$\mathrm{ord}(x)$ for $x \in (\mathbb{Z}/p\mathbb{Z})^\times$ (tools/ord_mod.hpp)

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• Last update: 2025-02-10 20:05:30+09:00
• Include: #include "tools/ord_mod.hpp"

For a fixed prime $p$, it returns the minimum positive integer $k$ such that $x^k \equiv 1 \pmod{p}$ in $\left\langle O\left(p^\frac{1}{4}\right), O\left((\log p)(\log \log p)\right) \right\rangle$ time.

References

• 解説 - AtCoder Beginner Contest 335（Sponsored by Mynavi）

License

• CC0

Author

• anqooqie

Constructor

ord_mod<T> ord(T p);

It constructs a data structure that allows $\mathrm{ord}(x)$ to be returned in $O((\log p)(\log \log p))$ time.

Constraints

• $p$ is a prime.
• $2 \leq p \leq 10^{18}$

Time Complexity

• Supposed to be $O\left(p^\frac{1}{4}\right)$ expected

query

T ord.query(T x);

It returns $\mathrm{ord}(x)$, the minimum positive integer $k$ such that $x^k \equiv 1 \pmod{p}$.

Constraints

• $1 \leq x < p$

Time Complexity

• $O((\log p)(\log \log p))$

count

std::map<T, T> ord.count();

It returns pairs of two integers $(k, v)$ such that

\[\begin{align*} \left\{\begin{array}{l} v = |\{x \in \mathbb{N} \mid 1 \leq x < p \land \mathrm{ord}(x) = k\}|\\ v > 0 \end{array}\right.& \end{align*}\]

Constraints

• None

Time Complexity

• $O\left(2^\frac{\log p}{\log \log p} \frac{\log p}{\log \log p}\right)$

Depends on

• std::abs(x) extended for my library (tools/abs.hpp)
• $2^{\left\lceil \log_2(x) \right\rceil}$ (tools/bit_ceil.hpp)
• $2^{\left\lfloor \log_2(x) \right\rfloor}$ (tools/bit_floor.hpp)
• Bit width required to represent a given integer (tools/bit_width.hpp)
• Number of trailing zeros (tools/countr_zero.hpp)
• tools/detail/int128_t.hpp
• Fixed point combinator (tools/fix.hpp)
• $\left\lfloor \log_2(x) \right\rfloor$ (tools/floor_log2.hpp)
• std::gcd(m, n) extended for my library (tools/gcd.hpp)
• Combine hash values (tools/hash_combine.hpp)
• std::is_integral extended for tools::int128_t and tools::uint128_t (tools/is_integral.hpp)
• Check whether T is a monoid (tools/is_monoid.hpp)
• Miller-Rabin primality test (tools/is_prime.hpp)
• std::is_signed extended for tools::int128_t and tools::uint128_t (tools/is_signed.hpp)
• std::is_unsigned extended for tools::int128_t and tools::uint128_t (tools/is_unsigned.hpp)
• std::make_signed extended for tools::int128_t and tools::uint128_t (tools/make_signed.hpp)
• std::make_unsigned extended for tools::int128_t and tools::uint128_t (tools/make_unsigned.hpp)
• Minimum non-negative reminder (tools/mod.hpp)
• Typical monoids (tools/monoid.hpp)
• The number of nanoseconds that have elapsed since epoch (tools/now.hpp)
• $b^n$ under a given monoid, and std::pow(b, n) extended for my library (tools/pow.hpp)
• $2^x$ (tools/pow2.hpp)
• $x^y \pmod{M}$ (tools/pow_mod.hpp)
• Pollard's rho algorithm (tools/prime_factorization.hpp)
• $x \cdot y \pmod{M}$ (tools/prod_mod.hpp)
• $x^2$ under a given monoid (tools/square.hpp)
• 128-bit unsigned integer (tools/uint128_t.hpp)

Verified with

• tests/ord_mod/count.test.cpp
• tests/ord_mod/query.test.cpp

Code

#ifndef TOOLS_ORD_MOD_HPP
#define TOOLS_ORD_MOD_HPP

#include <vector>
#include <cassert>
#include <cstddef>
#include <map>
#include "tools/is_prime.hpp"
#include "tools/prime_factorization.hpp"
#include "tools/pow.hpp"
#include "tools/fix.hpp"
#include "tools/pow_mod.hpp"

namespace tools {
  template <typename T>
  class ord_mod {
  private:
    T m_P;
    ::std::vector<T> m_p;
    ::std::vector<T> m_e;
    ::std::vector<::std::vector<T>> m_E;

    ::std::size_t n() const {
      return this->m_p.size();
    }

  public:
    ord_mod() = default;
    ord_mod(const T P) : m_P(P) {
      assert(::tools::is_prime(P));

      const auto factors = ::tools::prime_factorization(P - 1);
      for (::std::size_t l = 0, r = 0; l < factors.size(); l = r) {
        for (; r < factors.size() && factors[l] == factors[r]; ++r);
        this->m_p.push_back(factors[l]);
        this->m_e.push_back(r - l);
      }

      this->m_E.resize(this->n() + 1);
      for (::std::size_t l = 0; l <= this->n(); ++l) {
        this->m_E[l].resize(this->n() + 1);
        this->m_E[l][l] = 1;
      }
      for (::std::size_t l = 0; l + 1 <= this->n(); ++l) {
        this->m_E[l][l + 1] = ::tools::pow(this->m_p[l], this->m_e[l]);
      }
      for (::std::size_t l = 0; l + 2 <= this->n(); ++l) {
        for (::std::size_t r = l + 2; r <= this->n(); ++r) {
          this->m_E[l][r] = this->m_E[l][r - 1] * this->m_E[r - 1][r];
        }
      }
    }

    T query(const T x) const {
      assert(1 <= x && x < this->m_P);

      T m = 1;
      ::tools::fix([&](auto&& dfs, const ::std::size_t l, const ::std::size_t r, const T Xbar_l_r) -> void {
        switch (r - l) {
        case 0:
          return;
        case 1:
          {
            ::std::size_t f_l;
            T v;
            for (f_l = 0, v = Xbar_l_r; v != 1; ++f_l, v = ::tools::pow_mod(v, this->m_p[l], this->m_P));
            m *= ::tools::pow(this->m_p[l], f_l);
          }
          return;
        default:
          {
            const auto m = (l + r) / 2;
            dfs(l, m, ::tools::pow_mod(Xbar_l_r, this->m_E[m][r], this->m_P));
            dfs(m, r, ::tools::pow_mod(Xbar_l_r, this->m_E[l][m], this->m_P));
          }
          return;
        }
      })(0, this->n(), x);

      return m;
    }

    ::std::map<T, T> count() const {
      ::std::vector<T> E{1};
      for (const auto e_i : this->m_e) {
        E.push_back(E.back() * (e_i + 1));
      }

      ::std::vector<T> A(E.back());
      A[0] = 1;
      for (::std::size_t i = 0; i < this->n(); ++i) {
        for (T f = 1; f <= this->m_e[i]; ++f) {
          for (T s = 0; s < E[i]; ++s) {
            A[f * E[i] + s] = A[(f - 1) * E[i] + s] * this->m_p[i];
          }
        }
      }
      for (auto&& A_i : A) A_i = (this->m_P - 1) / A_i;

      auto B = A;
      for (::std::size_t i = 0; i < this->n(); ++i) {
        for (T s = 0, s_end = E.back() / E[i + 1]; s < s_end; ++s) {
          for (T t = 0; t < E[i]; ++t) {
            for (T f = 0; f < this->m_e[i]; ++f) {
              B[s * E[i + 1] + f * E[i] + t] -= B[s * E[i + 1] + (f + 1) * E[i] + t];
            }
          }
        }
      }

      ::std::map<T, T> result;
      for (T i = 0; i < E.back(); ++i) {
        result.emplace(A[i], B[i]);
      }
      return result;
    }
  };
}

#endif

#line 1 "tools/ord_mod.hpp"



#include <vector>
#include <cassert>
#include <cstddef>
#include <map>
#line 1 "tools/is_prime.hpp"



#include <array>
#line 1 "tools/prod_mod.hpp"



#line 1 "tools/uint128_t.hpp"



#line 1 "tools/detail/int128_t.hpp"



#include <algorithm>
#line 7 "tools/detail/int128_t.hpp"
#include <cstdint>
#include <functional>
#include <iostream>
#include <limits>
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"



namespace tools {
  constexpr float abs(const float x) {
    return x < 0 ? -x : x;
  }
  constexpr double abs(const double x) {
    return x < 0 ? -x : x;
  }
  constexpr long double abs(const long double x) {
    return x < 0 ? -x : x;
  }
  constexpr int abs(const int x) {
    return x < 0 ? -x : x;
  }
  constexpr long abs(const long x) {
    return x < 0 ? -x : x;
  }
  constexpr long long abs(const long long x) {
    return x < 0 ? -x : x;
  }
  constexpr unsigned int abs(const unsigned int x) {
    return x;
  }
  constexpr unsigned long abs(const unsigned long x) {
    return x;
  }
  constexpr unsigned long long abs(const unsigned long long x) {
    return x;
  }
}


#line 1 "tools/bit_ceil.hpp"



#include <bit>
#line 6 "tools/bit_ceil.hpp"
#include <type_traits>
#line 1 "tools/is_integral.hpp"



#line 5 "tools/is_integral.hpp"

namespace tools {
  template <typename T>
  struct is_integral : ::std::is_integral<T> {};

  template <typename T>
  inline constexpr bool is_integral_v = ::tools::is_integral<T>::value;
}


#line 1 "tools/is_signed.hpp"



#line 5 "tools/is_signed.hpp"

namespace tools {
  template <typename T>
  struct is_signed : ::std::is_signed<T> {};

  template <typename T>
  inline constexpr bool is_signed_v = ::tools::is_signed<T>::value;
}


#line 1 "tools/make_unsigned.hpp"



#line 5 "tools/make_unsigned.hpp"

namespace tools {
  template <typename T>
  struct make_unsigned : ::std::make_unsigned<T> {};

  template <typename T>
  using make_unsigned_t = typename ::tools::make_unsigned<T>::type;
}


#line 10 "tools/bit_ceil.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_ceil(T) noexcept;

  template <typename T>
  constexpr T bit_ceil(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_ceil<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_ceil(x);
    }
  }
}


#line 1 "tools/bit_floor.hpp"



#line 10 "tools/bit_floor.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_floor(T) noexcept;

  template <typename T>
  constexpr T bit_floor(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_floor<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_floor(x);
    }
  }
}


#line 1 "tools/bit_width.hpp"



#line 10 "tools/bit_width.hpp"

namespace tools {
  template <typename T>
  constexpr int bit_width(T) noexcept;

  template <typename T>
  constexpr int bit_width(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_width<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_width(x);
    }
  }
}


#line 1 "tools/countr_zero.hpp"



#line 12 "tools/countr_zero.hpp"

namespace tools {
  template <typename T>
  constexpr int countr_zero(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::std::min(::tools::countr_zero<::tools::make_unsigned_t<T>>(x), ::std::numeric_limits<T>::digits);
    } else {
      return ::std::countr_zero(x);
    }
  }
}


#line 1 "tools/is_unsigned.hpp"



#line 5 "tools/is_unsigned.hpp"

namespace tools {
  template <typename T>
  struct is_unsigned : ::std::is_unsigned<T> {};

  template <typename T>
  inline constexpr bool is_unsigned_v = ::tools::is_unsigned<T>::value;
}


#line 1 "tools/hash_combine.hpp"



#line 6 "tools/hash_combine.hpp"

// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.

// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

namespace tools {
  template <typename T>
  void hash_combine(::std::size_t& seed, const T& v) {
    static const ::std::hash<T> hasher;
    static constexpr ::std::size_t k_mul = 0x9ddfea08eb382d69ULL;
    ::std::size_t a = (hasher(v) ^ seed) * k_mul;
    a ^= (a >> 47);
    ::std::size_t b = (seed ^ a) * k_mul;
    b ^= (b >> 47);
    seed = b * k_mul;
  }
}


#line 1 "tools/make_signed.hpp"



#line 5 "tools/make_signed.hpp"

namespace tools {
  template <typename T>
  struct make_signed : ::std::make_signed<T> {};

  template <typename T>
  using make_signed_t = typename ::tools::make_signed<T>::type;
}


#line 1 "tools/now.hpp"



#include <chrono>

namespace tools {
  inline long long now() {
    return ::std::chrono::duration_cast<::std::chrono::nanoseconds>(::std::chrono::high_resolution_clock::now().time_since_epoch()).count();
  }
}


#line 25 "tools/detail/int128_t.hpp"

namespace tools {
  using uint128_t = unsigned __int128;
  using int128_t = __int128;

  namespace detail {
    namespace int128_t {
      constexpr ::tools::uint128_t parse_unsigned(const ::std::string_view s) noexcept {
        assert(!s.empty());
        ::tools::uint128_t x = 0;
        ::std::size_t i = s[0] == '+';
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x <<= 4;
            if ('0' <= s[i] && s[i] <= '9') {
              x |= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x |= s[i] - 'a' + 10;
            } else {
              x |= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x += s[i] - '0';
          }
        }
        return x;
      }

      constexpr ::tools::int128_t parse_signed(const ::std::string_view s) noexcept {
        assert(!s.empty());
        ::tools::int128_t x = 0;
        if (s[0] == '-') {
          ::std::size_t i = 1;
          if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
            for (i += 2; i < s.size(); ++i) {
              assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
              x *= 16;
              if ('0' <= s[i] && s[i] <= '9') {
                x -= s[i] - '0';
              } else if ('a' <= s[i] && s[i] <= 'f') {
                x -= s[i] - 'a' + 10;
              } else {
                x -= s[i] - 'A' + 10;
              }
            }
          } else {
            for (; i < s.size(); ++i) {
              assert('0' <= s[i] && s[i] <= '9');
              x *= 10;
              x -= s[i] - '0';
            }
          }
        } else {
          ::std::size_t i = s[0] == '+';
          if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
            for (i += 2; i < s.size(); ++i) {
              assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
              x <<= 4;
              if ('0' <= s[i] && s[i] <= '9') {
                x |= s[i] - '0';
              } else if ('a' <= s[i] && s[i] <= 'f') {
                x |= s[i] - 'a' + 10;
              } else {
                x |= s[i] - 'A' + 10;
              }
            }
          } else {
            for (; i < s.size(); ++i) {
              assert('0' <= s[i] && s[i] <= '9');
              x *= 10;
              x += s[i] - '0';
            }
          }
        }
        return x;
      }
    }
  }

  constexpr ::tools::uint128_t abs(const ::tools::uint128_t& x) noexcept {
    return x;
  }
  constexpr ::tools::int128_t abs(const ::tools::int128_t& x) {
    return x >= 0 ? x : -x;
  }
}

#define UINT128_C(c) ::tools::detail::int128_t::parse_unsigned(#c)
#define INT128_C(c) ::tools::detail::int128_t::parse_signed(#c)

inline ::std::istream& operator>>(::std::istream& is, ::tools::uint128_t& x) {
  ::std::string s;
  is >> s;
  x = ::tools::detail::int128_t::parse_unsigned(s);
  return is;
}
inline ::std::istream& operator>>(::std::istream& is, ::tools::int128_t& x) {
  ::std::string s;
  is >> s;
  x = ::tools::detail::int128_t::parse_signed(s);
  return is;
}

inline ::std::ostream& operator<<(::std::ostream& os, ::tools::uint128_t x) {
  ::std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else {
    s.push_back('0');
  }

  ::std::ranges::reverse(s);
  return os << s;
}
inline ::std::ostream& operator<<(::std::ostream& os, ::tools::int128_t x) {
  ::std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else if (x < 0) {
    while (x < 0) {
      s.push_back('0' + (-(x % 10)));
      x /= 10;
    }
    s.push_back('-');
  } else {
    s.push_back('0');
  }

  ::std::ranges::reverse(s);
  return os << s;
}

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
namespace std {
  template <>
  struct hash<::tools::uint128_t> {
    ::std::size_t operator()(const ::tools::uint128_t& x) const {
      static const ::std::size_t seed = ::tools::now();

      ::std::size_t hash = seed;
      ::tools::hash_combine(hash, static_cast<::std::uint64_t>(x >> 64));
      ::tools::hash_combine(hash, static_cast<::std::uint64_t>(x & ((UINT128_C(1) << 64) - 1)));
      return hash;
    }
  };
  template <>
  struct hash<::tools::int128_t> {
    ::std::size_t operator()(const ::tools::int128_t& x) const {
      static ::std::hash<::tools::uint128_t> hasher;
      return hasher(static_cast<::tools::uint128_t>(x));
    }
  };
}
#endif

namespace tools {
  template <>
  struct is_integral<::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<const ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<const ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<volatile ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<const volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<const volatile ::tools::uint128_t> : ::std::true_type {};

  template <>
  struct is_signed<::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<const ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<const ::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<volatile ::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<const volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<const volatile ::tools::uint128_t> : ::std::false_type {};

  template <>
  struct is_unsigned<::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<const ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<const ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<volatile ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<volatile ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<const volatile ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<const volatile ::tools::uint128_t> : ::std::true_type {};

  template <>
  struct make_signed<::tools::int128_t> {
    using type = ::tools::int128_t;
  };
  template <>
  struct make_signed<::tools::uint128_t> {
    using type = ::tools::int128_t;
  };
  template <>
  struct make_signed<const ::tools::int128_t> {
    using type = const ::tools::int128_t;
  };
  template <>
  struct make_signed<const ::tools::uint128_t> {
    using type = const ::tools::int128_t;
  };
  template <>
  struct make_signed<volatile ::tools::int128_t> {
    using type = volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<volatile ::tools::uint128_t> {
    using type = volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<const volatile ::tools::int128_t> {
    using type = const volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<const volatile ::tools::uint128_t> {
    using type = const volatile ::tools::int128_t;
  };

  template <>
  struct make_unsigned<::tools::int128_t> {
    using type = ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<::tools::uint128_t> {
    using type = ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const ::tools::int128_t> {
    using type = const ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const ::tools::uint128_t> {
    using type = const ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile ::tools::int128_t> {
    using type = volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile ::tools::uint128_t> {
    using type = volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile ::tools::int128_t> {
    using type = const volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile ::tools::uint128_t> {
    using type = const volatile ::tools::uint128_t;
  };

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  constexpr ::tools::uint128_t bit_ceil<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    if (x <= 1) return 1;
    --x;
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    x |= x >> 32;
    x |= x >> 64;
    return ++x;
  }

  template <>
  constexpr ::tools::uint128_t bit_floor<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    x |= x >> 32;
    x |= x >> 64;
    return x & ~(x >> 1);
  }

  template <>
  constexpr int bit_width<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    int w = 0;
    if (x & UINT128_C(0xffffffffffffffff0000000000000000)) {
      x >>= 64;
      w += 64;
    }
    if (x & UINT128_C(0xffffffff00000000)) {
      x >>= 32;
      w += 32;
    }
    if (x & UINT128_C(0xffff0000)) {
      x >>= 16;
      w += 16;
    }
    if (x & UINT128_C(0xff00)) {
      x >>= 8;
      w += 8;
    }
    if (x & UINT128_C(0xf0)) {
      x >>= 4;
      w += 4;
    }
    if (x & UINT128_C(0xc)) {
      x >>= 2;
      w += 2;
    }
    if (x & UINT128_C(0x2)) {
      x >>= 1;
      w += 1;
    }
    w += x;
    return w;
  }

  namespace detail {
    namespace countr_zero {
      template <::std::size_t N>
      struct ntz_traits;

      template <>
      struct ntz_traits<128> {
        using type = ::tools::uint128_t;
        static constexpr int shift = 120;
        static constexpr type magic = UINT128_C(0x01061438916347932a5cd9d3ead7b77f);
        static constexpr int ntz_table[255] = {
          128,   0,   1,  -1,   2,  -1,   8,  -1,   3,  -1,  15,  -1,   9,  -1,  22,  -1,
            4,  -1,  29,  -1,  16,  -1,  36,  -1,  10,  -1,  43,  -1,  23,  -1,  50,  -1,
            5,  -1,  33,  -1,  30,  -1,  57,  -1,  17,  -1,  64,  -1,  37,  -1,  71,  -1,
           11,  -1,  60,  -1,  44,  -1,  78,  -1,  24,  -1,  85,  -1,  51,  -1,  92,  -1,
           -1,   6,  -1,  20,  -1,  34,  -1,  48,  31,  -1,  -1,  69,  58,  -1,  -1,  90,
           18,  -1,  67,  -1,  65,  -1,  99,  -1,  38,  -1, 101,  -1,  72,  -1, 106,  -1,
           -1,  12,  -1,  40,  -1,  61,  -1,  82,  45,  -1,  -1, 103,  79,  -1, 113,  -1,
           -1,  25,  -1,  74,  86,  -1,  -1, 116,  -1,  52,  -1, 108,  -1,  93,  -1, 120,
          127,  -1,  -1,   7,  -1,  14,  -1,  21,  -1,  28,  -1,  35,  -1,  42,  -1,  49,
           -1,  32,  -1,  56,  -1,  63,  -1,  70,  -1,  59,  -1,  77,  -1,  84,  -1,  91,
           -1,  19,  -1,  47,  -1,  68,  -1,  89,  -1,  66,  -1,  98,  -1, 100,  -1, 105,
           -1,  39,  -1,  81,  -1, 102,  -1, 112,  -1,  73,  -1, 115,  -1, 107,  -1, 119,
          126,  -1,  13,  -1,  27,  -1,  41,  -1,  -1,  55,  62,  -1,  -1,  76,  83,  -1,
           -1,  46,  -1,  88,  -1,  97,  -1, 104,  -1,  80,  -1, 111,  -1, 114,  -1, 118,
          125,  -1,  26,  -1,  54,  -1,  75,  -1,  -1,  87,  96,  -1,  -1, 110,  -1, 117,
          124,  -1,  53,  -1,  -1,  95, 109,  -1, 123,  -1,  94,  -1, 122,  -1, 121
        };
      };

      template <typename T>
      constexpr int impl(const T x) noexcept {
        using tr = ::tools::detail::countr_zero::ntz_traits<::std::numeric_limits<T>::digits>;
        using type = typename tr::type;
        return tr::ntz_table[static_cast<type>(tr::magic * static_cast<type>(x & -x)) >> tr::shift];
      }
    }
  }

  template <>
  constexpr int countr_zero<::tools::uint128_t>(const ::tools::uint128_t x) noexcept {
    return ::tools::detail::countr_zero::impl(x);
  }
#endif
}


#line 5 "tools/uint128_t.hpp"


#line 5 "tools/prod_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 prod_mod(const T1 x, const T2 y, const T3 m) {
    using u128 = ::tools::uint128_t;
    u128 prod_mod = u128(x >= 0 ? x : -x) * u128(y >= 0 ? y : -y) % u128(m);
    if ((x >= 0) ^ (y >= 0)) prod_mod = u128(m) - prod_mod;
    return prod_mod;
  }
}


#line 1 "tools/pow_mod.hpp"



#line 1 "tools/mod.hpp"



#line 7 "tools/mod.hpp"

namespace tools {

  template <typename M, typename N> requires (
    ::tools::is_integral_v<M> && !::std::is_same_v<::std::remove_cv_t<M>, bool> &&
    ::tools::is_integral_v<N> && !::std::is_same_v<::std::remove_cv_t<N>, bool>)
  constexpr ::std::common_type_t<M, N> mod(const M a, const N b) noexcept {
    assert(b != 0);

    using UM = ::std::make_unsigned_t<M>;
    using UN = ::std::make_unsigned_t<N>;
    const UM ua = a >= 0 ? a : static_cast<UM>(-(a + 1)) + 1;
    const UN ub = b >= 0 ? b : static_cast<UN>(-(b + 1)) + 1;
    auto r = ua % ub;
    if (a < 0 && r > 0) {
      r = ub - r;
    }
    return r;
  }
}


#line 6 "tools/pow_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 pow_mod(const T1 x, T2 n, const T3 m) {
    if (m == 1) return 0;
    T3 r = 1;
    T3 y = ::tools::mod(x, m);
    while (n > 0) {
      if ((n & 1) > 0) {
        r = ::tools::prod_mod(r, y, m);
      }
      y = ::tools::prod_mod(y, y, m);
      n /= 2;
    }
    return r;
  }
}


#line 7 "tools/is_prime.hpp"

namespace tools {

  constexpr bool is_prime(const unsigned long long n) {
    constexpr ::std::array<unsigned long long, 7> bases = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};

    if (n <= 1) return false;
    if (n == 2) return true;
    if (n % 2 == 0) return false;

    auto d = n - 1;
    for (; d % 2 == 0; d /= 2);

    for (const auto a : bases) {
      if (a % n == 0) return true;

      auto power = d;
      auto target = ::tools::pow_mod(a, power, n);

      bool is_composite = true;
      if (target == 1) is_composite = false;
      for (; is_composite && power != n - 1; power *= 2, target = ::tools::prod_mod(target, target, n)) {
        if (target == n - 1) is_composite = false;
      }

      if (is_composite) {
        return false;
      }
    }

    return true;
  }
}


#line 1 "tools/prime_factorization.hpp"



#line 6 "tools/prime_factorization.hpp"
#include <queue>
#include <utility>
#line 9 "tools/prime_factorization.hpp"
#include <cmath>
#include <numeric>
#line 1 "tools/pow2.hpp"



#line 6 "tools/pow2.hpp"

namespace tools {

  template <typename T, typename ::std::enable_if<::std::is_unsigned<T>::value, ::std::nullptr_t>::type = nullptr>
  constexpr T pow2(const T x) {
    return static_cast<T>(1) << x;
  }

  template <typename T, typename ::std::enable_if<::std::is_signed<T>::value, ::std::nullptr_t>::type = nullptr>
  constexpr T pow2(const T x) {
    return static_cast<T>(static_cast<typename ::std::make_unsigned<T>::type>(1) << static_cast<typename ::std::make_unsigned<T>::type>(x));
  }
}


#line 1 "tools/floor_log2.hpp"



#line 6 "tools/floor_log2.hpp"

namespace tools {
  template <typename T>
  constexpr T floor_log2(T x) noexcept {
    assert(x > 0);
    return ::tools::bit_width(x) - 1;
  }
}


#line 15 "tools/prime_factorization.hpp"

namespace tools {

  template <typename T>
  ::std::vector<T> prime_factorization(T n) {
    assert(1 <= n && n <= 1000000000000000000);
    ::std::vector<T> result;

    if (n == 1) return result;

    ::std::queue<::std::pair<T, T>> factors({::std::pair<T, T>(n, 1)});
    while (!factors.empty()) {
      const T factor = factors.front().first;
      const T occurrences = factors.front().second;
      factors.pop();
      if (::tools::is_prime(factor)) {
        for (T i = 0; i < occurrences; ++i) {
          result.push_back(factor);
        }
      } else {
        const T m = ::tools::pow2((::tools::floor_log2(factor) + 1) / 8);
        for (T c = 1; ; ++c) {
          const auto f = [&](T& x) {
            x = ::tools::prod_mod(x, x, factor);
            x += c;
            if (x >= factor) x -= factor;
          };
          T y = 2;
          T r = 1;
          T q = 1;
          T x, g, ys;
          do {
            x = y;
            for (T i = 0; i < r; ++i) {
              f(y);
            }
            T k = 0;
            do {
              ys = y;
              for (T i = 0; i < ::std::min(m, r - k); ++i) {
                f(y);
                q = ::tools::prod_mod(q, ::std::abs(x - y), factor);
              }
              g = ::std::gcd(q, factor);
              k += m;
            } while (k < r && g == 1);
            r *= 2;
          } while (g == 1);
          if (g == factor) {
            do {
              f(ys);
              g = ::std::gcd(::std::abs(x - ys), factor);
            } while (g == 1);
          }
          if (g < factor) {
            T h = factor / g;
            if (h < g) ::std::swap(g, h);
            T n = 1;
            while (h % g == 0) {
              h /= g;
              ++n;
            }
            factors.emplace(g, occurrences * n);
            if (h > 1) factors.emplace(h, occurrences);
            break;
          }
        }
      }
    }

    ::std::sort(result.begin(), result.end());
    return result;
  }
}


#line 1 "tools/pow.hpp"



#line 1 "tools/monoid.hpp"



#line 1 "tools/gcd.hpp"



#line 6 "tools/gcd.hpp"

namespace tools {
  template <typename M, typename N>
  constexpr ::std::common_type_t<M, N> gcd(const M m, const N n) {
    return ::std::gcd(m, n);
  }
}


#line 9 "tools/monoid.hpp"

namespace tools {
  namespace monoid {
    template <typename M, M ...dummy>
    struct max;

    template <typename M>
    struct max<M> {
      static_assert(::std::is_arithmetic_v<M>, "M must be a built-in arithmetic type.");

      using T = M;
      static T op(const T lhs, const T rhs) {
        return ::std::max(lhs, rhs);
      }
      static T e() {
        if constexpr (::std::is_integral_v<M>) {
          return ::std::numeric_limits<M>::min();
        } else {
          return -::std::numeric_limits<M>::infinity();
        }
      }
    };

    template <typename M, M E>
    struct max<M, E> {
      static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");

      using T = M;
      static T op(const T lhs, const T rhs) {
        assert(E <= lhs);
        assert(E <= rhs);
        return ::std::max(lhs, rhs);
      }
      static T e() {
        return E;
      }
    };

    template <typename M, M ...dummy>
    struct min;

    template <typename M>
    struct min<M> {
      static_assert(::std::is_arithmetic_v<M>, "M must be a built-in arithmetic type.");

      using T = M;
      static T op(const T lhs, const T rhs) {
        return ::std::min(lhs, rhs);
      }
      static T e() {
        if constexpr (::std::is_integral_v<M>) {
          return ::std::numeric_limits<M>::max();
        } else {
          return ::std::numeric_limits<M>::infinity();
        }
      }
    };

    template <typename M, M E>
    struct min<M, E> {
      static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");

      using T = M;
      static T op(const T lhs, const T rhs) {
        assert(lhs <= E);
        assert(rhs <= E);
        return ::std::min(lhs, rhs);
      }
      static T e() {
        return E;
      }
    };

    template <typename M>
    struct multiplies {
    private:
      using VR = ::std::conditional_t<::std::is_arithmetic_v<M>, const M, const M&>;

    public:
      using T = M;
      static T op(VR lhs, VR rhs) {
        return lhs * rhs;
      }
      static T e() {
        return T(1);
      }
    };

    template <>
    struct multiplies<bool> {
      using T = bool;
      static T op(const bool lhs, const bool rhs) {
        return lhs && rhs;
      }
      static T e() {
        return true;
      }
    };

    template <typename M>
    struct gcd {
    private:
      static_assert(!::std::is_arithmetic_v<M> || (::std::is_integral_v<M> && !::std::is_same_v<M, bool>), "If M is a built-in arithmetic type, it must be integral except for bool.");
      using VR = ::std::conditional_t<::std::is_arithmetic_v<M>, const M, const M&>;

    public:
      using T = M;
      static T op(VR lhs, VR rhs) {
        return ::tools::gcd(lhs, rhs);
      }
      static T e() {
        return T(0);
      }
    };

    template <typename M, M E>
    struct update {
      static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");

      using T = M;
      static T op(const T lhs, const T rhs) {
        return lhs == E ? rhs : lhs;
      }
      static T e() {
        return E;
      }
    };
  }
}


#line 1 "tools/square.hpp"



#line 1 "tools/is_monoid.hpp"



#line 6 "tools/is_monoid.hpp"

namespace tools {

  template <typename M, typename = void>
  struct is_monoid : ::std::false_type {};

  template <typename M>
  struct is_monoid<M, ::std::enable_if_t<
    ::std::is_same_v<typename M::T, decltype(M::op(::std::declval<typename M::T>(), ::std::declval<typename M::T>()))> &&
    ::std::is_same_v<typename M::T, decltype(M::e())>
  , void>> : ::std::true_type {};

  template <typename M>
  inline constexpr bool is_monoid_v = ::tools::is_monoid<M>::value;
}


#line 6 "tools/square.hpp"

namespace tools {

  template <typename M>
  ::std::enable_if_t<::tools::is_monoid_v<M>, typename M::T> square(const typename M::T& x) {
    return M::op(x, x);
  }

  template <typename T>
  ::std::enable_if_t<!::tools::is_monoid_v<T>, T> square(const T& x) {
    return x * x;
  }
}


#line 9 "tools/pow.hpp"

namespace tools {

  template <typename M, typename E>
  ::std::enable_if_t<::std::is_integral_v<E>, typename M::T> pow(const typename M::T& base, const E exponent) {
    assert(exponent >= 0);
    return exponent == 0
      ? M::e()
      : exponent % 2 == 0
        ? ::tools::square<M>(::tools::pow<M>(base, exponent / 2))
        : M::op(::tools::pow<M>(base, exponent - 1), base);
  }

  template <typename T, typename E>
  ::std::enable_if_t<::std::is_integral_v<E>, T> pow(const T& base, const E exponent) {
    assert(exponent >= 0);
    return ::tools::pow<::tools::monoid::multiplies<T>>(base, exponent);
  }

  template <typename T, typename E>
  auto pow(const T base, const E exponent) -> ::std::enable_if_t<!::std::is_integral_v<E>, decltype(::std::pow(base, exponent))> {
    return ::std::pow(base, exponent);
  }
}


#line 1 "tools/fix.hpp"



#line 6 "tools/fix.hpp"

namespace tools {
  template <typename F>
  struct fix : F {
    template <typename G>
    fix(G&& g) : F({::std::forward<G>(g)}) {
    }

    template <typename... Args>
    decltype(auto) operator()(Args&&... args) const {
      return F::operator()(*this, ::std::forward<Args>(args)...);
    }
  };

  template <typename F>
  fix(F&&) -> fix<::std::decay_t<F>>;
}


#line 13 "tools/ord_mod.hpp"

namespace tools {
  template <typename T>
  class ord_mod {
  private:
    T m_P;
    ::std::vector<T> m_p;
    ::std::vector<T> m_e;
    ::std::vector<::std::vector<T>> m_E;

    ::std::size_t n() const {
      return this->m_p.size();
    }

  public:
    ord_mod() = default;
    ord_mod(const T P) : m_P(P) {
      assert(::tools::is_prime(P));

      const auto factors = ::tools::prime_factorization(P - 1);
      for (::std::size_t l = 0, r = 0; l < factors.size(); l = r) {
        for (; r < factors.size() && factors[l] == factors[r]; ++r);
        this->m_p.push_back(factors[l]);
        this->m_e.push_back(r - l);
      }

      this->m_E.resize(this->n() + 1);
      for (::std::size_t l = 0; l <= this->n(); ++l) {
        this->m_E[l].resize(this->n() + 1);
        this->m_E[l][l] = 1;
      }
      for (::std::size_t l = 0; l + 1 <= this->n(); ++l) {
        this->m_E[l][l + 1] = ::tools::pow(this->m_p[l], this->m_e[l]);
      }
      for (::std::size_t l = 0; l + 2 <= this->n(); ++l) {
        for (::std::size_t r = l + 2; r <= this->n(); ++r) {
          this->m_E[l][r] = this->m_E[l][r - 1] * this->m_E[r - 1][r];
        }
      }
    }

    T query(const T x) const {
      assert(1 <= x && x < this->m_P);

      T m = 1;
      ::tools::fix([&](auto&& dfs, const ::std::size_t l, const ::std::size_t r, const T Xbar_l_r) -> void {
        switch (r - l) {
        case 0:
          return;
        case 1:
          {
            ::std::size_t f_l;
            T v;
            for (f_l = 0, v = Xbar_l_r; v != 1; ++f_l, v = ::tools::pow_mod(v, this->m_p[l], this->m_P));
            m *= ::tools::pow(this->m_p[l], f_l);
          }
          return;
        default:
          {
            const auto m = (l + r) / 2;
            dfs(l, m, ::tools::pow_mod(Xbar_l_r, this->m_E[m][r], this->m_P));
            dfs(m, r, ::tools::pow_mod(Xbar_l_r, this->m_E[l][m], this->m_P));
          }
          return;
        }
      })(0, this->n(), x);

      return m;
    }

    ::std::map<T, T> count() const {
      ::std::vector<T> E{1};
      for (const auto e_i : this->m_e) {
        E.push_back(E.back() * (e_i + 1));
      }

      ::std::vector<T> A(E.back());
      A[0] = 1;
      for (::std::size_t i = 0; i < this->n(); ++i) {
        for (T f = 1; f <= this->m_e[i]; ++f) {
          for (T s = 0; s < E[i]; ++s) {
            A[f * E[i] + s] = A[(f - 1) * E[i] + s] * this->m_p[i];
          }
        }
      }
      for (auto&& A_i : A) A_i = (this->m_P - 1) / A_i;

      auto B = A;
      for (::std::size_t i = 0; i < this->n(); ++i) {
        for (T s = 0, s_end = E.back() / E[i + 1]; s < s_end; ++s) {
          for (T t = 0; t < E[i]; ++t) {
            for (T f = 0; f < this->m_e[i]; ++f) {
              B[s * E[i + 1] + f * E[i] + t] -= B[s * E[i + 1] + (f + 1) * E[i] + t];
            }
          }
        }
      }

      ::std::map<T, T> result;
      for (T i = 0; i < E.back(); ++i) {
        result.emplace(A[i], B[i]);
      }
      return result;
    }
  };
}

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