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 10:40:06 |
| 言語 | C++14 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 543 ms / 2,000 ms |
| コード長 | 15,484 bytes |
| コンパイル時間 | 2,785 ms |
| コンパイル使用メモリ | 213,364 KB |
| 実行使用メモリ | 14,592 KB |
| 最終ジャッジ日時 | 2024-06-28 03:04:56 |
| 合計ジャッジ時間 | 9,783 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 57 |
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 T> istream &operator>>(istream &is, vector<T> &vec){ for (auto &v : vec) is >> v; return is; }
template<typename T> ostream &operator<<(ostream &os, const vector<T> &vec){ os << "["; for (auto v : vec) os << v << ","; os << "]"; return os; }
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl;
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>
struct matrix
{
int H, W;
std::vector<T> elem;
typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
inline T &at(int i, int j) { return elem[i * W + j]; }
inline T get(int i, int j) const { return elem[i * W + j]; }
operator std::vector<std::vector<T>>() const {
std::vector<std::vector<T>> ret(H);
for (int i = 0; i < H; i++) std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
return ret;
}
matrix(int H = 0, int W = 0) : H(H), W(W), elem(H * W) {}
matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
}
static matrix Identity(int N) {
matrix ret(N, N);
for (int i = 0; i < N; i++) ret.at(i, i) = 1;
return ret;
}
matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; }
matrix operator+(const matrix &r) const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; }
matrix operator-(const matrix &r) const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; }
matrix operator*(const matrix &r) const {
matrix ret(H, r.W);
for (int i = 0; i < H; i++) {
for (int k = 0; k < W; k++) {
for (int j = 0; j < r.W; j++) {
ret.at(i, j) += this->get(i, k) * r.get(k, j);
}
}
}
return ret;
}
matrix &operator+=(const matrix &r) { return *this = *this + r; }
matrix &operator-=(const matrix &r) { return *this = *this - r; }
matrix &operator*=(const matrix &r) { return *this = *this * r; }
bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
bool operator<(const matrix &r) const { return elem < r.elem; }
matrix pow(int64_t n) const {
matrix ret = Identity(H);
if (n == 0) return ret;
for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
ret *= ret;
if ((n >> i) & 1) ret *= (*this);
}
return ret;
}
matrix transpose() const {
matrix ret(W, H);
for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
return ret;
}
// Gauss-Jordan elimination
// - Require inverse for every non-zero element
// - Complexity: O(H^2 W)
matrix gauss_jordan() const {
int c = 0;
matrix mtr(*this);
for (int h = 0; h < H; h++) {
if (c == W) break;
int piv = -1;
for (int j = h; j < H; j++) if (mtr.get(j, c)) {
piv = j;
break;
}
if (piv == -1) { c++; h--; continue; }
if (h != piv) {
for (int w = 0; w < W; w++) {
std::swap(mtr[piv][w], mtr[h][w]);
mtr.at(piv, w) *= -1; // To preserve sign of determinant
}
}
for (int hh = h + 1; hh < H; hh++) {
for (int w = W - 1; w >= c; w--) {
mtr.at(hh, w) -= mtr.at(h, w) * mtr.at(hh, c) * mtr.at(h, c).inv();
}
}
c++;
}
return mtr;
}
int rank_of_gauss_jordan() const {
for (int i = H * W - 1; i >= 0; i--) if (elem[i]) return i / W + 1;
return 0;
}
T determinant_of_upper_triangle() const {
T ret = 1;
for (int i = 0; i < H; i++) ret *= get(i, i);
return ret;
}
friend std::ostream &operator<<(std::ostream &os, const matrix &x) {
os << "[(" << x.H << " * " << x.W << " matrix)";
for (int i = 0; i < x.H; i++) {
os << "\n[";
for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
os << "]";
}
os << "]\n";
return os;
}
friend std::istream &operator>>(std::istream &is, matrix &x) {
for (auto &v : x.elem) is >> v;
return is;
}
};
using mint = ModIntRuntime;
using M = matrix<mint>;
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(M A, M 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;
}
M inverse(M m)
{
mint det = m.gauss_jordan().determinant_of_upper_triangle();
return M({{m[1][1] / det, -m[0][1] / det}, {-m[1][0] / det, m[0][0] / det}});
}
int main()
{
int m;
cin >> m;
mint::set_mod(m);
M A(2, 2);
M B(2, 2);
cin >> A >> B;
FOR(i, 1, 100) if (A.pow(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 / A[0][0]).val) + 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 = (A[0][1] !=0 ? B[0][1] / A[0][1] : 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();
M 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();
M trans_inv(2, 2);
if (trans[1][0]) trans_inv = M({{0, 1 / trans[1][0]}, {1, -trans[0][0] / trans[1][0]}});
const lint BN = 100000;
if (trans[1][0] == 0)
{
mint v = 1;
map<mint, lint> ma;
FOR(d, 1, BN + 2)
{
if (!ma.count(v)) ma[v] = d;
v = trans[0][0] * v;
}
REP(e, BN)
{
mint tgt = p / (mint(trans[0][0]) ^ (BN * e));
if (ma.count(tgt))
{
cout << ma[tgt] + e * BN << endl;
return 0;
}
}
NO();
}
mint detA = A.gauss_jordan().determinant_of_upper_triangle();
lint i_det = DiscreteLogarithm(m, detA.val).log(B.gauss_jordan().determinant_of_upper_triangle().val);
if (i_det < 0) NO();
lint p_det = DiscreteLogarithm(m, detA.val).log((1 / detA).val) + 1;
map<M, lint> ma;
M tmp = A.pow(i_det);
M tmp_p = A.pow(p_det);
REP(i, BN + 2)
{
if (!ma.count(tmp) and i_det + p_det * i > 0) ma[tmp] = i_det + p_det * i;
tmp = tmp * tmp_p;
}
REP(i, BN + 1)
{
M q = inverse(tmp_p.pow(i * BN)) * B;
if (ma.count(q))
{
cout << ma[q] + i * BN * p_det << endl;
return 0;
}
}
NO();
}