GEN
F2xqE_sub(GEN P, GEN Q, GEN a2, GEN T)
{
pari_sp av = avma;
GEN slope;
return gerepileupto(av, F2xqE_add_slope(P, F2xqE_neg_i(Q, a2), a2, T, &slope));
}
GEN
bnflogdegree(GEN nf, GEN A, GEN ell)
{
pari_sp av = avma;
GEN AZ, A0Z, NA0;
long vAZ;
if (typ(ell) != t_INT) pari_err_TYPE("bnflogdegree", ell);
nf = checknf(nf);
A = idealhnf(nf, A);
AZ = gcoeff(A,1,1);
vAZ = Z_pvalrem(AZ, ell, &A0Z);
if (is_pm1(A0Z))
NA0 = gen_1;
else
(void)Z_pvalrem(idealnorm(nf,A), ell, &NA0);
if (vAZ)
{
GEN Aell = ZM_hnfmodid(A, powiu(ell,vAZ));
GEN S = idealprimedec(nf, ell), T;
long l, i, s = 0;
T = padicfact(nf, S, 100);
l = lg(S);
for (i = 1; i < l; i++)
{
GEN P = gel(S,i);
long v = idealval(nf, Aell, P);
if (v) s += v * ftilde(nf, P, gel(T,i));
}
if (s) NA0 = gmul(NA0, gpowgs(ell1(ell), s));
}
return gerepileupto(av, NA0);
}
GEN
F2xqE_add(GEN P, GEN Q, GEN a, GEN T)
{
pari_sp av = avma;
GEN slope;
return gerepileupto(av, F2xqE_add_slope(P, Q, a, T, &slope));
}
GEN
F2xqE_dbl(GEN P, GEN a, GEN T)
{
pari_sp av = avma;
GEN slope;
return gerepileupto(av, F2xqE_dbl_slope(P, a, T,&slope));
}
GEN
readseq(char *t)
{
pari_sp av = avma;
GEN x;
if (gp_meta(t,0)) return gnil;
x = pari_compile_str(t);
return gerepileupto(av, closure_evalres(x));
}
static GEN _can_lin(void *E, GEN F, GEN V, long N)
{
pari_sp av=avma;
GEN d0, d1, z;
(void) E;
RgX_even_odd(V, &d0, &d1);
z = ZX_sub(V, ZX_sub(ZX_mul(gel(F,1), d0), ZX_mul(gel(F,2), d1)));
return gerepileupto(av, ZX_remi2n(z, N));
}
static GEN
_F2xqE_mul(void *E, GEN P, GEN n)
{
pari_sp av = avma;
struct _F2xqE *e=(struct _F2xqE *) E;
long s = signe(n);
if (!s || ell_is_inf(P)) return ellinf();
if (s<0) P = F2xqE_neg(P, e->a2, e->T);
if (is_pm1(n)) return s>0? gcopy(P): P;
return gerepileupto(av, gen_pow(P, n, e, &_F2xqE_dbl, &_F2xqE_add));
}
GEN
nfrootsall_and_pr(GEN nf, GEN pol)
{
GEN J1,J2, pr, T;
pari_sp av = avma;
GEN z = gerepileupto(av, nfsqff(checknf(nf), pol, 2));
if (lg(z) == 1) return NULL;
(void)nf_pick_prime(1, nf, unifpol(nf, pol, t_COL), 2,
&J1, &J2, &pr, &T);
return mkvec2(z, pr);
}
static void
bestlift_init(long a, GEN nf, GEN pr, GEN C, nflift_t *L)
{
const long D = 100;
const double alpha = ((double)D-1) / D; /* LLL parameter */
const long d = degpol(nf[1]);
pari_sp av = avma;
GEN prk, PRK, B, GSmin, pk;
pari_timer ti;
TIMERstart(&ti);
if (!a) a = (long)bestlift_bound(C, d, alpha, pr_norm(pr));
for (;; avma = av, a<<=1)
{
if (DEBUGLEVEL>2) fprintferr("exponent: %ld\n",a);
PRK = prk = idealpows(nf, pr, a);
pk = gcoeff(prk,1,1);
/* reduce size first, "scramble" matrix */
PRK = lllintpartial_ip(PRK);
/* now floating point reduction is fast */
PRK = lllint_fp_ip(PRK, 4);
PRK = lllint_i(PRK, D, 0, NULL, NULL, &B);
if (!PRK) { PRK = prk; GSmin = pk; } /* nf = Q */
else
{
pari_sp av2 = avma;
GEN S = invmat( get_R(PRK) ), BB = GS_norms(B, DEFAULTPREC);
GEN smax = gen_0;
long i, j;
for (i=1; i<=d; i++)
{
GEN s = gen_0;
for (j=1; j<=d; j++)
s = gadd(s, gdiv( gsqr(gcoeff(S,i,j)), gel(BB,j)));
if (gcmp(s, smax) > 0) smax = s;
}
GSmin = gerepileupto(av2, ginv(gmul2n(smax, 2)));
}
if (gcmp(GSmin, C) >= 0) break;
}
if (DEBUGLEVEL>2)
fprintferr("for this exponent, GSmin = %Z\nTime reduction: %ld\n",
GSmin, TIMER(&ti));
L->k = a;
L->den = L->pk = pk;
L->prk = PRK;
L->iprk = ZM_inv(PRK, pk);
L->GSmin= GSmin;
L->prkHNF = prk;
init_proj(L, gel(nf,1), gel(pr,1));
}
static GEN
nf_factor_bound(GEN nf, GEN polbase)
{
pari_sp av = avma;
GEN a = nf_Mignotte_bound(nf, polbase);
GEN b = nf_Beauzamy_bound(nf, polbase);
if (DEBUGLEVEL>2)
{
fprintferr("Mignotte bound: %Z\n",a);
fprintferr("Beauzamy bound: %Z\n",b);
}
return gerepileupto(av, gmin(a, b));
}
/* return the characteristic polynomial of alpha over nf, where alpha
is an element of the algebra nf[X]/(T) given as a polynomial in X */
GEN
rnfcharpoly(GEN nf, GEN T, GEN alpha, long v)
{
long vnf, vT, lT;
pari_sp av = avma;
GEN p1;
nf=checknf(nf); vnf = varn(nf[1]);
if (v<0) v = 0;
T = fix_relative_pol(nf,T,1);
if (typ(alpha) == t_POLMOD) alpha = lift_to_pol(alpha);
lT = lg(T);
if (typ(alpha) != t_POL || varn(alpha) == vnf)
return gerepileupto(av, gpowgs(gsub(pol_x[v], alpha), lT - 3));
vT = varn(T);
if (varn(alpha) != vT || varncmp(v, vnf)>=0)
pari_err(talker,"incorrect variables in rnfcharpoly");
if (lg(alpha) >= lT) alpha = RgX_rem(alpha, T);
if (lT <= 4)
return gerepileupto(av, gsub(pol_x[v], alpha));
p1 = caract2(T, unifpol(nf,alpha, t_POLMOD), v);
return gerepileupto(av, unifpol(nf, p1, t_POLMOD));
}
GEN
vecsum(GEN v)
{
pari_sp av = avma;
long i, l;
GEN p;
if (!is_vec_t(typ(v)))
pari_err_TYPE("vecsum", v);
l = lg(v);
if (l == 1) return gen_0;
p = gel(v,1);
if (l == 2) return gcopy(p);
for (i=2; i<l; i++)
{
p = gadd(p, gel(v,i));
if (gc_needed(av, 2))
{
if (DEBUGMEM>1) pari_warn(warnmem,"sum");
p = gerepileupto(av, p);
}
}
return gerepileupto(av, p);
}
static GEN
_powpolmodsimple(Cache *C, Red *R, GEN jac)
{
pari_sp av = avma;
GEN w = mulmat_pol(C->matvite, jac);
long j, ph = lg(w);
R->red = &_redsimple;
for (j=1; j<ph; j++)
gel(w,j) = _powpolmod(C, centermodii(gel(w,j), R->N, R->N2), R, &sqrmod);
w = centermod_i( gmul(C->matinvvite, w), R->N, R->N2 );
w = gerepileupto(av, w);
return RgV_to_RgX(w, 0);
}
GEN
F2xqE_weilpairing(GEN P, GEN Q, GEN m, GEN a2, GEN T)
{
pari_sp ltop = avma;
GEN num, denom, result;
if (ell_is_inf(P) || ell_is_inf(Q) || F2x_equal(P,Q))
return pol1_F2x(T[1]);
num = F2xqE_Miller(P, Q, m, a2, T);
if (!num) return pol1_F2x(T[1]);
denom = F2xqE_Miller(Q, P, m, a2, T);
if (!denom) { avma=ltop; return pol1_F2x(T[1]); }
result = F2xq_div(num, denom, T);
return gerepileupto(ltop, result);
}
static GEN
F2xqE_Miller(GEN Q, GEN P, GEN m, GEN a2, GEN T)
{
pari_sp ltop = avma;
struct _F2xqE_miller d;
GEN v, num, denom, g1;
d.a2 = a2; d.T = T; d.P = P;
g1 = pol1_F2x(T[1]);
v = gen_pow(mkvec3(g1,g1,Q), m, (void*)&d, F2xqE_Miller_dbl, F2xqE_Miller_add);
num = gel(v,1); denom = gel(v,2);
if (!lgpol(num) || !lgpol(denom)) { avma = ltop; return NULL; }
return gerepileupto(ltop, F2xq_div(num, denom, T));
}
static GEN
vecgroup_idxlist(GEN L, long order)
{
pari_sp ltop=avma;
GEN V;
long i,j,n;
for (n=0,i=1; i<lg(L); i++)
if (group_order(gel(L,i))==order)
n++;
V=cgetg(n+1,t_VECSMALL);
for(i=1,j=1; j<=n; i++)
if (group_order(gel(L,i))==order)
V[j++]=group_ident(gel(L,i),NULL);
return gerepileupto(ltop,vecsmall_uniq(V));
}
GEN
F2xqE_changepoint(GEN x, GEN ch, GEN T)
{
pari_sp av = avma;
GEN p1,z,u,r,s,t,v,v2,v3;
if (ell_is_inf(x)) return x;
u = gel(ch,1); r = gel(ch,2);
s = gel(ch,3); t = gel(ch,4);
v = F2xq_inv(u, T); v2 = F2xq_sqr(v, T); v3 = F2xq_mul(v,v2, T);
p1 = F2x_add(gel(x,1),r);
z = cgetg(3,t_VEC);
gel(z,1) = F2xq_mul(v2, p1, T);
gel(z,2) = F2xq_mul(v3, F2x_add(gel(x,2), F2x_add(F2xq_mul(s, p1, T),t)), T);
return gerepileupto(av, z);
}
static GEN
gen_Z2x_Dixon(GEN F, GEN V, long N, void *E, GEN lin(void *E, GEN F, GEN d, long N), GEN invl(void *E, GEN d))
{
pari_sp av = avma;
long N2, M;
GEN VN2, V2, VM, bil;
ulong q = 1UL<<N;
if (N == 1) return invl(E, V);
V = Flx_red(V, q);
N2 = (N + 1)>>1; M = N - N2;
F = FlxT_red(F, q);
VN2 = gen_Z2x_Dixon(F, V, N2, E, lin, invl);
bil = lin(E, F, VN2, N);
V2 = Z2x_rshift(Flx_sub(V, bil, q), N2);
VM = gen_Z2x_Dixon(F, V2, M, E, lin, invl);
return gerepileupto(av, Flx_add(VN2, Flx_Fl_mul(VM, 1UL<<N2, q), q));
}
/* 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
gen_Z2X_Dixon(GEN F, GEN V, long N, void *E,
GEN lin(void *E, GEN F, GEN d, long N),
GEN lins(void *E, GEN F, GEN d, long N),
GEN invls(void *E, GEN d))
{
pari_sp av = avma;
long n, m;
GEN Xn, Xm, FXn, Vm;
if (N<BITS_IN_LONG)
{
ulong q = 1UL<<N;
return Flx_to_ZX(gen_Z2x_Dixon(ZXT_to_FlxT(F,q), ZX_to_Flx(V,q),N,E,lins,invls));
}
V = ZX_remi2n(V, N);
n = (N + 1)>>1; m = N - n;
F = ZXT_remi2n(F, N);
Xn = gen_Z2X_Dixon(F, V, n, E, lin, lins, invls);
FXn = lin(E, F, Xn, N);
Vm = ZX_shifti(ZX_sub(V, FXn), -n);
Xm = gen_Z2X_Dixon(F, Vm, m, E, lin, lins, invls);
return gerepileupto(av, ZX_remi2n(ZX_add(Xn, ZX_shifti(Xm, n)), N));
}
/* return the roots of pol in nf */
GEN
nfroots(GEN nf,GEN pol)
{
pari_sp av = avma;
GEN A,g, T;
long d;
if (!nf) return nfrootsQ(pol);
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nfroots");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nfroots");
d = degpol(pol);
if (d == 0) return cgetg(1,t_VEC);
if (d == 1)
{
A = gneg_i(gdiv(gel(pol,2),gel(pol,3)));
return gerepilecopy(av, mkvec( basistoalg(nf,A) ));
}
A = fix_relative_pol(nf,pol,0);
A = Q_primpart( lift_intern(A) );
if (DEBUGLEVEL>3) fprintferr("test if polynomial is square-free\n");
g = nfgcd(A, derivpol(A), T, gel(nf,4));
if (degpol(g))
{ /* not squarefree */
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g,T);
}
A = QXQX_normalize(A, T);
A = Q_primpart(A);
A = nfsqff(nf,A,1);
A = RgXQV_to_mod(A, T);
return gerepileupto(av, gen_sort(A, 0, cmp_pol));
}
/* return the factorization of x in nf */
GEN
nffactor(GEN nf,GEN pol)
{
GEN A,g,y,p1,T, rep = cgetg(3, t_MAT);
long l, j, dA;
pari_sp av = avma;
pari_timer ti;
if (DEBUGLEVEL>2) { TIMERstart(&ti); fprintferr("\nEntering nffactor:\n"); }
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nffactor");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactor");
A = fix_relative_pol(nf,pol,0);
dA = degpol(A);
if (dA <= 0) {
avma = (pari_sp)(rep + 3);
return dA == 0? trivfact(): zerofact(varn(pol));
}
A = Q_primpart( QXQX_normalize(A, T) );
if (dA == 1) {
GEN c;
A = gerepilecopy(av, A); c = gel(A,2);
if (typ(c) == t_POL && degpol(c) > 0) gel(A,2) = mkpolmod(c, gcopy(T));
gel(rep,1) = mkcol(A);
gel(rep,2) = mkcol(gen_1); return rep;
}
if (degpol(T) == 1)
return gerepileupto(av, factpol(Q_primpart(simplify(pol)), 0));
A = Q_primpart( lift_intern(A) );
g = nfgcd(A, derivpol(A), T, gel(nf,4));
A = QXQX_normalize(A, T);
A = Q_primpart(A);
if (DEBUGLEVEL>2) msgTIMER(&ti, "squarefree test");
if (degpol(g))
{ /* not squarefree */
pari_sp av1;
GEN ex;
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g, T);
y = nfsqff(nf,A,0); av1 = avma;
l = lg(y);
ex=(GEN)gpmalloc(l * sizeof(long));
for (j=l-1; j>=1; j--)
{
GEN fact = lift(gel(y,j)), quo = g, q;
long e = 0;
for(e = 1;; e++)
{
q = RgXQX_divrem(quo,fact,T, ONLY_DIVIDES);
if (!q) break;
quo = q;
}
ex[j] = e;
}
avma = av1; y = gerepileupto(av, RgXQXV_to_mod(y,T));
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = utoipos(ex[j]);
free(ex);
}
else
{
y = gerepileupto(av, RgXQXV_to_mod(nfsqff(nf,A,0), T));
l = lg(y);
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = gen_1;
}
if (DEBUGLEVEL>3)
fprintferr("number of factor(s) found: %ld\n", lg(y)-1);
gel(rep,1) = y;
gel(rep,2) = p1; return sort_factor(rep, cmp_pol);
}
