discrete log / 離散対数 (the baby-step giant-step, $O(\sqrt{m})$)
(modulus/modlog.hpp)

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• Last update: 2021-08-30 04:35:37+09:00
• Include: #include "modulus/modlog.hpp"

Depends on

• modulus/modinv.hpp
• modulus/modpow.hpp
• utils/macros.hpp

Verified with

• modulus/modlog.yosupo.test.cpp
• modulus/modlog.yuki1255.test.cpp

Code

#pragma once
#include <algorithm>
#include <climits>
#include <cmath>
#include <cstdint>
#include <unordered_map>
#include "../modulus/modinv.hpp"
#include "../modulus/modpow.hpp"
#include "../utils/macros.hpp"
#include <iostream>

/**
 * @brief discrete log / 離散対数 (the baby-step giant-step, $O(\sqrt{m})$)
 * @description find the smallest $x \ge 0$ s.t. $g^x \equiv y \pmod{m}$
 * @param m is a positive integer
 * @note -1 if not found
 */
inline int modlog(int g, int y, int m) {
    assert (0 <= g and g < m);
    assert (0 <= y and y < m);
    if (m == 1) return 0;
    if (y == 1) return 0;
    if (g == 0 and y == 0) return 1;

    // meet-in-the-middle; let x = a \sqrt{m} + b
    int sqrt_m = sqrt(m) + 100;  // + 100 is required to bruteforce g^b for b < 100; this avoids problems with g != 0 and y = 0
    assert (sqrt_m >= 0);

    // baby-step: list (y, gy, g^2 y, ...) = (g^x, g^{x + 1}, g^{x + 2}, ...)
    std::unordered_map<int, int> table;
    int baby = 1;
    REP (b, sqrt_m) {
        if (baby == y) return b;
        table[(int64_t)baby * y % m] = b;
        baby = (int64_t)baby * g % m;
    }

    // giant-step: list (g^{sqrt(m)}, g^{2 sqrt(m)}, g^{3 sqrt(m)}, ...)
    int giant = 1;
    REP3 (a, 1, sqrt_m + 3) {
        giant = (int64_t)giant * baby % m;
        auto it = table.find(giant);
        if (it != table.end()) {
            int b = it->second;
            int x = (int64_t)a * sqrt_m - b;
            assert (x >= 0);
            return (modpow(g, x, m) == y ? x : -1);
        }
    }
    return -1;
}

#line 2 "modulus/modlog.hpp"
#include <algorithm>
#include <climits>
#include <cmath>
#include <cstdint>
#include <unordered_map>
#line 3 "modulus/modinv.hpp"
#include <cassert>
#line 5 "modulus/modinv.hpp"

inline int32_t modinv_nocheck(int32_t value, int32_t MOD) {
    assert (0 <= value and value < MOD);
    if (value == 0) return -1;
    int64_t a = value, b = MOD;
    int64_t x = 0, y = 1;
    for (int64_t u = 1, v = 0; a; ) {
        int64_t q = b / a;
        x -= q * u; std::swap(x, u);
        y -= q * v; std::swap(y, v);
        b -= q * a; std::swap(b, a);
    }
    if (not (value * x + MOD * y == b and b == 1)) return -1;
    if (x < 0) x += MOD;
    assert (0 <= x and x < MOD);
    return x;
}

inline int32_t modinv(int32_t x, int32_t MOD) {
    int32_t y = modinv_nocheck(x, MOD);
    assert (y != -1);
    return y;
}
#line 4 "modulus/modpow.hpp"

inline int32_t modpow(uint_fast64_t x, uint64_t k, int32_t MOD) {
    assert (/* 0 <= x and */ x < (uint_fast64_t)MOD);
    uint_fast64_t y = 1;
    for (; k; k >>= 1) {
        if (k & 1) (y *= x) %= MOD;
        (x *= x) %= MOD;
    }
    assert (/* 0 <= y and */ y < (uint_fast64_t)MOD);
    return y;
}
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 10 "modulus/modlog.hpp"
#include <iostream>

/**
 * @brief discrete log / 離散対数 (the baby-step giant-step, $O(\sqrt{m})$)
 * @description find the smallest $x \ge 0$ s.t. $g^x \equiv y \pmod{m}$
 * @param m is a positive integer
 * @note -1 if not found
 */
inline int modlog(int g, int y, int m) {
    assert (0 <= g and g < m);
    assert (0 <= y and y < m);
    if (m == 1) return 0;
    if (y == 1) return 0;
    if (g == 0 and y == 0) return 1;

    // meet-in-the-middle; let x = a \sqrt{m} + b
    int sqrt_m = sqrt(m) + 100;  // + 100 is required to bruteforce g^b for b < 100; this avoids problems with g != 0 and y = 0
    assert (sqrt_m >= 0);

    // baby-step: list (y, gy, g^2 y, ...) = (g^x, g^{x + 1}, g^{x + 2}, ...)
    std::unordered_map<int, int> table;
    int baby = 1;
    REP (b, sqrt_m) {
        if (baby == y) return b;
        table[(int64_t)baby * y % m] = b;
        baby = (int64_t)baby * g % m;
    }

    // giant-step: list (g^{sqrt(m)}, g^{2 sqrt(m)}, g^{3 sqrt(m)}, ...)
    int giant = 1;
    REP3 (a, 1, sqrt_m + 3) {
        giant = (int64_t)giant * baby % m;
        auto it = table.find(giant);
        if (it != table.end()) {
            int b = it->second;
            int x = (int64_t)a * sqrt_m - b;
            assert (x >= 0);
            return (modpow(g, x, m) == y ? x : -1);
        }
    }
    return -1;
}

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