#line 1 "ModularArithmetic/operations.cpp"
/**
* @brief 合同算術の基本演算
* @author えびちゃん
*/
#include <stdexcept>
#include <vector>
#line 1 "integer/bit.cpp"
/**
* @brief ビット演算
* @author えびちゃん
*/
// XXX integral promotion 関連の注意をあまりしていません
#include <climits>
#include <type_traits>
template <typename Tp>
constexpr auto countl_zero(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{
using value_type = typename std::make_unsigned<Tp>::type;
int bits = (sizeof(value_type) * CHAR_BIT);
if (n == 0) return bits;
int res = 0;
for (int i = bits / 2; i > 0; i /= 2) {
value_type mask = ((static_cast<value_type>(1) << i) - 1) << i;
if (n & mask) n >>= i;
else res += i;
}
return res;
}
template <typename Tp>
constexpr auto countl_one(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{
using value_type = typename std::make_unsigned<Tp>::type;
return countl_zero(static_cast<value_type>(~n));
}
template <typename Tp>
constexpr auto countr_zero(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{
using value_type = typename std::make_unsigned<Tp>::type;
int bits = (sizeof(value_type) * CHAR_BIT);
if (n == 0) return bits;
int res = 0;
for (int i = bits / 2; i > 0; i /= 2) {
value_type mask = ((static_cast<value_type>(1) << i) - 1);
if (!(n & mask)) res += i, n >>= i;
}
return res;
}
template <typename Tp>
constexpr auto countr_one(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{
using value_type = typename std::make_unsigned<Tp>::type;
return countr_zero(static_cast<value_type>(~n));
}
constexpr unsigned long long half_mask[] = {
0x5555555555555555uLL, 0x3333333333333333uLL, 0x0F0F0F0F0F0F0F0FuLL,
0x00FF00FF00FF00FFuLL, 0x0000FFFF0000FFFFuLL, 0x00000000FFFFFFFFuLL
};
template <typename Tp>
constexpr auto popcount(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{
int bits = static_cast<int>((sizeof n) * CHAR_BIT);
for (int i = 0, j = 1; j < bits; ++i, j *= 2) {
if (j <= 8) n = (n & half_mask[i]) + ((n >> j) & half_mask[i]);
else n += n >> j;
}
return n & 0xFF;
}
template <typename Tp>
constexpr auto parity(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, int>::type
{ return popcount(n) & 1; }
template <typename Tp>
int clz(Tp n) { return countl_zero(static_cast<typename std::make_unsigned<Tp>::type>(n)); }
template <typename Tp>
int ctz(Tp n) { return countr_zero(static_cast<typename std::make_unsigned<Tp>::type>(n)); }
template <typename Tp>
int ilog2(Tp n) {
return (CHAR_BIT * sizeof(Tp) - 1) - clz(static_cast<typename std::make_unsigned<Tp>::type>(n));
}
template <typename Tp>
bool is_pow2(Tp n) { return (n > 0) && ((n & (n-1)) == 0); }
template <typename Tp>
Tp floor2(Tp n) { return is_pow2(n)? n: static_cast<Tp>(1) << ilog2(n); }
template <typename Tp>
Tp ceil2(Tp n) { return is_pow2(n)? n: static_cast<Tp>(2) << ilog2(n); }
template <typename Tp>
constexpr auto reverse(Tp n)
-> typename std::enable_if<std::is_unsigned<Tp>::value, Tp>::type
{
int bits = static_cast<int>((sizeof n) * CHAR_BIT);
for (int i = 0, j = 1; j < bits; ++i, j *= 2) {
n = ((n & half_mask[i]) << j) | ((n >> j) & half_mask[i]);
}
return n;
}
#line 13 "ModularArithmetic/operations.cpp"
template <typename ModInt>
ModInt pow(ModInt const& n, intmax_t iexp) {
ModInt res(1);
for (ModInt dbl = n; iexp; iexp >>= 1) {
if (iexp & 1) res *= dbl;
dbl *= dbl;
}
return res;
}
template <typename ModInt>
ModInt sqrt(ModInt const& n) {
if (n == 0) return n;
using value_type = typename ModInt::value_type;
intmax_t const p = n.get_modulo();
if (p % 4 == 3) {
ModInt r = pow(n, (p+1) / 4);
if (r * r == n) return r;
throw std::logic_error("quadratic nonresidue");
}
value_type s = ctz(p-1);
value_type q = (p-1) >> s;
ModInt z;
for (value_type z0 = 2; z0 < p; ++z0) {
z = ModInt(z0);
if (pow(z, (p-1) / 2) == -1) break;
}
value_type m = s;
ModInt c = pow(z, q);
ModInt t = pow(n, q);
ModInt r = pow(n, (q+1) / 2);
while (true) {
if (t == 0) return 0;
if (t == 1) return r;
value_type i = 0;
for (auto tt = t; tt != 1; ++i) tt *= tt;
if (i == m) throw std::logic_error("quadratic nonresidue");
auto b = c;
for (value_type j = 0; j < m-i-1; ++j) b *= b;
m = i;
c = b * b;
t *= c;
r *= b;
}
}
template <typename ModInt>
std::vector<ModInt> sqrt_all(ModInt const& n) {
try {
auto r = sqrt(n);
if (r == -r) return {r};
return {r, -r};
} catch (std::logic_error const&) {
return {};
}
}
template <typename ModPolynomial>
ModPolynomial log(ModPolynomial const& f) {
auto g = f;
g.differentiate();
g *= f.inverse(f.degree()+1);
g.integrate();
return g;
}