Euler's Phi Function
(math/number-theory/EulersPhiFunction.hpp)

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• Last update: 2025-05-20 02:16:37+09:00
• Include: #include "math/number-theory/EulersPhiFunction.hpp"

概要

todo

計算量

todo

Depends on

• Factorize (math/number-theory/Factorize.hpp)
• Primality Test (math/number-theory/PrimalityTest.hpp)

Required by

• Tetration Mod (math/number-theory/TetrationMod.hpp)

Verified with

• verify/AizuOnlineJudge/math/number-theory/EulersPhiFunction.test.cpp
• verify/LibraryChecker/math/number-theory/TetrationMod.test.cpp

Code

#include "Factorize.hpp"

long long Eulers_phi_function(long long n) {
  __uint128_t upper = n;
  __uint128_t lower = 1;
  for (const long long p : factorize(n, true)) {
    upper *= (p - 1);
    lower *= p;
  }
  return upper / lower;
}

#line 1 "math/number-theory/PrimalityTest.hpp"
__int128_t mod_pow(__int128_t a, long long n, long long m) {
    __int128_t res = 1;
    a %= m;
    while (n) {
        if (n & 1) res = (res * a) % m;
        a = (a * a) % m;
        n >>= 1;
    }
    return res;
}

constexpr long long MR[] = {2, 7, 61};
constexpr long long MRl[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};

bool Miller_Rabin(long long n) {
    long long s = 0;
    long long d = n - 1;
    while ((d & 1) == 0) {
        s++;
        d >>= 1;
    }
    for (int i = 0; i < 3; i++) {
        if (n <= MR[i]) return true;
        __int128_t x = mod_pow(MR[i], d, n);
        if (x != 1) {
            bool ok = false;
            for (int t = 0; t < s; t++) {
                if (x == n - 1) {
                    ok = true;
                    break;
                }
                x = x * x % n;
            }
            if (!ok) return false;
        }
    }
    return true;
}

bool Miller_Rabinl(long long n) {
    long long s = 0;
    long long d = n - 1;
    while ((d & 1) == 0) {
        s++;
        d >>= 1;
    }
    for (int i = 0; i < 7; i++) {
        if (n <= MRl[i]) return true;
        __int128_t x = mod_pow(MRl[i], d, n);
        if (x != 1) {
            bool ok = false;
            for (int t = 0; t < s; t++) {
                if (x == n - 1) {
                    ok = true;
                    break;
                }
                x = x * x % n;
            }
            if (!ok) return false;
        }
    }
    return true;
}

bool brute_force(long long n) {
    for (int i = 2; i * i <= n; i++) {
        if (n % i == 0) return false;
    }
    return true;
}

bool is_prime(long long n) {
    if (n == 2) return true;
    if ((n & 1) == 0 or n < 2) return false;
    if (n < 1000) return brute_force(n);
    if (n < 4759123141LL) {
        return Miller_Rabin(n);
    }
    return Miller_Rabinl(n);
}
#line 2 "math/number-theory/Factorize.hpp"

long long find_prime_factor(long long n) {
  if ((n & 1) == 0) return 2;
  long long m = int64_t(powl(n, 0.125)) + 1;
  for (int i = 1; i < n; i++) {
    long long y = 0;
    long long g = 1;
    long long q = 1;
    long long r = 1;
    long long k = 0;
    long long ys = 0;
    long long x = 0;
    while (g == 1) {
      x = y;
      while (k < 3ll * r / 4) {
        y = (__int128_t(y) * y + i) % n;
        k++;
      }
      while (k < r and g == 1) {
        ys = y;
        for (int j = 0; j < min(m, r - k); j++) {
          y = (__int128_t(y) * y + i) % n;
          q = (__int128_t(q) * abs(x - y)) % n;
        }
        g = gcd(q, n);
        k += m;
      }
      k = r;
      r <<= 1;
    }
    if (g == n) {
      g = 1;
      y = ys;
      while (g == 1) {
        y = (__int128_t(y) * y + i) % n;
        g = gcd(abs(x - y), n);
      }
    }
    if (g == n) continue;
    if (is_prime(g)) return g;
    if (is_prime(n / g)) return n / g;
    return find_prime_factor(g);
  }
  return -1;
}

vector<long long> factorize(long long n, bool set = false) {
  vector<long long> res;
  while (!is_prime(n) and n > 1) {
    long long p = find_prime_factor(n);
    if (set) res.emplace_back(p);
    while (n % p == 0) {
      n /= p;
      if (!set) res.emplace_back(p);
    }
  }
  if (n > 1) {
    res.emplace_back(n);
  }
  sort(res.begin(), res.end());
  return res;
}
#line 2 "math/number-theory/EulersPhiFunction.hpp"

long long Eulers_phi_function(long long n) {
  __uint128_t upper = n;
  __uint128_t lower = 1;
  for (const long long p : factorize(n, true)) {
    upper *= (p - 1);
    lower *= p;
  }
  return upper / lower;
}

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