algorithm
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test/yosupo/kth_term_of_linearly_recurrent_sequence.test.cpp
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- Last update: 2022-03-17 11:21:22+09:00
- Problem: https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence
前提
- $p - 1$ が十分大きい $2$ べきで割り切れる場合に $\mod p$ で積を計算する
- modint ライブラリと併用する
例題
2D NTT
-
https://yukicoder.me/problems/no/1241
-
https://atcoder.jp/contests/jag2013spring/tasks/icpc2013spring_f (まだ)
Depends on
Code
#define PROBLEM \
"https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
//#pragma GCC optimize("Ofast")
//#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using ll = int64_t;
using ull = uint64_t;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
explicit operator int() const { return v; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
friend istream& operator>>(istream& in, M& x) {
ll v_;
in >> v_;
x = M(v_);
return in;
}
bool operator<(const M& r) const { return v < r.v; }
bool operator>(const M& r) const { return v < *this; }
bool operator<=(const M& r) const { return !(r < *this); }
bool operator>=(const M& r) const { return !(*this < r); }
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
static uint get_mod() { return MOD; }
};
using Mint = ModInt<998244353>;
/**
* @docs docs/ntt.md
*/
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
int v = D(n).inv().v;
if (inv) {
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}
// depends on FFT libs
// basically use with ModInt
NumberTheoreticTransform<Mint> ntt;
template <class D>
struct Poly : public V<D> {
template <class... Args>
Poly(Args... args) : V<D>(args...) {}
Poly(initializer_list<D> init) : V<D>(init.begin(), init.end()) {}
int size() const { return V<D>::size(); }
D at(int p) const { return (p < this->size() ? (*this)[p] : D(0)); }
// first len terms
Poly pref(int len) const {
return Poly(this->begin(), this->begin() + min(this->size(), len));
}
// for polynomial division
Poly rev() const {
Poly res = *this;
reverse(res.begin(), res.end());
return res;
}
Poly shiftr(int d) const {
int n = max(size() + d, 0);
Poly res(n);
for (int i = 0; i < size(); ++i) {
if (i + d >= 0) {
res[i + d] = at(i);
}
}
return res;
}
Poly operator+(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) + r.at(i);
}
return tmp;
}
Poly operator-(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) - r.at(i);
}
return tmp;
}
// scalar
Poly operator*(const D& k) const {
int n = size();
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) * k;
}
return tmp;
}
Poly operator*(const Poly& r) const {
Poly a = *this;
Poly b = r;
auto v = ntt.mul(a, b);
return v;
}
// scalar
Poly operator/(const D& k) const { return *this * k.inv(); }
Poly operator/(const Poly& r) const {
if (size() < r.size()) {
return {{}};
}
int d = size() - r.size() + 1;
return (rev().pref(d) * r.rev().inv(d)).pref(d).rev();
}
Poly operator%(const Poly& r) const {
auto res = *this - *this / r * r;
while (res.size() && !res.back()) {
res.pop_back();
}
return res;
}
Poly diff() const {
V<D> res(max(0, size() - 1));
for (int i = 1; i < size(); ++i) {
res[i - 1] = at(i) * i;
}
return res;
}
Poly inte() const {
V<D> res(size() + 1);
for (int i = 0; i < size(); ++i) {
res[i + 1] = at(i) / (D)(i + 1);
}
return res;
}
// f * f.inv(m) === 1 mod (x^m)
// f_0 ^ -1 must exist
Poly inv(int m) const {
Poly res = Poly({D(1) / at(0)});
for (int i = 1; i < m; i *= 2) {
res = (res * D(2) - res * res * pref(i * 2)).pref(i * 2);
}
return res.pref(m);
}
// f_0 = 1 must hold
Poly log(int n) const {
auto f = pref(n);
return (f.diff() * f.inv(n - 1)).pref(n - 1).inte();
}
// f_0 = 0 must hold
Poly exp(int n) const {
auto h = diff();
Poly f({1}), g({1});
for (int m = 1; m < n; m *= 2) {
g = (g * D(2) - f * g * g).pref(m);
auto q = h.pref(m - 1);
auto w = (q + g * (f.diff() - f * q)).pref(m * 2 - 1);
f = (f + f * (*this - w.inte()).pref(m * 2)).pref(m * 2);
}
return f.pref(n);
}
// be careful when k = 0
Poly pow(int n, ll k) const { return (log(n) * (D)k).exp(n); }
// f_0 = 1 must hold (use it with modular sqrt)
// CF250E
Poly sqrt(int n) const {
Poly f = pref(n);
Poly g({1});
for (int i = 1; i < n; i *= 2) {
g = (g + f.pref(i * 2) * g.inv(i * 2)) * D(2).inv();
}
return g.pref(n);
}
D eval(D x) const {
D res = 0, c = 1;
for (auto a : *this) {
res += a * c;
c *= x;
}
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D& r) { return *this = *this / r; }
Poly& operator%=(const Poly& r) { return *this = *this % r; }
friend ostream& operator<<(ostream& os, const Poly& pl) {
if (pl.size() == 0) return os << "0";
for (int i = 0; i < pl.size(); ++i) {
if (pl[i]) {
os << pl[i] << "x^" << i;
if (i + 1 != pl.size()) os << ",";
}
}
return os;
}
explicit operator bool() const {
bool f = false;
for (int i = 0; i < size(); ++i) {
if (at(i)) {
f = true;
}
}
return f;
}
};
// calculate characteristic polynomial
// c_0 * s_i + c_1 * s_{i+1} + ... + c_k * s_{i+k} = 0
// c_k = -1
template <class T>
Poly<T> berlekamp_massey(const V<T>& s) {
int n = int(s.size());
V<T> b = {T(-1)}, c = {T(-1)};
T y = Mint(1);
for (int ed = 1; ed <= n; ed++) {
int l = int(c.size()), m = int(b.size());
T x = 0;
for (int i = 0; i < l; i++) {
x += c[i] * s[ed - l + i];
}
b.push_back(0);
m++;
if (!x) {
continue;
}
T freq = x / y;
if (l < m) {
auto tmp = c;
c.insert(begin(c), m - l, Mint(0));
for (int i = 0; i < m; i++) {
c[m - 1 - i] -= freq * b[m - 1 - i];
}
b = tmp;
y = x;
} else {
for (int i = 0; i < m; i++) {
c[l - 1 - i] -= freq * b[m - 1 - i];
}
}
}
return c;
}
// HUPC 2020 day3 K
// calculate vec[0] * vec[1] * ...
// deg(result) must be bounded
template <class T>
Poly<T> prod(const V<Poly<T>>& vec) {
auto comp = [](const auto& a, const auto& b) -> bool {
return a.size() > b.size();
};
priority_queue<Poly<T>, V<Poly<T>>, decltype(comp)> que(comp);
que.push(Poly<T>{1});
for (auto& pl : vec) que.push(pl);
while (que.size() > 1) {
auto va = que.top();
que.pop();
auto vb = que.top();
que.pop();
que.push(va * vb);
}
return que.top();
}
#define call_from_test
#include "../../cpp_src/math/BostanMori.hpp"
#undef call_from_test
int main() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
ntt.init();
int d;
ll k;
cin >> d >> k;
Poly<Mint> a(d);
rep(i, d) cin >> a[i];
Poly<Mint> c(d);
rep(i, d) cin >> c[i];
auto v = bostan_mori(a, c, k);
cout << v << '\n';
return 0;
}#line 1 "test/yosupo/kth_term_of_linearly_recurrent_sequence.test.cpp"
#define PROBLEM \
"https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
//#pragma GCC optimize("Ofast")
//#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using ll = int64_t;
using ull = uint64_t;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
explicit operator int() const { return v; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
friend istream& operator>>(istream& in, M& x) {
ll v_;
in >> v_;
x = M(v_);
return in;
}
bool operator<(const M& r) const { return v < r.v; }
bool operator>(const M& r) const { return v < *this; }
bool operator<=(const M& r) const { return !(r < *this); }
bool operator>=(const M& r) const { return !(*this < r); }
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
static uint get_mod() { return MOD; }
};
using Mint = ModInt<998244353>;
/**
* @docs docs/ntt.md
*/
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
int v = D(n).inv().v;
if (inv) {
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}
// depends on FFT libs
// basically use with ModInt
NumberTheoreticTransform<Mint> ntt;
template <class D>
struct Poly : public V<D> {
template <class... Args>
Poly(Args... args) : V<D>(args...) {}
Poly(initializer_list<D> init) : V<D>(init.begin(), init.end()) {}
int size() const { return V<D>::size(); }
D at(int p) const { return (p < this->size() ? (*this)[p] : D(0)); }
// first len terms
Poly pref(int len) const {
return Poly(this->begin(), this->begin() + min(this->size(), len));
}
// for polynomial division
Poly rev() const {
Poly res = *this;
reverse(res.begin(), res.end());
return res;
}
Poly shiftr(int d) const {
int n = max(size() + d, 0);
Poly res(n);
for (int i = 0; i < size(); ++i) {
if (i + d >= 0) {
res[i + d] = at(i);
}
}
return res;
}
Poly operator+(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) + r.at(i);
}
return tmp;
}
Poly operator-(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) - r.at(i);
}
return tmp;
}
// scalar
Poly operator*(const D& k) const {
int n = size();
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) * k;
}
return tmp;
}
Poly operator*(const Poly& r) const {
Poly a = *this;
Poly b = r;
auto v = ntt.mul(a, b);
return v;
}
// scalar
Poly operator/(const D& k) const { return *this * k.inv(); }
Poly operator/(const Poly& r) const {
if (size() < r.size()) {
return {{}};
}
int d = size() - r.size() + 1;
return (rev().pref(d) * r.rev().inv(d)).pref(d).rev();
}
Poly operator%(const Poly& r) const {
auto res = *this - *this / r * r;
while (res.size() && !res.back()) {
res.pop_back();
}
return res;
}
Poly diff() const {
V<D> res(max(0, size() - 1));
for (int i = 1; i < size(); ++i) {
res[i - 1] = at(i) * i;
}
return res;
}
Poly inte() const {
V<D> res(size() + 1);
for (int i = 0; i < size(); ++i) {
res[i + 1] = at(i) / (D)(i + 1);
}
return res;
}
// f * f.inv(m) === 1 mod (x^m)
// f_0 ^ -1 must exist
Poly inv(int m) const {
Poly res = Poly({D(1) / at(0)});
for (int i = 1; i < m; i *= 2) {
res = (res * D(2) - res * res * pref(i * 2)).pref(i * 2);
}
return res.pref(m);
}
// f_0 = 1 must hold
Poly log(int n) const {
auto f = pref(n);
return (f.diff() * f.inv(n - 1)).pref(n - 1).inte();
}
// f_0 = 0 must hold
Poly exp(int n) const {
auto h = diff();
Poly f({1}), g({1});
for (int m = 1; m < n; m *= 2) {
g = (g * D(2) - f * g * g).pref(m);
auto q = h.pref(m - 1);
auto w = (q + g * (f.diff() - f * q)).pref(m * 2 - 1);
f = (f + f * (*this - w.inte()).pref(m * 2)).pref(m * 2);
}
return f.pref(n);
}
// be careful when k = 0
Poly pow(int n, ll k) const { return (log(n) * (D)k).exp(n); }
// f_0 = 1 must hold (use it with modular sqrt)
// CF250E
Poly sqrt(int n) const {
Poly f = pref(n);
Poly g({1});
for (int i = 1; i < n; i *= 2) {
g = (g + f.pref(i * 2) * g.inv(i * 2)) * D(2).inv();
}
return g.pref(n);
}
D eval(D x) const {
D res = 0, c = 1;
for (auto a : *this) {
res += a * c;
c *= x;
}
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D& r) { return *this = *this / r; }
Poly& operator%=(const Poly& r) { return *this = *this % r; }
friend ostream& operator<<(ostream& os, const Poly& pl) {
if (pl.size() == 0) return os << "0";
for (int i = 0; i < pl.size(); ++i) {
if (pl[i]) {
os << pl[i] << "x^" << i;
if (i + 1 != pl.size()) os << ",";
}
}
return os;
}
explicit operator bool() const {
bool f = false;
for (int i = 0; i < size(); ++i) {
if (at(i)) {
f = true;
}
}
return f;
}
};
// calculate characteristic polynomial
// c_0 * s_i + c_1 * s_{i+1} + ... + c_k * s_{i+k} = 0
// c_k = -1
template <class T>
Poly<T> berlekamp_massey(const V<T>& s) {
int n = int(s.size());
V<T> b = {T(-1)}, c = {T(-1)};
T y = Mint(1);
for (int ed = 1; ed <= n; ed++) {
int l = int(c.size()), m = int(b.size());
T x = 0;
for (int i = 0; i < l; i++) {
x += c[i] * s[ed - l + i];
}
b.push_back(0);
m++;
if (!x) {
continue;
}
T freq = x / y;
if (l < m) {
auto tmp = c;
c.insert(begin(c), m - l, Mint(0));
for (int i = 0; i < m; i++) {
c[m - 1 - i] -= freq * b[m - 1 - i];
}
b = tmp;
y = x;
} else {
for (int i = 0; i < m; i++) {
c[l - 1 - i] -= freq * b[m - 1 - i];
}
}
}
return c;
}
// HUPC 2020 day3 K
// calculate vec[0] * vec[1] * ...
// deg(result) must be bounded
template <class T>
Poly<T> prod(const V<Poly<T>>& vec) {
auto comp = [](const auto& a, const auto& b) -> bool {
return a.size() > b.size();
};
priority_queue<Poly<T>, V<Poly<T>>, decltype(comp)> que(comp);
que.push(Poly<T>{1});
for (auto& pl : vec) que.push(pl);
while (que.size() > 1) {
auto va = que.top();
que.pop();
auto vb = que.top();
que.pop();
que.push(va * vb);
}
return que.top();
}
#define call_from_test
#line 1 "cpp_src/math/BostanMori.hpp"
// ref :
// https://qiita.com/ryuhe1/items/da5acbcce4ac1911f47a#%E5%BD%A2%E5%BC%8F%E7%9A%84%E3%81%B9%E3%81%8D%E7%B4%9A%E6%95%B0
// a_i = \sum_{j=1}^d c_j * a_{i-j}
// input
// a_0, a_1, a_2, ..., a_{d-1}
// c_1, c_2, c_3, ..., c_d
// n
// calculate a_n
template <class T>
T bostan_mori(Poly<T> a, Poly<T> c, ll n) {
if (n < a.size()) return a[n];
using P = Poly<T>;
auto even = [&](const P& a) {
int sz = SZ(a);
P b((sz + 1) / 2);
for (int i = 0; i < SZ(a); i += 2) {
b[i / 2] = a[i];
}
return b;
};
auto odd = [&](const P& a) {
int sz = SZ(a);
P b(sz / 2);
for (int i = 1; i < SZ(a); i += 2) {
b[i / 2] = a[i];
}
return b;
};
// a(x) -> a(-x)
auto neg = [&](const P& a) {
auto b = a;
for (int i = 1; i < SZ(b); i += 2) {
b[i] = -b[i];
}
return b;
};
int d = SZ(c);
P q(d + 1);
q[0] = 1;
rep(i, SZ(c)) q[i + 1] = -c[i];
P p = a * q;
p = p.pref(d);
while (n > 0) {
debug(p, q);
auto u = p * neg(q);
if (n % 2 == 0) {
p = even(u);
} else {
p = odd(u);
}
q = even(q * neg(q));
n /= 2;
}
return p[0] / q[0];
}
#line 523 "test/yosupo/kth_term_of_linearly_recurrent_sequence.test.cpp"
#undef call_from_test
int main() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
ntt.init();
int d;
ll k;
cin >> d >> k;
Poly<Mint> a(d);
rep(i, d) cin >> a[i];
Poly<Mint> c(d);
rep(i, d) cin >> c[i];
auto v = bostan_mori(a, c, k);
cout << v << '\n';
return 0;
}