Primitive Root (verify/number_theory/primitive_root.test.cpp)

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Last update: 2024-06-29 00:06:31+02:00
Problem: https://judge.yosupo.jp/problem/primitive_root

Depends on

Code

// @brief Primitive Root
#define PROBLEM "https://judge.yosupo.jp/problem/primitive_root"
#pragma GCC optimize("Ofast,unroll-loops")
#pragma GCC target("tune=native")
#include "cp-algo/math/number_theory.hpp"
#include <bits/stdc++.h>

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

void solve() {
    int64_t p;
    cin >> p;
    cout << primitive_root(p) << "\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/primitive_root.test.cpp"
// @brief Primitive Root
#define PROBLEM "https://judge.yosupo.jp/problem/primitive_root"
#pragma GCC optimize("Ofast,unroll-loops")
#pragma GCC target("tune=native")
#line 1 "cp-algo/math/number_theory.hpp"


#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 1 "cp-algo/math/affine.hpp"


#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math {
    // a * x + b
    template<typename base>
    struct lin {
        base a = 1, b = 0;
        std::optional<base> c;
        lin() {}
        lin(base b): a(0), b(b) {}
        lin(base a, base b): a(a), b(b) {}
        lin(base a, base b, base _c): a(a), b(b), c(_c) {}

        // polynomial product modulo x^2 - c
        lin operator * (const lin& t) {
            assert(c && t.c && *c == *t.c);
            return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
        }

        // a * (t.a * x + t.b) + b
        lin apply(lin const& t) const {
            return {a * t.a, a * t.b + b};
        }

        void prepend(lin const& t) {
            *this = t.apply(*this);
        }

        base eval(base x) const {
            return a * x + b;
        }
    };

    // (ax+b) / (cx+d)
    template<typename base>
    struct linfrac {
        base a, b, c, d;
        linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
        linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
        linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}

        // composition of two linfracs
        linfrac operator * (linfrac t) const {
            return t.prepend(linfrac(*this));
        }

        linfrac operator-() const {
            return {-a, -b, -c, -d};
        }

        linfrac adj() const {
            return {d, -b, -c, a};
        }
        
        linfrac& prepend(linfrac const& t) {
            t.apply(a, c);
            t.apply(b, d);
            return *this;
        }

        // apply linfrac to A/B
        void apply(base &A, base &B) const {
            std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
        }
    };
}

#line 1 "cp-algo/math/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, int64_t 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, int64_t n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, int64_t n) {
        return bpow(x, n, T(1));
    }
}

#line 4 "cp-algo/math/modint.hpp"
#include <iostream>
namespace cp_algo::math {
    template<typename modint>
    struct modint_base {
        static int64_t mod() {
            return modint::mod();
        }
        modint_base(): r(0) {}
        modint_base(int64_t rr): r(rr % mod()) {
            r = std::min(r, r + mod());
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {return std::min(-r, mod() - r);}
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            if(mod() <= uint32_t(-1)) {
                r = r * t.r % mod();
            } else {
                r = __int128(r) * t.r % mod();
            }
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - mod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + 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;}
        auto operator <=> (const modint_base &t) const = default;
        int64_t rem() const {return 2 * r > (uint64_t)mod() ? r - mod() : r;}

        // Only use if you really know what you're doing!
        uint64_t modmod() const {return 8ULL * mod() * mod();};
        void add_unsafe(uint64_t t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod());}
        modint const& normalize() {
            if(r >= (uint64_t)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        uint64_t& setr() {return r;}
        uint64_t getr() const {return r;}
    private:
        uint64_t r;
        modint& to_modint() {return static_cast<modint&>(*this);}
        modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    std::istream& operator >> (std::istream &in, modint_base<modint> &x) {
        return in >> x.setr();
    }
    template<typename modint>
    std::ostream& operator << (std::ostream &out, modint_base<modint> const& x) {
        return out << x.getr();
    }

    template<typename modint>
    concept modint_type = std::is_base_of_v<modint_base<modint>, modint>;

    template<int64_t m>
    struct modint: modint_base<modint<m>> {
        static constexpr int64_t mod() {return m;}
        using Base = modint_base<modint<m>>;
        using Base::Base;
    };

    struct dynamic_modint: modint_base<dynamic_modint> {
        static int64_t mod() {return m;}
        static void switch_mod(int64_t nm) {m = nm;}
        using Base = modint_base<dynamic_modint>;
        using Base::Base;

        // Wrapper for temp switching
        auto static with_mod(int64_t tmp, auto callback) {
            struct scoped {
                int64_t prev = mod();
                ~scoped() {switch_mod(prev);}
            } _;
            switch_mod(tmp);
            return callback();
        }
    private:
        static int64_t m;
    };
    int64_t dynamic_modint::m = 0;
}

#line 6 "cp-algo/math/number_theory.hpp"
#include <algorithm>
#line 8 "cp-algo/math/number_theory.hpp"
#include <vector>
#include <bit>
namespace cp_algo::math {
    std::vector<int64_t> factorize(int64_t m);

    int64_t euler_phi(int64_t m) {
        auto primes = factorize(m);
        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;
    }
    // Find min non-negative x s.t. a*b^x = c (mod m)
    std::optional<uint64_t> discrete_log(int64_t b, int64_t c, uint64_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> {
            size_t sqrtmod = std::max<size_t>(1, std::sqrt(m) / 2);
            std::unordered_map<int64_t, int> small;
            base cur = a;
            for(size_t i = 0; i < sqrtmod; i++) {
                small[cur.getr()] = i;
                cur *= b;
            }
            base step = bpow(base(b), sqrtmod);
            cur = 1;
            for(size_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;
        });
    }
    // https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
    template<modint_type base>
    std::optional<base> sqrt(base b) {
        if(b == base(0)) {
            return base(0);
        } else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
            return std::nullopt;
        } else {
            while(true) {
                base z = random::rng();
                if(z * z == b) {
                    return z;
                }
                lin<base> x(1, z, b); // x + z (mod x^2 - b)
                x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
                if(x.a != base(0)) {
                    return x.a.inv();
                }
            }
        }
    }
    // 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);
        });
    }
    // https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
    void factorize(uint64_t m, std::vector<int64_t> &res) {
        if(m % 2 == 0) {
            factorize(m / 2, res);
            res.push_back(2);
        } else if(is_prime(m)) {
            res.push_back(m);
        } else if(m > 1) {
            using base = dynamic_modint;
            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);
                }
                factorize(g.getr(), res);
                factorize(m / g.getr(), res);
            });
        }
    }
    std::vector<int64_t> factorize(int64_t m) {
        std::vector<int64_t> res;
        factorize(m, res);
        return res;
    }
    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 6 "verify/number_theory/primitive_root.test.cpp"
#include <bits/stdc++.h>

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

void solve() {
    int64_t p;
    cin >> p;
    cout << primitive_root(p) << "\n";
}

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

Test cases

Env	Name	Status	Elapsed	Memory
g++	223092870x_plus_1_00	AC	27 ms	3 MB
g++	example_00	AC	5 ms	3 MB
g++	example_01	AC	5 ms	3 MB
g++	large_least_primitive_root_00	AC	21 ms	3 MB
g++	less_1000000000_00	AC	8 ms	3 MB
g++	less_1000000000_01	AC	8 ms	3 MB
g++	less_1000000000_02	AC	8 ms	3 MB
g++	random_00	AC	20 ms	3 MB
g++	random_01	AC	18 ms	3 MB
g++	random_02	AC	15 ms	3 MB
g++	safe_prime_00	AC	8 ms	3 MB
g++	small_00	AC	6 ms	3 MB
g++	small_01	AC	6 ms	3 MB
g++	small_02	AC	7 ms	3 MB