結果

問題 No.981 一般冪乗根
ユーザー heno239
提出日時 2020-02-11 20:33:56
言語 C++14
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 5,188 ms / 6,000 ms
コード長 4,911 bytes
コンパイル時間 1,363 ms
コンパイル使用メモリ 124,216 KB
実行使用メモリ 6,820 KB
最終ジャッジ日時 2025-02-07 08:55:50
合計ジャッジ時間 148,252 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge4
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ファイルパターン 結果
other AC * 30 MLE * 14
権限があれば一括ダウンロードができます

ソースコード

diff # 

#include<iostream>
#include<string>
#include<cstdio>
#include<vector>
#include<cmath>
#include<algorithm>
#include<functional>
#include<iomanip>
#include<queue>
#include<ciso646>
#include<random>
#include<map>
#include<set>
#include<bitset>
#include<stack>
#include<unordered_map>
#include<utility>
#include<cassert>
#include<complex>
#include<numeric>
using namespace std;

//#define int long long
typedef long long ll;

typedef unsigned long long ul;
typedef unsigned int ui;
const ll mod = 998244353;
const ll INF = (1e+18) + 7;
typedef pair<int, int>P;
#define stop char nyaa;cin>>nyaa;
#define rep(i,n) for(int i=0;i<n;i++)
#define per(i,n) for(int i=n-1;i>=0;i--)
#define Rep(i,sta,n) for(int i=sta;i<n;i++)
#define rep1(i,n) for(int i=1;i<=n;i++)
#define per1(i,n) for(int i=n;i>=1;i--)
#define Rep1(i,sta,n) for(int i=sta;i<=n;i++)
#define all(v) (v).begin(),(v).end()
typedef pair<ll, ll> LP;
typedef long double ld;
typedef pair<ld, ld> LDP;
const ld eps = 1e-6;
const ld pi = acos(-1.0);
//typedef vector<vector<ll>> mat;
typedef vector<int> vec;

ll mod_pow(ll a, ll n, ll m) {
	ll res = 1;
	while (n) {
		if (n & 1)res = res * a%m;
		a = a * a%m; n >>= 1;
	}
	return res;
}
ll mod_inv(ll a, ll m) {
	return mod_pow(a, m - 2, m);
}

struct modint {
	ll n;
	modint() :n(0) { ; }
	modint(ll m) :n(m) {
		if (n >= mod)n %= mod;
		else if (n < 0)n = (n%mod + mod) % mod;
	}
	operator int() { return n; }
};
bool operator==(modint a, modint b) { return a.n == b.n; }
modint operator+=(modint &a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= mod; return a; }
modint operator-=(modint &a, modint b) { a.n -= b.n; if (a.n < 0)a.n += mod; return a; }
modint operator*=(modint &a, modint b) { a.n = ((ll)a.n*b.n) % mod; return a; }
modint operator+(modint a, modint b) { return a += b; }
modint operator-(modint a, modint b) { return a -= b; }
modint operator*(modint a, modint b) { return a *= b; }
modint operator^(modint a, int n) {
	if (n == 0)return modint(1);
	modint res = (a*a) ^ (n / 2);
	if (n % 2)res = res * a;
	return res;
}

ll inv(ll a, ll p) {
	return (a == 1 ? 1 : (1 - p * inv(p%a, a)) / a + p);
}
modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); }

const int max_n = 500;
modint fact[max_n], factinv[max_n];
void init_f() {
	fact[0] = modint(1);
	for (int i = 0; i < max_n - 1; i++) {
		fact[i + 1] = fact[i] * modint(i + 1);
	}
	factinv[max_n - 1] = modint(1) / fact[max_n - 1];
	for (int i = max_n - 2; i >= 0; i--) {
		factinv[i] = factinv[i + 1] * modint(i + 1);
	}
}
modint comb(int a, int b) {
	if (a < 0 || b < 0 || a < b)return 0;
	return fact[a] * factinv[b] * factinv[a - b];
}

int dx[4] = { 0,1,0,-1 };
int dy[4] = { 1,0,-1,0 };

ll calc_proot(ll p) {
	vector<ll> ps;
	int k = sqrt(p + 0.1);
	int cop = p - 1;
	Rep1(i, 2, k) {
		if (cop%i == 0) {
			ps.push_back(i);
			while (cop%i == 0)cop /= i;
		}
	}
	if (cop > 1)ps.push_back(cop);
	random_device rnd;
	mt19937 mt(rnd());
	uniform_int_distribution<> ud(0, p - 1);
	while (true) {
		ll g = ud(mt);
		if (g == 0)continue;
		bool f = true;
		rep(i, ps.size()) {
			ll z = ps[i];
			ll u = mod_pow(g, (p - 1) / z, p);
			//cout << g << " " <<z<<" "<<p<<" "<< u << endl;
			if (u == 1) {
				f = false; break;
			}
		}
		if (f)return g;
	}
	return -1;
}
//a^x=b(mod p)
ll calc_kata(ll p, ll a, ll b) {
	int k = sqrt(p + 0.1);
	//map<int, int> mp;
	vector<P> v;
	ll cop = b;
	ll inva = mod_inv(a, p);
	rep(i, k + 2) {
		v.push_back({ cop,i });
		cop = cop * inva%p;
	}
	sort(all(v));
	ll sa = mod_pow(a, k, p);
	cop = 1;
	rep(i, k + 2) {
		int id = lower_bound(all(v), P{ cop,-1 }) - v.begin();
		if (id < v.size() && v[id].first == cop) {
			int res = i * k + v[id].second;
			return res % (p - 1);
		}
		cop = cop * sa%p;
	}
	return -1;
}
ll gcd(ll a, ll b) {
	if (a < b)swap(a, b);
	while (b) {
		ll r = a % b; a = b; b = r;
	}
	return a;
}
ll extgcd(ll a, ll b, ll& x, ll& y) {
	ll d = a;
	if (b != 0) {
		d = extgcd(b, a%b, y, x);
		y -= (a / b)*x;
	}
	else {
		x = 1; y = 0;
	}
	return d;
}
bool hantei(ll p, ll r) {
	ll c = 1;
	vector<bool> exi(p);
	for (int i = 0; i < p - 1; i++) {
		if (exi[c])return false;
		exi[c] = true;
		c = c * r%p;
	}
	return true;
}
void solve() {
	ll p, k, a; cin >> p >> k >> a;
	ll r = calc_proot(p);
	ll b = calc_kata(p, r, a);
	//cout << "!! " << r << endl;
	//assert(hantei(p,r));
	//cout << "?? " << b << endl;
	assert(b >= 0);
	ll g = gcd(k, p - 1);
	if (b%g != 0) {
		cout << -1 << endl; return;
	}
	ll ca = k, cb = p - 1;
	ll x, y;
	extgcd(ca, cb, x, y);
	while (x < 0)x += p - 1;
	x = x * (b / g);
	x %= (p - 1);
	ll ans = mod_pow(r, x, p);
	ll z = mod_pow(ans, k, p);
	//cout << ans << " ! " << z << " " << a << endl;
	cout << ans << endl;
}


signed main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	cout << fixed << setprecision(10);
	//init_f(); init();
	int t; cin >> t; rep(i, t)solve();
	//while (cin >> n, n)solve();
	//solve();
	stop
		return 0;
}
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