static trace_data * init_trace(trace_data *T, GEN S, nflift_t *L, GEN q) { long e = gexpo(S), i,j, l,h; GEN qgood, S1, invd; if (e < 0) return NULL; /* S = 0 */ qgood = int2n(e - 32); /* single precision check */ if (cmpii(qgood, q) > 0) q = qgood; S1 = gdivround(S, q); if (gcmp0(S1)) return NULL; invd = ginv(itor(L->den, DEFAULTPREC)); T->dPinvS = gmul(L->iprk, S); l = lg(S); h = lg(T->dPinvS[1]); T->PinvSdbl = (double**)cgetg(l, t_MAT); init_dalloc(); for (j = 1; j < l; j++) { double *t = dalloc(h * sizeof(double)); GEN c = gel(T->dPinvS,j); pari_sp av = avma; T->PinvSdbl[j] = t; for (i=1; i < h; i++) t[i] = rtodbl(mpmul(invd, gel(c,i))); avma = av; } T->d = L->den; T->P1 = gdivround(L->prk, q); T->S1 = S1; return T; }
/* Let X = Tra * M_L, Y = bestlift(X) return V s.t Y = X - PRK V * and set *eT2 = gexpo(Y) [cf nf_bestlift, but memory efficient] */ static GEN get_V(GEN Tra, GEN M_L, GEN PRK, GEN PRKinv, GEN pk, long *eT2) { long i, e = 0, l = lg(M_L); GEN V = cgetg(l, t_MAT); *eT2 = 0; for (i = 1; i < l; i++) { /* cf nf_bestlift(Tra * c) */ pari_sp av = avma, av2; GEN v, T2 = gmul(Tra, gel(M_L,i)); v = gdivround(gmul(PRKinv, T2), pk); /* small */ av2 = avma; T2 = gsub(T2, gmul(PRK, v)); e = gexpo(T2); if (e > *eT2) *eT2 = e; avma = av2; gel(V,i) = gerepileupto(av, v); /* small */ } return V; }
static GEN nf_LLL_cmbf(nfcmbf_t *T, GEN p, long k, long rec) { nflift_t *L = T->L; GEN pk = L->pk, PRK = L->prk, PRKinv = L->iprk, GSmin = L->GSmin; GEN Tpk = L->Tpk; GEN famod = T->fact, nf = T->nf, ZC = T->ZC, Br = T->Br; GEN Pbase = T->polbase, P = T->pol, dn = T->dn; GEN nfT = gel(nf,1); GEN Btra; long dnf = degpol(nfT), dP = degpol(P); double BitPerFactor = 0.5; /* nb bits / modular factor */ long i, C, tmax, n0; GEN lP, Bnorm, Tra, T2, TT, CM_L, m, list, ZERO; double Bhigh; pari_sp av, av2, lim; long ti_LLL = 0, ti_CF = 0; pari_timer ti2, TI; lP = absi(leading_term(P)); if (is_pm1(lP)) lP = NULL; n0 = lg(famod) - 1; /* Lattice: (S PRK), small vector (vS vP). To find k bound for the image, * write S = S1 q + S0, P = P1 q + P0 * |S1 vS + P1 vP|^2 <= Bhigh for all (vS,vP) assoc. to true factors */ Btra = mulrr(ZC, mulsr(dP*dP, normlp(Br, 2, dnf))); Bhigh = get_Bhigh(n0, dnf); C = (long)ceil(sqrt(Bhigh/n0)) + 1; /* C^2 n0 ~ Bhigh */ Bnorm = dbltor( n0 * C * C + Bhigh ); ZERO = zeromat(n0, dnf); av = avma; lim = stack_lim(av, 1); TT = cgetg(n0+1, t_VEC); Tra = cgetg(n0+1, t_MAT); for (i=1; i<=n0; i++) TT[i] = 0; CM_L = gscalsmat(C, n0); /* tmax = current number of traces used (and computed so far) */ for(tmax = 0;; tmax++) { long a, b, bmin, bgood, delta, tnew = tmax + 1, r = lg(CM_L)-1; GEN oldCM_L, M_L, q, S1, P1, VV; int first = 1; /* bound for f . S_k(genuine factor) = ZC * bound for T_2(S_tnew) */ Btra = mulrr(ZC, mulsr(dP*dP, normlp(Br, 2*tnew, dnf))); bmin = logint(ceil_safe(sqrtr(Btra)), gen_2, NULL); if (DEBUGLEVEL>2) fprintferr("\nLLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)\n", r, tmax, bmin); /* compute Newton sums (possibly relifting first) */ if (gcmp(GSmin, Btra) < 0) { nflift_t L1; GEN polred; bestlift_init(k<<1, nf, T->pr, Btra, &L1); polred = ZqX_normalize(Pbase, lP, &L1); k = L1.k; pk = L1.pk; PRK = L1.prk; PRKinv = L1.iprk; GSmin = L1.GSmin; Tpk = L1.Tpk; famod = hensel_lift_fact(polred, famod, Tpk, p, pk, k); for (i=1; i<=n0; i++) TT[i] = 0; } for (i=1; i<=n0; i++) { GEN h, lPpow = lP? gpowgs(lP, tnew): NULL; GEN z = polsym_gen(gel(famod,i), gel(TT,i), tnew, Tpk, pk); gel(TT,i) = z; h = gel(z,tnew+1); /* make Newton sums integral */ lPpow = mul_content(lPpow, dn); if (lPpow) h = FpX_red(gmul(h,lPpow), pk); gel(Tra,i) = nf_bestlift(h, NULL, L); /* S_tnew(famod) */ } /* compute truncation parameter */ if (DEBUGLEVEL>2) { TIMERstart(&ti2); TIMERstart(&TI); } oldCM_L = CM_L; av2 = avma; b = delta = 0; /* -Wall */ AGAIN: M_L = Q_div_to_int(CM_L, utoipos(C)); VV = get_V(Tra, M_L, PRK, PRKinv, pk, &a); if (first) { /* initialize lattice, using few p-adic digits for traces */ bgood = (long)(a - max(32, BitPerFactor * r)); b = max(bmin, bgood); delta = a - b; } else { /* add more p-adic digits and continue reduction */ if (a < b) b = a; b = max(b-delta, bmin); if (b - delta/2 < bmin) b = bmin; /* near there. Go all the way */ } /* restart with truncated entries */ q = int2n(b); P1 = gdivround(PRK, q); S1 = gdivround(Tra, q); T2 = gsub(gmul(S1, M_L), gmul(P1, VV)); m = vconcat( CM_L, T2 ); if (first) { first = 0; m = shallowconcat( m, vconcat(ZERO, P1) ); /* [ C M_L 0 ] * m = [ ] square matrix * [ T2' PRK ] T2' = Tra * M_L truncated */ } CM_L = LLL_check_progress(Bnorm, n0, m, b == bmin, /*dbg:*/ &ti_LLL); if (DEBUGLEVEL>2) fprintferr("LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld\n", a,b, lg(m)-1, CM_L? lg(CM_L)-1: 1, TIMER(&TI)); if (!CM_L) { list = mkcol(QXQX_normalize(P,nfT)); break; } if (b > bmin) { CM_L = gerepilecopy(av2, CM_L); goto AGAIN; } if (DEBUGLEVEL>2) msgTIMER(&ti2, "for this trace"); i = lg(CM_L) - 1; if (i == r && gequal(CM_L, oldCM_L)) { CM_L = oldCM_L; avma = av2; continue; } if (i <= r && i*rec < n0) { pari_timer ti; if (DEBUGLEVEL>2) TIMERstart(&ti); list = nf_chk_factors(T, P, Q_div_to_int(CM_L,utoipos(C)), famod, pk); if (DEBUGLEVEL>2) ti_CF += TIMER(&ti); if (list) break; CM_L = gerepilecopy(av2, CM_L); } if (low_stack(lim, stack_lim(av,1))) { if(DEBUGMEM>1) pari_warn(warnmem,"nf_LLL_cmbf"); gerepileall(av, Tpk? 9: 8, &CM_L,&TT,&Tra,&famod,&pk,&GSmin,&PRK,&PRKinv,&Tpk); } } if (DEBUGLEVEL>2) fprintferr("* Time LLL: %ld\n* Time Check Factor: %ld\n",ti_LLL,ti_CF); return list; }
GEN gdivrounde(GEN *x, GEN y) { *x=gdivround(*x,y); return *x; }