/* 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); }
/* d = requested degree for subfield. Return DATA, valid for given pol, S and d * If DATA != NULL, translate pol [ --> pol(X+1) ] and update DATA * 1: polynomial pol * 2: p^e (for Hensel lifts) such that p^e > max(M), * 3: Hensel lift to precision p^e of DATA[4] * 4: roots of pol in F_(p^S->lcm), * 5: number of polynomial changes (translations) * 6: Bezout coefficients associated to the S->ff[i] * 7: Hadamard bound for coefficients of h(x) such that g o h = 0 mod pol. * 8: bound M for polynomials defining subfields x PD->den * 9: *[i] = interpolation polynomial for S->ff[i] [= 1 on the first root S->firstroot[i], 0 on the others] */ static void compute_data(blockdata *B) { GEN ffL, roo, pe, p1, p2, fk, fhk, MM, maxroot, pol; primedata *S = B->S; GEN p = S->p, T = S->T, ff = S->ff, DATA = B->DATA; long i, j, l, e, N, lff = lg(ff); if (DEBUGLEVEL>1) fprintferr("Entering compute_data()\n\n"); pol = B->PD->pol; N = degpol(pol); roo = B->PD->roo; if (DATA) /* update (translate) an existing DATA */ { GEN Xm1 = gsub(pol_x[varn(pol)], gen_1); GEN TR = addis(gel(DATA,5), 1); GEN mTR = negi(TR), interp, bezoutC; gel(DATA,5) = TR; pol = translate_pol(gel(DATA,1), gen_m1); l = lg(roo); p1 = cgetg(l, t_VEC); for (i=1; i<l; i++) gel(p1,i) = gadd(TR, gel(roo,i)); roo = p1; fk = gel(DATA,4); l = lg(fk); for (i=1; i<l; i++) gel(fk,i) = gsub(Xm1, gel(fk,i)); bezoutC = gel(DATA,6); l = lg(bezoutC); interp = gel(DATA,9); for (i=1; i<l; i++) { if (degpol(interp[i]) > 0) /* do not turn pol_1[0] into gen_1 */ { p1 = translate_pol(gel(interp,i), gen_m1); gel(interp,i) = FpXX_red(p1, p); } if (degpol(bezoutC[i]) > 0) { p1 = translate_pol(gel(bezoutC,i), gen_m1); gel(bezoutC,i) = FpXX_red(p1, p); } } ff = cgetg(lff, t_VEC); /* copy, don't overwrite! */ for (i=1; i<lff; i++) gel(ff,i) = FpX_red(translate_pol((GEN)S->ff[i], mTR), p); } else { DATA = cgetg(10,t_VEC); fk = S->fk; gel(DATA,5) = gen_0; gel(DATA,6) = shallowcopy(S->bezoutC); gel(DATA,9) = shallowcopy(S->interp); } gel(DATA,1) = pol; MM = gmul2n(bound_for_coeff(B->d, roo, &maxroot), 1); gel(DATA,8) = MM; e = logint(shifti(vecmax(MM),20), p, &pe); /* overlift 2^20 [for d-1 test] */ gel(DATA,2) = pe; gel(DATA,4) = roots_from_deg1(fk); /* compute fhk = hensel_lift_fact(pol,fk,T,p,pe,e) in 2 steps * 1) lift in Zp to precision p^e */ ffL = hensel_lift_fact(pol, ff, NULL, p, pe, e); fhk = NULL; for (l=i=1; i<lff; i++) { /* 2) lift factorization of ff[i] in Qp[X] / T */ GEN F, L = gel(ffL,i); long di = degpol(L); F = cgetg(di+1, t_VEC); for (j=1; j<=di; j++) F[j] = fk[l++]; L = hensel_lift_fact(L, F, T, p, pe, e); fhk = fhk? shallowconcat(fhk, L): L; } gel(DATA,3) = roots_from_deg1(fhk); p1 = mulsr(N, gsqrt(gpowgs(utoipos(N-1),N-1),DEFAULTPREC)); p2 = gpowgs(maxroot, B->size + N*(N-1)/2); p1 = gdiv(gmul(p1,p2), gsqrt(B->PD->dis,DEFAULTPREC)); gel(DATA,7) = mulii(shifti(ceil_safe(p1), 1), B->PD->den); if (DEBUGLEVEL>1) { fprintferr("f = %Z\n",DATA[1]); fprintferr("p = %Z, lift to p^%ld\n", p, e); fprintferr("2 * Hadamard bound * ind = %Z\n",DATA[7]); fprintferr("2 * M = %Z\n",DATA[8]); } if (B->DATA) { DATA = gclone(DATA); if (isclone(B->DATA)) gunclone(B->DATA); } B->DATA = DATA; }