// References : Cohen H., A course in computational algebraic number theory
// (1996), page 25.
bool multiplicative_order(const Ptr<RCP<const Integer>> &o,
const RCP<const Integer> &a,
const RCP<const Integer> &n)
{
integer_class order, p, t;
integer_class _a = a->as_integer_class(),
_n = mp_abs(n->as_integer_class());
mp_gcd(t, _a, _n);
if (t != 1)
return false;
RCP<const Integer> lambda = carmichael(n);
map_integer_uint prime_mul;
prime_factor_multiplicities(prime_mul, *lambda);
_a %= _n;
order = lambda->as_integer_class();
for (const auto it : prime_mul) {
p = it.first->as_integer_class();
mp_pow_ui(t, p, it.second);
mp_divexact(order, order, t);
mp_powm(t, _a, order, _n);
while (t != 1) {
mp_powm(t, t, p, _n);
order *= p;
}
}
*o = integer(std::move(order));
return true;
}
RCP<const Integer> carmichael(const RCP<const Integer> &n)
{
if (n->is_zero())
return integer(1);
map_integer_uint prime_mul;
integer_class lambda, t, p;
unsigned multiplicity;
prime_factor_multiplicities(prime_mul, *n);
lambda = 1;
for (const auto it : prime_mul) {
p = it.first->as_integer_class();
multiplicity = it.second;
if (p == 2
and multiplicity
> 2) { // For powers of 2 greater than 4 divide by 2.
multiplicity--;
}
t = p - 1;
mp_lcm(lambda, lambda, t);
mp_pow_ui(t, p, multiplicity - 1);
// lambda and p are relatively prime.
lambda = lambda * t;
}
return integer(std::move(lambda));
}
integer_class UnivariateIntPolynomial::eval(const integer_class &x) const
{
unsigned int last_deg = dict_.rbegin()->first;
integer_class result(0), x_pow;
for (auto it = dict_.rbegin(); it != dict_.rend(); ++it) {
mp_pow_ui(x_pow, x, last_deg - (*it).first);
last_deg = (*it).first;
result = (*it).second + x_pow * result;
}
mp_pow_ui(x_pow, x, last_deg);
result *= x_pow;
return result;
}
RCP<const Number> harmonic(unsigned long n, long m)
{
rational_class res(0);
if (m == 1) {
for (unsigned i = 1; i <= n; ++i) {
res += rational_class(1, i);
}
return Rational::from_mpq(res);
} else {
for (unsigned i = 1; i <= n; ++i) {
if (m > 0) {
rational_class t(1u, i);
mp_pow_ui(get_den(t), get_den(t), m);
res += t;
} else {
integer_class t(i);
mp_pow_ui(t, t, -m);
res += t;
}
}
return Rational::from_mpq(res);
}
}
integer_class MIntPoly::eval(
std::map<RCP<const Basic>, integer_class, RCPBasicKeyLess> &vals) const
{
// TODO : handle missing values
integer_class ans(0), temp, term;
for (auto bucket : poly_.dict_) {
term = bucket.second;
unsigned int whichvar = 0;
for (auto sym : vars_) {
mp_pow_ui(temp, vals.find(sym)->second, bucket.first[whichvar]);
term *= temp;
whichvar++;
}
ans += term;
}
return ans;
}
