFULONG; }
5052 : else
5053 : {
5054 3761 : long l = lgefint(z) - 2;
5055 3761 : if (nbits2nlong(d*b) == d*l) b = l*BITS_IN_LONG;
5056 : }
5057 :
5058 74264 : switch (b) {
5059 16928 : case BITS_IN_HALFULONG:
5060 16928 : pack = kron_pack_Flx_spec_half;
5061 16928 : unpack = int_to_Flx_half;
5062 16928 : break;
5063 655 : case BITS_IN_LONG:
5064 655 : pack = kron_pack_Flx_spec;
5065 655 : unpack = kron_unpack_Flx;
5066 655 : break;
5067 49 : case 2*BITS_IN_LONG:
5068 49 : pack = kron_pack_Flx_spec_2;
5069 49 : unpack = kron_unpack_Flx_2;
5070 49 : break;
5071 0 : case 3*BITS_IN_LONG:
5072 0 : pack = kron_pack_Flx_spec_3;
5073 0 : unpack = kron_unpack_Flx_3;
5074 0 : break;
5075 56632 : default:
5076 56632 : A = FlxM_pack_ZM_bits(A, b);
5077 56632 : B = is_sqr? A: FlxM_pack_ZM_bits(B, b);
5078 56632 : C = ZM_mul(A, B);
5079 56632 : return ZM_unpack_FlxM_bits(C, b, p, pi, sv);
5080 : }
5081 17632 : A = FlxM_pack_ZM(A, pack);
5082 17632 : B = is_sqr? A: FlxM_pack_ZM(B, pack);
5083 17632 : C = ZM_mul(A, B);
5084 17632 : return ZM_unpack_FlxM(C, p, sv, unpack);
5085 : }
5086 :
5087 : GEN
5088 74264 : FlxqM_mul_Kronecker(GEN A, GEN B, GEN T, ulong p)
5089 : {
5090 74264 : pari_sp av = avma;
5091 74264 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
5092 74264 : long sv = get_Flx_var(T), d = get_Flx_degree(T);
5093 74264 : GEN C = FlxM_mul_Kronecker_i(A, B, p, pi, d, sv);
5094 74264 : C = FlxqM_red_pre(C, T, p, pi);
5095 74264 : return gc_upto(av, C);
5096 : }
5097 :
5098 : /* assume m > 1 */
5099 : static long
5100 0 : FlxV_max_degree_i(GEN x, long m)
5101 : {
5102 0 : long i, l = degpol(gel(x,1));
5103 0 : for (i = 2; i < m; i++) l = maxss(l, degpol(gel(x,i)));
5104 0 : return l;
5105 : }
5106 :
5107 : /* assume n > 1 and m > 1 */
5108 : static long
5109 0 : FlxM_max_degree_i(GEN x, long n, long m)
5110 : {
5111 0 : long j, l = FlxV_max_degree_i(gel(x,1), m);
5112 0 : for (j = 2; j < n; j++) l = maxss(l, FlxV_max_degree_i(gel(x,j), m));
5113 0 : return l;
5114 : }
5115 :
5116 : static long
5117 0 : FlxM_max_degree(GEN x)
5118 : {
5119 0 : long n = lg(x), m;
5120 0 : if (n == 1) return -1;
5121 0 : m = lgcols(x); return m == 1? -1: FlxM_max_degree_i(x, n, m);
5122 : }
5123 :
5124 : GEN
5125 0 : FlxM_mul(GEN x, GEN y, ulong p)
5126 : {
5127 0 : pari_sp av = avma;
5128 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
5129 : long sv, d;
5130 0 : if (lg(x) == 1) return cgetg(1,t_MAT);
5131 0 : if (lg(gel(x,1))==1) return FlxqM_mul(x, y, NULL, p);
5132 0 : sv = mael3(x,1,1,1);
5133 0 : d = maxss(FlxM_max_degree(x), FlxM_max_degree(y));
5134 0 : return gc_GEN(av, FlxM_mul_Kronecker_i(x, y, p, pi, d+1, sv));
5135 : }
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