#include #include #include #include #include #include #include #include #include #define debug_value(x) cerr << "line" << __LINE__ << ":<" << __func__ << ">:" << #x << "=" << x << endl; #define debug(x) cerr << "line" << __LINE__ << ":<" << __func__ << ">:" << x << endl; template inline bool chmax(T& a, T b) { if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T& a, T b) { if (a > b) { a = b; return 1; } return 0; } using namespace std; typedef long long ll; const ll MOD = 998244353; class ModInt{ public: ll v; ModInt(ll _v = 0){ if(_v >= MOD) _v %= MOD; v = _v; } ModInt operator+(ll n){ return ModInt((v+n)%MOD); } ModInt operator-(ll n){ return ModInt((v-n+MOD)%MOD); } ModInt operator*(ll n){ if(n >= MOD) n %= MOD; return ModInt((v*n)%MOD); } ModInt operator/(ll n){ return ModInt((ModInt(n).inv()*v).v%MOD); } void operator+=(ll n){ v = (v+n)%MOD; } void operator-=(ll n){ v = (v-n+MOD)%MOD; } void operator*=(ll n){ v = (v*n+MOD)%MOD; } ModInt operator+(ModInt n){ return ModInt((v+n.v)%MOD); } ModInt operator-(ModInt n){ return ModInt((v-n.v+MOD)%MOD); } ModInt operator*(ModInt n){ return ModInt((v*n.v)%MOD); } ModInt operator/(ModInt n){ return ModInt((n.inv()*v).v%MOD); } void operator+=(ModInt n){ v = (v+n.v)%MOD; } void operator-=(ModInt n){ v = (v-n.v+MOD)%MOD; } void operator*=(ModInt n){ v = (v*n.v)%MOD; } void operator=(ModInt n){ v = n.v; } bool operator==(ModInt n){ return v == n.v; } bool operator!=(ModInt n){ return v != n.v; } void operator=(ll n){ v = n%MOD; } ModInt inv(){ if(v == 1) return ModInt(1); else return ModInt(MOD-ModInt(MOD%v).inv().v*(MOD/v)%MOD); } }; ostream& operator<<(ostream& os, const ModInt& m){ os << m.v; return os; } istream & operator >> (istream &in, ModInt &m){ in >> m.v; return in; } #define root 3 unsigned int add(const unsigned int x, const unsigned int y){ return (x + y < MOD) ? x + y : x + y - MOD; } unsigned int sub(const unsigned int x, const unsigned int y){ return (x >= y) ? (x - y) : (MOD - y + x); } unsigned int mul(const unsigned int x, const unsigned int y){ return (unsigned long long)x * y % MOD; } unsigned int mod_pow(unsigned int x, unsigned int n){ unsigned int res = 1; while(n > 0){ if(n & 1){ res = mul(res, x); } x = mul(x, x); n >>= 1; } return res; } unsigned int inverse(const unsigned int x){ return mod_pow(x, MOD - 2); } void ntt(vector& a, const bool rev = false){ unsigned int i, j, k, l, p, q, r, s; const unsigned int size = a.size(); if(size == 1) return; vector b(size); r = rev ? (MOD - 1 - (MOD - 1) / size) : (MOD - 1) / size; s = mod_pow(root, r); vector kp(size / 2 + 1, 1); for(i = 0; i < size / 2; ++i) kp[i + 1] = mul(kp[i], s); for(i = 1, l = size / 2; i < size; i <<= 1, l >>= 1){ for(j = 0, r = 0; j < l; ++j, r += i){ for(k = 0, s = kp[i * j]; k < i; ++k){ p = a[k + r], q = a[k + r + size / 2]; b[k + 2 * r] = add(p, q); b[k + 2 * r + i] = mul(sub(p, q), s); } } swap(a, b); } if(rev){ s = inverse(size); for(i = 0; i < size; i++){ a[i] = mul(a[i], s); } } } vector convolute(const vector& a, const vector& b){ const int size = (int)a.size() + (int)b.size() - 1; int t = 1; while(t < size){ t <<= 1; } vector A(t, 0), B(t, 0); for(int i = 0; i < (int)a.size(); i++){ A[i] = a[i]; } for(int i = 0; i < (int)b.size(); i++){ B[i] = b[i]; } ntt(A), ntt(B); for (int i = 0; i < t; i++){ A[i] = mul(A[i], B[i]); } ntt(A, true); A.resize(size); return A; } bool ok(int p, int x){ vector used(p, false); int cur = 1; for(int i = 1; i < p; i++){ cur = (cur*x)%p; if(used[cur]) return false; used[cur] = true; } } int primitive_root(int p){ for(int i = 1; i < p; i++){ if(ok(p, i)) return i; } throw "unchi"; } using mint = ModInt; int main(){ ios::sync_with_stdio(false); cin.tie(0); cout << setprecision(10) << fixed; int p; cin >> p; vector a(p), b(p); for(int i = 1; i < p; i++) cin >> a[i]; for(int i = 1; i < p; i++) cin >> b[i]; int r = primitive_root(p); vector log_r(p); int cur = 1; for(int i = 0; i < p-1; i++){ log_r[cur] = i; cur = (cur*r)%p; } vector u(p-1), v(p-1); for(int i = 1; i < p; i++){ u[log_r[i]] = a[i]; v[log_r[i]] = b[i]; } auto prod = convolute(u, v); vector c(p); for(int i = 1; i < p; i++){ c[i] += prod[log_r[i]]; c[i] += prod[log_r[i]+p-1]; c[i] %= MOD; } for(int i = 1; i < p; i++) cout << c[i] << ' '; cout << endl; }