#include using namespace std; using lint = long long int; using pint = pair; using plint = pair; 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##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector &vec, int len) { vec.resize(len); } template void ndarray(vector &vec, int len, Args... args) { vec.resize(len); for (auto &v : vec) ndarray(v, args...); } template bool mmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; } template bool mmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template istream &operator>>(istream &is, vector &vec){ for (auto &v : vec) is >> v; return is; } ///// This part below is only for debug, not used ///// template ostream &operator<<(ostream &os, const vector &vec){ os << "["; for (auto v : vec) os << v << ","; os << "]"; return os; } template ostream &operator<<(ostream &os, const deque &vec){ os << "deq["; for (auto v : vec) os << v << ","; os << "]"; return os; } template ostream &operator<<(ostream &os, const set &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; } template ostream &operator<<(ostream &os, const multiset &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; } template ostream &operator<<(ostream &os, const pair &pa){ os << "(" << pa.first << "," << pa.second << ")"; return os; } template ostream &operator<<(ostream &os, const map &mp){ os << "{"; for (auto v : mp) os << v.first << "=>" << v.second << ","; os << "}"; return os; } template ostream &operator<<(ostream &os, const unordered_map &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 ///// lint power(lint x, lint n, lint MOD) { lint ans = 1; while (n>0) { if (n & 1) (ans *= x) %= MOD; (x *= x) %= MOD; n >>= 1; } return ans %= MOD; } // Solve ax+by=gcd(a, b) lint extgcd(lint a, lint b, lint &x, lint &y) { lint d = a; if (b != 0) d = extgcd(b, a % b, y, x), y -= (a / b) * x; else x = 1, y = 0; return d; } // Calc a^(-1) (MOD m) lint mod_inverse(lint a, lint m) { lint x, y; extgcd(a, m, x, y); return (m + x % m) % m; } // mod: 素数, primitive_root: modの原始根 is_inverse: trueならば逆変換 void fft_mod(vector &a, lint mod, lint primitive_root, bool is_inverse=false) { int n = a.size(); lint h = power(primitive_root, (mod - 1) / n, mod); if (is_inverse) h = mod_inverse(h, mod); int i = 0; FOR(j, 1, n - 1) { for (int k = n >> 1; k > (i ^= k); k >>= 1); if (j < i) swap(a[i], a[j]); } for (int m = 1; m < n; m *= 2) { int m2 = 2 * m; lint base = power(h, n / m2, mod); lint w = 1; REP(x, m) { for (int s = x; s < n; s += m2) { lint u = a[s], d = a[s + m] * w % mod; a[s] = u + d - (u + d >= mod ? mod : 0), a[s + m] = u - d + (u - d < 0 ? mod : 0); } w = w * base % mod; } } for (auto &v : a) v = (v < 0 ? v + mod : v); if (is_inverse) { lint n_inv = mod_inverse(n, mod); for (auto &v : a) v = v * n_inv % mod; } } // MOD modにおける畳み込み演算 retval[i] = \sum_j a[j] b[i - j] vector convolution_mod(vector a, vector b, lint mod, lint primitive_root) { int sz = 1; while (sz < a.size() + b.size()) sz <<= 1; a.resize(sz), b.resize(sz); fft_mod(a, mod, primitive_root, false), fft_mod(b, mod, primitive_root, false); REP(i, sz) a[i] = a[i] * b[i] % mod; fft_mod(a, mod, primitive_root, true); return a; } constexpr lint MOD = 998244353; int find_primitive_root(lint p) { vector fac; lint pp = 2; lint v = p - 1; while (v >= pp * pp) // prime factorization of p - 1 { int e = 0; while (v % pp == 0) { e++; v /= pp; } if (e) fac.push_back(pp); pp++; } if (v > 1) fac.push_back(v); int g = 2; while (g < p) { if (power(g, p - 1, p) != 1) return -1; bool ok = true; for (auto pp : fac) { if (power(g, (p - 1) / pp, p) == 1) { ok = false; break; } } if (ok) return g; g++; } return -1; } int main() { int P; cin >> P; vector A(P - 1), B(P - 1); cin >> A >> B; if (P == 2) { cout << A[0] * B[0] % MOD << endl; return 0; } lint b = find_primitive_root(P); vector pp(P, 1), ppinv(P); FOR(i, 1, P) pp[i] = pp[i - 1] * b % P; REP(i, P) ppinv[pp[i]] = i; vector AS(P), BS(P); REP(i, P - 1) AS[ppinv[i + 1]] = A[i]; REP(i, P - 1) BS[ppinv[i + 1]] = B[i]; vector v = convolution_mod(AS, BS, MOD, 3); vector ret(P + 1); FOR(i, 1, v.size()) { (ret[power(b, i, P)] += v[i]) %= MOD; } FOR(i, 1, P) printf("%lld ", ret[i]); }