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Modint
(library/number/modint.hpp)
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- Last update: 2024-01-30 21:03:50+09:00
- Include:
#include "library/number/modint.hpp"
Modint
Code
#ifndef SUISEN_MODINT
#define SUISEN_MODINT
#include <cassert>
#include <cstdint>
#include <limits>
#include <optional>
#include <iostream>
namespace suisen {
namespace internal::modint {
constexpr long long safe_mod(long long x, long long m) { return (x %= m) < 0 ? x + m : x; }
constexpr long long pow_mod(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int um = m;
unsigned long long r = 1, y = safe_mod(x, m);
for (; n; n >>= 1) {
if (n & 1) r = (r * y) % um;
y = (y * y) % um;
}
return r;
}
constexpr bool is_prime(int n) {
if (n <= 1) return false;
if (n == 2 or n == 7 or n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = { 2, 7, 61 };
for (long long a : bases) {
long long t = d, y = pow_mod(a, t, n);
for (; t != n - 1 and y != 1 and y != n - 1; t <<= 1) y = y * y % n;
if (y != n - 1 and t % 2 == 0) return false;
}
return true;
}
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return { b, 0 };
long long s = b, t = a, m0 = 0, m1 = 1, tmp = 0;
while (t) {
long long u = s / t;
s -= t * u, m0 -= m1 * u;
tmp = s, s = t, t = tmp;
tmp = m0, m0 = m1, m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return { s, m0 };
}
struct barrett_K128 {
uint32_t M; // mod
__uint128_t L; // ceil(2^K / M), where K = 128
uint64_t dL, uL; // dL | (uL << 64) = L
constexpr barrett_K128(uint32_t M) : M(M), L(~__uint128_t(0) / M + 1), dL(L), uL(L >> 64) {}
constexpr uint32_t umod() const { return M; }
// c mod M (correctly works for all 0<=c<2^64)
template <bool care_M1 = true>
constexpr uint32_t rem(uint64_t c) const {
if constexpr (care_M1) if (M == 1) return 0;
// uint32_t q = (c * L) >> 128;
__uint128_t cu = __uint128_t(c) * uL;
uint64_t cd = (__uint128_t(c) * dL) >> 64;
uint32_t r = c - uint64_t(cu >> 64) * M;
return uint64_t(cu) > ~cd ? r - M : r;
}
// a*b mod M
constexpr uint32_t mul(uint32_t a, uint32_t b) const { return rem<false>(uint64_t(a) * b); }
};
}
template <int m, std::enable_if_t<(1 <= m), std::nullptr_t> = nullptr>
class static_modint {
using mint = static_modint;
struct raw_construct {};
constexpr static_modint(int v, raw_construct) : _v(v) {}
public:
static constexpr int mod() { return m; }
static constexpr unsigned int umod() { return m; }
static constexpr mint raw(int v) { return mint(v, raw_construct{}); }
constexpr static_modint() : _v(0) {}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_signed<T>>, std::nullptr_t> = nullptr>
constexpr static_modint(T v) : _v{} {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_unsigned<T>>, std::nullptr_t> = nullptr>
constexpr static_modint(T v) : _v(v % umod()) {}
constexpr unsigned int val() const { return _v; }
constexpr mint& operator++() {
++_v;
if (_v == umod()) _v = 0;
return *this;
}
constexpr mint& operator--() {
if (_v == 0) _v = umod();
--_v;
return *this;
}
constexpr mint operator++(int) { mint x = *this; ++*this; return x; }
constexpr mint operator--(int) { mint x = *this; --*this; return x; }
constexpr mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
constexpr mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
constexpr mint& operator*=(const mint& rhs) {
_v = (unsigned long long) _v * rhs._v % umod();
return *this;
}
constexpr mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
constexpr mint operator+() const { return *this; }
constexpr mint operator-() const { return _v == 0 ? *this : raw(umod() - _v); }
constexpr mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
for (; n; n >>= 1) {
if (n & 1) r *= x;
x *= x;
}
return r;
}
constexpr mint xpow(long long n) const { return n < 0 ? inv().pow(-n) : pow(n); }
constexpr mint inv() const {
if constexpr (is_prime_mod) {
assert(_v);
return pow(umod() - 2);
} else {
const auto [g, res] = internal::modint::inv_gcd(_v, mod());
assert(g == 1);
return res;
}
}
friend constexpr mint operator+(const mint& lhs, const mint& rhs) { mint res = lhs; res += rhs; return res; }
friend constexpr mint operator-(const mint& lhs, const mint& rhs) { mint res = lhs; res -= rhs; return res; }
friend constexpr mint operator*(const mint& lhs, const mint& rhs) { mint res = lhs; res *= rhs; return res; }
friend constexpr mint operator/(const mint& lhs, const mint& rhs) { mint res = lhs; res /= rhs; return res; }
friend constexpr bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend constexpr bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
private:
unsigned int _v;
static constexpr bool is_prime_mod = internal::modint::is_prime(mod());
};
template <int id>
class dynamic_modint {
using mint = dynamic_modint;
using barrett = internal::modint::barrett_K128;
struct raw_construct {};
constexpr dynamic_modint(int v, raw_construct) : _v(v) {}
public:
static int mod() { return bt.umod(); }
static unsigned int umod() { return bt.umod(); }
static void set_mod(int m) {
assert(1 <= m);
bt = barrett(m);
}
static mint raw(int v) { return dynamic_modint(v, raw_construct{}); }
dynamic_modint() : _v(0) {}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_signed<T>>, std::nullptr_t> = nullptr>
dynamic_modint(T v) {
if (v < 0) {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
} else _v = bt.rem(v);
}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_unsigned<T>>, std::nullptr_t> = nullptr>
dynamic_modint(T v) : _v(bt.rem(v)) {}
dynamic_modint(__uint128_t v) : _v(v % umod()) {}
dynamic_modint(__int128_t v) {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
}
unsigned int val() const { return _v; }
mint& operator++() {
++_v;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
--_v;
return *this;
}
mint operator++(int) { mint x = *this; ++*this; return x; }
mint operator--(int) { mint x = *this; --*this; return x; }
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
for (; n; n >>= 1) {
if (n & 1) r *= x;
x *= x;
}
return r;
}
mint xpow(long long n) const { return n < 0 ? inv().pow(-n) : pow(n); }
mint inv() const {
const auto [g, res] = internal::modint::inv_gcd(_v, mod());
assert(g == 1);
return res;
}
friend mint operator+(const mint& lhs, const mint& rhs) { mint res = lhs; res += rhs; return res; }
friend mint operator-(const mint& lhs, const mint& rhs) { mint res = lhs; res -= rhs; return res; }
friend mint operator*(const mint& lhs, const mint& rhs) { mint res = lhs; res *= rhs; return res; }
friend mint operator/(const mint& lhs, const mint& rhs) { mint res = lhs; res /= rhs; return res; }
friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
private:
unsigned int _v;
static inline barrett bt{ 998244353 };
};
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
template <typename T> struct is_modint : std::false_type {};
template <int m> struct is_modint<static_modint<m>> : std::true_type {};
template <int id> struct is_modint<dynamic_modint<id>> : std::true_type {};
template <typename T> constexpr bool is_modint_v = is_modint<T>::value;
template <typename T> struct is_static_modint : std::false_type {};
template <int m> struct is_static_modint<static_modint<m>> : std::true_type {};
template <typename T> constexpr bool is_static_modint_v = is_static_modint<T>::value;
template <typename T> struct is_dynamic_modint : std::false_type {};
template <int id> struct is_dynamic_modint<dynamic_modint<id>> : std::true_type {};
template <typename T> constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::optional<mint> mod_sqrt(mint a) {
const int p = mint::mod();
if (a == 0) return mint(0);
if (p == 2) return a;
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
const int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return x;
}
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint sqrt(mint a) { return *mod_sqrt(a); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint log(mint a) { assert(a == 1); return 0; }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint exp(mint a) { assert(a == 0); return 1; }
template <typename mint, typename T, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint pow(mint a, T b) { return a.xpow(b); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint inv(mint a) { return a.inv(); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::istream& operator>>(std::istream& is, mint& v) { long long val; is >> val, v = val; return is; }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::ostream& operator<<(std::ostream& os, const mint& v) { return os << v.val(); }
} // namespace suisen
#endif // SUISEN_MODINT#line 1 "library/number/modint.hpp"
#include <cassert>
#include <cstdint>
#include <limits>
#include <optional>
#include <iostream>
namespace suisen {
namespace internal::modint {
constexpr long long safe_mod(long long x, long long m) { return (x %= m) < 0 ? x + m : x; }
constexpr long long pow_mod(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int um = m;
unsigned long long r = 1, y = safe_mod(x, m);
for (; n; n >>= 1) {
if (n & 1) r = (r * y) % um;
y = (y * y) % um;
}
return r;
}
constexpr bool is_prime(int n) {
if (n <= 1) return false;
if (n == 2 or n == 7 or n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = { 2, 7, 61 };
for (long long a : bases) {
long long t = d, y = pow_mod(a, t, n);
for (; t != n - 1 and y != 1 and y != n - 1; t <<= 1) y = y * y % n;
if (y != n - 1 and t % 2 == 0) return false;
}
return true;
}
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return { b, 0 };
long long s = b, t = a, m0 = 0, m1 = 1, tmp = 0;
while (t) {
long long u = s / t;
s -= t * u, m0 -= m1 * u;
tmp = s, s = t, t = tmp;
tmp = m0, m0 = m1, m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return { s, m0 };
}
struct barrett_K128 {
uint32_t M; // mod
__uint128_t L; // ceil(2^K / M), where K = 128
uint64_t dL, uL; // dL | (uL << 64) = L
constexpr barrett_K128(uint32_t M) : M(M), L(~__uint128_t(0) / M + 1), dL(L), uL(L >> 64) {}
constexpr uint32_t umod() const { return M; }
// c mod M (correctly works for all 0<=c<2^64)
template <bool care_M1 = true>
constexpr uint32_t rem(uint64_t c) const {
if constexpr (care_M1) if (M == 1) return 0;
// uint32_t q = (c * L) >> 128;
__uint128_t cu = __uint128_t(c) * uL;
uint64_t cd = (__uint128_t(c) * dL) >> 64;
uint32_t r = c - uint64_t(cu >> 64) * M;
return uint64_t(cu) > ~cd ? r - M : r;
}
// a*b mod M
constexpr uint32_t mul(uint32_t a, uint32_t b) const { return rem<false>(uint64_t(a) * b); }
};
}
template <int m, std::enable_if_t<(1 <= m), std::nullptr_t> = nullptr>
class static_modint {
using mint = static_modint;
struct raw_construct {};
constexpr static_modint(int v, raw_construct) : _v(v) {}
public:
static constexpr int mod() { return m; }
static constexpr unsigned int umod() { return m; }
static constexpr mint raw(int v) { return mint(v, raw_construct{}); }
constexpr static_modint() : _v(0) {}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_signed<T>>, std::nullptr_t> = nullptr>
constexpr static_modint(T v) : _v{} {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_unsigned<T>>, std::nullptr_t> = nullptr>
constexpr static_modint(T v) : _v(v % umod()) {}
constexpr unsigned int val() const { return _v; }
constexpr mint& operator++() {
++_v;
if (_v == umod()) _v = 0;
return *this;
}
constexpr mint& operator--() {
if (_v == 0) _v = umod();
--_v;
return *this;
}
constexpr mint operator++(int) { mint x = *this; ++*this; return x; }
constexpr mint operator--(int) { mint x = *this; --*this; return x; }
constexpr mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
constexpr mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
constexpr mint& operator*=(const mint& rhs) {
_v = (unsigned long long) _v * rhs._v % umod();
return *this;
}
constexpr mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
constexpr mint operator+() const { return *this; }
constexpr mint operator-() const { return _v == 0 ? *this : raw(umod() - _v); }
constexpr mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
for (; n; n >>= 1) {
if (n & 1) r *= x;
x *= x;
}
return r;
}
constexpr mint xpow(long long n) const { return n < 0 ? inv().pow(-n) : pow(n); }
constexpr mint inv() const {
if constexpr (is_prime_mod) {
assert(_v);
return pow(umod() - 2);
} else {
const auto [g, res] = internal::modint::inv_gcd(_v, mod());
assert(g == 1);
return res;
}
}
friend constexpr mint operator+(const mint& lhs, const mint& rhs) { mint res = lhs; res += rhs; return res; }
friend constexpr mint operator-(const mint& lhs, const mint& rhs) { mint res = lhs; res -= rhs; return res; }
friend constexpr mint operator*(const mint& lhs, const mint& rhs) { mint res = lhs; res *= rhs; return res; }
friend constexpr mint operator/(const mint& lhs, const mint& rhs) { mint res = lhs; res /= rhs; return res; }
friend constexpr bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend constexpr bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
private:
unsigned int _v;
static constexpr bool is_prime_mod = internal::modint::is_prime(mod());
};
template <int id>
class dynamic_modint {
using mint = dynamic_modint;
using barrett = internal::modint::barrett_K128;
struct raw_construct {};
constexpr dynamic_modint(int v, raw_construct) : _v(v) {}
public:
static int mod() { return bt.umod(); }
static unsigned int umod() { return bt.umod(); }
static void set_mod(int m) {
assert(1 <= m);
bt = barrett(m);
}
static mint raw(int v) { return dynamic_modint(v, raw_construct{}); }
dynamic_modint() : _v(0) {}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_signed<T>>, std::nullptr_t> = nullptr>
dynamic_modint(T v) {
if (v < 0) {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
} else _v = bt.rem(v);
}
template <class T, std::enable_if_t<std::conjunction_v<std::is_integral<T>, std::is_unsigned<T>>, std::nullptr_t> = nullptr>
dynamic_modint(T v) : _v(bt.rem(v)) {}
dynamic_modint(__uint128_t v) : _v(v % umod()) {}
dynamic_modint(__int128_t v) {
int x = v % mod();
if (x < 0) x += mod();
_v = x;
}
unsigned int val() const { return _v; }
mint& operator++() {
++_v;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
--_v;
return *this;
}
mint operator++(int) { mint x = *this; ++*this; return x; }
mint operator--(int) { mint x = *this; --*this; return x; }
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
for (; n; n >>= 1) {
if (n & 1) r *= x;
x *= x;
}
return r;
}
mint xpow(long long n) const { return n < 0 ? inv().pow(-n) : pow(n); }
mint inv() const {
const auto [g, res] = internal::modint::inv_gcd(_v, mod());
assert(g == 1);
return res;
}
friend mint operator+(const mint& lhs, const mint& rhs) { mint res = lhs; res += rhs; return res; }
friend mint operator-(const mint& lhs, const mint& rhs) { mint res = lhs; res -= rhs; return res; }
friend mint operator*(const mint& lhs, const mint& rhs) { mint res = lhs; res *= rhs; return res; }
friend mint operator/(const mint& lhs, const mint& rhs) { mint res = lhs; res /= rhs; return res; }
friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
private:
unsigned int _v;
static inline barrett bt{ 998244353 };
};
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
template <typename T> struct is_modint : std::false_type {};
template <int m> struct is_modint<static_modint<m>> : std::true_type {};
template <int id> struct is_modint<dynamic_modint<id>> : std::true_type {};
template <typename T> constexpr bool is_modint_v = is_modint<T>::value;
template <typename T> struct is_static_modint : std::false_type {};
template <int m> struct is_static_modint<static_modint<m>> : std::true_type {};
template <typename T> constexpr bool is_static_modint_v = is_static_modint<T>::value;
template <typename T> struct is_dynamic_modint : std::false_type {};
template <int id> struct is_dynamic_modint<dynamic_modint<id>> : std::true_type {};
template <typename T> constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::optional<mint> mod_sqrt(mint a) {
const int p = mint::mod();
if (a == 0) return mint(0);
if (p == 2) return a;
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
const int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return x;
}
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint sqrt(mint a) { return *mod_sqrt(a); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint log(mint a) { assert(a == 1); return 0; }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint exp(mint a) { assert(a == 0); return 1; }
template <typename mint, typename T, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint pow(mint a, T b) { return a.xpow(b); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
mint inv(mint a) { return a.inv(); }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::istream& operator>>(std::istream& is, mint& v) { long long val; is >> val, v = val; return is; }
template <typename mint, std::enable_if_t<is_modint_v<mint>, std::nullptr_t> = nullptr>
std::ostream& operator<<(std::ostream& os, const mint& v) { return os << v.val(); }
} // namespace suisen