// includes {{{ #include #include #include #include #include #include #include #include #include #include #include #include #include #include // #include // #include // #include // #include // }}} using namespace std; using ll = long long; // #undef DEBUG // #define DEBUG // DEBUG {{{ #include #include #include #include #include #include #include #include #include template < int n, class... T > typename std::enable_if< (n >= sizeof...(T)) >::type __output_tuple( std::ostream &, std::tuple< T... > const &) {} template < int n, class... T > typename std::enable_if< (n < sizeof...(T)) >::type __output_tuple( std::ostream &os, std::tuple< T... > const &t) { os << (n == 0 ? "" : ", ") << std::get< n >(t); __output_tuple< n + 1 >(os, t); } template < class... T > std::ostream &operator<<(std::ostream &os, std::tuple< T... > const &t) { os << "("; __output_tuple< 0 >(os, t); os << ")"; return os; } template < class T, class U > std::ostream &operator<<(std::ostream &os, std::pair< T, U > const &p) { os << "(" << p.first << ", " << p.second << ")"; return os; } template < class T > std::ostream &operator<<(std::ostream &os, const std::stack< T > &a) { os << "{"; for(auto tmp = a; tmp.size(); tmp.pop()) os << (a.size() == tmp.size() ? "" : ", ") << tmp.top(); os << "}"; return os; } template < class T, class Container, class Compare > std::ostream &operator<<(std::ostream &os, std::priority_queue< T, Container, Compare > a) { os << "{ (top) "; while(a.size()) os << a.top() << (a.size() == 1 ? "" : ", "), a.pop(); os << " }"; return os; } template < class T, class Container > std::ostream &operator<<(std::ostream &os, std::queue< T, Container > a) { os << "{ "; while(a.size()) os << a.front() << (a.size() == 1 ? "" : ", "), a.pop(); os << " }"; return os; } #ifdef DEBUG #if !defined(DEBUG_OUT) #define DEBUG_OUT std::cerr #endif #define dump(...) \ [&]() { \ auto __debug_tap = std::make_tuple(__VA_ARGS__); \ DEBUG_OUT << "[" << __LINE__ << "] " << #__VA_ARGS__ << " = " << __debug_tap \ << std::endl; \ }() template < class T > inline void dump2D(T &d, size_t sizey, size_t sizex) { for(size_t i = 0; i < sizey; i++) { DEBUG_OUT << "\t"; for(size_t j = 0; j < sizex; j++) DEBUG_OUT << d[i][j] << (j + 1 == sizex ? "" : "\t"); DEBUG_OUT << std::endl; } } template < class T > inline void dump1D(T &d, size_t sizey) { for(size_t i = 0; i < sizey; i++) { DEBUG_OUT << d[i] << (i + 1 == sizey ? "" : " "); } DEBUG_OUT << std::endl; } template < class T, class = typename std::iterator_traits< decltype(begin(T())) >::value_type, class = typename std::enable_if< !std::is_same< T, std::string >::value >::type > std::ostream &operator<<(std::ostream &os, const T &a) { os << "{"; for(auto ite = begin(a); ite != end(a); ++ite) os << (ite == begin(a) ? "" : ", ") << *ite; os << "}"; return os; } #else #define dump(...) ((void) 42) #define dump2D(...) ((void) 42) #define dump1D(...) ((void) 42) template < class T, class = typename std::iterator_traits< decltype(begin(T())) >::value_type, class = typename std::enable_if< !std::is_same< T, std::string >::value >::type > std::ostream &operator<<(std::ostream &os, const T &a) { for(auto ite = begin(a); ite != end(a); ++ite) os << (ite == begin(a) ? "" : " ") << *ite; return os; } #endif // }}} // MAX_N is max size of OUTPUT, DOUBLED INPUT // MAX_RES_VALUE = MAX_VALUE^2 * MAX_N // if MAX_N > 2^20, comment out primes! // NTT {{{ #include #include #include namespace NTT { using uint = uint_fast32_t; // NTT_PRIMES {{{ constexpr ll NTT_PRIMES[][2] = { {1224736769, 3}, // 2^24 * 73 + 1, {1053818881, 7}, // 2^20 * 3 * 5 * 67 + 1 {1051721729, 6}, // 2^20 * 17 * 59 + 1 {1045430273, 3}, // 2^20 * 997 + 1 {1012924417, 5}, // 2^21 * 3 * 7 * 23 + 1 {1007681537, 3}, // 2^20 * 31^2 + 1 {1004535809, 3}, // 2^21 * 479 + 1 {998244353, 3}, // 2^23 * 7 * 17 + 1 {985661441, 3}, // 2^22 * 5 * 47 + 1 {976224257, 3}, // 2^20 * 7^2 * 19 + 1 {975175681, 17}, // 2^21 * 3 * 5 * 31 + 1 {962592769, 7}, // 2^21 * 3^3 * 17 + 1 {950009857, 7}, // 2^21 * 4 * 151 + 1 {943718401, 7}, // 2^22 * 3^2 * 5^2 + 1 {935329793, 3}, // 2^22 * 223 + 1 {924844033, 5}, // 2^21 * 3^2 * 7^2 + 1 {469762049, 3}, // 2^26 * 7 + 1 {167772161, 3}, // 2^25 * 5 + 1 }; // }}} // general math {{{ ll extgcd(ll a, ll b, ll &x, ll &y) { ll d; return b == 0 ? (x = a < 0 ? -1 : 1, y = 0, a < 0 ? -a : a) : (d = extgcd(b, a % b, y, x), y -= a / b * x, d); } ll modinv(ll a, ll mod) { ll x, y; extgcd(a, mod, x, y); x %= mod; return x < 0 ? x + mod : x; } ll modpow(ll a, ll b, ll mod) { ll r = 1; a %= mod; while(b) { if(b & 1) r = r * a % mod; a = a * a % mod; b >>= 1; } return r; } // }}} // NTT Core {{{ template < int MAX_H > struct Pool { static ll *tmp, *A, *B; }; template < int MAX_H > ll *Pool< MAX_H >::tmp = new ll[1 << MAX_H]; template < int MAX_H > ll *Pool< MAX_H >::A = new ll[1 << MAX_H]; template < int MAX_H > ll *Pool< MAX_H >::B = new ll[1 << MAX_H]; template < int MAX_H, ll mod, ll primitive > class Core { public: static_assert((mod & ((1 << MAX_H) - 1)) == 1, "mod is too small; comment out"); // ord zetaList[i] = 2^(i + 1) ll zetaList[MAX_H], zetaInvList[MAX_H]; // constexpr Core() { zetaList[MAX_H - 1] = modpow(primitive, (mod - 1) / (1 << MAX_H), mod); zetaInvList[MAX_H - 1] = modinv(zetaList[MAX_H - 1], mod); for(int ih = MAX_H - 2; ih >= 0; --ih) { zetaList[ih] = zetaList[ih + 1] * zetaList[ih + 1] % mod; zetaInvList[ih] = zetaInvList[ih + 1] * zetaInvList[ih + 1] % mod; } } void fft(ll *a, uint n, uint nh, bool inverse) const { ll *tmp = Pool< MAX_H >::tmp; uint mask = n - 1; for(uint i = n >> 1, ih = nh - 1; i >= 1; i >>= 1, --ih) { ll zeta = inverse ? zetaInvList[nh - 1 - ih] : zetaList[nh - 1 - ih]; ll powZeta = 1; for(uint j = 0; j < n; j += i) { for(uint k = 0; k < i; ++k) { tmp[j | k] = (a[((j << 1) & mask) | k] + powZeta * a[(((j << 1) | i) & mask) | k]) % mod; } powZeta = powZeta * zeta % mod; } swap(a, tmp); } if(nh & 1) { swap(a, tmp); for(uint i = 0; i < n; ++i) a[i] = tmp[i]; } if(inverse) { ll invN = modinv(n, mod); for(uint i = 0; i < n; ++i) a[i] = a[i] * invN % mod; } } vector< ll > conv(const vector< ll > &a, const vector< ll > &b) const { uint t = a.size() + b.size() - 1; uint n = 1, nh = 0; while(n < t) n <<= 1, ++nh; return convStrict(a, b, n, nh); } vector< ll > convStrict(const vector< ll > &a, const vector< ll > &b, uint n, uint nh) const { ll *A = Pool< MAX_H >::A, *B = Pool< MAX_H >::B; for(uint i = 0; i < n; ++i) A[i] = B[i] = 0; copy(a.begin(), a.end(), A); copy(b.begin(), b.end(), B); fft(A, n, nh, 0), fft(B, n, nh, 0); for(uint i = 0; i < n; ++i) A[i] = A[i] * B[i] % mod; fft(A, n, nh, 1); return vector< ll >(A, A + n); } }; // }}} // Convolution With Garner {{{ template < int MAX_H, int I > class ConvolutionWithGarnerCore { public: static void conv_for(uint n, uint nh, const vector< ll > &a, const vector< ll > &b, vector< ll > &mods, vector< ll > &coeffs, vector< vector< ll > > &constants) { static const Core< MAX_H, NTT_PRIMES[I][0], NTT_PRIMES[I][1] > ntt; auto c = ntt.convStrict(a, b, n, nh); mods[I] = NTT_PRIMES[I][0]; ConvolutionWithGarnerCore< MAX_H, I - 1 >::conv_for( n, nh, a, b, mods, coeffs, constants); // garner for(size_t i = 0; i < c.size(); ++i) { ll v = (c[i] - constants[I][i]) * modinv(coeffs[I], mods[I]) % mods[I]; if(v < 0) v += mods[I]; for(size_t j = I + 1; j < mods.size(); ++j) { constants[j][i] = (constants[j][i] + coeffs[j] * v) % mods[j]; } } for(size_t j = I + 1; j < mods.size(); ++j) { coeffs[j] = (coeffs[j] * mods[I]) % mods[j]; } } }; template < int MAX_H > class ConvolutionWithGarnerCore< MAX_H, -1 > { public: static void conv_for(uint, uint, const vector< ll > &, const vector< ll > &, vector< ll > &, vector< ll > &, vector< vector< ll > > &) {} }; template < int MAX_H > class ConvolutionWithGarner { public: template < int USE > static vector< ll > conv(const vector< ll > &a, const vector< ll > &b, ll mod) { static_assert(USE >= 1, "USE must be positive"); static_assert(USE <= sizeof(NTT_PRIMES) / sizeof(*NTT_PRIMES), "USE is too big"); uint nt = a.size() + b.size() - 1; uint n = 1, nh = 0; while(n < nt) n <<= 1, ++nh; vector< ll > coeffs(USE + 1, 1); vector< vector< ll > > constants(USE + 1, vector< ll >(n)); vector< ll > mods(USE + 1, mod); ConvolutionWithGarnerCore< MAX_H, USE - 1 >::conv_for( n, nh, a, b, mods, coeffs, constants); return constants.back(); } }; // }}} } // namespace NTT // }}} // 1st param is MAX_H NTT::Core< 18, NTT::NTT_PRIMES[0][0], NTT::NTT_PRIMES[0][1] > nttBig; NTT::Core< 18, 998244353, 5 > ntt; using nttconv = NTT::ConvolutionWithGarner< 18 >; // nttconv::conv< USE >(a, b, mod) constexpr ll mod = 998244353; int a[112345]; int b[112345]; int c[112345]; int modlog[112345]; int modexp[112345]; // O(N^.5) /// --- divisor {{{ /// #include vector< ll > divisor(ll n) { vector< ll > res; for(ll i = 1; i * i <= n; i++) { if(n % i == 0) { res.emplace_back(i); if(i != n / i) res.emplace_back(n / i); } } return res; } /// }}}--- /// /// --- math {{{ /// ll gcd(ll a, ll b) { return b == 0 ? a : gcd(b, a % b); } ll lcm(ll a, ll b) { return a / gcd(a, b) * b; } ll extgcd(ll a, ll b, ll &x, ll &y) { ll d; return b == 0 ? (x = a < 0 ? -1 : 1, y = 0, a < 0 ? -a : a) : (d = extgcd(b, a % b, y, x), y -= a / b * x, d); } ll modinv(ll a, ll mod) { ll x, y; extgcd(a, mod, x, y); if(x < 0) x += mod; return x; } ll modpow(ll a, ll b, ll mod) { ll r = 1; a %= mod; while(b) { if(b & 1) r = r * a % mod; a = a * a % mod; b >>= 1; } return r; } /// }}}--- /// // O(log p) /// --- isPrimitive {{{ /// bool isPrimitive(ll x, ll p) { auto ds = divisor(p - 1); for(ll d : ds) if(d != p - 1) { if(modpow(x, d, p) == 1) return false; } return true; } /// }}}--- /// int main() { std::ios::sync_with_stdio(false), std::cin.tie(0); int p; cin >> p; for(int i = 1; i < p; i++) cin >> a[i]; for(int i = 1; i < p; i++) cin >> b[i]; int g = 0; for(int i = 1; i < p; i++) if(isPrimitive(i, p)) g = i; dump(g); for(int i = 0, now = 1; i < p - 1; i++, now = (ll) now * g % p) modlog[now] = i, modexp[i] = now; vector A(p), B(p); for(int i = 1; i < p; i++) A[modlog[i]] = a[i]; for(int i = 1; i < p; i++) B[modlog[i]] = b[i]; auto C = ntt.conv(A, B); for(int i = p - 1; i < C.size(); i++) (C[i % (p - 1)] += C[i]) %= mod; dump(C); for(int i = 1; i < p; i++) { cout << C[modlog[i] % (p - 1)] << " \n"[i == p - 1]; } return 0; }