/* g in Z[X] potentially defines a subfield of Q[X]/f. It is a subfield iff A * (cf subfield) was a block system; then there * exists h in Q[X] such that f | g o h. listdelta determines h s.t f | g o h * in Fp[X] (cf chinese_retrieve_pol). Try to lift it; den is a * multiplicative bound for denominator of lift. */ static GEN embedding(GEN g, GEN DATA, primedata *S, GEN den, GEN listdelta) { GEN TR, w0_Q, w0, w1_Q, w1, wpow, h0, gp, T, q2, q, maxp, a, p = S->p; long rt; pari_sp av; T = gel(DATA,1); rt = brent_kung_optpow(degpol(T), 2); maxp= gel(DATA,7); gp = derivpol(g); av = avma; w0 = chinese_retrieve_pol(DATA, S, listdelta); w0_Q = centermod(gmul(w0,den), p); h0 = FpXQ_inv(FpX_FpXQ_compo(gp,w0, T,p), T,p); /* = 1/g'(w0) mod (T,p) */ wpow = NULL; q = sqri(p); for(;;) {/* Given g,w0,h0 in Z[x], s.t. h0.g'(w0) = 1 and g(w0) = 0 mod (T,p), find * [w1,h1] satisfying the same conditions mod p^2, [w1,h1] = [w0,h0] (mod p) * (cf. Dixon: J. Austral. Math. Soc., Series A, vol.49, 1990, p.445) */ if (DEBUGLEVEL>1) fprintferr("lifting embedding mod p^k = %Z^%ld\n",p, Z_pval(q,p)); /* w1 := w0 - h0 g(w0) mod (T,q) */ if (wpow) a = FpX_FpXQV_compo(g,wpow, T,q); else a = FpX_FpXQ_compo(g,w0, T,q); /* first time */ /* now, a = 0 (p) */ a = gmul(gneg(h0), gdivexact(a, p)); w1 = gadd(w0, gmul(p, FpX_rem(a, T,p))); w1_Q = centermod(gmul(w1, remii(den,q)), q); if (gequal(w1_Q, w0_Q) || cmpii(q,maxp) > 0) { GEN G = gcmp1(den)? g: RgX_rescale(g,den); if (gcmp0(RgX_RgXQ_compo(G, w1_Q, T))) break; } if (cmpii(q, maxp) > 0) { if (DEBUGLEVEL) fprintferr("coeff too big for embedding\n"); return NULL; } gerepileall(av, 5, &w1,&h0,&w1_Q,&q,&p); q2 = sqri(q); wpow = FpXQ_powers(w1, rt, T, q2); /* h0 := h0 * (2 - h0 g'(w1)) mod (T,q) * = h0 + h0 * (1 - h0 g'(w1)) */ a = gmul(gneg(h0), FpX_FpXQV_compo(gp, FpXV_red(wpow,q),T,q)); a = ZX_Z_add(FpX_rem(a, T,q), gen_1); /* 1 - h0 g'(w1) = 0 (p) */ a = gmul(h0, gdivexact(a, p)); h0 = gadd(h0, gmul(p, FpX_rem(a, T,p))); w0 = w1; w0_Q = w1_Q; p = q; q = q2; } TR = gel(DATA,5); if (!gcmp0(TR)) w1_Q = translate_pol(w1_Q, TR); return gdiv(w1_Q,den); }
// Digit reversal GEN rev(GEN n, long B) { pari_sp av = avma; if (typ(n) != t_INT) pari_err_TYPE("rev", n); GEN m = modis(n, B); n = divis(n, B); pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (signe(n)) { m = addis(mulis(m, B), smodis(n, B)); n = divis(n, B); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &m, &n); } m = gerepilecopy(av, m); return m; }
/* return d = gcd(a,b), sets u, v such that au + bv = gcd(a,b) */ GEN extgcd(GEN A, GEN B, GEN *U, GEN *V) { pari_sp av = avma; GEN ux = gen_1, vx = gen_0, a = A, b = B; if (typ(a) != t_INT) pari_err_TYPE("extgcd",a); if (typ(b) != t_INT) pari_err_TYPE("extgcd",b); if (signe(a) < 0) { a = negi(a); ux = negi(ux); } while (!gequal0(b)) { GEN r, q = dvmdii(a, b, &r), v = vx; vx = subii(ux, mulii(q, vx)); ux = v; a = b; b = r; } *U = ux; *V = diviiexact( subii(a, mulii(A,ux)), B ); gerepileall(av, 3, &a, U, V); return a; }
GEN bezout(GEN a, GEN b, GEN *pu, GEN *pv) { GEN t,u,u1,v,v1,d,d1,q,r; GEN *pt; pari_sp av, av1; long s, sa, sb; ulong g; ulong xu,xu1,xv,xv1; /* Lehmer stage recurrence matrix */ int lhmres; /* Lehmer stage return value */ s = abscmpii(a,b); if (s < 0) { t=b; b=a; a=t; pt=pu; pu=pv; pv=pt; } /* now |a| >= |b| */ sa = signe(a); sb = signe(b); if (!sb) { if (pv) *pv = gen_0; switch(sa) { case 0: if (pu) *pu = gen_0; return gen_0; case 1: if (pu) *pu = gen_1; return icopy(a); case -1: if (pu) *pu = gen_m1; return(negi(a)); } } if (s == 0) /* |a| == |b| != 0 */ { if (pu) *pu = gen_0; if (sb > 0) { if (pv) *pv = gen_1; return icopy(b); } else { if (pv) *pv = gen_m1; return(negi(b)); } } /* now |a| > |b| > 0 */ if (lgefint(a) == 3) /* single-word affair */ { g = xxgcduu(uel(a,2), uel(b,2), 0, &xu, &xu1, &xv, &xv1, &s); sa = s > 0 ? sa : -sa; sb = s > 0 ? -sb : sb; if (pu) { if (xu == 0) *pu = gen_0; /* can happen when b divides a */ else if (xu == 1) *pu = sa < 0 ? gen_m1 : gen_1; else if (xu == 2) *pu = sa < 0 ? gen_m2 : gen_2; else { *pu = cgeti(3); (*pu)[1] = evalsigne(sa)|evallgefint(3); (*pu)[2] = xu; } } if (pv) { if (xv == 1) *pv = sb < 0 ? gen_m1 : gen_1; else if (xv == 2) *pv = sb < 0 ? gen_m2 : gen_2; else { *pv = cgeti(3); (*pv)[1] = evalsigne(sb)|evallgefint(3); (*pv)[2] = xv; } } if (g == 1) return gen_1; else if (g == 2) return gen_2; else return utoipos(g); } /* general case */ av = avma; (void)new_chunk(lgefint(b) + (lgefint(a)<<1)); /* room for u,v,gcd */ /* if a is significantly larger than b, calling lgcdii() is not the best * way to start -- reduce a mod b first */ if (lgefint(a) > lgefint(b)) { d = absi(b), q = dvmdii(absi(a), d, &d1); if (!signe(d1)) /* a == qb */ { avma = av; if (pu) *pu = gen_0; if (pv) *pv = sb < 0 ? gen_m1 : gen_1; return (icopy(d)); } else { u = gen_0; u1 = v = gen_1; v1 = negi(q); } /* if this results in lgefint(d) == 3, will fall past main loop */ } else { d = absi(a); d1 = absi(b); u = v1 = gen_1; u1 = v = gen_0; } av1 = avma; /* main loop is almost identical to that of invmod() */ while (lgefint(d) > 3 && signe(d1)) { lhmres = lgcdii((ulong *)d, (ulong *)d1, &xu, &xu1, &xv, &xv1, ULONG_MAX); if (lhmres != 0) /* check progress */ { /* apply matrix */ if ((lhmres == 1) || (lhmres == -1)) { if (xv1 == 1) { r = subii(d,d1); d=d1; d1=r; a = subii(u,u1); u=u1; u1=a; a = subii(v,v1); v=v1; v1=a; } else { r = subii(d, mului(xv1,d1)); d=d1; d1=r; a = subii(u, mului(xv1,u1)); u=u1; u1=a; a = subii(v, mului(xv1,v1)); v=v1; v1=a; } } else { r = subii(muliu(d,xu), muliu(d1,xv)); d1 = subii(muliu(d,xu1), muliu(d1,xv1)); d = r; a = subii(muliu(u,xu), muliu(u1,xv)); u1 = subii(muliu(u,xu1), muliu(u1,xv1)); u = a; a = subii(muliu(v,xu), muliu(v1,xv)); v1 = subii(muliu(v,xu1), muliu(v1,xv1)); v = a; if (lhmres&1) { togglesign(d); togglesign(u); togglesign(v); } else { togglesign(d1); togglesign(u1); togglesign(v1); } } } if (lhmres <= 0 && signe(d1)) { q = dvmdii(d,d1,&r); a = subii(u,mulii(q,u1)); u=u1; u1=a; a = subii(v,mulii(q,v1)); v=v1; v1=a; d=d1; d1=r; } if (gc_needed(av,1)) { if(DEBUGMEM>1) pari_warn(warnmem,"bezout"); gerepileall(av1,6, &d,&d1,&u,&u1,&v,&v1); } } /* end while */ /* Postprocessing - final sprint */ if (signe(d1)) { /* Assertions: lgefint(d)==lgefint(d1)==3, and * gcd(d,d1) is nonzero and fits into one word */ g = xxgcduu(uel(d,2), uel(d1,2), 0, &xu, &xu1, &xv, &xv1, &s); u = subii(muliu(u,xu), muliu(u1, xv)); v = subii(muliu(v,xu), muliu(v1, xv)); if (s < 0) { sa = -sa; sb = -sb; } avma = av; if (pu) *pu = sa < 0 ? negi(u) : icopy(u); if (pv) *pv = sb < 0 ? negi(v) : icopy(v); if (g == 1) return gen_1; else if (g == 2) return gen_2; else return utoipos(g); } /* get here when the final sprint was skipped (d1 was zero already). * Now the matrix is final, and d contains the gcd. */ avma = av; if (pu) *pu = sa < 0 ? negi(u) : icopy(u); if (pv) *pv = sb < 0 ? negi(v) : icopy(v); return icopy(d); }
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; }