結果
| 問題 |
No.981 一般冪乗根
|
| ユーザー |
 tokusakurai
|
| 提出日時 | 2022-09-01 17:52:32 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 61 ms / 6,000 ms |
| コード長 | 11,339 bytes |
| コンパイル時間 | 2,350 ms |
| コンパイル使用メモリ | 216,372 KB |
| 最終ジャッジ日時 | 2025-02-07 00:23:44 |
|
ジャッジサーバーID （参考情報） |
judge5 / judge5 |
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| ファイルパターン | 結果 |
|---|---|
| other | AC * 30 WA * 14 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
struct io_setup {
io_setup() {
ios_base::sync_with_stdio(false);
cin.tie(NULL);
cout << fixed << setprecision(15);
}
} io_setup;
struct Random_Number_Generator {
mt19937_64 mt;
Random_Number_Generator() : mt(chrono::steady_clock::now().time_since_epoch().count()) {}
// [l,r) での一様乱数
int64_t operator()(int64_t l, int64_t r) {
uniform_int_distribution<int64_t> dist(l, r - 1);
return dist(mt);
}
// [0,r) での一様乱数
int64_t operator()(int64_t r) { return (*this)(0, r); }
} rng;
long long modpow(long long x, long long n, const int &m) {
x %= m;
long long ret = 1;
for (; n > 0; n >>= 1, x *= x, x %= m) {
if (n & 1) ret *= x, ret %= m;
}
return ret;
}
template <typename T>
T modinv(T a, const T &m) {
T b = m, u = 1, v = 0;
while (b > 0) {
T t = a / b;
swap(a -= t * b, b);
swap(u -= t * v, v);
}
return u >= 0 ? u % m : (m - (-u) % m) % m;
}
// オイラーの φ 関数 (x と m が互いに素ならば、x^φ(m) ≡ 1 (mod m))
template <typename T>
T Euler_totient(T m) {
T ret = m;
for (T i = 2; i * i <= m; i++) {
if (m % i == 0) ret /= i, ret *= i - 1;
while (m % i == 0) m /= i;
}
if (m > 1) ret /= m, ret *= m - 1;
return ret;
}
// x^k ≡ y (mod m) となる最小の非負整数 k (存在しなければ -1)
int modlog(int x, int y, int m, int max_ans = -1) {
if (max_ans == -1) max_ans = m;
long long g = 1;
for (int i = m; i > 0; i >>= 1) g *= x, g %= m;
g = gcd(g, m);
int c = 0;
long long t = 1;
for (; t % g != 0; c++) {
if (t == y) return c;
t *= x, t %= m;
}
if (y % g != 0) return -1;
t /= g, y /= g, m /= g;
int n = 0;
long long gs = 1;
for (; n * n < max_ans; n++) gs *= x, gs %= m;
unordered_map<int, int> mp;
long long e = y;
for (int i = 0; i < n; mp[e] = ++i) e *= x, e %= m;
e = t;
for (int i = 0; i < n; i++) {
e *= gs, e %= m;
if (mp.count(e)) return c + n * (i + 1) - mp[e];
}
return -1;
}
// x^k ≡ 1 (mod m) となる最小の正整数 k (x と m は互いに素)
template <typename T>
T order(T x, const T &m) {
T n = Euler_totient(m);
vector<T> ds;
for (T i = 1; i * i <= n; i++) {
if (n % i == 0) ds.push_back(i), ds.push_back(n / i);
}
sort(begin(ds), end(ds));
for (auto &e : ds) {
if (modpow(x, e, m) == 1) return e;
}
return -1;
}
// 素数 p の原始根
template <typename T>
T primitive_root(const T &p) {
vector<T> ds;
for (T i = 1; i * i <= p - 1; i++) {
if ((p - 1) % i == 0) ds.push_back(i), ds.push_back((p - 1) / i);
}
sort(begin(ds), end(ds));
while (true) {
T r = rng(1, p);
for (auto &e : ds) {
if (e == p - 1) return r;
if (modpow(r, e, p) == 1) break;
}
}
}
struct Montgomery_Mod_Int_64 {
using u64 = uint64_t;
using u128 = __uint128_t;
static u64 mod;
static u64 r; // m*r ≡ 1 (mod 2^64)
static u64 n2; // 2^128 (mod mod)
u64 x;
Montgomery_Mod_Int_64() : x(0) {}
Montgomery_Mod_Int_64(long long b) : x(reduce((u128(b) + mod) * n2)) {}
static u64 get_r() { // mod 2^64 での逆元
u64 ret = mod;
for (int i = 0; i < 5; i++) ret *= 2 - mod * ret;
return ret;
}
static u64 get_mod() { return mod; }
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 reduce(const u128 &b) { return (b + u128(u64(b) * u64(-r)) * mod) >> 64; }
Montgomery_Mod_Int_64 &operator+=(const Montgomery_Mod_Int_64 &p) {
if ((x += p.x) >= 2 * mod) x -= 2 * mod;
return *this;
}
Montgomery_Mod_Int_64 &operator-=(const Montgomery_Mod_Int_64 &p) {
if ((x += 2 * mod - p.x) >= 2 * mod) x -= 2 * mod;
return *this;
}
Montgomery_Mod_Int_64 &operator*=(const Montgomery_Mod_Int_64 &p) {
x = reduce(u128(x) * p.x);
return *this;
}
Montgomery_Mod_Int_64 &operator/=(const Montgomery_Mod_Int_64 &p) {
*this *= p.inverse();
return *this;
}
Montgomery_Mod_Int_64 &operator++() { return *this += Montgomery_Mod_Int_64(1); }
Montgomery_Mod_Int_64 operator++(int) {
Montgomery_Mod_Int_64 tmp = *this;
++*this;
return tmp;
}
Montgomery_Mod_Int_64 &operator--() { return *this -= Montgomery_Mod_Int_64(1); }
Montgomery_Mod_Int_64 operator--(int) {
Montgomery_Mod_Int_64 tmp = *this;
--*this;
return tmp;
}
Montgomery_Mod_Int_64 operator+(const Montgomery_Mod_Int_64 &p) const { return Montgomery_Mod_Int_64(*this) += p; };
Montgomery_Mod_Int_64 operator-(const Montgomery_Mod_Int_64 &p) const { return Montgomery_Mod_Int_64(*this) -= p; };
Montgomery_Mod_Int_64 operator*(const Montgomery_Mod_Int_64 &p) const { return Montgomery_Mod_Int_64(*this) *= p; };
Montgomery_Mod_Int_64 operator/(const Montgomery_Mod_Int_64 &p) const { return Montgomery_Mod_Int_64(*this) /= p; };
bool operator==(const Montgomery_Mod_Int_64 &p) const { return (x >= mod ? x - mod : x) == (p.x >= mod ? p.x - mod : p.x); };
bool operator!=(const Montgomery_Mod_Int_64 &p) const { return (x >= mod ? x - mod : x) != (p.x >= mod ? p.x - mod : p.x); };
Montgomery_Mod_Int_64 inverse() const {
assert(*this != Montgomery_Mod_Int_64(0));
return pow(mod - 2);
}
Montgomery_Mod_Int_64 pow(long long k) const {
Montgomery_Mod_Int_64 now = *this, ret = 1;
for (; k > 0; k >>= 1, now *= now) {
if (k & 1) ret *= now;
}
return ret;
}
u64 get() const {
u64 ret = reduce(x);
return ret >= mod ? ret - mod : ret;
}
friend ostream &operator<<(ostream &os, const Montgomery_Mod_Int_64 &p) { return os << p.get(); }
friend istream &operator>>(istream &is, Montgomery_Mod_Int_64 &p) {
long long a;
is >> a;
p = Montgomery_Mod_Int_64(a);
return is;
}
};
typename Montgomery_Mod_Int_64::u64 Montgomery_Mod_Int_64::mod, Montgomery_Mod_Int_64::r, Montgomery_Mod_Int_64::n2;
bool Miller_Rabin(unsigned long long n, vector<unsigned long long> as) {
using Mint = Montgomery_Mod_Int_64;
if (Mint::get_mod() != n) Mint::set_mod(n);
unsigned long long d = n - 1;
while (!(d & 1)) d >>= 1;
Mint e = 1, rev = n - 1;
for (unsigned long long a : as) {
if (n <= a) break;
unsigned long long t = d;
Mint y = Mint(a).pow(t);
while (t != n - 1 && y != e && y != rev) {
y *= y;
t <<= 1;
}
if (y != rev && (!(t & 1))) return false;
}
return true;
}
bool is_prime(unsigned long long n) {
if (!(n & 1)) return n == 2;
if (n <= 1) return false;
if (n < (1LL << 30)) return Miller_Rabin(n, {2, 7, 61});
return Miller_Rabin(n, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
}
unsigned long long Pollard_rho(unsigned long long n) {
using Mint = Montgomery_Mod_Int_64;
if (!(n & 1)) return 2;
if (is_prime(n)) return n;
if (Mint::get_mod() != n) Mint::set_mod(n);
Mint R, one = 1;
auto f = [&](Mint x) { return x * x + R; };
auto rnd = [&]() { return rng(n - 2) + 2; };
while (true) {
Mint x, y, ys, q = one;
R = rnd(), y = rnd();
unsigned long long g = 1;
int m = 128;
for (int r = 1; g == 1; r <<= 1) {
x = y;
for (int i = 0; i < r; i++) y = f(y);
for (int k = 0; g == 1 && k < r; k += m) {
ys = y;
for (int i = 0; i < m && i < r - k; i++) q *= x - (y = f(y));
g = gcd(q.get(), n);
}
}
if (g == n) {
do { g = gcd((x - (ys = f(ys))).get(), n); } while (g == 1);
}
if (g != n) return g;
}
return 0;
}
vector<unsigned long long> factorize(unsigned long long n) {
if (n <= 1) return {};
unsigned long long p = Pollard_rho(n);
if (p == n) return {n};
auto l = factorize(p);
auto r = factorize(n / p);
copy(begin(r), end(r), back_inserter(l));
return l;
}
vector<pair<unsigned long long, int>> prime_factor(unsigned long long n) {
auto ps = factorize(n);
sort(begin(ps), end(ps));
vector<pair<unsigned long long, int>> ret;
for (auto &e : ps) {
if (!ret.empty() && ret.back().first == e) {
ret.back().second++;
} else {
ret.emplace_back(e, 1);
}
}
return ret;
}
// 素数 p について、x^2 ≡ a (mod p) となる x を 1 つ求める (存在しなければ -1)
int sqrt_mod(int a, const int &p) {
if (a == 0) return 0;
if (p == 2) return a;
if (modpow(a, (p - 1) / 2, p) != 1) return -1;
int s = p - 1, t = 0;
while (s % 2 == 0) s /= 2, t++;
long long x = modpow(a, (s + 1) / 2, p);
long long u = 1;
while (modpow(u, (p - 1) / 2, p) == 1) u = rng(1, p);
u = modpow(u, s, p);
long long y = (1LL * x * x % p) * modinv(a, p) % p;
while (y != 1) {
int k = 0;
long long z = y;
while (z != 1) k++, z *= z, z %= p;
for (int i = 0; i < t - k - 1; i++) u *= u, u %= p;
x *= u, x %= p;
u *= u, u %= p;
y *= u, y %= p;
t = k;
}
return x;
}
// x^(m^e) ≡ a (mod p) (m が p-1 の素因数であり、解が存在する場合)
int prime_power_root_mod(int m, int e, int a, const int &p) {
int s = p - 1, t = 0;
while (s % m == 0) s /= m, t++;
vector<int> pw(t + 1, 1);
for (int i = 0; i < t; i++) pw[i + 1] = pw[i] * m;
int q = pw[e];
long long x = modpow(a, (1 + 1LL * s * (q - modinv(s, q))) / q, p);
long long u = 1;
while (modpow(u, (p - 1) / m, p) == 1) u = rng(1, p);
u = modpow(u, s, p);
long long y = modpow(x, q, p) * modinv(a, p) % p;
while (y != 1) {
int k = 0;
long long z = y;
while (z != 1) k++, z = modpow(z, m, p);
long long w = modpow(u, pw[t - k - e], p);
long long v = modpow(w, pw[e + k - 1], p);
long long inv = modinv((int)modpow(y, pw[k - 1], p), p);
// v^n ≡ inv (mod p) となる最小の n (m 未満であることに注意) を BSGS で求める
int n = modlog(v, inv, p, m);
w = modpow(w, n, p);
x *= w, x %= p;
y *= modpow(w, q, p), y %= p;
}
return x;
}
// 素数 p について、x^k ≡ a (mod p) となる x を 1 つ求める (存在しなければ -1)
int kth_root_mod(int k, int a, const int &p) {
if (a == 0) return k == 0 ? -1 : 0;
if (p == 2) return a;
int g = gcd(k, p - 1);
if (modpow(a, (p - 1) / g, p) != 1) return -1;
auto ps = prime_factor(g);
int x = modpow(a, modinv(k / g, (p - 1) / g), p);
for (auto [m, e] : ps) x = prime_power_root_mod(m, e, x, p);
return x;
}
int main() {
int T;
cin >> T;
while (T--) {
int K, A, P;
cin >> P >> K >> A;
cout << kth_root_mod(K, A, P) << '\n';
}
}