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);
}
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
gcdii(GEN a, GEN b)
{
long v, w;
pari_sp av;
GEN t;
switch (absi_cmp(a,b))
{
case 0: return absi(a);
case -1: swap(a,b);
}
if (!signe(b)) return absi(a);
/* here |a|>|b|>0. Try single precision first */
if (lgefint(a)==3)
return igcduu((ulong)a[2], (ulong)b[2]);
if (lgefint(b)==3)
{
ulong u = resiu(a,(ulong)b[2]);
if (!u) return absi(b);
return igcduu((ulong)b[2], u);
}
/* larger than gcd: "avma=av" gerepile (erasing t) is valid */
av = avma; (void)new_chunk(lgefint(b)+1); /* HACK */
t = remii(a,b);
if (!signe(t)) { avma=av; return absi(b); }
a = b; b = t;
v = vali(a); a = shifti(a,-v); setabssign(a);
w = vali(b); b = shifti(b,-w); setabssign(b);
if (w < v) v = w;
switch(absi_cmp(a,b))
{
case 0: avma=av; a=shifti(a,v); return a;
case -1: swap(a,b);
}
if (is_pm1(b)) { avma=av; return int2n(v); }
{
/* general case */
/*This serve two purposes: 1) mpn_gcd destroy its input and need an extra
* limb 2) this allows us to use icopy instead of gerepile later. NOTE: we
* must put u before d else the final icopy could fail.
*/
GEN res= cgeti(lgefint(a)+1);
GEN ca = icopy_ef(a,lgefint(a)+1);
GEN cb = icopy_ef(b,lgefint(b)+1);
long l = mpn_gcd(LIMBS(res), LIMBS(ca), NLIMBS(ca), LIMBS(cb), NLIMBS(cb));
res[1] = evalsigne(1)|evallgefint(l+2);
avma=av;
return shifti(res,v);
}
}
static GEN
nf_chk_factors(nfcmbf_t *T, GEN P, GEN M_L, GEN famod, GEN pk)
{
GEN nf = T->nf, bound = T->bound;
GEN nfT = gel(nf,1);
long i, r;
GEN pol = P, list, piv, y;
GEN C2ltpol, C = T->L->topowden, Tpk = T->L->Tpk;
GEN lc = absi(leading_term(pol)), lt = is_pm1(lc)? NULL: lc;
GEN Clt = mul_content(C, lt);
GEN C2lt = mul_content(C,Clt);
piv = special_pivot(M_L);
if (!piv) return NULL;
if (DEBUGLEVEL>3) fprintferr("special_pivot output:\n%Z\n",piv);
r = lg(piv)-1;
list = cgetg(r+1, t_COL);
C2ltpol = C2lt? gmul(C2lt,pol): pol;
for (i = 1;;)
{
pari_sp av = avma;
if (DEBUGLEVEL)
fprintferr("nf_LLL_cmbf: checking factor %ld (avma - bot = %lu)\n",
i, avma - bot);
y = chk_factors_get(lt, famod, gel(piv,i), Tpk, pk);
if (DEBUGLEVEL>2) fprintferr("... mod p^k (avma - bot = %lu)\n", avma-bot);
if (! (y = nf_pol_lift(y, bound, T)) ) return NULL;
if (DEBUGLEVEL>2) fprintferr("... lifted (avma - bot = %lu)\n", avma-bot);
y = gerepilecopy(av, y);
/* y is the candidate factor */
pol = RgXQX_divrem(C2ltpol, y, nfT, ONLY_DIVIDES);
if (!pol) return NULL;
y = Q_primpart(y);
gel(list,i) = QXQX_normalize(y, nfT);
if (++i >= r) break;
if (C2lt) pol = Q_primpart(pol);
if (lt) lt = absi(leading_term(pol));
Clt = mul_content(C, lt);
C2lt = mul_content(C,Clt);
C2ltpol = C2lt? gmul(C2lt,pol): pol;
}
y = Q_primpart(pol);
gel(list,i) = QXQX_normalize(y, nfT); return list;
}
/* Naive recombination of modular factors: combine up to maxK modular
* factors, degree <= klim and divisible by hint
*
* target = polynomial we want to factor
* famod = array of modular factors. Product should be congruent to
* target/lc(target) modulo p^a
* For true factors: S1,S2 <= p^b, with b <= a and p^(b-a) < 2^31 */
static GEN
nfcmbf(nfcmbf_t *T, GEN p, long a, long maxK, long klim)
{
GEN pol = T->pol, nf = T->nf, famod = T->fact, dn = T->dn;
GEN bound = T->bound;
GEN nfpol = gel(nf,1);
long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1, dnf = degpol(nfpol);
GEN res = cgetg(3, t_VEC);
pari_sp av0 = avma;
GEN pk = gpowgs(p,a), pks2 = shifti(pk,-1);
GEN ind = cgetg(lfamod+1, t_VECSMALL);
GEN degpol = cgetg(lfamod+1, t_VECSMALL);
GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
GEN listmod = cgetg(lfamod+1, t_COL);
GEN fa = cgetg(lfamod+1, t_COL);
GEN lc = absi(leading_term(pol)), lt = is_pm1(lc)? NULL: lc;
GEN C2ltpol, C = T->L->topowden, Tpk = T->L->Tpk;
GEN Clt = mul_content(C, lt);
GEN C2lt = mul_content(C,Clt);
const double Bhigh = get_Bhigh(lfamod, dnf);
trace_data _T1, _T2, *T1, *T2;
pari_timer ti;
TIMERstart(&ti);
if (maxK < 0) maxK = lfamod-1;
C2ltpol = C2lt? gmul(C2lt,pol): pol;
{
GEN q = ceil_safe(sqrtr(T->BS_2));
GEN t1,t2, ltdn, lt2dn;
GEN trace1 = cgetg(lfamod+1, t_MAT);
GEN trace2 = cgetg(lfamod+1, t_MAT);
ltdn = mul_content(lt, dn);
lt2dn= mul_content(ltdn, lt);
for (i=1; i <= lfamod; i++)
{
pari_sp av = avma;
GEN P = gel(famod,i);
long d = degpol(P);
degpol[i] = d; P += 2;
t1 = gel(P,d-1);/* = - S_1 */
t2 = gsqr(t1);
if (d > 1) t2 = gsub(t2, gmul2n(gel(P,d-2), 1));
/* t2 = S_2 Newton sum */
t2 = typ(t2)!=t_INT? FpX_rem(t2, Tpk, pk): modii(t2, pk);
if (lt)
{
if (typ(t2)!=t_INT) {
t1 = FpX_red(gmul(ltdn, t1), pk);
t2 = FpX_red(gmul(lt2dn,t2), pk);
} else {
t1 = remii(mulii(ltdn, t1), pk);
t2 = remii(mulii(lt2dn,t2), pk);
}
}
gel(trace1,i) = gclone( nf_bestlift(t1, NULL, T->L) );
gel(trace2,i) = gclone( nf_bestlift(t2, NULL, T->L) ); avma = av;
}
T1 = init_trace(&_T1, trace1, T->L, q);
T2 = init_trace(&_T2, trace2, T->L, q);
for (i=1; i <= lfamod; i++) {
gunclone(gel(trace1,i));
gunclone(gel(trace2,i));
}
}
degsofar[0] = 0; /* sentinel */
/* ind runs through strictly increasing sequences of length K,
* 1 <= ind[i] <= lfamod */
nextK:
if (K > maxK || 2*K > lfamod) goto END;
if (DEBUGLEVEL > 3)
fprintferr("\n### K = %d, %Z combinations\n", K,binomial(utoipos(lfamod), K));
setlg(ind, K+1); ind[1] = 1;
i = 1; curdeg = degpol[ind[1]];
for(;;)
{ /* try all combinations of K factors */
for (j = i; j < K; j++)
{
degsofar[j] = curdeg;
ind[j+1] = ind[j]+1; curdeg += degpol[ind[j+1]];
}
if (curdeg <= klim && curdeg % T->hint == 0) /* trial divide */
{
GEN t, y, q, list;
pari_sp av;
av = avma;
/* d - 1 test */
if (T1)
{
t = get_trace(ind, T1);
if (rtodbl(QuickNormL2(t,DEFAULTPREC)) > Bhigh)
{
if (DEBUGLEVEL>6) fprintferr(".");
avma = av; goto NEXT;
}
}
/* d - 2 test */
if (T2)
{
t = get_trace(ind, T2);
if (rtodbl(QuickNormL2(t,DEFAULTPREC)) > Bhigh)
{
if (DEBUGLEVEL>3) fprintferr("|");
avma = av; goto NEXT;
}
}
avma = av;
y = lt; /* full computation */
for (i=1; i<=K; i++)
{
GEN q = gel(famod, ind[i]);
if (y) q = gmul(y, q);
y = FqX_centermod(q, Tpk, pk, pks2);
}
y = nf_pol_lift(y, bound, T);
if (!y)
{
if (DEBUGLEVEL>3) fprintferr("@");
avma = av; goto NEXT;
}
/* try out the new combination: y is the candidate factor */
q = RgXQX_divrem(C2ltpol, y, nfpol, ONLY_DIVIDES);
if (!q)
{
if (DEBUGLEVEL>3) fprintferr("*");
avma = av; goto NEXT;
}
/* found a factor */
list = cgetg(K+1, t_VEC);
gel(listmod,cnt) = list;
for (i=1; i<=K; i++) list[i] = famod[ind[i]];
y = Q_primpart(y);
gel(fa,cnt++) = QXQX_normalize(y, nfpol);
/* fix up pol */
pol = q;
for (i=j=k=1; i <= lfamod; i++)
{ /* remove used factors */
if (j <= K && i == ind[j]) j++;
else
{
famod[k] = famod[i];
update_trace(T1, k, i);
update_trace(T2, k, i);
degpol[k] = degpol[i]; k++;
}
}
lfamod -= K;
if (lfamod < 2*K) goto END;
i = 1; curdeg = degpol[ind[1]];
if (C2lt) pol = Q_primpart(pol);
if (lt) lt = absi(leading_term(pol));
Clt = mul_content(C, lt);
C2lt = mul_content(C,Clt);
C2ltpol = C2lt? gmul(C2lt,pol): pol;
if (DEBUGLEVEL > 2)
{
fprintferr("\n"); msgTIMER(&ti, "to find factor %Z",y);
fprintferr("remaining modular factor(s): %ld\n", lfamod);
}
continue;
}
NEXT:
for (i = K+1;;)
{
if (--i == 0) { K++; goto nextK; }
if (++ind[i] <= lfamod - K + i)
{
curdeg = degsofar[i-1] + degpol[ind[i]];
if (curdeg <= klim) break;
}
}
}
END:
if (degpol(pol) > 0)
{ /* leftover factor */
if (signe(leading_term(pol)) < 0) pol = gneg_i(pol);
if (C2lt && lfamod < 2*K) pol = QXQX_normalize(Q_primpart(pol), nfpol);
setlg(famod, lfamod+1);
gel(listmod,cnt) = shallowcopy(famod);
gel(fa,cnt++) = pol;
}
if (DEBUGLEVEL>6) fprintferr("\n");
if (cnt == 2) {
avma = av0;
gel(res,1) = mkvec(T->pol);
gel(res,2) = mkvec(T->fact);
}
else
{
setlg(listmod, cnt); setlg(fa, cnt);
gel(res,1) = fa;
gel(res,2) = listmod;
res = gerepilecopy(av0, res);
}
return res;
}
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;
}
