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order.cpp
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order.cpp
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//
// calc order
// min: x s.t. a^x \equiv 1 (mod p)
//
// verified:
// AtCoder ABC 335 G - Discrete Logarithm Problems
// https://atcoder.jp/contests/abc335/tasks/abc335_g
//
// AOJ 3062 Product
// https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=3062
//
#include <bits/stdc++.h>
using namespace std;
// mod_pow, mod_inv
template<class T> T mod_pow(T a, T n, T m) {
T res = 1;
while (n > 0) {
if (n % 2 == 1) res = res * a % m;
a = a * a % m;
n >>= 1;
}
return res;
};
template<class T> T mod_inv(T a, T m) {
T b = m, u = 1, v = 0;
while (b > 0) {
T t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
u %= m;
if (u < 0) u += m;
return u;
};
// montgomery modint (MOD < 2^62, MOD is odd)
struct MontgomeryModInt64 {
using mint = MontgomeryModInt64;
using u64 = uint64_t;
using u128 = __uint128_t;
// static menber
static u64 MOD;
static u64 INV_MOD; // INV_MOD * MOD ≡ 1 (mod 2^64)
static u64 T128; // 2^128 (mod MOD)
// inner value
u64 val;
// constructor
MontgomeryModInt64() : val(0) { }
MontgomeryModInt64(long long v) : val(reduce((u128(v) + MOD) * T128)) { }
u64 get() const {
u64 res = reduce(val);
return res >= MOD ? res - MOD : res;
}
// mod getter and setter
static u64 get_mod() { return MOD; }
static void set_mod(u64 mod) {
assert(mod < (1LL << 62));
assert((mod & 1));
MOD = mod;
T128 = -u128(mod) % mod;
INV_MOD = get_inv_mod();
}
static u64 get_inv_mod() {
u64 res = MOD;
for (int i = 0; i < 5; ++i) res *= 2 - MOD * res;
return res;
}
static u64 reduce(const u128 &v) {
return (v + u128(u64(v) * u64(-INV_MOD)) * MOD) >> 64;
}
// arithmetic operators
mint operator + () const { return mint(*this); }
mint operator - () const { return mint() - mint(*this); }
mint operator + (const mint &r) const { return mint(*this) += r; }
mint operator - (const mint &r) const { return mint(*this) -= r; }
mint operator * (const mint &r) const { return mint(*this) *= r; }
mint operator / (const mint &r) const { return mint(*this) /= r; }
mint& operator += (const mint &r) {
if ((val += r.val) >= 2 * MOD) val -= 2 * MOD;
return *this;
}
mint& operator -= (const mint &r) {
if ((val += 2 * MOD - r.val) >= 2 * MOD) val -= 2 * MOD;
return *this;
}
mint& operator *= (const mint &r) {
val = reduce(u128(val) * r.val);
return *this;
}
mint& operator /= (const mint &r) {
*this *= r.inv();
return *this;
}
mint inv() const { return pow(MOD - 2); }
mint pow(u128 n) const {
mint res(1), mul(*this);
while (n > 0) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
// other operators
bool operator == (const mint &r) const {
return (val >= MOD ? val - MOD : val) == (r.val >= MOD ? r.val - MOD : r.val);
}
bool operator != (const mint &r) const {
return (val >= MOD ? val - MOD : val) != (r.val >= MOD ? r.val - MOD : r.val);
}
mint& operator ++ () {
++val;
if (val >= MOD) val -= MOD;
return *this;
}
mint& operator -- () {
if (val == 0) val += MOD;
--val;
return *this;
}
mint operator ++ (int) {
mint res = *this;
++*this;
return res;
}
mint operator -- (int) {
mint res = *this;
--*this;
return res;
}
friend istream& operator >> (istream &is, mint &x) {
long long t;
is >> t;
x = mint(t);
return is;
}
friend ostream& operator << (ostream &os, const mint &x) {
return os << x.get();
}
friend mint pow(const mint &r, long long n) {
return r.pow(n);
}
friend mint inv(const mint &r) {
return r.inv();
}
};
typename MontgomeryModInt64::u64
MontgomeryModInt64::MOD, MontgomeryModInt64::INV_MOD, MontgomeryModInt64::T128;
// Miller-Rabin
bool MillerRabin(long long N, const vector<long long> &A) {
assert(N % 2 == 1);
assert(N < (1LL<<62));
using mint = MontgomeryModInt64;
mint::set_mod(N);
long long s = 0, d = N - 1;
while (d % 2 == 0) {
++s;
d >>= 1;
}
for (auto a : A) {
if (N <= a) return true;
mint x = mint(a).pow(d);
if (x != 1) {
long long t;
for (t = 0; t < s; ++t) {
if (x == N - 1) break;
x *= x;
}
if (t == s) return false;
}
}
return true;
}
bool is_prime(long long N) {
if (N <= 1) return false;
else if (N == 2) return true;
else if (N % 2 == 0) return false;
else if (N < 4759123141LL)
return MillerRabin(N, {2, 7, 61});
else
return MillerRabin(N, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
}
// Pollard's Rho
unsigned int xor_shift_rng() {
static unsigned int tx = 123456789, ty=362436069, tz=521288629, tw=88675123;
unsigned int tt = (tx^(tx<<11));
tx = ty, ty = tz, tz = tw;
return ( tw=(tw^(tw>>19))^(tt^(tt>>8)) );
}
long long pollard(long long N) {
if (N % 2 == 0) return 2;
if (is_prime(N)) return N;
assert(N < (1LL<<62));
using mint = MontgomeryModInt64;
mint::set_mod(N);
long long step = 0;
while (true) {
mint r = xor_shift_rng(); // random r
auto f = [&](mint x) -> mint { return x * x + r; };
mint x = ++step, y = f(x);
while (true) {
long long p = gcd((y - x).get(), N);
if (p == 0 || p == N) break;
if (p != 1) return p;
x = f(x);
y = f(f(y));
}
}
}
vector<long long> pollard_prime_factorize(long long N) {
if (N == 1) return {};
long long p = pollard(N);
if (p == N) return {p};
vector<long long> left = pollard_prime_factorize(p);
vector<long long> right = pollard_prime_factorize(N / p);
if (left.size() > right.size()) swap(left, right);
left.insert(left.end(), right.begin(), right.end());
sort(left.begin(), left.end());
return left;
}
vector<pair<long long, long long>> prime_factorize(long long N) {
vector<pair<long long, long long>> res;
const auto &prs = pollard_prime_factorize(N);
long long prev = -1, num = 0;
for (const auto &pr : prs) {
if (pr == prev) ++num;
else {
if (prev != -1) res.emplace_back(prev, num);
prev = pr, num = 1;
}
}
if (prev != -1) res.emplace_back(prev, num);
return res;
}
// various methods mod prime P
struct PrimeProcessor {
using mint = MontgomeryModInt64;
// input prime
long long prime;
vector<pair<long long, long long>> pf; // prime factorization of p-1
// constructors
PrimeProcessor() {}
PrimeProcessor(long long p) : prime(p) {
init(p);
}
// initializer
void init(long long p) {
assert(is_prime(p));
prime = p;
if (p % 2 == 1) {
assert(p < (1LL<<62));
prime = p;
pf = prime_factorize(prime - 1);
mint::set_mod(prime);
}
}
// min: x s.t. a^x \equiv 1 (mod prime)
long long calc_order(long long a) {
assert(a != 0);
if (prime == 2) return 1;
long long res = prime - 1;
for (const auto &[p, num] : pf) {
while (res % p == 0 && mint(a).pow(res / p) == 1) res /= p;
}
return res;
}
};
//------------------------------//
// Examples
//------------------------------//
void ABC_335_G() {
long long N, P;
cin >> N >> P;
vector<long long> A(N);
for (int i = 0; i < N; ++i) cin >> A[i];
PrimeProcessor pp(P);
map<long long, long long> ma;
for (auto a : A) {
long long order = pp.calc_order(a);
++ma[order];
}
long long res = 0;
for (auto [v1, num1] : ma) {
for (auto [v2, num2] : ma) {
if (v2 % v1 == 0) res += num1 * num2;
}
}
cout << res << endl;
}
void AOJ_3062() {
int P, T;
cin >> P >> T;
PrimeProcessor pp(P);
while (T--) {
int N, A;
cin >> N;
int g = P - 1;
for (int i = 0; i < N; ++i) {
int G;
cin >> G;
g = gcd(g, (P - 1) / pp.calc_order(G));
}
cin >> A;
int ga = (P - 1) / pp.calc_order(A);
cout << (ga % g ? 0 : 1) << endl;
}
}
int main() {
ABC_335_G();
//AOJ_3062();
}