#include #define rep(i,a,b) for(int i=a;i=b;i--) #define fore(i,a) for(auto &i:a) #define all(x) (x).begin(),(x).end() //#pragma GCC optimize ("-O3") using namespace std; void _main(); int main() { cin.tie(0); ios::sync_with_stdio(false); _main(); } typedef long long ll; const int inf = INT_MAX / 2; const ll infl = 1LL << 60; templatebool chmax(T& a, const T& b) { if (a < b) { a = b; return 1; } return 0; } templatebool chmin(T& a, const T& b) { if (b < a) { a = b; return 1; } return 0; } //--------------------------------------------------------------------------------------------------- #define FOR(i,n) for(int i = 0; i < (n); i++) #define sz(c) ((int)(c).size()) #define ten(x) ((int)1e##x) template T extgcd(T a, T b, T & x, T & y) { for (T u = y = 1, v = x = 0; a;) { T q = b / a; swap(x -= q * u, u); swap(y -= q * v, v); swap(b -= q * a, a); } return b; } template T mod_inv(T a, T m) { T x, y; extgcd(a, m, x, y); return (m + x % m) % m; } ll mod_pow(ll a, ll n, ll mod) { ll ret = 1; ll p = a % mod; while (n) { if (n & 1) ret = ret * p % mod; p = p * p % mod; n >>= 1; } return ret; } struct MathsNTTModAny { template class NTT { public: int get_mod() const { return mod; } void _ntt(vector& a, int sign) { const int n = sz(a); assert((n ^ (n & -n)) == 0); //n = 2^k const int g = 3; //g is primitive root of mod int h = (int)mod_pow(g, (mod - 1) / n, mod); // h^n = 1 if (sign == -1) h = (int)mod_inv(h, mod); //h = h^-1 % mod //bit reverse int i = 0; for (int j = 1; j < n - 1; ++j) { 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) { const int m2 = 2 * m; const ll base = mod_pow(h, n / m2, mod); ll w = 1; FOR(x, m) { for (int s = x; s < n; s += m2) { ll u = a[s]; ll d = a[s + m] * w % mod; a[s] = u + d; if (a[s] >= mod) a[s] -= mod; a[s + m] = u - d; if (a[s + m] < 0) a[s + m] += mod; } w = w * base % mod; } } for (auto& x : a) if (x < 0) x += mod; } void ntt(vector & input) { _ntt(input, 1); } void intt(vector & input) { _ntt(input, -1); const int n_inv = mod_inv(sz(input), mod); for (auto& x : input) x = x * n_inv % mod; } vector convolution(const vector & a, const vector & b) { int ntt_size = 1; while (ntt_size < sz(a) + sz(b)) ntt_size *= 2; vector _a = a, _b = b; _a.resize(ntt_size); _b.resize(ntt_size); ntt(_a); ntt(_b); FOR(i, ntt_size) { (_a[i] *= _b[i]) %= mod; } intt(_a); return _a; } }; ll garner(vector> mr, int mod) { mr.emplace_back(mod, 0); vector coffs(sz(mr), 1); vector constants(sz(mr), 0); FOR(i, sz(mr) - 1) { // coffs[i] * v + constants[i] == mr[i].second (mod mr[i].first) を解く ll v = (mr[i].second - constants[i]) * mod_inv(coffs[i], mr[i].first) % mr[i].first; if (v < 0) v += mr[i].first; for (int j = i + 1; j < sz(mr); j++) { (constants[j] += coffs[j] * v) %= mr[j].first; (coffs[j] *= mr[i].first) %= mr[j].first; } } return constants[sz(mr) - 1]; } typedef NTT<167772161, 3> NTT_1; typedef NTT<469762049, 3> NTT_2; typedef NTT<1224736769, 3> NTT_3; vector solve(vector a, vector b, int mod = 1000000007) { for (auto& x : a) x %= mod; for (auto& x : b) x %= mod; NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3; assert(ntt1.get_mod() < ntt2.get_mod() && ntt2.get_mod() < ntt3.get_mod()); auto x = ntt1.convolution(a, b); auto y = ntt2.convolution(a, b); auto z = ntt3.convolution(a, b); const ll m1 = ntt1.get_mod(), m2 = ntt2.get_mod(), m3 = ntt3.get_mod(); const ll m1_inv_m2 = mod_inv(m1, m2); const ll m12_inv_m3 = mod_inv(m1 * m2, m3); const ll m12_mod = m1 * m2 % mod; vector ret(sz(x)); FOR(i, sz(x)) { ll v1 = (y[i] - x[i]) * m1_inv_m2 % m2; if (v1 < 0) v1 += m2; ll v2 = (z[i] - (x[i] + m1 * v1) % m3) * m12_inv_m3 % m3; if (v2 < 0) v2 += m3; ll constants3 = (x[i] + m1 * v1 + m12_mod * v2) % mod; if (constants3 < 0) constants3 += mod; ret[i] = constants3; } return ret; } vector solve(vector a, vector b, int mod = 1000000007) { vector x(all(a)); vector y(all(b)); auto z = solve(x, y, mod); vector res; fore(aa, z) res.push_back(aa % mod); return res; } }; // return primitive root of P ll get_root(ll p) { for (int i = 2; i < p; i++) { set S; int x = 1, j; while (1) { if (S.size() == p - 1) return i; if (S.count(x)) break; S.insert(x); x = x * i % p; } } assert(0); } // C[k] = sum_{1<=i,j

convolutionMulModP(vector A, vector B, int P, int mod) { if (P == 2) return { 0, (int)((1LL * A[1] * B[1]) % mod) }; ll root = get_root(P); vector a(1 << 17), b(1 << 17); ll v = 1; rep(i, 0, P - 1) { a[i] = A[v]; b[i] = B[v]; v = (v * root) % P; } MathsNTTModAny ntt; auto c = ntt.solve(a, b, mod); vector res(P); v = 1; rep(i, 0, c.size()) { res[v] = (res[v] + c[i]) % mod; v = (v * root) % P; } return res; } /*---------------------------------------------------------------------------------------------------             ∧_∧       ∧_∧  (´<_` )  Welcome to My Coding Space!      ( ´_ゝ`) /  ⌒i @hamayanhamayan     /   \    | |     /   / ̄ ̄ ̄ ̄/  |   __(__ニつ/  _/ .| .|____      \/____/ (u ⊃ ---------------------------------------------------------------------------------------------------*/ int P; //--------------------------------------------------------------------------------------------------- void _main() { cin >> P; vector A = { 0 }; rep(i, 0, P - 1) { int a; cin >> a; A.push_back(a); } vector B = { 0 }; rep(i, 0, P - 1) { int a; cin >> a; B.push_back(a); } auto ans = convolutionMulModP(A, B, P, 998244353); rep(i, 1, P) { if (i != 1) printf(" "); printf("%d", ans[i]); } printf("\n"); }