/* Warning: return L->topowden * (best lift) */
static GEN
nf_bestlift_to_pol(GEN elt, GEN bound, nflift_t *L)
{
pari_sp av = avma;
GEN v, u = nf_bestlift(elt,bound,L);
if (!u) return NULL;
v = gclone(u); avma = av;
u = gmul(L->topow, v);
gunclone(v); return u;
}
/* assume A or B is a t_LIST */
static GEN
listconcat(GEN A, GEN B)
{
long i, l1, lx;
GEN L, z, L1, L2;
if (typ(A) != t_LIST) {
if (list_typ(B)!=t_LIST_RAW) pari_err_TYPE("listconcat",B);
L2 = list_data(B);
if (!L2) return mklistcopy(A);
lx = lg(L2) + 1;
z = listcreate();
list_data(z) = L = cgetg(lx, t_VEC);
for (i = 2; i < lx; i++) gel(L,i) = gcopy(gel(L2,i-1));
gel(L,1) = gcopy(A); return z;
} else if (typ(B) != t_LIST) {
if (list_typ(A)!=t_LIST_RAW) pari_err_TYPE("listconcat",A);
L1 = list_data(A);
if (!L1) return mklistcopy(B);
lx = lg(L1) + 1;
z = listcreate();
list_data(z) = L = cgetg(lx, t_VEC);
for (i = 1; i < lx-1; i++) gel(L,i) = gcopy(gel(L1,i));
gel(L,i) = gcopy(B); return z;
}
/* A, B both t_LISTs */
if (list_typ(A)!=t_LIST_RAW) pari_err_TYPE("listconcat",A);
if (list_typ(B)!=t_LIST_RAW) pari_err_TYPE("listconcat",B);
L1 = list_data(A); if (!L1) return listcopy(B);
L2 = list_data(B); if (!L2) return listcopy(A);
l1 = lg(L1);
lx = l1-1 + lg(L2);
z = cgetg(3, t_LIST);
z[1] = 0UL;
list_data(z) = L = cgetg(lx, t_VEC);
L2 -= l1-1;
for (i=1; i<l1; i++) gel(L,i) = gclone(gel(L1,i));
for ( ; i<lx; i++) gel(L,i) = gclone(gel(L2,i));
return z;
}
/* L list of current subfields, test whether potential block D is a block,
* if so, append corresponding subfield */
static GEN
test_block(blockdata *B, GEN L, GEN D)
{
pari_sp av = avma;
GEN sub = subfield(D, B);
if (sub) {
GEN old = L;
L = gclone( L? shallowconcat(L, sub): sub );
if (old) gunclone(old);
}
avma = av; return L;
}
void
ut_slice(void)
{
useclass(Slice);
UTEST_START("Slice")
// equality and default args
UTEST( Slice_isEqual(atSlice(10) , atSlice(0,10, 1)) );
UTEST( Slice_isEqual(atSlice(0,10), atSlice(0,10, 1)) );
UTEST(!Slice_isEqual(atSlice(0,10), atSlice(0,10,-1)) );
UTEST( Slice_isEqual(atSlice(0,10), atSlice(0,10, 0)) );
UTEST( gisEqual(aSlice(10) , aSlice(0,10, 1)) == True );
UTEST( gisEqual(aSlice(0,10), aSlice(0,10, 1)) == True );
UTEST( gisEqual(aSlice(0,10), aSlice(0,10,-1)) == False );
// new vs auto
UTEST( isEq(gnewSlc(Slice, 0, 10, 1), aSlice(0,10,1)) );
UTEST(!isEq(gnewSlc(Slice, 0, 10, 1), aSlice(0,10,-1)) );
// clone vs auto
UTEST( isEq(gclone(aSlice(10)), aSlice(10)) );
UTEST( isEq(gclone(aSlice(0,10)), aSlice(0,10)) );
UTEST( isEq(gclone(aSlice(0,10,-1)), aSlice(0,10,-1)) );
// eval
UTEST( Slice_eval(atSlice(10),0) == 0 );
UTEST( Slice_eval(atSlice(10),1) == 1 );
UTEST( Slice_eval(atSlice(10),10) == 10 );
UTEST( Slice_eval(atSlice(1,10),0) == 1 );
UTEST( Slice_eval(atSlice(1,10),1) == 2 );
UTEST( Slice_eval(atSlice(1,10),10) == 11 );
UTEST( Slice_eval(atSlice(1,10,2),0) == 1 );
UTEST( Slice_eval(atSlice(1,10,2),1) == 3 );
UTEST( Slice_eval(atSlice(1,10,2),10) == 21 );
UTEST( Slice_eval(atSlice(10,1,-2),0) == 10 );
UTEST( Slice_eval(atSlice(10,1,-2),1) == 8 );
UTEST( Slice_eval(atSlice(10,1,-2),10) == -10 );
// first
UTEST( Slice_first(atSlice(10)) == 0 );
UTEST( Slice_first(atSlice(1,10)) == 1 );
UTEST( Slice_first(atSlice(-1,10)) == (U32)-1 );
UTEST( Slice_first(atSlice(-1,-10)) == (U32)-1 );
// last
UTEST( Slice_last(atSlice(10)) == 9 );
UTEST( Slice_last(atSlice(1,10)) == 10 );
UTEST( Slice_last(atSlice(1,10,2)) == 19 );
UTEST( Slice_last(atSlice(0,-10)) == (U32)-11 );
UTEST( Slice_last(atSlice(-1,-10)) == (U32)-12 );
// size
UTEST( Slice_size(atSlice(0,9,1)) == 9 );
UTEST( Slice_size(atSlice(1,10,1)) == 10 );
UTEST( Slice_size(atSlice(1,10,2)) == 10 );
UTEST( Slice_size(atSlice(1,10,3)) == 10 );
UTEST( Slice_size(atSlice(1,9,3)) == 9 );
UTEST( Slice_size(atSlice(9,0,-1)) == 0 );
UTEST( Slice_size(atSlice(10,1,-1)) == 1 );
UTEST( Slice_size(atSlice(10,1,-2)) == 1 );
UTEST( Slice_size(atSlice(10,1,-3)) == 1 );
UTEST( Slice_size(atSlice(9,1,-3)) == 1 );
// slice vs range
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,1,1), 0), atSlice(-1,3,1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,1,2), 0), atSlice(-1,2,2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,5,3), 0), atSlice(-1,3,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,4,3), 0), atSlice(-1,2,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange( 1,5,3), 0), atSlice( 1,2,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,0,-1), 0), atSlice(9,10,-1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,1,-2), 0), atSlice(9,5,-2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,0,-3), 0), atSlice(9,4,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,3,-3), 0), atSlice(9,3,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(0,9,1) , 0), atSlice(0,10,1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(1,10,1), 0), atSlice(1,10,1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(1,9,2) , 0), atSlice(1,5,2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(1,10,3), 0), atSlice(1,4,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(1,7,3) , 0), atSlice(1,3,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,0,-1) , 0), atSlice(9,10,-1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(10,1,-1), 0), atSlice(10,10,-1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(10,2,-2), 0), atSlice(10,5,-2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(10,1,-3), 0), atSlice(10,4,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(9,3,-3) , 0), atSlice(9,3,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,-10,-1), 0), atSlice(-1,10,-1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,-9,-2) , 0), atSlice(-1,5,-2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,-10,-3), 0), atSlice(-1,4,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-1,-7,-3) , 0), atSlice(-1,3,-3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-10,-1,1), 0), atSlice(-10,10,1)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-10,-2,2), 0), atSlice(-10,5,2)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-10,-1,3), 0), atSlice(-10,4,3)) );
UTEST( eq(Slice_fromRange(atSlice(0), atRange(-9,-3,3) , 0), atSlice(-9,3,3)) );
UTEST_END
}
/* 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;
}
/* 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;
}
