long ProbIrredTest(const ZZ_pEX& f, long iter)
{
long n = deg(f);
if (n <= 0) return 0;
if (n == 1) return 1;
ZZ_pEXModulus F;
build(F, f);
ZZ_pEX b, r, s;
FrobeniusMap(b, F);
long all_zero = 1;
long i;
for (i = 0; i < iter; i++) {
random(r, n);
TraceMap(s, r, n, F, b);
all_zero = all_zero && IsZero(s);
if (deg(s) > 0) return 0;
}
if (!all_zero || (n & 1)) return 1;
PowerCompose(s, b, n/2, F);
return !IsX(s);
}
static
void RecFindRoots(vec_GF2E& x, const GF2EX& f)
{
if (deg(f) == 0) return;
if (deg(f) == 1) {
long k = x.length();
x.SetLength(k+1);
x[k] = ConstTerm(f);
return;
}
GF2EX h;
GF2E r;
{
GF2EXModulus F;
build(F, f);
do {
random(r);
clear(h);
SetCoeff(h, 1, r);
TraceMap(h, h, F);
GCD(h, h, f);
} while (deg(h) <= 0 || deg(h) == deg(f));
}
RecFindRoots(x, h);
div(h, f, h);
RecFindRoots(x, h);
}
void FindRoot(GF2E& root, const GF2EX& ff)
// finds a root of ff.
// assumes that ff is monic and splits into distinct linear factors
{
GF2EXModulus F;
GF2EX h, h1, f;
GF2E r;
f = ff;
if (!IsOne(LeadCoeff(f)))
Error("FindRoot: bad args");
if (deg(f) == 0)
Error("FindRoot: bad args");
while (deg(f) > 1) {
build(F, f);
random(r);
clear(h);
SetCoeff(h, 1, r);
TraceMap(h, h, F);
GCD(h, h, f);
if (deg(h) > 0 && deg(h) < deg(f)) {
if (deg(h) > deg(f)/2)
div(f, f, h);
else
f = h;
}
}
root = ConstTerm(f);
}
void EDFSplit(vec_ZZ_pEX& v, const ZZ_pEX& f, const ZZ_pEX& b, long d)
{
ZZ_pEX a, g, h;
ZZ_pEXModulus F;
vec_ZZ_pE roots;
build(F, f);
long n = F.n;
long r = n/d;
random(a, n);
TraceMap(g, a, d, F, b);
MinPolyMod(h, g, F, r);
FindRoots(roots, h);
FindFactors(v, f, g, roots);
}
static
void EDFSplit(GF2X& f1, GF2X& f2, const GF2X& f, long d)
{
GF2X a, g;
GF2XModulus F;
build(F, f);
long n = F.n;
do {
random(a, n);
TraceMap(g, a, d, F);
} while (deg(g) <= 0);
GCD(f1, f, g);
div(f2, f, f1);
}
long ProbComputeDegree(const ZZ_pX& h, const ZZ_pXModulus& F)
{
if (F.n == 1 || IsX(h))
return 1;
long n = F.n;
ZZ_pX P1, P2, P3;
random(P1, n);
TraceMap(P2, P1, n, F, h);
ProbMinPolyMod(P3, P2, F, n/2);
long r = deg(P3);
if (r <= 0 || n % r != 0)
return 0;
else
return n/r;
}
long ProbIrredTest(const ZZ_pX& f, long iter)
{
long n = deg(f);
if (n <= 0) return 0;
if (n == 1) return 1;
const ZZ& p = ZZ_p::modulus();
ZZ_pXModulus F;
build(F, f);
ZZ_pX b, r, s;
PowerXMod(b, p, F);
long i;
for (i = 0; i < iter; i++) {
random(r, n);
TraceMap(s, r, n, F, b);
if (deg(s) > 0) return 0;
}
if (p >= n) return 1;
long pp;
conv(pp, p);
if (n % pp != 0) return 1;
PowerCompose(s, b, n/pp, F);
return !IsX(s);
}