#include // created [2019/11/22] 22:11:34 #pragma GCC diagnostic ignored "-Wsign-compare" #pragma GCC diagnostic ignored "-Wsign-conversion" using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; using uint = unsigned int; using usize = std::size_t; using ll = long long; using ull = unsigned long long; using ld = long double; template constexpr T popcount(const T u) { return u ? static_cast(__builtin_popcountll(static_cast(u))) : static_cast(0); } template constexpr T log2p1(const T u) { return u ? static_cast(64 - __builtin_clzll(static_cast(u))) : static_cast(0); } template constexpr T msbp1(const T u) { return log2p1(u); } template constexpr T lsbp1(const T u) { return __builtin_ffsll(u); } template constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast(u); } template constexpr bool ispow2(const T u) { return u and (static_cast(u) & static_cast(u - 1)) == 0; } template constexpr T ceil2(const T u) { return static_cast(1) << clog(u); } template constexpr T floor2(const T u) { return u == 0 ? static_cast(0) : static_cast(1) << (log2p1(u) - 1); } template constexpr bool btest(const T mask, const usize ind) { return static_cast((static_cast(mask) >> ind) & static_cast(1)); } template void bset(T& mask, const usize ind) { mask |= (static_cast(1) << ind); } template void breset(T& mask, const usize ind) { mask &= ~(static_cast(1) << ind); } template void bflip(T& mask, const usize ind) { mask ^= (static_cast(1) << ind); } template void bset(T& mask, const usize ind, const bool b) { (b ? bset(mask, ind) : breset(mask, ind)); } template constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast(0) : static_cast((static_cast(mask) << (64 - ind)) >> (64 - ind)); } template bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); } template bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); } constexpr unsigned int mod = 1000000007; template constexpr T inf_v = std::numeric_limits::max() / 4; template constexpr Real pi_v = Real{3.141592653589793238462643383279502884}; template T read() { T v; return std::cin >> v, v; } template auto read(const usize size, Args... args) { std::vector(args...))> ans(size); for (usize i = 0; i < size; i++) { ans[i] = read(args...); } return ans; } template auto reads() { return std::tuple...>{read()...}; } # define SHOW(...) static_cast(0) template std::vector make_v(const usize size, const T v) { return std::vector(size, v); } template auto make_v(const usize size, Args... args) { return std::vector(size, make_v(args...)); } template T gcd(const T& a, const T& b) { return a < 0 ? gcd(-a, b) : b < 0 ? gcd(a, -b) : (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); } template T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; } template constexpr std::pair extgcd(const T a, const T b) { if (b == 0) { return std::pair{1, 0}; } const auto g = gcd(a, b), da = std::abs(b) / g; const auto p = extgcd(b, a % b); const auto x = (da + p.second % da) % da, y = (g - a * x) / b; return {x, y}; } template constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; } template class modint_base { public: template static std::enable_if_t mod() { return mod_ref(); } template static constexpr std::enable_if_t mod() { return mod_value; } template static void set_mod(const std::enable_if_t mod) { mod_ref() = mod, inv_ref() = {1, 1}; } modint_base() : v{0} {} modint_base(const ll val) : v{norm(static_cast(val % static_cast(mod()) + static_cast(mod())))} {} modint_base(const modint_base& n) : v{n()} {} explicit operator bool() const { return v != 0; } bool operator!() const { return not static_cast(*this); } modint_base& operator=(const modint_base& m) { return v = m(), (*this); } modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast(mod()) + static_cast(mod()))), (*this); } friend modint_base operator+(const modint_base& m) { return m; } friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); } friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); } friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); } friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast(static_cast(m1.v) * static_cast(m2.v) % static_cast(mod()))); } friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); } friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) + val}; } friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) - val}; } friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) * (val % static_cast(mod()))}; } friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) * inv(val)}; } friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast(m.v) + val}; } friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast(m.v) + val}; } friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast(m.v) * (val % static_cast(mod()))}; } friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast(m.v))}; } friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; } friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; } friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; } friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; } friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; } friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; } friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; } friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; } friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); } friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; } friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; } friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); } friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast(static_cast(mod()) + val % static_cast(mod()))); } friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); } friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast(static_cast(mod()) + val % static_cast(mod()))); } friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); } friend std::istream& operator>>(std::istream& is, modint_base& m) { ll v; return is >> v, m = v, is; } friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); } uint operator()() const { return v; } static modint_base small_inv(const usize n) { auto& in = inv_ref(); if (n < in.size()) { return in[n]; } for (usize i = in.size(); i <= n; i++) { in.push_back(-in[modint_base::mod() % i] * (modint_base::mod() / i)); } return in.back(); } private: template static std::enable_if_t mod_ref() { static UInt mod = 0; return mod; } static uint norm(const uint x) { return x < mod() ? x : x - mod(); } static modint_base make(const uint x) { modint_base m; return m.v = x, m; } static modint_base power(modint_base x, ull n) { modint_base ans = 1; for (; n; n >>= 1, x *= x) { if (n & 1) { ans *= x; } } return ans; } static modint_base inv(const ll v) { return v < 1000000 ? small_inv(static_cast(v)) : modint_base{inverse(v, static_cast(mod()))}; } static std::vector& inv_ref() { static std::vector in{1, 1}; return in; } uint v; }; template using modint = modint_base; template using dynamic_modint = modint_base; template std::vector> prime_factors(T n) { std::vector> ans; for (T i = 2; i * i <= n; i++) { usize cnt = 0; for (; n % i == 0; n /= i, cnt++) {} if (cnt > 0) { ans.push_back({i, cnt}); } } if (n > 1) { ans.push_back({n, 1}); } return ans; } template class ntt { private: using value_type = modint; static constexpr usize depth = 30; static void transform(std::vector& a, const usize lg, const bool rev) { const usize N = a.size(); assert(1UL << lg == N); static std::vector R[depth]; if (R[lg].empty()) { R[lg].reserve(N), R[lg].resize(N, value_type(1)); const value_type r = value_type(root) ^ ((mod - 1) / N); for (usize i = 1; i < N; i++) { R[lg][i] = R[lg][i - 1] * r; } } std::vector tmp(N); for (usize w = (N >> 1); w > 0; w >>= 1) { for (usize y = 0; y < (N >> 1); y += w) { const value_type r = rev ? R[lg][y == 0 ? 0 : N - y] : R[lg][y]; for (usize x = 0; x < w; x++) { const auto u = a[y << 1 | x], v = a[y << 1 | x | w]() * r; tmp[y | x] = u + v, tmp[y | x | (N >> 1)] = u - v; } } std::swap(tmp, a); } if (rev) { for (usize i = 0; i < N; i++) { a[i] /= value_type(N); } } } public: ntt() = delete; static std::vector convolute(const std::vector& a, const std::vector& b) { const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg; std::vector A(sz, 0), B(sz, 0); for (usize i = 0; i < a.size(); i++) { A[i] = a[i](); } for (usize i = 0; i < b.size(); i++) { B[i] = b[i](); } transform(A, lg, false), transform(B, lg, false); for (usize i = 0; i < sz; i++) { A[i] *= B[i]; } transform(A, lg, true); std::vector ans(need); for (usize i = 0; i < need; i++) { ans[i] = int(A[i]()); } return ans; } }; int main() { using mint = dynamic_modint<0>; const auto P = read(); mint::set_mod(P); std::vector A(P, 0), B(P, 0); for (int i = 1; i < P; i++) { std::cin >> A[i]; } for (int i = 1; i < P; i++) { std::cin >> B[i]; } uint g = 2; const auto fs = prime_factors(P - 1); for (; g < P; g++) { bool ok = true; for (const auto& p : fs) { const uint q = p.first; const uint per = (P - 1) / q; if ((mint(g) ^ per) == 1) { ok = false; break; } } if (ok) { break; } } SHOW(g); std::vector ps(P - 1, 1); for (int i = 1; i < P - 1; i++) { ps[i] = ps[i - 1] * g; } constexpr uint mod = 998244353; std::vector> a(P), b(P); for (int i = 0; i < P - 1; i++) { a[i] = A[ps[i]()]; } for (int i = 0; i < P - 1; i++) { b[i] = B[ps[i]()]; } SHOW(a, b); auto c = ntt::convolute(a, b); SHOW(c); std::vector> d(P - 1, 0); for (int i = 0; i < c.size(); i++) { d[i % (P - 1)] += c[i]; } SHOW(d); std::vector> C(P, 0); for (int i = 0; i < P - 1; i++) { C[ps[i]()] = d[i]; } for (int i = 1; i < P; i++) { std::cout << C[i] << " "; } std::cout << std::endl; return 0; }