// factor n into a*b using Pollard Rho method
// pre-condition: n>1
inline void PollardRho(ZZ& a, ZZ& b, const ZZ& n,
const ZZ& bnd=ZZ::zero()) {
ZZ_pBak bak;
bak.save();
ZZ_p::init(n);
ZZ d;
ZZ_p x1;
random(x1);
ZZ_p x2(x1);
ZZ end(IsZero(bnd)?5*SqrRoot(SqrRoot(n)):2*SqrRoot(bnd));
for (; !IsZero(end); --end) {
x1 = x1*x1 + 1;
x2 = x2*x2 + 1;
x2 = x2*x2 + 1;
GCD(d,n,rep(x2-x1));
if ((d>1)&&(d<n)) {
a=d; b=n/d;
return;
}
}
// failure
a=1; b=n;
}
void ECM(ZZ& q, const ZZ& N, long ncurves, long B1, long B2, long D,
bool verbose) {
// initialize ZZ_p
ZZ_pBak bak;
bak.save();
ZZ_p::init(N);
PrimeSeq seq;
for (long i=0; i<ncurves; ++i) {
if (verbose) {
if (ncurves<NTL_MAX_LONG)
std::cout<<"ECM: "<<NumBits(N)<<" bits; "
<<i<<"/"<<ncurves<<" curves\r"<<std::flush;
else
std::cout<<"ECM: "<<NumBits(N)<<" bits; "
<<i<<" curves\r"<<std::flush;
}
// try to find a factor
ECM_one_curve(q,seq,B1,B2,D);
if (!IsOne(q)) {
if (verbose) {
std::cout<<"ECM: "<<NumBits(N)<<" bits; "
<<(i+1)<<" curves; found "<<q<<" "<<std::endl;
}
return;
}
}
if (verbose)
std::cout<<"ECM: "<<NumBits(N)<<" bits; FAILED! "<<std::endl;
set(q);
}
void inv(ZZ& d_out, mat_ZZ& x_out, const mat_ZZ& A, long deterministic)
{
long n = A.NumRows();
if (A.NumCols() != n)
Error("solve: nonsquare matrix");
if (n == 0) {
set(d_out);
x_out.SetDims(0, 0);
return;
}
zz_pBak zbak;
zbak.save();
ZZ_pBak Zbak;
Zbak.save();
mat_ZZ x(INIT_SIZE, n, n);
ZZ d, d1;
ZZ d_prod, x_prod;
set(d_prod);
set(x_prod);
long d_instable = 1;
long x_instable = 1;
long gp_cnt = 0;
long check = 0;
mat_ZZ y;
long i;
long bound = 2+DetBound(A);
for (i = 0; ; i++) {
if ((check || IsZero(d)) && !d_instable) {
if (NumBits(d_prod) > bound) {
break;
}
else if (!deterministic &&
bound > 1000 && NumBits(d_prod) < 0.25*bound) {
ZZ P;
long plen = 90 + NumBits(max(bound, NumBits(d)));
GenPrime(P, plen, 90 + 2*NumBits(gp_cnt++));
ZZ_p::init(P);
mat_ZZ_p AA;
conv(AA, A);
ZZ_p dd;
determinant(dd, AA);
if (CRT(d, d_prod, rep(dd), P))
d_instable = 1;
else
break;
}
}
zz_p::FFTInit(i);
long p = zz_p::modulus();
mat_zz_p AA;
conv(AA, A);
if (!check) {
mat_zz_p xx;
zz_p dd;
inv(dd, xx, AA);
d_instable = CRT(d, d_prod, rep(dd), p);
if (!IsZero(dd)) {
mul(xx, xx, dd);
x_instable = CRT(x, x_prod, xx);
}
else
x_instable = 1;
if (!d_instable && !x_instable) {
mul(y, x, A);
if (IsDiag(y, n, d)) {
d1 = d;
check = 1;
}
}
}
else {
zz_p dd;
determinant(dd, AA);
d_instable = CRT(d, d_prod, rep(dd), p);
}
}
if (check && d1 != d) {
mul(x, x, d);
ExactDiv(x, d1);
}
d_out = d;
if (check) x_out = x;
zbak.restore();
Zbak.restore();
}
void determinant(ZZ& rres, const mat_ZZ& a, long deterministic)
{
long n = a.NumRows();
if (a.NumCols() != n)
Error("determinant: nonsquare matrix");
if (n == 0) {
set(rres);
return;
}
zz_pBak zbak;
zbak.save();
ZZ_pBak Zbak;
Zbak.save();
long instable = 1;
long gp_cnt = 0;
long bound = 2+DetBound(a);
ZZ res, prod;
clear(res);
set(prod);
long i;
for (i = 0; ; i++) {
if (NumBits(prod) > bound)
break;
if (!deterministic &&
!instable && bound > 1000 && NumBits(prod) < 0.25*bound) {
ZZ P;
long plen = 90 + NumBits(max(bound, NumBits(res)));
GenPrime(P, plen, 90 + 2*NumBits(gp_cnt++));
ZZ_p::init(P);
mat_ZZ_p A;
conv(A, a);
ZZ_p t;
determinant(t, A);
if (CRT(res, prod, rep(t), P))
instable = 1;
else
break;
}
zz_p::FFTInit(i);
long p = zz_p::modulus();
mat_zz_p A;
conv(A, a);
zz_p t;
determinant(t, A);
instable = CRT(res, prod, rep(t), p);
}
rres = res;
zbak.restore();
Zbak.restore();
}