Discrete Logarithm (verify/number_theory/discrete_log.test.cpp)

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Last update: 2024-11-21 04:36:37+01:00
Link: https://judge.yosupo.jp/problem/discrete_logarithm_mod

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Code

// @brief Discrete Logarithm
#define PROBLEM "https://judge.yosupo.jp/problem/discrete_logarithm_mod"
#pragma GCC optimize("Ofast,unroll-loops")
#include "cp-algo/number_theory/discrete_log.hpp"
#include <bits/stdc++.h>

using namespace std;
using namespace cp_algo;
using namespace math;
using base = dynamic_modint<>;

void solve() {
    int x, y, m;
    cin >> x >> y >> m;
    auto res = discrete_log(x, y, m);
    if(res) {
        cout << *res << "\n";
    } else {
        cout << -1 << "\n";
    }
}

signed main() {
    //freopen("input.txt", "r", stdin);
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t = 1;
    cin >> t;
    while(t--) {
        solve();
    }
}

#line 1 "verify/number_theory/discrete_log.test.cpp"
// @brief Discrete Logarithm
#define PROBLEM "https://judge.yosupo.jp/problem/discrete_logarithm_mod"
#pragma GCC optimize("Ofast,unroll-loops")
#line 1 "cp-algo/number_theory/discrete_log.hpp"


#line 1 "cp-algo/number_theory/euler.hpp"


#line 1 "cp-algo/number_theory/factorize.hpp"


#line 1 "cp-algo/number_theory/primality.hpp"


#line 1 "cp-algo/number_theory/modint.hpp"


#line 1 "cp-algo/math/common.hpp"


#include <functional>
#include <cstdint>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
    const int maxn = CP_ALGO_MAXN;
#else
    const int maxn = 1 << 19;
#endif
    const int magic = 64; // threshold for sizes to run the naive algo

    auto bpow(auto const& x, auto n, auto const& one, auto op) {
        if(n == 0) {
            return one;
        } else {
            auto t = bpow(x, n / 2, one, op);
            t = op(t, t);
            if(n % 2) {
                t = op(t, x);
            }
            return t;
        }
    }
    auto bpow(auto x, auto n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, auto n) {
        return bpow(x, n, T(1));
    }
}

#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#include <cassert>
namespace cp_algo::math {
    inline constexpr auto inv2(auto x) {
        assert(x % 2);
        std::make_unsigned_t<decltype(x)> y = 1;
        while(y * x != 1) {
            y *= 2 - x * y;
        }
        return y;
    }

    template<typename modint, typename _Int>
    struct modint_base {
        using Int = _Int;
        using UInt = std::make_unsigned_t<Int>;
        static constexpr size_t bits = sizeof(Int) * 8;
        using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
        using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
        static Int mod() {
            return modint::mod();
        }
        static UInt imod() {
            return modint::imod();
        }
        static UInt2 pw128() {
            return modint::pw128();
        }
        static UInt m_reduce(UInt2 ab) {
            if(mod() % 2 == 0) [[unlikely]] {
                return UInt(ab % mod());
            } else {
                UInt2 m = (UInt)ab * imod();
                return UInt((ab + m * mod()) >> bits);
            }
        }
        static UInt m_transform(UInt a) {
            if(mod() % 2 == 0) [[unlikely]] {
                return a;
            } else {
                return m_reduce(a * pw128());
            }
        }
        modint_base(): r(0) {}
        modint_base(Int2 rr): r(UInt(rr % mod())) {
            r = std::min(r, r + mod());
            r = m_transform(r);
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {
            modint neg;
            neg.r = std::min(-r, 2 * mod() - r);
            return neg;
        }
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            r = m_reduce((UInt2)r * t.r);
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - 2 * mod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + 2 * mod());
            return to_modint();
        }
        modint operator + (const modint &t) const {return modint(to_modint()) += t;}
        modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
        modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
        modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
        // Why <=> doesn't work?..
        auto operator == (const modint_base &t) const {return getr() == t.getr();}
        auto operator != (const modint_base &t) const {return getr() != t.getr();}
        auto operator <= (const modint_base &t) const {return getr() <= t.getr();}
        auto operator >= (const modint_base &t) const {return getr() >= t.getr();}
        auto operator < (const modint_base &t) const {return getr() < t.getr();}
        auto operator > (const modint_base &t) const {return getr() > t.getr();}
        Int rem() const {
            UInt R = getr();
            return 2 * R > (UInt)mod() ? R - mod() : R;
        }

        // Only use if you really know what you're doing!
        UInt modmod() const {return (UInt)8 * mod() * mod();};
        void add_unsafe(UInt t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod());}
        modint const& normalize() {
            if(r >= (UInt)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        void setr(UInt rr) {r = m_transform(rr);}
        UInt getr() const {
            UInt res = m_reduce(r);
            return std::min(res, res - mod());
        }
        void setr_direct(UInt rr) {r = rr;}
        UInt getr_direct() const {return r;}
    private:
        UInt r;
        modint& to_modint() {return static_cast<modint&>(*this);}
        modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
    template<modint_type modint>
    std::istream& operator >> (std::istream &in, modint &x) {
        typename modint::UInt r;
        auto &res = in >> r;
        x.setr(r);
        return res;
    }
    template<modint_type modint>
    std::ostream& operator << (std::ostream &out, modint const& x) {
        return out << x.getr();
    }

    template<auto m>
    struct modint: modint_base<modint<m>, decltype(m)> {
        using Base = modint_base<modint<m>, decltype(m)>;
        using Base::Base;
        static constexpr Base::UInt im = m % 2 ? inv2(-m) : 0;
        static constexpr Base::UInt r2 = (typename Base::UInt2)(-1) % m + 1;
        static constexpr Base::Int mod() {return m;}
        static constexpr Base::UInt imod() {return im;}
        static constexpr Base::UInt2 pw128() {return r2;}
    };

    template<typename Int = int64_t>
    struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
        using Base = modint_base<dynamic_modint<Int>, Int>;
        using Base::Base;
        static Int mod() {return m;}
        static Base::UInt imod() {return im;}
        static Base::UInt2 pw128() {return r2;}
        static void switch_mod(Int nm) {
            m = nm;
            im = m % 2 ? inv2(-m) : 0;
            r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
        }

        // Wrapper for temp switching
        auto static with_mod(Int tmp, auto callback) {
            struct scoped {
                Int prev = mod();
                ~scoped() {switch_mod(prev);}
            } _;
            switch_mod(tmp);
            return callback();
        }
    private:
        static thread_local Int m;
        static thread_local Base::UInt im, r2;
    };
    template<typename Int>
    Int thread_local dynamic_modint<Int>::m = 1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}

#line 4 "cp-algo/number_theory/primality.hpp"
#include <algorithm>
#include <bit>
namespace cp_algo::math {
    // https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
    bool is_prime(uint64_t m) {
        if(m == 1 || m % 2 == 0) {
            return m == 2;
        }
        // m - 1 = 2^s * d
        int s = std::countr_zero(m - 1);
        auto d = (m - 1) >> s;
        using base = dynamic_modint<>;
        auto test = [&](base x) {
            x = bpow(x, d);
            if(std::abs(x.rem()) <= 1) {
                return true;
            }
            for(int i = 1; i < s && x != -1; i++) {
                x *= x;
            }
            return x == -1;
        };
        return base::with_mod(m, [&](){
            // Works for all m < 2^64: https://miller-rabin.appspot.com
            return std::ranges::all_of(std::array{
                2, 325, 9375, 28178, 450775, 9780504, 1795265022
            }, test);
        });
    }
}

#line 1 "cp-algo/random/rng.hpp"


#include <chrono>
#include <random>
namespace cp_algo::random {
    uint64_t rng() {
        static std::mt19937_64 rng(
            std::chrono::steady_clock::now().time_since_epoch().count()
        );
        return rng();
    }
}

#line 5 "cp-algo/number_theory/factorize.hpp"
#include <generator>
namespace cp_algo::math {
    // https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
    auto proper_divisor(uint64_t m) {
        using base = dynamic_modint<>;
        return m % 2 == 0 ? 2 : base::with_mod(m, [&]() {
            base t = random::rng();
            auto f = [&](auto x) {
                return x * x + t;
            };
            base x, y;
            base g = 1;
            while(g == 1) {
                for(int i = 0; i < 64; i++) {
                    x = f(x);
                    y = f(f(y));
                    if(x == y) [[unlikely]] {
                        t = random::rng();
                        x = y = 0;
                    } else {
                        base t = g * (x - y);
                        g = t == 0 ? g : t;
                    }
                }
                g = std::gcd(g.getr(), m);
            }
            return g.getr();
        });
    }
    std::generator<uint64_t> factorize(uint64_t m) {
        if(is_prime(m)) {
            co_yield m;
        } else if(m > 1) {
            auto g = proper_divisor(m);
            co_yield std::ranges::elements_of(factorize(g));
            co_yield std::ranges::elements_of(factorize(m / g));
        }
    }
}

#line 4 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math {
    int64_t euler_phi(int64_t m) {
        auto primes = to<std::vector>(factorize(m));
        std::ranges::sort(primes);
        auto [from, to] = std::ranges::unique(primes);
        primes.erase(from, to);
        int64_t ans = m;
        for(auto it: primes) {
            ans -= ans / it;
        }
        return ans;
    }
    template<modint_type base>
    int64_t period(base x) {
        auto ans = euler_phi(base::mod());
        base x0 = bpow(x, ans);
        for(auto t: factorize(ans)) {
            while(ans % t == 0 && x0 * bpow(x, ans / t) == x0) {
                ans /= t;
            }
        }
        return ans;
    }
    int64_t primitive_root(int64_t p) {
        using base = dynamic_modint<>;
        return base::with_mod(p, [p](){
            base t = 1;
            while(period(t) != p - 1) {
                t = random::rng();
            }
            return t.getr();
        });
    }
}

#line 4 "cp-algo/number_theory/discrete_log.hpp"
#include <optional>
namespace cp_algo::math {
    // Find min non-negative x s.t. a*b^x = c (mod m)
    std::optional<int64_t> discrete_log(int64_t b, int64_t c, int64_t m, int64_t a = 1) {
        if(std::abs(a - c) % m == 0) {
            return 0;
        }
        if(std::gcd(a, m) != std::gcd(a * b, m)) {
            auto res = discrete_log(b, c, m, a * b % m);
            return res ? std::optional(*res + 1) : res;
        }
        // a * b^x is periodic here
        using base = dynamic_modint<>;
        return base::with_mod(m, [&]() -> std::optional<uint64_t> {
            int sqrtmod = std::max(1, (int)std::sqrt(m) / 2);
            std::unordered_map<int64_t, int> small;
            base cur = a;
            for(int i = 0; i < sqrtmod; i++) {
                small[cur.getr()] = i;
                cur *= b;
            }
            base step = bpow(base(b), sqrtmod);
            cur = 1;
            for(ptrdiff_t k = 0; k < m; k += sqrtmod) {
                auto it = small.find((base(c) * cur).getr());
                if(it != end(small)) {
                    auto cand = base::with_mod(period(base(b)), [&]() {
                        return base(it->second - k).getr();
                    });
                    if(base(a) * bpow(base(b), cand) == base(c)) {
                        return cand;
                    } else {
                        return std::nullopt;
                    }
                }
                cur *= step;
            }
            return std::nullopt;
        });
    }
}

#line 5 "verify/number_theory/discrete_log.test.cpp"
#include <bits/stdc++.h>

using namespace std;
using namespace cp_algo;
using namespace math;
using base = dynamic_modint<>;

void solve() {
    int x, y, m;
    cin >> x >> y >> m;
    auto res = discrete_log(x, y, m);
    if(res) {
        cout << *res << "\n";
    } else {
        cout << -1 << "\n";
    }
}

signed main() {
    //freopen("input.txt", "r", stdin);
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t = 1;
    cin >> t;
    while(t--) {
        solve();
    }
}