/* 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;
}
