#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)); } }; namespace FFT { using Real = double; struct Complex { Real re, im; Complex(Real re = 0, Real im = 0) : re(re), im(im) {} inline Complex operator+(const Complex &rhs) const { return Complex(re + rhs.re, im + rhs.im); } inline Complex operator-(const Complex &rhs) const { return Complex(re - rhs.re, im - rhs.im); } inline Complex operator*(const Complex &rhs) const { return Complex(re * rhs.re - im * rhs.im, re * rhs.im + im * rhs.re); } inline Complex mul_real(Real r) const { return Complex(re * r, im * r); } inline Complex mul_pin(Real r) const { return Complex(-im * r, re * r); } inline Complex conj() const { return Complex(re, -im); } }; vector butterfly{0}; vector > zeta{{{1, 0}}}; void calc(int n) { int prev_n = butterfly.size(); if (n <= prev_n) return; butterfly.resize(n); int prev_lg = zeta.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)); zeta.resize(lg); FOR(i, prev_lg, lg) { zeta[i].resize(1 << i); Real angle = -M_PI * 2 / (1 << (i + 1)); REP(j, 1 << (i - 1)) { zeta[i][j << 1] = zeta[i - 1][j]; Real theta = angle * ((j << 1) + 1); zeta[i][(j << 1) + 1] = Complex(cos(theta), sin(theta)); } } } 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) { Complex tmp = a[i + j + block] * zeta[den][j]; a[i + j + block] = a[i + j] - tmp; a[i + j] = a[i + j] + tmp; } } } template vector dft(const vector &a) { int sz = a.size(), lg = 1; while ((1 << lg) < sz) ++lg; vector c(1 << lg); REP(i, sz) c[i].re = a[i]; sub_dft(c); return c; } vector real_idft(vector &a) { int n = a.size(), half = n >> 1, quarter = half >> 1; // assert(__builtin_popcount(n) == 1); calc(n); a[0] = (a[0] + a[half] + (a[0] - a[half]).mul_pin(1)).mul_real(0.5); int den = __builtin_ctz(half); FOR(i, 1, quarter) { int j = half - i; Complex tmp1 = a[i] + a[j].conj(), tmp2 = ((a[i] - a[j].conj()) * zeta[den][j]).mul_pin(1); a[i] = (tmp1 - tmp2).mul_real(0.5); a[j] = (tmp1 + tmp2).mul_real(0.5).conj(); } if (quarter > 0) a[quarter] = a[quarter].conj(); a.resize(half); sub_dft(a); reverse(a.begin() + 1, a.end()); Real r = 1.0 / half; vector res(n); REP(i, n) res[i] = (i & 1 ? a[i >> 1].im : a[i >> 1].re) * r; return res; } void idft(vector &a) { int n = a.size(); sub_dft(a); reverse(a.begin() + 1, a.end()); Real r = 1.0 / n; REP(i, n) a[i] = a[i].mul_real(r); } 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 c(n); REP(i, a_sz) c[i].re = a[i]; REP(i, b_sz) c[i].im = b[i]; sub_dft(c); int half = n >> 1; c[0] = Complex(c[0].re * c[0].im, 0); FOR(i, 1, half) { Complex i_square = c[i] * c[i], j_square = c[n - i] * c[n - i]; c[i] = (j_square.conj() - i_square).mul_pin(0.25); c[n - i] = (i_square.conj() - j_square).mul_pin(0.25); } c[half] = Complex(c[half].re * c[half].im, 0); vector res = real_idft(c); res.resize(sz); return res; } }; vector convolution(const vector &a, const vector &b) { using Complex = FFT::Complex; const int pre = 15; 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) { int ai = a[i].val; A[i] = Complex(ai & ((1 << pre) - 1), ai >> pre); } REP(i, b_sz) { int bi = b[i].val; B[i] = Complex(bi & ((1 << pre) - 1), bi >> pre); } FFT::sub_dft(A); FFT::sub_dft(B); int half = n >> 1; Complex tmp_a = A[0], tmp_b = B[0]; A[0] = Complex(tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); B[0] = Complex(tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); FOR(i, 1, half) { int j = n - i; Complex a_l_i = (A[i] + A[j].conj()).mul_real(0.5), a_h_i = (A[j].conj() - A[i]).mul_pin(0.5); Complex b_l_i = (B[i] + B[j].conj()).mul_real(0.5), b_h_i = (B[j].conj() - B[i]).mul_pin(0.5); Complex a_l_j = (A[j] + A[i].conj()).mul_real(0.5), a_h_j = (A[i].conj() - A[j]).mul_pin(0.5); Complex b_l_j = (B[j] + B[i].conj()).mul_real(0.5), b_h_j = (B[i].conj() - B[j]).mul_pin(0.5); A[i] = a_l_i * b_l_i + (a_h_i * b_h_i).mul_pin(1); B[i] = a_l_i * b_h_i + a_h_i * b_l_i; A[j] = a_l_j * b_l_j + (a_h_j * b_h_j).mul_pin(1); B[j] = a_l_j * b_h_j + a_h_j * b_l_j; } tmp_a = A[half]; tmp_b = B[half]; A[half] = Complex(tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); B[half] = Complex(tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); FFT::idft(A); FFT::idft(B); vector res(sz); ModInt tmp1 = 1 << pre, tmp2 = 1LL << (pre << 1); REP(i, sz) { res[i] += static_cast(A[i].re + 0.5); res[i] += tmp1 * static_cast(B[i].re + 0.5); res[i] += tmp2 * static_cast(A[i].im + 0.5); } return res; } 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 memo(p - 1); REP(i, p - 1) memo[i] = ModInt(root).pow(i).val; mod = MOD; 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[memo[i]]; B[i] = b[memo[i]]; } vector C = 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] << (i + 1 == p ? '\n' : ' '); return 0; }