verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp

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• Last update: 2025-10-21 21:13:36+09:00
• Problem: https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence

Depends on

• NTT (数論変換) (fft/ntt.hpp)
• Berlekamp-Massey (fps/berlekamp-massey.hpp)
• Bostan-Mori (fps/bostan-mori.hpp)
• fps/formal-power-series.hpp
• fps/fps-ntt-friendly.hpp
• Inverse の次数シフト (fps/inverse-shift.hpp)
• 線形漸化式用 (fps/linearly-recurrent-sequence.hpp)
• modint/modint.hpp
• template/debug.hpp
• template/inout.hpp
• template/macro.hpp
• template/template.hpp
• template/util.hpp

Code

#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"

#include "template/template.hpp"
#include "modint/modint.hpp"
using mint = ModInt<998244353>;
#include "fps/fps-ntt-friendly.hpp"
using fps = FormalPowerSeries<mint>;
#include "fps/linearly-recurrent-sequence.hpp"

int main() {
  int d;
  ll k;
  in(d, k);
  fps a(d), c(d);
  in(a, c);
  out(LinearyRecurrentSequence(a, c, k));
}

#line 1 "verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"

#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;

#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"

#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

template <class T, class S = T>
S SUM(const vector<T> &a) {
  return accumulate(ALL(a), S(0));
}
template <class T>
inline bool chmin(T &a, T b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}
template <class T>
inline bool chmax(T &a, T b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}

template <class T>
int popcnt(T x) {
  return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
  return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
  return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"

#line 2 "template/inout.hpp"
struct Fast {
  Fast() {
    cin.tie(nullptr);
    ios_base::sync_with_stdio(false);
    cout << fixed << setprecision(15);
  }
} fast;

template <class T1, class T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
  return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
  return os << p.first << " " << p.second;
}
template <class T>
istream &operator>>(istream &is, vector<T> &a) {
  for (auto &v : a) is >> v;
  return is;
}
template <class T>
ostream &operator<<(ostream &os, const vector<T> &a) {
  for (auto it = a.begin(); it != a.end();) {
    os << *it;
    if (++it != a.end()) os << " ";
  }
  return os;
}
template <class T>
ostream &operator<<(ostream &os, const set<T> &st) {
  os << "{";
  for (auto it = st.begin(); it != st.end();) {
    os << *it;
    if (++it != st.end()) os << ",";
  }
  os << "}";
  return os;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const map<T1, T2> &mp) {
  os << "{";
  for (auto it = mp.begin(); it != mp.end();) {
    os << it->first << ":" << it->second;
    if (++it != mp.end()) os << ",";
  }
  os << "}";
  return os;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
  cin >> t;
  in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
  cout << t;
  if (sizeof...(u)) cout << sep;
  out(u...);
}
#line 10 "template/template.hpp"

#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int i, T name) {
  cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
  for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
  cerr << ":" << b << " ";
  _show(i + 1, a, c...);
}
#line 2 "modint/modint.hpp"

template <unsigned int m = 998244353>
struct ModInt {
  using mint = ModInt;
  unsigned int _v;
  static constexpr unsigned int get_mod() { return m; }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }
  ModInt() : _v(0) {}
  ModInt(int64_t v) {
    long long x = (long long)(v % (long long)(umod()));
    if (x < 0) x += umod();
    _v = (unsigned int)(x);
  }
  unsigned int val() const { return _v; }
  mint &operator++() {
    _v++;
    if (_v == umod()) _v = 0;
    return *this;
  }
  mint &operator--() {
    if (_v == 0) _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }

  mint &operator+=(const mint &rhs) {
    _v += rhs._v;
    if (_v >= umod()) _v -= umod();
    return *this;
  }
  mint &operator-=(const mint &rhs) {
    _v -= rhs._v;
    if (_v >= umod()) _v += umod();
    return *this;
  }
  mint &operator*=(const mint &rhs) {
    unsigned long long z = _v;
    z *= rhs._v;
    _v = (unsigned int)(z % umod());
    return *this;
  }
  mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1) r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    assert(_v);
    return pow(umod() - 2);
  }

  friend mint operator+(const mint &lhs, const mint &rhs) {
    return mint(lhs) += rhs;
  }
  friend mint operator-(const mint &lhs, const mint &rhs) {
    return mint(lhs) -= rhs;
  }
  friend mint operator*(const mint &lhs, const mint &rhs) {
    return mint(lhs) *= rhs;
  }
  friend mint operator/(const mint &lhs, const mint &rhs) {
    return mint(lhs) /= rhs;
  }
  friend bool operator==(const mint &lhs, const mint &rhs) {
    return lhs._v == rhs._v;
  }
  friend bool operator!=(const mint &lhs, const mint &rhs) {
    return lhs._v != rhs._v;
  }
  friend istream &operator>>(istream &is, mint &x) {
    return is >> x._v;
  }
  friend ostream &operator<<(ostream &os, const mint &x) {
    return os << x.val();
  }

 private:
  static constexpr unsigned int umod() { return m; }
};
#line 5 "verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp"
using mint = ModInt<998244353>;
#line 2 "fps/fps-ntt-friendly.hpp"

#line 2 "fft/ntt.hpp"

template <class mint>
struct NTT {
  static constexpr unsigned int mod = mint::get_mod();
  static constexpr unsigned long long pow_constexpr(unsigned long long x, unsigned long long n, unsigned long long m) {
    unsigned long long y = 1;
    while (n) {
      if (n & 1) y = y * x % m;
      x = x * x % m;
      n >>= 1;
    }
    return y;
  }
  static constexpr unsigned int get_g() {
    unsigned long long x = 2;
    while (pow_constexpr(x, (mod - 1) >> 1, mod) == 1) x += 1;
    return x;
  }
  static constexpr unsigned int g = get_g();
  static constexpr int rank2 = __builtin_ctzll(mod - 1);
  array<mint, rank2 + 1> root;
  array<mint, rank2 + 1> iroot;
  array<mint, max(0, rank2 - 2 + 1)> rate2;
  array<mint, max(0, rank2 - 2 + 1)> irate2;
  array<mint, max(0, rank2 - 3 + 1)> rate3;
  array<mint, max(0, rank2 - 3 + 1)> irate3;

  NTT() {
    root[rank2] = mint(g).pow((mod - 1) >> rank2);
    iroot[rank2] = root[rank2].inv();
    for (int i = rank2 - 1; i >= 0; i--) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 2; i++) {
        rate2[i] = root[i + 2] * prod;
        irate2[i] = iroot[i + 2] * iprod;
        prod *= iroot[i + 2];
        iprod *= root[i + 2];
      }
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 3; i++) {
        rate3[i] = root[i + 3] * prod;
        irate3[i] = iroot[i + 3] * iprod;
        prod *= iroot[i + 3];
        iprod *= root[i + 3];
      }
    }
  }
  void ntt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    a.resize(1 << h);
    int len = 0;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        for (int s = 0; s < (1 << len); s++) {
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          if (s + 1 != (1 << len)) rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len++;
      } else {
        // 4-base
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto mod2 = 1ULL * mint::get_mod() * mint::get_mod();
            auto a0 = 1ULL * a[i + offset].val();
            auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
            auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
            auto na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          if (s + 1 != (1 << len)) rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len += 2;
      }
    }
  }
  void intt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    a.resize(1 << h);

    int len = h;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        for (int s = 0; s < (1 << (len - 1)); s++) {
          int offset = s << (h - len + 1);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p];
            a[i + offset] = l + r;
            a[i + offset + p] = (unsigned long long)(mint::get_mod() + l.val() - r.val()) * irot.val();
          }
          if (s + 1 != (1 << (len - 1))) irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len--;
      } else {
        // 4-base
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        for (int s = 0; s < (1 << (len - 2)); s++) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            auto a0 = 1ULL * a[i + offset + 0 * p].val();
            auto a1 = 1ULL * a[i + offset + 1 * p].val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val();
            auto a2na3iimag = 1ULL * mint((mint::get_mod() + a2 - a3) * iimag.val()).val();
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + (mint::get_mod() - a1) + a2na3iimag) * irot.val();
            a[i + offset + 2 * p] = (a0 + a1 + (mint::get_mod() - a2) + (mint::get_mod() - a3)) * irot2.val();
            a[i + offset + 3 * p] = (a0 + (mint::get_mod() - a1) + (mint::get_mod() - a2na3iimag)) * irot3.val();
          }
          if (s + 1 != (1 << (len - 2))) irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len -= 2;
      }
    }
    mint e = mint(n).inv();
    for (auto& x : a) x *= e;
  }
  vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
    if (a.empty() || b.empty()) return vector<mint>();
    int n = a.size(), m = b.size();
    int sz = n + m - 1;
    if (n <= 30 || m <= 30) {
      if (n > 30) return multiply(b, a);
      vector<mint> res(sz);
      for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < sz) sz1 <<= 1;
    vector<mint> res(sz1);
    for (int i = 0; i < n; i++) res[i] = a[i];
    ntt(res);
    if (a == b)
      for (int i = 0; i < sz1; i++) res[i] *= res[i];
    else {
      vector<mint> c(sz1);
      for (int i = 0; i < m; i++) c[i] = b[i];
      ntt(c);
      for (int i = 0; i < sz1; i++) res[i] *= c[i];
    }
    intt(res);
    res.resize(sz);
    return res;
  }
  // c[i]=sum[j]a[j]b[i+j]
  vector<mint> middle_product(const vector<mint>& a, const vector<mint>& b) {
    if (b.empty() || a.size() > b.size()) return {};
    int n = a.size(), m = b.size();
    int sz = m - n + 1;
    if (n <= 30 || sz <= 30) {
      vector<mint> res(sz);
      for (int i = 0; i < sz; i++)
        for (int j = 0; j < n; j++) res[i] += a[j] * b[i + j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < m) sz1 <<= 1;
    vector<mint> res(sz1), b2(sz1);
    reverse_copy(a.begin(), a.end(), res.begin());
    copy(b.begin(), b.end(), b2.begin());
    ntt(res);
    ntt(b2);
    for (int i = 0; i < res.size(); i++) res[i] *= b2[i];
    intt(res);
    res.resize(m);
    res.erase(res.begin(), res.begin() + n - 1);
    return res;
  }
  void ntt_doubling(vector<mint>& a) {
    int n = (int)a.size();
    auto b = a;
    intt(b);
    mint r = 1, zeta = mint(g).pow((mint::get_mod() - 1) / (n << 1));
    for (int i = 0; i < n; i++) b[i] *= r, r *= zeta;
    ntt(b);
    copy(b.begin(), b.end(), back_inserter(a));
  }
};
/**
 * @brief NTT (数論変換)
 * @docs docs/fft/ntt.md
 */
#line 2 "fps/formal-power-series.hpp"

template <class mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;
  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }
  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }
  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }
  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }
  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }
  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.at(g.size() - 1).inv();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }
  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }
  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }
  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }
  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }
  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz) return {};
    this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }
  FPS operator<<=(int sz) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }
  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }
  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }
  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }
  FPS log(int deg = -1) const {
    assert((*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }
  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      FPS ret(deg);
      if (deg) ret[0] = 1;
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_ntt();
  FPS &operator*=(const FPS &r);
  FPS middle_product(const FPS &r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 5 "fps/fps-ntt-friendly.hpp"

template <class mint>
void FormalPowerSeries<mint>::set_ntt() {
  if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}
template <class mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint>& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  set_ntt();
  auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::middle_product(const FormalPowerSeries<mint>& r) const {
  set_ntt();
  auto ret = static_cast<NTT<mint>*>(ntt_ptr)->middle_product(*this, r);
  return FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
void FormalPowerSeries<mint>::ntt() {
  set_ntt();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <class mint>
void FormalPowerSeries<mint>::intt() {
  set_ntt();
  static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <class mint>
void FormalPowerSeries<mint>::ntt_doubling() {
  set_ntt();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_root() {
  set_ntt();
  return static_cast<NTT<mint>*>(ntt_ptr)->g;
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1) / (*this)[0]};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
  return ret.pre(deg);
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  assert((*this)[0] == mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1)};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret * ((*this).pre(i << 1) - ret.log(i << 1) + 1)).pre(i << 1);
  return ret.pre(deg);
}
#line 7 "verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp"
using fps = FormalPowerSeries<mint>;
#line 1 "fps/linearly-recurrent-sequence.hpp"
#pragma
#line 3 "fps/berlekamp-massey.hpp"

template <class mint>
FormalPowerSeries<mint> BerlekampMassey(const FormalPowerSeries<mint>& a) {
  int n = a.size();
  FormalPowerSeries<mint> b, c;
  b.reserve(n + 1), c.reserve(n + 1);
  b.push_back(1), c.push_back(1);
  mint y = 1;
  for (int k = 0; k < n; k++) {
    mint x = 0;
    for (int i = 0; i < c.size(); i++) x += c[i] * a[k - i];
    b.insert(b.begin(), 0);
    if (x == 0) continue;
    mint v = x / y;
    if (b.size() > c.size()) {
      for (int i = 0; i < b.size(); i++) b[i] *= -v;
      for (int i = 0; i < c.size(); i++) b[i] += c[i];
      swap(b, c);
      y = x;
    } else {
      for (int i = 0; i < b.size(); i++) c[i] -= v * b[i];
    }
  }
  return c;
}

/**
 * @brief Berlekamp-Massey
 * @docs docs/fps/berlekamp-massey.md
 */
#line 3 "fps/bostan-mori.hpp"

// [x^n]f(x)/g(x)
template <class mint>
mint BostanMori(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, long long n) {
  g.shrink();
  mint ret = 0;
  if (f.size() >= g.size()) {
    auto q = f / g;
    if (n < q.size()) ret += q[n];
    f -= q * g;
    f.shrink();
  }
  if (f.empty()) return ret;
  FormalPowerSeries<mint>::set_ntt();
  if (!FormalPowerSeries<mint>::ntt_ptr) {
    f.resize(g.size() - 1);
    for (; n > 0; n >>= 1) {
      auto g1 = g;
      for (int i = 1; i < g1.size(); i += 2) g1[i] = -g1[i];
      auto p = f * g1, q = g * g1;
      if (n & 1) {
        for (int i = 0; i < f.size(); i++) f[i] = p[(i << 1) | 1];
      } else {
        for (int i = 0; i < f.size(); i++) f[i] = p[i << 1];
      }
      for (int i = 0; i < g.size(); i++) g[i] = q[i << 1];
    }
    return ret + f[0] / g[0];
  } else {
    int m = 1, log = 0;
    while (m < g.size()) m <<= 1, log++;
    mint wi = mint(FormalPowerSeries<mint>::ntt_root()).inv().pow((mint::get_mod() - 1) >> (log + 1));
    vector<int> rev(m);
    for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
    vector<mint> pow(m, 1);
    for (int i = 1; i < m; i++) pow[rev[i]] = pow[rev[i - 1]] * wi;
    f.resize(m), g.resize(m);
    f.ntt(), g.ntt();
    mint inv2 = mint(2).inv();
    for (; n >= m; n >>= 1) {
      f.ntt_doubling(), g.ntt_doubling();
      if (n & 1) {
        for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] - f[(i << 1) | 1] * g[i << 1]) * inv2 * pow[i];
      } else {
        for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] + f[(i << 1) | 1] * g[i << 1]) * inv2;
      }
      for (int i = 0; i < m; i++) g[i] = g[i << 1] * g[(i << 1) | 1];
      f.resize(m), g.resize(m);
    }
    f.intt(), g.intt();
    return ret + (f * g.inv())[n];
  }
}
/**
 * @brief Bostan-Mori
 * @docs docs/fps/bostan-mori.md
 */
#line 3 "fps/inverse-shift.hpp"

// [x^n,...,x^{n+k-1}]1/f(x)
template <class mint>
FormalPowerSeries<mint> InverseShift(FormalPowerSeries<mint> f, long long n, int k = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(f[0] != 0);
  if (k == -1) k = f.size();
  int m = 1;
  while (m < k) m <<= 1;

  int log = __builtin_ctz((unsigned int)m);
  mint w = mint(fps::ntt_root()).pow((mint::get_mod() - 1) >> (log + 1));
  mint wi = w.inv();
  vector<int> rev(m);
  for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
  mint inv2 = mint(2).inv();

  f.resize(m);
  f.ntt();
  auto rec = [&](auto& rec, long long n) -> void {
    if (n < m) {
      f.intt();
      f = f.inv(n + m);
      f >>= n;
      f.ntt();
      return;
    }
    f.ntt_doubling();
    assert(f.size() == 2 * m);
    auto f1 = f;
    for (int i = 0; i < m; i++) swap(f1[i << 1], f1[(i << 1) | 1]);
    for (int i = 0; i < m; i++) f[i] = f[i << 1] * f[(i << 1) | 1];
    f.resize(m);
    rec(rec, (n - m + 1) / 2);
    if (((n - m) & 1) == 0) {
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = f[i];
        f[i << 1] = f[i];
      }
    } else {
      mint p = 1;
      for (auto i : rev) f[i] *= p, p *= w;
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = -f[i];
        f[i << 1] = f[i];
      }
    }
    for (int i = 0; i < 2 * m; i++) f[i] *= f1[i];
    auto odd = fps(f.begin() + m, f.end());
    odd.intt();
    mint p = 1;
    for (int i = 0; i < m; i++) odd[i] *= p, p *= wi;
    odd.ntt();
    f.resize(m);
    for (int i = 0; i < m; i++) f[i] = (f[i] - odd[i]) * inv2;
  };
  rec(rec, n);
  f.intt();
  f.resize(k);
  return f;
}
/**
 * @brief Inverse の次数シフト
 * @docs docs/fps/inverse-shift.md
 */
#line 6 "fps/linearly-recurrent-sequence.hpp"

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n]
template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n) {
  assert(a.size() == c.size());
  if (n < a.size()) return a[n];
  while (!c.empty() && c.back() == 0) c.pop_back();
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    n -= z;
    a.erase(a.begin(), a.begin() + z);
  }
  int d = c.size();
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  return BostanMori(p, q, n);
}

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n],a[n+1],...,a[n+k-1]
template <class mint>
FormalPowerSeries<mint> LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n, int k) {
  assert(a.size() == c.size());
  if (n + k < a.size()) {
    a >>= (int)n;
    return a.pre(k);
  }
  while (!c.empty() && c.back() == 0) c.pop_back();
  int d = c.size();
  FormalPowerSeries<mint> ret{};
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    if (n < z) {
      ret.reserve(k);
      for (int i = n; i < z; i++) ret.push_back(a[i]);
    }
    n -= z;
    if (n < 0) {
      k += n;
      n = 0;
    }
    a >>= z;
  }
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  if (n < a.size()) {
    p *= q.inv(n + k);
    p >>= n;
  } else {
    p *= (InverseShift(q, n) * q).pre(d);
    p %= q;
    p *= q.inv(k);
  }
  p.resize(k);
  if (ret.empty()) {
    return p;
  } else {
    for (auto v : p) ret.push_back(v);
    return ret;
  }
}

template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, long long n) {
  if (n < a.size()) return a[n];
  auto b = BerlekampMassey(a);
  int d = b.size() - 1;
  a.resize(d);
  int z = 0;
  while (b.back() == 0) b.pop_back(), z++;
  a >>= z;
  n -= z;
  d -= z;
  return BostanMori((a * b).pre(b.size()), b, n);
}

/**
 * @brief 線形漸化式用
 * @docs docs/fps/linearly-recurrent-sequence.md
 */
#line 9 "verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp"

int main() {
  int d;
  ll k;
  in(d, k);
  fps a(d), c(d);
  in(a, c);
  out(LinearyRecurrentSequence(a, c, k));
}

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