number_theory/ax_by_c.hpp

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• Last update: 2024-07-18 16:56:22+09:00
• Include: #include "number_theory/ax_by_c.hpp"

Depends on

• number_theory/utils.hpp

Verified with

• number_theory/test/ax_by_c_stress.test.cpp

Code

#pragma once

#include "utils.hpp"
#include <optional>

// solve a * x + b * y = c
// |x| <= |b * c|, |y| <= |a * c| (|a|, |b|, |c| > 0)
template <typename T>
std::optional<std::pair<T, T>> ax_by_c(T a, T b, T c) {
    if (c == 0) {
        return std::pair<T, T>(0, 0);
    }
    if (a == 0 && b == 0) {
        return std::nullopt;
    }
    if (a == 0) {
        if (c % b) {
            return std::nullopt;
        }
        return std::pair<T, T>(0, c / b);
    }
    if (b == 0) {
        if (c % a) {
            return std::nullopt;
        }
        return std::pair<T, T>(c / a, 0);
    }
    if (b < 0) {
        a = -a;
        b = -b;
        c = -c;
    }
    auto [g, x, y] = extgcd2(a, b);
    if (c % g) {
        return std::nullopt;
    }
    T mult = c / g;
    x *= mult;
    y *= mult;
    return std::pair<T, T>(x, y);
}

#line 2 "number_theory/ax_by_c.hpp"

#line 2 "number_theory/utils.hpp"

#include <utility>

constexpr bool is_prime(unsigned n) {
    if (n == 0 || n == 1) {
        return false;
    }
    for (unsigned i = 2; i * i <= n; ++i) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

constexpr unsigned mod_pow(unsigned x, unsigned y, unsigned mod) {
    unsigned ret = 1, self = x;
    while (y != 0) {
        if (y & 1) {
            ret = (unsigned)((unsigned long long)ret * self % mod);
        }
        self = (unsigned)((unsigned long long)self * self % mod);
        y /= 2;
    }
    return ret;
}

template <unsigned mod>
constexpr unsigned primitive_root() {
    static_assert(is_prime(mod), "`mod` must be a prime number.");
    if (mod == 2) {
        return 1;
    }

    unsigned primes[32] = {};
    int it = 0;
    {
        unsigned m = mod - 1;
        for (unsigned i = 2; i * i <= m; ++i) {
            if (m % i == 0) {
                primes[it++] = i;
                while (m % i == 0) {
                    m /= i;
                }
            }
        }
        if (m != 1) {
            primes[it++] = m;
        }
    }
    for (unsigned i = 2; i < mod; ++i) {
        bool ok = true;
        for (int j = 0; j < it; ++j) {
            if (mod_pow(i, (mod - 1) / primes[j], mod) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return i;
    }
    return 0;
}

// y >= 1
template <typename T>
constexpr T safe_mod(T x, T y) {
    x %= y;
    if (x < 0) {
        x += y;
    }
    return x;
}

// y != 0
template <typename T>
constexpr T floor_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return x / y;
    } else {
        return -((-x + y - 1) / y);
    }
}

// y != 0
template <typename T>
constexpr T ceil_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return (x + y - 1) / y;
    } else {
        return -(-x / y);
    }
}

// b >= 1
// returns (g, x) s.t. g = gcd(a, b), a * x = g (mod b), 0 <= x < b / g
// from ACL
template <typename T>
std::pair<T, T> extgcd(T a, T b) {
    a = safe_mod(a, b);
    T s = b, t = a, m0 = 0, m1 = 1;
    while (t) {
        T u = s / t;
        s -= t * u;
        m0 -= m1 * u;
        std::swap(s, t);
        std::swap(m0, m1);
    }
    if (m0 < 0) {
        m0 += b / s;
    }
    return std::pair<T, T>(s, m0);
}

// b >= 1
// returns (g, x, y) s.t. g = gcd(a, b), a * x + b * y = g, 0 <= x < b / g, |y| < max(2, |a| / g)
template <typename T>
std::tuple<T, T, T> extgcd2(T a, T b) {
    T _a = safe_mod(a, b);
    T quot = (a - _a) / b;
    T x00 = 0, x01 = 1, y0 = b;
    T x10 = 1, x11 = -quot, y1 = _a;
    while (y1) {
        T u = y0 / y1;
        x00 -= u * x10;
        x01 -= u * x11;
        y0 -= u * y1;
        std::swap(x00, x10);
        std::swap(x01, x11);
        std::swap(y0, y1);
    }
    if (x00 < 0) {
        x00 += b / y0;
        x01 -= a / y0;
    }
    return std::tuple<T, T, T>(y0, x00, x01);
}

// gcd(x, m) == 1
template <typename T>
T inv_mod(T x, T m) {
    return extgcd(x, m).second;
}
#line 4 "number_theory/ax_by_c.hpp"
#include <optional>

// solve a * x + b * y = c
// |x| <= |b * c|, |y| <= |a * c| (|a|, |b|, |c| > 0)
template <typename T>
std::optional<std::pair<T, T>> ax_by_c(T a, T b, T c) {
    if (c == 0) {
        return std::pair<T, T>(0, 0);
    }
    if (a == 0 && b == 0) {
        return std::nullopt;
    }
    if (a == 0) {
        if (c % b) {
            return std::nullopt;
        }
        return std::pair<T, T>(0, c / b);
    }
    if (b == 0) {
        if (c % a) {
            return std::nullopt;
        }
        return std::pair<T, T>(c / a, 0);
    }
    if (b < 0) {
        a = -a;
        b = -b;
        c = -c;
    }
    auto [g, x, y] = extgcd2(a, b);
    if (c % g) {
        return std::nullopt;
    }
    T mult = c / g;
    x *= mult;
    y *= mult;
    return std::pair<T, T>(x, y);
}

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