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#include "Math/pollard.hpp"
#pragma once #include "Math/miller.hpp" #include "Utility/random.hpp" vector<ll> Pollard(ll n) { if (n <= 1) return {}; if (Miller(n)) return {n}; if ((n & 1) == 0) { vector<ll> v = Pollard(n >> 1); v.push_back(2); return v; } for (ll x = 2, y = 2, d;;) { ll c = Random::get(2LL, n - 1); do { x = (__int128_t(x) * x + c) % n; y = (__int128_t(y) * y + c) % n; y = (__int128_t(y) * y + c) % n; d = __gcd(x - y + n, n); } while (d == 1); if (d < n) { vector<ll> lb = Pollard(d), rb = Pollard(n / d); lb.insert(lb.end(), ALL(rb)); return lb; } } } /** * @brief Pollard-Rho */
#line 2 "Math/miller.hpp" struct m64 { using i64 = int64_t; using u64 = uint64_t; using u128 = __uint128_t; static u64 mod; static u64 r; static u64 n2; static u64 get_r() { u64 ret = mod; rep(_,0,5) ret *= 2 - mod * ret; return ret; } static void set_mod(u64 m) { assert(m < (1LL << 62)); assert((m & 1) == 1); mod = m; n2 = -u128(m) % m; r = get_r(); assert(r * mod == 1); } static u64 get_mod() { return mod; } u64 a; m64() : a(0) {} m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){}; static u64 reduce(const u128 &b) { return (b + u128(u64(b) * u64(-r)) * mod) >> 64; } u64 get() const { u64 ret = reduce(a); return ret >= mod ? ret - mod : ret; } m64 &operator*=(const m64 &b) { a = reduce(u128(a) * b.a); return *this; } m64 operator*(const m64 &b) const { return m64(*this) *= b; } bool operator==(const m64 &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } bool operator!=(const m64 &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } m64 pow(u128 n) const { m64 ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } }; typename m64::u64 m64::mod, m64::r, m64::n2; bool Miller(ll n){ if(n<2 or (n&1)==0)return (n==2); m64::set_mod(n); ll d=n-1; while((d&1)==0)d>>=1; vector<ll> seeds; if(n<(1<<30))seeds={2, 7, 61}; else seeds={2, 325, 9375, 28178, 450775, 9780504}; for(auto& x:seeds){ if(n<=x)break; ll t=d; m64 y=m64(x).pow(t); while(t!=n-1 and y!=1 and y!=n-1){ y*=y; t<<=1; } if(y!=n-1 and (t&1)==0)return 0; } return 1; } /** * @brief Miller-Rabin */ #line 2 "Utility/random.hpp" namespace Random { mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count()); using u64 = unsigned long long; u64 get() { return randgen(); } template <typename T> T get(T L) { // [0,L] return get() % (L + 1); } template <typename T> T get(T L, T R) { // [L,R] return get(R - L) + L; } double uniform() { return double(get(1000000000)) / 1000000000; } string str(int n) { string ret; rep(i, 0, n) ret += get('a', 'z'); return ret; } template <typename Iter> void shuffle(Iter first, Iter last) { if (first == last) return; int len = 1; for (auto it = first + 1; it != last; it++) { len++; int j = get(0, len - 1); if (j != len - 1) iter_swap(it, first + j); } } template <typename T> vector<T> select(int n, T L, T R) { // [L,R] if (n * 2 >= R - L + 1) { vector<T> ret(R - L + 1); iota(ALL(ret), L); shuffle(ALL(ret)); ret.resize(n); return ret; } else { unordered_set<T> used; vector<T> ret; while (SZ(used) < n) { T x = get(L, R); if (!used.count(x)) { used.insert(x); ret.push_back(x); } } return ret; } } void relabel(int n, vector<pair<int, int>> &es) { shuffle(ALL(es)); vector<int> ord(n); iota(ALL(ord), 0); shuffle(ALL(ord)); for (auto &[u, v] : es) u = ord[u], v = ord[v]; } template <bool directed, bool simple> vector<pair<int, int>> genGraph(int n) { vector<pair<int, int>> cand, es; rep(u, 0, n) rep(v, 0, n) { if (simple and u == v) continue; if (!directed and u > v) continue; cand.push_back({u, v}); } int m = get(SZ(cand)); vector<int> ord; if (simple) ord = select(m, 0, SZ(cand) - 1); else { rep(_, 0, m) ord.push_back(get(SZ(cand) - 1)); } for (auto &i : ord) es.push_back(cand[i]); relabel(n, es); return es; } vector<pair<int, int>> genTree(int n) { vector<pair<int, int>> es; rep(i, 1, n) es.push_back({get(i - 1), i}); relabel(n, es); return es; } }; // namespace Random /** * @brief Random */ #line 4 "Math/pollard.hpp" vector<ll> Pollard(ll n) { if (n <= 1) return {}; if (Miller(n)) return {n}; if ((n & 1) == 0) { vector<ll> v = Pollard(n >> 1); v.push_back(2); return v; } for (ll x = 2, y = 2, d;;) { ll c = Random::get(2LL, n - 1); do { x = (__int128_t(x) * x + c) % n; y = (__int128_t(y) * y + c) % n; y = (__int128_t(y) * y + c) % n; d = __gcd(x - y + n, n); } while (d == 1); if (d < n) { vector<ll> lb = Pollard(d), rb = Pollard(n / d); lb.insert(lb.end(), ALL(rb)); return lb; } } } /** * @brief Pollard-Rho */