Line data Source code
1 : /* Copyright (C) 2000-2004 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 : /*******************************************************************/
16 : /* */
17 : /* POLYNOMIAL FACTORIZATION IN A NUMBER FIELD */
18 : /* */
19 : /*******************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_nffactor
24 :
25 : static GEN nfsqff(GEN nf,GEN pol,long fl,GEN den);
26 : static int nfsqff_use_Trager(long n, long dpol);
27 :
28 : enum { FACTORS = 0, ROOTS, ROOTS_SPLIT };
29 :
30 : /* for nf_bestlift: reconstruction of algebraic integers known mod P^k,
31 : * P maximal ideal above p */
32 : typedef struct {
33 : long k; /* input known mod P^k */
34 : GEN p, pk; /* p^k = denom(prk^-1) [ assume pr unramified ]*/
35 : GEN prk; /* |.|^2 LLL-reduced basis (b_i) of P^k (NOT T2-reduced) */
36 : GEN iprk; /* den * prk^-1 */
37 : GEN GSmin; /* min |b_i^*|^2 */
38 :
39 : GEN Tp; /* Tpk mod p */
40 : GEN Tpk;
41 : GEN ZqProj;/* projector to Zp / P^k = Z/p^k[X] / Tpk */
42 :
43 : GEN tozk;
44 : GEN topow;
45 : GEN topowden; /* topow x / topowden = basistoalg(x) */
46 : GEN dn; /* NULL (we trust nf.zk) or a t_INT > 1 (an alg. integer has
47 : denominator dividing dn, when expressed on nf.zk */
48 : } nflift_t;
49 :
50 : typedef struct
51 : {
52 : nflift_t *L;
53 : GEN nf;
54 : GEN pol, polbase; /* leading coeff is a t_INT */
55 : GEN fact;
56 : GEN Br, bound, ZC, BS_2;
57 : } nfcmbf_t;
58 :
59 : /*******************************************************************/
60 : /* RATIONAL RECONSTRUCTION (use ratlift) */
61 : /*******************************************************************/
62 : /* NOT stack clean. a, b stay on the stack */
63 : static GEN
64 26721747 : lift_to_frac(GEN t, GEN N, GEN amax, GEN bmax, GEN den, GEN tden)
65 : {
66 : GEN a, b;
67 26721747 : if (signe(t) < 0) t = addii(t, N); /* in case t is a centerlift */
68 26721716 : if (tden)
69 : {
70 7209680 : pari_sp av = avma;
71 7209680 : a = Fp_center_i(Fp_mul(t, tden, N), N, shifti(N,-1));
72 7205636 : if (abscmpii(a, amax) < 0) return gerepileupto(av, Qdivii(a, tden));
73 840084 : set_avma(av);
74 : }
75 20352123 : if (!Fp_ratlift(t, N, amax,bmax, &a,&b)
76 20053932 : || (den && !dvdii(den,b))
77 20355217 : || !is_pm1(gcdii(a,b))) return NULL;
78 20047272 : if (is_pm1(b)) { cgiv(b); return a; }
79 3109541 : return mkfrac(a, b);
80 : }
81 :
82 : /* Compute rational lifting for all the components of P modulo N. Assume
83 : * Fp_ratlift preconditions are met; we allow centerlifts. If one component
84 : * fails, return NULL. If den != NULL, check that the deninators divide den;
85 : * assume (N, den) = 1. */
86 : GEN
87 5261969 : FpC_ratlift(GEN P, GEN N, GEN amax, GEN bmax, GEN den)
88 : {
89 5261969 : pari_sp av = avma;
90 : long j, l;
91 5261969 : GEN tden = NULL, Q = cgetg_copy(P, &l);
92 5261970 : if (l==1) return Q;
93 5261970 : if (den && cmpii(bmax, den) > 0) bmax = den;
94 30833261 : for (j = 1; j < l; ++j)
95 : {
96 25769907 : GEN a = lift_to_frac(gel(P,j), N, amax, bmax, den, tden);
97 25770518 : if (!a) return gc_NULL(av);
98 25571732 : if (typ(a) == t_FRAC)
99 : {
100 6491170 : GEN d = gel(a,2);
101 6491170 : tden = tden? (cmpii(tden, d) < 0? d: tden): d;
102 : }
103 25571291 : gel(Q,j) = a;
104 : }
105 5063354 : return Q;
106 : }
107 : GEN
108 360536 : FpX_ratlift(GEN P, GEN N, GEN amax, GEN bmax, GEN den)
109 : {
110 360536 : pari_sp av = avma;
111 : long j, l;
112 360536 : GEN tden = NULL, Q = cgetg_copy(P, &l);
113 360537 : Q[1] = P[1];
114 360537 : if (den && cmpii(bmax, den) > 0) bmax = den;
115 1202603 : for (j = 2; j < l; ++j)
116 : {
117 951488 : GEN a = lift_to_frac(gel(P,j), N, amax, bmax, den, tden);
118 951485 : if (!a) return gc_NULL(av);
119 842063 : if (typ(a) == t_FRAC)
120 : {
121 576106 : GEN d = gel(a,2);
122 576106 : tden = tden? (cmpii(tden, d) < 0? d: tden): d;
123 : }
124 842063 : gel(Q,j) = a;
125 : }
126 251115 : return Q;
127 : }
128 :
129 : GEN
130 2740103 : FpM_ratlift(GEN M, GEN mod, GEN amax, GEN bmax, GEN den)
131 : {
132 2740103 : pari_sp av = avma;
133 2740103 : long j, l = lg(M);
134 2740103 : GEN N = cgetg_copy(M, &l);
135 2740131 : if (l == 1) return N;
136 7723578 : for (j = 1; j < l; ++j)
137 : {
138 5172953 : GEN a = FpC_ratlift(gel(M, j), mod, amax, bmax, den);
139 5172948 : if (!a) return gc_NULL(av);
140 4983447 : gel(N,j) = a;
141 : }
142 2550625 : return N;
143 : }
144 :
145 : /*******************************************************************/
146 : /* GCD in K[X], K NUMBER FIELD */
147 : /*******************************************************************/
148 : /* P a nonzero ZXQX */
149 : static GEN
150 113946 : lead_simplify(GEN P)
151 : {
152 113946 : GEN x = leading_coeff(P); /* x a nonzero ZX or t_INT */
153 113948 : if (typ(x) == t_POL)
154 : {
155 3724 : if (degpol(x)) return x;
156 3437 : x = gel(x,2);
157 : }
158 113661 : return is_pm1(x)? NULL: x;
159 : }
160 : /* P,Q in Z[X,Y], T in Z[Y] irreducible. compute GCD in Q[Y]/(T)[X].
161 : *
162 : * M. Encarnacion "On a modular Algorithm for computing GCDs of polynomials
163 : * over number fields" (ISSAC'94).
164 : *
165 : * We procede as follows
166 : * 1:compute the gcd modulo primes discarding bad primes as they are detected.
167 : * 2:reconstruct the result via FpM_ratlift, stoping as soon as we get weird
168 : * denominators.
169 : * 3:if FpM_ratlift succeeds, try the full division.
170 : * Suppose accuracy is insufficient to get the result right: FpM_ratlift will
171 : * rarely succeed, and even if it does the polynomial we get has sensible
172 : * coefficients, so the full division will not be too costly.
173 : *
174 : * If not NULL, den must be a multiple of the denominator of the gcd,
175 : * for example the discriminant of T.
176 : *
177 : * NOTE: if T is not irreducible, nfgcd may loop forever, esp. if gcd | T */
178 : GEN
179 83778 : nfgcd_all(GEN P, GEN Q, GEN T, GEN den, GEN *Pnew)
180 : {
181 83778 : pari_sp btop, ltop = avma;
182 83778 : GEN lP, lQ, M, dsol, R, bo, sol, mod = NULL, lden = NULL;
183 83778 : long vP = varn(P), vT = varn(T), dT = degpol(T), dM = 0, dR;
184 : forprime_t S;
185 :
186 83777 : if (!signe(P)) { if (Pnew) *Pnew = pol_0(vT); return gcopy(Q); }
187 83777 : if (!signe(Q)) { if (Pnew) *Pnew = pol_1(vT); return gcopy(P); }
188 : /* Compute denominators */
189 83679 : if ((lP = lead_simplify(P)) && (lQ = lead_simplify(Q)))
190 : {
191 14070 : if (typ(lP) == t_INT && typ(lQ) == t_INT)
192 13923 : lden = powiu(gcdii(lP, lQ), dT);
193 147 : else if (typ(lP) == t_INT)
194 7 : lden = gcdii(powiu(lP, dT), ZX_resultant(lQ, T));
195 140 : else if (typ(lQ) == t_INT)
196 0 : lden = gcdii(powiu(lQ, dT), ZX_resultant(lP, T));
197 : else
198 140 : lden = gcdii(ZX_resultant(lP, T), ZX_resultant(lQ, T));
199 14070 : if (is_pm1(lden)) lden = NULL;
200 14070 : if (den && lden) den = mulii(den, lden);
201 : }
202 83681 : init_modular_small(&S);
203 83681 : btop = avma;
204 : for(;;)
205 5451 : {
206 89132 : ulong p = u_forprime_next(&S);
207 : GEN Tp;
208 89132 : if (!p) pari_err_OVERFLOW("nfgcd [ran out of primes]");
209 : /*Discard primes dividing disc(T) or lc(PQ) */
210 89132 : if (lden && !umodiu(lden, p)) continue;
211 89132 : Tp = ZX_to_Flx(T,p);
212 89132 : if (!Flx_is_squarefree(Tp, p)) continue;
213 : /*Discard primes when modular gcd does not exist*/
214 89131 : if ((R = FlxqX_safegcd(ZXX_to_FlxX(P,p,vT),
215 : ZXX_to_FlxX(Q,p,vT),
216 0 : Tp, p)) == NULL) continue;
217 89132 : dR = degpol(R);
218 89132 : if (dR == 0) { set_avma(ltop); if (Pnew) *Pnew = P; return pol_1(vP); }
219 8195 : if (mod && dR > dM) continue; /* p divides Res(P/gcd, Q/gcd). Discard. */
220 :
221 8195 : R = FlxX_to_Flm(R, dT);
222 : /* previous primes divided Res(P/gcd, Q/gcd)? Discard them. */
223 8195 : if (!mod || dR < dM) { M = ZM_init_CRT(R, p); mod = utoipos(p); dM = dR; continue; }
224 5451 : (void)ZM_incremental_CRT(&M,R, &mod,p);
225 5451 : if (gc_needed(btop, 1))
226 : {
227 0 : if (DEBUGMEM>1) pari_warn(warnmem,"nfgcd");
228 0 : gerepileall(btop, 2, &M, &mod);
229 : }
230 : /* I suspect it must be better to take amax > bmax*/
231 5451 : bo = sqrti(shifti(mod, -1));
232 5451 : if ((sol = FpM_ratlift(M, mod, bo, bo, den)) == NULL) continue;
233 2782 : sol = RgM_to_RgXX(sol,vP,vT);
234 2782 : dsol = Q_primpart(sol);
235 :
236 2782 : if (!ZXQX_dvd(Q, dsol, T)) continue;
237 2744 : if (Pnew)
238 : {
239 238 : *Pnew = RgXQX_pseudodivrem(P, dsol, T, &R);
240 238 : if (signe(R)) continue;
241 : }
242 2506 : else if (!ZXQX_dvd(P, dsol, T)) continue;
243 2744 : return gc_all(ltop, Pnew? 2: 1, &dsol, Pnew); /* both remainders are 0 */
244 : }
245 : }
246 : GEN
247 47414 : nfgcd(GEN P, GEN Q, GEN T, GEN den)
248 47414 : { return nfgcd_all(P, Q, T, den, NULL); }
249 :
250 : GEN
251 5425 : ZXQX_gcd(GEN P, GEN Q, GEN T)
252 5425 : { return nfgcd_all(P, Q, T, NULL, NULL); }
253 :
254 : GEN
255 3136 : QXQX_gcd(GEN P, GEN Q, GEN T)
256 : {
257 3136 : pari_sp av = avma;
258 3136 : GEN P1 = Q_remove_denom(P, NULL);
259 3136 : GEN Q1 = Q_remove_denom(Q, NULL);
260 3136 : return gerepileupto(av, ZXQX_gcd(P1, Q1, T));
261 : }
262 :
263 : int
264 49055 : nfissquarefree(GEN nf, GEN x)
265 : {
266 49055 : pari_sp av = avma;
267 49055 : GEN g, y = RgX_deriv(x);
268 49054 : if (RgX_is_rational(x)) g = QX_gcd(x, y);
269 : else
270 : {
271 46364 : GEN T = get_nfpol(nf,&nf);
272 46364 : x = Q_primpart( liftpol_shallow(x) );
273 46364 : y = Q_primpart( liftpol_shallow(y) );
274 46364 : g = nfgcd(x, y, T, nf? nf_get_index(nf): NULL);
275 : }
276 49056 : return gc_bool(av, degpol(g) == 0);
277 : }
278 :
279 : /*******************************************************************/
280 : /* FACTOR OVER (Z_K/pr)[X] --> FqX_factor */
281 : /*******************************************************************/
282 : GEN
283 7 : nffactormod(GEN nf, GEN x, GEN pr)
284 : {
285 7 : long j, l, vx = varn(x), vn;
286 7 : pari_sp av = avma;
287 : GEN F, E, rep, xrd, modpr, T, p;
288 :
289 7 : nf = checknf(nf);
290 7 : vn = nf_get_varn(nf);
291 7 : if (typ(x)!=t_POL) pari_err_TYPE("nffactormod",x);
292 7 : if (varncmp(vx,vn) >= 0) pari_err_PRIORITY("nffactormod", x, ">=", vn);
293 :
294 7 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
295 7 : xrd = nfX_to_FqX(x, nf, modpr);
296 7 : rep = FqX_factor(xrd,T,p);
297 7 : settyp(rep, t_MAT);
298 7 : F = gel(rep,1); l = lg(F);
299 7 : E = gel(rep,2); settyp(E, t_COL);
300 14 : for (j = 1; j < l; j++) {
301 7 : gel(F,j) = FqX_to_nfX(gel(F,j), modpr);
302 7 : gel(E,j) = stoi(E[j]);
303 : }
304 7 : return gerepilecopy(av, rep);
305 : }
306 :
307 : /*******************************************************************/
308 : /* MAIN ROUTINES nfroots / nffactor */
309 : /*******************************************************************/
310 : static GEN
311 32059 : QXQX_normalize(GEN P, GEN T)
312 : {
313 32059 : GEN P0 = leading_coeff(P);
314 32059 : long t = typ(P0);
315 32059 : if (t == t_POL)
316 : {
317 490 : if (degpol(P0)) return RgXQX_RgXQ_mul(P, QXQ_inv(P0,T), T);
318 455 : P0 = gel(P0,2); t = typ(P0);
319 : }
320 : /* t = t_INT/t_FRAC */
321 32024 : if (t == t_INT && is_pm1(P0) && signe(P0) > 0) return P; /* monic */
322 5845 : return RgX_Rg_div(P, P0);
323 : }
324 : /* assume leading term of P is an integer */
325 : static GEN
326 36868 : RgX_int_normalize(GEN P)
327 : {
328 36868 : GEN P0 = leading_coeff(P);
329 : /* cater for t_POL */
330 36868 : if (typ(P0) == t_POL)
331 : {
332 0 : P0 = gel(P0,2); /* nonzero constant */
333 0 : P = shallowcopy(P);
334 0 : gel(P,lg(P)-1) = P0; /* now leading term is a t_INT */
335 : }
336 36868 : if (typ(P0) != t_INT) pari_err_BUG("RgX_int_normalize");
337 36868 : if (is_pm1(P0)) return signe(P0) > 0? P: RgX_neg(P);
338 16688 : return RgX_Rg_div(P, P0);
339 : }
340 :
341 : /* discard change of variable if nf is of the form [nf,c] as return by nfinit
342 : * for nonmonic polynomials */
343 : static GEN
344 12404 : proper_nf(GEN nf)
345 12404 : { return (lg(nf) == 3)? gel(nf,1): nf; }
346 :
347 : /* if *pnf = NULL replace if by a "quick" K = nfinit(T), ensuring maximality
348 : * by small primes only. Return a multiplicative bound for the denominator of
349 : * algebraic integers in Z_K in terms of K.zk */
350 : static GEN
351 30659 : fix_nf(GEN *pnf, GEN *pT, GEN *pA)
352 : {
353 30659 : GEN nf, NF, P, q, D, T = *pT;
354 : nfmaxord_t S;
355 : long i, l, lim;
356 :
357 30659 : if (*pnf) return gen_1;
358 12404 : lim = GP_DATA->factorlimit + 1;
359 12404 : nfmaxord(&S, mkvec2(T, utoipos(lim)), nf_PARTIALFACT);
360 12404 : NF = nfinit_complete(&S, 0, DEFAULTPREC);
361 12404 : *pnf = nf = proper_nf(NF);
362 12404 : if (nf != NF) { /* t_POL defining base field changed (not monic) */
363 14 : GEN A = *pA, a = cgetg_copy(A, &l);
364 14 : GEN rev = gel(NF,2), pow, dpow;
365 :
366 14 : *pT = T = nf_get_pol(nf); /* need to update T */
367 14 : pow = QXQ_powers(lift_shallow(rev), degpol(T)-1, T);
368 14 : pow = Q_remove_denom(pow, &dpow);
369 14 : a[1] = A[1];
370 56 : for (i=2; i<l; i++) {
371 42 : GEN c = gel(A,i);
372 42 : if (typ(c) == t_POL) c = QX_ZXQV_eval(c, pow, dpow);
373 42 : gel(a,i) = c;
374 : }
375 14 : *pA = Q_primpart(a); /* need to update A */
376 : }
377 :
378 12404 : D = nf_get_disc(nf); if (is_pm1(D)) return gen_1;
379 : /* D may be incorrect */
380 12404 : P = nf_get_ramified_primes(nf); l = lg(P);
381 31381 : for (i = 1, q = gen_1; i < l; i++)
382 : {
383 18977 : GEN p = gel(P,i);
384 18977 : if (cmpiu(p, lim) >= 0 && !BPSW_psp(p)) q = mulii(q, p);
385 : }
386 12404 : return q;
387 : }
388 :
389 : /* lt(A) is an integer; ensure it is not a constant t_POL. In place */
390 : static void
391 30820 : ensure_lt_INT(GEN A)
392 : {
393 30820 : long n = lg(A)-1;
394 30820 : GEN lt = gel(A,n);
395 30932 : while (typ(lt) != t_INT) gel(A,n) = lt = gel(lt,2);
396 30820 : }
397 :
398 : /* set B = A/gcd(A,A'), squarefree */
399 : static GEN
400 30806 : get_nfsqff_data(GEN *pnf, GEN *pT, GEN *pA, GEN *pB, GEN *ptbad)
401 : {
402 30806 : GEN den, bad, D, B, A = *pA, T = *pT;
403 30806 : long n = degpol(T);
404 :
405 30806 : A = Q_primpart( QXQX_normalize(A, T) );
406 30806 : if (nfsqff_use_Trager(n, degpol(A)))
407 : {
408 168 : *pnf = T;
409 168 : bad = den = absi_shallow(ZX_disc(T));
410 168 : if (is_pm1(leading_coeff(T))) den = indexpartial(T, den);
411 : }
412 : else
413 : {
414 30638 : den = fix_nf(pnf, &T, &A);
415 30638 : bad = nf_get_index(*pnf);
416 30638 : if (den != gen_1) bad = mulii(bad, den);
417 : }
418 30806 : D = nfgcd_all(A, RgX_deriv(A), T, bad, &B);
419 30806 : if (degpol(D)) B = Q_primpart( QXQX_normalize(B, T) );
420 30806 : if (ptbad) *ptbad = bad;
421 30806 : *pA = A;
422 30806 : *pB = B; ensure_lt_INT(B);
423 30806 : *pT = T; return den;
424 : }
425 :
426 : /* return the roots of pol in nf */
427 : GEN
428 36014 : nfroots(GEN nf,GEN pol)
429 : {
430 36014 : pari_sp av = avma;
431 : GEN z, A, B, T, den;
432 : long d, dT;
433 :
434 36014 : if (!nf) return nfrootsQ(pol);
435 23876 : T = get_nfpol(nf, &nf);
436 23876 : RgX_check_ZX(T,"nfroots");
437 23876 : A = RgX_nffix("nfroots", T,pol,1);
438 23876 : d = degpol(A);
439 23876 : if (d < 0) pari_err_ROOTS0("nfroots");
440 23876 : if (d == 0) return cgetg(1,t_COL);
441 23876 : if (d == 1)
442 : {
443 7 : A = QXQX_normalize(A,T);
444 7 : A = mkpolmod(gneg_i(gel(A,2)), T);
445 7 : return gerepilecopy(av, mkcol(A));
446 : }
447 23869 : dT = degpol(T);
448 23869 : if (dT == 1) return gerepileupto(av, nfrootsQ(simplify_shallow(A)));
449 :
450 18234 : den = get_nfsqff_data(&nf, &T, &A, &B, NULL);
451 18234 : if (RgX_is_ZX(B))
452 : {
453 6034 : GEN v = gel(ZX_factor(B), 1);
454 6034 : long i, l = lg(v), p = mael(factoru(dT),1,1); /* smallest prime divisor */
455 6034 : z = cgetg(1, t_VEC);
456 16051 : for (i = 1; i < l; i++)
457 : {
458 10017 : GEN b = gel(v,i); /* irreducible / Q */
459 10017 : long db = degpol(b);
460 10017 : if (db != 1 && degpol(b) < p) continue;
461 9905 : z = shallowconcat(z, nfsqff(nf, b, ROOTS, den));
462 : }
463 : }
464 : else
465 12200 : z = nfsqff(nf,B, ROOTS, den);
466 18234 : z = gerepileupto(av, QXQV_to_mod(z, T));
467 18234 : gen_sort_inplace(z, (void*)&cmp_RgX, &cmp_nodata, NULL);
468 18234 : settyp(z, t_COL); return z;
469 : }
470 :
471 : static GEN
472 478050 : _norml2(GEN x) { return RgC_fpnorml2(x, DEFAULTPREC); }
473 :
474 : /* return a minimal lift of elt modulo id, as a ZC */
475 : static GEN
476 262148 : nf_bestlift(GEN elt, GEN bound, nflift_t *L)
477 : {
478 : GEN u;
479 262148 : long i,l = lg(L->prk), t = typ(elt);
480 262148 : if (t != t_INT)
481 : {
482 15649 : if (t == t_POL) elt = ZM_ZX_mul(L->tozk, elt);
483 15649 : u = ZM_ZC_mul(L->iprk,elt);
484 310017 : for (i=1; i<l; i++) gel(u,i) = diviiround(gel(u,i), L->pk);
485 : }
486 : else
487 : {
488 246499 : u = ZC_Z_mul(gel(L->iprk,1), elt);
489 1919924 : for (i=1; i<l; i++) gel(u,i) = diviiround(gel(u,i), L->pk);
490 246479 : elt = scalarcol(elt, l-1);
491 : }
492 262140 : u = ZC_sub(elt, ZM_ZC_mul(L->prk, u));
493 262132 : if (bound && gcmp(_norml2(u), bound) > 0) u = NULL;
494 262143 : return u;
495 : }
496 :
497 : /* Warning: return L->topowden * (best lift). */
498 : static GEN
499 156974 : nf_bestlift_to_pol(GEN elt, GEN bound, nflift_t *L)
500 : {
501 156974 : pari_sp av = avma;
502 156974 : GEN u,v = nf_bestlift(elt,bound,L);
503 156972 : if (!v) return NULL;
504 147475 : if (ZV_isscalar(v))
505 : {
506 65168 : if (L->topowden)
507 65168 : u = mulii(L->topowden, gel(v,1));
508 : else
509 0 : u = icopy(gel(v,1));
510 65168 : u = gerepileuptoint(av, u);
511 : }
512 : else
513 : {
514 82307 : v = gclone(v); set_avma(av);
515 82309 : u = RgV_dotproduct(L->topow, v);
516 82308 : gunclone(v);
517 : }
518 147476 : return u;
519 : }
520 :
521 : /* return the T->powden * (lift of pol with coefficients of T2-norm <= C)
522 : * if it exists. */
523 : static GEN
524 36722 : nf_pol_lift(GEN pol, GEN bound, nflift_t *L)
525 : {
526 36722 : long i, l = lg(pol);
527 36722 : GEN x = cgetg(l,t_POL);
528 :
529 36722 : x[1] = pol[1];
530 172204 : for (i=l-1; i>1; i--)
531 : {
532 144980 : GEN t = nf_bestlift_to_pol(gel(pol,i), bound, L);
533 144979 : if (!t) return NULL;
534 135482 : gel(x,i) = t;
535 : }
536 27224 : return x;
537 : }
538 :
539 : static GEN
540 0 : zerofact(long v)
541 : {
542 0 : GEN z = cgetg(3, t_MAT);
543 0 : gel(z,1) = mkcol(pol_0(v));
544 0 : gel(z,2) = mkcol(gen_1); return z;
545 : }
546 :
547 : /* Return the factorization of A in Q[X]/(T) in rep [pre-allocated with
548 : * cgetg(3,t_MAT)], reclaiming all memory between avma and rep.
549 : * y is the vector of irreducible factors of B = Q_primpart( A/gcd(A,A') ).
550 : * Bad primes divide 'bad' */
551 : static void
552 12586 : fact_from_sqff(GEN rep, GEN A, GEN B, GEN y, GEN T, GEN bad)
553 : {
554 12586 : pari_sp av = (pari_sp)rep;
555 12586 : long n = lg(y)-1;
556 : GEN ex;
557 :
558 12586 : if (A != B)
559 : { /* not squarefree */
560 133 : if (n == 1)
561 : { /* perfect power, simple ! */
562 28 : long e = degpol(A) / degpol(gel(y,1));
563 28 : y = gerepileupto(av, QXQXV_to_mod(y, T));
564 28 : ex = mkcol(utoipos(e));
565 : }
566 : else
567 : { /* compute valuations mod a prime of degree 1 (avoid coeff explosion) */
568 105 : GEN quo, p, r, Bp, lb = leading_coeff(B), E = cgetalloc(n+1,t_VECSMALL);
569 105 : pari_sp av1 = avma;
570 : ulong pp;
571 : long j;
572 : forprime_t S;
573 105 : u_forprime_init(&S, degpol(T), ULONG_MAX);
574 266 : for (; ; set_avma(av1))
575 : {
576 371 : pp = u_forprime_next(&S);
577 371 : if (! umodiu(bad,pp) || !umodiu(lb, pp)) continue;
578 350 : p = utoipos(pp);
579 350 : r = FpX_oneroot(T, p);
580 350 : if (!r) continue;
581 189 : Bp = FpXY_evalx(B, r, p);
582 189 : if (FpX_is_squarefree(Bp, p)) break;
583 : }
584 :
585 105 : quo = FpXY_evalx(Q_primpart(A), r, p);
586 245 : for (j=n; j>=2; j--)
587 : {
588 140 : GEN junk, fact = Q_remove_denom(gel(y,j), &junk);
589 140 : long e = 0;
590 140 : fact = FpXY_evalx(fact, r, p);
591 287 : for(;; e++)
592 287 : {
593 427 : GEN q = FpX_divrem(quo,fact,p,ONLY_DIVIDES);
594 427 : if (!q) break;
595 287 : quo = q;
596 : }
597 140 : E[j] = e;
598 : }
599 105 : E[1] = degpol(quo) / degpol(gel(y,1));
600 105 : y = gerepileupto(av, QXQXV_to_mod(y, T));
601 105 : ex = zc_to_ZC(E); pari_free((void*)E);
602 : }
603 : }
604 : else
605 : {
606 12453 : y = gerepileupto(av, QXQXV_to_mod(y, T));
607 12453 : ex = const_col(n, gen_1);
608 : }
609 12586 : gel(rep,1) = y; settyp(y, t_COL);
610 12586 : gel(rep,2) = ex;
611 12586 : }
612 :
613 : /* return the factorization of polynomial pol in nf */
614 : static GEN
615 12887 : nffactor_i(GEN nf,GEN T,GEN pol)
616 : {
617 12887 : GEN bad, A, B, y, den, rep = cgetg(3, t_MAT);
618 12887 : pari_sp av = avma;
619 : long dA;
620 : pari_timer ti;
621 :
622 12887 : if (DEBUGLEVEL>2) { timer_start(&ti); err_printf("\nEntering nffactor:\n"); }
623 12887 : A = RgX_nffix("nffactor",T,pol,1);
624 12887 : dA = degpol(A);
625 12887 : if (dA <= 0) {
626 7 : set_avma((pari_sp)(rep + 3));
627 7 : return (dA == 0)? trivial_fact(): zerofact(varn(pol));
628 : }
629 12880 : if (dA == 1) {
630 : GEN c;
631 196 : A = Q_primpart( QXQX_normalize(A, T) );
632 196 : A = gerepilecopy(av, A); c = gel(A,2);
633 196 : if (typ(c) == t_POL && degpol(c) > 0) gel(A,2) = mkpolmod(c, ZX_copy(T));
634 196 : gel(rep,1) = mkcol(A);
635 196 : gel(rep,2) = mkcol(gen_1); return rep;
636 : }
637 12684 : if (degpol(T) == 1) return gerepileupto(av, QX_factor(simplify_shallow(A)));
638 :
639 12572 : den = get_nfsqff_data(&nf, &T, &A, &B, &bad);
640 12572 : if (DEBUGLEVEL>2) timer_printf(&ti, "squarefree test");
641 12572 : if (RgX_is_ZX(B))
642 : {
643 11998 : GEN v = gel(ZX_factor(B), 1);
644 11998 : long i, l = lg(v);
645 11998 : y = cgetg(1, t_VEC);
646 24052 : for (i = 1; i < l; i++)
647 : {
648 12054 : GEN b = gel(v,i); /* irreducible / Q */
649 12054 : y = shallowconcat(y, nfsqff(nf, b, 0, den));
650 : }
651 : }
652 : else
653 574 : y = nfsqff(nf,B, 0, den);
654 12572 : if (DEBUGLEVEL>3) err_printf("number of factor(s) found: %ld\n", lg(y)-1);
655 :
656 12572 : fact_from_sqff(rep, A, B, y, T, bad);
657 12572 : return rep;
658 : }
659 :
660 : /* return the factorization of P in nf */
661 : GEN
662 12880 : nffactor(GEN nf, GEN P)
663 : {
664 12880 : GEN y, T = get_nfpol(nf, &nf);
665 12880 : if (!nf) RgX_check_ZX(T,"nffactor");
666 12880 : if (typ(P) == t_RFRAC)
667 : {
668 14 : pari_sp av = avma;
669 14 : GEN a = gel(P, 1), b = gel(P, 2);
670 14 : y = famat_inv_shallow(nffactor_i(nf, T, b));
671 14 : if (typ(a) == t_POL && varn(a) == varn(b))
672 7 : y = famat_mul_shallow(nffactor_i(nf, T, a), y);
673 14 : y = gerepilecopy(av, y);
674 : }
675 : else
676 12866 : y = nffactor_i(nf, T, P);
677 12880 : return sort_factor_pol(y, cmp_RgX);
678 : }
679 :
680 : /* assume x scalar or t_COL, G t_MAT */
681 : static GEN
682 202937 : arch_for_T2(GEN G, GEN x)
683 : {
684 2380 : return (typ(x) == t_COL)? RgM_RgC_mul(G,x)
685 205317 : : RgC_Rg_mul(gel(G,1),x);
686 : }
687 :
688 : /* polbase a zkX with t_INT leading coeff; return a bound for T_2(P),
689 : * P | polbase in C[X]. NB: Mignotte bound: A | S ==>
690 : * |a_i| <= binom(d-1, i-1) || S ||_2 + binom(d-1, i) lc(S)
691 : *
692 : * Apply to sigma(S) for all embeddings sigma, then take the L_2 norm over
693 : * sigma, then take the sup over i */
694 : static GEN
695 12159 : nf_Mignotte_bound(GEN nf, GEN polbase)
696 12159 : { GEN lS = leading_coeff(polbase); /* t_INT */
697 : GEN p1, C, N2, binlS, bin;
698 12159 : long prec = nf_get_prec(nf), n = nf_get_degree(nf), r1 = nf_get_r1(nf);
699 12159 : long i, j, d = degpol(polbase);
700 :
701 12159 : binlS = bin = vecbinomial(d-1);
702 12159 : if (!isint1(lS)) binlS = ZC_Z_mul(bin,lS);
703 :
704 12159 : N2 = cgetg(n+1, t_VEC);
705 : for (;;)
706 0 : {
707 12159 : GEN G = nf_get_G(nf), matGS = cgetg(d+2, t_MAT);
708 :
709 134117 : for (j=0; j<=d; j++) gel(matGS,j+1) = arch_for_T2(G, gel(polbase,j+2));
710 12159 : matGS = shallowtrans(matGS);
711 20104 : for (j=1; j <= r1; j++) /* N2[j] = || sigma_j(S) ||_2 */
712 : {
713 7945 : GEN c = sqrtr( _norml2(gel(matGS,j)) );
714 7945 : gel(N2,j) = c; if (!signe(c)) goto PRECPB;
715 : }
716 31241 : for ( ; j <= n; j+=2)
717 : {
718 19082 : GEN q1 = _norml2(gel(matGS, j));
719 19082 : GEN q2 = _norml2(gel(matGS, j+1));
720 19082 : GEN c = sqrtr( gmul2n(addrr(q1, q2), -1) );
721 19082 : gel(N2,j) = gel(N2,j+1) = c; if (!signe(c)) goto PRECPB;
722 : }
723 12159 : break; /* done */
724 0 : PRECPB:
725 0 : prec = precdbl(prec);
726 0 : nf = nfnewprec_shallow(nf, prec);
727 0 : if (DEBUGLEVEL>1) pari_warn(warnprec, "nf_factor_bound", prec);
728 : }
729 :
730 : /* Take sup over 0 <= i <= d of
731 : * sum_j | binom(d-1, i-1) ||sigma_j(S)||_2 + binom(d-1,i) lc(S) |^2 */
732 :
733 : /* i = 0: n lc(S)^2 */
734 12159 : C = mului(n, sqri(lS));
735 : /* i = d: sum_sigma ||sigma(S)||_2^2 */
736 12159 : p1 = gnorml2(N2); if (gcmp(C, p1) < 0) C = p1;
737 109801 : for (i = 1; i < d; i++)
738 : {
739 97642 : GEN B = gel(bin,i), L = gel(binlS,i+1);
740 97642 : GEN s = sqrr(addri(mulir(B, gel(N2,1)), L)); /* j=1 */
741 627001 : for (j = 2; j <= n; j++) s = addrr(s, sqrr(addri(mulir(B, gel(N2,j)), L)));
742 97641 : if (mpcmp(C, s) < 0) C = s;
743 : }
744 12159 : return C;
745 : }
746 :
747 : /* return a bound for T_2(P), P | polbase
748 : * max |b_i|^2 <= 3^{3/2 + d} / (4 \pi d) [P]_2,
749 : * where [P]_2 is Bombieri's 2-norm
750 : * Sum over conjugates */
751 : static GEN
752 12159 : nf_Beauzamy_bound(GEN nf, GEN polbase)
753 : {
754 : GEN lt, C, s, POL, bin;
755 12159 : long d = degpol(polbase), n = nf_get_degree(nf), prec = nf_get_prec(nf);
756 12159 : bin = vecbinomial(d);
757 12159 : POL = polbase + 2;
758 : /* compute [POL]_2 */
759 : for (;;)
760 0 : {
761 12159 : GEN G = nf_get_G(nf);
762 : long i;
763 :
764 12159 : s = real_0(prec);
765 134112 : for (i=0; i<=d; i++)
766 : {
767 121954 : GEN c = gel(POL,i);
768 121954 : if (gequal0(c)) continue;
769 80982 : c = _norml2(arch_for_T2(G,c));
770 80985 : if (!signe(c)) goto PRECPB;
771 : /* s += T2(POL[i]) / binomial(d,i) */
772 80985 : s = addrr(s, divri(c, gel(bin,i+1)));
773 : }
774 12158 : break;
775 0 : PRECPB:
776 0 : prec = precdbl(prec);
777 0 : nf = nfnewprec_shallow(nf, prec);
778 0 : if (DEBUGLEVEL>1) pari_warn(warnprec, "nf_factor_bound", prec);
779 : }
780 12158 : lt = leading_coeff(polbase);
781 12158 : s = mulri(s, muliu(sqri(lt), n));
782 12158 : C = powruhalf(utor(3,DEFAULTPREC), 3 + 2*d); /* 3^{3/2 + d} */
783 12159 : return divrr(mulrr(C, s), mulur(d, mppi(DEFAULTPREC)));
784 : }
785 :
786 : static GEN
787 12159 : nf_factor_bound(GEN nf, GEN polbase)
788 : {
789 12159 : pari_sp av = avma;
790 12159 : GEN a = nf_Mignotte_bound(nf, polbase);
791 12159 : GEN b = nf_Beauzamy_bound(nf, polbase);
792 12158 : if (DEBUGLEVEL>2)
793 : {
794 0 : err_printf("Mignotte bound: %Ps\n",a);
795 0 : err_printf("Beauzamy bound: %Ps\n",b);
796 : }
797 12158 : return gerepileupto(av, gmin(a, b));
798 : }
799 :
800 : /* True nf; return Bs: if r a root of sigma_i(P), |r| < Bs[i] */
801 : static GEN
802 18408 : nf_root_bounds(GEN nf, GEN P)
803 : {
804 : long lR, i, j, l, prec, r1;
805 : GEN Ps, R, V;
806 :
807 18408 : if (RgX_is_rational(P)) return polrootsbound(P, NULL);
808 5381 : r1 = nf_get_r1(nf);
809 5381 : P = Q_primpart(P);
810 5381 : prec = ZXX_max_lg(P) + 1;
811 5381 : l = lg(P);
812 5381 : if (nf_get_prec(nf) >= prec)
813 5373 : R = nf_get_roots(nf);
814 : else
815 8 : R = QX_complex_roots(nf_get_pol(nf), prec);
816 5381 : lR = lg(R);
817 5381 : V = cgetg(lR, t_VEC);
818 5381 : Ps = cgetg(l, t_POL); /* sigma (P) */
819 5381 : Ps[1] = P[1];
820 19335 : for (j=1; j<lg(R); j++)
821 : {
822 13954 : GEN r = gel(R,j);
823 100518 : for (i=2; i<l; i++) gel(Ps,i) = poleval(gel(P,i), r);
824 13954 : gel(V,j) = polrootsbound(Ps, NULL);
825 : }
826 5381 : return mkvec2(vecslice(V,1,r1), vecslice(V,r1+1,lg(V)-1));
827 : }
828 :
829 : /* return B such that, if x = sum x_i K.zk[i] in O_K, then ||x||_2^2 <= B T_2(x)
830 : * den = multiplicative bound for denom(x) [usually NULL, for 1, but when we
831 : * use nf_PARTIALFACT K.zk may not generate O_K] */
832 : GEN
833 22183 : nf_L2_bound(GEN nf, GEN den, GEN *pL)
834 : {
835 22183 : GEN M, L, prep, T = nf_get_pol(nf), tozk = nf_get_invzk(nf);
836 22183 : long prec = ZM_max_lg(tozk) + ZX_max_lg(T) + nbits2prec(degpol(T));
837 22184 : (void)initgaloisborne(nf, den? den: gen_1, prec, &L, &prep, NULL);
838 22184 : M = vandermondeinverse(L, RgX_gtofp(T,prec), den, prep);
839 22184 : if (pL) *pL = L;
840 22184 : return RgM_fpnorml2(RgM_mul(tozk,M), DEFAULTPREC);
841 : }
842 :
843 : /* sum_i L[i]^p */
844 : static GEN
845 20866 : normlp(GEN L, long p)
846 : {
847 20866 : long i, l = lg(L);
848 : GEN z;
849 20866 : if (l == 1) return gen_0;
850 10902 : z = gpowgs(gel(L,1), p);
851 27257 : for (i=2; i<l; i++) z = gadd(z, gpowgs(gel(L,i), p));
852 10902 : return z;
853 : }
854 : /* \sum_i deg(sigma_i) L[i]^p in dimension n (L may be a scalar
855 : * or [L1,L2], where Ld corresponds to the archimedean places of degree d) */
856 : static GEN
857 25070 : normTp(GEN L, long p, long n)
858 : {
859 25070 : if (typ(L) != t_VEC) return gmulsg(n, gpowgs(L, p));
860 10433 : return gadd(normlp(gel(L,1),p), gmul2n(normlp(gel(L,2),p), 1));
861 : }
862 :
863 : /* S = S0 + tS1, P = P0 + tP1 (Euclidean div. by t integer). For a true
864 : * factor (vS, vP), we have:
865 : * | S vS + P vP |^2 < Btra
866 : * This implies | S1 vS + P1 vP |^2 < Bhigh, assuming t > sqrt(Btra).
867 : * d = dimension of low part (= [nf:Q])
868 : * n0 = bound for |vS|^2
869 : * */
870 : static double
871 12250 : get_Bhigh(long n0, long d)
872 : {
873 12250 : double sqrtd = sqrt((double)d);
874 12250 : double z = n0*sqrtd + sqrtd/2 * (d * (n0+1));
875 12250 : z = 1. + 0.5 * z; return z * z;
876 : }
877 :
878 : typedef struct {
879 : GEN d;
880 : GEN dPinvS; /* d P^(-1) S [ integral ] */
881 : double **PinvSdbl; /* P^(-1) S as double */
882 : GEN S1, P1; /* S = S0 + S1 q, idem P */
883 : } trace_data;
884 :
885 : /* S1 * u - P1 * round(P^-1 S u). K nonzero coords in u given by ind */
886 : static GEN
887 205988 : get_trace(GEN ind, trace_data *T)
888 : {
889 205988 : long i, j, l, K = lg(ind)-1;
890 : GEN z, s, v;
891 :
892 205988 : s = gel(T->S1, ind[1]);
893 205988 : if (K == 1) return s;
894 :
895 : /* compute s = S1 u */
896 502987 : for (j=2; j<=K; j++) s = ZC_add(s, gel(T->S1, ind[j]));
897 :
898 : /* compute v := - round(P^1 S u) */
899 179096 : l = lg(s);
900 179096 : v = cgetg(l, t_VECSMALL);
901 2620943 : for (i=1; i<l; i++)
902 : {
903 2441847 : double r, t = 0.;
904 : /* quick approximate computation */
905 9301585 : for (j=1; j<=K; j++) t += T->PinvSdbl[ ind[j] ][i];
906 2441847 : r = floor(t + 0.5);
907 2441847 : if (fabs(t + 0.5 - r) < 0.0001)
908 : { /* dubious, compute exactly */
909 362 : z = gen_0;
910 1391 : for (j=1; j<=K; j++) z = addii(z, ((GEN**)T->dPinvS)[ ind[j] ][i]);
911 362 : v[i] = - itos( diviiround(z, T->d) );
912 : }
913 : else
914 2441485 : v[i] = - (long)r;
915 : }
916 179096 : return ZC_add(s, ZM_zc_mul(T->P1, v));
917 : }
918 :
919 : static trace_data *
920 24318 : init_trace(trace_data *T, GEN S, nflift_t *L, GEN q)
921 : {
922 24318 : long e = gexpo(S), i,j, l,h;
923 : GEN qgood, S1, invd;
924 :
925 24318 : if (e < 0) return NULL; /* S = 0 */
926 :
927 20590 : qgood = int2n(e - 32); /* single precision check */
928 20590 : if (cmpii(qgood, q) > 0) q = qgood;
929 :
930 20590 : S1 = gdivround(S, q);
931 20590 : if (gequal0(S1)) return NULL;
932 :
933 3944 : invd = invr(itor(L->pk, DEFAULTPREC));
934 :
935 3944 : T->dPinvS = ZM_mul(L->iprk, S);
936 3944 : l = lg(S);
937 3944 : h = lgcols(T->dPinvS);
938 3944 : T->PinvSdbl = (double**)cgetg(l, t_MAT);
939 34382 : for (j = 1; j < l; j++)
940 : {
941 30438 : double *t = (double *) stack_malloc_align(h * sizeof(double), sizeof(double));
942 30438 : GEN c = gel(T->dPinvS,j);
943 30438 : pari_sp av = avma;
944 30438 : T->PinvSdbl[j] = t;
945 321740 : for (i=1; i < h; i++) t[i] = rtodbl(mulri(invd, gel(c,i)));
946 30438 : set_avma(av);
947 : }
948 :
949 3944 : T->d = L->pk;
950 3944 : T->P1 = gdivround(L->prk, q);
951 3944 : T->S1 = S1; return T;
952 : }
953 :
954 : static void
955 209152 : update_trace(trace_data *T, long k, long i)
956 : {
957 209152 : if (!T) return;
958 92224 : gel(T->S1,k) = gel(T->S1,i);
959 92224 : gel(T->dPinvS,k) = gel(T->dPinvS,i);
960 92224 : T->PinvSdbl[k] = T->PinvSdbl[i];
961 : }
962 :
963 : /* reduce coeffs mod (T,pk), then center mod pk */
964 : static GEN
965 53046 : FqX_centermod(GEN z, GEN T, GEN pk, GEN pks2)
966 : {
967 : long i, l;
968 : GEN y;
969 53046 : if (!T) return centermod_i(z, pk, pks2);
970 15435 : y = FpXQX_red(z, T, pk); l = lg(y);
971 138537 : for (i = 2; i < l; i++)
972 : {
973 123102 : GEN c = gel(y,i);
974 123102 : if (typ(c) == t_INT)
975 81417 : c = Fp_center_i(c, pk, pks2);
976 : else
977 41685 : c = FpX_center_i(c, pk, pks2);
978 123102 : gel(y,i) = c;
979 : }
980 15435 : return y;
981 : }
982 :
983 : typedef struct {
984 : GEN lt, C, Clt, C2lt, C2ltpol;
985 : } div_data;
986 :
987 : static void
988 18485 : init_div_data(div_data *D, GEN pol, nflift_t *L)
989 : {
990 18485 : GEN C2lt, Clt, C = mul_content(L->topowden, L->dn);
991 18485 : GEN lc = leading_coeff(pol), lt = is_pm1(lc)? NULL: absi_shallow(lc);
992 18485 : if (C)
993 : {
994 18485 : GEN C2 = sqri(C);
995 18485 : if (lt) {
996 3405 : C2lt = mulii(C2, lt);
997 3405 : Clt = mulii(C,lt);
998 : } else {
999 15080 : C2lt = C2;
1000 15080 : Clt = C;
1001 : }
1002 : }
1003 : else
1004 0 : C2lt = Clt = lt;
1005 18485 : D->lt = lt;
1006 18485 : D->C = C;
1007 18485 : D->Clt = Clt;
1008 18485 : D->C2lt = C2lt;
1009 18485 : D->C2ltpol = C2lt? RgX_Rg_mul(pol, C2lt): pol;
1010 18485 : }
1011 : static void
1012 24549 : update_target(div_data *D, GEN pol)
1013 24549 : { D->C2ltpol = D->Clt? RgX_Rg_mul(pol, D->Clt): pol; }
1014 :
1015 : /* nb = number of modular factors; return a "good" K such that naive
1016 : * recombination of up to maxK modular factors is not too costly */
1017 : long
1018 111006 : cmbf_maxK(long nb)
1019 : {
1020 111006 : if (nb > 10) return 3;
1021 107037 : return nb-1;
1022 : }
1023 : /* Naive recombination of modular factors: combine up to maxK modular
1024 : * factors, degree <= klim
1025 : *
1026 : * target = polynomial we want to factor
1027 : * famod = array of modular factors. Product should be congruent to
1028 : * target/lc(target) modulo p^a
1029 : * For true factors: S1,S2 <= p^b, with b <= a and p^(b-a) < 2^31 */
1030 : /* set *done = 1 if factorisation is known to be complete */
1031 : static GEN
1032 12159 : nfcmbf(nfcmbf_t *T, long klim, long *pmaxK, int *done)
1033 : {
1034 12159 : GEN nf = T->nf, famod = T->fact, bound = T->bound;
1035 12159 : GEN ltdn, nfpol = nf_get_pol(nf);
1036 12159 : long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1, dnf = degpol(nfpol);
1037 12159 : pari_sp av0 = avma;
1038 12159 : GEN Tpk = T->L->Tpk, pk = T->L->pk, pks2 = shifti(pk,-1);
1039 12159 : GEN ind = cgetg(lfamod+1, t_VECSMALL);
1040 12159 : GEN deg = cgetg(lfamod+1, t_VECSMALL);
1041 12159 : GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
1042 12159 : GEN fa = cgetg(lfamod+1, t_VEC);
1043 12159 : const double Bhigh = get_Bhigh(lfamod, dnf);
1044 : trace_data _T1, _T2, *T1, *T2;
1045 : div_data D;
1046 : pari_timer ti;
1047 :
1048 12159 : timer_start(&ti);
1049 :
1050 12159 : *pmaxK = cmbf_maxK(lfamod);
1051 12159 : init_div_data(&D, T->pol, T->L);
1052 12159 : ltdn = mul_content(D.lt, T->L->dn);
1053 : {
1054 12159 : GEN q = ceil_safe(sqrtr(T->BS_2));
1055 12159 : GEN t1,t2, lt2dn = mul_content(ltdn, D.lt);
1056 12159 : GEN trace1 = cgetg(lfamod+1, t_MAT);
1057 12159 : GEN trace2 = cgetg(lfamod+1, t_MAT);
1058 61519 : for (i=1; i <= lfamod; i++)
1059 : {
1060 49360 : pari_sp av = avma;
1061 49360 : GEN P = gel(famod,i);
1062 49360 : long d = degpol(P);
1063 :
1064 49360 : deg[i] = d; P += 2;
1065 49360 : t1 = gel(P,d-1);/* = - S_1 */
1066 49360 : t2 = Fq_sqr(t1, Tpk, pk);
1067 49360 : if (d > 1) t2 = Fq_sub(t2, gmul2n(gel(P,d-2), 1), Tpk, pk);
1068 : /* t2 = S_2 Newton sum */
1069 49360 : if (ltdn)
1070 : {
1071 476 : t1 = Fq_Fp_mul(t1, ltdn, Tpk, pk);
1072 476 : t2 = Fq_Fp_mul(t2, lt2dn, Tpk, pk);
1073 : }
1074 49360 : gel(trace1,i) = gclone( nf_bestlift(t1, NULL, T->L) );
1075 49360 : gel(trace2,i) = gclone( nf_bestlift(t2, NULL, T->L) ); set_avma(av);
1076 : }
1077 12159 : T1 = init_trace(&_T1, trace1, T->L, q);
1078 12159 : T2 = init_trace(&_T2, trace2, T->L, q);
1079 61519 : for (i=1; i <= lfamod; i++) {
1080 49360 : gunclone(gel(trace1,i));
1081 49360 : gunclone(gel(trace2,i));
1082 : }
1083 : }
1084 12159 : degsofar[0] = 0; /* sentinel */
1085 :
1086 : /* ind runs through strictly increasing sequences of length K,
1087 : * 1 <= ind[i] <= lfamod */
1088 16376 : nextK:
1089 16376 : if (K > *pmaxK || 2*K > lfamod) goto END;
1090 14098 : if (DEBUGLEVEL > 3)
1091 0 : err_printf("\n### K = %d, %Ps combinations\n", K,binomial(utoipos(lfamod), K));
1092 14098 : setlg(ind, K+1); ind[1] = 1;
1093 14098 : i = 1; curdeg = deg[ind[1]];
1094 : for(;;)
1095 : { /* try all combinations of K factors */
1096 253630 : for (j = i; j < K; j++)
1097 : {
1098 29708 : degsofar[j] = curdeg;
1099 29708 : ind[j+1] = ind[j]+1; curdeg += deg[ind[j+1]];
1100 : }
1101 223922 : if (curdeg <= klim) /* trial divide */
1102 14556 : {
1103 : GEN t, y, q;
1104 : pari_sp av;
1105 :
1106 223922 : av = avma;
1107 223922 : if (T1)
1108 : { /* d-1 test */
1109 100578 : t = get_trace(ind, T1);
1110 100578 : if (rtodbl(_norml2(t)) > Bhigh)
1111 : {
1112 92248 : if (DEBUGLEVEL>6) err_printf(".");
1113 92248 : set_avma(av); goto NEXT;
1114 : }
1115 : }
1116 131674 : if (T2)
1117 : { /* d-2 test */
1118 105410 : t = get_trace(ind, T2);
1119 105410 : if (rtodbl(_norml2(t)) > Bhigh)
1120 : {
1121 95176 : if (DEBUGLEVEL>3) err_printf("|");
1122 95176 : set_avma(av); goto NEXT;
1123 : }
1124 : }
1125 36498 : set_avma(av);
1126 36498 : y = ltdn; /* full computation */
1127 89544 : for (i=1; i<=K; i++)
1128 : {
1129 53046 : GEN q = gel(famod, ind[i]);
1130 53046 : if (y) q = gmul(y, q);
1131 53046 : y = FqX_centermod(q, Tpk, pk, pks2);
1132 : }
1133 36498 : y = nf_pol_lift(y, bound, T->L);
1134 36498 : if (!y)
1135 : {
1136 9483 : if (DEBUGLEVEL>3) err_printf("@");
1137 9483 : set_avma(av); goto NEXT;
1138 : }
1139 : /* y = topowden*dn*lt*\prod_{i in ind} famod[i] is apparently in O_K[X],
1140 : * in fact in (Z[Y]/nf.pol)[X] due to multiplication by C = topowden*dn.
1141 : * Try out this candidate factor */
1142 27015 : q = RgXQX_divrem(D.C2ltpol, y, nfpol, ONLY_DIVIDES);
1143 27015 : if (!q)
1144 : {
1145 2578 : if (DEBUGLEVEL>3) err_printf("*");
1146 2578 : set_avma(av); goto NEXT;
1147 : }
1148 : /* Original T->pol in O_K[X] with leading coeff lt in Z,
1149 : * y = C*lt \prod famod[i] is in O_K[X] with leading coeff in Z
1150 : * q = C^2*lt*pol / y = C * (lt*pol) / (lt*\prod famod[i]) is a
1151 : * K-rational factor, in fact in Z[Y]/nf.pol)[X] as above, with
1152 : * leading term C*lt. */
1153 24437 : update_target(&D, q);
1154 24437 : gel(fa,cnt++) = D.C2lt? RgX_int_normalize(y): y; /* make monic */
1155 160541 : for (i=j=k=1; i <= lfamod; i++)
1156 : { /* remove used factors */
1157 136104 : if (j <= K && i == ind[j]) j++;
1158 : else
1159 : {
1160 104576 : gel(famod,k) = gel(famod,i);
1161 104576 : update_trace(T1, k, i);
1162 104576 : update_trace(T2, k, i);
1163 104576 : deg[k] = deg[i]; k++;
1164 : }
1165 : }
1166 24437 : lfamod -= K;
1167 24437 : *pmaxK = cmbf_maxK(lfamod);
1168 24437 : if (lfamod < 2*K) goto END;
1169 14556 : i = 1; curdeg = deg[ind[1]];
1170 14556 : if (DEBUGLEVEL > 2)
1171 : {
1172 0 : err_printf("\n"); timer_printf(&ti, "to find factor %Ps",gel(fa,cnt-1));
1173 0 : err_printf("remaining modular factor(s): %ld\n", lfamod);
1174 : }
1175 14556 : continue;
1176 : }
1177 :
1178 0 : NEXT:
1179 199485 : for (i = K+1;;)
1180 : {
1181 226319 : if (--i == 0) { K++; goto nextK; }
1182 222102 : if (++ind[i] <= lfamod - K + i)
1183 : {
1184 195268 : curdeg = degsofar[i-1] + deg[ind[i]];
1185 195268 : if (curdeg <= klim) break;
1186 : }
1187 : }
1188 : }
1189 12159 : END:
1190 12159 : *done = 1;
1191 12159 : if (degpol(D.C2ltpol) > 0)
1192 : { /* leftover factor */
1193 12159 : GEN q = D.C2ltpol;
1194 12159 : if (D.C2lt) q = RgX_int_normalize(q);
1195 12159 : if (lfamod >= 2*K)
1196 : { /* restore leading coefficient [#930] */
1197 91 : if (D.lt) q = RgX_Rg_mul(q, D.lt);
1198 91 : *done = 0; /* ... may still be reducible */
1199 : }
1200 12159 : setlg(famod, lfamod+1);
1201 12159 : gel(fa,cnt++) = q;
1202 : }
1203 12159 : if (DEBUGLEVEL>6) err_printf("\n");
1204 12159 : setlg(fa, cnt);
1205 12159 : return gerepilecopy(av0, fa);
1206 : }
1207 :
1208 : static GEN
1209 98 : nf_chk_factors(nfcmbf_t *T, GEN P, GEN M_L, GEN famod, GEN pk)
1210 : {
1211 98 : GEN nf = T->nf, bound = T->bound;
1212 98 : GEN nfT = nf_get_pol(nf);
1213 : long i, r;
1214 98 : GEN pol = P, list, piv, y;
1215 98 : GEN Tpk = T->L->Tpk;
1216 : div_data D;
1217 :
1218 98 : piv = ZM_hnf_knapsack(M_L);
1219 98 : if (!piv) return NULL;
1220 77 : if (DEBUGLEVEL>3) err_printf("ZM_hnf_knapsack output:\n%Ps\n",piv);
1221 :
1222 77 : r = lg(piv)-1;
1223 77 : list = cgetg(r+1, t_VEC);
1224 77 : init_div_data(&D, pol, T->L);
1225 77 : for (i = 1;;)
1226 112 : {
1227 189 : pari_sp av = avma;
1228 189 : if (DEBUGLEVEL) err_printf("nf_LLL_cmbf: checking factor %ld\n", i);
1229 189 : y = chk_factors_get(D.lt, famod, gel(piv,i), Tpk, pk);
1230 :
1231 189 : if (! (y = nf_pol_lift(y, bound, T->L)) ) return NULL;
1232 182 : y = gerepilecopy(av, y);
1233 : /* y is the candidate factor */
1234 182 : pol = RgXQX_divrem(D.C2ltpol, y, nfT, ONLY_DIVIDES);
1235 182 : if (!pol) return NULL;
1236 :
1237 182 : if (D.C2lt) y = RgX_int_normalize(y);
1238 182 : gel(list,i) = y;
1239 182 : if (++i >= r) break;
1240 :
1241 112 : update_target(&D, pol);
1242 : }
1243 70 : gel(list,i) = RgX_int_normalize(pol); return list;
1244 : }
1245 :
1246 : static GEN
1247 161538 : nf_to_Zq(GEN x, GEN T, GEN pk, GEN pks2, GEN proj)
1248 : {
1249 : GEN y;
1250 161538 : if (typ(x) != t_COL) return centermodii(x, pk, pks2);
1251 23042 : if (!T)
1252 : {
1253 22769 : y = ZV_dotproduct(proj, x);
1254 22769 : return centermodii(y, pk, pks2);
1255 : }
1256 273 : y = ZM_ZC_mul(proj, x);
1257 273 : y = RgV_to_RgX(y, varn(T));
1258 273 : return FpX_center_i(FpX_rem(y, T, pk), pk, pks2);
1259 : }
1260 :
1261 : /* Assume P in nfX form, lc(P) != 0 mod p. Reduce P to Zp[X]/(T) mod p^a */
1262 : static GEN
1263 18450 : ZqX(GEN P, GEN pk, GEN T, GEN proj)
1264 : {
1265 18450 : long i, l = lg(P);
1266 18450 : GEN z, pks2 = shifti(pk,-1);
1267 :
1268 18450 : z = cgetg(l,t_POL); z[1] = P[1];
1269 179952 : for (i=2; i<l; i++) gel(z,i) = nf_to_Zq(gel(P,i),T,pk,pks2,proj);
1270 18449 : return normalizepol_lg(z, l);
1271 : }
1272 :
1273 : static GEN
1274 18415 : ZqX_normalize(GEN P, GEN lt, nflift_t *L)
1275 : {
1276 18415 : GEN R = lt? RgX_Rg_mul(P, Fp_inv(lt, L->pk)): P;
1277 18415 : return ZqX(R, L->pk, L->Tpk, L->ZqProj);
1278 : }
1279 :
1280 : /* k allowing to reconstruct x, T_2(x)^2 < C, from x mod pr^k */
1281 : /* return log [ 2sqrt(C/d) * ( (3/2)sqrt(gamma) )^(d-1) ] ^d / log N(pr)
1282 : * cf. Belabas relative van Hoeij algorithm, lemma 3.12 */
1283 : static double
1284 18450 : bestlift_bound(GEN C, long d, double alpha, GEN p, long f)
1285 : {
1286 18450 : const double g = 1 / (alpha - 0.25); /* = 2 if alpha = 3/4 */
1287 18450 : GEN C4 = shiftr(gtofp(C,DEFAULTPREC), 2);
1288 18450 : double t, logp = log(gtodouble(p));
1289 18450 : if (f == d)
1290 : { /* p inert, no LLL fudge factor: p^(2k) / 4 > C */
1291 35 : t = 0.5 * rtodbl(mplog(C4));
1292 35 : return ceil(t / logp);
1293 : }
1294 : /* (1/2)log (4C/d) + (d-1)(log 3/2 sqrt(gamma)) */
1295 18415 : t = 0.5 * rtodbl(mplog(divru(C4,d))) + (d-1) * log(1.5 * sqrt(g));
1296 18415 : return ceil((t * d) / (logp * f));
1297 : }
1298 :
1299 : static GEN
1300 18682 : get_R(GEN M)
1301 : {
1302 : GEN R;
1303 18682 : long i, l, prec = nbits2prec( gexpo(M) + 64 );
1304 :
1305 : for(;;)
1306 : {
1307 18682 : R = gaussred_from_QR(M, prec);
1308 18682 : if (R) break;
1309 0 : prec = precdbl(prec);
1310 : }
1311 18682 : l = lg(R);
1312 88939 : for (i=1; i<l; i++) gcoeff(R,i,i) = gen_1;
1313 18682 : return R;
1314 : }
1315 :
1316 : static void
1317 18450 : init_proj(nflift_t *L, GEN prkHNF, GEN nfT)
1318 : {
1319 18450 : if (degpol(L->Tp)>1)
1320 : {
1321 203 : GEN coTp = FpX_div(FpX_red(nfT, L->p), L->Tp, L->p); /* Tp's cofactor */
1322 : GEN z, proj;
1323 203 : z = ZpX_liftfact(nfT, mkvec2(L->Tp, coTp), L->pk, L->p, L->k);
1324 203 : L->Tpk = gel(z,1);
1325 203 : proj = QXQV_to_FpM(L->topow, L->Tpk, L->pk);
1326 203 : if (L->topowden)
1327 203 : proj = FpM_red(ZM_Z_mul(proj, Fp_inv(L->topowden, L->pk)), L->pk);
1328 203 : L->ZqProj = proj;
1329 : }
1330 : else
1331 : {
1332 18247 : L->Tpk = NULL;
1333 18247 : L->ZqProj = dim1proj(prkHNF);
1334 : }
1335 18450 : }
1336 :
1337 : /* Square of the radius of largest ball inscript in PRK's fundamental domain,
1338 : * whose orthogonalized vector's norms are the Bi
1339 : * Rmax ^2 = min 1/4T_i where T_i = sum_j ( s_ij^2 / B_j)
1340 : * For p inert, S = Id, T_i = 1 / p^{2k} and Rmax = p^k / 2 */
1341 : static GEN
1342 18682 : max_radius(GEN PRK, GEN B)
1343 : {
1344 18682 : GEN S, smax = gen_0;
1345 18682 : pari_sp av = avma;
1346 18682 : long i, j, d = lg(PRK)-1;
1347 :
1348 18682 : S = RgM_inv( get_R(PRK) ); if (!S) pari_err_PREC("max_radius");
1349 88938 : for (i=1; i<=d; i++)
1350 : {
1351 70256 : GEN s = gen_0;
1352 513057 : for (j=1; j<=d; j++)
1353 442801 : s = mpadd(s, mpdiv( mpsqr(gcoeff(S,i,j)), gel(B,j)));
1354 70256 : if (mpcmp(s, smax) > 0) smax = s;
1355 : }
1356 18682 : return gerepileupto(av, ginv(gmul2n(smax, 2)));
1357 : }
1358 :
1359 : static void
1360 18450 : bestlift_init(long a, GEN nf, GEN C, nflift_t *L)
1361 : {
1362 18450 : const double alpha = 0.99; /* LLL parameter */
1363 18450 : const long d = nf_get_degree(nf);
1364 18450 : pari_sp av = avma, av2;
1365 18450 : GEN prk, PRK, iPRK, GSmin, T = L->Tp, p = L->p;
1366 18450 : long f = degpol(T);
1367 : pari_timer ti;
1368 :
1369 18450 : if (f == d)
1370 : { /* inert p, much simpler */
1371 35 : long a0 = bestlift_bound(C, d, alpha, p, f);
1372 : GEN q;
1373 35 : if (a < a0) a = a0; /* guarantees GSmin >= C */
1374 35 : if (DEBUGLEVEL>2) err_printf("exponent %ld\n",a);
1375 35 : q = powiu(p,a);
1376 35 : PRK = prk = scalarmat_shallow(q, d);
1377 35 : GSmin = shiftr(itor(q, DEFAULTPREC), -1);
1378 35 : iPRK = matid(d); goto END;
1379 : }
1380 18415 : timer_start(&ti);
1381 18415 : if (!a) a = (long)bestlift_bound(C, d, alpha, p, f);
1382 267 : for (;; set_avma(av), a += (a==1)? 1: (a>>1)) /* roughly a *= 1.5 */
1383 267 : {
1384 18682 : GEN B, q = powiu(p,a), Tq = FpXQ_powu(T, a, FpX_red(nf_get_pol(nf), q), q);
1385 18682 : if (DEBUGLEVEL>2) err_printf("exponent %ld\n",a);
1386 18682 : prk = idealhnf_two(nf, mkvec2(q, Tq));
1387 18682 : av2 = avma;
1388 18682 : PRK = ZM_lll_norms(prk, alpha, LLL_INPLACE, &B);
1389 18682 : GSmin = max_radius(PRK, B);
1390 18682 : if (gcmp(GSmin, C) >= 0) break;
1391 : }
1392 18415 : gerepileall(av2, 2, &PRK, &GSmin);
1393 18415 : iPRK = ZM_inv(PRK, NULL);
1394 18415 : if (DEBUGLEVEL>2)
1395 0 : err_printf("for this exponent, GSmin = %Ps\nTime reduction: %ld\n",
1396 : GSmin, timer_delay(&ti));
1397 18415 : END:
1398 18450 : L->k = a;
1399 18450 : L->pk = gcoeff(prk,1,1);
1400 18450 : L->prk = PRK;
1401 18450 : L->iprk = iPRK;
1402 18450 : L->GSmin= GSmin;
1403 18450 : init_proj(L, prk, nf_get_pol(nf));
1404 18450 : }
1405 :
1406 : /* Let X = Tra * M_L, Y = bestlift(X) return V s.t Y = X - PRK V
1407 : * and set *eT2 = gexpo(Y) [cf nf_bestlift, but memory efficient] */
1408 : static GEN
1409 476 : get_V(GEN Tra, GEN M_L, GEN PRK, GEN PRKinv, GEN pk, long *eT2)
1410 : {
1411 476 : long i, e = 0, l = lg(M_L);
1412 476 : GEN V = cgetg(l, t_MAT);
1413 476 : *eT2 = 0;
1414 6013 : for (i = 1; i < l; i++)
1415 : { /* cf nf_bestlift(Tra * c) */
1416 5537 : pari_sp av = avma, av2;
1417 5537 : GEN v, T2 = ZM_ZC_mul(Tra, gel(M_L,i));
1418 :
1419 5537 : v = gdivround(ZM_ZC_mul(PRKinv, T2), pk); /* small */
1420 5537 : av2 = avma;
1421 5537 : T2 = ZC_sub(T2, ZM_ZC_mul(PRK, v));
1422 5537 : e = gexpo(T2); if (e > *eT2) *eT2 = e;
1423 5537 : set_avma(av2);
1424 5537 : gel(V,i) = gerepileupto(av, v); /* small */
1425 : }
1426 476 : return V;
1427 : }
1428 :
1429 : static GEN
1430 91 : nf_LLL_cmbf(nfcmbf_t *T, long rec)
1431 : {
1432 91 : const double BitPerFactor = 0.4; /* nb bits / modular factor */
1433 91 : nflift_t *L = T->L;
1434 91 : GEN famod = T->fact, ZC = T->ZC, Br = T->Br, P = T->pol, dn = T->L->dn;
1435 91 : long dnf = nf_get_degree(T->nf), dP = degpol(P);
1436 : long i, C, tmax, n0;
1437 : GEN lP, Bnorm, Tra, T2, TT, CM_L, m, list, ZERO, Btra;
1438 : double Bhigh;
1439 : pari_sp av, av2;
1440 91 : long ti_LLL = 0, ti_CF = 0;
1441 : pari_timer ti2, TI;
1442 :
1443 91 : lP = absi_shallow(leading_coeff(P));
1444 91 : if (is_pm1(lP)) lP = NULL;
1445 :
1446 91 : n0 = lg(famod) - 1;
1447 : /* Lattice: (S PRK), small vector (vS vP). To find k bound for the image,
1448 : * write S = S1 q + S0, P = P1 q + P0
1449 : * |S1 vS + P1 vP|^2 <= Bhigh for all (vS,vP) assoc. to true factors */
1450 91 : Btra = mulrr(ZC, mulur(dP*dP, normTp(Br, 2, dnf)));
1451 91 : Bhigh = get_Bhigh(n0, dnf);
1452 91 : C = (long)ceil(sqrt(Bhigh/n0)) + 1; /* C^2 n0 ~ Bhigh */
1453 91 : Bnorm = dbltor( n0 * C * C + Bhigh );
1454 91 : ZERO = zeromat(n0, dnf);
1455 :
1456 91 : av = avma;
1457 91 : TT = const_vec(n0, NULL);
1458 91 : Tra = cgetg(n0+1, t_MAT);
1459 91 : CM_L = scalarmat_s(C, n0);
1460 : /* tmax = current number of traces used (and computed so far) */
1461 91 : for(tmax = 0;; tmax++)
1462 231 : {
1463 322 : long a, b, bmin, bgood, delta, tnew = tmax + 1, r = lg(CM_L)-1;
1464 : GEN M_L, q, CM_Lp, oldCM_L, S1, P1, VV;
1465 322 : int first = 1;
1466 :
1467 : /* bound for f . S_k(genuine factor) = ZC * bound for T_2(S_tnew) */
1468 322 : Btra = mulrr(ZC, mulur(dP*dP, normTp(Br, 2*tnew, dnf)));
1469 322 : bmin = logint(ceil_safe(sqrtr(Btra)), gen_2) + 1;
1470 322 : if (DEBUGLEVEL>2)
1471 0 : err_printf("\nLLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)\n",
1472 : r, tmax, bmin);
1473 :
1474 : /* compute Newton sums (possibly relifting first) */
1475 322 : if (gcmp(L->GSmin, Btra) < 0)
1476 : {
1477 : GEN polred;
1478 :
1479 7 : bestlift_init((L->k)<<1, T->nf, Btra, L);
1480 7 : polred = ZqX_normalize(T->polbase, lP, L);
1481 7 : famod = ZqX_liftfact(polred, famod, L->Tpk, L->pk, L->p, L->k);
1482 133 : for (i=1; i<=n0; i++) gel(TT,i) = NULL;
1483 : }
1484 6776 : for (i=1; i<=n0; i++)
1485 : {
1486 6454 : GEN h, lPpow = lP? powiu(lP, tnew): NULL;
1487 6454 : GEN z = polsym_gen(gel(famod,i), gel(TT,i), tnew, L->Tpk, L->pk);
1488 6454 : gel(TT,i) = z;
1489 6454 : h = gel(z,tnew+1);
1490 : /* make Newton sums integral */
1491 6454 : lPpow = mul_content(lPpow, dn);
1492 6454 : if (lPpow)
1493 126 : h = (typ(h) == t_INT)? Fp_mul(h, lPpow, L->pk): FpX_Fp_mul(h, lPpow, L->pk);
1494 6454 : gel(Tra,i) = nf_bestlift(h, NULL, L); /* S_tnew(famod) */
1495 : }
1496 :
1497 : /* compute truncation parameter */
1498 322 : if (DEBUGLEVEL>2) { timer_start(&ti2); timer_start(&TI); }
1499 322 : oldCM_L = CM_L;
1500 322 : av2 = avma;
1501 322 : b = delta = 0; /* -Wall */
1502 476 : AGAIN:
1503 476 : M_L = Q_div_to_int(CM_L, utoipos(C));
1504 476 : VV = get_V(Tra, M_L, L->prk, L->iprk, L->pk, &a);
1505 476 : if (first)
1506 : { /* initialize lattice, using few p-adic digits for traces */
1507 322 : bgood = (long)(a - maxss(32, (long)(BitPerFactor * r)));
1508 322 : b = maxss(bmin, bgood);
1509 322 : delta = a - b;
1510 : }
1511 : else
1512 : { /* add more p-adic digits and continue reduction */
1513 154 : if (a < b) b = a;
1514 154 : b = maxss(b-delta, bmin);
1515 154 : if (b - delta/2 < bmin) b = bmin; /* near there. Go all the way */
1516 : }
1517 :
1518 : /* restart with truncated entries */
1519 476 : q = int2n(b);
1520 476 : P1 = gdivround(L->prk, q);
1521 476 : S1 = gdivround(Tra, q);
1522 476 : T2 = ZM_sub(ZM_mul(S1, M_L), ZM_mul(P1, VV));
1523 476 : m = vconcat( CM_L, T2 );
1524 476 : if (first)
1525 : {
1526 322 : first = 0;
1527 322 : m = shallowconcat( m, vconcat(ZERO, P1) );
1528 : /* [ C M_L 0 ]
1529 : * m = [ ] square matrix
1530 : * [ T2' PRK ] T2' = Tra * M_L truncated
1531 : */
1532 : }
1533 476 : CM_L = LLL_check_progress(Bnorm, n0, m, b == bmin, /*dbg:*/ &ti_LLL);
1534 476 : if (DEBUGLEVEL>2)
1535 0 : err_printf("LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld\n",
1536 0 : a,b, lg(m)-1, CM_L? lg(CM_L)-1: 1, timer_delay(&TI));
1537 476 : if (!CM_L) { list = mkcol(RgX_int_normalize(P)); break; }
1538 455 : if (b > bmin)
1539 : {
1540 154 : CM_L = gerepilecopy(av2, CM_L);
1541 154 : goto AGAIN;
1542 : }
1543 301 : if (DEBUGLEVEL>2) timer_printf(&ti2, "for this trace");
1544 :
1545 301 : i = lg(CM_L) - 1;
1546 301 : if (i == r && ZM_equal(CM_L, oldCM_L))
1547 : {
1548 91 : CM_L = oldCM_L;
1549 91 : set_avma(av2); continue;
1550 : }
1551 :
1552 210 : CM_Lp = FpM_image(CM_L, utoipos(27449)); /* inexpensive test */
1553 210 : if (lg(CM_Lp) != lg(CM_L))
1554 : {
1555 0 : if (DEBUGLEVEL>2) err_printf("LLL_cmbf: rank decrease\n");
1556 0 : CM_L = ZM_hnf(CM_L);
1557 : }
1558 :
1559 210 : if (i <= r && i*rec < n0)
1560 : {
1561 : pari_timer ti;
1562 98 : if (DEBUGLEVEL>2) timer_start(&ti);
1563 98 : list = nf_chk_factors(T, P, Q_div_to_int(CM_L,utoipos(C)), famod, L->pk);
1564 98 : if (DEBUGLEVEL>2) ti_CF += timer_delay(&ti);
1565 98 : if (list) break;
1566 : }
1567 140 : if (gc_needed(av,1))
1568 : {
1569 0 : if(DEBUGMEM>1) pari_warn(warnmem,"nf_LLL_cmbf");
1570 0 : gerepileall(av, L->Tpk? 9: 8,
1571 : &CM_L,&TT,&Tra,&famod,&L->GSmin,&L->pk,&L->prk,&L->iprk,
1572 : &L->Tpk);
1573 : }
1574 140 : else CM_L = gerepilecopy(av2, CM_L);
1575 : }
1576 91 : if (DEBUGLEVEL>2)
1577 0 : err_printf("* Time LLL: %ld\n* Time Check Factor: %ld\n",ti_LLL,ti_CF);
1578 91 : return list;
1579 : }
1580 :
1581 : static GEN
1582 12159 : nf_combine_factors(nfcmbf_t *T, GEN polred, long klim)
1583 : {
1584 12159 : nflift_t *L = T->L;
1585 : GEN res;
1586 : long maxK;
1587 : int done;
1588 : pari_timer ti;
1589 :
1590 12159 : if (DEBUGLEVEL>2) timer_start(&ti);
1591 12159 : T->fact = ZqX_liftfact(polred, T->fact, L->Tpk, L->pk, L->p, L->k);
1592 12159 : if (DEBUGLEVEL>2) timer_printf(&ti, "Hensel lift");
1593 12159 : res = nfcmbf(T, klim, &maxK, &done);
1594 12159 : if (DEBUGLEVEL>2) timer_printf(&ti, "Naive recombination");
1595 12159 : if (!done)
1596 : {
1597 91 : long l = lg(res)-1;
1598 : GEN v;
1599 91 : if (l > 1)
1600 : {
1601 : GEN den;
1602 49 : T->pol = gel(res,l);
1603 49 : T->polbase = Q_remove_denom(RgX_to_nfX(T->nf, T->pol), &den);
1604 49 : if (den) { T->Br = gmul(T->Br, den); T->pol = RgX_Rg_mul(T->pol, den); }
1605 : }
1606 91 : v = nf_LLL_cmbf(T, maxK);
1607 : /* remove last elt, possibly unfactored. Add all new ones. */
1608 91 : setlg(res, l); res = shallowconcat(res, v);
1609 : }
1610 12159 : return res;
1611 : }
1612 :
1613 : static GEN
1614 6249 : nf_DDF_roots(GEN pol, GEN polred, GEN nfpol, long fl, nflift_t *L)
1615 : {
1616 : GEN z, Cltx_r, ltdn;
1617 : long i, m, lz;
1618 : div_data D;
1619 :
1620 6249 : init_div_data(&D, pol, L);
1621 6249 : ltdn = mul_content(D.lt, L->dn);
1622 6249 : z = ZqX_roots(polred, L->Tpk, L->p, L->k);
1623 6249 : Cltx_r = deg1pol_shallow(D.Clt? D.Clt: gen_1, NULL, varn(pol));
1624 6249 : lz = lg(z);
1625 6249 : if (DEBUGLEVEL > 3) err_printf("Checking %ld roots:",lz-1);
1626 18243 : for (m=1,i=1; i<lz; i++)
1627 : {
1628 11994 : GEN r = gel(z,i);
1629 : int dvd;
1630 : pari_sp av;
1631 11994 : if (DEBUGLEVEL > 3) err_printf(" %ld",i);
1632 : /* lt*dn*topowden * r = Clt * r */
1633 11994 : r = nf_bestlift_to_pol(ltdn? gmul(ltdn,r): r, NULL, L);
1634 11994 : av = avma;
1635 11994 : gel(Cltx_r,2) = gneg(r); /* check P(r) == 0 */
1636 11994 : dvd = ZXQX_dvd(D.C2ltpol, Cltx_r, nfpol); /* integral */
1637 11994 : set_avma(av);
1638 : /* don't go on with q, usually much larger that C2ltpol */
1639 11994 : if (dvd) {
1640 10899 : if (D.Clt) r = gdiv(r, D.Clt);
1641 10899 : gel(z,m++) = r;
1642 : }
1643 1095 : else if (fl == ROOTS_SPLIT) return cgetg(1, t_VEC);
1644 : }
1645 6249 : if (DEBUGLEVEL > 3) err_printf(" done\n");
1646 6249 : z[0] = evaltyp(t_VEC) | _evallg(m);
1647 6249 : return z;
1648 : }
1649 :
1650 : /* returns a few factors of T in Fp of degree <= maxf, NULL if none exist */
1651 : static GEN
1652 428415 : get_good_factor(GEN T, ulong p, long maxf)
1653 : {
1654 428415 : pari_sp av = avma;
1655 428415 : GEN r, R = gel(Flx_factor(ZX_to_Flx(T,p),p), 1);
1656 428413 : if (maxf == 1)
1657 : { /* degree 1 factors are best */
1658 419173 : r = gel(R,1);
1659 419173 : if (degpol(r) == 1) return mkvec(r);
1660 : }
1661 : else
1662 : { /* otherwise, pick factor of largish degree */
1663 9240 : long i, j, dr, d = 0, l = lg(R);
1664 : GEN v;
1665 9240 : if (l == 2) return mkvec(gel(R,1)); /* inert is fine */
1666 8358 : v = cgetg(l, t_VEC);
1667 61004 : for (i = j = 1; i < l; i++)
1668 : {
1669 54480 : r = gel(R,i); dr = degpol(r);
1670 54480 : if (dr > maxf) break;
1671 52646 : if (dr != d) { gel(v,j++) = r; d = dr; }
1672 : }
1673 8358 : setlg(v,j); if (j > 1) return v;
1674 : }
1675 279692 : return gc_NULL(av); /* failure */
1676 : }
1677 :
1678 : /* n = number of modular factors, f = residue degree; nold/fold current best
1679 : * return 1 if new values are "better" than old ones */
1680 : static int
1681 123047 : record(long nold, long n, long fold, long f)
1682 : {
1683 123047 : if (!nold) return 1; /* uninitialized */
1684 100418 : if (fold == f) return n < nold;
1685 : /* if f increases, allow increasing n a little */
1686 7572 : if (fold < f) return n <= 20 || n < 1.1*nold;
1687 : /* f decreases, only allow if decreasing n a lot */
1688 3488 : return n < 0.7*nold;
1689 : }
1690 : /* Optimization problem: factorization of polynomials over large Fq is slow,
1691 : * BUT bestlift correspondingly faster.
1692 : * Return maximal residue degree to be considered when picking a prime ideal */
1693 : static long
1694 27439 : get_maxf(long nfdeg)
1695 : {
1696 27439 : long maxf = 1;
1697 27439 : if (nfdeg >= 45) maxf =32;
1698 27439 : else if (nfdeg >= 30) maxf =16;
1699 27390 : else if (nfdeg >= 15) maxf = 8;
1700 27439 : return maxf;
1701 : }
1702 : /* number of maximal ideals to test before settling on best prime and number
1703 : * of factors; B = [K:Q]*deg(P) */
1704 : static long
1705 27439 : get_nbprimes(long B)
1706 : {
1707 27439 : if (B <= 128) return 5;
1708 1155 : if (B <= 1024) return 20;
1709 56 : if (B <= 2048) return 65;
1710 0 : return 100;
1711 : }
1712 : /* Select a prime ideal pr over which to factor pol.
1713 : * Return the number of factors (or roots, according to flag fl) mod pr.
1714 : * Set:
1715 : * lt: leading term of polbase (t_INT or NULL [ for 1 ])
1716 : * pr: a suitable maximal ideal
1717 : * Fa: factors found mod pr
1718 : * Tp: polynomial defining Fq/Fp */
1719 : static long
1720 27439 : nf_pick_prime(GEN nf, GEN pol, long fl, GEN *lt, GEN *Tp, ulong *pp)
1721 : {
1722 27439 : GEN nfpol = nf_get_pol(nf), bad = mulii(nf_get_disc(nf), nf_get_index(nf));
1723 27439 : long nfdeg = degpol(nfpol), dpol = degpol(pol), nold = 0, fold = 1;
1724 27439 : long maxf = get_maxf(nfdeg), ct = get_nbprimes(nfdeg * dpol);
1725 : ulong p;
1726 : forprime_t S;
1727 : pari_timer ti_pr;
1728 :
1729 27439 : if (DEBUGLEVEL>3) timer_start(&ti_pr);
1730 : /* if field degree is large, try hard to find an inert prime */
1731 27439 : if (nfdeg > 30) ct = maxss(ct, 2*nfdeg);
1732 27439 : *lt = leading_coeff(pol); /* t_INT */
1733 27439 : if (gequal1(*lt)) *lt = NULL;
1734 27439 : *pp = 0;
1735 27439 : *Tp = NULL;
1736 27439 : (void)u_forprime_init(&S, 2, ULONG_MAX);
1737 : /* select pr such that pol has the smallest number of factors, ct attempts */
1738 482292 : while ((p = u_forprime_next(&S)))
1739 : {
1740 : GEN vT;
1741 482292 : long n, i, l, ok = 0;
1742 482292 : ulong ltp = 0;
1743 :
1744 482292 : if (! umodiu(bad,p)) continue;
1745 432259 : if (*lt) { ltp = umodiu(*lt, p); if (!ltp) continue; }
1746 428416 : vT = get_good_factor(nfpol, p, maxf);
1747 428413 : if (!vT) continue;
1748 148720 : l = lg(vT);
1749 291278 : for (i = 1; i < l; i++)
1750 : {
1751 151600 : pari_sp av2 = avma;
1752 151600 : GEN T = gel(vT,i), red = RgX_to_FlxqX(pol, T, p);
1753 151600 : long f = degpol(T);
1754 151600 : if (f == 1)
1755 : { /* degree 1 */
1756 141621 : red = FlxX_to_Flx(red);
1757 141624 : if (ltp) red = Flx_normalize(red, p);
1758 141624 : if (!Flx_is_squarefree(red, p)) { set_avma(av2); continue; }
1759 122463 : ok = 1;
1760 122463 : n = (fl == FACTORS)? Flx_nbfact(red,p): Flx_nbroots(red,p);
1761 : }
1762 : else
1763 : {
1764 9979 : if (ltp) red = FlxqX_normalize(red, T, p);
1765 9979 : if (!FlxqX_is_squarefree(red, T, p)) { set_avma(av2); continue; }
1766 9629 : ok = 1;
1767 9629 : n = (fl == FACTORS)? FlxqX_nbfact(red,T,p): FlxqX_nbroots(red,T,p);
1768 : }
1769 132092 : if (fl == ROOTS_SPLIT && n < dpol) return n; /* not split */
1770 132071 : if (n <= 1)
1771 : {
1772 20245 : if (fl == FACTORS) return n; /* irreducible */
1773 19867 : if (!n) return 0; /* no root */
1774 : }
1775 123061 : if (DEBUGLEVEL>3)
1776 0 : err_printf("%3ld %s at prime (%ld,x^%ld+...)\n Time: %ld\n",
1777 : n, (fl == FACTORS)? "factors": "roots", p,f, timer_delay(&ti_pr));
1778 :
1779 123061 : if (fl == ROOTS && f==nfdeg) { *Tp = T; *pp = p; return n; }
1780 123047 : if (record(nold, n, fold, f)) { nold = n; fold = f; *Tp = T; *pp = p; }
1781 93118 : else set_avma(av2);
1782 : }
1783 139678 : if (ok && --ct <= 0) break;
1784 : }
1785 18394 : if (!nold) pari_err_OVERFLOW("nf_pick_prime [ran out of primes]");
1786 18394 : return nold;
1787 : }
1788 :
1789 : /* Assume lt(T) is a t_INT and T square free. Return t_VEC of irred. factors */
1790 : static GEN
1791 168 : nfsqff_trager(GEN u, GEN T, GEN dent, long fl)
1792 : {
1793 168 : long k = 0, i, lx;
1794 168 : GEN U, P, x0, mx0, fa, n = ZX_ZXY_rnfequation(T, u, &k);
1795 : int tmonic;
1796 168 : if (DEBUGLEVEL>4) err_printf("nfsqff_trager: choosing k = %ld\n",k);
1797 :
1798 : /* n guaranteed to be squarefree */
1799 168 : fa = ZX_DDF_max(Q_primpart(n),fl==ROOTS || fl==ROOTS_SPLIT ? degpol(T): 0); lx = lg(fa);
1800 168 : if (lx == 2) return mkvec(u);
1801 :
1802 133 : tmonic = is_pm1(leading_coeff(T));
1803 133 : P = cgetg(lx,t_VEC);
1804 133 : x0 = deg1pol_shallow(stoi(-k), gen_0, varn(T));
1805 133 : mx0 = deg1pol_shallow(stoi(k), gen_0, varn(T));
1806 133 : U = RgXQX_translate(u, mx0, T);
1807 133 : if (!tmonic) U = Q_primpart(U);
1808 511 : for (i=lx-1; i>0; i--)
1809 : {
1810 378 : GEN f = gel(fa,i), F = nfgcd(U, f, T, dent);
1811 378 : F = RgXQX_translate(F, x0, T);
1812 : /* F = gcd(f, u(t - x0)) [t + x0] = gcd(f(t + x0), u), more efficient */
1813 378 : if (typ(F) != t_POL || degpol(F) == 0)
1814 0 : pari_err_IRREDPOL("factornf [modulus]",T);
1815 378 : gel(P,i) = QXQX_normalize(F, T);
1816 : }
1817 133 : gen_sort_inplace(P, (void*)&cmp_RgX, &gen_cmp_RgX, NULL);
1818 133 : return P;
1819 : }
1820 :
1821 : /* Factor polynomial a on the number field defined by polynomial T, using
1822 : * Trager's trick */
1823 : GEN
1824 14 : polfnf(GEN a, GEN T)
1825 : {
1826 14 : GEN rep = cgetg(3, t_MAT), A, B, y, dent, bad;
1827 : long dA;
1828 : int tmonic;
1829 :
1830 14 : if (typ(a)!=t_POL) pari_err_TYPE("polfnf",a);
1831 14 : if (typ(T)!=t_POL) pari_err_TYPE("polfnf",T);
1832 14 : T = Q_primpart(T); tmonic = is_pm1(leading_coeff(T));
1833 14 : RgX_check_ZX(T,"polfnf");
1834 14 : A = Q_primpart( QXQX_normalize(RgX_nffix("polfnf",T,a,1), T) );
1835 14 : dA = degpol(A);
1836 14 : if (dA <= 0)
1837 : {
1838 0 : set_avma((pari_sp)(rep + 3));
1839 0 : return (dA == 0)? trivial_fact(): zerofact(varn(A));
1840 : }
1841 14 : bad = dent = absi_shallow(ZX_disc(T));
1842 14 : if (tmonic) dent = indexpartial(T, dent);
1843 14 : (void)nfgcd_all(A,RgX_deriv(A), T, dent, &B);
1844 14 : if (degpol(B) != dA) B = Q_primpart( QXQX_normalize(B, T) );
1845 14 : ensure_lt_INT(B);
1846 14 : y = nfsqff_trager(B, T, dent, FACTORS);
1847 14 : fact_from_sqff(rep, A, B, y, T, bad);
1848 14 : return sort_factor_pol(rep, cmp_RgX);
1849 : }
1850 :
1851 : static int
1852 58266 : nfsqff_use_Trager(long n, long dpol)
1853 : {
1854 58266 : return dpol*3<n;
1855 : }
1856 :
1857 : /* return the factorization of the square-free polynomial pol. Not memory-clean
1858 : The coeffs of pol are in Z_nf and its leading term is a rational integer.
1859 : deg(pol) > 0, deg(nfpol) > 1
1860 : fl is either FACTORS (return factors), or ROOTS / ROOTS_SPLIT (return roots):
1861 : - ROOTS, return only the roots of x in nf
1862 : - ROOTS_SPLIT, as ROOTS if pol splits, [] otherwise
1863 : den is usually 1, otherwise nf.zk is doubtful, and den bounds the
1864 : denominator of an arbitrary element of Z_nf on nf.zk */
1865 : static GEN
1866 34754 : nfsqff(GEN nf, GEN pol, long fl, GEN den)
1867 : {
1868 34754 : long n, nbf, dpol = degpol(pol);
1869 : GEN C0, polbase;
1870 34754 : GEN N2, res, polred, lt, nfpol = typ(nf)==t_POL?nf:nf_get_pol(nf);
1871 : ulong pp;
1872 : nfcmbf_t T;
1873 : nflift_t L;
1874 : pari_timer ti, ti_tot;
1875 :
1876 34754 : if (DEBUGLEVEL>2) { timer_start(&ti); timer_start(&ti_tot); }
1877 34754 : n = degpol(nfpol);
1878 : /* deg = 1 => irreducible */
1879 34754 : if (dpol == 1) {
1880 7161 : if (fl == FACTORS) return mkvec(QXQX_normalize(pol, nfpol));
1881 7105 : return mkvec(gneg(gdiv(gel(pol,2),gel(pol,3))));
1882 : }
1883 27593 : if (typ(nf)==t_POL || nfsqff_use_Trager(n,dpol))
1884 : {
1885 : GEN z;
1886 154 : if (DEBUGLEVEL>2) err_printf("Using Trager's method\n");
1887 154 : if (typ(nf) != t_POL) den = mulii(den, nf_get_index(nf));
1888 154 : z = nfsqff_trager(Q_primpart(pol), nfpol, den, fl);
1889 154 : if (fl != FACTORS) {
1890 119 : long i, l = lg(z);
1891 329 : for (i = 1; i < l; i++)
1892 : {
1893 252 : GEN LT, t = gel(z,i); if (degpol(t) > 1) break;
1894 210 : LT = gel(t,3);
1895 210 : if (typ(LT) == t_POL) LT = gel(LT,2); /* constant */
1896 210 : gel(z,i) = gdiv(gel(t,2), negi(LT));
1897 : }
1898 119 : setlg(z, i);
1899 119 : if (fl == ROOTS_SPLIT && i != l) return cgetg(1,t_VEC);
1900 : }
1901 154 : return z;
1902 : }
1903 :
1904 27439 : polbase = RgX_to_nfX(nf, pol);
1905 27439 : nbf = nf_pick_prime(nf, pol, fl, <, &L.Tp, &pp);
1906 27439 : if (L.Tp)
1907 : {
1908 22636 : L.Tp = Flx_to_ZX(L.Tp);
1909 22635 : L.p = utoi(pp);
1910 : }
1911 :
1912 27438 : if (fl == ROOTS_SPLIT && nbf < dpol) return cgetg(1,t_VEC);
1913 27417 : if (nbf <= 1)
1914 : {
1915 12125 : if (fl == FACTORS) return mkvec(QXQX_normalize(pol, nfpol)); /* irred. */
1916 11747 : if (!nbf) return cgetg(1,t_VEC); /* no root */
1917 : }
1918 :
1919 18407 : if (DEBUGLEVEL>2) {
1920 0 : timer_printf(&ti, "choice of a prime ideal");
1921 0 : err_printf("Prime ideal chosen: (%lu,x^%ld+...)\n", pp, degpol(L.Tp));
1922 : }
1923 18407 : L.tozk = nf_get_invzk(nf);
1924 18407 : L.topow= nf_get_zkprimpart(nf);
1925 18407 : L.topowden = nf_get_zkden(nf);
1926 18407 : if (is_pm1(den)) den = NULL;
1927 18407 : L.dn = den;
1928 18407 : T.ZC = nf_L2_bound(nf, den, NULL);
1929 18408 : T.Br = nf_root_bounds(nf, pol); if (lt) T.Br = gmul(T.Br, lt);
1930 :
1931 : /* C0 = bound for T_2(Q_i), Q | P */
1932 18408 : if (fl != FACTORS) C0 = normTp(T.Br, 2, n);
1933 12159 : else C0 = nf_factor_bound(nf, polbase);
1934 18408 : T.bound = mulrr(T.ZC, C0); /* bound for |Q_i|^2 in Z^n on chosen Z-basis */
1935 :
1936 18408 : N2 = mulur(dpol*dpol, normTp(T.Br, 4, n)); /* bound for T_2(lt * S_2) */
1937 18407 : T.BS_2 = mulrr(T.ZC, N2); /* bound for |S_2|^2 on chosen Z-basis */
1938 :
1939 18407 : if (DEBUGLEVEL>2) {
1940 0 : timer_printf(&ti, "bound computation");
1941 0 : err_printf(" 1) T_2 bound for %s: %Ps\n",
1942 : fl == FACTORS?"factor": "root", C0);
1943 0 : err_printf(" 2) Conversion from T_2 --> | |^2 bound : %Ps\n", T.ZC);
1944 0 : err_printf(" 3) Final bound: %Ps\n", T.bound);
1945 : }
1946 :
1947 18407 : bestlift_init(0, nf, T.bound, &L);
1948 18408 : if (DEBUGLEVEL>2) timer_start(&ti);
1949 18408 : polred = ZqX_normalize(polbase, lt, &L); /* monic */
1950 :
1951 18407 : if (fl != FACTORS) {
1952 6249 : GEN z = nf_DDF_roots(pol, polred, nfpol, fl, &L);
1953 6249 : if (lg(z) == 1) return cgetg(1, t_VEC);
1954 5866 : return z;
1955 : }
1956 :
1957 12158 : T.fact = gel(FqX_factor(polred, L.Tp, L.p), 1);
1958 12159 : if (DEBUGLEVEL>2)
1959 0 : timer_printf(&ti, "splitting mod %Ps^%ld", L.p, degpol(L.Tp));
1960 12159 : T.L = &L;
1961 12159 : T.polbase = polbase;
1962 12159 : T.pol = pol;
1963 12159 : T.nf = nf;
1964 12159 : res = nf_combine_factors(&T, polred, dpol-1);
1965 12159 : if (DEBUGLEVEL>2)
1966 0 : err_printf("Total Time: %ld\n===========\n", timer_delay(&ti_tot));
1967 12159 : return res;
1968 : }
1969 :
1970 : /* assume pol monic in nf.zk[X] */
1971 : GEN
1972 21 : nfroots_if_split(GEN *pnf, GEN pol)
1973 : {
1974 21 : GEN T = get_nfpol(*pnf,pnf), den = fix_nf(pnf, &T, &pol);
1975 21 : pari_sp av = avma;
1976 21 : GEN z = nfsqff(*pnf, pol, ROOTS_SPLIT, den);
1977 21 : if (lg(z) == 1) return gc_NULL(av);
1978 0 : return gerepilecopy(av, z);
1979 : }
1980 :
1981 : /*******************************************************************/
1982 : /* */
1983 : /* Roots of unity in a number field */
1984 : /* (alternative to nfrootsof1 using factorization in K[X]) */
1985 : /* */
1986 : /*******************************************************************/
1987 : /* Code adapted from nffactor. Structure of the algorithm; only step 1 is
1988 : * specific to roots of unity.
1989 : *
1990 : * [Step 1]: guess roots via ramification. If trivial output this.
1991 : * [Step 2]: select prime [p] unramified and ideal [pr] above
1992 : * [Step 4]: factor the cyclotomic polynomial mod [pr],
1993 : */
1994 :
1995 : /* Guesses the number of roots of unity in number field [nf].
1996 : * Computes gcd of N(P)-1 for some primes. The value returned is a proven
1997 : * multiple of the correct value. */
1998 : static long
1999 25032 : guess_roots(GEN T, GEN D, GEN index)
2000 : {
2001 25032 : long c = 0, nfdegree = degpol(T), B = nfdegree + 20, l;
2002 25032 : ulong p = 2;
2003 25032 : GEN nbroots = NULL;
2004 : forprime_t S;
2005 : pari_sp av;
2006 :
2007 25032 : (void)u_forprime_init(&S, 3, ULONG_MAX);
2008 25032 : av = avma;
2009 : /* result must be stationary (counter c) for at least B loops */
2010 744498 : for (l=1; (p = u_forprime_next(&S)); l++)
2011 : {
2012 : GEN old, F, pf_1, Tp;
2013 744496 : ulong i, gcdf = 0;
2014 : long nb;
2015 :
2016 744496 : if (!umodiu(D,p) || !umodiu(index,p)) continue;
2017 707688 : Tp = ZX_to_Flx(T,p); /* squarefree */
2018 707720 : F = Flx_nbfact_by_degree(Tp, &nb, p);
2019 : /* the gcd of the p^f - 1 is p^(gcd of the f's) - 1 */
2020 2582915 : for (i = 1; i <= (ulong) nfdegree; i++)
2021 2142865 : if (F[i]) {
2022 708562 : gcdf = gcdf? ugcd(gcdf, i): i;
2023 708622 : if (gcdf == 1) break;
2024 : }
2025 707796 : pf_1 = subiu(powuu(p, gcdf), 1);
2026 707394 : old = nbroots;
2027 707394 : nbroots = nbroots? gcdii(pf_1, nbroots): pf_1;
2028 707333 : if (DEBUGLEVEL>5)
2029 0 : err_printf("p=%lu; gcf(f(P/p))=%ld; nbroots | %Ps",p, gcdf, nbroots);
2030 : /* if same result go on else reset the stop counter [c] */
2031 707372 : if (old && equalii(nbroots,old))
2032 652256 : { if (!is_bigint(nbroots) && ++c > B) break; }
2033 : else
2034 55080 : c = 0;
2035 : }
2036 25007 : if (!nbroots) pari_err_OVERFLOW("guess_roots [ran out of primes]");
2037 25007 : if (DEBUGLEVEL>5) err_printf("%ld loops\n",l);
2038 25007 : return gc_long(av, itos(nbroots));
2039 : }
2040 :
2041 : /* T(x) an irreducible ZX. Is it of the form Phi_n(c \pm x) ?
2042 : * Return NULL if not, and a root of 1 of maximal order in Z[x]/(T) otherwise
2043 : *
2044 : * N.B. Set n_squarefree = 1 if n is squarefree, and 0 otherwise.
2045 : * This last parameter is inconvenient, but it allows a cheap
2046 : * stringent test. (n guessed from guess_roots())*/
2047 : static GEN
2048 2919 : ZXirred_is_cyclo_translate(GEN T, long n_squarefree)
2049 : {
2050 2919 : long r, m, d = degpol(T);
2051 2919 : GEN T1, c = divis_rem(gel(T, d+1), d, &r); /* d-1 th coeff divisible by d ? */
2052 : /* The trace coefficient of polcyclo(n) is \pm1 if n is square free, and 0
2053 : * otherwise. */
2054 2919 : if (!n_squarefree)
2055 1155 : { if (r) return NULL; }
2056 : else
2057 : {
2058 1764 : if (r < -1)
2059 : {
2060 0 : r += d;
2061 0 : c = subiu(c, 1);
2062 : }
2063 1764 : else if (r == d-1)
2064 : {
2065 49 : r = -1;
2066 49 : c = addiu(c, 1);
2067 : }
2068 1764 : if (r != 1 && r != -1) return NULL;
2069 : }
2070 2653 : if (signe(c)) /* presumably Phi_guess(c \pm x) */
2071 49 : T = RgX_translate(T, negi(c));
2072 2653 : if (!n_squarefree) T = RgX_deflate_max(T, &m);
2073 : /* presumably Phi_core(guess)(\pm x), cyclotomic iff original T was */
2074 2653 : T1 = ZX_graeffe(T);
2075 2653 : if (ZX_equal(T, T1)) /* T = Phi_n, n odd */
2076 385 : return deg1pol_shallow(gen_m1, negi(c), varn(T));
2077 2268 : else if (ZX_equal(T1, ZX_z_unscale(T, -1))) /* T = Phi_{2n}, nodd */
2078 2114 : return deg1pol_shallow(gen_1, c, varn(T));
2079 154 : return NULL;
2080 : }
2081 :
2082 : static GEN
2083 55292 : trivroots(void) { return mkvec2(gen_2, gen_m1); }
2084 : /* Number of roots of unity in number field [nf]. */
2085 : GEN
2086 64427 : nfrootsof1(GEN nf)
2087 : {
2088 : GEN T, fa, LP, LE, z, disc, index;
2089 : pari_timer ti;
2090 : long i, l, nbguessed, nbroots, nfdegree;
2091 : pari_sp av;
2092 :
2093 64427 : T = get_nfpol(nf, &nf); nfdegree = degpol(T);
2094 64427 : RgX_check_ZX(T, "nfrootsof1");
2095 64427 : if (nfdegree <= 0) pari_err_CONSTPOL("nfrootsof1");
2096 64420 : if (nf)
2097 : {
2098 64406 : if (nf_get_r1(nf)) return trivroots();
2099 25032 : disc = nf_get_disc(nf);
2100 25032 : index = nf_get_index(nf);
2101 : }
2102 : else
2103 : {
2104 14 : if (odd(nfdegree) || signe(leading_coeff(T)) != signe(constant_coeff(T)))
2105 14 : return trivroots();
2106 0 : disc = ZX_disc(T);
2107 0 : index = gen_1;
2108 : }
2109 : /* Step 1 : guess number of roots and discard trivial case 2 */
2110 25032 : if (DEBUGLEVEL>2) timer_start(&ti);
2111 25032 : nbguessed = guess_roots(T, disc, index);
2112 25032 : if (DEBUGLEVEL>2)
2113 0 : timer_printf(&ti, "guessing roots of 1 [guess = %ld]", nbguessed);
2114 25032 : if (nbguessed == 2) return trivroots();
2115 :
2116 9135 : fa = factoru(nbguessed);
2117 9135 : LP = gel(fa,1); l = lg(LP);
2118 9135 : LE = gel(fa,2);
2119 25403 : for (i = 1; i < l; i++)
2120 : {
2121 16268 : long p = LP[i];
2122 : /* Degree and ramification test: find largest k such that Q(zeta_{p^k})
2123 : * may be a subfield of K. Q(zeta_p^k) has degree (p-1)p^(k-1)
2124 : * and v_p(discriminant) = ((p-1)k-1)p^(k-1); so we must have
2125 : * v_p(disc_K) >= ((p-1)k-1) * n / (p-1) = kn - q, where q = n/(p-1) */
2126 16268 : if (p == 2)
2127 : { /* the test simplifies a little in that case */
2128 : long v, vnf, k;
2129 9135 : if (LE[i] == 1) continue;
2130 2919 : vnf = vals(nfdegree);
2131 2919 : v = vali(disc);
2132 3031 : for (k = minss(LE[i], vnf+1); k >= 1; k--)
2133 3031 : if (v >= nfdegree*(k-1)) { nbguessed >>= LE[i]-k; LE[i] = k; break; }
2134 : /* N.B the test above always works for k = 1: LE[i] >= 1 */
2135 : }
2136 : else
2137 : {
2138 : long v, vnf, k;
2139 7133 : ulong r, q = udivuu_rem(nfdegree, p-1, &r);
2140 7133 : if (r) { nbguessed /= upowuu(p, LE[i]); LE[i] = 0; continue; }
2141 : /* q = n/(p-1) */
2142 7133 : vnf = u_lval(q, p);
2143 7133 : v = Z_lval(disc, p);
2144 7154 : for (k = minss(LE[i], vnf+1); k >= 0; k--)
2145 7154 : if (v >= nfdegree*k-(long)q)
2146 7133 : { nbguessed /= upowuu(p, LE[i]-k); LE[i] = k; break; }
2147 : /* N.B the test above always works for k = 0: LE[i] >= 0 */
2148 : }
2149 : }
2150 9135 : if (DEBUGLEVEL>2)
2151 0 : timer_printf(&ti, "after ramification conditions [guess = %ld]", nbguessed);
2152 9135 : if (nbguessed == 2) return trivroots();
2153 9128 : av = avma;
2154 :
2155 : /* Step 1.5 : test if nf.pol == subst(polcyclo(nbguessed), x, \pm x+c) */
2156 9128 : if (eulerphiu_fact(fa) == (ulong)nfdegree)
2157 : {
2158 2919 : z = ZXirred_is_cyclo_translate(T, uissquarefree_fact(fa));
2159 2919 : if (DEBUGLEVEL>2) timer_printf(&ti, "checking for cyclotomic polynomial");
2160 2919 : if (z)
2161 : {
2162 2499 : if (nf) z = nf_to_scalar_or_basis(nf,z);
2163 2499 : return gerepilecopy(av, mkvec2(utoipos(nbguessed), z));
2164 : }
2165 420 : set_avma(av);
2166 : }
2167 :
2168 : /* Step 2 : actual computation of roots */
2169 6629 : nbroots = 2; z = scalarpol(gen_m1, varn(T));
2170 18438 : for (i = 1; i < l; i++)
2171 : { /* for all prime power factors of nbguessed, find a p^k-th root of unity */
2172 11809 : long k, p = LP[i];
2173 16618 : for (k = LE[i]; k > 0; k--)
2174 : { /* find p^k-th roots */
2175 11823 : pari_sp av = avma;
2176 11823 : long pk = upowuu(p,k);
2177 : GEN r;
2178 11823 : if (pk==2) continue; /* no need to test second roots ! */
2179 7091 : r = nfisincl0(polcyclo(pk, 0), T, 1);
2180 7091 : if (!isintzero(r))
2181 : {
2182 7014 : if (DEBUGLEVEL>2) err_printf(" %s root of unity found\n",uordinal(pk));
2183 7014 : if (p==2) { nbroots = pk; z = r; }
2184 : else
2185 : {
2186 5117 : nbroots *= pk;
2187 5117 : z = nf ? nfmul(nf, z, r): QXQ_mul(z, r, T);
2188 : }
2189 7014 : break;
2190 : }
2191 77 : set_avma(av);
2192 77 : if (DEBUGLEVEL) pari_warn(warner,"nfrootsof1: wrong guess");
2193 : }
2194 : }
2195 6629 : if (nf) z = nf_to_scalar_or_basis(nf,z);
2196 6629 : return gerepilecopy(av, mkvec2(utoi(nbroots), z));
2197 : }
2198 :
2199 : /*******************************************************************/
2200 : /* */
2201 : /* rnfgaloisconj */
2202 : /* */
2203 : /*******************************************************************/
2204 :
2205 : static GEN
2206 35 : rnffrobeniuslift(GEN nf, GEN P, GEN Tp, GEN p, GEN bound, GEN den, GEN iden, GEN T)
2207 : {
2208 : GEN q, Tq, Pp, Pq, s, S;
2209 35 : long v = varn(P), e;
2210 : nflift_t L;
2211 35 : L.p = p; L.Tp = Tp;
2212 35 : L.tozk = nf_get_invzk(nf);
2213 35 : L.topow= nf_get_zkprimpart(nf);
2214 35 : L.topowden = nf_get_zkden(nf);
2215 35 : bestlift_init(0, nf, bound, &L);
2216 35 : q = L.pk; Tq = L.Tpk; e = L.k;
2217 35 : P = gmul(P,L.topowden);
2218 35 : Pq = ZqX(RgX_to_nfX(nf, P), q, Tq, L.ZqProj);
2219 35 : Pp = FqX_red(Pq, Tp, p);
2220 35 : s = FpXQXQ_pow(pol_x(v), powiu(p, degpol(Tp)), Pp, Tp, p);
2221 35 : S = ZqX_ZqXQ_liftroot(Pq, s, Pq, Tq, p, e);
2222 35 : den = nf_to_Zq(den, Tq, q, shifti(q,-1), L.ZqProj);
2223 35 : S = FqX_Fq_mul(S, den, Tq, q);
2224 35 : P = nf_pol_lift(S, bound, &L);
2225 35 : if (!P) return NULL;
2226 28 : return gdiv(QXQX_QXQ_mul(P, iden, T), L.topowden);
2227 : }
2228 :
2229 : static GEN
2230 63 : RgX_galconj_bound(GEN T, long prec)
2231 : {
2232 63 : GEN L = QX_complex_roots(T, prec);
2233 63 : GEN prep = vandermondeinverseinit(L);
2234 63 : GEN M = vandermondeinverse(L, RgX_gtofp(T, prec), gen_1, prep);
2235 63 : return gmul(matrixnorm(M, prec), gsupnorm(L, prec));
2236 : }
2237 :
2238 : static GEN
2239 42 : rnfgaloisconj_bound(GEN P, GEN z, GEN den, long prec)
2240 : {
2241 42 : pari_sp av = avma;
2242 42 : long i, l = lg(z);
2243 42 : GEN s = gen_0;
2244 105 : for (i = 1; i < l; i++)
2245 : {
2246 63 : GEN a = gabs(RgX_cxeval(den, gel(z,i), NULL),prec);
2247 63 : GEN b = RgX_galconj_bound(RgXY_cxevalx(P, gel(z,i), NULL), prec);
2248 63 : s = gerepileupto(av, gadd(s, gsqr(gmul(a,b))));
2249 : }
2250 42 : return gerepileupto(av, ceil_safe(s));
2251 : }
2252 :
2253 : static GEN
2254 0 : FlxqXQV_fixedalg(GEN aut, GEN S, GEN T, ulong p)
2255 : {
2256 0 : pari_sp av = avma;
2257 0 : long d = get_FlxqX_degree(S), sv = get_Flx_var(T), v = get_FlxqX_var(S);
2258 0 : long i, l = lg(aut);
2259 0 : GEN A = RgX_to_FlxqX(gel(aut,1), T, p);
2260 0 : GEN M = FlxXV_to_FlxM(FlxqXQ_powers(A, d-1, S, T, p), d, sv);
2261 0 : GEN M2 = FlxM_Flx_add_shallow(M, Fl_to_Flx(p-1, sv), p);
2262 0 : GEN C = FlxM_to_FlxXV(FlxqM_ker(M2, T, p), v);
2263 0 : for (i = 2; i < l; i++)
2264 : {
2265 0 : GEN A = RgX_to_FlxqX(gel(aut,i), T, p);
2266 0 : GEN M = FlxXC_sub(FlxqXC_FlxqXQ_eval(C, A, S, T, p), C, p);
2267 0 : GEN K = FlxqM_ker(FlxXV_to_FlxM(M, d, sv), T, p);
2268 0 : C = FlxM_to_FlxXV(FlxqM_mul(FlxXV_to_FlxM(C, d, sv), K, T, p), v);
2269 : }
2270 0 : return gerepilecopy(av, C);
2271 : }
2272 :
2273 : static long
2274 0 : fixedalg_order(GEN aut, GEN S, GEN T, ulong p)
2275 : {
2276 0 : pari_sp av = avma;
2277 0 : long e, i, d = get_Flx_degree(T), l = lg(aut);
2278 0 : for (e = 1;; e++, set_avma(av))
2279 : {
2280 0 : for (i = 1; i < l; i++, set_avma(av))
2281 0 : if (!gequal(FlxqXQ_pow(gel(aut,i), powuu(p, d*e), S, T, p), gel(aut,i)))
2282 0 : break;
2283 0 : if (i == l) return e;
2284 : }
2285 : }
2286 :
2287 : static GEN
2288 35 : rnfgaloisanalysis(GEN nf, GEN P, GEN aut, long m, GEN d, long *pt_o)
2289 : {
2290 35 : GEN T = nf_get_pol(nf), R = NULL, den = nf_get_zkden(nf);
2291 35 : long n = degpol(P), omax = 0, ntry = 0;
2292 35 : GEN daut = lg(aut) > 1 ? Q_denom(aut) : NULL;
2293 : forprime_t S;
2294 35 : u_forprime_init(&S, 2, ULONG_MAX);
2295 182 : while (ntry <= n || omax < 2)
2296 : {
2297 182 : ulong p = u_forprime_next(&S);
2298 : GEN F, Tp;
2299 : long i, lF;
2300 182 : if ((d && dvdiu(d,p)) || dvdiu(den, p) || (daut && dvdiu(daut,p))) continue;
2301 161 : Tp = ZX_to_Flx(T, p);
2302 161 : if (!Flx_is_squarefree(Tp,p)) continue;
2303 140 : F = gel(Flx_factor(Tp, p), 1); lF = lg(F);
2304 294 : for (i = 1; i < lF; i++)
2305 : {
2306 189 : pari_sp av = avma;
2307 : long o, d;
2308 189 : GEN D, Fi = gel(F,i), Pp = RgX_to_FlxqX(P, Fi, p);
2309 189 : if (degpol(Pp) < n || !FlxqX_is_squarefree(Pp, Fi, p)) continue;
2310 147 : D = FlxqX_nbfact_by_degree(Pp, &d, Fi, p);
2311 147 : o = n / d; /* d factors, all should have degree o */
2312 147 : if (D[o] != d)
2313 : {
2314 0 : if(DEBUGLEVEL) err_printf("rnfisabelian: not Galois at %lu: %Ps\n",p,D);
2315 35 : return NULL;
2316 : }
2317 147 : if (lg(aut) > 1)
2318 : {
2319 0 : GEN U = FlxqXQV_fixedalg(aut, Pp, Fi, p);
2320 0 : o = fixedalg_order(U, Pp, Fi, p);
2321 0 : d = m / o;
2322 : }
2323 147 : if (d == 1) { *pt_o = o; return mkvec2(Flx_to_ZX(Fi), utoi(p)); }
2324 112 : if (o > omax) { omax = o; R = mkvec2(Flx_to_ZX(Fi), utoi(p)); }
2325 84 : else set_avma(av);
2326 : }
2327 105 : ntry++;
2328 : }
2329 0 : *pt_o = omax; return R;
2330 : }
2331 :
2332 : static GEN
2333 42 : rnfabelianconjgen_i(GEN nf, GEN P)
2334 : {
2335 : long prec, m, i, l;
2336 : GEN bnd, den, iden, T, Pr, gen, orders, d;
2337 42 : nf = checknf(nf);
2338 42 : T = nf_get_pol(nf);
2339 42 : P = RgX_nffix("rnfgaloisconj", T, P, 1);
2340 42 : P = Q_remove_denom(P, &d);
2341 42 : Pr = RgX_to_nfX(nf, P);
2342 42 : prec = DEFAULTPREC + nbits2extraprec(gexpo(Pr) * degpol(T));
2343 42 : nf = nfnewprec_shallow(nf, prec);
2344 42 : den = nfX_disc(nf, P);
2345 42 : bnd = rnfgaloisconj_bound(P, nf_get_roots(nf), den, prec);
2346 42 : iden = QXQ_inv(den, T);
2347 42 : den = nf_to_scalar_or_basis(nf, den);
2348 42 : m = degpol(P); l = expu(m) + 1;
2349 42 : gen = cgetg(l, t_VEC); orders = cgetg(l,t_VECSMALL);
2350 70 : for (i = 1; m > 1; i++)
2351 : {
2352 : long o;
2353 : GEN S, Tp;
2354 35 : setlg(gen, i);
2355 35 : Tp = rnfgaloisanalysis(nf, P, gen, m, d, &o);
2356 42 : if (!Tp) return NULL;
2357 35 : S = rnffrobeniuslift(nf, P, gel(Tp,1), gel(Tp,2), bnd, den, iden, T);
2358 35 : if (!S) return NULL;
2359 28 : gel(gen,i) = S; orders[i] = o; m /= o;
2360 : }
2361 35 : setlg(gen,i); setlg(orders,i); return mkvec2(gen, orders);
2362 : }
2363 : GEN
2364 0 : rnfabelianconjgen(GEN nf, GEN P)
2365 : {
2366 0 : pari_sp av = avma;
2367 0 : GEN G = rnfabelianconjgen_i(nf, P);
2368 0 : return G? gerepilecopy(av, G): gc_const(av, gen_0);
2369 : }
2370 :
2371 : long
2372 42 : rnfisabelian(GEN nf, GEN pol)
2373 : {
2374 42 : pari_sp av = avma;
2375 42 : GEN G = rnfabelianconjgen_i(nf, pol);
2376 42 : return gc_long(av, G? 1: 0);
2377 : }
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