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number_theory/test/primitive_root.test.cpp
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- Last update: 2025-06-28 10:05:47+09:00
- Problem: https://judge.yosupo.jp/problem/primitive_root
Depends on
- number_theory/factorize.hpp
- number_theory/montgomery_64.hpp
- number_theory/primality.hpp
- number_theory/primitive_root.hpp
- template/random.hpp
- template/template.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/primitive_root"
#define FAST_IO
#include "../../number_theory/primitive_root.hpp"
#include "../../template/template.hpp"
void solve() {
U64(p);
cout << find_primitive_root(p) << '\n';
}
int main() {
i32 t = 1;
cin >> t;
while (t--) {
solve();
}
}#line 1 "number_theory/test/primitive_root.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/primitive_root"
#define FAST_IO
#line 2 "number_theory/factorize.hpp"
#include <algorithm>
#include <vector>
#line 2 "template/random.hpp"
#include <chrono>
#include <random>
#if defined(LOCAL) || defined(FIX_SEED)
std::mt19937_64 mt(123456789);
#else
std::mt19937_64 mt(std::chrono::steady_clock::now().time_since_epoch().count());
#endif
template <typename T>
T uniform(T l, T r) {
return std::uniform_int_distribution<T>(l, r - 1)(mt);
}
template <typename T>
T uniform(T n) {
return std::uniform_int_distribution<T>(0, n - 1)(mt);
}
#line 2 "number_theory/primality.hpp"
namespace primality {
using u64 = unsigned long long;
using u128 = __uint128_t;
u64 inv_64(u64 n) {
u64 r = n;
for (int i = 0; i < 5; ++i) {
r *= 2 - n * r;
}
return r;
}
// n: odd, < 2^{62}
struct Montgomery64 {
u64 n, mni, p;
Montgomery64(u64 n) : n(n), mni(-inv_64(n)), p(-1ULL % n + 1) {}
u64 mulmr(u64 xr, u64 yr) const {
u128 z = (u128)xr * yr;
u64 ret = (z + (u128)((u64)z * mni) * n) >> 64;
if (ret >= n) {
ret -= n;
}
return ret;
}
u64 mr(u64 xr) const {
u64 ret = (xr + (u128)(xr * mni) * n) >> 64;
if (ret >= n) {
ret -= n;
}
return ret;
}
u64 pow(u64 xr, u64 t) const {
u64 ret = p;
while (t) {
if (t & 1) {
ret = mulmr(ret, xr);
}
xr = mulmr(xr, xr);
t >>= 1;
}
return ret;
}
};
bool is_prime(u64 n) {
if (n == 2) {
return true;
}
if (n == 1 || n % 2 == 0) {
return false;
}
u64 s = __builtin_ctzll(n - 1);
u64 d = (n - 1) >> s;
u64 base[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
Montgomery64 mont(n);
u64 fl = n - mont.p;
for (u64 b : base) {
b = mont.mr(b);
if (!b) {
continue;
}
u64 t = mont.pow(b, d);
if (t == mont.p) {
continue;
}
u64 i = 0;
for (; i < s; ++i) {
if (t == fl) {
break;
}
t = mont.mulmr(t, t);
}
if (i == s) {
return false;
}
}
return true;
}
} // namespace primality
bool is_prime(unsigned long long n) {
return primality::is_prime(n);
}
#line 2 "number_theory/montgomery_64.hpp"
#include <cassert>
// mod: odd, < 2^{63}
template <int id>
struct MontgomeryModInt64 {
using u64 = unsigned long long;
using u128 = __uint128_t;
static u64 inv_64(u64 n) {
u64 r = n;
for (int i = 0; i < 5; ++i) {
r *= 2 - n * r;
}
return r;
}
static u64 mod, neg_inv, sq;
static void set_mod(u64 m) {
assert(m % 2 == 1 && m < (1ULL << 63));
mod = m;
neg_inv = -inv_64(m);
sq = -u128(mod) % mod;
}
static u64 get_mod() { return mod; }
static u64 reduce(u128 xr) {
u64 ret = (xr + u128(u64(xr) * neg_inv) * mod) >> 64;
if (ret >= mod) {
ret -= mod;
}
return ret;
}
using M = MontgomeryModInt64<id>;
u64 x;
MontgomeryModInt64() : x(0) {}
MontgomeryModInt64(u64 _x) : x(reduce(u128(_x) * sq)) {}
u64 val() const { return reduce(u128(x)); }
M &operator+=(M rhs) {
if ((x += rhs.x) >= mod) {
x -= mod;
}
return *this;
}
M &operator-=(M rhs) {
if ((x -= rhs.x) >= mod) {
x += mod;
}
return *this;
}
M &operator*=(M rhs) {
x = reduce(u128(x) * rhs.x);
return *this;
}
M operator+(M rhs) const { return M(*this) += rhs; }
M operator-(M rhs) const { return M(*this) -= rhs; }
M operator*(M rhs) const { return M(*this) *= rhs; }
M pow(u64 t) const {
M ret(1);
M self = *this;
while (t) {
if (t & 1) {
ret *= self;
}
self *= self;
t >>= 1;
}
return ret;
}
M inv() const {
assert(x);
return this->pow(mod - 2);
}
M &operator/=(M rhs) {
*this /= rhs.inv();
return *this;
}
M operator/(M rhs) const { return M(*this) /= rhs; }
};
template <int id> unsigned long long MontgomeryModInt64<id>::mod = 1;
template <int id> unsigned long long MontgomeryModInt64<id>::neg_inv = 1;
template <int id> unsigned long long MontgomeryModInt64<id>::sq = 1;
#line 7 "number_theory/factorize.hpp"
namespace factorize_impl {
unsigned long long bgcd(unsigned long long x, unsigned long long y) {
if (x == 0) {
return y;
}
if (y == 0) {
return x;
}
int n = __builtin_ctzll(x);
int m = __builtin_ctzll(y);
x >>= n;
y >>= m;
while (x != y) {
if (x > y) {
x = (x - y) >> __builtin_ctzll(x - y);
} else {
y = (y - x) >> __builtin_ctzll(y - x);
}
}
return x << (n < m ? n : m);
}
template <typename T>
unsigned long long rho(unsigned long long n, unsigned long long c) {
T cc(c);
auto f = [cc](T x) -> T {
return x * x + cc;
};
T y(2);
T x = y;
T z = y;
T p(1);
unsigned long long g = 1;
constexpr int M = 128;
for (int r = 1; g == 1; r *= 2) {
x = y;
for (int i = 0; i < r && g == 1; i += M) {
z = y;
for (int j = 0; j < r - i && j < M; ++j) {
y = f(y);
p *= y - x;
}
g = bgcd(p.val(), n);
}
}
if (g == n) {
do {
z = f(z);
g = bgcd((z - x).val(), n);
} while (g == 1);
}
return g;
}
unsigned long long find_factor(unsigned long long n) {
using M = MontgomeryModInt64<20250127>;
M::set_mod(n);
while (true) {
unsigned long long c = uniform(n);
unsigned long long g = rho<M>(n, c);
if (g != n) {
return g;
}
}
return 0;
}
void factor_inner(unsigned long long n, std::vector<unsigned long long> &ps) {
if (is_prime(n)) {
ps.push_back(n);
return;
}
if (n % 2 == 0) {
ps.push_back(2);
factor_inner(n / 2, ps);
return;
}
unsigned long long m = find_factor(n);
factor_inner(m, ps);
factor_inner(n / m, ps);
}
}
std::vector<unsigned long long> factorize(unsigned long long n) {
if (n <= 1) {
return std::vector<unsigned long long>();
}
std::vector<unsigned long long> ps;
factorize_impl::factor_inner(n, ps);
std::sort(ps.begin(), ps.end());
return ps;
}
#line 3 "number_theory/primitive_root.hpp"
// p: prime
unsigned long long find_primitive_root(unsigned long long p) {
using M = MontgomeryModInt64<20250128>;
assert(is_prime(p));
if (p == 2) {
return 1;
}
M::set_mod(p);
std::vector<unsigned long long> ps = factorize(p - 1);
ps.erase(std::unique(ps.begin(), ps.end()), ps.end());
while (true) {
unsigned long long x = uniform<unsigned long long>(1, p);
M x_(x), one(1ULL);
bool ok = true;
for (unsigned long long q : ps) {
if (x_.pow((p - 1) / q).x == one.x) {
ok = false;
break;
}
}
if (ok) {
return x;
}
}
return 0;
}
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
#define OVERRIDE(a, b, c, d, ...) d
#define REP2(i, n) for (i32 i = 0; i < (i32)(n); ++i)
#define REP3(i, m, n) for (i32 i = (i32)(m); i < (i32)(n); ++i)
#define REP(...) OVERRIDE(__VA_ARGS__, REP3, REP2)(__VA_ARGS__)
#define PER2(i, n) for (i32 i = (i32)(n)-1; i >= 0; --i)
#define PER3(i, m, n) for (i32 i = (i32)(n)-1; i >= (i32)(m); --i)
#define PER(...) OVERRIDE(__VA_ARGS__, PER3, PER2)(__VA_ARGS__)
#define ALL(x) begin(x), end(x)
#define LEN(x) (i32)(x.size())
using namespace std;
using u32 = unsigned int;
using u64 = unsigned long long;
using i32 = signed int;
using i64 = signed long long;
using f64 = double;
using f80 = long double;
using pi = pair<i32, i32>;
using pl = pair<i64, i64>;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = V<V<T>>;
template <typename T>
using VVV = V<V<V<T>>>;
template <typename T>
using VVVV = V<V<V<V<T>>>>;
template <typename T>
using PQR = priority_queue<T, V<T>, greater<T>>;
template <typename T>
bool chmin(T &x, const T &y) {
if (x > y) {
x = y;
return true;
}
return false;
}
template <typename T>
bool chmax(T &x, const T &y) {
if (x < y) {
x = y;
return true;
}
return false;
}
template <typename T>
i32 lob(const V<T> &arr, const T &v) {
return (i32)(lower_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
i32 upb(const V<T> &arr, const T &v) {
return (i32)(upper_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
V<i32> argsort(const V<T> &arr) {
V<i32> ret(arr.size());
iota(ALL(ret), 0);
sort(ALL(ret), [&](i32 i, i32 j) -> bool {
if (arr[i] == arr[j]) {
return i < j;
} else {
return arr[i] < arr[j];
}
});
return ret;
}
#ifdef INT128
using u128 = __uint128_t;
using i128 = __int128_t;
#endif
[[maybe_unused]] constexpr i32 INF = 1000000100;
[[maybe_unused]] constexpr i64 INF64 = 3000000000000000100;
struct SetUpIO {
SetUpIO() {
#ifdef FAST_IO
ios::sync_with_stdio(false);
cin.tie(nullptr);
#endif
cout << fixed << setprecision(15);
}
} set_up_io;
void scan(char &x) { cin >> x; }
void scan(u32 &x) { cin >> x; }
void scan(u64 &x) { cin >> x; }
void scan(i32 &x) { cin >> x; }
void scan(i64 &x) { cin >> x; }
void scan(f64 &x) { cin >> x; }
void scan(string &x) { cin >> x; }
template <typename T>
void scan(V<T> &x) {
for (T &ele : x) {
scan(ele);
}
}
void read() {}
template <typename Head, typename... Tail>
void read(Head &head, Tail &...tail) {
scan(head);
read(tail...);
}
#define CHAR(...) \
char __VA_ARGS__; \
read(__VA_ARGS__);
#define U32(...) \
u32 __VA_ARGS__; \
read(__VA_ARGS__);
#define U64(...) \
u64 __VA_ARGS__; \
read(__VA_ARGS__);
#define I32(...) \
i32 __VA_ARGS__; \
read(__VA_ARGS__);
#define I64(...) \
i64 __VA_ARGS__; \
read(__VA_ARGS__);
#define F64(...) \
f64 __VA_ARGS__; \
read(__VA_ARGS__);
#define STR(...) \
string __VA_ARGS__; \
read(__VA_ARGS__);
#define VEC(type, name, size) \
V<type> name(size); \
read(name);
#define VVEC(type, name, size1, size2) \
VV<type> name(size1, V<type>(size2)); \
read(name);
#line 5 "number_theory/test/primitive_root.test.cpp"
void solve() {
U64(p);
cout << find_primitive_root(p) << '\n';
}
int main() {
i32 t = 1;
cin >> t;
while (t--) {
solve();
}
}