#include #include #include #include #include #define _USE_MATH_DEFINES #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) #define ALL(v) (v).begin(),(v).end() const int INF = 0x3f3f3f3f; const long long LINF = 0x3f3f3f3f3f3f3f3fLL; const double EPS = 1e-8; // const int MOD = 1000000007; const int MOD = 998244353; const int dy[] = {1, 0, -1, 0}, dx[] = {0, -1, 0, 1}; // const int dy[] = {1, 1, 0, -1, -1, -1, 0, 1}, // dx[] = {0, -1, -1, -1, 0, 1, 1, 1}; struct IOSetup { IOSetup() { cin.tie(nullptr); ios_base::sync_with_stdio(false); cout << fixed << setprecision(20); cerr << fixed << setprecision(10); } } iosetup; /*-------------------------------------------------*/ int mod = MOD; struct ModInt { unsigned val; ModInt(): val(0) {} ModInt(long long x) : val(x >= 0 ? x % mod : x % mod + mod) {} ModInt pow(long long exponent) { ModInt tmp = *this, res = 1; while (exponent > 0) { if (exponent & 1) res *= tmp; tmp *= tmp; exponent >>= 1; } return res; } ModInt &operator+=(const ModInt &rhs) { if((val += rhs.val) >= mod) val -= mod; return *this; } ModInt &operator-=(const ModInt &rhs) { if((val += mod - rhs.val) >= mod) val -= mod; return *this; } ModInt &operator*=(const ModInt &rhs) { val = static_cast(val) * rhs.val % mod; return *this; } ModInt &operator/=(const ModInt &rhs) { return *this *= rhs.inv(); } bool operator==(const ModInt &rhs) const { return val == rhs.val; } bool operator!=(const ModInt &rhs) const { return val != rhs.val; } bool operator<(const ModInt &rhs) const { return val < rhs.val; } bool operator<=(const ModInt &rhs) const { return val <= rhs.val; } bool operator>(const ModInt &rhs) const { return val > rhs.val; } bool operator>=(const ModInt &rhs) const { return val >= rhs.val; } ModInt operator-() const { return ModInt(val ? mod - val : 0); } ModInt operator+(const ModInt &rhs) const { return ModInt(*this) += rhs; } ModInt operator-(const ModInt &rhs) const { return ModInt(*this) -= rhs; } ModInt operator*(const ModInt &rhs) const { return ModInt(*this) *= rhs; } ModInt operator/(const ModInt &rhs) const { return ModInt(*this) /= rhs; } friend ostream &operator<<(ostream &os, const ModInt &rhs) { return os << rhs.val; } friend istream &operator>>(istream &is, ModInt &rhs) { long long x; is >> x; rhs = ModInt(x); return is; } private: ModInt inv() const { // if (__gcd(val, mod) != 1) assert(false); unsigned a = val, b = mod; int x = 1, y = 0; while (b) { unsigned tmp = a / b; swap(a -= tmp * b, b); swap(x -= tmp * y, y); } return ModInt(x); } }; int abs(const ModInt &x) { return x.val; } struct Combinatorics { int val; vector fact, fact_inv, inv; Combinatorics(int val = 10000000) : val(val), fact(val + 1), fact_inv(val + 1), inv(val + 1) { fact[0] = 1; FOR(i, 1, val + 1) fact[i] = fact[i - 1] * i; fact_inv[val] = ModInt(1) / fact[val]; for (int i = val; i > 0; --i) fact_inv[i - 1] = fact_inv[i] * i; FOR(i, 1, val + 1) inv[i] = fact[i - 1] * fact_inv[i]; } ModInt nCk(int n, int k) { if (n < 0 || n < k || k < 0) return ModInt(0); // assert(n <= val && k <= val); return fact[n] * fact_inv[k] * fact_inv[n - k]; } ModInt nPk(int n, int k) { if (n < 0 || n < k || k < 0) return ModInt(0); // assert(n <= val); return fact[n] * fact_inv[n - k]; } ModInt nHk(int n, int k) { if (n < 0 || k < 0) return ModInt(0); return (k == 0 ? ModInt(1) : nCk(n + k - 1, k)); } }; struct NTT { NTT(int mod_) { mod = mod_; REP(i, 23 + 1) { // if (i == 23) assert(false); if (primes[i][0] == mod) { n_max = 1 << primes[i][2]; root = ModInt(primes[i][1]).pow((mod - 1) >> primes[i][2]); break; } } } void sub_dft(vector &a) { int n = a.size(); // assert(__builtin_popcount(n) == 1); calc(n); int shift = __builtin_ctz(butterfly.size()) - __builtin_ctz(n); REP(i, n) { int j = butterfly[i] >> shift; if (i < j) swap(a[i], a[j]); } for (int block = 1; block < n; block <<= 1) { int den = __builtin_ctz(block); for (int i = 0; i < n; i += (block << 1)) REP(j, block) { ModInt tmp = a[i + j + block] * omega[den][j]; a[i + j + block] = a[i + j] - tmp; a[i + j] += tmp; } } } template vector dft(const vector &a) { int sz = a.size(), lg = 1; while ((1 << lg) < sz) ++lg; vector A(1 << lg, 0); REP(i, sz) A[i] = a[i]; sub_dft(A); return A; } void idft(vector &a) { int n = a.size(); // assert(__builtin_popcount(n) == 1); sub_dft(a); reverse(a.begin() + 1, a.end()); ModInt inv_n = ModInt(1) / n; REP(i, n) a[i] *= inv_n; } template vector convolution(const vector &a, const vector &b) { int a_sz = a.size(), b_sz = b.size(), sz = a_sz + b_sz - 1, lg = 1; while ((1 << lg) < sz) ++lg; int n = 1 << lg; vector A(n, 0), B(n, 0); REP(i, a_sz) A[i] = a[i]; REP(i, b_sz) B[i] = b[i]; sub_dft(A); sub_dft(B); REP(i, n) A[i] *= B[i]; idft(A); A.resize(sz); return A; } private: const int primes[23][3] = { {16957441, 329, 14}, {17006593, 26, 15}, {19529729, 770, 17}, {167772161, 3, 25}, {469762049, 3, 26}, {645922817, 3, 23}, {897581057, 3, 23}, {924844033, 5, 21}, {935329793, 3, 22}, {943718401, 7, 22}, {950009857, 7, 21}, {962592769, 7, 21}, {975175681, 17, 21}, {976224257, 3, 20}, {985661441, 3, 22}, {998244353, 3, 23}, {1004535809, 3, 21}, {1007681537, 3, 20}, {1012924417, 5, 21}, {1045430273, 3, 20}, {1051721729, 6, 20}, {1053818881, 7, 20}, {1224736769, 3, 24} }; int n_max; ModInt root; vector butterfly{0}; vector > omega{{1}}; void calc(int n) { int prev_n = butterfly.size(); if (n <= prev_n) return; // assert(n <= n_max); butterfly.resize(n); int prev_lg = omega.size(), lg = __builtin_ctz(n); FOR(i, 1, prev_n) butterfly[i] <<= lg - prev_lg; FOR(i, prev_n, n) butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1)); omega.resize(lg); FOR(i, prev_lg, lg) { omega[i].resize(1 << i); ModInt tmp = root.pow((mod - 1) / (1 << (i + 1))); REP(j, 1 << (i - 1)) { omega[i][j << 1] = omega[i - 1][j]; omega[i][(j << 1) + 1] = omega[i - 1][j] * tmp; } } } }; int main() { int p; cin >> p; mod = p; int root = 2; while (true) { set st; FOR(i, 1, p) st.emplace(ModInt(root).pow(i)); if (st.size() == p - 1) break; ++root; } vector a(p, 0), b(p, 0); FOR(i, 1, p) cin >> a[i]; FOR(i, 1, p) cin >> b[i]; vector A(p - 1, 0), B(p - 1, 0); REP(i, p - 1) { A[i] = a[ModInt(root).pow(i).val]; B[i] = b[ModInt(root).pow(i).val]; } NTT ntt(MOD); vector C = ntt.convolution(A, B); FOR(i, p - 1, C.size()) C[i % (p - 1)] += C[i]; mod = p; vector ans(p, 0); REP(i, p - 1) ans[ModInt(root).pow(i).val] = C[i]; FOR(i, 1, p) cout << ans[i] << (i + 1 == p ? '\n' : ' '); return 0; }