$\mathbb{Z}$ の乗除算を $\mathbb{Z}/n\mathbb{Z}$ の上でやるデータ構造
(modulus/mint_with_zero.hpp)

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• Last update: 2021-08-30 04:35:37+09:00
• Include: #include "modulus/mint_with_zero.hpp"
• Link: https://kimiyuki.net/blog/2021/04/23/modulo-with-zero/

Depends on

• quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$ (modulus/mint.hpp)
• modulus/modinv.hpp
• modulus/modpow.hpp

Verified with

• modulus/mint_with_zero.test.cpp

Code

#pragma once
#include <cstdint>
#include <iostream>
#include "../modulus/modpow.hpp"
#include "../modulus/modinv.hpp"
#include "../modulus/mint.hpp"

/**
 * @brief $\mathbb{Z}$ の乗除算を $\mathbb{Z}/n\mathbb{Z}$ の上でやるデータ構造
 * @sa https://kimiyuki.net/blog/2021/04/23/modulo-with-zero/
 */
template <int32_t MOD>
struct zmint {
    int32_t nonzero;
    int32_t zero;
    zmint() : nonzero(1) {}
    zmint(int64_t nonzero_, int32_t zero_ = 0) : nonzero(nonzero_), zero(zero_) { assert (nonzero != 0); while (nonzero % MOD == 0) { nonzero /= MOD; zero += 1; } if (nonzero < 0) nonzero = nonzero % MOD + MOD; }
    zmint(int32_t nonzero_, int32_t zero_, std::nullptr_t) : nonzero(nonzero_), zero(zero_) {}
    explicit operator bool() const { return nonzero; }
    inline zmint<MOD> operator * (zmint<MOD> other) const { return zmint<MOD>(*this) *= other; }
    inline zmint<MOD> & operator *= (zmint<MOD> other) { nonzero = static_cast<uint_fast64_t>(this->nonzero) * other.nonzero % MOD; zero += other.zero; return *this; }
    inline zmint<MOD> operator / (zmint<MOD> other) const { return *this * other.inv(); }
    inline zmint<MOD> & operator /= (zmint<MOD> other) { return *this *= other.inv(); }
    inline zmint<MOD> operator - () const { return zmint<MOD>(nonzero ? MOD - nonzero : 0, zero, nullptr); }
    inline bool operator == (zmint<MOD> other) const { return nonzero == other.nonzero and zero == other.zero; }
    inline bool operator != (zmint<MOD> other) const { return nonzero != other.nonzero or zero != other.zero; }
    inline zmint<MOD> pow(uint64_t k) const { return zmint<MOD>(modpow(nonzero, k, MOD), k * zero, nullptr); }
    inline zmint<MOD> inv() const { return zmint<MOD>(modinv(nonzero, MOD), - zero, nullptr); }
    inline mint<MOD> to_mint() const { assert (nonzero >= 0); return zero ? 0 : nonzero; }
};
template <int32_t MOD> zmint<MOD> operator * (int64_t nonzero, zmint<MOD> n) { return zmint<MOD>(nonzero) * n; }
template <int32_t MOD> zmint<MOD> operator / (int64_t nonzero, zmint<MOD> n) { return zmint<MOD>(nonzero) / n; }

#line 2 "modulus/mint_with_zero.hpp"
#include <cstdint>
#include <iostream>
#line 2 "modulus/modpow.hpp"
#include <cassert>
#line 4 "modulus/modpow.hpp"

inline int32_t modpow(uint_fast64_t x, uint64_t k, int32_t MOD) {
    assert (/* 0 <= x and */ x < (uint_fast64_t)MOD);
    uint_fast64_t y = 1;
    for (; k; k >>= 1) {
        if (k & 1) (y *= x) %= MOD;
        (x *= x) %= MOD;
    }
    assert (/* 0 <= y and */ y < (uint_fast64_t)MOD);
    return y;
}
#line 2 "modulus/modinv.hpp"
#include <algorithm>
#line 5 "modulus/modinv.hpp"

inline int32_t modinv_nocheck(int32_t value, int32_t MOD) {
    assert (0 <= value and value < MOD);
    if (value == 0) return -1;
    int64_t a = value, b = MOD;
    int64_t x = 0, y = 1;
    for (int64_t u = 1, v = 0; a; ) {
        int64_t q = b / a;
        x -= q * u; std::swap(x, u);
        y -= q * v; std::swap(y, v);
        b -= q * a; std::swap(b, a);
    }
    if (not (value * x + MOD * y == b and b == 1)) return -1;
    if (x < 0) x += MOD;
    assert (0 <= x and x < MOD);
    return x;
}

inline int32_t modinv(int32_t x, int32_t MOD) {
    int32_t y = modinv_nocheck(x, MOD);
    assert (y != -1);
    return y;
}
#line 6 "modulus/mint.hpp"

/**
 * @brief quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$
 */
template <int32_t MOD>
struct mint {
    int32_t value;
    mint() : value() {}
    mint(int64_t value_) : value(value_ < 0 ? value_ % MOD + MOD : value_ >= MOD ? value_ % MOD : value_) {}
    mint(int32_t value_, std::nullptr_t) : value(value_) {}
    explicit operator bool() const { return value; }
    inline mint<MOD> operator + (mint<MOD> other) const { return mint<MOD>(*this) += other; }
    inline mint<MOD> operator - (mint<MOD> other) const { return mint<MOD>(*this) -= other; }
    inline mint<MOD> operator * (mint<MOD> other) const { return mint<MOD>(*this) *= other; }
    inline mint<MOD> & operator += (mint<MOD> other) { this->value += other.value; if (this->value >= MOD) this->value -= MOD; return *this; }
    inline mint<MOD> & operator -= (mint<MOD> other) { this->value -= other.value; if (this->value <    0) this->value += MOD; return *this; }
    inline mint<MOD> & operator *= (mint<MOD> other) { this->value = (uint_fast64_t)this->value * other.value % MOD; return *this; }
    inline mint<MOD> operator - () const { return mint<MOD>(this->value ? MOD - this->value : 0, nullptr); }
    inline bool operator == (mint<MOD> other) const { return value == other.value; }
    inline bool operator != (mint<MOD> other) const { return value != other.value; }
    inline mint<MOD> pow(uint64_t k) const { return mint<MOD>(modpow(value, k, MOD), nullptr); }
    inline mint<MOD> inv() const { return mint<MOD>(modinv(value, MOD), nullptr); }
    inline mint<MOD> operator / (mint<MOD> other) const { return *this * other.inv(); }
    inline mint<MOD> & operator /= (mint<MOD> other) { return *this *= other.inv(); }
};
template <int32_t MOD> mint<MOD> operator + (int64_t value, mint<MOD> n) { return mint<MOD>(value) + n; }
template <int32_t MOD> mint<MOD> operator - (int64_t value, mint<MOD> n) { return mint<MOD>(value) - n; }
template <int32_t MOD> mint<MOD> operator * (int64_t value, mint<MOD> n) { return mint<MOD>(value) * n; }
template <int32_t MOD> mint<MOD> operator / (int64_t value, mint<MOD> n) { return mint<MOD>(value) / n; }
template <int32_t MOD> std::istream & operator >> (std::istream & in, mint<MOD> & n) { int64_t value; in >> value; n = value; return in; }
template <int32_t MOD> std::ostream & operator << (std::ostream & out, mint<MOD> n) { return out << n.value; }
#line 7 "modulus/mint_with_zero.hpp"

/**
 * @brief $\mathbb{Z}$ の乗除算を $\mathbb{Z}/n\mathbb{Z}$ の上でやるデータ構造
 * @sa https://kimiyuki.net/blog/2021/04/23/modulo-with-zero/
 */
template <int32_t MOD>
struct zmint {
    int32_t nonzero;
    int32_t zero;
    zmint() : nonzero(1) {}
    zmint(int64_t nonzero_, int32_t zero_ = 0) : nonzero(nonzero_), zero(zero_) { assert (nonzero != 0); while (nonzero % MOD == 0) { nonzero /= MOD; zero += 1; } if (nonzero < 0) nonzero = nonzero % MOD + MOD; }
    zmint(int32_t nonzero_, int32_t zero_, std::nullptr_t) : nonzero(nonzero_), zero(zero_) {}
    explicit operator bool() const { return nonzero; }
    inline zmint<MOD> operator * (zmint<MOD> other) const { return zmint<MOD>(*this) *= other; }
    inline zmint<MOD> & operator *= (zmint<MOD> other) { nonzero = static_cast<uint_fast64_t>(this->nonzero) * other.nonzero % MOD; zero += other.zero; return *this; }
    inline zmint<MOD> operator / (zmint<MOD> other) const { return *this * other.inv(); }
    inline zmint<MOD> & operator /= (zmint<MOD> other) { return *this *= other.inv(); }
    inline zmint<MOD> operator - () const { return zmint<MOD>(nonzero ? MOD - nonzero : 0, zero, nullptr); }
    inline bool operator == (zmint<MOD> other) const { return nonzero == other.nonzero and zero == other.zero; }
    inline bool operator != (zmint<MOD> other) const { return nonzero != other.nonzero or zero != other.zero; }
    inline zmint<MOD> pow(uint64_t k) const { return zmint<MOD>(modpow(nonzero, k, MOD), k * zero, nullptr); }
    inline zmint<MOD> inv() const { return zmint<MOD>(modinv(nonzero, MOD), - zero, nullptr); }
    inline mint<MOD> to_mint() const { assert (nonzero >= 0); return zero ? 0 : nonzero; }
};
template <int32_t MOD> zmint<MOD> operator * (int64_t nonzero, zmint<MOD> n) { return zmint<MOD>(nonzero) * n; }
template <int32_t MOD> zmint<MOD> operator / (int64_t nonzero, zmint<MOD> n) { return zmint<MOD>(nonzero) / n; }

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