#define _USE_MATH_DEFINES #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() using ll = long long; template using posteriority_queue = priority_queue, greater >; const int INF = 0x3f3f3f3f; const ll 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}; template inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T &a, U b) { return a > b ? (a = b, true) : false; } template void unique(vector &a) { a.erase(unique(ALL(a)), a.end()); } struct IOSetup { IOSetup() { cin.tie(nullptr); ios_base::sync_with_stdio(false); cout << fixed << setprecision(20); } } iosetup; template vector divisor(T val) { vector res; for (T i = 1; i * i <= val; ++i) { if (val % i == 0) { res.emplace_back(i); if (i * i != val) res.emplace_back(val / i); } } sort(ALL(res)); return res; } ll mod_pow(ll base, ll exponent, int md = MOD) { base %= md; ll res = 1; while (exponent > 0) { if (exponent & 1) (res *= base) %= md; (base *= base) %= md; exponent >>= 1; } return res; } bool is_primitive_root(int primitive_root, int md) { vector d = divisor(md - 1); d.pop_back(); for (int e : d) { if (mod_pow(primitive_root, e, md) == 1) return false; } return true; } int mod = MOD; struct ModInt { unsigned val; ModInt(): val(0) {} ModInt(ll x) : val(x >= 0 ? x % mod : x % mod + mod) {} ModInt pow(ll exponent) { ModInt tmp = *this, res = 1; while (exponent > 0) { if (exponent & 1) res *= tmp; tmp *= tmp; exponent >>= 1; } return res; } ModInt &operator+=(const ModInt &x) { if((val += x.val) >= mod) val -= mod; return *this; } ModInt &operator-=(const ModInt &x) { if((val += mod - x.val) >= mod) val -= mod; return *this; } ModInt &operator*=(const ModInt &x) { val = static_cast(val) * x.val % mod; return *this; } ModInt &operator/=(const ModInt &x) { return *this *= x.inv(); } bool operator==(const ModInt &x) const { return val == x.val; } bool operator!=(const ModInt &x) const { return val != x.val; } bool operator<(const ModInt &x) const { return val < x.val; } bool operator<=(const ModInt &x) const { return val <= x.val; } bool operator>(const ModInt &x) const { return val > x.val; } bool operator>=(const ModInt &x) const { return val >= x.val; } ModInt &operator++() { if (++val == mod) val = 0; return *this; } ModInt operator++(int) { ModInt res = *this; ++*this; return res; } ModInt &operator--() { val = (val == 0 ? mod : val) - 1; return *this; } ModInt operator--(int) { ModInt res = *this; --*this; return res; } ModInt operator+() const { return *this; } ModInt operator-() const { return ModInt(val ? mod - val : 0); } ModInt operator+(const ModInt &x) const { return ModInt(*this) += x; } ModInt operator-(const ModInt &x) const { return ModInt(*this) -= x; } ModInt operator*(const ModInt &x) const { return ModInt(*this) *= x; } ModInt operator/(const ModInt &x) const { return ModInt(*this) /= x; } friend ostream &operator<<(ostream &os, const ModInt &x) { return os << x.val; } friend istream &operator>>(istream &is, ModInt &x) { ll val; is >> val; x = ModInt(val); return is; } private: ModInt inv() const { // assert(__gcd(val, mod) == 1); 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); } }; ModInt abs(const ModInt &x) { return x; } struct Combinatorics { int val; // "val!" and "mod" must be disjoint. 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_) { for (int i = 0; ; ++i) { assert(i < 23); if (primes[i][0] == mod_) { mod = 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 n = a.size(), lg = 1; while ((1 << lg) < n) ++lg; vector A(1 << lg, 0); REP(i, n) 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; vector memo(p - 1); for (int root = 2; ; ++root) { if (is_primitive_root(root, p)) { REP(i, p - 1) memo[i] = ModInt(root).pow(i).val; break; } } vector a(p, 0), b(p, 0); FOR(i, 1, p) cin >> a[i]; FOR(i, 1, p) cin >> b[i]; NTT ntt(998244353); vector A(p - 1, 0), B(p - 1, 0); REP(i, p - 1) { A[i] = a[memo[i]]; B[i] = b[memo[i]]; } vector C = ntt.convolution(A, B); FOR(i, p - 1, C.size()) C[i % (p - 1)] += C[i]; vector ans(p, 0); REP(i, p - 1) ans[memo[i]] = C[i]; FOR(i, 1, p) cout << ans[i] << " \n"[i + 1 == p]; return 0; }