Mod logarithm
(Mylib/Number/Mod/mod_log.cpp)

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• Last update: 2021-04-23 23:44:44+09:00
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Operations

• mod_log(a, b, n)
  • $a^x = b \pmod m$を満たすxを求める。
  • Time complexity $O(\sqrt m)$

Requirements

Notes

Problems

References

Depends on

• Mod inverse (Mylib/Number/Mod/mod_inv.cpp)
• Mod pow (Mylib/Number/Mod/mod_pow.cpp)

Verified with

• test/yosupo-judge/discrete_logarithm_mod/main.test.cpp

Code

#pragma once
#include <cmath>
#include <numeric>
#include <optional>
#include <unordered_map>
#include "Mylib/Number/Mod/mod_inv.cpp"
#include "Mylib/Number/Mod/mod_pow.cpp"

namespace haar_lib {
  std::optional<int64_t> mod_log(int64_t a, int64_t b, int64_t m) {
    if (b == 1) return 0;

    int64_t d = 0;

    while (1) {
      if (auto g = std::gcd(a, m); g != 1) {
        if (b % g != 0) return {};

        d += 1;
        m /= g;
        b /= g;
        (b *= mod_inv(a / g, m)) %= m;

        if (b == 1) return d;
      } else {
        break;
      }
    }

    const int64_t sq = std::sqrt(m) + 1;

    std::unordered_map<int64_t, int64_t> mp;
    {
      int64_t t = 1 % m;

      for (int i = 0; i < sq; ++i) {
        if (mp.find(t) == mp.end()) mp[t] = i;
        (t *= a) %= m;
      }
    }

    {
      int64_t A = mod_pow(mod_inv(a, m), sq, m);
      int64_t t = b % m;

      for (int i = 0; i < sq; ++i) {
        if (mp.find(t) != mp.end()) {
          int64_t ret = i * sq + mp[t] + d;

          return ret;
        }

        (t *= A) %= m;
      }
    }

    return {};
  }
}  // namespace haar_lib

#line 2 "Mylib/Number/Mod/mod_log.cpp"
#include <cmath>
#include <numeric>
#include <optional>
#include <unordered_map>
#line 2 "Mylib/Number/Mod/mod_inv.cpp"
#include <cstdint>
#include <utility>

namespace haar_lib {
  constexpr int64_t mod_inv(int64_t a, int64_t m) {
    int64_t b = m, u = 1, v = 0;

    while (b) {
      int64_t t = a / b;
      a -= t * b;
      a = a ^ b;
      b = a ^ b;
      a = a ^ b;

      u -= t * v;
      u = u ^ v;
      v = u ^ v;
      u = u ^ v;
    }

    u %= m;
    if (u < 0) u += m;

    return u;
  }
}  // namespace haar_lib
#line 3 "Mylib/Number/Mod/mod_pow.cpp"

namespace haar_lib {
  constexpr int64_t mod_pow(int64_t n, int64_t p, int64_t m) {
    int64_t ret = 1;
    while (p > 0) {
      if (p & 1) (ret *= n) %= m;
      (n *= n) %= m;
      p >>= 1;
    }
    return ret;
  }
}  // namespace haar_lib
#line 8 "Mylib/Number/Mod/mod_log.cpp"

namespace haar_lib {
  std::optional<int64_t> mod_log(int64_t a, int64_t b, int64_t m) {
    if (b == 1) return 0;

    int64_t d = 0;

    while (1) {
      if (auto g = std::gcd(a, m); g != 1) {
        if (b % g != 0) return {};

        d += 1;
        m /= g;
        b /= g;
        (b *= mod_inv(a / g, m)) %= m;

        if (b == 1) return d;
      } else {
        break;
      }
    }

    const int64_t sq = std::sqrt(m) + 1;

    std::unordered_map<int64_t, int64_t> mp;
    {
      int64_t t = 1 % m;

      for (int i = 0; i < sq; ++i) {
        if (mp.find(t) == mp.end()) mp[t] = i;
        (t *= a) %= m;
      }
    }

    {
      int64_t A = mod_pow(mod_inv(a, m), sq, m);
      int64_t t = b % m;

      for (int i = 0; i < sq; ++i) {
        if (mp.find(t) != mp.end()) {
          int64_t ret = i * sq + mp[t] + d;

          return ret;
        }

        (t *= A) %= m;
      }
    }

    return {};
  }
}  // namespace haar_lib

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