結果
| 問題 |
No.950 行列累乗
|
| コンテスト | |
| ユーザー |
 hitonanode
|
| 提出日時 | 2019-12-14 00:29:19 |
| 言語 | C++14 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 13,964 bytes |
| コンパイル時間 | 2,641 ms |
| コンパイル使用メモリ | 205,116 KB |
| 実行使用メモリ | 14,592 KB |
| 最終ジャッジ日時 | 2024-06-27 23:31:13 |
| 合計ジャッジ時間 | 24,618 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 34 WA * 23 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using lint = long long int;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(0); ios::sync_with_stdio(false); cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((lint)(x).size())
#define POW2(n) (1LL << (n))
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template<typename T> void ndarray(vector<T> &vec, int len) { vec.resize(len); }
template<typename T, typename... Args> void ndarray(vector<T> &vec, int len, Args... args) { vec.resize(len); for (auto &v : vec) ndarray(v, args...); }
template<typename T> bool mmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; }
template<typename T> bool mmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; }
template<typename T1, typename T2> pair<T1, T2> operator+(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first + r.first, l.second + r.second); }
template<typename T1, typename T2> pair<T1, T2> operator-(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first - r.first, l.second - r.second); }
template<typename T> istream &operator>>(istream &is, vector<T> &vec){ for (auto &v : vec) is >> v; return is; }
///// This part below is only for debug, not used /////
template<typename T> ostream &operator<<(ostream &os, const vector<T> &vec){ os << "["; for (auto v : vec) os << v << ","; os << "]"; return os; }
template<typename T> ostream &operator<<(ostream &os, const deque<T> &vec){ os << "deq["; for (auto v : vec) os << v << ","; os << "]"; return os; }
template<typename T> ostream &operator<<(ostream &os, const set<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const unordered_set<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const multiset<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const unordered_multiset<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &pa){ os << "(" << pa.first << "," << pa.second << ")"; return os; }
template<typename TK, typename TV> ostream &operator<<(ostream &os, const map<TK, TV> &mp){ os << "{"; for (auto v : mp) os << v.first << "=>" << v.second << ","; os << "}"; return os; }
template<typename TK, typename TV> ostream &operator<<(ostream &os, const unordered_map<TK, TV> &mp){ os << "{"; for (auto v : mp) os << v.first << "=>" << v.second << ","; os << "}"; return os; }
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl;
///// END /////
struct ModIntRuntime
{
using lint = long long int;
static int get_mod() { return mod; }
int val;
static int mod;
static vector<ModIntRuntime> &facs()
{
static vector<ModIntRuntime> facs_;
return facs_;
}
static int &get_primitive_root() {
static int primitive_root_ = 0;
if (!primitive_root_) {
primitive_root_ = [&](){
set<int> fac;
int v = mod - 1;
for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < mod; g++) {
bool ok = true;
for (auto i : fac) if (ModIntRuntime(g).power((mod - 1) / i) == 1) { ok = false; break; }
if (ok) return g;
}
return -1;
}();
}
return primitive_root_;
}
static void set_mod(const int &m) {
if (mod != m) facs().clear();
mod = m;
get_primitive_root() = 0;
}
ModIntRuntime &_setval(lint v) { val = (v >= mod ? v - mod : v); return *this; }
ModIntRuntime() : val(0) {}
ModIntRuntime(lint v) { _setval(v % mod + mod); }
explicit operator bool() const { return val != 0; }
ModIntRuntime operator+(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val + x.val); }
ModIntRuntime operator-(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val - x.val + mod); }
ModIntRuntime operator*(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val * x.val % mod); }
ModIntRuntime operator/(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val * x.inv() % mod); }
ModIntRuntime operator-() const { return ModIntRuntime()._setval(mod - val); }
ModIntRuntime &operator+=(const ModIntRuntime &x) { return *this = *this + x; }
ModIntRuntime &operator-=(const ModIntRuntime &x) { return *this = *this - x; }
ModIntRuntime &operator*=(const ModIntRuntime &x) { return *this = *this * x; }
ModIntRuntime &operator/=(const ModIntRuntime &x) { return *this = *this / x; }
friend ModIntRuntime operator+(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % mod + x.val); }
friend ModIntRuntime operator-(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % mod - x.val + mod); }
friend ModIntRuntime operator*(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % mod * x.val % mod); }
friend ModIntRuntime operator/(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % mod * x.inv() % mod); }
bool operator==(const ModIntRuntime &x) const { return val == x.val; }
bool operator!=(const ModIntRuntime &x) const { return val != x.val; }
bool operator<(const ModIntRuntime &x) const { return val < x.val; }
friend istream &operator>>(istream &is, ModIntRuntime &x) { lint t; is >> t; x = ModIntRuntime(t); return is; }
friend ostream &operator<<(ostream &os, const ModIntRuntime &x) { os << x.val; return os; }
lint power(lint n) const {
lint ans = 1, tmp = this->val;
while (n) {
if (n & 1) ans = ans * tmp % mod;
tmp = tmp * tmp % mod;
n /= 2;
}
return ans;
}
lint inv() const { return this->power(mod - 2); }
ModIntRuntime operator^(lint n) const { return ModIntRuntime(this->power(n)); }
ModIntRuntime &operator^=(lint n) { return *this = *this ^ n; }
ModIntRuntime fac() const {
int l0 = facs().size();
if (l0 > this->val) return facs()[this->val];
facs().resize(this->val + 1);
for (int i = l0; i <= this->val; i++) facs()[i] = (i == 0 ? ModIntRuntime(1) : facs()[i - 1] * ModIntRuntime(i));
return facs()[this->val];
}
ModIntRuntime doublefac() const {
lint k = (this->val + 1) / 2;
if (this->val & 1) return ModIntRuntime(k * 2).fac() / ModIntRuntime(2).power(k) / ModIntRuntime(k).fac();
else return ModIntRuntime(k).fac() * ModIntRuntime(2).power(k);
}
ModIntRuntime nCr(const ModIntRuntime &r) const {
if (this->val < r.val) return ModIntRuntime(0);
return this->fac() / ((*this - r).fac() * r.fac());
}
ModIntRuntime sqrt() const {
if (val == 0) return 0;
if (mod == 2) return val;
if (power((mod - 1) / 2) != 1) return 0;
ModIntRuntime b = 1;
while (b.power((mod - 1) / 2) == 1) b += 1;
int e = 0, m = mod - 1;
while (m % 2 == 0) m >>= 1, e++;
ModIntRuntime x = power((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModIntRuntime z = b.power(m);
while (y != 1) {
int j = 0;
ModIntRuntime t = y;
while (t != 1) j++, t *= t;
z = z.power(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModIntRuntime(min(x.val, mod - x.val));
}
};
int ModIntRuntime::mod = 1;
template<typename T>
T mod_determinant(vector<vector<T>> mtr)
{
if (mtr.empty()) return 1;
assert(mtr.size() == mtr[0].size());
lint ans = 1;
mtr = gauss_jordan(mtr);
for (int i = 0; i < (int)mtr.size(); i++) ans = ans * mtr[i][i];
return ans;
}
template<typename T>
vector<vector<T>> matmul(const vector<vector<T>> &A, const vector<vector<T>> &B)
{
int H = A.size(), W = B[0].size(), K = B.size();
vector<vector<T>> C(H, vector<T>(W));
for (int i = 0; i < H; i++) {
for (int j = 0; j < W; j++) {
for (int k = 0; k < K; k++) {
C[i][j] += A[i][k] * B[k][j];
}
}
}
return C;
}
template<typename T>
vector<T> matmul(const vector<vector<T>> &A, const vector<T> &v)
{
vector<T> res(A.size());
for (int i = 0; i < (int)A.size(); i++) {
for (int j = 0; j < (int)v.size(); j++) {
res[i] += A[i][j] * v[j];
}
}
return res;
}
template<typename T>
vector<vector<T>> matpower(vector<vector<T>> X, lint n)
{
vector<vector<T>> res(X.size(), vector<T>(X.size()));
for (int i = 0; i < (int)res.size(); i++) res[i][i] = 1;
while (n)
{
if (n & 1) res = matmul(res, X);
X = matmul(X, X); n >>= 1;
}
return res;
}
template<typename T>
vector<T> eigen_vec_cand(vector<vector<T>> X, T a)
{
X[0][0] -= a;
X[1][1] -= a;
return vector<T>{X[0][1], -X[0][0]};
}
using mint = ModIntRuntime;
using Matrix = vector<vector<mint>>;
template<typename T>
T eigval_cand(vector<vector<T>> M)
{
// x^2 - 2ax - b = 0
T a = (M[0][0] + M[1][1]) / 2;
T b = M[0][1] * M[1][0] - M[0][0] * M[1][1];
T u = b + a * a;
T squ = u.sqrt();
dbg(a);
dbg(squ);
return (a + squ == 0 ? a - squ : a + squ);
}
struct DiscreteLogarithm
{
using lint = long long int;
int M, stepsize;
lint baby_a, giant_a, g;
std::unordered_map<lint, int> baby_log_dict;
lint inverse(lint a) {
lint b = M / g, u = 1, v = 0;
while (b) {
lint t = a / b;
a -= t * b; std::swap(a, b);
u -= t * v; std::swap(u, v);
}
u %= M / g;
return u >= 0 ? u : u + M / g;
}
DiscreteLogarithm(int mod, int a_new) : M(mod), baby_a(a_new % mod), giant_a(1) {
g = 1;
while (std::__gcd(baby_a, M / g) > 1) g *= std::__gcd(baby_a, M / g);
stepsize = 32; // lg(MAX_M)
while (stepsize * stepsize < M / g) stepsize++;
lint now = 1 % (M / g), inv_g = inverse(baby_a);
for (int n = 0; n < stepsize; n++) {
if (!baby_log_dict.count(now)) baby_log_dict[now] = n;
(now *= baby_a) %= M / g;
(giant_a *= inv_g) %= M / g;
}
}
// log(): returns the smallest nonnegative x that satisfies a^x = b mod M, or -1 if there's no solution
lint log(lint b) {
b %= M;
lint acc = 1 % M;
for (int i = 0; i < stepsize; i++) {
if (acc == b) return i;
(acc *= baby_a) %= M;
}
if (b % g) return -1; // No solution
lint now = b * giant_a % (M / g);
for (lint q = 1; q <= M / stepsize + 1; q++) {
if (baby_log_dict.count(now)) return q * stepsize + baby_log_dict[now];
(now *= giant_a) %= M / g;
}
return -1;
}
};
void NO()
{
puts("-1");
exit(0);
}
bool valid_check(Matrix A, Matrix B)
{
if (A[0][1] == 0 and B[0][1] != 0) NO();
if (A[1][0] == 0 and B[1][0] != 0) NO();
return true;
}
int main()
{
int m;
cin >> m;
mint::set_mod(m);
vector<vector<mint>> A(2, vector<mint>(2));
vector<vector<mint>> B(2, vector<mint>(2));
cin >> A >> B;
FOR(i, 1, 100)
{
if (matpower(A, i) == B) {
cout << i << endl;
return 0;
}
}
valid_check(A, B);
if (A[0][1] == 0 and A[1][0] == 0)
{
// Aが対角行列：コーナーケース
REP(d, 2) if ((A[d][d] == 0) ^ (B[d][d] == 0)) NO();
lint n0 = DiscreteLogarithm(m, A[0][0].val).log(B[0][0].val);
lint n1 = DiscreteLogarithm(m, A[1][1].val).log(B[1][1].val);
if (A[0][0] == 0 and A[1][1] == 0) NO();
else if (A[0][0] == 0)
{
cout << n1 << endl;
}
else if (A[1][1] == 0)
{
cout << n0 << endl;
}
else if (n0 < 0 or n1 < 0) NO();
else
{
lint m0 = DiscreteLogarithm(m, A[0][0].val).log(1);
lint t1 = DiscreteLogarithm(m, A[1][1].power(m0)).log((B[1][1] / A[1][1].power(n0)).val);
if (t1 < 0) NO();
else cout << n0 + m0 * t1 << endl;
}
return 0;
}
mint p;
if (A[0][1]) p = B[0][1] / A[0][1];
else p = B[1][0] / A[1][0];
mint q = B[0][0] - p * A[0][0];
if (A[1][1] * p + q != B[1][1] or A[1][0] * p != B[1][0] or A[0][1] * p != B[0][1]) NO();
Matrix trans{{A[0][0] + A[1][1], 1}, {A[0][1] * A[1][0] - A[0][0] * A[1][1], 0}};
if (trans[0][0] == 0 and trans[1][0] == 0) NO();
Matrix trans_inv(2, vector<mint>(2));
if (trans[1][0] != 0) trans_inv = {{0, 1 / trans[1][0]}, {1, trans[0][0] / trans[1][0]}};
vector<mint> v{0, 1};
map<vector<mint>, lint> ma;
const lint BN = 100000;
REP(d, BN)
{
if (!ma.count(v)) ma[v] = d;
v = matmul(trans, v);
}
REP(e, BN)
{
vector<mint> tgt(2);
if (trans[1][0] == 0)
{
tgt[0] = p / (mint(trans[0][0])^(BN * e));
}
else
{
tgt = matmul(matpower(trans_inv, e * BN), vector<mint>{p, q});
}
if (ma.count(tgt))
{
cout << ma[tgt] + e * BN << endl;
return 0;
}
}
NO();
}