log_mod
(src/math/log_mod.hpp)

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• Last update: 2024-11-09 02:03:28+09:00
• Include: #include "src/math/log_mod.hpp"

log_mod

ll log_mod(ll a, ll b, ll mod)

$a^x \equiv b \pmod{\mathrm{mod}}$ を満たす非負整数 $x$ の最小値を返します．
存在しない場合は $-1$ を返します．

制約

• $1 \leq \mathrm{mod}$

計算量

• $O(\sqrt{mod})$

Depends on

• policy_based_data_structure (src/template/policy_based_data_structure.hpp)
• template (src/template/template.hpp)

Verified with

• verify/library_checker/number_theory/discrete_logarithm.test.cpp

Code

#pragma once
#include "../template/template.hpp"
#include "../template/policy_based_data_structure.hpp"
long long log_mod(const long long a, long long b, const long long mod) {
    assert(mod >= 1);
    long long g = 1;
    for(long long i = mod; i; i /= 2) (g *= a) %= mod;
    g = gcd(g, mod);
    long long t = 1, c = 0;
    for(; t % g; ++c) {
        if(t == b) return c;
        (t *= a) %= mod;
    }
    if(b % g) return -1;
    t /= g;
    b /= g;
    const long long n = mod / g;
    long long h = 0, gs = 1;
    for(; h * h < n; ++h) (gs *= a) %= n;
    gp_hash_table<long long, long long> ht;
    for(long long s = 0, e = b; s < h; ht[e] = ++s) {
        (e *= a) %= n;
    }
    for(long long s = 0, e = t; s < n;) {
        (e *= gs) %= n;
        s += h;
        if(ht.find(e) != ht.end()) return c + s - ht[e];
    }
    return -1;
}

#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 2 "src/template/policy_based_data_structure.hpp"
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
using namespace __gnu_pbds;
#line 4 "src/math/log_mod.hpp"
long long log_mod(const long long a, long long b, const long long mod) {
    assert(mod >= 1);
    long long g = 1;
    for(long long i = mod; i; i /= 2) (g *= a) %= mod;
    g = gcd(g, mod);
    long long t = 1, c = 0;
    for(; t % g; ++c) {
        if(t == b) return c;
        (t *= a) %= mod;
    }
    if(b % g) return -1;
    t /= g;
    b /= g;
    const long long n = mod / g;
    long long h = 0, gs = 1;
    for(; h * h < n; ++h) (gs *= a) %= n;
    gp_hash_table<long long, long long> ht;
    for(long long s = 0, e = b; s < h; ht[e] = ++s) {
        (e *= a) %= n;
    }
    for(long long s = 0, e = t; s < n;) {
        (e *= gs) %= n;
        s += h;
        if(ht.find(e) != ht.end()) return c + s - ht[e];
    }
    return -1;
}

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