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 981283704 : get_Flx_red(GEN T, GEN *B)
22 : {
23 981283704 : if (typ(T)!=t_VEC) { *B=NULL; return T; }
24 677815 : *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 37062880 : Flx_to_ZX(GEN z)
43 : {
44 37062880 : long i, l = lg(z);
45 37062880 : GEN x = cgetg(l,t_POL);
46 241965832 : for (i=2; i<l; i++) gel(x,i) = utoi(z[i]);
47 37047749 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
48 : }
49 :
50 : GEN
51 71517 : Flx_to_FlxX(GEN z, long sv)
52 : {
53 71517 : long i, l = lg(z);
54 71517 : GEN x = cgetg(l,t_POL);
55 278633 : for (i=2; i<l; i++) gel(x,i) = Fl_to_Flx(z[i], sv);
56 71517 : x[1] = evalsigne(l-2!=0)| z[1]; return x;
57 : }
58 :
59 : /* same as Flx_to_ZX, in place */
60 : GEN
61 36789311 : Flx_to_ZX_inplace(GEN z)
62 : {
63 36789311 : long i, l = lg(z);
64 228561228 : for (i=2; i<l; i++) gel(z,i) = utoi(z[i]);
65 36780968 : 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 63881197 : Flx_to_Flv(GEN x, long N)
71 : {
72 63881197 : GEN z = cgetg(N+1,t_VECSMALL);
73 63875598 : long i, l = lg(x)-1;
74 63875598 : x++;
75 695547351 : for (i=1; i<l ; i++) z[i]=x[i];
76 325066473 : for ( ; i<=N; i++) z[i]=0;
77 63875598 : return z;
78 : }
79 :
80 : /*Flv_to_Flx=zv_to_zx*/
81 : GEN
82 26918172 : Flv_to_Flx(GEN x, long sv)
83 : {
84 26918172 : long i, l=lg(x)+1;
85 26918172 : GEN z = cgetg(l,t_VECSMALL); z[1]=sv;
86 26913671 : x--;
87 292311786 : for (i=2; i<l ; i++) z[i]=x[i];
88 26913671 : return Flx_renormalize(z,l);
89 : }
90 :
91 : /*Flm_to_FlxV=zm_to_zxV*/
92 : GEN
93 2296 : Flm_to_FlxV(GEN x, long sv)
94 6272 : { pari_APPLY_type(t_VEC, Flv_to_Flx(gel(x,i), sv)) }
95 :
96 : /*FlxC_to_ZXC=zxC_to_ZXC*/
97 : GEN
98 109566 : FlxC_to_ZXC(GEN x)
99 564027 : { pari_APPLY_type(t_COL, Flx_to_ZX(gel(x,i))) }
100 :
101 : /*FlxC_to_ZXC=zxV_to_ZXV*/
102 : GEN
103 600403 : FlxV_to_ZXV(GEN x)
104 2429173 : { pari_APPLY_type(t_VEC, Flx_to_ZX(gel(x,i))) }
105 :
106 : void
107 2923709 : FlxV_to_ZXV_inplace(GEN v)
108 : {
109 : long i;
110 7768819 : for(i=1;i<lg(v);i++) gel(v,i)= Flx_to_ZX(gel(v,i));
111 2923591 : }
112 :
113 : /*FlxM_to_ZXM=zxM_to_ZXM*/
114 : GEN
115 4611 : FlxM_to_ZXM(GEN x)
116 15935 : { pari_APPLY_same(FlxC_to_ZXC(gel(x,i))) }
117 :
118 : GEN
119 396559 : FlxV_to_FlxX(GEN x, long v)
120 : {
121 396559 : long i, l = lg(x)+1;
122 396559 : GEN z = cgetg(l,t_POL); z[1] = evalvarn(v);
123 396559 : x--;
124 4970439 : for (i=2; i<l ; i++) gel(z,i) = gel(x,i);
125 396559 : 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 19843144 : Fl_to_Flx(ulong x, long sv) { return x? mkvecsmall2(sv, x): pol0_Flx(sv); }
156 :
157 : /* a X^d */
158 : GEN
159 912158 : monomial_Flx(ulong a, long d, long vs)
160 : {
161 : GEN P;
162 912158 : if (a==0) return pol0_Flx(vs);
163 912158 : P = const_vecsmall(d+2, 0);
164 912163 : P[1] = vs; P[d+2] = a; return P;
165 : }
166 :
167 : GEN
168 2673501 : Z_to_Flx(GEN x, ulong p, long sv)
169 : {
170 2673501 : long u = umodiu(x,p);
171 2673489 : 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 167549364 : ZX_to_Flx(GEN x, ulong p)
177 : {
178 167549364 : long i, lx = lg(x);
179 167549364 : GEN a = cgetg(lx, t_VECSMALL);
180 167486493 : a[1]=((ulong)x[1])&VARNBITS;
181 1110017298 : for (i=2; i<lx; i++) a[i] = umodiu(gel(x,i), p);
182 167494142 : 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 6082710 : zx_to_Flx(GEN x, ulong p)
188 : {
189 6082710 : long i, lx = lg(x);
190 6082710 : GEN a = cgetg(lx, t_VECSMALL);
191 6077942 : a[1] = x[1];
192 18665618 : for (i=2; i<lx; i++) uel(a,i) = umodsu(x[i], p);
193 6077432 : return Flx_renormalize(a,lx);
194 : }
195 :
196 : ulong
197 74783797 : Rg_to_Fl(GEN x, ulong p)
198 : {
199 74783797 : switch(typ(x))
200 : {
201 47741194 : case t_INT: return umodiu(x, p);
202 457606 : case t_FRAC: {
203 457606 : ulong z = umodiu(gel(x,1), p);
204 457606 : if (!z) return 0;
205 446235 : return Fl_div(z, umodiu(gel(x,2), p), p);
206 : }
207 205950 : case t_PADIC: return padic_to_Fl(x, p);
208 26379059 : case t_INTMOD: {
209 26379059 : GEN q = gel(x,1), a = gel(x,2);
210 26379059 : 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 1707084 : Rg_to_F2(GEN x)
221 : {
222 1707084 : switch(typ(x))
223 : {
224 274275 : case t_INT: return mpodd(x);
225 0 : case t_FRAC:
226 0 : if (!mpodd(gel(x,2))) (void)Fl_inv(0,2); /* error */
227 0 : return mpodd(gel(x,1));
228 0 : case t_PADIC:
229 0 : if (!absequaliu(gel(x,2),2)) pari_err_OP("",x, mkintmodu(1,2));
230 0 : if (valp(x) < 0) (void)Fl_inv(0,2);
231 0 : return valp(x) & 1;
232 1432809 : case t_INTMOD: {
233 1432809 : GEN q = gel(x,1), a = gel(x,2);
234 1432809 : if (mpodd(q)) pari_err_MODULUS("Rg_to_F2", q, gen_2);
235 1432809 : return mpodd(a);
236 : }
237 0 : default: pari_err_TYPE("Rg_to_F2",x);
238 : return 0; /* LCOV_EXCL_LINE */
239 : }
240 : }
241 :
242 : GEN
243 2826388 : RgX_to_Flx(GEN x, ulong p)
244 : {
245 2826388 : long i, lx = lg(x);
246 2826388 : GEN a = cgetg(lx, t_VECSMALL);
247 2826388 : a[1]=((ulong)x[1])&VARNBITS;
248 22238931 : for (i=2; i<lx; i++) a[i] = Rg_to_Fl(gel(x,i), p);
249 2826388 : 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 3565080 : Rg_to_Flxq(GEN x, GEN T, ulong p)
259 : {
260 3565080 : long ta, tx = typ(x), v = get_Flx_var(T);
261 : ulong pi;
262 : GEN a, b;
263 3565079 : if (is_const_t(tx))
264 : {
265 3314563 : if (tx == t_FFELT) return FF_to_Flxq(x);
266 2583555 : return Fl_to_Flx(Rg_to_Fl(x, p), v);
267 : }
268 250515 : 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 241939 : case t_POL:
280 241939 : x = RgX_to_Flx(x,p);
281 241939 : if (x[1] != v) break;
282 241939 : 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 2128151390 : Flx_renormalize(GEN /*in place*/ x, long lx)
298 : {
299 : long i;
300 2385883639 : for (i = lx-1; i>1; i--)
301 2290249853 : if (x[i]) break;
302 2128151390 : stackdummy((pari_sp)(x + lg(x)), (pari_sp)(x + i+1));
303 2127082095 : setlg(x, i+1); return x;
304 : }
305 :
306 : GEN
307 1876934 : Flx_red(GEN z, ulong p)
308 : {
309 1876934 : long i, l = lg(z);
310 1876934 : GEN x = cgetg(l, t_VECSMALL);
311 1876749 : x[1] = z[1];
312 33591732 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
313 1876749 : return Flx_renormalize(x,l);
314 : }
315 :
316 : int
317 29474776 : Flx_equal(GEN V, GEN W)
318 : {
319 29474776 : long l = lg(V);
320 29474776 : if (lg(W) != l) return 0;
321 30635041 : while (--l > 1) /* do not compare variables, V[1] */
322 29360911 : if (V[l] != W[l]) return 0;
323 1274130 : return 1;
324 : }
325 :
326 : GEN
327 2663691 : random_Flx(long d1, long vs, ulong p)
328 : {
329 2663691 : long i, d = d1+2;
330 2663691 : GEN y = cgetg(d,t_VECSMALL); y[1] = vs;
331 18389894 : for (i=2; i<d; i++) y[i] = random_Fl(p);
332 2663857 : return Flx_renormalize(y,d);
333 : }
334 :
335 : static GEN
336 7159055 : Flx_addspec(GEN x, GEN y, ulong p, long lx, long ly)
337 : {
338 : long i,lz;
339 : GEN z;
340 :
341 7159055 : if (ly>lx) swapspec(x,y, lx,ly);
342 7159055 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
343 106787875 : for (i=0; i<ly; i++) z[i+2] = Fl_add(x[i], y[i], p);
344 90414955 : for ( ; i<lx; i++) z[i+2] = x[i];
345 7159055 : z[1] = 0; return Flx_renormalize(z, lz);
346 : }
347 :
348 : GEN
349 63595293 : Flx_add(GEN x, GEN y, ulong p)
350 : {
351 : long i,lz;
352 : GEN z;
353 63595293 : long lx=lg(x);
354 63595293 : long ly=lg(y);
355 63595293 : if (ly>lx) swapspec(x,y, lx,ly);
356 63595293 : lz = lx; z = cgetg(lz, t_VECSMALL); z[1]=x[1];
357 574815905 : for (i=2; i<ly; i++) z[i] = Fl_add(x[i], y[i], p);
358 132252594 : for ( ; i<lx; i++) z[i] = x[i];
359 63572419 : return Flx_renormalize(z, lz);
360 : }
361 :
362 : GEN
363 9694134 : Flx_Fl_add(GEN y, ulong x, ulong p)
364 : {
365 : GEN z;
366 : long lz, i;
367 9694134 : if (!lgpol(y))
368 229128 : return Fl_to_Flx(x,y[1]);
369 9465797 : lz=lg(y);
370 9465797 : z=cgetg(lz,t_VECSMALL);
371 9465404 : z[1]=y[1];
372 9465404 : z[2] = Fl_add(y[2],x,p);
373 46100575 : for(i=3;i<lz;i++)
374 36635518 : z[i] = y[i];
375 9465057 : if (lz==3) z = Flx_renormalize(z,lz);
376 9464787 : return z;
377 : }
378 :
379 : static GEN
380 897216 : Flx_subspec(GEN x, GEN y, ulong p, long lx, long ly)
381 : {
382 : long i,lz;
383 : GEN z;
384 :
385 897216 : if (ly <= lx)
386 : {
387 897233 : lz = lx+2; z = cgetg(lz, t_VECSMALL);
388 53818020 : for (i=0; i<ly; i++) z[i+2] = Fl_sub(x[i],y[i],p);
389 1447505 : 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 896809 : z[1] = 0; return Flx_renormalize(z, lz);
398 : }
399 :
400 : GEN
401 138091857 : Flx_sub(GEN x, GEN y, ulong p)
402 : {
403 138091857 : long i,lz,lx = lg(x), ly = lg(y);
404 : GEN z;
405 :
406 138091857 : if (ly <= lx)
407 : {
408 87759989 : lz = lx; z = cgetg(lz, t_VECSMALL);
409 457611717 : for (i=2; i<ly; i++) z[i] = Fl_sub(x[i],y[i],p);
410 175642291 : for ( ; i<lx; i++) z[i] = x[i];
411 : }
412 : else
413 : {
414 50331868 : lz = ly; z = cgetg(lz, t_VECSMALL);
415 257287271 : for (i=2; i<lx; i++) z[i] = Fl_sub(x[i],y[i],p);
416 234538824 : for ( ; i<ly; i++) z[i] = y[i]? (long)(p - y[i]): y[i];
417 : }
418 138079437 : z[1]=x[1]; return Flx_renormalize(z, lz);
419 : }
420 :
421 : GEN
422 151013 : Flx_Fl_sub(GEN y, ulong x, ulong p)
423 : {
424 : GEN z;
425 151013 : long lz = lg(y), i;
426 151013 : if (lz==2)
427 513 : return Fl_to_Flx(Fl_neg(x, p),y[1]);
428 150500 : z = cgetg(lz, t_VECSMALL);
429 150500 : z[1] = y[1];
430 150500 : z[2] = Fl_sub(uel(y,2), x, p);
431 748428 : for(i=3; i<lz; i++)
432 597928 : z[i] = y[i];
433 150500 : if (lz==3) z = Flx_renormalize(z,lz);
434 150500 : return z;
435 : }
436 :
437 : static GEN
438 3452469 : Flx_negspec(GEN x, ulong p, long l)
439 : {
440 : long i;
441 3452469 : GEN z = cgetg(l+2, t_VECSMALL) + 2;
442 21241500 : for (i=0; i<l; i++) z[i] = Fl_neg(x[i], p);
443 3452456 : return z-2;
444 : }
445 :
446 : GEN
447 3452473 : Flx_neg(GEN x, ulong p)
448 : {
449 3452473 : GEN z = Flx_negspec(x+2, p, lgpol(x));
450 3452532 : z[1] = x[1];
451 3452532 : return z;
452 : }
453 :
454 : GEN
455 1754809 : Flx_neg_inplace(GEN x, ulong p)
456 : {
457 1754809 : long i, l = lg(x);
458 52449381 : for (i=2; i<l; i++)
459 50694572 : if (x[i]) x[i] = p - x[i];
460 1754809 : return x;
461 : }
462 :
463 : GEN
464 2396554 : Flx_double(GEN y, ulong p)
465 : {
466 : long i, l;
467 2396554 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
468 19409541 : for(i=2; i<l; i++) z[i] = Fl_double(y[i], p);
469 2396554 : return Flx_renormalize(z, l);
470 : }
471 : GEN
472 1024658 : Flx_triple(GEN y, ulong p)
473 : {
474 : long i, l;
475 1024658 : GEN z = cgetg_copy(y, &l); z[1] = y[1];
476 7776091 : for(i=2; i<l; i++) z[i] = Fl_triple(y[i], p);
477 1024658 : return Flx_renormalize(z, l);
478 : }
479 :
480 : GEN
481 18797087 : Flx_Fl_mul_pre(GEN y, ulong x, ulong p, ulong pi)
482 : {
483 : GEN z;
484 : long i, l;
485 18797087 : if (!x) return pol0_Flx(y[1]);
486 18016090 : z = cgetg_copy(y, &l); z[1] = y[1];
487 18016151 : if (pi==0)
488 : {
489 15857888 : if (HIGHWORD(x | p))
490 0 : for(i=2; i<l; i++) z[i] = Fl_mul(uel(y,i), x, p);
491 : else
492 94175373 : for(i=2; i<l; i++) z[i] = (uel(y,i) * x) % p;
493 : } else
494 17911208 : for(i=2; i<l; i++) z[i] = Fl_mul_pre(uel(y,i), x, p, pi);
495 18016972 : return Flx_renormalize(z, l);
496 : }
497 :
498 : GEN
499 7116176 : Flx_Fl_mul(GEN x, ulong y, ulong p)
500 7116176 : { 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 11945915 : Flx_Fl_mul_to_monic(GEN y, ulong x, ulong p)
515 : {
516 : GEN z;
517 : long i, l;
518 11945915 : z = cgetg_copy(y, &l); z[1] = y[1];
519 11942970 : if (HIGHWORD(x | p))
520 5398687 : for(i=2; i<l-1; i++) z[i] = Fl_mul(y[i], x, p);
521 : else
522 26796498 : for(i=2; i<l-1; i++) z[i] = (y[i] * x) % p;
523 11942962 : 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 26731584 : Flx_shift(GEN a, long n)
529 : {
530 26731584 : long i, l = lg(a);
531 : GEN b;
532 26731584 : if (l==2 || !n) return Flx_copy(a);
533 26389495 : if (l+n<=2) return pol0_Flx(a[1]);
534 26175130 : b = cgetg(l+n, t_VECSMALL);
535 26172915 : b[1] = a[1];
536 26172915 : if (n < 0)
537 71628706 : for (i=2-n; i<l; i++) b[i+n] = a[i];
538 : else
539 : {
540 50752966 : for (i=0; i<n; i++) b[2+i] = 0;
541 147739966 : for (i=2; i<l; i++) b[i+n] = a[i];
542 : }
543 26172915 : return b;
544 : }
545 :
546 : GEN
547 62132565 : Flx_normalize(GEN z, ulong p)
548 : {
549 62132565 : long l = lg(z)-1;
550 62132565 : ulong p1 = z[l]; /* leading term */
551 62132565 : if (p1 == 1) return z;
552 11924287 : 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 3678827 : Flx_addshift(GEN x, GEN y, ulong p, long d)
558 : {
559 3678827 : GEN xd,yd,zd = (GEN)avma;
560 3678827 : long a,lz,ny = lgpol(y), nx = lgpol(x);
561 3678827 : long vs = x[1];
562 3678827 : if (nx == 0) return y;
563 3676975 : x += 2; y += 2; a = ny-d;
564 3676975 : if (a <= 0)
565 : {
566 85126 : lz = (a>nx)? ny+2: nx+d+2;
567 85126 : (void)new_chunk(lz); xd = x+nx; yd = y+ny;
568 1733197 : while (xd > x) *--zd = *--xd;
569 85126 : x = zd + a;
570 166548 : while (zd > x) *--zd = 0;
571 : }
572 : else
573 : {
574 3591849 : xd = new_chunk(d); yd = y+d;
575 3591849 : x = Flx_addspec(x,yd,p, nx,a);
576 3591849 : lz = (a>nx)? ny+2: lg(x)+d;
577 133014663 : x += 2; while (xd > x) *--zd = *--xd;
578 : }
579 60465298 : while (yd > y) *--zd = *--yd;
580 3676975 : *--zd = vs;
581 3676975 : *--zd = evaltyp(t_VECSMALL) | evallg(lz); return zd;
582 : }
583 :
584 : /* shift polynomial + gerepile */
585 : /* Do not set evalvarn*/
586 : static GEN
587 634566673 : Flx_shiftip(pari_sp av, GEN x, long v)
588 : {
589 634566673 : long i, lx = lg(x), ly;
590 : GEN y;
591 634566673 : if (!v || lx==2) return gerepileuptoleaf(av, x);
592 175404015 : ly = lx + v; /* result length */
593 175404015 : (void)new_chunk(ly); /* check that result fits */
594 175296608 : x += lx; y = (GEN)av;
595 1237105793 : for (i = 2; i<lx; i++) *--y = *--x;
596 706848594 : for (i = 0; i< v; i++) *--y = 0;
597 175296608 : y -= 2; y[0] = evaltyp(t_VECSMALL) | evallg(ly);
598 175417061 : return gc_const((pari_sp)y, y);
599 : }
600 :
601 : static long
602 2319949041 : get_Fl_threshold(ulong p, long mul, long mul2)
603 : {
604 2319949041 : return SMALL_ULONG(p) ? mul: mul2;
605 : }
606 :
607 : #define BITS_IN_QUARTULONG (BITS_IN_HALFULONG >> 1)
608 : #define QUARTMASK ((1UL<<BITS_IN_QUARTULONG)-1UL)
609 : #define LLQUARTWORD(x) ((x) & QUARTMASK)
610 : #define HLQUARTWORD(x) (((x) >> BITS_IN_QUARTULONG) & QUARTMASK)
611 : #define LHQUARTWORD(x) (((x) >> (2*BITS_IN_QUARTULONG)) & QUARTMASK)
612 : #define HHQUARTWORD(x) (((x) >> (3*BITS_IN_QUARTULONG)) & QUARTMASK)
613 : INLINE long
614 8364860 : maxbitcoeffpol(ulong p, long n)
615 : {
616 8364860 : GEN z = muliu(sqru(p - 1), n);
617 8362309 : long b = expi(z) + 1;
618 : /* only do expensive bit-packing if it saves at least 1 limb */
619 8362849 : if (b <= BITS_IN_QUARTULONG)
620 : {
621 905154 : if (nbits2nlong(n*b) == (n + 3)>>2)
622 107344 : b = BITS_IN_QUARTULONG;
623 : }
624 7457695 : else if (b <= BITS_IN_HALFULONG)
625 : {
626 1549328 : if (nbits2nlong(n*b) == (n + 1)>>1)
627 5590 : b = BITS_IN_HALFULONG;
628 : }
629 : else
630 : {
631 5908367 : long l = lgefint(z) - 2;
632 5908367 : if (nbits2nlong(n*b) == n*l)
633 307183 : b = l*BITS_IN_LONG;
634 : }
635 8362742 : return b;
636 : }
637 :
638 : INLINE ulong
639 3407596066 : Flx_mullimb_ok(GEN x, GEN y, ulong p, long a, long b)
640 : { /* Assume OK_ULONG*/
641 3407596066 : ulong p1 = 0;
642 : long i;
643 16159652646 : for (i=a; i<b; i++)
644 12752056580 : if (y[i])
645 : {
646 10702969921 : p1 += y[i] * x[-i];
647 10702969921 : if (p1 & HIGHBIT) p1 %= p;
648 : }
649 3407596066 : return p1 % p;
650 : }
651 :
652 : INLINE ulong
653 1149924623 : Flx_mullimb(GEN x, GEN y, ulong p, ulong pi, long a, long b)
654 : {
655 1149924623 : ulong p1 = 0;
656 : long i;
657 3615381277 : for (i=a; i<b; i++)
658 2465640424 : if (y[i])
659 2441337624 : p1 = Fl_addmul_pre(p1, y[i], x[-i], p, pi);
660 1149740853 : return p1;
661 : }
662 :
663 : /* assume nx >= ny > 0 */
664 : static GEN
665 343848921 : Flx_mulspec_basecase(GEN x, GEN y, ulong p, ulong pi, long nx, long ny)
666 : {
667 : long i,lz,nz;
668 : GEN z;
669 :
670 343848921 : lz = nx+ny+1; nz = lz-2;
671 343848921 : z = cgetg(lz, t_VECSMALL) + 2; /* x:y:z [i] = term of degree i */
672 343671709 : if (!pi)
673 : {
674 1152406001 : for (i=0; i<ny; i++)z[i] = Flx_mullimb_ok(x+i,y,p,0,i+1);
675 728729970 : for ( ; i<nx; i++) z[i] = Flx_mullimb_ok(x+i,y,p,0,ny);
676 898059689 : for ( ; i<nz; i++) z[i] = Flx_mullimb_ok(x+i,y,p,i-nx+1,ny);
677 : }
678 : else
679 : {
680 306007500 : for (i=0; i<ny; i++)z[i] = Flx_mullimb(x+i,y,p,pi,0,i+1);
681 213143532 : for ( ; i<nx; i++) z[i] = Flx_mullimb(x+i,y,p,pi,0,ny);
682 218247938 : for ( ; i<nz; i++) z[i] = Flx_mullimb(x+i,y,p,pi,i-nx+1,ny);
683 : }
684 343585068 : z -= 2; return Flx_renormalize(z, lz);
685 : }
686 :
687 : static GEN
688 12302 : int_to_Flx(GEN z, ulong p)
689 : {
690 12302 : long i, l = lgefint(z);
691 12302 : GEN x = cgetg(l, t_VECSMALL);
692 1059535 : for (i=2; i<l; i++) x[i] = uel(z,i)%p;
693 12293 : return Flx_renormalize(x, l);
694 : }
695 :
696 : INLINE GEN
697 10031 : Flx_mulspec_mulii(GEN a, GEN b, ulong p, long na, long nb)
698 : {
699 10031 : GEN z=muliispec(a,b,na,nb);
700 10039 : return int_to_Flx(z,p);
701 : }
702 :
703 : static GEN
704 509899 : Flx_to_int_halfspec(GEN a, long na)
705 : {
706 : long j;
707 509899 : long n = (na+1)>>1UL;
708 509899 : GEN V = cgetipos(2+n);
709 : GEN w;
710 1438576 : for (w = int_LSW(V), j=0; j+1<na; j+=2, w=int_nextW(w))
711 928677 : *w = a[j]|(a[j+1]<<BITS_IN_HALFULONG);
712 509899 : if (j<na)
713 340551 : *w = a[j];
714 509899 : return V;
715 : }
716 :
717 : static GEN
718 594101 : int_to_Flx_half(GEN z, ulong p)
719 : {
720 : long i;
721 594101 : long lx = (lgefint(z)-2)*2+2;
722 594101 : GEN w, x = cgetg(lx, t_VECSMALL);
723 2022823 : for (w = int_LSW(z), i=2; i<lx; i+=2, w=int_nextW(w))
724 : {
725 1428722 : x[i] = LOWWORD((ulong)*w)%p;
726 1428722 : x[i+1] = HIGHWORD((ulong)*w)%p;
727 : }
728 594101 : return Flx_renormalize(x, lx);
729 : }
730 :
731 : static GEN
732 5454 : Flx_mulspec_halfmulii(GEN a, GEN b, ulong p, long na, long nb)
733 : {
734 5454 : GEN A = Flx_to_int_halfspec(a,na);
735 5454 : GEN B = Flx_to_int_halfspec(b,nb);
736 5454 : GEN z = mulii(A,B);
737 5454 : return int_to_Flx_half(z,p);
738 : }
739 :
740 : static GEN
741 204471 : Flx_to_int_quartspec(GEN a, long na)
742 : {
743 : long j;
744 204471 : long n = (na+3)>>2UL;
745 204471 : GEN V = cgetipos(2+n);
746 : GEN w;
747 4377588 : for (w = int_LSW(V), j=0; j+3<na; j+=4, w=int_nextW(w))
748 4173117 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG))|(a[j+3]<<(3*BITS_IN_QUARTULONG));
749 204471 : switch (na-j)
750 : {
751 116411 : case 3:
752 116411 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG)|(a[j+2]<<(2*BITS_IN_QUARTULONG));
753 116411 : break;
754 34450 : case 2:
755 34450 : *w = a[j]|(a[j+1]<<BITS_IN_QUARTULONG);
756 34450 : break;
757 27223 : case 1:
758 27223 : *w = a[j];
759 27223 : break;
760 26387 : case 0:
761 26387 : break;
762 : }
763 204471 : return V;
764 : }
765 :
766 : static GEN
767 107346 : int_to_Flx_quart(GEN z, ulong p)
768 : {
769 : long i;
770 107346 : long lx = (lgefint(z)-2)*4+2;
771 107346 : GEN w, x = cgetg(lx, t_VECSMALL);
772 4873748 : for (w = int_LSW(z), i=2; i<lx; i+=4, w=int_nextW(w))
773 : {
774 4766402 : x[i] = LLQUARTWORD((ulong)*w)%p;
775 4766402 : x[i+1] = HLQUARTWORD((ulong)*w)%p;
776 4766402 : x[i+2] = LHQUARTWORD((ulong)*w)%p;
777 4766402 : x[i+3] = HHQUARTWORD((ulong)*w)%p;
778 : }
779 107346 : return Flx_renormalize(x, lx);
780 : }
781 :
782 : static GEN
783 97125 : Flx_mulspec_quartmulii(GEN a, GEN b, ulong p, long na, long nb)
784 : {
785 97125 : GEN A = Flx_to_int_quartspec(a,na);
786 97125 : GEN B = Flx_to_int_quartspec(b,nb);
787 97125 : GEN z = mulii(A,B);
788 97125 : return int_to_Flx_quart(z,p);
789 : }
790 :
791 : /*Eval x in 2^(k*BIL) in linear time, k==2 or 3*/
792 : static GEN
793 581670 : Flx_eval2BILspec(GEN x, long k, long l)
794 : {
795 581670 : long i, lz = k*l, ki;
796 581670 : GEN pz = cgetipos(2+lz);
797 16356426 : for (i=0; i < lz; i++)
798 15774756 : *int_W(pz,i) = 0UL;
799 8469048 : for (i=0, ki=0; i<l; i++, ki+=k)
800 7887378 : *int_W(pz,ki) = x[i];
801 581670 : return int_normalize(pz,0);
802 : }
803 :
804 : static GEN
805 297821 : Z_mod2BIL_Flx_2(GEN x, long d, ulong p)
806 : {
807 297821 : long i, offset, lm = lgefint(x)-2, l = d+3;
808 297821 : ulong pi = get_Fl_red(p);
809 297821 : GEN pol = cgetg(l, t_VECSMALL);
810 297821 : pol[1] = 0;
811 8003982 : for (i=0, offset=0; offset+1 < lm; i++, offset += 2)
812 7706161 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
813 297821 : if (offset < lm)
814 224978 : pol[i+2] = (*int_W(x,offset)) % p;
815 297821 : return Flx_renormalize(pol,l);
816 : }
817 :
818 : static GEN
819 0 : Z_mod2BIL_Flx_3(GEN x, long d, ulong p)
820 : {
821 0 : long i, offset, lm = lgefint(x)-2, l = d+3;
822 0 : ulong pi = get_Fl_red(p);
823 0 : GEN pol = cgetg(l, t_VECSMALL);
824 0 : pol[1] = 0;
825 0 : for (i=0, offset=0; offset+2 < lm; i++, offset += 3)
826 0 : pol[i+2] = remlll_pre(*int_W(x,offset+2), *int_W(x,offset+1),
827 0 : *int_W(x,offset), p, pi);
828 0 : if (offset+1 < lm)
829 0 : pol[i+2] = remll_pre(*int_W(x,offset+1), *int_W(x,offset), p, pi);
830 0 : else if (offset < lm)
831 0 : pol[i+2] = (*int_W(x,offset)) % p;
832 0 : return Flx_renormalize(pol,l);
833 : }
834 :
835 : static GEN
836 294891 : Z_mod2BIL_Flx(GEN x, long bs, long d, ulong p)
837 : {
838 294891 : return bs==2 ? Z_mod2BIL_Flx_2(x, d, p): Z_mod2BIL_Flx_3(x, d, p);
839 : }
840 :
841 : static GEN
842 283390 : Flx_mulspec_mulii_inflate(GEN x, GEN y, long N, ulong p, long nx, long ny)
843 : {
844 283390 : pari_sp av = avma;
845 283390 : GEN z = mulii(Flx_eval2BILspec(x,N,nx), Flx_eval2BILspec(y,N,ny));
846 283390 : return gerepileupto(av, Z_mod2BIL_Flx(z, N, nx+ny-2, p));
847 : }
848 :
849 : static GEN
850 20820157 : kron_pack_Flx_spec_bits(GEN x, long b, long l) {
851 : GEN y;
852 : long i;
853 20820157 : if (l == 0)
854 3428683 : return gen_0;
855 17391474 : y = cgetg(l + 1, t_VECSMALL);
856 813917122 : for(i = 1; i <= l; i++)
857 796529830 : y[i] = x[l - i];
858 17387292 : return nv_fromdigits_2k(y, b);
859 : }
860 :
861 : /* assume b < BITS_IN_LONG */
862 : static GEN
863 5664912 : kron_unpack_Flx_bits_narrow(GEN z, long b, ulong p) {
864 5664912 : GEN v = binary_2k_nv(z, b), x;
865 5664983 : long i, l = lg(v) + 1;
866 5664983 : x = cgetg(l, t_VECSMALL);
867 621701302 : for (i = 2; i < l; i++)
868 616036215 : x[i] = v[l - i] % p;
869 5665087 : return Flx_renormalize(x, l);
870 : }
871 :
872 : static GEN
873 5541116 : kron_unpack_Flx_bits_wide(GEN z, long b, ulong p, ulong pi) {
874 5541116 : GEN v = binary_2k(z, b), x, y;
875 5539910 : long i, l = lg(v) + 1, ly;
876 5539910 : x = cgetg(l, t_VECSMALL);
877 232973951 : for (i = 2; i < l; i++) {
878 227436113 : y = gel(v, l - i);
879 227436113 : ly = lgefint(y);
880 227436113 : switch (ly) {
881 6287419 : case 2: x[i] = 0; break;
882 29299407 : case 3: x[i] = *int_W_lg(y, 0, ly) % p; break;
883 175979748 : case 4: x[i] = remll_pre(*int_W_lg(y, 1, ly), *int_W_lg(y, 0, ly), p, pi); break;
884 31738699 : case 5: x[i] = remlll_pre(*int_W_lg(y, 2, ly), *int_W_lg(y, 1, ly),
885 15869539 : *int_W_lg(y, 0, ly), p, pi); break;
886 0 : default: x[i] = umodiu(gel(v, l - i), p);
887 : }
888 : }
889 5537838 : return Flx_renormalize(x, l);
890 : }
891 :
892 : static GEN
893 7262739 : Flx_mulspec_Kronecker(GEN A, GEN B, long b, ulong p, long lA, long lB)
894 : {
895 : GEN C, D;
896 7262739 : pari_sp av = avma;
897 7262739 : A = kron_pack_Flx_spec_bits(A, b, lA);
898 7268970 : B = kron_pack_Flx_spec_bits(B, b, lB);
899 7269027 : C = gerepileuptoint(av, mulii(A, B));
900 7267516 : if (b < BITS_IN_LONG)
901 2100346 : D = kron_unpack_Flx_bits_narrow(C, b, p);
902 : else
903 : {
904 5167170 : ulong pi = get_Fl_red(p);
905 5166117 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
906 : }
907 7265202 : return D;
908 : }
909 :
910 : static GEN
911 678900 : Flx_sqrspec_Kronecker(GEN A, long b, ulong p, long lA)
912 : {
913 : GEN C, D;
914 678900 : A = kron_pack_Flx_spec_bits(A, b, lA);
915 678963 : C = sqri(A);
916 678985 : if (b < BITS_IN_LONG)
917 470803 : D = kron_unpack_Flx_bits_narrow(C, b, p);
918 : else
919 : {
920 208182 : ulong pi = get_Fl_red(p);
921 208176 : D = kron_unpack_Flx_bits_wide(C, b, p, pi);
922 : }
923 678955 : return D;
924 : }
925 :
926 : /* fast product (Karatsuba) of polynomials a,b. These are not real GENs, a+2,
927 : * b+2 were sent instead. na, nb = number of terms of a, b.
928 : * Only c, c0, c1, c2 are genuine GEN.
929 : */
930 : static GEN
931 381300077 : Flx_mulspec(GEN a, GEN b, ulong p, ulong pi, long na, long nb)
932 : {
933 : GEN a0,c,c0;
934 381300077 : long n0, n0a, i, v = 0;
935 : pari_sp av;
936 :
937 486936044 : while (na && !a[0]) { a++; na--; v++; }
938 568697208 : while (nb && !b[0]) { b++; nb--; v++; }
939 381300077 : if (na < nb) swapspec(a,b, na,nb);
940 381300077 : if (!nb) return pol0_Flx(0);
941 :
942 353079572 : av = avma;
943 353079572 : if (nb >= get_Fl_threshold(p, Flx_MUL_MULII_LIMIT, Flx_MUL2_MULII_LIMIT))
944 : {
945 7662199 : long m = maxbitcoeffpol(p,nb);
946 7658457 : switch (m)
947 : {
948 97123 : case BITS_IN_QUARTULONG:
949 97123 : return Flx_shiftip(av,Flx_mulspec_quartmulii(a,b,p,na,nb), v);
950 5454 : case BITS_IN_HALFULONG:
951 5454 : return Flx_shiftip(av,Flx_mulspec_halfmulii(a,b,p,na,nb), v);
952 10031 : case BITS_IN_LONG:
953 10031 : return Flx_shiftip(av,Flx_mulspec_mulii(a,b,p,na,nb), v);
954 283390 : case 2*BITS_IN_LONG:
955 283390 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,2,p,na,nb), v);
956 0 : case 3*BITS_IN_LONG:
957 0 : return Flx_shiftip(av,Flx_mulspec_mulii_inflate(a,b,3,p,na,nb), v);
958 7262459 : default:
959 7262459 : return Flx_shiftip(av,Flx_mulspec_Kronecker(a,b,m,p,na,nb), v);
960 : }
961 : }
962 345575787 : if (nb < get_Fl_threshold(p, Flx_MUL_KARATSUBA_LIMIT, Flx_MUL2_KARATSUBA_LIMIT))
963 343782892 : return Flx_shiftip(av,Flx_mulspec_basecase(a,b,p,pi,na,nb), v);
964 1808823 : i=(na>>1); n0=na-i; na=i;
965 1808823 : a0=a+n0; n0a=n0;
966 2583261 : while (n0a && !a[n0a-1]) n0a--;
967 :
968 1808823 : if (nb > n0)
969 : {
970 : GEN b0,c1,c2;
971 : long n0b;
972 :
973 1754809 : nb -= n0; b0 = b+n0; n0b = n0;
974 2838614 : while (n0b && !b[n0b-1]) n0b--;
975 1754809 : c = Flx_mulspec(a,b,p,pi,n0a,n0b);
976 1754809 : c0 = Flx_mulspec(a0,b0,p,pi,na,nb);
977 :
978 1754809 : c2 = Flx_addspec(a0,a,p,na,n0a);
979 1754809 : c1 = Flx_addspec(b0,b,p,nb,n0b);
980 :
981 1754809 : c1 = Flx_mul_pre(c1,c2,p,pi);
982 1754809 : c2 = Flx_add(c0,c,p);
983 :
984 1754809 : c2 = Flx_neg_inplace(c2,p);
985 1754809 : c2 = Flx_add(c1,c2,p);
986 1754809 : c0 = Flx_addshift(c0,c2 ,p, n0);
987 : }
988 : else
989 : {
990 54014 : c = Flx_mulspec(a,b,p,pi,n0a,nb);
991 54014 : c0 = Flx_mulspec(a0,b,p,pi,na,nb);
992 : }
993 1808823 : c0 = Flx_addshift(c0,c,p,n0);
994 1808823 : return Flx_shiftip(av,c0, v);
995 : }
996 :
997 : GEN
998 375676819 : Flx_mul_pre(GEN x, GEN y, ulong p, ulong pi)
999 : {
1000 375676819 : GEN z = Flx_mulspec(x+2,y+2,p, pi, lgpol(x),lgpol(y));
1001 375814903 : z[1] = x[1]; return z;
1002 : }
1003 : GEN
1004 27222781 : Flx_mul(GEN x, GEN y, ulong p)
1005 27222781 : { return Flx_mul_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1006 :
1007 : static GEN
1008 281949832 : Flx_sqrspec_basecase(GEN x, ulong p, ulong pi, long nx)
1009 : {
1010 : long i, lz, nz;
1011 : ulong p1;
1012 : GEN z;
1013 :
1014 281949832 : if (!nx) return pol0_Flx(0);
1015 281949832 : lz = (nx << 1) + 1, nz = lz-2;
1016 281949832 : z = cgetg(lz, t_VECSMALL) + 2;
1017 281456010 : if (!pi)
1018 : {
1019 216405891 : z[0] = x[0]*x[0]%p;
1020 926179181 : for (i=1; i<nx; i++)
1021 : {
1022 709895125 : p1 = Flx_mullimb_ok(x+i,x,p,0, (i+1)>>1);
1023 709773290 : p1 <<= 1;
1024 709773290 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1025 709773290 : z[i] = p1 % p;
1026 : }
1027 929780293 : for ( ; i<nz; i++)
1028 : {
1029 712825668 : p1 = Flx_mullimb_ok(x+i,x,p,i-nx+1, (i+1)>>1);
1030 713496237 : p1 <<= 1;
1031 713496237 : if ((i&1) == 0) p1 += x[i>>1] * x[i>>1];
1032 713496237 : z[i] = p1 % p;
1033 : }
1034 : }
1035 : else
1036 : {
1037 65050119 : z[0] = Fl_sqr_pre(x[0], p, pi);
1038 409294337 : for (i=1; i<nx; i++)
1039 : {
1040 344362029 : p1 = Flx_mullimb(x+i,x,p,pi,0, (i+1)>>1);
1041 344615920 : p1 = Fl_add(p1, p1, p);
1042 344148532 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1043 344168797 : z[i] = p1;
1044 : }
1045 409423520 : for ( ; i<nz; i++)
1046 : {
1047 344375362 : p1 = Flx_mullimb(x+i,x,p,pi,i-nx+1, (i+1)>>1);
1048 345129835 : p1 = Fl_add(p1, p1, p);
1049 344743789 : if ((i&1) == 0) p1 = Fl_add(p1, Fl_sqr_pre(x[i>>1], p, pi), p);
1050 344491212 : z[i] = p1;
1051 : }
1052 : }
1053 282002783 : z -= 2; return Flx_renormalize(z, lz);
1054 : }
1055 :
1056 : static GEN
1057 2261 : Flx_sqrspec_sqri(GEN a, ulong p, long na)
1058 : {
1059 2261 : GEN z=sqrispec(a,na);
1060 2265 : return int_to_Flx(z,p);
1061 : }
1062 :
1063 : static GEN
1064 136 : Flx_sqrspec_halfsqri(GEN a, ulong p, long na)
1065 : {
1066 136 : GEN z = sqri(Flx_to_int_halfspec(a,na));
1067 136 : return int_to_Flx_half(z,p);
1068 : }
1069 :
1070 : static GEN
1071 10221 : Flx_sqrspec_quartsqri(GEN a, ulong p, long na)
1072 : {
1073 10221 : GEN z = sqri(Flx_to_int_quartspec(a,na));
1074 10221 : return int_to_Flx_quart(z,p);
1075 : }
1076 :
1077 : static GEN
1078 11501 : Flx_sqrspec_sqri_inflate(GEN x, long N, ulong p, long nx)
1079 : {
1080 11501 : pari_sp av = avma;
1081 11501 : GEN z = sqri(Flx_eval2BILspec(x,N,nx));
1082 11501 : return gerepileupto(av, Z_mod2BIL_Flx(z, N, (nx-1)*2, p));
1083 : }
1084 :
1085 : static GEN
1086 282158866 : Flx_sqrspec(GEN a, ulong p, ulong pi, long na)
1087 : {
1088 : GEN a0, c, c0;
1089 282158866 : long n0, n0a, i, v = 0, m;
1090 : pari_sp av;
1091 :
1092 403356312 : while (na && !a[0]) { a++; na--; v += 2; }
1093 282158866 : if (!na) return pol0_Flx(0);
1094 :
1095 281911204 : av = avma;
1096 281911204 : if (na >= get_Fl_threshold(p, Flx_SQR_SQRI_LIMIT, Flx_SQR2_SQRI_LIMIT))
1097 : {
1098 702999 : m = maxbitcoeffpol(p,na);
1099 703017 : switch(m)
1100 : {
1101 10221 : case BITS_IN_QUARTULONG:
1102 10221 : return Flx_shiftip(av, Flx_sqrspec_quartsqri(a,p,na), v);
1103 136 : case BITS_IN_HALFULONG:
1104 136 : return Flx_shiftip(av, Flx_sqrspec_halfsqri(a,p,na), v);
1105 2261 : case BITS_IN_LONG:
1106 2261 : return Flx_shiftip(av, Flx_sqrspec_sqri(a,p,na), v);
1107 11501 : case 2*BITS_IN_LONG:
1108 11501 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,2,p,na), v);
1109 0 : case 3*BITS_IN_LONG:
1110 0 : return Flx_shiftip(av, Flx_sqrspec_sqri_inflate(a,3,p,na), v);
1111 678898 : default:
1112 678898 : return Flx_shiftip(av, Flx_sqrspec_Kronecker(a,m,p,na), v);
1113 : }
1114 : }
1115 281671794 : if (na < get_Fl_threshold(p, Flx_SQR_KARATSUBA_LIMIT, Flx_SQR2_KARATSUBA_LIMIT))
1116 281626434 : return Flx_shiftip(av, Flx_sqrspec_basecase(a,p,pi,na), v);
1117 57607 : i=(na>>1); n0=na-i; na=i;
1118 57607 : a0=a+n0; n0a=n0;
1119 72398 : while (n0a && !a[n0a-1]) n0a--;
1120 :
1121 57607 : c = Flx_sqrspec(a,p,pi,n0a);
1122 57607 : c0= Flx_sqrspec(a0,p,pi,na);
1123 57607 : if (p == 2) n0 *= 2;
1124 : else
1125 : {
1126 57588 : GEN c1, t = Flx_addspec(a0,a,p,na,n0a);
1127 57588 : t = Flx_sqr_pre(t,p,pi);
1128 57588 : c1= Flx_add(c0,c, p);
1129 57588 : c1= Flx_sub(t, c1, p);
1130 57588 : c0 = Flx_addshift(c0,c1,p,n0);
1131 : }
1132 57607 : c0 = Flx_addshift(c0,c,p,n0);
1133 57607 : return Flx_shiftip(av,c0,v);
1134 : }
1135 :
1136 : GEN
1137 281906375 : Flx_sqr_pre(GEN x, ulong p, ulong pi)
1138 : {
1139 281906375 : GEN z = Flx_sqrspec(x+2,p, pi, lgpol(x));
1140 283071400 : z[1] = x[1]; return z;
1141 : }
1142 : GEN
1143 356076 : Flx_sqr(GEN x, ulong p)
1144 356076 : { return Flx_sqr_pre(x, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1145 :
1146 : GEN
1147 7939 : Flx_powu_pre(GEN x, ulong n, ulong p, ulong pi)
1148 : {
1149 7939 : GEN y = pol1_Flx(x[1]), z;
1150 : ulong m;
1151 7937 : if (n == 0) return y;
1152 7937 : m = n; z = x;
1153 : for (;;)
1154 : {
1155 30613 : if (m&1UL) y = Flx_mul_pre(y,z, p, pi);
1156 30616 : m >>= 1; if (!m) return y;
1157 22678 : z = Flx_sqr_pre(z, p, pi);
1158 : }
1159 : }
1160 : GEN
1161 0 : Flx_powu(GEN x, ulong n, ulong p)
1162 : {
1163 0 : if (n == 0) return pol1_Flx(x[1]);
1164 0 : return Flx_powu_pre(x, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p));
1165 : }
1166 :
1167 : GEN
1168 14265 : Flx_halve(GEN y, ulong p)
1169 : {
1170 : GEN z;
1171 : long i, l;
1172 14265 : z = cgetg_copy(y, &l); z[1] = y[1];
1173 59438 : for(i=2; i<l; i++) uel(z,i) = Fl_halve(uel(y,i), p);
1174 14265 : return z;
1175 : }
1176 :
1177 : static GEN
1178 7118336 : Flx_recipspec(GEN x, long l, long n)
1179 : {
1180 : long i;
1181 7118336 : GEN z=cgetg(n+2,t_VECSMALL)+2;
1182 115641921 : for(i=0; i<l; i++)
1183 108525196 : z[n-i-1] = x[i];
1184 15564383 : for( ; i<n; i++)
1185 8447658 : z[n-i-1] = 0;
1186 7116725 : return Flx_renormalize(z-2,n+2);
1187 : }
1188 :
1189 : GEN
1190 0 : Flx_recip(GEN x)
1191 : {
1192 0 : GEN z=Flx_recipspec(x+2,lgpol(x),lgpol(x));
1193 0 : z[1]=x[1];
1194 0 : return z;
1195 : }
1196 :
1197 : /* Return h^degpol(P) P(x / h) */
1198 : GEN
1199 1117 : Flx_rescale(GEN P, ulong h, ulong p)
1200 : {
1201 1117 : long i, l = lg(P);
1202 1117 : GEN Q = cgetg(l,t_VECSMALL);
1203 1117 : ulong hi = h;
1204 1117 : Q[l-1] = P[l-1];
1205 12538 : for (i=l-2; i>=2; i--)
1206 : {
1207 12538 : Q[i] = Fl_mul(P[i], hi, p);
1208 12538 : if (i == 2) break;
1209 11421 : hi = Fl_mul(hi,h, p);
1210 : }
1211 1117 : Q[1] = P[1]; return Q;
1212 : }
1213 :
1214 : /* x/polrecip(P)+O(x^n); allow pi = 0 */
1215 : static GEN
1216 134215 : Flx_invBarrett_basecase(GEN T, ulong p, ulong pi)
1217 : {
1218 134215 : long i, l=lg(T)-1, lr=l-1, k;
1219 134215 : GEN r=cgetg(lr,t_VECSMALL); r[1] = T[1];
1220 134215 : r[2] = 1;
1221 134215 : if (!pi)
1222 764047 : for (i=3;i<lr;i++)
1223 : {
1224 757056 : ulong u = uel(T, l-i+2);
1225 45376663 : for (k=3; k<i; k++)
1226 44619607 : { u += uel(T,l-i+k) * uel(r, k); if (u & HIGHBIT) u %= p; }
1227 757056 : r[i] = Fl_neg(u % p, p);
1228 : }
1229 : else
1230 2109569 : for (i=3;i<lr;i++)
1231 : {
1232 1982347 : ulong u = Fl_neg(uel(T,l-i+2), p);
1233 59521574 : for (k=3; k<i; k++)
1234 : {
1235 57539228 : ulong t = Fl_neg(uel(T,l-i+k), p);
1236 57539224 : u = Fl_addmul_pre(u, t, uel(r,k), p, pi);
1237 : }
1238 1982346 : r[i] = u;
1239 : }
1240 134213 : return Flx_renormalize(r,lr);
1241 : }
1242 :
1243 : /* Return new lgpol */
1244 : static long
1245 2131183 : Flx_lgrenormalizespec(GEN x, long lx)
1246 : {
1247 : long i;
1248 7436456 : for (i = lx-1; i>=0; i--)
1249 7435624 : if (x[i]) break;
1250 2131183 : return i+1;
1251 : }
1252 : /* allow pi = 0 */
1253 : static GEN
1254 23116 : Flx_invBarrett_Newton(GEN T, ulong p, ulong pi)
1255 : {
1256 23116 : long nold, lx, lz, lq, l = degpol(T), lQ;
1257 23116 : GEN q, y, z, x = zero_zv(l+1) + 2;
1258 23116 : ulong mask = quadratic_prec_mask(l-2); /* assume l > 2 */
1259 : pari_sp av;
1260 :
1261 23116 : y = T+2;
1262 23116 : q = Flx_recipspec(y,l+1,l+1); lQ = lgpol(q); q+=2;
1263 23116 : av = avma;
1264 : /* We work on _spec_ Flx's, all the l[xzq12] below are lgpol's */
1265 :
1266 : /* initialize */
1267 23116 : x[0] = Fl_inv(q[0], p);
1268 23116 : if (lQ>1 && q[1])
1269 5109 : {
1270 5109 : ulong u = q[1];
1271 5109 : if (x[0] != 1) u = Fl_mul(u, Fl_sqr(x[0],p), p);
1272 5109 : x[1] = p - u; lx = 2;
1273 : }
1274 : else
1275 18007 : lx = 1;
1276 23116 : nold = 1;
1277 158705 : for (; mask > 1; set_avma(av))
1278 : { /* set x -= x(x*q - 1) + O(t^(nnew + 1)), knowing x*q = 1 + O(t^(nold+1)) */
1279 135598 : long i, lnew, nnew = nold << 1;
1280 :
1281 135598 : if (mask & 1) nnew--;
1282 135598 : mask >>= 1;
1283 :
1284 135598 : lnew = nnew + 1;
1285 135598 : lq = Flx_lgrenormalizespec(q, minss(lQ, lnew));
1286 135604 : z = Flx_mulspec(x, q, p, pi, lx, lq); /* FIXME: high product */
1287 135588 : lz = lgpol(z); if (lz > lnew) lz = lnew;
1288 135588 : z += 2;
1289 : /* subtract 1 [=>first nold words are 0]: renormalize so that z(0) != 0 */
1290 290672 : for (i = nold; i < lz; i++) if (z[i]) break;
1291 135588 : nold = nnew;
1292 135588 : if (i >= lz) continue; /* z-1 = 0(t^(nnew + 1)) */
1293 :
1294 : /* z + i represents (x*q - 1) / t^i */
1295 100753 : lz = Flx_lgrenormalizespec (z+i, lz-i);
1296 100753 : z = Flx_mulspec(x, z+i, p, pi, lx, lz); /* FIXME: low product */
1297 100754 : lz = lgpol(z); z += 2;
1298 100754 : if (lz > lnew-i) lz = Flx_lgrenormalizespec(z, lnew-i);
1299 :
1300 100754 : lx = lz+ i;
1301 100754 : y = x + i; /* x -= z * t^i, in place */
1302 915393 : for (i = 0; i < lz; i++) y[i] = Fl_neg(z[i], p);
1303 : }
1304 23116 : x -= 2; setlg(x, lx + 2); x[1] = T[1];
1305 23116 : return x;
1306 : }
1307 :
1308 : /* allow pi = 0 */
1309 : static GEN
1310 158721 : Flx_invBarrett_pre(GEN T, ulong p, ulong pi)
1311 : {
1312 158721 : pari_sp ltop = avma;
1313 158721 : long l = lgpol(T);
1314 : GEN r;
1315 158721 : if (l < 3) return pol0_Flx(T[1]);
1316 157331 : if (l < get_Fl_threshold(p, Flx_INVBARRETT_LIMIT, Flx_INVBARRETT2_LIMIT))
1317 : {
1318 134215 : ulong c = T[l+1];
1319 134215 : if (c != 1)
1320 : {
1321 98118 : ulong ci = Fl_inv(c,p);
1322 98118 : T = Flx_Fl_mul_pre(T, ci, p, pi);
1323 98118 : r = Flx_invBarrett_basecase(T, p, pi);
1324 98118 : r = Flx_Fl_mul_pre(r, ci, p, pi);
1325 : }
1326 : else
1327 36097 : r = Flx_invBarrett_basecase(T, p, pi);
1328 : }
1329 : else
1330 23116 : r = Flx_invBarrett_Newton(T, p, pi);
1331 157330 : return gerepileuptoleaf(ltop, r);
1332 : }
1333 : GEN
1334 0 : Flx_invBarrett(GEN T, ulong p)
1335 0 : { return Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1336 :
1337 : /* allow pi = 0 */
1338 : GEN
1339 101481541 : Flx_get_red_pre(GEN T, ulong p, ulong pi)
1340 : {
1341 101481541 : if (typ(T)!=t_VECSMALL
1342 101445313 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1343 : Flx_BARRETT2_LIMIT))
1344 101461428 : return T;
1345 7611 : retmkvec2(Flx_invBarrett_pre(T, p, pi),T);
1346 : }
1347 : GEN
1348 14030531 : Flx_get_red(GEN T, ulong p)
1349 : {
1350 14030531 : if (typ(T)!=t_VECSMALL
1351 14030434 : || lgpol(T) < get_Fl_threshold(p, Flx_BARRETT_LIMIT,
1352 : Flx_BARRETT2_LIMIT))
1353 14024684 : return T;
1354 5194 : retmkvec2(Flx_invBarrett_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)),T);
1355 : }
1356 :
1357 : /* separate from Flx_divrem for maximal speed. */
1358 : static GEN
1359 792523083 : Flx_rem_basecase(GEN x, GEN y, ulong p, ulong pi)
1360 : {
1361 : pari_sp av;
1362 : GEN z, c;
1363 : long dx,dy,dy1,dz,i,j;
1364 : ulong p1,inv;
1365 792523083 : long vs=x[1];
1366 :
1367 792523083 : dy = degpol(y); if (!dy) return pol0_Flx(x[1]);
1368 757037926 : dx = degpol(x);
1369 757026486 : dz = dx-dy; if (dz < 0) return Flx_copy(x);
1370 757026486 : x += 2; y += 2;
1371 757026486 : inv = y[dy];
1372 757026486 : if (inv != 1UL) inv = Fl_inv(inv,p);
1373 911516252 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1374 :
1375 758596639 : c = cgetg(dy+3, t_VECSMALL); c[1]=vs; c += 2; av=avma;
1376 757270267 : z = cgetg(dz+3, t_VECSMALL); z[1]=vs; z += 2;
1377 :
1378 755611923 : if (!pi)
1379 : {
1380 483960928 : z[dz] = (inv*x[dx]) % p;
1381 1817777608 : for (i=dx-1; i>=dy; --i)
1382 : {
1383 1333816680 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1384 10517964910 : for (j=i-dy1; j<=i && j<=dz; j++)
1385 : {
1386 9184148230 : p1 += z[j]*y[i-j];
1387 9184148230 : if (p1 & HIGHBIT) p1 %= p;
1388 : }
1389 1333816680 : p1 %= p;
1390 1333816680 : z[i-dy] = p1? ((p - p1)*inv) % p: 0;
1391 : }
1392 3299872746 : for (i=0; i<dy; i++)
1393 : {
1394 2816087927 : p1 = z[0]*y[i];
1395 14530929754 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1396 : {
1397 11714841827 : p1 += z[j]*y[i-j];
1398 11714841827 : if (p1 & HIGHBIT) p1 %= p;
1399 : }
1400 2816189138 : c[i] = Fl_sub(x[i], p1%p, p);
1401 : }
1402 : }
1403 : else
1404 : {
1405 271650995 : z[dz] = Fl_mul_pre(inv, x[dx], p, pi);
1406 824244416 : for (i=dx-1; i>=dy; --i)
1407 : {
1408 552237538 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1409 2324341862 : for (j=i-dy1; j<=i && j<=dz; j++)
1410 1771322703 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1411 553019159 : z[i-dy] = p1? Fl_mul_pre(p - p1, inv, p, pi): 0;
1412 : }
1413 2004631206 : for (i=0; i<dy; i++)
1414 : {
1415 1733498154 : p1 = Fl_mul_pre(z[0],y[i],p,pi);
1416 4651927946 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1417 2910822256 : p1 = Fl_addmul_pre(p1, z[j], y[i - j], p, pi);
1418 1721043588 : c[i] = Fl_sub(x[i], p1, p);
1419 : }
1420 : }
1421 923996729 : i = dy-1; while (i>=0 && !c[i]) i--;
1422 754917871 : set_avma(av); return Flx_renormalize(c-2, i+3);
1423 : }
1424 :
1425 : /* as FpX_divrem but working only on ulong types.
1426 : * if relevant, *pr is the last object on stack */
1427 : static GEN
1428 62459735 : Flx_divrem_basecase(GEN x, GEN y, ulong p, ulong pi, GEN *pr)
1429 : {
1430 : GEN z,q,c;
1431 : long dx,dy,dy1,dz,i,j;
1432 : ulong p1,inv;
1433 62459735 : long sv=x[1];
1434 :
1435 62459735 : dy = degpol(y);
1436 62456945 : if (dy<0) pari_err_INV("Flx_divrem",y);
1437 62457115 : if (pr == ONLY_REM) return Flx_rem_basecase(x, y, p, pi);
1438 62456717 : if (!dy)
1439 : {
1440 7595917 : if (pr && pr != ONLY_DIVIDES) *pr = pol0_Flx(sv);
1441 7595915 : if (y[2] == 1UL) return Flx_copy(x);
1442 5407493 : return Flx_Fl_mul_pre(x, Fl_inv(y[2], p), p, pi);
1443 : }
1444 54860800 : dx = degpol(x);
1445 54863868 : dz = dx-dy;
1446 54863868 : if (dz < 0)
1447 : {
1448 1026097 : q = pol0_Flx(sv);
1449 1026094 : if (pr && pr != ONLY_DIVIDES) *pr = Flx_copy(x);
1450 1026094 : return q;
1451 : }
1452 53837771 : x += 2;
1453 53837771 : y += 2;
1454 53837771 : z = cgetg(dz + 3, t_VECSMALL); z[1] = sv; z += 2;
1455 53836244 : inv = uel(y, dy);
1456 53836244 : if (inv != 1UL) inv = Fl_inv(inv,p);
1457 79409960 : for (dy1=dy-1; dy1>=0 && !y[dy1]; dy1--);
1458 :
1459 53839168 : if (SMALL_ULONG(p))
1460 : {
1461 51966355 : z[dz] = (inv*x[dx]) % p;
1462 132008389 : for (i=dx-1; i>=dy; --i)
1463 : {
1464 80042034 : p1 = p - x[i]; /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1465 257910185 : for (j=i-dy1; j<=i && j<=dz; j++)
1466 : {
1467 177868151 : p1 += z[j]*y[i-j];
1468 177868151 : if (p1 & HIGHBIT) p1 %= p;
1469 : }
1470 80042034 : p1 %= p;
1471 80042034 : z[i-dy] = p1? (long) ((p - p1)*inv) % p: 0;
1472 : }
1473 : }
1474 : else
1475 : {
1476 1872813 : z[dz] = Fl_mul(inv, x[dx], p);
1477 9234101 : for (i=dx-1; i>=dy; --i)
1478 : { /* compute -p1 instead of p1 (pb with ulongs otherwise) */
1479 7361168 : p1 = p - uel(x,i);
1480 26338201 : for (j=i-dy1; j<=i && j<=dz; j++)
1481 18977035 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1482 7361166 : z[i-dy] = p1? Fl_mul(p - p1, inv, p): 0;
1483 : }
1484 : }
1485 53839288 : q = Flx_renormalize(z-2, dz+3);
1486 53837911 : if (!pr) return q;
1487 :
1488 26738723 : c = cgetg(dy + 3, t_VECSMALL); c[1] = sv; c += 2;
1489 26741441 : if (SMALL_ULONG(p))
1490 : {
1491 219708597 : for (i=0; i<dy; i++)
1492 : {
1493 194601184 : p1 = (ulong)z[0]*y[i];
1494 458279895 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1495 : {
1496 263678711 : p1 += (ulong)z[j]*y[i-j];
1497 263678711 : if (p1 & HIGHBIT) p1 %= p;
1498 : }
1499 194600892 : c[i] = Fl_sub(x[i], p1%p, p);
1500 : }
1501 : }
1502 : else
1503 : {
1504 16034460 : for (i=0; i<dy; i++)
1505 : {
1506 14401047 : p1 = Fl_mul(z[0],y[i],p);
1507 50241366 : for (j=maxss(1,i-dy1); j<=i && j<=dz; j++)
1508 35840321 : p1 = Fl_add(p1, Fl_mul(z[j],y[i-j],p), p);
1509 14401045 : c[i] = Fl_sub(x[i], p1, p);
1510 : }
1511 : }
1512 36077820 : i=dy-1; while (i>=0 && !c[i]) i--;
1513 26740826 : c = Flx_renormalize(c-2, i+3);
1514 26741392 : if (pr == ONLY_DIVIDES)
1515 427 : { if (lg(c) != 2) return NULL; }
1516 : else
1517 26740965 : *pr = c;
1518 26741252 : return q;
1519 : }
1520 :
1521 : /* Compute x mod T where 2 <= degpol(T) <= l+1 <= 2*(degpol(T)-1)
1522 : * and mg is the Barrett inverse of T. */
1523 : static GEN
1524 904769 : Flx_divrem_Barrettspec(GEN x, long l, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1525 : {
1526 : GEN q, r;
1527 904769 : long lt = degpol(T); /*We discard the leading term*/
1528 : long ld, lm, lT, lmg;
1529 904715 : ld = l-lt;
1530 904715 : lm = minss(ld, lgpol(mg));
1531 905071 : lT = Flx_lgrenormalizespec(T+2,lt);
1532 905217 : lmg = Flx_lgrenormalizespec(mg+2,lm);
1533 905060 : q = Flx_recipspec(x+lt,ld,ld); /* q = rec(x) lz<=ld*/
1534 904296 : q = Flx_mulspec(q+2,mg+2,p,pi,lgpol(q),lmg); /* q = rec(x) * mg lz<=ld+lm*/
1535 905050 : q = Flx_recipspec(q+2,minss(ld,lgpol(q)),ld);/* q = rec (rec(x) * mg) lz<=ld*/
1536 904117 : if (!pr) return q;
1537 896426 : r = Flx_mulspec(q+2,T+2,p,pi,lgpol(q),lT); /* r = q*pol lz<=ld+lt*/
1538 897360 : r = Flx_subspec(x,r+2,p,lt,minss(lt,lgpol(r)));/* r = x - q*pol lz<=lt */
1539 897145 : if (pr == ONLY_REM) return r;
1540 427859 : *pr = r; return q;
1541 : }
1542 :
1543 : static GEN
1544 604450 : Flx_divrem_Barrett(GEN x, GEN mg, GEN T, ulong p, ulong pi, GEN *pr)
1545 : {
1546 604450 : GEN q = NULL, r = Flx_copy(x);
1547 604474 : long l = lgpol(x), lt = degpol(T), lm = 2*lt-1, v = T[1];
1548 : long i;
1549 604472 : if (l <= lt)
1550 : {
1551 0 : if (pr == ONLY_REM) return Flx_copy(x);
1552 0 : if (pr == ONLY_DIVIDES) return lgpol(x)? NULL: pol0_Flx(v);
1553 0 : if (pr) *pr = Flx_copy(x);
1554 0 : return pol0_Flx(v);
1555 : }
1556 604472 : if (lt <= 1)
1557 1390 : return Flx_divrem_basecase(x,T,p,pi,pr);
1558 603082 : if (pr != ONLY_REM && l>lm)
1559 28900 : { q = zero_zv(l-lt+1); q[1] = T[1]; }
1560 906384 : while (l>lm)
1561 : {
1562 303345 : GEN zr, zq = Flx_divrem_Barrettspec(r+2+l-lm,lm,mg,T,p,pi,&zr);
1563 303357 : long lz = lgpol(zr);
1564 303302 : if (pr != ONLY_REM)
1565 : {
1566 57980 : long lq = lgpol(zq);
1567 872366 : for(i=0; i<lq; i++) q[2+l-lm+i] = zq[2+i];
1568 : }
1569 4391906 : for(i=0; i<lz; i++) r[2+l-lm+i] = zr[2+i];
1570 303302 : l = l-lm+lz;
1571 : }
1572 603039 : if (pr == ONLY_REM)
1573 : {
1574 469338 : if (l > lt)
1575 469296 : r = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi,ONLY_REM);
1576 : else
1577 42 : r = Flx_renormalize(r, l+2);
1578 469328 : r[1] = v; return r;
1579 : }
1580 133701 : if (l > lt)
1581 : {
1582 132170 : GEN zq = Flx_divrem_Barrettspec(r+2,l,mg,T,p,pi, pr ? &r: NULL);
1583 132170 : if (!q) q = zq;
1584 : else
1585 : {
1586 27326 : long lq = lgpol(zq);
1587 158629 : for(i=0; i<lq; i++) q[2+i] = zq[2+i];
1588 : }
1589 : }
1590 1531 : else if (pr)
1591 1535 : r = Flx_renormalize(r, l+2);
1592 133701 : q[1] = v; q = Flx_renormalize(q, lg(q));
1593 133744 : if (pr == ONLY_DIVIDES) return lgpol(r)? NULL: q;
1594 133744 : if (pr) { r[1] = v; *pr = r; }
1595 133744 : return q;
1596 : }
1597 :
1598 : /* allow pi = 0 (SMALL_ULONG) */
1599 : GEN
1600 79871591 : Flx_divrem_pre(GEN x, GEN T, ulong p, ulong pi, GEN *pr)
1601 : {
1602 : GEN B, y;
1603 : long dy, dx, d;
1604 79871591 : if (pr==ONLY_REM) return Flx_rem_pre(x, T, p, pi);
1605 62581537 : y = get_Flx_red(T, &B);
1606 62595432 : dy = degpol(y); dx = degpol(x); d = dx-dy;
1607 62590904 : if (!B && d+3 < get_Fl_threshold(p, Flx_DIVREM_BARRETT_LIMIT,Flx_DIVREM2_BARRETT_LIMIT))
1608 62456010 : return Flx_divrem_basecase(x,y,p,pi,pr);
1609 : else
1610 : {
1611 134736 : pari_sp av = avma;
1612 134736 : GEN mg = B? B: Flx_invBarrett_pre(y, p, pi);
1613 134736 : GEN q1 = Flx_divrem_Barrett(x,mg,y,p,pi,pr);
1614 134736 : if (!q1) return gc_NULL(av);
1615 134736 : if (!pr || pr==ONLY_DIVIDES) return gerepileuptoleaf(av, q1);
1616 126445 : return gc_all(av, 2, &q1, pr);
1617 : }
1618 : }
1619 : GEN
1620 30353647 : Flx_divrem(GEN x, GEN T, ulong p, GEN *pr)
1621 30353647 : { return Flx_divrem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p), pr); }
1622 :
1623 : GEN
1624 917197373 : Flx_rem_pre(GEN x, GEN T, ulong p, ulong pi)
1625 : {
1626 917197373 : GEN B, y = get_Flx_red(T, &B);
1627 917083041 : long d = degpol(x) - degpol(y);
1628 916898646 : if (d < 0) return Flx_copy(x);
1629 793048935 : if (!B && d+3 < get_Fl_threshold(p, Flx_REM_BARRETT_LIMIT,Flx_REM2_BARRETT_LIMIT))
1630 792506300 : return Flx_rem_basecase(x,y,p, pi);
1631 : else
1632 : {
1633 469714 : pari_sp av=avma;
1634 469714 : GEN mg = B ? B: Flx_invBarrett_pre(y, p, pi);
1635 469714 : GEN r = Flx_divrem_Barrett(x, mg, y, p, pi, ONLY_REM);
1636 469726 : return gerepileuptoleaf(av, r);
1637 : }
1638 : }
1639 : GEN
1640 41748776 : Flx_rem(GEN x, GEN T, ulong p)
1641 41748776 : { return Flx_rem_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1642 :
1643 : /* reduce T mod (X^n - 1, p). Shallow function */
1644 : GEN
1645 5098710 : Flx_mod_Xnm1(GEN T, ulong n, ulong p)
1646 : {
1647 5098710 : long i, j, L = lg(T), l = n+2;
1648 : GEN S;
1649 5098710 : if (L <= l || n & ~LGBITS) return T;
1650 3450 : S = cgetg(l, t_VECSMALL);
1651 3450 : S[1] = T[1];
1652 14013 : for (i = 2; i < l; i++) S[i] = T[i];
1653 9420 : for (j = 2; i < L; i++) {
1654 5970 : S[j] = Fl_add(S[j], T[i], p);
1655 5970 : if (++j == l) j = 2;
1656 : }
1657 3450 : return Flx_renormalize(S, l);
1658 : }
1659 : /* reduce T mod (X^n + 1, p). Shallow function */
1660 : GEN
1661 30282 : Flx_mod_Xn1(GEN T, ulong n, ulong p)
1662 : {
1663 30282 : long i, j, L = lg(T), l = n+2;
1664 : GEN S;
1665 30282 : if (L <= l || n & ~LGBITS) return T;
1666 2682 : S = cgetg(l, t_VECSMALL);
1667 2682 : S[1] = T[1];
1668 11347 : for (i = 2; i < l; i++) S[i] = T[i];
1669 6974 : for (j = 2; i < L; i++) {
1670 4292 : S[j] = Fl_sub(S[j], T[i], p);
1671 4292 : if (++j == l) j = 2;
1672 : }
1673 2682 : return Flx_renormalize(S, l);
1674 : }
1675 :
1676 : struct _Flxq {
1677 : GEN aut, T;
1678 : ulong p, pi;
1679 : };
1680 : /* allow pi = 0 */
1681 : static void
1682 73811919 : set_Flxq_pre(struct _Flxq *D, GEN T, ulong p, ulong pi)
1683 : {
1684 73811919 : D->p = p;
1685 73811919 : D->pi = pi;
1686 73811919 : D->T = Flx_get_red_pre(T, p, pi);
1687 73805832 : }
1688 : static void
1689 73628 : set_Flxq(struct _Flxq *D, GEN T, ulong p)
1690 73628 : { set_Flxq_pre(D, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
1691 :
1692 : static GEN
1693 0 : _Flx_divrem(void * E, GEN x, GEN y, GEN *r)
1694 : {
1695 0 : struct _Flxq *D = (struct _Flxq*) E;
1696 0 : return Flx_divrem_pre(x, y, D->p, D->pi, r);
1697 : }
1698 : static GEN
1699 578752 : _Flx_add(void * E, GEN x, GEN y) {
1700 578752 : struct _Flxq *D = (struct _Flxq*) E;
1701 578752 : return Flx_add(x, y, D->p);
1702 : }
1703 : static GEN
1704 10483880 : _Flx_mul(void *E, GEN x, GEN y) {
1705 10483880 : struct _Flxq *D = (struct _Flxq*) E;
1706 10483880 : return Flx_mul_pre(x, y, D->p, D->pi);
1707 : }
1708 : static GEN
1709 0 : _Flx_sqr(void *E, GEN x) {
1710 0 : struct _Flxq *D = (struct _Flxq*) E;
1711 0 : return Flx_sqr_pre(x, D->p, D->pi);
1712 : }
1713 :
1714 : static struct bb_ring Flx_ring = { _Flx_add,_Flx_mul,_Flx_sqr };
1715 :
1716 : GEN
1717 0 : Flx_digits(GEN x, GEN T, ulong p)
1718 : {
1719 : struct _Flxq D;
1720 0 : long d = degpol(T), n = (lgpol(x)+d-1)/d;
1721 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1722 0 : return gen_digits(x,T,n,(void *)&D, &Flx_ring, _Flx_divrem);
1723 : }
1724 :
1725 : GEN
1726 0 : FlxV_Flx_fromdigits(GEN x, GEN T, ulong p)
1727 : {
1728 : struct _Flxq D;
1729 0 : D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1730 0 : return gen_fromdigits(x,T,(void *)&D, &Flx_ring);
1731 : }
1732 :
1733 : long
1734 4156406 : Flx_val(GEN x)
1735 : {
1736 4156406 : long i, l=lg(x);
1737 4156406 : if (l==2) return LONG_MAX;
1738 4165325 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1739 4156406 : return i-2;
1740 : }
1741 : long
1742 26298149 : Flx_valrem(GEN x, GEN *Z)
1743 : {
1744 26298149 : long v, i, l=lg(x);
1745 : GEN y;
1746 26298149 : if (l==2) { *Z = Flx_copy(x); return LONG_MAX; }
1747 28466115 : for (i=2; i<l && x[i]==0; i++) /*empty*/;
1748 26298149 : v = i-2;
1749 26298149 : if (v == 0) { *Z = x; return 0; }
1750 1017186 : l -= v;
1751 1017186 : y = cgetg(l, t_VECSMALL); y[1] = x[1];
1752 2612015 : for (i=2; i<l; i++) y[i] = x[i+v];
1753 1017849 : *Z = y; return v;
1754 : }
1755 :
1756 : GEN
1757 21062537 : Flx_deriv(GEN z, ulong p)
1758 : {
1759 21062537 : long i,l = lg(z)-1;
1760 : GEN x;
1761 21062537 : if (l < 2) l = 2;
1762 21062537 : x = cgetg(l, t_VECSMALL); x[1] = z[1]; z++;
1763 21060409 : if (HIGHWORD(l | p))
1764 56937716 : for (i=2; i<l; i++) x[i] = Fl_mul((ulong)i-1, z[i], p);
1765 : else
1766 85287633 : for (i=2; i<l; i++) x[i] = ((i-1) * z[i]) % p;
1767 21061629 : return Flx_renormalize(x,l);
1768 : }
1769 :
1770 : static GEN
1771 422578 : Flx_integXn(GEN x, long n, ulong p)
1772 : {
1773 422578 : long i, lx = lg(x);
1774 : GEN y;
1775 422578 : if (lx == 2) return Flx_copy(x);
1776 412766 : y = cgetg(lx, t_VECSMALL); y[1] = x[1];
1777 2096086 : for (i=2; i<lx; i++)
1778 : {
1779 1682815 : ulong xi = uel(x,i);
1780 1682815 : if (xi == 0)
1781 13345 : uel(y,i) = 0;
1782 : else
1783 : {
1784 1669470 : ulong j = n+i-1;
1785 1669470 : ulong d = ugcd(j, xi);
1786 1669423 : if (d==1)
1787 1018058 : uel(y,i) = Fl_div(xi, j, p);
1788 : else
1789 651365 : uel(y,i) = Fl_div(xi/d, j/d, p);
1790 : }
1791 : }
1792 413271 : return Flx_renormalize(y, lx);;
1793 : }
1794 :
1795 : GEN
1796 0 : Flx_integ(GEN x, ulong p)
1797 : {
1798 0 : long i, lx = lg(x);
1799 : GEN y;
1800 0 : if (lx == 2) return Flx_copy(x);
1801 0 : y = cgetg(lx+1, t_VECSMALL); y[1] = x[1];
1802 0 : uel(y,2) = 0;
1803 0 : for (i=3; i<=lx; i++)
1804 0 : uel(y,i) = uel(x,i-1) ? Fl_div(uel(x,i-1), (i-2)%p, p): 0UL;
1805 0 : return Flx_renormalize(y, lx+1);;
1806 : }
1807 :
1808 : /* assume p prime */
1809 : GEN
1810 13447 : Flx_diff1(GEN P, ulong p)
1811 : {
1812 13447 : return Flx_sub(Flx_translate1(P, p), P, p);
1813 : }
1814 :
1815 : GEN
1816 419209 : Flx_deflate(GEN x0, long d)
1817 : {
1818 : GEN z, y, x;
1819 419209 : long i,id, dy, dx = degpol(x0);
1820 419209 : if (d == 1 || dx <= 0) return Flx_copy(x0);
1821 355958 : dy = dx/d;
1822 355958 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1823 355954 : z = y + 2;
1824 355954 : x = x0+ 2;
1825 1156882 : for (i=id=0; i<=dy; i++,id+=d) z[i] = x[id];
1826 355954 : return y;
1827 : }
1828 :
1829 : GEN
1830 158490 : Flx_inflate(GEN x0, long d)
1831 : {
1832 158490 : long i, id, dy, dx = degpol(x0);
1833 158488 : GEN x = x0 + 2, z, y;
1834 158488 : if (dx <= 0) return Flx_copy(x0);
1835 157414 : dy = dx*d;
1836 157414 : y = cgetg(dy+3, t_VECSMALL); y[1] = x0[1];
1837 157414 : z = y + 2;
1838 8854944 : for (i=0; i<=dy; i++) z[i] = 0;
1839 4312488 : for (i=id=0; i<=dx; i++,id+=d) z[id] = x[i];
1840 157414 : return y;
1841 : }
1842 :
1843 : /* write p(X) = a_0(X^k) + X*a_1(X^k) + ... + X^(k-1)*a_{k-1}(X^k) */
1844 : GEN
1845 147667 : Flx_splitting(GEN p, long k)
1846 : {
1847 147667 : long n = degpol(p), v = p[1], m, i, j, l;
1848 : GEN r;
1849 :
1850 147665 : m = n/k;
1851 147665 : r = cgetg(k+1,t_VEC);
1852 680313 : for(i=1; i<=k; i++)
1853 : {
1854 532651 : gel(r,i) = cgetg(m+3, t_VECSMALL);
1855 532639 : mael(r,i,1) = v;
1856 : }
1857 4502432 : for (j=1, i=0, l=2; i<=n; i++)
1858 : {
1859 4354770 : mael(r,j,l) = p[2+i];
1860 4354770 : if (j==k) { j=1; l++; } else j++;
1861 : }
1862 680310 : for(i=1; i<=k; i++)
1863 532655 : gel(r,i) = Flx_renormalize(gel(r,i),i<j?l+1:l);
1864 147655 : return r;
1865 : }
1866 :
1867 : /* ux + vy */
1868 : static GEN
1869 386367 : Flx_addmulmul(GEN u, GEN v, GEN x, GEN y, ulong p, ulong pi)
1870 386367 : { return Flx_add(Flx_mul_pre(u,x, p,pi), Flx_mul_pre(v,y, p,pi), p); }
1871 :
1872 : static GEN
1873 24751 : FlxM_Flx_mul2(GEN M, GEN x, GEN y, ulong p, ulong pi)
1874 : {
1875 24751 : GEN res = cgetg(3, t_COL);
1876 24751 : gel(res, 1) = Flx_addmulmul(gcoeff(M,1,1), gcoeff(M,1,2), x, y, p, pi);
1877 24752 : gel(res, 2) = Flx_addmulmul(gcoeff(M,2,1), gcoeff(M,2,2), x, y, p, pi);
1878 24752 : return res;
1879 : }
1880 :
1881 : #if 0
1882 : static GEN
1883 : FlxM_mul2_old(GEN M, GEN N, ulong p)
1884 : {
1885 : GEN res = cgetg(3, t_MAT);
1886 : gel(res, 1) = FlxM_Flx_mul2(M,gcoeff(N,1,1),gcoeff(N,2,1),p);
1887 : gel(res, 2) = FlxM_Flx_mul2(M,gcoeff(N,1,2),gcoeff(N,2,2),p);
1888 : return res;
1889 : }
1890 : #endif
1891 : /* A,B are 2x2 matrices, Flx entries. Return A x B using Strassen 7M formula */
1892 : static GEN
1893 6517 : FlxM_mul2(GEN A, GEN B, ulong p, ulong pi)
1894 : {
1895 6517 : GEN A11=gcoeff(A,1,1),A12=gcoeff(A,1,2), B11=gcoeff(B,1,1),B12=gcoeff(B,1,2);
1896 6517 : GEN A21=gcoeff(A,2,1),A22=gcoeff(A,2,2), B21=gcoeff(B,2,1),B22=gcoeff(B,2,2);
1897 6517 : GEN M1 = Flx_mul_pre(Flx_add(A11,A22, p), Flx_add(B11,B22, p), p, pi);
1898 6517 : GEN M2 = Flx_mul_pre(Flx_add(A21,A22, p), B11, p, pi);
1899 6517 : GEN M3 = Flx_mul_pre(A11, Flx_sub(B12,B22, p), p, pi);
1900 6517 : GEN M4 = Flx_mul_pre(A22, Flx_sub(B21,B11, p), p, pi);
1901 6517 : GEN M5 = Flx_mul_pre(Flx_add(A11,A12, p), B22, p, pi);
1902 6517 : GEN M6 = Flx_mul_pre(Flx_sub(A21,A11, p), Flx_add(B11,B12, p), p, pi);
1903 6517 : GEN M7 = Flx_mul_pre(Flx_sub(A12,A22, p), Flx_add(B21,B22, p), p, pi);
1904 6517 : GEN T1 = Flx_add(M1,M4, p), T2 = Flx_sub(M7,M5, p);
1905 6517 : GEN T3 = Flx_sub(M1,M2, p), T4 = Flx_add(M3,M6, p);
1906 6517 : retmkmat22(Flx_add(T1,T2, p), Flx_add(M3,M5, p),
1907 : Flx_add(M2,M4, p), Flx_add(T3,T4, p));
1908 : }
1909 :
1910 : /* Return [0,1;1,-q]*M */
1911 : static GEN
1912 6345 : Flx_FlxM_qmul(GEN q, GEN M, ulong p, ulong pi)
1913 : {
1914 6345 : GEN u = Flx_mul_pre(gcoeff(M,2,1), q, p, pi);
1915 6345 : GEN v = Flx_mul_pre(gcoeff(M,2,2), q, p, pi);
1916 6345 : retmkmat22(gcoeff(M,2,1), gcoeff(M,2,2),
1917 : Flx_sub(gcoeff(M,1,1), u, p), Flx_sub(gcoeff(M,1,2), v, p));
1918 : }
1919 :
1920 : static GEN
1921 895 : matid2_FlxM(long v)
1922 895 : { retmkmat22(pol1_Flx(v),pol0_Flx(v),pol0_Flx(v),pol1_Flx(v)); }
1923 :
1924 : static GEN
1925 13 : matJ2_FlxM(long v)
1926 13 : { retmkmat22(pol0_Flx(v),pol1_Flx(v),pol1_Flx(v),pol0_Flx(v)); }
1927 :
1928 : struct Flx_res
1929 : {
1930 : ulong res, lc;
1931 : long deg0, deg1, off;
1932 : };
1933 :
1934 : INLINE void
1935 9405 : Flx_halfres_update_pre(long da, long db, long dr, ulong p, ulong pi, struct Flx_res *res)
1936 : {
1937 9405 : if (dr >= 0)
1938 : {
1939 9405 : if (res->lc != 1)
1940 : {
1941 7596 : if (pi)
1942 : {
1943 3127 : res->lc = Fl_powu_pre(res->lc, da - dr, p, pi);
1944 3127 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1945 : } else
1946 : {
1947 4469 : res->lc = Fl_powu(res->lc, da - dr, p);
1948 4469 : res->res = Fl_mul(res->res, res->lc, p);
1949 : }
1950 : }
1951 9405 : if (both_odd(da + res->off, db + res->off))
1952 63 : res->res = Fl_neg(res->res, p);
1953 : } else
1954 : {
1955 0 : if (db == 0)
1956 : {
1957 0 : if (res->lc != 1)
1958 : {
1959 0 : if (pi)
1960 : {
1961 0 : res->lc = Fl_powu_pre(res->lc, da, p, pi);
1962 0 : res->res = Fl_mul_pre(res->res, res->lc, p, pi);
1963 : } else
1964 : {
1965 0 : res->lc = Fl_powu(res->lc, da, p);
1966 0 : res->res = Fl_mul(res->res, res->lc, p);
1967 : }
1968 : }
1969 : } else
1970 0 : res->res = 0;
1971 : }
1972 9405 : }
1973 :
1974 : static GEN
1975 1087101 : Flx_halfres_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *pa, GEN *pb, struct Flx_res *res)
1976 : {
1977 1087101 : pari_sp av = avma;
1978 : GEN u, u1, v, v1, M;
1979 1087101 : long vx = a[1], n = lgpol(a)>>1;
1980 1087099 : u1 = v = pol0_Flx(vx);
1981 1087088 : u = v1 = pol1_Flx(vx);
1982 6538229 : while (lgpol(b)>n)
1983 : {
1984 : GEN r, q;
1985 5451154 : q = Flx_divrem_pre(a,b,p,pi, &r);
1986 5451233 : if (res)
1987 : {
1988 8362 : long da = degpol(a), db=degpol(b), dr = degpol(r);
1989 8362 : res->lc = b[db+2];
1990 8362 : if (dr >= n)
1991 7133 : Flx_halfres_update_pre(da, db, dr, p, pi, res);
1992 : else
1993 : {
1994 1229 : res->deg0 = da;
1995 1229 : res->deg1 = db;
1996 : }
1997 : }
1998 5451233 : a = b; b = r; swap(u,u1); swap(v,v1);
1999 5451233 : u1 = Flx_sub(u1, Flx_mul(u, q, p), p);
2000 5451124 : v1 = Flx_sub(v1, Flx_mul(v, q, p), p);
2001 5451145 : if (gc_needed(av,2))
2002 : {
2003 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_halfgcd (d = %ld)",degpol(b));
2004 0 : gerepileall(av,6, &a,&b,&u1,&v1,&u,&v);
2005 : }
2006 : }
2007 1086924 : M = mkmat22(u,v,u1,v1); *pa = a; *pb = b;
2008 1087067 : return gc_all(av,3, &M, pa, pb);
2009 : }
2010 :
2011 : static GEN Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res);
2012 :
2013 : static GEN
2014 19282 : Flx_halfres_split(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2015 : {
2016 19282 : pari_sp av = avma;
2017 : GEN R, S, T, V1, V2;
2018 : GEN x1, y1, r, q;
2019 19282 : long l = lgpol(x), n = l>>1, k;
2020 19282 : if (lgpol(y) <= n)
2021 855 : { *a = Flx_copy(x); *b = Flx_copy(y); return matid2_FlxM(x[1]); }
2022 18427 : if (res)
2023 : {
2024 3263 : res->lc = Flx_lead(y);
2025 3263 : res->deg0 -= n;
2026 3263 : res->deg1 -= n;
2027 3263 : res->off += n;
2028 : }
2029 18427 : R = Flx_halfres_i(Flx_shift(x,-n),Flx_shift(y,-n),p,pi,a,b,res);
2030 18427 : if (res)
2031 : {
2032 3263 : res->off -= n;
2033 3263 : res->deg0 += n;
2034 3263 : res->deg1 += n;
2035 : }
2036 18427 : V1 = FlxM_Flx_mul2(R, Flxn_red(x,n), Flxn_red(y,n), p, pi);
2037 18427 : x1 = Flx_add(Flx_shift(*a,n), gel(V1,1), p);
2038 18427 : y1 = Flx_add(Flx_shift(*b,n), gel(V1,2), p);
2039 18427 : if (lgpol(y1) <= n)
2040 12102 : { *a = x1; *b = y1; return gc_all(av, 3, &R, a, b); }
2041 6325 : k = 2*n-degpol(y1);
2042 6325 : q = Flx_divrem_pre(x1, y1, p, pi, &r);
2043 6325 : if (res)
2044 : {
2045 1043 : long dx1 = degpol(x1), dy1 = degpol(y1), dr = degpol(r);
2046 1043 : if (dy1 < degpol(y))
2047 185 : Flx_halfres_update_pre(res->deg0, res->deg1, dy1, p, pi, res);
2048 1043 : res->lc = uel(y1, dy1+2);
2049 1043 : res->deg0 = dx1;
2050 1043 : res->deg1 = dy1;
2051 1043 : if (dr >= n)
2052 : {
2053 1043 : Flx_halfres_update_pre(dx1, dy1, dr, p, pi, res);
2054 1043 : res->deg0 = dy1;
2055 1043 : res->deg1 = dr;
2056 : }
2057 1043 : res->deg0 -= k;
2058 1043 : res->deg1 -= k;
2059 1043 : res->off += k;
2060 : }
2061 6325 : S = Flx_halfres_i(Flx_shift(y1,-k), Flx_shift(r,-k), p, pi, a, b, res);
2062 6325 : if (res)
2063 : {
2064 1043 : res->deg0 += k;
2065 1043 : res->deg1 += k;
2066 1043 : res->off -= k;
2067 : }
2068 6325 : T = FlxM_mul2(S, Flx_FlxM_qmul(q, R, p,pi), p, pi);
2069 6325 : V2 = FlxM_Flx_mul2(S, Flxn_red(y1,k), Flxn_red(r,k), p, pi);
2070 6325 : *a = Flx_add(Flx_shift(*a,k), gel(V2,1), p);
2071 6325 : *b = Flx_add(Flx_shift(*b,k), gel(V2,2), p);
2072 6325 : return gc_all(av, 3, &T, a, b);
2073 : }
2074 :
2075 : static GEN
2076 1106384 : Flx_halfres_i(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, struct Flx_res *res)
2077 : {
2078 1106384 : if (lgpol(x) < get_Fl_threshold(p, Flx_HALFGCD_LIMIT, Flx_HALFGCD2_LIMIT))
2079 1087100 : return Flx_halfres_basecase(x, y, p, pi, a, b, res);
2080 19282 : return Flx_halfres_split(x, y, p, pi, a, b, res);
2081 : }
2082 :
2083 : static GEN
2084 1080589 : Flx_halfgcd_all_i(GEN x, GEN y, ulong p, ulong pi, GEN *pa, GEN *pb)
2085 : {
2086 : GEN a, b, R;
2087 1080589 : R = Flx_halfres_i(x, y, p, pi, &a, &b, NULL);
2088 1080595 : if (pa) *pa = a;
2089 1080595 : if (pb) *pb = b;
2090 1080595 : return R;
2091 : }
2092 :
2093 : /* Return M in GL_2(Fl[X]) such that:
2094 : if [a',b']~=M*[a,b]~ then degpol(a')>= (lgpol(a)>>1) >degpol(b')
2095 : */
2096 :
2097 : GEN
2098 1080593 : Flx_halfgcd_all_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b)
2099 : {
2100 : pari_sp av;
2101 : GEN R, q ,r;
2102 1080593 : long lx = lgpol(x), ly = lgpol(y);
2103 1080589 : if (!lx)
2104 : {
2105 0 : if (a) *a = Flx_copy(y);
2106 0 : if (b) *b = Flx_copy(x);
2107 0 : return matJ2_FlxM(x[1]);
2108 : }
2109 1080589 : if (ly < lx) return Flx_halfgcd_all_i(x, y, p, pi, a, b);
2110 8585 : av = avma;
2111 8585 : q = Flx_divrem(y,x,p,&r);
2112 8585 : R = Flx_halfgcd_all_i(x, r, p, pi, a, b);
2113 8585 : gcoeff(R,1,1) = Flx_sub(gcoeff(R,1,1), Flx_mul_pre(q,gcoeff(R,1,2), p,pi), p);
2114 8585 : gcoeff(R,2,1) = Flx_sub(gcoeff(R,2,1), Flx_mul_pre(q,gcoeff(R,2,2), p,pi), p);
2115 8585 : return !a && b ? gc_all(av, 2, &R, b): gc_all(av, 1+!!a+!!b, &R, a, b);
2116 : }
2117 :
2118 : GEN
2119 154 : Flx_halfgcd_all(GEN x, GEN y, ulong p, GEN *a, GEN *b)
2120 154 : { return Flx_halfgcd_all_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), a, b); }
2121 :
2122 : GEN
2123 845370 : Flx_halfgcd_pre(GEN x, GEN y, ulong p, ulong pi)
2124 845370 : { return Flx_halfgcd_all_pre(x, y, p, pi, NULL, NULL); }
2125 :
2126 : GEN
2127 0 : Flx_halfgcd(GEN x, GEN y, ulong p)
2128 0 : { return Flx_halfgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2129 :
2130 : /*Do not garbage collect*/
2131 : static GEN
2132 82947083 : Flx_gcd_basecase(GEN a, GEN b, ulong p, ulong pi)
2133 : {
2134 82947083 : pari_sp av = avma;
2135 82947083 : ulong iter = 0;
2136 82947083 : if (lg(b) > lg(a)) swap(a, b);
2137 286268666 : while (lgpol(b))
2138 : {
2139 202880288 : GEN c = Flx_rem_pre(a,b,p,pi);
2140 203321583 : iter++; a = b; b = c;
2141 203321583 : if (gc_needed(av,2))
2142 : {
2143 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (d = %ld)",degpol(c));
2144 0 : gerepileall(av,2, &a,&b);
2145 : }
2146 : }
2147 82895120 : return iter < 2 ? Flx_copy(a) : a;
2148 : }
2149 :
2150 : GEN
2151 84590674 : Flx_gcd_pre(GEN x, GEN y, ulong p, ulong pi)
2152 : {
2153 84590674 : pari_sp av = avma;
2154 : long lim;
2155 84590674 : if (!lgpol(x)) return Flx_copy(y);
2156 82950890 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2157 82955152 : while (lgpol(y) >= lim)
2158 : {
2159 150 : if (lgpol(y)<=(lgpol(x)>>1))
2160 : {
2161 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2162 0 : x = y; y = r;
2163 : }
2164 150 : (void) Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2165 150 : if (gc_needed(av,2))
2166 : {
2167 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_gcd (y = %ld)",degpol(y));
2168 0 : gerepileall(av,2,&x,&y);
2169 : }
2170 : }
2171 82943907 : return gerepileuptoleaf(av, Flx_gcd_basecase(x,y,p,pi));
2172 : }
2173 : GEN
2174 32496759 : Flx_gcd(GEN x, GEN y, ulong p)
2175 32496759 : { return Flx_gcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2176 :
2177 : int
2178 8524548 : Flx_is_squarefree(GEN z, ulong p)
2179 : {
2180 8524548 : pari_sp av = avma;
2181 8524548 : GEN d = Flx_gcd(z, Flx_deriv(z,p) , p);
2182 8524313 : return gc_bool(av, degpol(d) == 0);
2183 : }
2184 :
2185 : static long
2186 126734 : Flx_is_smooth_squarefree(GEN f, long r, ulong p, ulong pi)
2187 : {
2188 126734 : pari_sp av = avma;
2189 : long i;
2190 126734 : GEN sx = polx_Flx(f[1]), a = sx;
2191 531645 : for(i=1;;i++)
2192 : {
2193 531645 : if (degpol(f)<=r) return gc_long(av,1);
2194 509561 : a = Flxq_powu_pre(Flx_rem_pre(a,f,p,pi), p, f, p, pi);
2195 509425 : if (Flx_equal(a, sx)) return gc_long(av,1);
2196 505771 : if (i==r) return gc_long(av,0);
2197 404617 : f = Flx_div_pre(f, Flx_gcd_pre(Flx_sub(a,sx,p),f,p,pi),p,pi);
2198 : }
2199 : }
2200 :
2201 : static long
2202 8374 : Flx_is_l_pow(GEN x, ulong p)
2203 : {
2204 8374 : ulong i, lx = lgpol(x);
2205 16619 : for (i=1; i<lx; i++)
2206 14924 : if (x[i+2] && i%p) return 0;
2207 1695 : return 1;
2208 : }
2209 :
2210 : int
2211 126740 : Flx_is_smooth_pre(GEN g, long r, ulong p, ulong pi)
2212 : {
2213 : while (1)
2214 8375 : {
2215 126740 : GEN f = Flx_gcd_pre(g, Flx_deriv(g, p), p, pi);
2216 126572 : if (!Flx_is_smooth_squarefree(Flx_div_pre(g, f, p, pi), r, p, pi))
2217 101163 : return 0;
2218 25573 : if (degpol(f)==0) return 1;
2219 8367 : g = Flx_is_l_pow(f,p) ? Flx_deflate(f, p): f;
2220 : }
2221 : }
2222 : int
2223 74256 : Flx_is_smooth(GEN g, long r, ulong p)
2224 74256 : { return Flx_is_smooth_pre(g, r, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2225 :
2226 : static GEN
2227 6737747 : Flx_extgcd_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2228 : {
2229 6737747 : pari_sp av=avma;
2230 : GEN u,v,u1,v1;
2231 6737747 : long vx = a[1];
2232 6737747 : v = pol0_Flx(vx); v1 = pol1_Flx(vx);
2233 6737569 : if (ptu) { u = pol1_Flx(vx); u1 = pol0_Flx(vx); }
2234 29456097 : while (lgpol(b))
2235 : {
2236 22717503 : GEN r, q = Flx_divrem_pre(a,b,p,pi, &r);
2237 22718905 : a = b; b = r;
2238 22718905 : if (ptu)
2239 : {
2240 2316226 : swap(u,u1);
2241 2316226 : u1 = Flx_sub(u1, Flx_mul_pre(u, q, p, pi), p);
2242 : }
2243 22718899 : swap(v,v1);
2244 22718899 : v1 = Flx_sub(v1, Flx_mul_pre(v, q, p, pi), p);
2245 22718532 : if (gc_needed(av,2))
2246 : {
2247 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_extgcd (d = %ld)",degpol(a));
2248 0 : gerepileall(av,ptu ? 6: 4, &a,&b,&v,&v1,&u,&u1);
2249 : }
2250 : }
2251 6737624 : if (ptu) *ptu = u;
2252 6737624 : *ptv = v;
2253 6737624 : return a;
2254 : }
2255 :
2256 : static GEN
2257 136247 : Flx_extgcd_halfgcd(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2258 : {
2259 : GEN u, v;
2260 136247 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2261 136247 : GEN V = cgetg(expu(lgpol(y))+2,t_VEC);
2262 136247 : long i, n = 0, vs = x[1];
2263 369480 : while (lgpol(y) >= lim)
2264 : {
2265 233233 : if (lgpol(y)<=(lgpol(x)>>1))
2266 : {
2267 26 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2268 26 : x = y; y = r;
2269 26 : gel(V,++n) = mkmat22(pol0_Flx(vs),pol1_Flx(vs),pol1_Flx(vs),Flx_neg(q,p));
2270 : } else
2271 233207 : gel(V,++n) = Flx_halfgcd_all_pre(x, y, p, pi, &x, &y);
2272 : }
2273 136247 : y = Flx_extgcd_basecase(x,y,p,pi,&u,&v);
2274 233232 : for (i = n; i>1; i--)
2275 : {
2276 96986 : GEN R = gel(V,i);
2277 96986 : GEN u1 = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2278 96986 : GEN v1 = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2279 96986 : u = u1; v = v1;
2280 : }
2281 : {
2282 136246 : GEN R = gel(V,1);
2283 136246 : if (ptu)
2284 6543 : *ptu = Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi);
2285 136246 : *ptv = Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi);
2286 : }
2287 136247 : return y;
2288 : }
2289 :
2290 : /* x and y in Z[X], return lift(gcd(x mod p, y mod p)). Set u and v st
2291 : * ux + vy = gcd (mod p) */
2292 : GEN
2293 6737752 : Flx_extgcd_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptu, GEN *ptv)
2294 : {
2295 6737752 : pari_sp av = avma;
2296 : GEN d;
2297 6737752 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2298 6737747 : if (lgpol(y) >= lim)
2299 136247 : d = Flx_extgcd_halfgcd(x, y, p, pi, ptu, ptv);
2300 : else
2301 6601490 : d = Flx_extgcd_basecase(x, y, p, pi, ptu, ptv);
2302 6737626 : return gc_all(av, ptu?3:2, &d, ptv, ptu);
2303 : }
2304 : GEN
2305 852669 : Flx_extgcd(GEN x, GEN y, ulong p, GEN *ptu, GEN *ptv)
2306 852669 : { return Flx_extgcd_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptu, ptv); }
2307 :
2308 : static GEN
2309 1044 : Flx_halfres_pre(GEN x, GEN y, ulong p, ulong pi, GEN *a, GEN *b, ulong *r)
2310 : {
2311 : struct Flx_res res;
2312 : GEN R;
2313 : long dB;
2314 :
2315 1044 : res.res = *r;
2316 1044 : res.lc = Flx_lead(y);
2317 1044 : res.deg0 = degpol(x);
2318 1044 : res.deg1 = degpol(y);
2319 1044 : res.off = 0;
2320 1044 : R = Flx_halfres_i(x, y, p, pi, a, b, &res);
2321 1044 : dB = degpol(*b);
2322 1044 : if (dB < degpol(y))
2323 1044 : Flx_halfres_update_pre(res.deg0, res.deg1, dB, p, pi, &res);
2324 1044 : *r = res.res;
2325 1044 : return R;
2326 : }
2327 :
2328 : static ulong
2329 10188498 : Flx_resultant_basecase_pre(GEN a, GEN b, ulong p, ulong pi)
2330 : {
2331 : pari_sp av;
2332 : long da,db,dc;
2333 10188498 : ulong lb, res = 1UL;
2334 : GEN c;
2335 :
2336 10188498 : da = degpol(a);
2337 10188338 : db = degpol(b);
2338 10188512 : if (db > da)
2339 : {
2340 0 : swapspec(a,b, da,db);
2341 0 : if (both_odd(da,db)) res = p-res;
2342 : }
2343 10188512 : else if (!da) return 1; /* = res * a[2] ^ db, since 0 <= db <= da = 0 */
2344 10188512 : av = avma;
2345 106820395 : while (db)
2346 : {
2347 96654544 : lb = b[db+2];
2348 96654544 : c = Flx_rem_pre(a,b, p,pi);
2349 96466359 : a = b; b = c; dc = degpol(c);
2350 96445070 : if (dc < 0) return gc_long(av,0);
2351 :
2352 96439592 : if (both_odd(da,db)) res = p - res;
2353 96428791 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, da - dc, p, pi), p);
2354 96629944 : if (gc_needed(av,2))
2355 : {
2356 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant (da = %ld)",da);
2357 0 : gerepileall(av,2, &a,&b);
2358 : }
2359 96631883 : da = db; /* = degpol(a) */
2360 96631883 : db = dc; /* = degpol(b) */
2361 : }
2362 10165851 : return gc_ulong(av, Fl_mul(res, Fl_powu_pre(b[2], da, p, pi), p));
2363 : }
2364 :
2365 : ulong
2366 10190436 : Flx_resultant_pre(GEN x, GEN y, ulong p, ulong pi)
2367 : {
2368 10190436 : pari_sp av = avma;
2369 : long lim;
2370 10190436 : ulong res = 1;
2371 10190436 : long dx = degpol(x), dy = degpol(y);
2372 10190013 : if (dx < 0 || dy < 0) return 0;
2373 10188571 : if (dx < dy)
2374 : {
2375 1065971 : swap(x,y);
2376 1065971 : if (both_odd(dx, dy))
2377 1906 : res = Fl_neg(res, p);
2378 : }
2379 10188570 : lim = get_Fl_threshold(p, Flx_GCD_LIMIT, Flx_GCD2_LIMIT);
2380 10189450 : while (lgpol(y) >= lim)
2381 : {
2382 852 : if (lgpol(y)<=(lgpol(x)>>1))
2383 : {
2384 0 : GEN r = Flx_rem_pre(x, y, p, pi);
2385 0 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2386 0 : ulong ly = y[dy+2];
2387 0 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2388 0 : if (both_odd(dx, dy))
2389 0 : res = Fl_neg(res, p);
2390 0 : x = y; y = r;
2391 : }
2392 852 : (void) Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2393 852 : if (gc_needed(av,2))
2394 : {
2395 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_res (y = %ld)",degpol(y));
2396 0 : gerepileall(av,2,&x,&y);
2397 : }
2398 : }
2399 10188508 : return gc_ulong(av, Fl_mul(res, Flx_resultant_basecase_pre(x, y, p, pi), p));
2400 : }
2401 :
2402 : ulong
2403 4654831 : Flx_resultant(GEN a, GEN b, ulong p)
2404 4654831 : { return Flx_resultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2405 :
2406 : /* If resultant is 0, *ptU and *ptV are not set */
2407 : static ulong
2408 53 : Flx_extresultant_basecase(GEN a, GEN b, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2409 : {
2410 53 : GEN z,q,u,v, x = a, y = b;
2411 53 : ulong lb, res = 1UL;
2412 53 : pari_sp av = avma;
2413 : long dx, dy, dz;
2414 53 : long vs = a[1];
2415 :
2416 53 : u = pol0_Flx(vs);
2417 53 : v = pol1_Flx(vs); /* v = 1 */
2418 53 : dx = degpol(x);
2419 53 : dy = degpol(y);
2420 764 : while (dy)
2421 : { /* b u = x (a), b v = y (a) */
2422 711 : lb = y[dy+2];
2423 711 : q = Flx_divrem_pre(x,y, p, pi, &z);
2424 711 : x = y; y = z; /* (x,y) = (y, x - q y) */
2425 711 : dz = degpol(z); if (dz < 0) return gc_ulong(av,0);
2426 711 : z = Flx_sub(u, Flx_mul_pre(q,v, p, pi), p);
2427 711 : u = v; v = z; /* (u,v) = (v, u - q v) */
2428 :
2429 711 : if (both_odd(dx,dy)) res = p - res;
2430 711 : if (lb != 1) res = Fl_mul(res, Fl_powu_pre(lb, dx-dz, p, pi), p);
2431 711 : dx = dy; /* = degpol(x) */
2432 711 : dy = dz; /* = degpol(y) */
2433 : }
2434 53 : res = Fl_mul(res, Fl_powu_pre(y[2], dx, p, pi), p);
2435 53 : lb = Fl_mul(res, Fl_inv(y[2],p), p);
2436 53 : v = gerepileuptoleaf(av, Flx_Fl_mul_pre(v, lb, p, pi));
2437 53 : av = avma;
2438 53 : u = Flx_sub(Fl_to_Flx(res,vs), Flx_mul_pre(b,v,p,pi), p);
2439 53 : u = gerepileuptoleaf(av, Flx_div_pre(u,a,p,pi)); /* = (res - b v) / a */
2440 53 : *ptU = u;
2441 53 : *ptV = v; return res;
2442 : }
2443 :
2444 : ulong
2445 53 : Flx_extresultant_pre(GEN x, GEN y, ulong p, ulong pi, GEN *ptU, GEN *ptV)
2446 : {
2447 53 : pari_sp av=avma;
2448 : GEN u, v, R;
2449 53 : long lim = get_Fl_threshold(p, Flx_EXTGCD_LIMIT, Flx_EXTGCD2_LIMIT);
2450 53 : ulong res = 1, res1;
2451 53 : long dx = degpol(x), dy = degpol(y);
2452 53 : if (dy > dx)
2453 : {
2454 13 : swap(x,y); lswap(dx,dy);
2455 13 : if (both_odd(dx,dy)) res = p-res;
2456 13 : R = matJ2_FlxM(x[1]);
2457 40 : } else R = matid2_FlxM(x[1]);
2458 53 : if (dy < 0) return 0;
2459 245 : while (lgpol(y) >= lim)
2460 : {
2461 : GEN M;
2462 192 : if (lgpol(y)<=(lgpol(x)>>1))
2463 : {
2464 20 : GEN r, q = Flx_divrem_pre(x, y, p, pi, &r);
2465 20 : long dx = degpol(x), dy = degpol(y), dr = degpol(r);
2466 20 : ulong ly = y[dy+2];
2467 20 : if (ly != 1) res = Fl_mul(res, Fl_powu_pre(ly, dx - dr, p, pi), p);
2468 20 : if (both_odd(dx, dy))
2469 0 : res = Fl_neg(res, p);
2470 20 : x = y; y = r;
2471 20 : R = Flx_FlxM_qmul(q, R, p,pi);
2472 : }
2473 192 : M = Flx_halfres_pre(x, y, p, pi, &x, &y, &res);
2474 192 : if (!res) return gc_ulong(av, 0);
2475 192 : R = FlxM_mul2(M, R, p, pi);
2476 192 : gerepileall(av,3,&x,&y,&R);
2477 : }
2478 53 : res1 = Flx_extresultant_basecase(x,y,p,pi,&u,&v);
2479 53 : if (!res1) return gc_ulong(av, 0);
2480 53 : *ptU = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,1), gcoeff(R,2,1), p, pi), res, p, pi);
2481 53 : *ptV = Flx_Fl_mul_pre(Flx_addmulmul(u, v, gcoeff(R,1,2), gcoeff(R,2,2), p, pi), res, p, pi);
2482 53 : gerepileall(av, 2, ptU, ptV);
2483 53 : return Fl_mul(res1,res,p);
2484 : }
2485 :
2486 : ulong
2487 53 : Flx_extresultant(GEN a, GEN b, ulong p, GEN *ptU, GEN *ptV)
2488 53 : { return Flx_extresultant_pre(a, b, p, SMALL_ULONG(p)? 0: get_Fl_red(p), ptU, ptV); }
2489 :
2490 : /* allow pi = 0 (SMALL_ULONG) */
2491 : ulong
2492 43757790 : Flx_eval_powers_pre(GEN x, GEN y, ulong p, ulong pi)
2493 : {
2494 43757790 : ulong l0, l1, h0, h1, v1, i = 1, lx = lg(x)-1;
2495 :
2496 43757790 : if (lx == 1) return 0;
2497 41000338 : x++;
2498 41000338 : if (pi)
2499 : {
2500 : LOCAL_OVERFLOW;
2501 : LOCAL_HIREMAINDER;
2502 40936104 : l1 = mulll(uel(x,i), uel(y,i)); h1 = hiremainder; v1 = 0;
2503 97765979 : while (++i < lx)
2504 : {
2505 56829875 : l0 = mulll(uel(x,i), uel(y,i)); h0 = hiremainder;
2506 56829875 : l1 = addll(l0, l1); h1 = addllx(h0, h1); v1 += overflow;
2507 : }
2508 81118 : return v1? remlll_pre(v1, h1, l1, p, pi)
2509 41017222 : : remll_pre(h1, l1, p, pi);
2510 : }
2511 : else
2512 : {
2513 64234 : l1 = x[i] * y[i];
2514 30923797 : while (++i < lx) { l1 += x[i] * y[i]; if (l1 & HIGHBIT) l1 %= p; }
2515 64234 : return l1 % p;
2516 : }
2517 : }
2518 :
2519 : /* allow pi = 0 (SMALL_ULONG) */
2520 : ulong
2521 100597344 : Flx_eval_pre(GEN x, ulong y, ulong p, ulong pi)
2522 : {
2523 100597344 : long i, n = degpol(x);
2524 : ulong t;
2525 100600872 : if (n <= 0) return n? 0: x[2];
2526 32884723 : if (n > 15)
2527 : {
2528 180421 : pari_sp av = avma;
2529 180421 : GEN v = Fl_powers_pre(y, n, p, pi);
2530 180404 : return gc_ulong(av, Flx_eval_powers_pre(x, v, p, pi));
2531 : }
2532 32704302 : i = n+2; t = x[i];
2533 32704302 : if (pi)
2534 : {
2535 122808605 : for (i--; i>=2; i--) t = Fl_addmul_pre(uel(x, i), t, y, p, pi);
2536 31604172 : return t;
2537 : }
2538 2667071 : for (i--; i>=2; i--) t = (t * y + x[i]) % p;
2539 1111464 : return t %= p;
2540 : }
2541 : ulong
2542 20350563 : Flx_eval(GEN x, ulong y, ulong p)
2543 20350563 : { return Flx_eval_pre(x, y, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2544 :
2545 : ulong
2546 3255 : Flv_prod_pre(GEN x, ulong p, ulong pi)
2547 : {
2548 3255 : pari_sp ltop = avma;
2549 : GEN v;
2550 3255 : long i,k,lx = lg(x);
2551 3255 : if (lx == 1) return 1UL;
2552 3255 : if (lx == 2) return uel(x,1);
2553 3003 : v = cgetg(1+(lx << 1), t_VECSMALL);
2554 3003 : k = 1;
2555 28593 : for (i=1; i<lx-1; i+=2)
2556 25590 : uel(v,k++) = Fl_mul_pre(uel(x,i), uel(x,i+1), p, pi);
2557 3003 : if (i < lx) uel(v,k++) = uel(x,i);
2558 13529 : while (k > 2)
2559 : {
2560 10526 : lx = k; k = 1;
2561 36116 : for (i=1; i<lx-1; i+=2)
2562 25590 : uel(v,k++) = Fl_mul_pre(uel(v,i), uel(v,i+1), p, pi);
2563 10526 : if (i < lx) uel(v,k++) = uel(v,i);
2564 : }
2565 3003 : return gc_ulong(ltop, uel(v,1));
2566 : }
2567 :
2568 : ulong
2569 0 : Flv_prod(GEN v, ulong p)
2570 : {
2571 0 : return Flv_prod_pre(v, p, get_Fl_red(p));
2572 : }
2573 :
2574 : GEN
2575 0 : FlxV_prod(GEN V, ulong p)
2576 : {
2577 : struct _Flxq D;
2578 0 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2579 0 : return gen_product(V, (void *)&D, &_Flx_mul);
2580 : }
2581 :
2582 : /* compute prod (x - a[i]) */
2583 : GEN
2584 737475 : Flv_roots_to_pol(GEN a, ulong p, long vs)
2585 : {
2586 : struct _Flxq D;
2587 737475 : long i,k,lx = lg(a);
2588 : GEN p1;
2589 737475 : if (lx == 1) return pol1_Flx(vs);
2590 737475 : p1 = cgetg(lx, t_VEC);
2591 11905071 : for (k=1,i=1; i<lx-1; i+=2)
2592 11166390 : gel(p1,k++) = mkvecsmall4(vs, Fl_mul(a[i], a[i+1], p),
2593 11167864 : Fl_neg(Fl_add(a[i],a[i+1],p),p), 1);
2594 737207 : if (i < lx)
2595 58112 : gel(p1,k++) = mkvecsmall3(vs, Fl_neg(a[i],p), 1);
2596 737206 : D.T = NULL; D.aut = NULL; D.p = p; D.pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2597 737200 : setlg(p1, k); return gen_product(p1, (void *)&D, _Flx_mul);
2598 : }
2599 :
2600 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for "large" p */
2601 : INLINE void
2602 18976053 : Flv_inv_pre_indir(GEN w, GEN v, ulong p, ulong pi)
2603 : {
2604 18976053 : pari_sp av = avma;
2605 18976053 : long n = lg(w), i;
2606 : ulong u;
2607 : GEN c;
2608 :
2609 18976053 : if (n == 1) return;
2610 18976053 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2611 80615188 : for (i = 2; i < n; ++i) c[i] = Fl_mul_pre(w[i], c[i-1], p, pi);
2612 19142049 : i = n-1; u = Fl_inv(c[i], p);
2613 80976646 : for ( ; i > 1; --i)
2614 : {
2615 61755581 : ulong t = Fl_mul_pre(u, c[i-1], p, pi);
2616 61705162 : u = Fl_mul_pre(u, w[i], p, pi); v[i] = t;
2617 : }
2618 19221065 : v[1] = u; set_avma(av);
2619 : }
2620 :
2621 : void
2622 18370839 : Flv_inv_pre_inplace(GEN v, ulong p, ulong pi) { Flv_inv_pre_indir(v,v, p, pi); }
2623 :
2624 : GEN
2625 10850 : Flv_inv_pre(GEN w, ulong p, ulong pi)
2626 10850 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_pre_indir(w, v, p, pi); return v; }
2627 :
2628 : /* set v[i] = w[i]^{-1}; may be called with w = v, suitable for SMALL_ULONG p */
2629 : INLINE void
2630 49733 : Flv_inv_indir(GEN w, GEN v, ulong p)
2631 : {
2632 49733 : pari_sp av = avma;
2633 49733 : long n = lg(w), i;
2634 : ulong u;
2635 : GEN c;
2636 :
2637 49733 : if (n == 1) return;
2638 49733 : c = cgetg(n, t_VECSMALL); c[1] = w[1];
2639 1718514 : for (i = 2; i < n; ++i) c[i] = Fl_mul(w[i], c[i-1], p);
2640 49746 : i = n-1; u = Fl_inv(c[i], p);
2641 1718572 : for ( ; i > 1; --i)
2642 : {
2643 1668838 : ulong t = Fl_mul(u, c[i-1], p);
2644 1668834 : u = Fl_mul(u, w[i], p); v[i] = t;
2645 : }
2646 49734 : v[1] = u; set_avma(av);
2647 : }
2648 : static void
2649 635646 : Flv_inv_i(GEN v, GEN w, ulong p)
2650 : {
2651 635646 : if (SMALL_ULONG(p)) Flv_inv_indir(w, v, p);
2652 585913 : else Flv_inv_pre_indir(w, v, p, get_Fl_red(p));
2653 635649 : }
2654 : void
2655 12017 : Flv_inv_inplace(GEN v, ulong p) { Flv_inv_i(v, v, p); }
2656 : GEN
2657 623630 : Flv_inv(GEN w, ulong p)
2658 623630 : { GEN v = cgetg(lg(w), t_VECSMALL); Flv_inv_i(v, w, p); return v; }
2659 :
2660 : GEN
2661 33019352 : Flx_div_by_X_x(GEN a, ulong x, ulong p, ulong *rem)
2662 : {
2663 33019352 : long l = lg(a), i;
2664 : GEN a0, z0, z;
2665 33019352 : if (l <= 3)
2666 : {
2667 0 : if (rem) *rem = l == 2? 0: a[2];
2668 0 : return zero_Flx(a[1]);
2669 : }
2670 33019352 : z = cgetg(l-1,t_VECSMALL); z[1] = a[1];
2671 32896533 : a0 = a + l-1;
2672 32896533 : z0 = z + l-2; *z0 = *a0--;
2673 32896533 : if (SMALL_ULONG(p))
2674 : {
2675 79762676 : for (i=l-3; i>1; i--) /* z[i] = (a[i+1] + x*z[i+1]) % p */
2676 : {
2677 59096051 : ulong t = (*a0-- + x * *z0--) % p;
2678 59096051 : *z0 = (long)t;
2679 : }
2680 20666625 : if (rem) *rem = (*a0 + x * *z0) % p;
2681 : }
2682 : else
2683 : {
2684 48247801 : for (i=l-3; i>1; i--)
2685 : {
2686 35997119 : ulong t = Fl_add((ulong)*a0--, Fl_mul(x, *z0--, p), p);
2687 36017893 : *z0 = (long)t;
2688 : }
2689 12250682 : if (rem) *rem = Fl_add((ulong)*a0, Fl_mul(x, *z0, p), p);
2690 : }
2691 32920044 : return z;
2692 : }
2693 :
2694 : /* xa, ya = t_VECSMALL */
2695 : static GEN
2696 624829 : Flv_producttree(GEN xa, GEN s, ulong p, ulong pi, long vs)
2697 : {
2698 624829 : long n = lg(xa)-1;
2699 624829 : long m = n==1 ? 1: expu(n-1)+1;
2700 624828 : long i, j, k, ls = lg(s);
2701 624828 : GEN T = cgetg(m+1, t_VEC);
2702 624819 : GEN t = cgetg(ls, t_VEC);
2703 7831817 : for (j=1, k=1; j<ls; k+=s[j++])
2704 7206829 : gel(t, j) = s[j] == 1 ?
2705 7206992 : mkvecsmall3(vs, Fl_neg(xa[k], p), 1):
2706 1515895 : mkvecsmall4(vs, Fl_mul(xa[k], xa[k+1], p),
2707 1515902 : Fl_neg(Fl_add(xa[k],xa[k+1],p),p), 1);
2708 624825 : gel(T,1) = t;
2709 2355785 : for (i=2; i<=m; i++)
2710 : {
2711 1731064 : GEN u = gel(T, i-1);
2712 1731064 : long n = lg(u)-1;
2713 1731064 : GEN t = cgetg(((n+1)>>1)+1, t_VEC);
2714 8312176 : for (j=1, k=1; k<n; j++, k+=2)
2715 6581216 : gel(t, j) = Flx_mul_pre(gel(u, k), gel(u, k+1), p, pi);
2716 1730960 : gel(T, i) = t;
2717 : }
2718 624721 : return T;
2719 : }
2720 :
2721 : static GEN
2722 665134 : Flx_Flv_multieval_tree(GEN P, GEN xa, GEN T, ulong p, ulong pi)
2723 : {
2724 : long i,j,k;
2725 665134 : long m = lg(T)-1;
2726 665134 : GEN R = cgetg(lg(xa), t_VECSMALL);
2727 665125 : GEN Tp = cgetg(m+1, t_VEC), t;
2728 665115 : gel(Tp, m) = mkvec(P);
2729 2581480 : for (i=m-1; i>=1; i--)
2730 : {
2731 1916355 : GEN u = gel(T, i), v = gel(Tp, i+1);
2732 1916355 : long n = lg(u)-1;
2733 1916355 : t = cgetg(n+1, t_VEC);
2734 9526740 : for (j=1, k=1; k<n; j++, k+=2)
2735 : {
2736 7610410 : gel(t, k) = Flx_rem_pre(gel(v, j), gel(u, k), p, pi);
2737 7610446 : gel(t, k+1) = Flx_rem_pre(gel(v, j), gel(u, k+1), p, pi);
2738 : }
2739 1916330 : gel(Tp, i) = t;
2740 : }
2741 : {
2742 665125 : GEN u = gel(T, i+1), v = gel(Tp, i+1);
2743 665125 : long n = lg(u)-1;
2744 8944761 : for (j=1, k=1; j<=n; j++)
2745 : {
2746 8279607 : long c, d = degpol(gel(u,j));
2747 18324264 : for (c=1; c<=d; c++, k++) R[k] = Flx_eval_pre(gel(v, j), xa[k], p, pi);
2748 : }
2749 665154 : return gc_const((pari_sp)R, R);
2750 : }
2751 : }
2752 :
2753 : static GEN
2754 1386228 : FlvV_polint_tree(GEN T, GEN R, GEN s, GEN xa, GEN ya, ulong p, ulong pi, long vs)
2755 : {
2756 1386228 : pari_sp av = avma;
2757 1386228 : long m = lg(T)-1;
2758 1386228 : long i, j, k, ls = lg(s);
2759 1386228 : GEN Tp = cgetg(m+1, t_VEC);
2760 1385839 : GEN t = cgetg(ls, t_VEC);
2761 24933975 : for (j=1, k=1; j<ls; k+=s[j++])
2762 23548274 : if (s[j]==2)
2763 : {
2764 6938967 : ulong a = Fl_mul(ya[k], R[k], p);
2765 6938575 : ulong b = Fl_mul(ya[k+1], R[k+1], p);
2766 6944825 : gel(t, j) = mkvecsmall3(vs, Fl_neg(Fl_add(Fl_mul(xa[k], b, p ),
2767 6939019 : Fl_mul(xa[k+1], a, p), p), p), Fl_add(a, b, p));
2768 6942629 : gel(t, j) = Flx_renormalize(gel(t, j), 4);
2769 : }
2770 : else
2771 16609307 : gel(t, j) = Fl_to_Flx(Fl_mul(ya[k], R[k], p), vs);
2772 1385701 : gel(Tp, 1) = t;
2773 6386828 : for (i=2; i<=m; i++)
2774 : {
2775 5001119 : GEN u = gel(T, i-1);
2776 5001119 : GEN t = cgetg(lg(gel(T,i)), t_VEC);
2777 4998324 : GEN v = gel(Tp, i-1);
2778 4998324 : long n = lg(v)-1;
2779 27110070 : for (j=1, k=1; k<n; j++, k+=2)
2780 22096270 : gel(t, j) = Flx_add(Flx_mul_pre(gel(u, k), gel(v, k+1), p, pi),
2781 22108943 : Flx_mul_pre(gel(u, k+1), gel(v, k), p, pi), p);
2782 5001127 : gel(Tp, i) = t;
2783 : }
2784 1385709 : return gerepileuptoleaf(av, gmael(Tp,m,1));
2785 : }
2786 :
2787 : GEN
2788 0 : Flx_Flv_multieval(GEN P, GEN xa, ulong p)
2789 : {
2790 0 : pari_sp av = avma;
2791 0 : GEN s = producttree_scheme(lg(xa)-1);
2792 0 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2793 0 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2794 0 : return gerepileuptoleaf(av, Flx_Flv_multieval_tree(P, xa, T, p, pi));
2795 : }
2796 :
2797 : static GEN
2798 2471 : FlxV_Flv_multieval_tree(GEN x, GEN xa, GEN T, ulong p, ulong pi)
2799 45246 : { pari_APPLY_same(Flx_Flv_multieval_tree(gel(x,i), xa, T, p, pi)) }
2800 :
2801 : GEN
2802 2471 : FlxV_Flv_multieval(GEN P, GEN xa, ulong p)
2803 : {
2804 2471 : pari_sp av = avma;
2805 2471 : GEN s = producttree_scheme(lg(xa)-1);
2806 2471 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2807 2471 : GEN T = Flv_producttree(xa, s, p, pi, P[1]);
2808 2471 : return gerepileupto(av, FlxV_Flv_multieval_tree(P, xa, T, p, pi));
2809 : }
2810 :
2811 : GEN
2812 368282 : Flv_polint(GEN xa, GEN ya, ulong p, long vs)
2813 : {
2814 368282 : pari_sp av = avma;
2815 368282 : GEN s = producttree_scheme(lg(xa)-1);
2816 368288 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2817 368286 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2818 368285 : long m = lg(T)-1;
2819 368285 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2820 368292 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2821 368291 : return gerepileuptoleaf(av, FlvV_polint_tree(T, R, s, xa, ya, p, pi, vs));
2822 : }
2823 :
2824 : GEN
2825 101076 : Flv_Flm_polint(GEN xa, GEN ya, ulong p, long vs)
2826 : {
2827 101076 : pari_sp av = avma;
2828 101076 : GEN s = producttree_scheme(lg(xa)-1);
2829 101077 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2830 101077 : GEN T = Flv_producttree(xa, s, p, pi, vs);
2831 101072 : long i, m = lg(T)-1, l = lg(ya)-1;
2832 101072 : GEN P = Flx_deriv(gmael(T, m, 1), p);
2833 101072 : GEN R = Flv_inv(Flx_Flv_multieval_tree(P, xa, T, p, pi), p);
2834 101074 : GEN M = cgetg(l+1, t_VEC);
2835 1118844 : for (i=1; i<=l; i++)
2836 1017773 : gel(M,i) = FlvV_polint_tree(T, R, s, xa, gel(ya,i), p, pi, vs);
2837 101071 : return gerepileupto(av, M);
2838 : }
2839 :
2840 : GEN
2841 152995 : Flv_invVandermonde(GEN L, ulong den, ulong p)
2842 : {
2843 152995 : pari_sp av = avma;
2844 152995 : long i, n = lg(L);
2845 : GEN M, R;
2846 152995 : GEN s = producttree_scheme(n-1);
2847 152995 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2848 152995 : GEN tree = Flv_producttree(L, s, p, pi, 0);
2849 152995 : long m = lg(tree)-1;
2850 152995 : GEN T = gmael(tree, m, 1);
2851 152995 : R = Flv_inv(Flx_Flv_multieval_tree(Flx_deriv(T, p), L, tree, p, pi), p);
2852 152995 : if (den!=1) R = Flv_Fl_mul(R, den, p);
2853 152995 : M = cgetg(n, t_MAT);
2854 600537 : for (i = 1; i < n; i++)
2855 : {
2856 447542 : GEN P = Flx_Fl_mul(Flx_div_by_X_x(T, uel(L,i), p, NULL), uel(R,i), p);
2857 447542 : gel(M,i) = Flx_to_Flv(P, n-1);
2858 : }
2859 152995 : return gerepilecopy(av, M);
2860 : }
2861 :
2862 : /***********************************************************************/
2863 : /** Flxq **/
2864 : /***********************************************************************/
2865 : /* Flxq objects are Flx modulo another Flx called q. */
2866 :
2867 : /* Product of y and x in Z/pZ[X]/(T), as t_VECSMALL. */
2868 : GEN
2869 194258369 : Flxq_mul_pre(GEN x,GEN y,GEN T,ulong p,ulong pi)
2870 194258369 : { return Flx_rem_pre(Flx_mul_pre(x,y,p,pi),T,p,pi); }
2871 : GEN
2872 13180472 : Flxq_mul(GEN x,GEN y,GEN T,ulong p)
2873 13180472 : { return Flxq_mul_pre(x,y,T,p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2874 :
2875 : GEN
2876 280949805 : Flxq_sqr_pre(GEN x,GEN T,ulong p,ulong pi)
2877 280949805 : { return Flx_rem_pre(Flx_sqr_pre(x, p,pi), T, p,pi); }
2878 : /* Square of y in Z/pZ[X]/(T), as t_VECSMALL. */
2879 : GEN
2880 2746546 : Flxq_sqr(GEN x,GEN T,ulong p)
2881 2746546 : { return Flxq_sqr_pre(x,T,p,SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2882 :
2883 : static GEN
2884 1773388 : _Flxq_red(void *E, GEN x)
2885 1773388 : { struct _Flxq *s = (struct _Flxq *)E;
2886 1773388 : return Flx_rem_pre(x, s->T, s->p, s->pi); }
2887 : #if 0
2888 : static GEN
2889 : _Flx_sub(void *E, GEN x, GEN y)
2890 : { struct _Flxq *s = (struct _Flxq *)E;
2891 : return Flx_sub(x,y,s->p); }
2892 : #endif
2893 : static GEN
2894 273542715 : _Flxq_sqr(void *data, GEN x)
2895 : {
2896 273542715 : struct _Flxq *D = (struct _Flxq*)data;
2897 273542715 : return Flxq_sqr_pre(x, D->T, D->p, D->pi);
2898 : }
2899 : static GEN
2900 152838693 : _Flxq_mul(void *data, GEN x, GEN y)
2901 : {
2902 152838693 : struct _Flxq *D = (struct _Flxq*)data;
2903 152838693 : return Flxq_mul_pre(x,y, D->T, D->p, D->pi);
2904 : }
2905 : static GEN
2906 22674689 : _Flxq_one(void *data)
2907 : {
2908 22674689 : struct _Flxq *D = (struct _Flxq*)data;
2909 22674689 : return pol1_Flx(get_Flx_var(D->T));
2910 : }
2911 :
2912 : static GEN
2913 22868444 : _Flxq_powu_i(struct _Flxq *D, GEN x, ulong n)
2914 22868444 : { return gen_powu_i(x, n, (void*)D, &_Flxq_sqr, &_Flxq_mul); }
2915 : static GEN
2916 68 : _Flxq_powu(struct _Flxq *D, GEN x, ulong n)
2917 68 : { pari_sp av = avma; return gerepileuptoleaf(av, _Flxq_powu_i(D, x, n)); }
2918 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2919 : GEN
2920 23912298 : Flxq_powu_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
2921 : {
2922 : pari_sp av;
2923 : struct _Flxq D;
2924 23912298 : switch(n)
2925 : {
2926 0 : case 0: return pol1_Flx(get_Flx_var(T));
2927 144454 : case 1: return Flx_copy(x);
2928 899441 : case 2: return Flxq_sqr_pre(x, T, p, pi);
2929 : }
2930 22868403 : av = avma; set_Flxq_pre(&D, T, p, pi);
2931 22868122 : return gerepileuptoleaf(av, _Flxq_powu_i(&D, x, n));
2932 : }
2933 : GEN
2934 486933 : Flxq_powu(GEN x, ulong n, GEN T, ulong p)
2935 486933 : { return Flxq_powu_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2936 :
2937 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL. */
2938 : GEN
2939 28061267 : Flxq_pow_pre(GEN x, GEN n, GEN T, ulong p, ulong pi)
2940 : {
2941 28061267 : pari_sp av = avma;
2942 : struct _Flxq D;
2943 : GEN y;
2944 28061267 : long s = signe(n);
2945 28061267 : if (!s) return pol1_Flx(get_Flx_var(T));
2946 27825624 : if (s < 0) x = Flxq_inv_pre(x,T,p,pi);
2947 27825625 : if (is_pm1(n)) return s < 0 ? x : Flx_copy(x);
2948 26650285 : set_Flxq_pre(&D, T, p, pi);
2949 26650323 : y = gen_pow_i(x, n, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2950 26650258 : return gerepileuptoleaf(av, y);
2951 : }
2952 : GEN
2953 930763 : Flxq_pow(GEN x, GEN n, GEN T, ulong p)
2954 930763 : { return Flxq_pow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2955 :
2956 : GEN
2957 28 : Flxq_pow_init_pre(GEN x, GEN n, long k, GEN T, ulong p, ulong pi)
2958 : {
2959 28 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2960 28 : return gen_pow_init(x, n, k, (void*)&D, &_Flxq_sqr, &_Flxq_mul);
2961 : }
2962 : GEN
2963 0 : Flxq_pow_init(GEN x, GEN n, long k, GEN T, ulong p)
2964 0 : { return Flxq_pow_init_pre(x, n, k, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2965 :
2966 : GEN
2967 4393 : Flxq_pow_table_pre(GEN R, GEN n, GEN T, ulong p, ulong pi)
2968 : {
2969 4393 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
2970 4393 : return gen_pow_table(R, n, (void*)&D, &_Flxq_one, &_Flxq_mul);
2971 : }
2972 : GEN
2973 0 : Flxq_pow_table(GEN R, GEN n, GEN T, ulong p)
2974 0 : { return Flxq_pow_table_pre(R, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2975 :
2976 : /* Inverse of x in Z/lZ[X]/(T) or NULL if inverse doesn't exist
2977 : * not stack clean. */
2978 : GEN
2979 5885097 : Flxq_invsafe_pre(GEN x, GEN T, ulong p, ulong pi)
2980 : {
2981 5885097 : GEN V, z = Flx_extgcd_pre(get_Flx_mod(T), x, p, pi, NULL, &V);
2982 : ulong iz;
2983 5885208 : if (degpol(z)) return NULL;
2984 5884549 : iz = Fl_inv(uel(z,2), p);
2985 5884553 : return Flx_Fl_mul_pre(V, iz, p, pi);
2986 : }
2987 : GEN
2988 656243 : Flxq_invsafe(GEN x, GEN T, ulong p)
2989 656243 : { return Flxq_invsafe_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
2990 :
2991 : GEN
2992 4783190 : Flxq_inv_pre(GEN x, GEN T, ulong p, ulong pi)
2993 : {
2994 4783190 : pari_sp av=avma;
2995 4783190 : GEN U = Flxq_invsafe_pre(x, T, p, pi);
2996 4783188 : if (!U) pari_err_INV("Flxq_inv",Flx_to_ZX(x));
2997 4783181 : return gerepileuptoleaf(av, U);
2998 : }
2999 : GEN
3000 345508 : Flxq_inv(GEN x, GEN T, ulong p)
3001 345508 : { return Flxq_inv_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3002 :
3003 : GEN
3004 2388876 : Flxq_div_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
3005 : {
3006 2388876 : pari_sp av = avma;
3007 2388876 : return gerepileuptoleaf(av, Flxq_mul_pre(x,Flxq_inv_pre(y,T,p,pi),T,p,pi));
3008 : }
3009 : GEN
3010 237395 : Flxq_div(GEN x, GEN y, GEN T, ulong p)
3011 237395 : { return Flxq_div_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3012 :
3013 : GEN
3014 22677745 : Flxq_powers_pre(GEN x, long l, GEN T, ulong p, ulong pi)
3015 : {
3016 22677745 : int use_sqr = 2*degpol(x) >= get_Flx_degree(T);
3017 22673930 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3018 22671396 : return gen_powers(x, l, use_sqr, (void*)&D, &_Flxq_sqr, &_Flxq_mul, &_Flxq_one);
3019 : }
3020 : GEN
3021 232191 : Flxq_powers(GEN x, long l, GEN T, ulong p)
3022 232191 : { return Flxq_powers_pre(x, l, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3023 :
3024 : GEN
3025 170490 : Flxq_matrix_pow_pre(GEN y, long n, long m, GEN P, ulong l, ulong li)
3026 170490 : { return FlxV_to_Flm(Flxq_powers_pre(y,m-1,P,l,li),n); }
3027 : GEN
3028 399 : Flxq_matrix_pow(GEN y, long n, long m, GEN P, ulong l)
3029 399 : { return Flxq_matrix_pow_pre(y, n, m, P, l, SMALL_ULONG(l)? 0: get_Fl_red(l)); }
3030 :
3031 : GEN
3032 13665140 : Flx_Frobenius_pre(GEN T, ulong p, ulong pi)
3033 13665140 : { return Flxq_powu_pre(polx_Flx(get_Flx_var(T)), p, T, p, pi); }
3034 : GEN
3035 86486 : Flx_Frobenius(GEN T, ulong p)
3036 86486 : { return Flx_Frobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3037 :
3038 : GEN
3039 86455 : Flx_matFrobenius_pre(GEN T, ulong p, ulong pi)
3040 : {
3041 86455 : long n = get_Flx_degree(T);
3042 86455 : return Flxq_matrix_pow_pre(Flx_Frobenius_pre(T, p, pi), n, n, T, p, pi);
3043 : }
3044 : GEN
3045 0 : Flx_matFrobenius(GEN T, ulong p)
3046 0 : { return Flx_matFrobenius_pre(T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3047 :
3048 : static GEN
3049 13017845 : Flx_blocks_Flm(GEN P, long n, long m)
3050 : {
3051 13017845 : GEN z = cgetg(m+1,t_MAT);
3052 13017584 : long i,j, k=2, l = lg(P);
3053 38571339 : for(i=1; i<=m; i++)
3054 : {
3055 25557759 : GEN zi = cgetg(n+1,t_VECSMALL);
3056 25553755 : gel(z,i) = zi;
3057 113461349 : for(j=1; j<=n; j++)
3058 87907594 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3059 : }
3060 13013580 : return z;
3061 : }
3062 :
3063 : GEN
3064 516391 : Flx_blocks(GEN P, long n, long m)
3065 : {
3066 516391 : GEN z = cgetg(m+1,t_VEC);
3067 516114 : long i,j, k=2, l = lg(P);
3068 1546632 : for(i=1; i<=m; i++)
3069 : {
3070 1030827 : GEN zi = cgetg(n+2,t_VECSMALL);
3071 1030138 : zi[1] = P[1];
3072 1030138 : gel(z,i) = zi;
3073 6459077 : for(j=2; j<n+2; j++)
3074 5428939 : uel(zi, j) = k==l ? 0 : uel(P,k++);
3075 1030138 : zi = Flx_renormalize(zi, n+2);
3076 : }
3077 515805 : return z;
3078 : }
3079 :
3080 : static GEN
3081 13019056 : FlxV_to_Flm_lg(GEN x, long m, long n)
3082 : {
3083 : long i;
3084 13019056 : GEN y = cgetg(n+1, t_MAT);
3085 59189056 : for (i=1; i<=n; i++) gel(y,i) = Flx_to_Flv(gel(x,i), m);
3086 13015710 : return y;
3087 : }
3088 :
3089 : /* allow pi = 0 (SMALL_ULONG) */
3090 : GEN
3091 13216833 : Flx_FlxqV_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3092 : {
3093 13216833 : pari_sp btop, av = avma;
3094 13216833 : long sv = get_Flx_var(T), m = get_Flx_degree(T);
3095 13217152 : long i, l = lg(x)-1, lQ = lgpol(Q), n, d;
3096 : GEN A, B, C, S, g;
3097 13217975 : if (lQ == 0) return pol0_Flx(sv);
3098 13019830 : if (lQ <= l)
3099 : {
3100 5803644 : n = l;
3101 5803644 : d = 1;
3102 : }
3103 : else
3104 : {
3105 7216186 : n = l-1;
3106 7216186 : d = (lQ+n-1)/n;
3107 : }
3108 13019830 : A = FlxV_to_Flm_lg(x, m, n);
3109 13017729 : B = Flx_blocks_Flm(Q, n, d);
3110 13016961 : C = gerepileupto(av, Flm_mul(A, B, p));
3111 13020351 : g = gel(x, l);
3112 13020351 : if (pi && SMALL_ULONG(p)) pi = 0;
3113 13020351 : T = Flx_get_red_pre(T, p, pi);
3114 13019913 : btop = avma;
3115 13019913 : S = Flv_to_Flx(gel(C, d), sv);
3116 25562912 : for (i = d-1; i>0; i--)
3117 : {
3118 12544217 : S = Flx_add(Flxq_mul_pre(S, g, T, p, pi), Flv_to_Flx(gel(C,i), sv), p);
3119 12543830 : if (gc_needed(btop,1))
3120 0 : S = gerepileuptoleaf(btop, S);
3121 : }
3122 13018695 : return gerepileuptoleaf(av, S);
3123 : }
3124 : GEN
3125 5103 : Flx_FlxqV_eval(GEN Q, GEN x, GEN T, ulong p)
3126 5103 : { return Flx_FlxqV_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3127 :
3128 : /* allow pi = 0 (SMALL_ULONG) */
3129 : GEN
3130 2625881 : Flx_Flxq_eval_pre(GEN Q, GEN x, GEN T, ulong p, ulong pi)
3131 : {
3132 2625881 : pari_sp av = avma;
3133 : GEN z, V;
3134 2625881 : long d = degpol(Q), rtd;
3135 2625880 : if (d < 0) return pol0_Flx(get_Flx_var(T));
3136 2625789 : rtd = (long) sqrt((double)d);
3137 2625789 : T = Flx_get_red_pre(T, p, pi);
3138 2625801 : V = Flxq_powers_pre(x, rtd, T, p, pi);
3139 2625841 : z = Flx_FlxqV_eval_pre(Q, V, T, p, pi);
3140 2625806 : return gerepileupto(av, z);
3141 : }
3142 : GEN
3143 786372 : Flx_Flxq_eval(GEN Q, GEN x, GEN T, ulong p)
3144 786372 : { return Flx_Flxq_eval_pre(Q, x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3145 :
3146 : /* allow pi = 0 (SMALL_ULONG) */
3147 : GEN
3148 0 : FlxC_FlxqV_eval_pre(GEN x, GEN v, GEN T, ulong p, ulong pi)
3149 0 : { pari_APPLY_type(t_COL, Flx_FlxqV_eval_pre(gel(x,i), v, T, p, pi)) }
3150 : GEN
3151 0 : FlxC_FlxqV_eval(GEN x, GEN v, GEN T, ulong p)
3152 0 : { return FlxC_FlxqV_eval_pre(x, v, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3153 :
3154 : /* allow pi = 0 (SMALL_ULONG) */
3155 : GEN
3156 0 : FlxC_Flxq_eval_pre(GEN x, GEN F, GEN T, ulong p, ulong pi)
3157 : {
3158 0 : long d = brent_kung_optpow(get_Flx_degree(T)-1,lg(x)-1,1);
3159 0 : GEN Fp = Flxq_powers_pre(F, d, T, p, pi);
3160 0 : return FlxC_FlxqV_eval_pre(x, Fp, T, p, pi);
3161 : }
3162 : GEN
3163 0 : FlxC_Flxq_eval(GEN x, GEN F, GEN T, ulong p)
3164 0 : { return FlxC_Flxq_eval_pre(x, F, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3165 :
3166 : #if 0
3167 : static struct bb_algebra Flxq_algebra = { _Flxq_red, _Flx_add, _Flx_sub,
3168 : _Flxq_mul, _Flxq_sqr, _Flxq_one, _Flxq_zero};
3169 : #endif
3170 :
3171 : static GEN
3172 385179 : Flxq_autpow_sqr(void *E, GEN x)
3173 : {
3174 385179 : struct _Flxq *D = (struct _Flxq*)E;
3175 385179 : return Flx_Flxq_eval_pre(x, x, D->T, D->p, D->pi);
3176 : }
3177 : static GEN
3178 20711 : Flxq_autpow_msqr(void *E, GEN x)
3179 : {
3180 20711 : struct _Flxq *D = (struct _Flxq*)E;
3181 20711 : return Flx_FlxqV_eval_pre(Flxq_autpow_sqr(E, x), D->aut, D->T, D->p, D->pi);
3182 : }
3183 :
3184 : GEN
3185 303786 : Flxq_autpow_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3186 : {
3187 303786 : pari_sp av = avma;
3188 : struct _Flxq D;
3189 : long d;
3190 303786 : if (n==0) return Flx_rem_pre(polx_Flx(x[1]), T, p, pi);
3191 303779 : if (n==1) return Flx_rem_pre(x, T, p, pi);
3192 303289 : set_Flxq_pre(&D, T, p, pi);
3193 303289 : d = brent_kung_optpow(get_Flx_degree(T), hammingl(n)-1, 1);
3194 303289 : D.aut = Flxq_powers_pre(x, d, T, p, D.pi);
3195 303289 : x = gen_powu_fold_i(x,n,(void*)&D,Flxq_autpow_sqr,Flxq_autpow_msqr);
3196 303289 : return gerepilecopy(av, x);
3197 : }
3198 : GEN
3199 7 : Flxq_autpow(GEN x, ulong n, GEN T, ulong p)
3200 7 : { return Flxq_autpow_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3201 :
3202 : GEN
3203 1667 : Flxq_autpowers(GEN x, ulong l, GEN T, ulong p)
3204 : {
3205 1667 : long d, vT = get_Flx_var(T), dT = get_Flx_degree(T);
3206 : ulong i, pi;
3207 1667 : pari_sp av = avma;
3208 1667 : GEN xp, V = cgetg(l+2,t_VEC);
3209 1667 : gel(V,1) = polx_Flx(vT); if (l==0) return V;
3210 1667 : gel(V,2) = gcopy(x); if (l==1) return V;
3211 1667 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3212 1667 : T = Flx_get_red_pre(T, p, pi);
3213 1667 : d = brent_kung_optpow(dT-1, l-1, 1);
3214 1667 : xp = Flxq_powers_pre(x, d, T, p, pi);
3215 6998 : for(i = 3; i < l+2; i++)
3216 5331 : gel(V,i) = Flx_FlxqV_eval_pre(gel(V,i-1), xp, T, p, pi);
3217 1667 : return gerepilecopy(av, V);
3218 : }
3219 :
3220 : static GEN
3221 588709 : Flxq_autsum_mul(void *E, GEN x, GEN y)
3222 : {
3223 588709 : struct _Flxq *D = (struct _Flxq*)E;
3224 588709 : GEN T = D->T;
3225 588709 : ulong p = D->p, pi = D->pi;
3226 588709 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3227 588709 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3228 588709 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3229 588709 : GEN V2 = Flxq_powers_pre(phi2, d, T, p, pi);
3230 588709 : GEN phi3 = Flx_FlxqV_eval_pre(phi1, V2, T, p, pi);
3231 588709 : GEN aphi = Flx_FlxqV_eval_pre(a1, V2, T, p, pi);
3232 588709 : GEN a3 = Flxq_mul_pre(aphi, a2, T, p, pi);
3233 588709 : return mkvec2(phi3, a3);
3234 : }
3235 : static GEN
3236 379851 : Flxq_autsum_sqr(void *E, GEN x)
3237 379851 : { return Flxq_autsum_mul(E, x, x); }
3238 :
3239 : static GEN
3240 315752 : Flxq_autsum_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3241 : {
3242 315752 : pari_sp av = avma;
3243 315752 : struct _Flxq D; set_Flxq_pre(&D, T, p, pi);
3244 315752 : x = gen_powu_i(x,n,(void*)&D,Flxq_autsum_sqr,Flxq_autsum_mul);
3245 315752 : return gerepilecopy(av, x);
3246 : }
3247 : GEN
3248 0 : Flxq_autsum(GEN x, ulong n, GEN T, ulong p)
3249 0 : { return Flxq_autsum_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3250 :
3251 : static GEN
3252 762697 : Flxq_auttrace_mul(void *E, GEN x, GEN y)
3253 : {
3254 762697 : struct _Flxq *D = (struct _Flxq*)E;
3255 762697 : GEN T = D->T;
3256 762697 : ulong p = D->p, pi = D->pi;
3257 762697 : GEN phi1 = gel(x,1), a1 = gel(x,2);
3258 762697 : GEN phi2 = gel(y,1), a2 = gel(y,2);
3259 762697 : ulong d = brent_kung_optpow(maxss(degpol(phi1),degpol(a1)),2,1);
3260 762709 : GEN V1 = Flxq_powers_pre(phi1, d, T, p, pi);
3261 762665 : GEN phi3 = Flx_FlxqV_eval_pre(phi2, V1, T, p, pi);
3262 762678 : GEN aphi = Flx_FlxqV_eval_pre(a2, V1, T, p, pi);
3263 762689 : GEN a3 = Flx_add(a1, aphi, p);
3264 762705 : return mkvec2(phi3, a3);
3265 : }
3266 :
3267 : static GEN
3268 635692 : Flxq_auttrace_sqr(void *E, GEN x)
3269 635692 : { return Flxq_auttrace_mul(E, x, x); }
3270 :
3271 : GEN
3272 934980 : Flxq_auttrace_pre(GEN x, ulong n, GEN T, ulong p, ulong pi)
3273 : {
3274 934980 : pari_sp av = avma;
3275 : struct _Flxq D;
3276 934980 : set_Flxq_pre(&D, T, p, pi);
3277 934983 : x = gen_powu_i(x,n,(void*)&D,Flxq_auttrace_sqr,Flxq_auttrace_mul);
3278 934971 : return gerepilecopy(av, x);
3279 : }
3280 : GEN
3281 0 : Flxq_auttrace(GEN x, ulong n, GEN T, ulong p)
3282 0 : { return Flxq_auttrace_pre(x, n, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3283 :
3284 : static long
3285 849195 : bounded_order(ulong p, GEN b, long k)
3286 : {
3287 849195 : GEN a = modii(utoipos(p), b);
3288 : long i;
3289 2071848 : for(i = 1; i < k; i++)
3290 : {
3291 1606483 : if (equali1(a)) return i;
3292 1222652 : a = modii(muliu(a,p),b);
3293 : }
3294 465365 : return 0;
3295 : }
3296 :
3297 : /* n = (p^d-a)\b
3298 : * b = bb*p^vb
3299 : * p^k = 1 [bb]
3300 : * d = m*k+r+vb
3301 : * u = (p^k-1)/bb;
3302 : * v = (p^(r+vb)-a)/b;
3303 : * w = (p^(m*k)-1)/(p^k-1)
3304 : * n = p^r*w*u+v
3305 : * w*u = p^vb*(p^(m*k)-1)/b
3306 : * n = p^(r+vb)*(p^(m*k)-1)/b+(p^(r+vb)-a)/b */
3307 : static GEN
3308 27296099 : Flxq_pow_Frobenius(GEN x, GEN n, GEN aut, GEN T, ulong p, ulong pi)
3309 : {
3310 27296099 : pari_sp av=avma;
3311 27296099 : long d = get_Flx_degree(T);
3312 27296097 : GEN an = absi_shallow(n), z, q;
3313 27296099 : if (abscmpiu(an,p)<0 || cmpis(an,d)<=0) return Flxq_pow_pre(x, n, T, p, pi);
3314 849590 : q = powuu(p, d);
3315 849590 : if (dvdii(q, n))
3316 : {
3317 332 : long vn = logint(an, utoipos(p));
3318 332 : GEN autvn = vn==1 ? aut: Flxq_autpow_pre(aut,vn,T,p,pi);
3319 332 : z = Flx_Flxq_eval_pre(x,autvn,T,p,pi);
3320 : } else
3321 : {
3322 849257 : GEN b = diviiround(q, an), a = subii(q, mulii(an,b));
3323 : GEN bb, u, v, autk;
3324 849257 : long vb = Z_lvalrem(b,p,&bb);
3325 849257 : long m, r, k = is_pm1(bb)? 1: bounded_order(p,bb,d);
3326 849258 : if (!k || d-vb < k) return Flxq_pow_pre(x,n, T,p,pi);
3327 383886 : m = (d-vb)/k; r = (d-vb)%k;
3328 383886 : u = diviiexact(subiu(powuu(p,k),1),bb);
3329 383886 : v = diviiexact(subii(powuu(p,r+vb),a),b);
3330 383886 : autk = k==1 ? aut: Flxq_autpow_pre(aut,k,T,p,pi);
3331 383886 : if (r)
3332 : {
3333 94181 : GEN autr = r==1 ? aut: Flxq_autpow_pre(aut,r,T,p,pi);
3334 94181 : z = Flx_Flxq_eval_pre(x,autr,T,p,pi);
3335 289705 : } else z = x;
3336 383886 : if (m > 1) z = gel(Flxq_autsum_pre(mkvec2(autk, z), m, T, p, pi), 2);
3337 383886 : if (!is_pm1(u)) z = Flxq_pow_pre(z, u, T, p, pi);
3338 383886 : if (signe(v)) z = Flxq_mul_pre(z, Flxq_pow_pre(x, v, T, p, pi), T, p, pi);
3339 : }
3340 384218 : return gerepileupto(av,signe(n)>0 ? z : Flxq_inv_pre(z,T,p,pi));
3341 : }
3342 :
3343 : static GEN
3344 27288724 : _Flxq_pow(void *data, GEN x, GEN n)
3345 : {
3346 27288724 : struct _Flxq *D = (struct _Flxq*)data;
3347 27288724 : return Flxq_pow_Frobenius(x, n, D->aut, D->T, D->p, D->pi);
3348 : }
3349 :
3350 : static GEN
3351 329857 : _Flxq_rand(void *data)
3352 : {
3353 329857 : pari_sp av=avma;
3354 329857 : struct _Flxq *D = (struct _Flxq*)data;
3355 : GEN z;
3356 : do
3357 : {
3358 331242 : set_avma(av);
3359 331242 : z = random_Flx(get_Flx_degree(D->T),get_Flx_var(D->T),D->p);
3360 331242 : } while (lgpol(z)==0);
3361 329857 : return z;
3362 : }
3363 :
3364 : /* discrete log in FpXQ for a in Fp^*, g in FpXQ^* of order ord */
3365 : static GEN
3366 35417 : Fl_Flxq_log(ulong a, GEN g, GEN o, GEN T, ulong p)
3367 : {
3368 35417 : pari_sp av = avma;
3369 : GEN q,n_q,ord,ordp, op;
3370 :
3371 35417 : if (a == 1UL) return gen_0;
3372 : /* p > 2 */
3373 :
3374 35417 : ordp = utoi(p - 1);
3375 35417 : ord = get_arith_Z(o);
3376 35417 : if (!ord) ord = T? subiu(powuu(p, get_FpX_degree(T)), 1): ordp;
3377 35417 : if (a == p - 1) /* -1 */
3378 7732 : return gerepileuptoint(av, shifti(ord,-1));
3379 27685 : ordp = gcdii(ordp, ord);
3380 27685 : op = typ(o)==t_MAT ? famat_Z_gcd(o, ordp) : ordp;
3381 :
3382 27685 : q = NULL;
3383 27685 : if (T)
3384 : { /* we want < g > = Fp^* */
3385 27685 : if (!equalii(ord,ordp)) {
3386 11906 : q = diviiexact(ord,ordp);
3387 11906 : g = Flxq_pow(g,q,T,p);
3388 : }
3389 : }
3390 27685 : n_q = Fp_log(utoi(a), utoipos(uel(g,2)), op, utoipos(p));
3391 27685 : if (lg(n_q)==1) return gerepileuptoleaf(av, n_q);
3392 27685 : if (q) n_q = mulii(q, n_q);
3393 27685 : return gerepileuptoint(av, n_q);
3394 : }
3395 :
3396 : static GEN
3397 691022 : Flxq_easylog(void* E, GEN a, GEN g, GEN ord)
3398 : {
3399 691022 : struct _Flxq *f = (struct _Flxq *)E;
3400 691022 : GEN T = f->T;
3401 691022 : ulong p = f->p;
3402 691022 : long d = get_Flx_degree(T);
3403 691022 : if (Flx_equal1(a)) return gen_0;
3404 531209 : if (Flx_equal(a,g)) return gen_1;
3405 174301 : if (!degpol(a))
3406 35417 : return Fl_Flxq_log(uel(a,2), g, ord, T, p);
3407 138884 : if (typ(ord)!=t_INT || d <= 4 || d == 6 || abscmpiu(ord,1UL<<27)<0)
3408 138856 : return NULL;
3409 28 : return Flxq_log_index(a, g, ord, T, p);
3410 : }
3411 :
3412 : static const struct bb_group Flxq_star={_Flxq_mul,_Flxq_pow,_Flxq_rand,hash_GEN,Flx_equal,Flx_equal1,Flxq_easylog};
3413 :
3414 : const struct bb_group *
3415 431496 : get_Flxq_star(void **E, GEN T, ulong p)
3416 : {
3417 431496 : struct _Flxq *e = (struct _Flxq *) stack_malloc(sizeof(struct _Flxq));
3418 431496 : e->T = T; e->p = p; e->pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3419 431496 : e->aut = Flx_Frobenius_pre(T, p, e->pi);
3420 431496 : *E = (void*)e; return &Flxq_star;
3421 : }
3422 :
3423 : GEN
3424 96500 : Flxq_order(GEN a, GEN ord, GEN T, ulong p)
3425 : {
3426 : void *E;
3427 96500 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3428 96500 : return gen_order(a,ord,E,S);
3429 : }
3430 :
3431 : GEN
3432 164469 : Flxq_log(GEN a, GEN g, GEN ord, GEN T, ulong p)
3433 : {
3434 : void *E;
3435 164469 : pari_sp av = avma;
3436 164469 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3437 164469 : GEN v = get_arith_ZZM(ord), F = gmael(v,2,1);
3438 164469 : if (lg(F) > 1 && Flxq_log_use_index(veclast(F), T, p))
3439 24381 : v = mkvec2(gel(v, 1), ZM_famat_limit(gel(v, 2), int2n(27)));
3440 164469 : return gerepileuptoleaf(av, gen_PH_log(a, g, v, E, S));
3441 : }
3442 :
3443 : GEN
3444 176624 : Flxq_sqrtn(GEN a, GEN n, GEN T, ulong p, GEN *zeta)
3445 : {
3446 176624 : if (!lgpol(a))
3447 : {
3448 6097 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
3449 6090 : if (zeta)
3450 0 : *zeta=pol1_Flx(get_Flx_var(T));
3451 6090 : return pol0_Flx(get_Flx_var(T));
3452 : }
3453 : else
3454 : {
3455 : void *E;
3456 170527 : pari_sp av = avma;
3457 170527 : const struct bb_group *S = get_Flxq_star(&E,T,p);
3458 170527 : GEN o = subiu(powuu(p,get_Flx_degree(T)), 1);
3459 170527 : GEN s = gen_Shanks_sqrtn(a,n,o,zeta,E,S);
3460 170528 : if (!s) return gc_NULL(av);
3461 170430 : return gc_all(av, zeta?2:1, &s, zeta);
3462 : }
3463 : }
3464 :
3465 : GEN
3466 224361 : Flxq_sqrt_pre(GEN z, GEN T, ulong p, ulong pi)
3467 : {
3468 224361 : pari_sp av = avma;
3469 224361 : if (p==2)
3470 : {
3471 0 : GEN r = F2xq_sqrt(Flx_to_F2x(z), Flx_to_F2x(get_Flx_mod(T)));
3472 0 : return gerepileupto(av, F2x_to_Flx(r));
3473 : }
3474 224361 : if (get_Flx_degree(T)==2)
3475 : {
3476 67842 : GEN P = get_Flx_mod(T), s;
3477 67842 : ulong c = uel(P,2), b = uel(P,3), a = uel(P,4);
3478 67842 : ulong y = degpol(z)<1 ? 0: uel(z,3);
3479 67842 : if (a==1 && b==0)
3480 14890 : {
3481 15670 : ulong x = degpol(z)<1 ? Flx_constant(z): uel(z,2);
3482 15670 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), Fl_neg(c, p), p, pi);
3483 15670 : if (!r) return gc_NULL(av);
3484 14890 : s = mkvecsmall3(P[1], uel(r,1), uel(r,2));
3485 : }
3486 : else
3487 : {
3488 52172 : ulong b2 = Fl_halve(b, p), t = Fl_div(b2, a, p);
3489 52172 : ulong D = Fl_sub(Fl_sqr(b2, p), Fl_mul(a, c, p), p);
3490 52172 : ulong x = degpol(z)<1 ? Flx_constant(z): Fl_sub(uel(z,2), Fl_mul(uel(z,3), t, p), p);
3491 52172 : GEN r = Fl2_sqrt_pre(mkvecsmall2(x, y), D, p, pi);
3492 52172 : if (!r) return gc_NULL(av);
3493 49778 : s = mkvecsmall3(P[1], Fl_add(uel(r,1), Fl_mul(uel(r,2),t,p), p), uel(r,2));
3494 : }
3495 64668 : return gerepileuptoleaf(av, Flx_renormalize(s, 4));
3496 : }
3497 : else
3498 156519 : return Flxq_sqrtn(z, gen_2, T, p, NULL);
3499 : }
3500 :
3501 : GEN
3502 224361 : Flxq_sqrt(GEN a, GEN T, ulong p)
3503 224361 : { return Flxq_sqrt_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3504 :
3505 : /* assume T irreducible mod p */
3506 : int
3507 399374 : Flxq_issquare(GEN x, GEN T, ulong p)
3508 : {
3509 399374 : if (lgpol(x) == 0 || p == 2) return 1;
3510 393032 : return krouu(Flxq_norm(x,T,p), p) == 1;
3511 : }
3512 :
3513 : /* assume T irreducible mod p */
3514 : int
3515 0 : Flxq_is2npower(GEN x, long n, GEN T, ulong p)
3516 : {
3517 : pari_sp av;
3518 : GEN m;
3519 0 : if (n==1) return Flxq_issquare(x, T, p);
3520 0 : if (lgpol(x) == 0 || p == 2) return 1;
3521 0 : av = avma;
3522 0 : m = shifti(subiu(powuu(p, get_Flx_degree(T)), 1), -n);
3523 0 : return gc_bool(av, Flx_equal1(Flxq_pow(x, m, T, p)));
3524 : }
3525 :
3526 : GEN
3527 113589 : Flxq_lroot_fast_pre(GEN a, GEN sqx, GEN T, long p, ulong pi)
3528 : {
3529 113589 : pari_sp av=avma;
3530 113589 : GEN A = Flx_splitting(a,p);
3531 113589 : return gerepileuptoleaf(av, FlxqV_dotproduct_pre(A,sqx,T,p,pi));
3532 : }
3533 : GEN
3534 0 : Flxq_lroot_fast(GEN a, GEN sqx, GEN T, long p)
3535 0 : { return Flxq_lroot_fast_pre(a, sqx, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3536 :
3537 : GEN
3538 25053 : Flxq_lroot_pre(GEN a, GEN T, long p, ulong pi)
3539 : {
3540 25053 : pari_sp av=avma;
3541 25053 : long n = get_Flx_degree(T), d = degpol(a);
3542 : GEN sqx, V;
3543 25053 : if (n==1) return leafcopy(a);
3544 25053 : if (n==2) return Flxq_powu_pre(a, p, T, p, pi);
3545 25053 : sqx = Flxq_autpow_pre(Flx_Frobenius_pre(T, p, pi), n-1, T, p, pi);
3546 25053 : if (d==1 && a[2]==0 && a[3]==1) return gerepileuptoleaf(av, sqx);
3547 0 : if (d>=p)
3548 : {
3549 0 : V = Flxq_powers_pre(sqx,p-1,T,p,pi);
3550 0 : return gerepileuptoleaf(av, Flxq_lroot_fast_pre(a,V,T,p,pi));
3551 : } else
3552 0 : return gerepileuptoleaf(av, Flx_Flxq_eval_pre(a,sqx,T,p,pi));
3553 : }
3554 : GEN
3555 0 : Flxq_lroot(GEN a, GEN T, long p)
3556 0 : { return Flxq_lroot_pre(a, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3557 :
3558 : ulong
3559 438305 : Flxq_norm(GEN x, GEN TB, ulong p)
3560 : {
3561 438305 : GEN T = get_Flx_mod(TB);
3562 438305 : ulong y = Flx_resultant(T, x, p), L = Flx_lead(T);
3563 438305 : if (L==1 || lgpol(x)==0) return y;
3564 0 : return Fl_div(y, Fl_powu(L, (ulong)degpol(x), p), p);
3565 : }
3566 :
3567 : ulong
3568 4402 : Flxq_trace(GEN x, GEN TB, ulong p)
3569 : {
3570 4402 : pari_sp av = avma;
3571 : ulong t;
3572 4402 : GEN T = get_Flx_mod(TB);
3573 4402 : long n = degpol(T)-1;
3574 4402 : GEN z = Flxq_mul(x, Flx_deriv(T, p), TB, p);
3575 4402 : t = degpol(z)<n ? 0 : Fl_div(z[2+n],T[3+n],p);
3576 4402 : return gc_ulong(av, t);
3577 : }
3578 :
3579 : /*x must be reduced*/
3580 : GEN
3581 3632 : Flxq_charpoly(GEN x, GEN TB, ulong p)
3582 : {
3583 3632 : pari_sp ltop=avma;
3584 3632 : GEN T = get_Flx_mod(TB);
3585 3632 : long vs = evalvarn(fetch_var());
3586 3632 : GEN xm1 = deg1pol_shallow(pol1_Flx(x[1]),Flx_neg(x,p),vs);
3587 3632 : GEN r = Flx_FlxY_resultant(T, xm1, p);
3588 3632 : r[1] = x[1];
3589 3632 : (void)delete_var(); return gerepileupto(ltop, r);
3590 : }
3591 :
3592 : /* Computing minimal polynomial : */
3593 : /* cf Shoup 'Efficient Computation of Minimal Polynomials */
3594 : /* in Algebraic Extensions of Finite Fields' */
3595 :
3596 : /* Let v a linear form, return the linear form z->v(tau*z)
3597 : that is, v*(M_tau) */
3598 :
3599 : static GEN
3600 1690703 : Flxq_transmul_init(GEN tau, GEN T, ulong p, ulong pi)
3601 : {
3602 : GEN bht;
3603 1690703 : GEN h, Tp = get_Flx_red(T, &h);
3604 1690695 : long n = degpol(Tp), vT = Tp[1];
3605 1690693 : GEN ft = Flx_recipspec(Tp+2, n+1, n+1);
3606 1690691 : GEN bt = Flx_recipspec(tau+2, lgpol(tau), n);
3607 1690691 : ft[1] = vT; bt[1] = vT;
3608 1690691 : if (h)
3609 2688 : bht = Flxn_mul_pre(bt, h, n-1, p, pi);
3610 : else
3611 : {
3612 1688003 : GEN bh = Flx_div_pre(Flx_shift(tau, n-1), T, p, pi);
3613 1688001 : bht = Flx_recipspec(bh+2, lgpol(bh), n-1);
3614 1688000 : bht[1] = vT;
3615 : }
3616 1690688 : return mkvec3(bt, bht, ft);
3617 : }
3618 :
3619 : static GEN
3620 4081495 : Flxq_transmul(GEN tau, GEN a, long n, ulong p, ulong pi)
3621 : {
3622 4081495 : pari_sp ltop = avma;
3623 : GEN t1, t2, t3, vec;
3624 4081495 : GEN bt = gel(tau, 1), bht = gel(tau, 2), ft = gel(tau, 3);
3625 4081495 : if (lgpol(a)==0) return pol0_Flx(a[1]);
3626 4050912 : t2 = Flx_shift(Flx_mul_pre(bt, a, p, pi),1-n);
3627 4050551 : if (lgpol(bht)==0) return gerepileuptoleaf(ltop, t2);
3628 3060072 : t1 = Flx_shift(Flx_mul_pre(ft, a, p, pi),-n);
3629 3060120 : t3 = Flxn_mul_pre(t1, bht, n-1, p, pi);
3630 3060106 : vec = Flx_sub(t2, Flx_shift(t3, 1), p);
3631 3060177 : return gerepileuptoleaf(ltop, vec);
3632 : }
3633 :
3634 : GEN
3635 783944 : Flxq_minpoly_pre(GEN x, GEN T, ulong p, ulong pi)
3636 : {
3637 783944 : pari_sp ltop = avma;
3638 783944 : long vT = get_Flx_var(T), n = get_Flx_degree(T);
3639 : GEN v_x;
3640 783937 : GEN g = pol1_Flx(vT), tau = pol1_Flx(vT);
3641 783912 : T = Flx_get_red_pre(T, p, pi);
3642 783913 : v_x = Flxq_powers_pre(x, usqrt(2*n), T, p, pi);
3643 1629265 : while (lgpol(tau) != 0)
3644 : {
3645 : long i, j, m, k1;
3646 : GEN M, v, tr, g_prime, c;
3647 845341 : if (degpol(g) == n) { tau = pol1_Flx(vT); g = pol1_Flx(vT); }
3648 845341 : v = random_Flx(n, vT, p);
3649 845363 : tr = Flxq_transmul_init(tau, T, p, pi);
3650 845334 : v = Flxq_transmul(tr, v, n, p, pi);
3651 845346 : m = 2*(n-degpol(g));
3652 845347 : k1 = usqrt(m);
3653 845349 : tr = Flxq_transmul_init(gel(v_x,k1+1), T, p, pi);
3654 845342 : c = cgetg(m+2,t_VECSMALL);
3655 845327 : c[1] = vT;
3656 4081382 : for (i=0; i<m; i+=k1)
3657 : {
3658 3236020 : long mj = minss(m-i, k1);
3659 12644590 : for (j=0; j<mj; j++)
3660 9408341 : uel(c,m+1-(i+j)) = Flx_dotproduct_pre(v, gel(v_x,j+1), p, pi);
3661 3236249 : v = Flxq_transmul(tr, v, n, p, pi);
3662 : }
3663 845362 : c = Flx_renormalize(c, m+2);
3664 : /* now c contains <v,x^i> , i = 0..m-1 */
3665 845361 : M = Flx_halfgcd_pre(monomial_Flx(1, m, vT), c, p, pi);
3666 845371 : g_prime = gmael(M, 2, 2);
3667 845371 : if (degpol(g_prime) < 1) continue;
3668 833664 : g = Flx_mul_pre(g, g_prime, p, pi);
3669 833640 : tau = Flxq_mul_pre(tau, Flx_FlxqV_eval_pre(g_prime, v_x, T,p,pi), T,p,pi);
3670 : }
3671 783874 : g = Flx_normalize(g,p);
3672 783935 : return gerepileuptoleaf(ltop,g);
3673 : }
3674 : GEN
3675 44415 : Flxq_minpoly(GEN x, GEN T, ulong p)
3676 44415 : { return Flxq_minpoly_pre(x, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3677 :
3678 : GEN
3679 20 : Flxq_conjvec(GEN x, GEN T, ulong p)
3680 : {
3681 20 : long i, l = 1+get_Flx_degree(T);
3682 20 : GEN z = cgetg(l,t_COL);
3683 20 : struct _Flxq D; set_Flxq(&D, T, p);
3684 20 : gel(z,1) = Flx_copy(x);
3685 88 : for (i=2; i<l; i++) gel(z,i) = _Flxq_powu(&D, gel(z,i-1), p);
3686 20 : return z;
3687 : }
3688 :
3689 : GEN
3690 7187 : gener_Flxq(GEN T, ulong p, GEN *po)
3691 : {
3692 7187 : long i, j, vT = get_Flx_var(T), f = get_Flx_degree(T);
3693 : ulong p_1, pi;
3694 : GEN g, L, L2, o, q, F;
3695 : pari_sp av0, av;
3696 :
3697 7187 : if (f == 1) {
3698 : GEN fa;
3699 28 : o = utoipos(p-1);
3700 28 : fa = Z_factor(o);
3701 28 : L = gel(fa,1);
3702 28 : L = vecslice(L, 2, lg(L)-1); /* remove 2 for efficiency */
3703 28 : g = Fl_to_Flx(pgener_Fl_local(p, vec_to_vecsmall(L)), vT);
3704 28 : if (po) *po = mkvec2(o, fa);
3705 28 : return g;
3706 : }
3707 :
3708 7159 : av0 = avma; p_1 = p - 1;
3709 7159 : q = diviuexact(subiu(powuu(p,f), 1), p_1);
3710 :
3711 7159 : L = cgetg(1, t_VECSMALL);
3712 7159 : if (p > 3)
3713 : {
3714 2357 : ulong t = p_1 >> vals(p_1);
3715 2357 : GEN P = gel(factoru(t), 1);
3716 2357 : L = cgetg_copy(P, &i);
3717 3773 : while (--i) L[i] = p_1 / P[i];
3718 : }
3719 7159 : o = factor_pn_1(utoipos(p),f);
3720 7159 : L2 = leafcopy( gel(o, 1) );
3721 19170 : for (i = j = 1; i < lg(L2); i++)
3722 : {
3723 12011 : if (umodui(p_1, gel(L2,i)) == 0) continue;
3724 6474 : gel(L2,j++) = diviiexact(q, gel(L2,i));
3725 : }
3726 7159 : setlg(L2, j); pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3727 7159 : F = Flx_Frobenius_pre(T, p, pi);
3728 17667 : for (av = avma;; set_avma(av))
3729 10508 : {
3730 : GEN tt;
3731 17667 : g = random_Flx(f, vT, p);
3732 17667 : if (degpol(g) < 1) continue;
3733 12033 : if (p == 2) tt = g;
3734 : else
3735 : {
3736 8834 : ulong t = Flxq_norm(g, T, p);
3737 8834 : if (t == 1 || !is_gener_Fl(t, p, p_1, L)) continue;
3738 4753 : tt = Flxq_powu_pre(g, p_1>>1, T, p, pi);
3739 : }
3740 14530 : for (i = 1; i < j; i++)
3741 : {
3742 7371 : GEN a = Flxq_pow_Frobenius(tt, gel(L2,i), F, T, p, pi);
3743 7371 : if (!degpol(a) && uel(a,2) == p_1) break;
3744 : }
3745 7952 : if (i == j) break;
3746 : }
3747 7159 : if (!po)
3748 : {
3749 187 : set_avma((pari_sp)g);
3750 187 : g = gerepileuptoleaf(av0, g);
3751 : }
3752 : else {
3753 6972 : *po = mkvec2(subiu(powuu(p,f), 1), o);
3754 6972 : gerepileall(av0, 2, &g, po);
3755 : }
3756 7159 : return g;
3757 : }
3758 :
3759 : static GEN
3760 553759 : _Flxq_neg(void *E, GEN x)
3761 553759 : { struct _Flxq *s = (struct _Flxq *)E;
3762 553759 : return Flx_neg(x,s->p); }
3763 :
3764 : static GEN
3765 1577269 : _Flxq_rmul(void *E, GEN x, GEN y)
3766 1577269 : { struct _Flxq *s = (struct _Flxq *)E;
3767 1577269 : return Flx_mul_pre(x,y,s->p,s->pi); }
3768 :
3769 : static GEN
3770 18980 : _Flxq_inv(void *E, GEN x)
3771 18980 : { struct _Flxq *s = (struct _Flxq *)E;
3772 18980 : return Flxq_inv(x,s->T,s->p); }
3773 :
3774 : static int
3775 165726 : _Flxq_equal0(GEN x) { return lgpol(x)==0; }
3776 :
3777 : static GEN
3778 16890 : _Flxq_s(void *E, long x)
3779 16890 : { struct _Flxq *s = (struct _Flxq *)E;
3780 16890 : ulong u = x<0 ? s->p+x: (ulong)x;
3781 16890 : return Fl_to_Flx(u, get_Flx_var(s->T));
3782 : }
3783 :
3784 : static const struct bb_field Flxq_field={_Flxq_red,_Flx_add,_Flxq_rmul,_Flxq_neg,
3785 : _Flxq_inv,_Flxq_equal0,_Flxq_s};
3786 :
3787 73608 : const struct bb_field *get_Flxq_field(void **E, GEN T, ulong p)
3788 : {
3789 73608 : GEN z = new_chunk(sizeof(struct _Flxq));
3790 73608 : set_Flxq((struct _Flxq *)z, T, p); *E = (void*)z; return &Flxq_field;
3791 : }
3792 :
3793 : /***********************************************************************/
3794 : /** Flxn **/
3795 : /***********************************************************************/
3796 :
3797 : GEN
3798 54238 : Flx_invLaplace(GEN x, ulong p)
3799 : {
3800 54238 : long i, d = degpol(x);
3801 : ulong t;
3802 : GEN y;
3803 54237 : if (d <= 1) return Flx_copy(x);
3804 54237 : t = Fl_inv(factorial_Fl(d, p), p);
3805 54298 : y = cgetg(d+3, t_VECSMALL);
3806 54250 : y[1] = x[1];
3807 1325687 : for (i=d; i>=2; i--)
3808 : {
3809 1271400 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
3810 1271411 : t = Fl_mul(t, i, p);
3811 : }
3812 54287 : uel(y,3) = uel(x,3);
3813 54287 : uel(y,2) = uel(x,2);
3814 54287 : return y;
3815 : }
3816 :
3817 : GEN
3818 27276 : Flx_Laplace(GEN x, ulong p)
3819 : {
3820 27276 : long i, d = degpol(x);
3821 27274 : ulong t = 1;
3822 : GEN y;
3823 27274 : if (d <= 1) return Flx_copy(x);
3824 27274 : y = cgetg(d+3, t_VECSMALL);
3825 27265 : y[1] = x[1];
3826 27265 : uel(y,2) = uel(x,2);
3827 27265 : uel(y,3) = uel(x,3);
3828 755031 : for (i=2; i<=d; i++)
3829 : {
3830 727737 : t = Fl_mul(t, i%p, p);
3831 727758 : uel(y,i+2) = Fl_mul(uel(x,i+2), t, p);
3832 : }
3833 27294 : return y;
3834 : }
3835 :
3836 : GEN
3837 6230878 : Flxn_red(GEN a, long n)
3838 : {
3839 6230878 : long i, L, l = lg(a);
3840 : GEN b;
3841 6230878 : if (l == 2 || !n) return zero_Flx(a[1]);
3842 5840772 : L = n+2; if (L > l) L = l;
3843 5840772 : b = cgetg(L, t_VECSMALL); b[1] = a[1];
3844 58567394 : for (i=2; i<L; i++) b[i] = a[i];
3845 5838556 : return Flx_renormalize(b,L);
3846 : }
3847 :
3848 : GEN
3849 5063292 : Flxn_mul_pre(GEN a, GEN b, long n, ulong p, ulong pi)
3850 5063292 : { return Flxn_red(Flx_mul_pre(a, b, p, pi), n); }
3851 : GEN
3852 75315 : Flxn_mul(GEN a, GEN b, long n, ulong p)
3853 75315 : { return Flxn_mul_pre(a, b, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3854 :
3855 : GEN
3856 0 : Flxn_sqr_pre(GEN a, long n, ulong p, ulong pi)
3857 0 : { return Flxn_red(Flx_sqr_pre(a, p, pi), n); }
3858 : GEN
3859 0 : Flxn_sqr(GEN a, long n, ulong p)
3860 0 : { return Flxn_sqr_pre(a, n, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3861 :
3862 : /* (f*g) \/ x^n */
3863 : static GEN
3864 937900 : Flx_mulhigh_i(GEN f, GEN g, long n, ulong p, ulong pi)
3865 937900 : { return Flx_shift(Flx_mul_pre(f, g, p, pi),-n); }
3866 :
3867 : static GEN
3868 516280 : Flxn_mulhigh(GEN f, GEN g, long n2, long n, ulong p, ulong pi)
3869 : {
3870 516280 : GEN F = Flx_blocks(f, n2, 2), fl = gel(F,1), fh = gel(F,2);
3871 515931 : return Flx_add(Flx_mulhigh_i(fl, g, n2, p, pi),
3872 : Flxn_mul_pre(fh, g, n - n2, p, pi), p);
3873 : }
3874 :
3875 : /* g==NULL -> assume g==1 */
3876 : GEN
3877 55041 : Flxn_div_pre(GEN g, GEN f, long e, ulong p, ulong pi)
3878 : {
3879 55041 : pari_sp av = avma, av2;
3880 : ulong mask;
3881 : GEN W;
3882 55041 : long n = 1;
3883 55041 : if (lg(f) <= 2) pari_err_INV("Flxn_inv",f);
3884 55041 : W = Fl_to_Flx(Fl_inv(uel(f,2),p), f[1]);
3885 55068 : mask = quadratic_prec_mask(e);
3886 55065 : av2 = avma;
3887 258468 : for (;mask>1;)
3888 : {
3889 : GEN u, fr;
3890 203377 : long n2 = n;
3891 203377 : n<<=1; if (mask & 1) n--;
3892 203377 : mask >>= 1;
3893 203377 : fr = Flxn_red(f, n);
3894 203239 : if (mask>1 || !g)
3895 : {
3896 149227 : u = Flxn_mul_pre(W, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
3897 149518 : W = Flx_sub(W, Flx_shift(u, n2), p);
3898 : } else
3899 : {
3900 54012 : GEN y = Flxn_mul_pre(g, W, n, p, pi), yt = Flxn_red(y, n-n2);
3901 53986 : u = Flxn_mul_pre(yt, Flxn_mulhigh(fr, W, n2, n, p, pi), n-n2, p, pi);
3902 54000 : W = Flx_sub(y, Flx_shift(u, n2), p);
3903 : }
3904 203400 : if (gc_needed(av2,2))
3905 : {
3906 0 : if(DEBUGMEM>1) pari_warn(warnmem,"Flxn_div, e = %ld", n);
3907 0 : W = gerepileupto(av2, W);
3908 : }
3909 : }
3910 55091 : return gerepileupto(av, W);
3911 : }
3912 : GEN
3913 55026 : Flxn_div(GEN g, GEN f, long e, ulong p)
3914 55026 : { return Flxn_div_pre(g, f, e, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
3915 :
3916 : GEN
3917 1030 : Flxn_inv(GEN f, long e, ulong p)
3918 1030 : { return Flxn_div(NULL, f, e, p); }
3919 :
3920 : GEN
3921 109341 : Flxn_expint(GEN h, long e, ulong p)
3922 : {
3923 109341 : pari_sp av = avma, av2;
3924 109341 : long v = h[1], n=1;
3925 109341 : GEN f = pol1_Flx(v), g = pol1_Flx(v);
3926 109305 : ulong mask = quadratic_prec_mask(e), pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
3927 109315 : av2 = avma;
3928 422603 : for (;mask>1;)
3929 : {
3930 : GEN u, w;
3931 422556 : long n2 = n;
3932 422556 : n<<=1; if (mask & 1) n--;
3933 422556 : mask >>= 1;
3934 422556 : u = Flxn_mul_pre(g, Flx_mulhigh_i(f, Flxn_red(h, n2-1), n2-1, p,pi), n-n2, p,pi);
3935 422515 : u = Flx_add(u, Flx_shift(Flxn_red(h, n-1), 1-n2), p);
3936 422558 : w = Flxn_mul_pre(f, Flx_integXn(u, n2-1, p), n-n2, p, pi);
3937 422500 : f = Flx_add(f, Flx_shift(w, n2), p);
3938 422635 : if (mask<=1) break;
3939 313301 : u = Flxn_mul_pre(g, Flxn_mulhigh(f, g, n2, n, p, pi), n-n2, p, pi);
3940 313282 : g = Flx_sub(g, Flx_shift(u, n2), p);
3941 313288 : if (gc_needed(av2,2))
3942 : {
3943 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flxn_exp, e = %ld", n);
3944 0 : gerepileall(av2, 2, &f, &g);
3945 : }
3946 : }
3947 109381 : return gerepileupto(av, f);
3948 : }
3949 :
3950 : GEN
3951 0 : Flxn_exp(GEN h, long e, ulong p)
3952 : {
3953 0 : if (degpol(h)<1 || uel(h,2)!=0)
3954 0 : pari_err_DOMAIN("Flxn_exp","valuation", "<", gen_1, h);
3955 0 : return Flxn_expint(Flx_deriv(h, p), e, p);
3956 : }
3957 :
3958 : INLINE GEN
3959 217054 : Flxn_recip(GEN x, long n)
3960 : {
3961 217054 : GEN z=Flx_recipspec(x+2,lgpol(x),n);
3962 216888 : z[1]=x[1];
3963 216888 : return z;
3964 : }
3965 :
3966 : GEN
3967 53992 : Flx_Newton(GEN P, long n, ulong p)
3968 : {
3969 53992 : pari_sp av = avma;
3970 53992 : long d = degpol(P);
3971 53987 : GEN dP = Flxn_recip(Flx_deriv(P, p), d);
3972 53888 : GEN Q = Flxn_div(dP, Flxn_recip(P, d+1), n, p);
3973 53944 : return gerepileuptoleaf(av, Q);
3974 : }
3975 :
3976 : GEN
3977 109343 : Flx_fromNewton(GEN P, ulong p)
3978 : {
3979 109343 : pari_sp av = avma;
3980 109343 : ulong n = Flx_constant(P)+1;
3981 109341 : GEN z = Flx_neg(Flx_shift(P, -1), p);
3982 109341 : GEN Q = Flxn_recip(Flxn_expint(z, n, p), n);
3983 109325 : return gerepileuptoleaf(av, Q);
3984 : }
3985 :
3986 : static void
3987 12514 : init_invlaplace(long d, ulong p, GEN *pt_P, GEN *pt_V)
3988 : {
3989 : long i;
3990 : ulong e;
3991 12514 : GEN P = cgetg(d+1, t_VECSMALL);
3992 12514 : GEN V = cgetg(d+1, t_VECSMALL);
3993 1396581 : for (i=1, e=1; i<=d; i++, e++)
3994 : {
3995 1384067 : if (e==p)
3996 : {
3997 459153 : e = 0;
3998 459153 : V[i] = u_lvalrem(i, p, &uel(P,i));
3999 : } else
4000 : {
4001 924914 : V[i] = 0; uel(P,i) = i;
4002 : }
4003 : }
4004 12514 : *pt_P = P; *pt_V = V;
4005 12514 : }
4006 :
4007 : /* return p^val * FpX_invLaplace(1+x+...x^(n-1), q), with q a power of p and
4008 : * val large enough to compensate for the power of p in the factorials */
4009 :
4010 : static GEN
4011 497 : ZpX_invLaplace_init(long n, GEN q, ulong p, long v, long sv)
4012 : {
4013 497 : pari_sp av = avma;
4014 497 : long i, d = n-1, w;
4015 : GEN y, W, E, t;
4016 497 : init_invlaplace(d, p, &E, &W);
4017 497 : t = Fp_inv(FpV_prod(Flv_to_ZV(E), q), q);
4018 497 : w = zv_sum(W);
4019 497 : if (v > w) t = Fp_mul(t, powuu(p, v-w), q);
4020 497 : y = cgetg(d+3,t_POL);
4021 497 : y[1] = evalsigne(1) | sv;
4022 28882 : for (i=d; i>=1; i--)
4023 : {
4024 28385 : gel(y,i+2) = t;
4025 28385 : t = Fp_mulu(t, uel(E,i), q);
4026 28385 : if (uel(W,i)) t = Fp_mul(t, powuu(p, uel(W,i)), q);
4027 : }
4028 497 : gel(y,2) = t;
4029 497 : return gerepilecopy(av, ZX_renormalize(y, d+3));
4030 : }
4031 :
4032 : GEN
4033 27493 : Flx_composedsum(GEN P, GEN Q, ulong p)
4034 : {
4035 27493 : pari_sp av = avma;
4036 27493 : long n = 1 + degpol(P)*degpol(Q);
4037 27490 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4038 27491 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4039 : GEN R;
4040 27497 : if (p >= (ulong)n)
4041 : {
4042 27000 : GEN Pl = Flx_invLaplace(Flx_Newton(P,n,p), p);
4043 27003 : GEN Ql = Flx_invLaplace(Flx_Newton(Q,n,p), p);
4044 27000 : GEN L = Flx_Laplace(Flxn_mul(Pl, Ql, n, p), p);
4045 26999 : R = Flx_fromNewton(L, p);
4046 : } else
4047 : {
4048 497 : long v = factorial_lval(n-1, p);
4049 497 : long w = 1 + ulogint(n-1, p);
4050 497 : GEN pv = powuu(p, v);
4051 497 : GEN qf = powuu(p, w), q = mulii(pv, qf), q2 = mulii(q, pv);
4052 497 : GEN iL = ZpX_invLaplace_init(n, q, p, v, P[1]);
4053 497 : GEN Pl = FpX_convol(iL, FpX_Newton(Flx_to_ZX(P), n, qf), q);
4054 497 : GEN Ql = FpX_convol(iL, FpX_Newton(Flx_to_ZX(Q), n, qf), q);
4055 497 : GEN Ln = ZX_Z_divexact(FpXn_mul(Pl, Ql, n, q2), pv);
4056 497 : GEN L = ZX_Z_divexact(FpX_Laplace(Ln, q), pv);
4057 497 : R = ZX_to_Flx(FpX_fromNewton(L, qf), p);
4058 : }
4059 27487 : return gerepileuptoleaf(av, Flx_Fl_mul(R, lead, p));
4060 : }
4061 :
4062 : static GEN
4063 3826 : _Flx_composedsum(void *E, GEN a, GEN b)
4064 3826 : { return Flx_composedsum(a, b, (ulong)E); }
4065 :
4066 : GEN
4067 28911 : FlxV_composedsum(GEN V, ulong p)
4068 28911 : { return gen_product(V, (void *)p, &_Flx_composedsum); }
4069 :
4070 : GEN
4071 0 : Flx_composedprod(GEN P, GEN Q, ulong p)
4072 : {
4073 0 : pari_sp av = avma;
4074 0 : long n = 1+ degpol(P)*degpol(Q);
4075 0 : ulong lead = Fl_mul(Fl_powu(Flx_lead(P), degpol(Q), p),
4076 0 : Fl_powu(Flx_lead(Q), degpol(P), p), p);
4077 : GEN R;
4078 0 : if (p >= (ulong)n)
4079 : {
4080 0 : GEN L = Flx_convol(Flx_Newton(P,n,p), Flx_Newton(Q,n,p), p);
4081 0 : R = Flx_fromNewton(L, p);
4082 : } else
4083 : {
4084 0 : long w = 1 + ulogint(n, p);
4085 0 : GEN qf = powuu(p, w);
4086 0 : GEN Pl = FpX_convol(FpX_Newton(Flx_to_ZX(P), n, qf), FpX_Newton(Flx_to_ZX(Q), n, qf), qf);
4087 0 : R = ZX_to_Flx(FpX_fromNewton(Pl, qf), p);
4088 : }
4089 0 : return gerepileuptoleaf(av, Flx_Fl_mul(R, lead, p));
4090 :
4091 : }
4092 :
4093 : /* (x+1)^n mod p; assume 2 <= n < 2p prime */
4094 : static GEN
4095 0 : Fl_Xp1_powu(ulong n, ulong p, long v)
4096 : {
4097 0 : ulong k, d = (n + 1) >> 1;
4098 0 : GEN C, V = identity_zv(d);
4099 :
4100 0 : Flv_inv_inplace(V, p); /* could restrict to odd integers in [3,d] */
4101 0 : C = cgetg(n+3, t_VECSMALL);
4102 0 : C[1] = v;
4103 0 : uel(C,2) = 1UL;
4104 0 : uel(C,3) = n%p;
4105 0 : uel(C,4) = Fl_mul(odd(n)? n: n-1, n >> 1, p);
4106 : /* binom(n,k) = binom(n,k-1) * (n-k+1) / k */
4107 0 : if (SMALL_ULONG(p))
4108 0 : for (k = 3; k <= d; k++)
4109 0 : uel(C,k+2) = Fl_mul(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p);
4110 : else
4111 : {
4112 0 : ulong pi = get_Fl_red(p);
4113 0 : for (k = 3; k <= d; k++)
4114 0 : uel(C,k+2) = Fl_mul_pre(Fl_mul(n-k+1, uel(C,k+1), p), uel(V,k), p, pi);
4115 : }
4116 0 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4117 0 : return C; /* normalized */
4118 : }
4119 :
4120 : /* p arbitrary */
4121 : GEN
4122 28201 : Flx_translate1_basecase(GEN P, ulong p)
4123 : {
4124 28201 : GEN R = Flx_copy(P);
4125 28201 : long i, k, n = degpol(P);
4126 654753 : for (i = 1; i <= n; i++)
4127 14846523 : for (k = n-i; k < n; k++) uel(R,k+2) = Fl_add(uel(R,k+2), uel(R,k+3), p);
4128 28201 : return R;
4129 : }
4130 :
4131 : static int
4132 41366 : translate_basecase(long n, ulong p)
4133 : {
4134 : #ifdef LONG_IS_64BIT
4135 36072 : if (p <= 19) return n < 40;
4136 29910 : if (p < 1UL<<30) return n < 58;
4137 0 : if (p < 1UL<<59) return n < 100;
4138 0 : if (p < 1UL<<62) return n < 120;
4139 0 : if (p < 1UL<<63) return n < 240;
4140 0 : return n < 250;
4141 : #else
4142 5294 : if (p <= 13) return n < 18;
4143 4136 : if (p <= 17) return n < 22;
4144 4078 : if (p <= 29) return n < 39;
4145 3886 : if (p <= 67) return n < 69;
4146 3667 : if (p < 1UL<< 15) return n < 80;
4147 2047 : if (p < 1UL<< 16) return n < 100;
4148 0 : if (p < 1UL<< 28) return n < 300;
4149 0 : return n < 650;
4150 : #endif
4151 : }
4152 : /* assume p prime */
4153 : GEN
4154 16107 : Flx_translate1(GEN P, ulong p)
4155 : {
4156 16107 : long d, n = degpol(P);
4157 : GEN R, Q, S;
4158 16107 : if (translate_basecase(n, p)) return Flx_translate1_basecase(P, p);
4159 : /* n > 0 */
4160 1148 : d = n >> 1;
4161 1148 : if ((ulong)n < p)
4162 : {
4163 0 : R = Flx_translate1(Flxn_red(P, d), p);
4164 0 : Q = Flx_translate1(Flx_shift(P, -d), p);
4165 0 : S = Fl_Xp1_powu(d, p, P[1]);
4166 0 : return Flx_add(Flx_mul(Q, S, p), R, p);
4167 : }
4168 : else
4169 : {
4170 : ulong q;
4171 1148 : if ((ulong)d > p) (void)ulogintall(d, p, &q); else q = p;
4172 1148 : R = Flx_translate1(Flxn_red(P, q), p);
4173 1148 : Q = Flx_translate1(Flx_shift(P, -q), p);
4174 1148 : S = Flx_add(Flx_shift(Q, q), Q, p);
4175 1148 : return Flx_add(S, R, p); /* P(x+1) = Q(x+1) (x^q+1) + R(x+1) */
4176 : }
4177 : }
4178 :
4179 : static GEN
4180 12017 : zl_Xp1_powu(ulong n, ulong p, ulong q, long e, long vs)
4181 : {
4182 12017 : ulong k, d = n >> 1, c, v = 0;
4183 12017 : GEN C, V, W, U = upowers(p, e-1);
4184 12017 : init_invlaplace(d, p, &V, &W);
4185 12017 : Flv_inv_inplace(V, q);
4186 12017 : C = cgetg(n+3, t_VECSMALL);
4187 12017 : C[1] = vs;
4188 12017 : uel(C,2) = 1UL;
4189 12017 : uel(C,3) = n%q;
4190 12017 : v = u_lvalrem(n, p, &c);
4191 1355682 : for (k = 2; k <= d; k++)
4192 : {
4193 : ulong w;
4194 1343665 : v += u_lvalrem(n-k+1, p, &w) - W[k];
4195 1343665 : c = Fl_mul(Fl_mul(w%q, c, q), uel(V,k), q);
4196 1343665 : uel(C,2+k) = v >= (ulong)e ? 0: v==0 ? c : Fl_mul(c, uel(U, v+1), q);
4197 : }
4198 1374521 : for ( ; k <= n; k++) uel(C,2+k) = uel(C,2+n-k);
4199 12017 : return C; /* normalized */
4200 : }
4201 :
4202 : GEN
4203 25259 : zlx_translate1(GEN P, ulong p, long e)
4204 : {
4205 25259 : ulong d, q = upowuu(p,e), n = degpol(P);
4206 : GEN R, Q, S;
4207 25259 : if (translate_basecase(n, q)) return Flx_translate1_basecase(P, q);
4208 : /* n > 0 */
4209 12017 : d = n >> 1;
4210 12017 : R = zlx_translate1(Flxn_red(P, d), p, e);
4211 12017 : Q = zlx_translate1(Flx_shift(P, -d), p, e);
4212 12017 : S = zl_Xp1_powu(d, p, q, e, P[1]);
4213 12017 : return Flx_add(Flx_mul(Q, S, q), R, q);
4214 : }
4215 :
4216 : /***********************************************************************/
4217 : /** Fl2 **/
4218 : /***********************************************************************/
4219 : /* Fl2 objects are Flv of length 2 [a,b] representing a+bsqrt(D) for
4220 : * a nonsquare D. */
4221 :
4222 : INLINE GEN
4223 7191026 : mkF2(ulong a, ulong b) { return mkvecsmall2(a,b); }
4224 :
4225 : /* allow pi = 0 */
4226 : GEN
4227 1916389 : Fl2_mul_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4228 : {
4229 : ulong xaya, xbyb, Db2, mid, z1, z2;
4230 1916389 : ulong x1 = x[1], x2 = x[2], y1 = y[1], y2 = y[2];
4231 1916389 : if (pi)
4232 : {
4233 1916409 : xaya = Fl_mul_pre(x1,y1,p,pi);
4234 1916854 : if (x2==0 && y2==0) return mkF2(xaya,0);
4235 1847865 : if (x2==0) return mkF2(xaya,Fl_mul_pre(x1,y2,p,pi));
4236 1823307 : if (y2==0) return mkF2(xaya,Fl_mul_pre(x2,y1,p,pi));
4237 1823002 : xbyb = Fl_mul_pre(x2,y2,p,pi);
4238 1822878 : mid = Fl_mul_pre(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p,pi);
4239 1823040 : Db2 = Fl_mul_pre(D, xbyb, p,pi);
4240 : }
4241 0 : else if (p & HIGHMASK)
4242 : {
4243 0 : xaya = Fl_mul(x1,y1,p);
4244 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4245 0 : if (x2==0) return mkF2(xaya,Fl_mul(x1,y2,p));
4246 0 : if (y2==0) return mkF2(xaya,Fl_mul(x2,y1,p));
4247 0 : xbyb = Fl_mul(x2,y2,p);
4248 0 : mid = Fl_mul(Fl_add(x1,x2,p), Fl_add(y1,y2,p),p);
4249 0 : Db2 = Fl_mul(D, xbyb, p);
4250 : }
4251 : else
4252 : {
4253 0 : xaya = (x1 * y1) % p;
4254 0 : if (x2==0 && y2==0) return mkF2(xaya,0);
4255 0 : if (x2==0) return mkF2(xaya, (x1 * y2) % p);
4256 0 : if (y2==0) return mkF2(xaya, (x2 * y1) % p);
4257 0 : xbyb = (x2 * y2) % p;
4258 0 : mid = (Fl_add(x1,x2,p) * Fl_add(y1,y2,p)) % p;
4259 0 : Db2 = (D * xbyb) % p;
4260 : }
4261 1822953 : z1 = Fl_add(xaya,Db2,p);
4262 1822933 : z2 = Fl_sub(mid,Fl_add(xaya,xbyb,p),p);
4263 1822910 : return mkF2(z1,z2);
4264 : }
4265 :
4266 : /* allow pi = 0 */
4267 : GEN
4268 4822613 : Fl2_sqr_pre(GEN x, ulong D, ulong p, ulong pi)
4269 : {
4270 4822613 : ulong a = x[1], b = x[2];
4271 : ulong a2, Db2, ab;
4272 4822613 : if (pi)
4273 : {
4274 4822640 : a2 = Fl_sqr_pre(a,p,pi);
4275 4824773 : if (b==0) return mkF2(a2,0);
4276 4612042 : Db2= Fl_mul_pre(D, Fl_sqr_pre(b,p,pi), p,pi);
4277 4612116 : ab = Fl_mul_pre(a,b,p,pi);
4278 : }
4279 0 : else if (p & HIGHMASK)
4280 : {
4281 0 : a2 = Fl_sqr(a,p);
4282 0 : if (b==0) return mkF2(a2,0);
4283 0 : Db2= Fl_mul(D, Fl_sqr(b,p), p);
4284 0 : ab = Fl_mul(a,b,p);
4285 : }
4286 : else
4287 : {
4288 0 : a2 = (a * a) % p;
4289 0 : if (b==0) return mkF2(a2,0);
4290 0 : Db2= (D * ((b * b) % p)) % p;
4291 0 : ab = (a * b) % p;
4292 : }
4293 4612143 : return mkF2(Fl_add(a2,Db2,p), Fl_double(ab,p));
4294 : }
4295 :
4296 : /* allow pi = 0 */
4297 : ulong
4298 126108 : Fl2_norm_pre(GEN x, ulong D, ulong p, ulong pi)
4299 : {
4300 126108 : ulong a = x[1], b = x[2], a2;
4301 126108 : if (pi)
4302 : {
4303 72316 : a2 = Fl_sqr_pre(a,p,pi);
4304 72316 : return b? Fl_sub(a2, Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p): a2;
4305 : }
4306 53792 : else if (p & HIGHMASK)
4307 : {
4308 0 : a2 = Fl_sqr(a,p);
4309 0 : return b? Fl_sub(a2, Fl_mul(D, Fl_sqr(b, p), p), p): a2;
4310 : }
4311 : else
4312 : {
4313 53792 : a2 = (a * a) % p;
4314 53792 : return b? Fl_sub(a2, (D * ((b * b) % p)) % p, p): a2;
4315 : }
4316 : }
4317 :
4318 : /* allow pi = 0 */
4319 : GEN
4320 193035 : Fl2_inv_pre(GEN x, ulong D, ulong p, ulong pi)
4321 : {
4322 193035 : ulong a = x[1], b = x[2], n, ni;
4323 193035 : if (b == 0) return mkF2(Fl_inv(a,p), 0);
4324 162295 : b = Fl_neg(b, p);
4325 162296 : if (pi)
4326 : {
4327 162296 : n = Fl_sub(Fl_sqr_pre(a, p,pi),
4328 : Fl_mul_pre(D, Fl_sqr_pre(b, p,pi), p,pi), p);
4329 162297 : ni = Fl_inv(n,p);
4330 162298 : return mkF2(Fl_mul_pre(a, ni, p,pi), Fl_mul_pre(b, ni, p,pi));
4331 : }
4332 0 : else if (p & HIGHMASK)
4333 : {
4334 0 : n = Fl_sub(Fl_sqr(a, p), Fl_mul(D, Fl_sqr(b, p), p), p);
4335 0 : ni = Fl_inv(n,p);
4336 0 : return mkF2(Fl_mul(a, ni, p), Fl_mul(b, ni, p));
4337 : }
4338 : else
4339 : {
4340 0 : n = Fl_sub((a * a) % p, (D * ((b * b) % p)) % p, p);
4341 0 : ni = Fl_inv(n,p);
4342 0 : return mkF2((a * ni) % p, (b * ni) % p);
4343 : }
4344 : }
4345 :
4346 : int
4347 439693 : Fl2_equal1(GEN x) { return x[1]==1 && x[2]==0; }
4348 :
4349 : struct _Fl2 {
4350 : ulong p, pi, D;
4351 : };
4352 :
4353 : static GEN
4354 4822709 : _Fl2_sqr(void *data, GEN x)
4355 : {
4356 4822709 : struct _Fl2 *D = (struct _Fl2*)data;
4357 4822709 : return Fl2_sqr_pre(x, D->D, D->p, D->pi);
4358 : }
4359 : static GEN
4360 1888278 : _Fl2_mul(void *data, GEN x, GEN y)
4361 : {
4362 1888278 : struct _Fl2 *D = (struct _Fl2*)data;
4363 1888278 : return Fl2_mul_pre(x,y, D->D, D->p, D->pi);
4364 : }
4365 :
4366 : /* n-Power of x in Z/pZ[X]/(T), as t_VECSMALL; allow pi = 0 */
4367 : GEN
4368 657110 : Fl2_pow_pre(GEN x, GEN n, ulong D, ulong p, ulong pi)
4369 : {
4370 657110 : pari_sp av = avma;
4371 : struct _Fl2 d;
4372 : GEN y;
4373 657110 : long s = signe(n);
4374 657110 : if (!s) return mkF2(1,0);
4375 582387 : if (s < 0)
4376 193036 : x = Fl2_inv_pre(x,D,p,pi);
4377 582381 : if (is_pm1(n)) return s < 0 ? x : zv_copy(x);
4378 428950 : d.p = p; d.pi = pi; d.D=D;
4379 428950 : y = gen_pow_i(x, n, (void*)&d, &_Fl2_sqr, &_Fl2_mul);
4380 428975 : return gerepileuptoleaf(av, y);
4381 : }
4382 :
4383 : static GEN
4384 657102 : _Fl2_pow(void *data, GEN x, GEN n)
4385 : {
4386 657102 : struct _Fl2 *D = (struct _Fl2*)data;
4387 657102 : return Fl2_pow_pre(x, n, D->D, D->p, D->pi);
4388 : }
4389 :
4390 : static GEN
4391 111125 : _Fl2_rand(void *data)
4392 : {
4393 111125 : struct _Fl2 *D = (struct _Fl2*)data;
4394 111125 : ulong a = random_Fl(D->p), b=random_Fl(D->p-1)+1;
4395 111129 : return mkF2(a,b);
4396 : }
4397 :
4398 : GEN
4399 67842 : Fl2_sqrt_pre(GEN z, ulong D, ulong p, ulong pi)
4400 : {
4401 67842 : ulong a = uel(z,1), b = uel(z,2), as2, u, v, s;
4402 67842 : ulong y = Fl_2gener_pre_i(D, p, pi);
4403 67842 : if (b == 0)
4404 19476 : return krouu(a, p)==1 ? mkF2(Fl_sqrt_pre_i(a, y, p, pi), 0)
4405 19476 : : mkF2(0, Fl_sqrt_pre_i(Fl_div(a, D, p), y, p, pi));
4406 54622 : s = Fl_sqrt_pre_i(Fl2_norm_pre(z, D, p, pi), y, p, pi);
4407 54622 : if (s==~0UL) return NULL;
4408 51448 : as2 = Fl_halve(Fl_add(a, s, p), p);
4409 51448 : if (krouu(as2, p)==-1) as2 = Fl_sub(as2, s, p);
4410 51448 : u = Fl_sqrt_pre_i(as2, y, p, pi);
4411 51448 : v = Fl_div(b, Fl_double(u, p), p);
4412 51448 : return mkF2(u,v);
4413 : }
4414 :
4415 : static const struct bb_group Fl2_star={_Fl2_mul, _Fl2_pow, _Fl2_rand,
4416 : hash_GEN, zv_equal, Fl2_equal1, NULL};
4417 :
4418 : /* allow pi = 0 */
4419 : GEN
4420 74724 : Fl2_sqrtn_pre(GEN a, GEN n, ulong D, ulong p, ulong pi, GEN *zeta)
4421 : {
4422 : struct _Fl2 E;
4423 : GEN o;
4424 74724 : if (a[1]==0 && a[2]==0)
4425 : {
4426 0 : if (signe(n) < 0) pari_err_INV("Flxq_sqrtn",a);
4427 0 : if (zeta) *zeta=mkF2(1,0);
4428 0 : return zv_copy(a);
4429 : }
4430 74724 : E.p=p; E.pi = pi; E.D = D;
4431 74724 : o = subiu(powuu(p,2), 1);
4432 74722 : return gen_Shanks_sqrtn(a,n,o,zeta,(void*)&E,&Fl2_star);
4433 : }
4434 :
4435 : /* allow pi = 0 */
4436 : GEN
4437 10402 : Flx_Fl2_eval_pre(GEN x, GEN y, ulong D, ulong p, ulong pi)
4438 : {
4439 : GEN p1;
4440 10402 : long i = lg(x)-1;
4441 10402 : if (i <= 2)
4442 2065 : return mkF2(i == 2? x[2]: 0, 0);
4443 8337 : p1 = mkF2(x[i], 0);
4444 36456 : for (i--; i>=2; i--)
4445 : {
4446 28119 : p1 = Fl2_mul_pre(p1, y, D, p, pi);
4447 28119 : uel(p1,1) = Fl_add(uel(p1,1), uel(x,i), p);
4448 : }
4449 8337 : return p1;
4450 : }
4451 :
4452 : /***********************************************************************/
4453 : /** FlxV **/
4454 : /***********************************************************************/
4455 : /* FlxV are t_VEC with Flx coefficients. */
4456 :
4457 : GEN
4458 34482 : FlxV_Flc_mul(GEN V, GEN W, ulong p)
4459 : {
4460 34482 : pari_sp ltop=avma;
4461 : long i;
4462 34482 : GEN z = Flx_Fl_mul(gel(V,1),W[1],p);
4463 257044 : for(i=2;i<lg(V);i++)
4464 222562 : z=Flx_add(z,Flx_Fl_mul(gel(V,i),W[i],p),p);
4465 34482 : return gerepileuptoleaf(ltop,z);
4466 : }
4467 :
4468 : GEN
4469 0 : ZXV_to_FlxV(GEN x, ulong p)
4470 0 : { pari_APPLY_type(t_VEC, ZX_to_Flx(gel(x,i), p)) }
4471 :
4472 : GEN
4473 3798359 : ZXT_to_FlxT(GEN x, ulong p)
4474 : {
4475 3798359 : if (typ(x) == t_POL)
4476 3739427 : return ZX_to_Flx(x, p);
4477 : else
4478 193432 : pari_APPLY_type(t_VEC, ZXT_to_FlxT(gel(x,i), p))
4479 : }
4480 :
4481 : GEN
4482 171699 : FlxV_to_Flm(GEN x, long n)
4483 926353 : { pari_APPLY_type(t_MAT, Flx_to_Flv(gel(x,i), n)) }
4484 :
4485 : GEN
4486 0 : FlxV_red(GEN x, ulong p)
4487 0 : { pari_APPLY_type(t_VEC, Flx_red(gel(x,i), p)) }
4488 :
4489 : GEN
4490 297179 : FlxT_red(GEN x, ulong p)
4491 : {
4492 297179 : if (typ(x) == t_VECSMALL)
4493 199925 : return Flx_red(x, p);
4494 : else
4495 326060 : pari_APPLY_type(t_VEC, FlxT_red(gel(x,i), p))
4496 : }
4497 :
4498 : GEN
4499 113589 : FlxqV_dotproduct_pre(GEN x, GEN y, GEN T, ulong p, ulong pi)
4500 : {
4501 113589 : long i, lx = lg(x);
4502 : pari_sp av;
4503 : GEN c;
4504 113589 : if (lx == 1) return pol0_Flx(get_Flx_var(T));
4505 113589 : av = avma; c = Flx_mul_pre(gel(x,1),gel(y,1), p, pi);
4506 464499 : for (i=2; i<lx; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4507 113589 : return gerepileuptoleaf(av, Flx_rem_pre(c,T,p,pi));
4508 : }
4509 : GEN
4510 0 : FlxqV_dotproduct(GEN x, GEN y, GEN T, ulong p)
4511 0 : { return FlxqV_dotproduct_pre(x, y, T, p, SMALL_ULONG(p)? 0: get_Fl_red(p)); }
4512 :
4513 : GEN
4514 2456 : FlxqX_dotproduct(GEN x, GEN y, GEN T, ulong p)
4515 : {
4516 2456 : long i, l = minss(lg(x), lg(y));
4517 : ulong pi;
4518 : pari_sp av;
4519 : GEN c;
4520 2456 : if (l == 2) return pol0_Flx(get_Flx_var(T));
4521 2387 : av = avma; pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4522 2387 : c = Flx_mul_pre(gel(x,2),gel(y,2), p, pi);
4523 7169 : for (i=3; i<l; i++) c = Flx_add(c, Flx_mul_pre(gel(x,i),gel(y,i), p, pi), p);
4524 2387 : return gerepileuptoleaf(av, Flx_rem_pre(c,T,p,pi));
4525 : }
4526 :
4527 : /* allow pi = 0 */
4528 : GEN
4529 251071 : FlxC_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4530 : {
4531 251071 : long i, l = lg(z);
4532 251071 : GEN y = cgetg(l, t_VECSMALL);
4533 12741610 : for (i=1; i<l; i++) uel(y,i) = Flx_eval_powers_pre(gel(z,i), x, p, pi);
4534 251062 : return y;
4535 : }
4536 :
4537 : /***********************************************************************/
4538 : /** FlxM **/
4539 : /***********************************************************************/
4540 : /* allow pi = 0 */
4541 : GEN
4542 19452 : FlxM_eval_powers_pre(GEN z, GEN x, ulong p, ulong pi)
4543 : {
4544 19452 : long i, l = lg(z);
4545 19452 : GEN y = cgetg(l, t_MAT);
4546 270523 : for (i=1; i<l; i++) gel(y,i) = FlxC_eval_powers_pre(gel(z,i), x, p, pi);
4547 19452 : return y;
4548 : }
4549 :
4550 : GEN
4551 0 : zero_FlxC(long n, long sv)
4552 : {
4553 0 : GEN x = cgetg(n + 1, t_COL), z = zero_Flx(sv);
4554 : long i;
4555 0 : for (i = 1; i <= n; i++) gel(x, i) = z;
4556 0 : return x;
4557 : }
4558 :
4559 : GEN
4560 0 : FlxC_neg(GEN x, ulong p)
4561 0 : { pari_APPLY_type(t_COL, Flx_neg(gel(x, i), p)) }
4562 :
4563 : GEN
4564 0 : FlxC_sub(GEN x, GEN y, ulong p)
4565 0 : { pari_APPLY_type(t_COL, Flx_sub(gel(x, i), gel(y, i), p)) }
4566 :
4567 : GEN
4568 0 : zero_FlxM(long r, long c, long sv)
4569 : {
4570 0 : GEN x = cgetg(c + 1, t_MAT), z = zero_FlxC(r, sv);
4571 : long j;
4572 0 : for (j = 1; j <= c; j++) gel(x, j) = z;
4573 0 : return x;
4574 : }
4575 :
4576 : GEN
4577 0 : FlxM_neg(GEN x, ulong p)
4578 0 : { pari_APPLY_same(FlxC_neg(gel(x, i), p)) }
4579 :
4580 : GEN
4581 0 : FlxM_sub(GEN x, GEN y, ulong p)
4582 0 : { pari_APPLY_same(FlxC_sub(gel(x, i), gel(y,i), p)) }
4583 :
4584 : GEN
4585 0 : FlxqC_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4586 0 : { pari_APPLY_type(t_COL, Flxq_mul(gel(x, i), y, T, p)) }
4587 :
4588 : GEN
4589 0 : FlxqM_Flxq_mul(GEN x, GEN y, GEN T, ulong p)
4590 0 : { pari_APPLY_same(FlxqC_Flxq_mul(gel(x, i), y, T, p)) }
4591 :
4592 : static GEN
4593 58683 : FlxM_pack_ZM(GEN M, GEN (*pack)(GEN, long)) {
4594 : long i, j, l, lc;
4595 58683 : GEN N = cgetg_copy(M, &l), x;
4596 58683 : if (l == 1)
4597 0 : return N;
4598 58683 : lc = lgcols(M);
4599 279934 : for (j = 1; j < l; j++) {
4600 221251 : gel(N, j) = cgetg(lc, t_COL);
4601 1135923 : for (i = 1; i < lc; i++) {
4602 914672 : x = gcoeff(M, i, j);
4603 914672 : gcoeff(N, i, j) = pack(x + 2, lgpol(x));
4604 : }
4605 : }
4606 58683 : return N;
4607 : }
4608 :
4609 : static GEN
4610 857451 : kron_pack_Flx_spec_half(GEN x, long l) {
4611 857451 : if (l == 0) return gen_0;
4612 498855 : return Flx_to_int_halfspec(x, l);
4613 : }
4614 :
4615 : static GEN
4616 53832 : kron_pack_Flx_spec(GEN x, long l) {
4617 : long i;
4618 : GEN w, y;
4619 53832 : if (l == 0)
4620 10072 : return gen_0;
4621 43760 : y = cgetipos(l + 2);
4622 159479 : for (i = 0, w = int_LSW(y); i < l; i++, w = int_nextW(w))
4623 115719 : *w = x[i];
4624 43760 : return y;
4625 : }
4626 :
4627 : static GEN
4628 3389 : kron_pack_Flx_spec_2(GEN x, long l) { return Flx_eval2BILspec(x, 2, l); }
4629 :
4630 : static GEN
4631 0 : kron_pack_Flx_spec_3(GEN x, long l) { return Flx_eval2BILspec(x, 3, l); }
4632 :
4633 : static GEN
4634 42953 : kron_unpack_Flx(GEN z, ulong p)
4635 : {
4636 42953 : long i, l = lgefint(z);
4637 42953 : GEN x = cgetg(l, t_VECSMALL), w;
4638 201969 : for (w = int_LSW(z), i = 2; i < l; w = int_nextW(w), i++)
4639 159016 : x[i] = ((ulong) *w) % p;
4640 42953 : return Flx_renormalize(x, l);
4641 : }
4642 :
4643 : static GEN
4644 2930 : kron_unpack_Flx_2(GEN x, ulong p) {
4645 2930 : long d = (lgefint(x)-1)/2 - 1;
4646 2930 : return Z_mod2BIL_Flx_2(x, d, p);
4647 : }
4648 :
4649 : static GEN
4650 0 : kron_unpack_Flx_3(GEN x, ulong p) {
4651 0 : long d = lgefint(x)/3 - 1;
4652 0 : return Z_mod2BIL_Flx_3(x, d, p);
4653 : }
4654 :
4655 : static GEN
4656 119615 : FlxM_pack_ZM_bits(GEN M, long b)
4657 : {
4658 : long i, j, l, lc;
4659 119615 : GEN N = cgetg_copy(M, &l), x;
4660 119615 : if (l == 1)
4661 0 : return N;
4662 119615 : lc = lgcols(M);
4663 493928 : for (j = 1; j < l; j++) {
4664 374313 : gel(N, j) = cgetg(lc, t_COL);
4665 5988006 : for (i = 1; i < lc; i++) {
4666 5613693 : x = gcoeff(M, i, j);
4667 5613693 : gcoeff(N, i, j) = kron_pack_Flx_spec_bits(x + 2, b, lgpol(x));
4668 : }
4669 : }
4670 119615 : return N;
4671 : }
4672 :
4673 : static GEN
4674 29345 : ZM_unpack_FlxqM(GEN M, GEN T, ulong p, ulong pi, GEN (*unpack)(GEN, ulong))
4675 : {
4676 29345 : long i, j, l, lc, sv = get_Flx_var(T);
4677 29345 : GEN N = cgetg_copy(M, &l), x;
4678 29345 : if (l == 1)
4679 0 : return N;
4680 29345 : lc = lgcols(M);
4681 163939 : for (j = 1; j < l; j++) {
4682 134594 : gel(N, j) = cgetg(lc, t_COL);
4683 768988 : for (i = 1; i < lc; i++) {
4684 634394 : x = unpack(gcoeff(M, i, j), p);
4685 634394 : x[1] = sv;
4686 634394 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4687 : }
4688 : }
4689 29345 : return N;
4690 : }
4691 :
4692 : static GEN
4693 59848 : ZM_unpack_FlxqM_bits(GEN M, long b, GEN T, ulong p, ulong pi)
4694 : {
4695 59848 : long i, j, l, lc, sv = get_Flx_var(T);
4696 59848 : GEN N = cgetg_copy(M, &l), x;
4697 59848 : if (l == 1)
4698 0 : return N;
4699 59848 : lc = lgcols(M);
4700 59848 : if (b < BITS_IN_LONG) {
4701 201798 : for (j = 1; j < l; j++) {
4702 143609 : gel(N, j) = cgetg(lc, t_COL);
4703 3237399 : for (i = 1; i < lc; i++) {
4704 3093790 : x = kron_unpack_Flx_bits_narrow(gcoeff(M, i, j), b, p);
4705 3093790 : x[1] = sv;
4706 3093790 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4707 : }
4708 : }
4709 : } else {
4710 1659 : ulong pi = get_Fl_red(p);
4711 9796 : for (j = 1; j < l; j++) {
4712 8137 : gel(N, j) = cgetg(lc, t_COL);
4713 175265 : for (i = 1; i < lc; i++) {
4714 167128 : x = kron_unpack_Flx_bits_wide(gcoeff(M, i, j), b, p, pi);
4715 167128 : x[1] = sv;
4716 167128 : gcoeff(N, i, j) = Flx_rem_pre(x, T, p, pi);
4717 : }
4718 : }
4719 : }
4720 59848 : return N;
4721 : }
4722 :
4723 : GEN
4724 89193 : FlxqM_mul_Kronecker(GEN A, GEN B, GEN T, ulong p)
4725 : {
4726 89193 : pari_sp av = avma;
4727 89193 : long b, d = get_Flx_degree(T), n = lg(A) - 1;
4728 : GEN C, D, z;
4729 : GEN (*pack)(GEN, long), (*unpack)(GEN, ulong);
4730 89193 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
4731 89193 : int is_sqr = A==B;
4732 :
4733 89193 : z = muliu(muliu(sqru(p - 1), d), n);
4734 89193 : b = expi(z) + 1;
4735 : /* only do expensive bit-packing if it saves at least 1 limb */
4736 89193 : if (b <= BITS_IN_HALFULONG)
4737 84784 : { if (nbits2nlong(d*b) == (d + 1)/2) b = BITS_IN_HALFULONG; }
4738 : else
4739 : {
4740 4409 : long l = lgefint(z) - 2;
4741 4409 : if (nbits2nlong(d*b) == d*l) b = l*BITS_IN_LONG;
4742 : }
4743 89193 : set_avma(av);
4744 :
4745 89193 : switch (b) {
4746 28278 : case BITS_IN_HALFULONG:
4747 28278 : pack = kron_pack_Flx_spec_half;
4748 28278 : unpack = int_to_Flx_half;
4749 28278 : break;
4750 1018 : case BITS_IN_LONG:
4751 1018 : pack = kron_pack_Flx_spec;
4752 1018 : unpack = kron_unpack_Flx;
4753 1018 : break;
4754 49 : case 2*BITS_IN_LONG:
4755 49 : pack = kron_pack_Flx_spec_2;
4756 49 : unpack = kron_unpack_Flx_2;
4757 49 : break;
4758 0 : case 3*BITS_IN_LONG:
4759 0 : pack = kron_pack_Flx_spec_3;
4760 0 : unpack = kron_unpack_Flx_3;
4761 0 : break;
4762 59848 : default:
4763 59848 : A = FlxM_pack_ZM_bits(A, b);
4764 59848 : B = is_sqr? A: FlxM_pack_ZM_bits(B, b);
4765 59848 : C = ZM_mul(A, B);
4766 59848 : D = ZM_unpack_FlxqM_bits(C, b, T, p, pi);
4767 59848 : return gerepilecopy(av, D);
4768 : }
4769 29345 : A = FlxM_pack_ZM(A, pack);
4770 29345 : B = is_sqr? A: FlxM_pack_ZM(B, pack);
4771 29345 : C = ZM_mul(A, B);
4772 29345 : D = ZM_unpack_FlxqM(C, T, p, pi, unpack);
4773 29345 : return gerepilecopy(av, D);
4774 : }
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