Math/NumberTheory/Pollard_factorize.h

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• Last update: 2023-10-22 16:42:38+08:00
• Link: https://judge.yosupo.jp/problem/factorize
• Link: https://judge.yosupo.jp/submission/61447
• Link: https://oj.vnoi.info/problem/icpc21_beta_l
• Link: https://www.spoj.com/problems/FACT0/

Required by

• Math/NumberTheory/cnt_divisors.h

Verified with

• Math/tests/aizu_ntl_1_a_factorize.test.cpp
• Math/tests/cnt_divisors_stress.test.cpp
• Math/tests/factorize.test.cpp
• Math/tests/is_prime_yukicoder.test.cpp

Code

// Copied from https://judge.yosupo.jp/submission/61447
// O(N^0.25)
//
// Tested:
// - (up to 10^18; 200 tests) https://judge.yosupo.jp/problem/factorize
// - https://oj.vnoi.info/problem/icpc21_beta_l
// - https://www.spoj.com/problems/FACT0/
//
// Pollard {{{
using ll = long long;
using ull = unsigned long long;
using ld = long double;
ll mult(ll x, ll y, ll md) {
    ull q = (ld)x * y / md;
    ll res = ((ull)x * y - q * md);
    if (res >= md) res -= md;
    if (res < 0) res += md;
    return res;
}

ll powMod(ll x, ll p, ll md) {
    if (p == 0) return 1;
    if (p & 1) return mult(x, powMod(x, p - 1, md), md);
    return powMod(mult(x, x, md), p / 2, md);
}

bool checkMillerRabin(ll x, ll md, ll s, int k) {
    x = powMod(x, s, md);
    if (x == 1) return true;
    while(k--) {
        if (x == md - 1) return true;
        x = mult(x, x, md);
        if (x == 1) return false;
    }
    return false;
}
bool isPrime(ll x) {
    if (x == 2 || x == 3 || x == 5 || x == 7) return true;
    if (x % 2 == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false;
    if (x < 121) return x > 1;
    ll s = x - 1;
    int k = 0;
    while(s % 2 == 0) {
        s >>= 1;
        k++;
    }
    if (x < 1LL << 32) {
        for (ll z : {2, 7, 61}) {
            if (!checkMillerRabin(z, x, s, k)) return false;
        }
    } else {
        for (ll z : {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
            if (!checkMillerRabin(z, x, s, k)) return false;
        }
    }
    return true;
}

ll gcd(ll x, ll y) {
    return y == 0 ? x : gcd(y, x % y);
}

void pollard(ll x, vector<ll> &ans) {
    if (isPrime(x)) {
        ans.push_back(x);
        return;
    }
    ll c = 1;
    while(true) {
        c = 1 + get_rand(x - 1);
        auto f = [&](ll y) {
            ll res = mult(y, y, x) + c;
            if (res >= x) res -= x;
            return res;
        };
        ll y = 2;
        int B = 100;
        int len = 1;
        ll g = 1;
        while(g == 1) {
            ll z = y;
            for (int i = 0; i < len; i++) {
                z = f(z);
            }
            ll zs = -1;
            int lft = len;
            while(g == 1 && lft > 0) {
                zs = z;
                ll p = 1;
                for (int i = 0; i < B && i < lft; i++) {
                    p = mult(p, abs(z - y), x);
                    z = f(z);
                }
                g = gcd(p, x);
                lft -= B;
            }
            if (g == 1) {
                y = z;
                len <<= 1;
                continue;
            }
            if (g == x) {
                g = 1;
                z = zs;
                while(g == 1) {
                    g = gcd(abs(z - y), x);
                    z = f(z);
                }
            }
            if (g == x) break;
            assert(g != 1);
            pollard(g, ans);
            pollard(x / g, ans);
            return;
        }
    }
}
// return list of all prime factors of x (can have duplicates)
vector<ll> factorize(ll x) {
    vector<ll> ans;
    for (ll p : {2, 3, 5, 7, 11, 13, 17, 19}) {
        while(x % p == 0) {
            x /= p;
            ans.push_back(p);
        }
    }
    if (x != 1) {
        pollard(x, ans);
    }
    sort(ans.begin(), ans.end());
    return ans;
}
// return pairs of (p, k) where x = product(p^k)
vector<pair<ll, int>> factorize_pk(ll x) {
    auto ps = factorize(x);
    ll last = -1, cnt = 0;
    vector<pair<ll, int>> res;
    for (auto p : ps) {
        if (p == last) ++cnt;
        else {
            if (last > 0) res.emplace_back(last, cnt);
            last = p;
            cnt = 1;
        }
    }
    if (cnt > 0) {
        res.emplace_back(last, cnt);
    }
    return res;
}
vector<ll> get_all_divisors(ll n) {
    auto pks = factorize_pk(n);

    vector<ll> res;
    function<void(int, ll)> gen = [&] (int i, ll prod) {
        if (i == static_cast<int>(pks.size())) {
            res.push_back(prod);
            return;
        }

        ll cur_power = 1;
        for (int cur = 0; cur <= pks[i].second; ++cur) {
            gen(i+1, prod * cur_power);
            cur_power *= pks[i].first;
        }
    };

    gen(0, 1LL);
    sort(res.begin(), res.end());
    return res;
}
// }}}

#line 1 "Math/NumberTheory/Pollard_factorize.h"
// Copied from https://judge.yosupo.jp/submission/61447
// O(N^0.25)
//
// Tested:
// - (up to 10^18; 200 tests) https://judge.yosupo.jp/problem/factorize
// - https://oj.vnoi.info/problem/icpc21_beta_l
// - https://www.spoj.com/problems/FACT0/
//
// Pollard {{{
using ll = long long;
using ull = unsigned long long;
using ld = long double;
ll mult(ll x, ll y, ll md) {
    ull q = (ld)x * y / md;
    ll res = ((ull)x * y - q * md);
    if (res >= md) res -= md;
    if (res < 0) res += md;
    return res;
}

ll powMod(ll x, ll p, ll md) {
    if (p == 0) return 1;
    if (p & 1) return mult(x, powMod(x, p - 1, md), md);
    return powMod(mult(x, x, md), p / 2, md);
}

bool checkMillerRabin(ll x, ll md, ll s, int k) {
    x = powMod(x, s, md);
    if (x == 1) return true;
    while(k--) {
        if (x == md - 1) return true;
        x = mult(x, x, md);
        if (x == 1) return false;
    }
    return false;
}
bool isPrime(ll x) {
    if (x == 2 || x == 3 || x == 5 || x == 7) return true;
    if (x % 2 == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false;
    if (x < 121) return x > 1;
    ll s = x - 1;
    int k = 0;
    while(s % 2 == 0) {
        s >>= 1;
        k++;
    }
    if (x < 1LL << 32) {
        for (ll z : {2, 7, 61}) {
            if (!checkMillerRabin(z, x, s, k)) return false;
        }
    } else {
        for (ll z : {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
            if (!checkMillerRabin(z, x, s, k)) return false;
        }
    }
    return true;
}

ll gcd(ll x, ll y) {
    return y == 0 ? x : gcd(y, x % y);
}

void pollard(ll x, vector<ll> &ans) {
    if (isPrime(x)) {
        ans.push_back(x);
        return;
    }
    ll c = 1;
    while(true) {
        c = 1 + get_rand(x - 1);
        auto f = [&](ll y) {
            ll res = mult(y, y, x) + c;
            if (res >= x) res -= x;
            return res;
        };
        ll y = 2;
        int B = 100;
        int len = 1;
        ll g = 1;
        while(g == 1) {
            ll z = y;
            for (int i = 0; i < len; i++) {
                z = f(z);
            }
            ll zs = -1;
            int lft = len;
            while(g == 1 && lft > 0) {
                zs = z;
                ll p = 1;
                for (int i = 0; i < B && i < lft; i++) {
                    p = mult(p, abs(z - y), x);
                    z = f(z);
                }
                g = gcd(p, x);
                lft -= B;
            }
            if (g == 1) {
                y = z;
                len <<= 1;
                continue;
            }
            if (g == x) {
                g = 1;
                z = zs;
                while(g == 1) {
                    g = gcd(abs(z - y), x);
                    z = f(z);
                }
            }
            if (g == x) break;
            assert(g != 1);
            pollard(g, ans);
            pollard(x / g, ans);
            return;
        }
    }
}
// return list of all prime factors of x (can have duplicates)
vector<ll> factorize(ll x) {
    vector<ll> ans;
    for (ll p : {2, 3, 5, 7, 11, 13, 17, 19}) {
        while(x % p == 0) {
            x /= p;
            ans.push_back(p);
        }
    }
    if (x != 1) {
        pollard(x, ans);
    }
    sort(ans.begin(), ans.end());
    return ans;
}
// return pairs of (p, k) where x = product(p^k)
vector<pair<ll, int>> factorize_pk(ll x) {
    auto ps = factorize(x);
    ll last = -1, cnt = 0;
    vector<pair<ll, int>> res;
    for (auto p : ps) {
        if (p == last) ++cnt;
        else {
            if (last > 0) res.emplace_back(last, cnt);
            last = p;
            cnt = 1;
        }
    }
    if (cnt > 0) {
        res.emplace_back(last, cnt);
    }
    return res;
}
vector<ll> get_all_divisors(ll n) {
    auto pks = factorize_pk(n);

    vector<ll> res;
    function<void(int, ll)> gen = [&] (int i, ll prod) {
        if (i == static_cast<int>(pks.size())) {
            res.push_back(prod);
            return;
        }

        ll cur_power = 1;
        for (int cur = 0; cur <= pks[i].second; ++cur) {
            gen(i+1, prod * cur_power);
            cur_power *= pks[i].first;
        }
    };

    gen(0, 1LL);
    sort(res.begin(), res.end());
    return res;
}
// }}}

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