void FastTraceVec(vec_zz_p& S, const zz_pX& f)
{
long n = deg(f);
if (n <= 0)
Error("FastTraceVec: bad args");
if (n == 0) {
S.SetLength(0);
return;
}
if (n == 1) {
S.SetLength(1);
set(S[0]);
return;
}
long i;
zz_pX f1;
f1.rep.SetLength(n-1);
for (i = 0; i <= n-2; i++)
f1.rep[i] = f.rep[n-i];
f1.normalize();
zz_pX f2;
f2.rep.SetLength(n-1);
for (i = 0; i <= n-2; i++)
mul(f2.rep[i], f.rep[n-1-i], i+1);
f2.normalize();
zz_pX f3;
InvTrunc(f3, f1, n-1);
MulTrunc(f3, f3, f2, n-1);
S.SetLength(n);
S[0] = n;
for (i = 1; i < n; i++)
negate(S[i], coeff(f3, i-1));
}
static
void compute_a_vals(Vec<ZZ>& a, long p, long e)
// computes a[m] = a(m)/m! for m = p..(e-1)(p-1)+1,
// as defined by Chen and Han.
// a.length() is set to (e-1)(p-1)+2
{
ZZ p_to_e = power_ZZ(p, e);
ZZ p_to_2e = power_ZZ(p, 2*e);
long len = (e-1)*(p-1)+2;
ZZ_pPush push(p_to_2e);
ZZ_pX x_plus_1_to_p = power(ZZ_pX(INIT_MONO, 1) + 1, p);
ZZ_pX denom = InvTrunc(x_plus_1_to_p - ZZ_pX(INIT_MONO, p), len);
ZZ_pX poly = MulTrunc(x_plus_1_to_p, denom, len);
poly *= p;
a.SetLength(len);
ZZ m_fac(1);
for (long m = 2; m < p; m++) {
m_fac = MulMod(m_fac, m, p_to_2e);
}
for (long m = p; m < len; m++) {
m_fac = MulMod(m_fac, m, p_to_2e);
ZZ c = rep(coeff(poly, m));
ZZ d = GCD(m_fac, p_to_2e);
if (d == 0 || d > p_to_e || c % d != 0) Error("cannot divide");
ZZ m_fac_deflated = (m_fac / d) % p_to_e;
ZZ c_deflated = (c / d) % p_to_e;
a[m] = MulMod(c_deflated, InvMod(m_fac_deflated, p_to_e), p_to_e);
}
}
static CYTHON_INLINE struct ZZX* ZZX_multiply_and_truncate(struct ZZX* x, struct ZZX* y, long m)
{
ZZX* t = new ZZX();
MulTrunc(*t, *x, *y, m);
return t;
}