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convolution/test/mul_modp_convolution.test.cpp
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- Last update: 2025-06-28 10:05:47+09:00
- Problem: https://judge.yosupo.jp/problem/mul_modp_convolution
Depends on
- convolution/mul_mod_p_conv.hpp
- number_theory/factorize.hpp
- number_theory/mod_int.hpp
- number_theory/montgomery_64.hpp
- number_theory/primality.hpp
- number_theory/primitive_root.hpp
- number_theory/utils.hpp
- poly/fft.hpp
- template/fastio.hpp
- template/random.hpp
- template/template.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/mul_modp_convolution"
#include "../../convolution/mul_mod_p_conv.hpp"
#include "../../template/fastio.hpp"
#include "../../template/template.hpp"
void solve() {
using M = ModInt<998244353>;
i32 p;
rd.read(p);
V<M> a(p), b(p);
REP(i, p) {
rd.read(a[i].val);
}
REP(i, p) {
rd.read(b[i].val);
}
V<M> c = mul_mod_p_convolution(a, b);
REP(i, p) {
wr.write(c[i].val);
if (i != p - 1) {
wr.write(' ');
}
}
wr.writeln();
}
int main() {
i32 t = 1;
// cin >> t;
while (t--) {
solve();
}
}#line 1 "convolution/test/mul_modp_convolution.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/mul_modp_convolution"
#line 2 "number_theory/factorize.hpp"
#include <algorithm>
#include <vector>
#line 2 "template/random.hpp"
#include <chrono>
#include <random>
#if defined(LOCAL) || defined(FIX_SEED)
std::mt19937_64 mt(123456789);
#else
std::mt19937_64 mt(std::chrono::steady_clock::now().time_since_epoch().count());
#endif
template <typename T>
T uniform(T l, T r) {
return std::uniform_int_distribution<T>(l, r - 1)(mt);
}
template <typename T>
T uniform(T n) {
return std::uniform_int_distribution<T>(0, n - 1)(mt);
}
#line 2 "number_theory/primality.hpp"
namespace primality {
using u64 = unsigned long long;
using u128 = __uint128_t;
u64 inv_64(u64 n) {
u64 r = n;
for (int i = 0; i < 5; ++i) {
r *= 2 - n * r;
}
return r;
}
// n: odd, < 2^{62}
struct Montgomery64 {
u64 n, mni, p;
Montgomery64(u64 n) : n(n), mni(-inv_64(n)), p(-1ULL % n + 1) {}
u64 mulmr(u64 xr, u64 yr) const {
u128 z = (u128)xr * yr;
u64 ret = (z + (u128)((u64)z * mni) * n) >> 64;
if (ret >= n) {
ret -= n;
}
return ret;
}
u64 mr(u64 xr) const {
u64 ret = (xr + (u128)(xr * mni) * n) >> 64;
if (ret >= n) {
ret -= n;
}
return ret;
}
u64 pow(u64 xr, u64 t) const {
u64 ret = p;
while (t) {
if (t & 1) {
ret = mulmr(ret, xr);
}
xr = mulmr(xr, xr);
t >>= 1;
}
return ret;
}
};
bool is_prime(u64 n) {
if (n == 2) {
return true;
}
if (n == 1 || n % 2 == 0) {
return false;
}
u64 s = __builtin_ctzll(n - 1);
u64 d = (n - 1) >> s;
u64 base[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
Montgomery64 mont(n);
u64 fl = n - mont.p;
for (u64 b : base) {
b = mont.mr(b);
if (!b) {
continue;
}
u64 t = mont.pow(b, d);
if (t == mont.p) {
continue;
}
u64 i = 0;
for (; i < s; ++i) {
if (t == fl) {
break;
}
t = mont.mulmr(t, t);
}
if (i == s) {
return false;
}
}
return true;
}
} // namespace primality
bool is_prime(unsigned long long n) {
return primality::is_prime(n);
}
#line 2 "number_theory/montgomery_64.hpp"
#include <cassert>
// mod: odd, < 2^{63}
template <int id>
struct MontgomeryModInt64 {
using u64 = unsigned long long;
using u128 = __uint128_t;
static u64 inv_64(u64 n) {
u64 r = n;
for (int i = 0; i < 5; ++i) {
r *= 2 - n * r;
}
return r;
}
static u64 mod, neg_inv, sq;
static void set_mod(u64 m) {
assert(m % 2 == 1 && m < (1ULL << 63));
mod = m;
neg_inv = -inv_64(m);
sq = -u128(mod) % mod;
}
static u64 get_mod() { return mod; }
static u64 reduce(u128 xr) {
u64 ret = (xr + u128(u64(xr) * neg_inv) * mod) >> 64;
if (ret >= mod) {
ret -= mod;
}
return ret;
}
using M = MontgomeryModInt64<id>;
u64 x;
MontgomeryModInt64() : x(0) {}
MontgomeryModInt64(u64 _x) : x(reduce(u128(_x) * sq)) {}
u64 val() const { return reduce(u128(x)); }
M &operator+=(M rhs) {
if ((x += rhs.x) >= mod) {
x -= mod;
}
return *this;
}
M &operator-=(M rhs) {
if ((x -= rhs.x) >= mod) {
x += mod;
}
return *this;
}
M &operator*=(M rhs) {
x = reduce(u128(x) * rhs.x);
return *this;
}
M operator+(M rhs) const { return M(*this) += rhs; }
M operator-(M rhs) const { return M(*this) -= rhs; }
M operator*(M rhs) const { return M(*this) *= rhs; }
M pow(u64 t) const {
M ret(1);
M self = *this;
while (t) {
if (t & 1) {
ret *= self;
}
self *= self;
t >>= 1;
}
return ret;
}
M inv() const {
assert(x);
return this->pow(mod - 2);
}
M &operator/=(M rhs) {
*this /= rhs.inv();
return *this;
}
M operator/(M rhs) const { return M(*this) /= rhs; }
};
template <int id> unsigned long long MontgomeryModInt64<id>::mod = 1;
template <int id> unsigned long long MontgomeryModInt64<id>::neg_inv = 1;
template <int id> unsigned long long MontgomeryModInt64<id>::sq = 1;
#line 7 "number_theory/factorize.hpp"
namespace factorize_impl {
unsigned long long bgcd(unsigned long long x, unsigned long long y) {
if (x == 0) {
return y;
}
if (y == 0) {
return x;
}
int n = __builtin_ctzll(x);
int m = __builtin_ctzll(y);
x >>= n;
y >>= m;
while (x != y) {
if (x > y) {
x = (x - y) >> __builtin_ctzll(x - y);
} else {
y = (y - x) >> __builtin_ctzll(y - x);
}
}
return x << (n < m ? n : m);
}
template <typename T>
unsigned long long rho(unsigned long long n, unsigned long long c) {
T cc(c);
auto f = [cc](T x) -> T {
return x * x + cc;
};
T y(2);
T x = y;
T z = y;
T p(1);
unsigned long long g = 1;
constexpr int M = 128;
for (int r = 1; g == 1; r *= 2) {
x = y;
for (int i = 0; i < r && g == 1; i += M) {
z = y;
for (int j = 0; j < r - i && j < M; ++j) {
y = f(y);
p *= y - x;
}
g = bgcd(p.val(), n);
}
}
if (g == n) {
do {
z = f(z);
g = bgcd((z - x).val(), n);
} while (g == 1);
}
return g;
}
unsigned long long find_factor(unsigned long long n) {
using M = MontgomeryModInt64<20250127>;
M::set_mod(n);
while (true) {
unsigned long long c = uniform(n);
unsigned long long g = rho<M>(n, c);
if (g != n) {
return g;
}
}
return 0;
}
void factor_inner(unsigned long long n, std::vector<unsigned long long> &ps) {
if (is_prime(n)) {
ps.push_back(n);
return;
}
if (n % 2 == 0) {
ps.push_back(2);
factor_inner(n / 2, ps);
return;
}
unsigned long long m = find_factor(n);
factor_inner(m, ps);
factor_inner(n / m, ps);
}
}
std::vector<unsigned long long> factorize(unsigned long long n) {
if (n <= 1) {
return std::vector<unsigned long long>();
}
std::vector<unsigned long long> ps;
factorize_impl::factor_inner(n, ps);
std::sort(ps.begin(), ps.end());
return ps;
}
#line 3 "number_theory/primitive_root.hpp"
// p: prime
unsigned long long find_primitive_root(unsigned long long p) {
using M = MontgomeryModInt64<20250128>;
assert(is_prime(p));
if (p == 2) {
return 1;
}
M::set_mod(p);
std::vector<unsigned long long> ps = factorize(p - 1);
ps.erase(std::unique(ps.begin(), ps.end()), ps.end());
while (true) {
unsigned long long x = uniform<unsigned long long>(1, p);
M x_(x), one(1ULL);
bool ok = true;
for (unsigned long long q : ps) {
if (x_.pow((p - 1) / q).x == one.x) {
ok = false;
break;
}
}
if (ok) {
return x;
}
}
return 0;
}
#line 2 "poly/fft.hpp"
#include <array>
#line 2 "number_theory/mod_int.hpp"
#line 4 "number_theory/mod_int.hpp"
#include <iostream>
#include <type_traits>
#line 2 "number_theory/utils.hpp"
#include <utility>
constexpr bool is_prime(unsigned n) {
if (n == 0 || n == 1) {
return false;
}
for (unsigned i = 2; i * i <= n; ++i) {
if (n % i == 0) {
return false;
}
}
return true;
}
constexpr unsigned mod_pow(unsigned x, unsigned y, unsigned mod) {
unsigned ret = 1, self = x;
while (y != 0) {
if (y & 1) {
ret = (unsigned)((unsigned long long)ret * self % mod);
}
self = (unsigned)((unsigned long long)self * self % mod);
y /= 2;
}
return ret;
}
template <unsigned mod>
constexpr unsigned primitive_root() {
static_assert(is_prime(mod), "`mod` must be a prime number.");
if (mod == 2) {
return 1;
}
unsigned primes[32] = {};
int it = 0;
{
unsigned m = mod - 1;
for (unsigned i = 2; i * i <= m; ++i) {
if (m % i == 0) {
primes[it++] = i;
while (m % i == 0) {
m /= i;
}
}
}
if (m != 1) {
primes[it++] = m;
}
}
for (unsigned i = 2; i < mod; ++i) {
bool ok = true;
for (int j = 0; j < it; ++j) {
if (mod_pow(i, (mod - 1) / primes[j], mod) == 1) {
ok = false;
break;
}
}
if (ok) return i;
}
return 0;
}
// y >= 1
template <typename T>
constexpr T safe_mod(T x, T y) {
x %= y;
if (x < 0) {
x += y;
}
return x;
}
// y != 0
template <typename T>
constexpr T floor_div(T x, T y) {
if (y < 0) {
x *= -1;
y *= -1;
}
if (x >= 0) {
return x / y;
} else {
return -((-x + y - 1) / y);
}
}
// y != 0
template <typename T>
constexpr T ceil_div(T x, T y) {
if (y < 0) {
x *= -1;
y *= -1;
}
if (x >= 0) {
return (x + y - 1) / y;
} else {
return -(-x / y);
}
}
// b >= 1
// returns (g, x) s.t. g = gcd(a, b), a * x = g (mod b), 0 <= x < b / g
// from ACL
template <typename T>
std::pair<T, T> extgcd(T a, T b) {
a = safe_mod(a, b);
T s = b, t = a, m0 = 0, m1 = 1;
while (t) {
T u = s / t;
s -= t * u;
m0 -= m1 * u;
std::swap(s, t);
std::swap(m0, m1);
}
if (m0 < 0) {
m0 += b / s;
}
return std::pair<T, T>(s, m0);
}
// b >= 1
// returns (g, x, y) s.t. g = gcd(a, b), a * x + b * y = g, 0 <= x < b / g, |y| < max(2, |a| / g)
template <typename T>
std::tuple<T, T, T> extgcd2(T a, T b) {
T _a = safe_mod(a, b);
T quot = (a - _a) / b;
T x00 = 0, x01 = 1, y0 = b;
T x10 = 1, x11 = -quot, y1 = _a;
while (y1) {
T u = y0 / y1;
x00 -= u * x10;
x01 -= u * x11;
y0 -= u * y1;
std::swap(x00, x10);
std::swap(x01, x11);
std::swap(y0, y1);
}
if (x00 < 0) {
x00 += b / y0;
x01 -= a / y0;
}
return std::tuple<T, T, T>(y0, x00, x01);
}
// gcd(x, m) == 1
template <typename T>
T inv_mod(T x, T m) {
return extgcd(x, m).second;
}
#line 7 "number_theory/mod_int.hpp"
template <unsigned mod>
struct ModInt {
static_assert(mod != 0, "`mod` must not be equal to 0.");
static_assert(mod < (1u << 31),
"`mod` must be less than (1u << 31) = 2147483648.");
unsigned val;
static constexpr unsigned get_mod() { return mod; }
constexpr ModInt() : val(0) {}
template <typename T, std::enable_if_t<std::is_signed_v<T>> * = nullptr>
constexpr ModInt(T x)
: val((unsigned)((long long)x % (long long)mod + (x < 0 ? mod : 0))) {}
template <typename T, std::enable_if_t<std::is_unsigned_v<T>> * = nullptr>
constexpr ModInt(T x) : val((unsigned)(x % mod)) {}
static constexpr ModInt raw(unsigned x) {
ModInt<mod> ret;
ret.val = x;
return ret;
}
constexpr unsigned get_val() const { return val; }
constexpr ModInt operator+() const { return *this; }
constexpr ModInt operator-() const { return ModInt<mod>(0u) - *this; }
constexpr ModInt &operator+=(const ModInt &rhs) {
val += rhs.val;
if (val >= mod) val -= mod;
return *this;
}
constexpr ModInt &operator-=(const ModInt &rhs) {
val -= rhs.val;
if (val >= mod) val += mod;
return *this;
}
constexpr ModInt &operator*=(const ModInt &rhs) {
val = (unsigned long long)val * rhs.val % mod;
return *this;
}
constexpr ModInt &operator/=(const ModInt &rhs) {
val = (unsigned long long)val * rhs.inv().val % mod;
return *this;
}
friend constexpr ModInt operator+(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) += rhs;
}
friend constexpr ModInt operator-(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) -= rhs;
}
friend constexpr ModInt operator*(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) *= rhs;
}
friend constexpr ModInt operator/(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) /= rhs;
}
constexpr ModInt pow(unsigned long long x) const {
ModInt<mod> ret = ModInt<mod>::raw(1);
ModInt<mod> self = *this;
while (x != 0) {
if (x & 1) ret *= self;
self *= self;
x >>= 1;
}
return ret;
}
constexpr ModInt inv() const {
static_assert(is_prime(mod), "`mod` must be a prime number.");
assert(val != 0);
return this->pow(mod - 2);
}
friend std::istream &operator>>(std::istream &is, ModInt<mod> &x) {
long long val;
is >> val;
x.val = val % mod + (val < 0 ? mod : 0);
return is;
}
friend std::ostream &operator<<(std::ostream &os, const ModInt<mod> &x) {
os << x.val;
return os;
}
friend bool operator==(const ModInt &lhs, const ModInt &rhs) {
return lhs.val == rhs.val;
}
friend bool operator!=(const ModInt &lhs, const ModInt &rhs) {
return lhs.val != rhs.val;
}
};
template <unsigned mod>
void debug(ModInt<mod> x) {
std::cerr << x.val;
}
#line 5 "poly/fft.hpp"
constexpr int ctz_constexpr(unsigned n) {
int x = 0;
while (!(n & (1u << x))) {
++x;
}
return x;
}
template <unsigned MOD>
struct FFTRoot {
static constexpr unsigned R = ctz_constexpr(MOD - 1);
std::array<ModInt<MOD>, R + 1> root, iroot;
std::array<ModInt<MOD>, R> rate2, irate2;
std::array<ModInt<MOD>, R - 1> rate3, irate3;
std::array<ModInt<MOD>, R + 1> inv2;
constexpr FFTRoot() : root{}, iroot{}, rate2{}, irate2{}, rate3{}, irate3{}, inv2{} {
unsigned pr = primitive_root<MOD>();
root[R] = ModInt<MOD>(pr).pow(MOD >> R);
iroot[R] = root[R].inv();
for (int i = R - 1; i >= 0; --i) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
ModInt<MOD> prod(1), iprod(1);
for (int i = 0; i < (int)R - 1; ++i) {
rate2[i] = prod * root[i + 2];
irate2[i] = iprod * iroot[i + 2];
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
prod = ModInt<MOD>(1);
iprod = ModInt<MOD>(1);
for (int i = 0; i < (int)R - 2; ++i) {
rate3[i] = prod * root[i + 3];
irate3[i] = iprod * iroot[i + 3];
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
ModInt<MOD> i2 = ModInt<MOD>(2).inv();
inv2[0] = ModInt<MOD>(1);
for (int i = 0; i < (int)R; ++i) {
inv2[i + 1] = inv2[i] * i2;
}
}
};
template <typename M>
void fft(M *a, int n) {
using ull = unsigned long long;
static_assert(M::get_mod() < (1u << 30));
static constexpr FFTRoot<M::get_mod()> fftroot;
static constexpr ull CEIL = 2ULL * M::get_mod() * M::get_mod();
int l = __builtin_ctz(n);
int ph = 0;
while (ph < l) {
if (ph + 1 == l) {
int b = 1 << ph;
M z = M::raw(1);
for (int i = 0; i < b; ++i) {
int offset = i << 1;
M x = a[offset];
M y = a[offset + 1] * z;
a[offset] = x + y;
a[offset + 1] = x - y;
z *= fftroot.rate2[__builtin_ctz(~i)];
}
++ph;
} else {
int bl = 1 << ph;
int wd = 1 << (l - 2 - ph);
M zeta = M::raw(1);
for (int i = 0; i < bl; ++i) {
int offset = i << (l - ph);
M zeta2 = zeta * zeta;
M zeta3 = zeta2 * zeta;
for (int j = 0; j < wd; ++j) {
ull w = a[offset + j].val;
ull x = (ull)a[offset + j + wd].val * zeta.val;
ull y = (ull)a[offset + j + 2 * wd].val * zeta2.val;
ull z = (ull)a[offset + j + 3 * wd].val * zeta3.val;
ull ix_m_iz = (CEIL + x - z) % M::get_mod() * fftroot.root[2].val;
a[offset + j] = M(w + x + y + z);
a[offset + j + wd] = M(CEIL + w - x + y - z);
a[offset + j + 2 * wd] = M(CEIL + w - y + ix_m_iz);
a[offset + j + 3 * wd] = M(CEIL + w - y - ix_m_iz);
}
zeta *= fftroot.rate3[__builtin_ctz(~i)];
}
ph += 2;
}
}
}
template <typename M>
void ifft(M *a, int n) {
using ull = unsigned long long;
static_assert(M::get_mod() < (1u << 30));
static constexpr FFTRoot<M::get_mod()> fftroot;
int l = __builtin_ctz(n);
int ph = l;
while (ph > 0) {
if (ph == 1) {
--ph;
int wd = 1 << (l - 1);
for (int i = 0; i < wd; ++i) {
M x = a[i];
M y = a[i + wd];
a[i] = x + y;
a[i + wd] = x - y;
}
} else {
ph -= 2;
int bl = 1 << ph;
int wd = 1 << (l - 2 - ph);
M zeta = M::raw(1);
for (int i = 0; i < bl; ++i) {
int offset = i << (l - ph);
M zeta2 = zeta * zeta;
M zeta3 = zeta2 * zeta;
for (int j = 0; j < wd; ++j) {
unsigned w = a[offset + j].val;
unsigned x = a[offset + j + wd].val;
unsigned y = a[offset + j + 2 * wd].val;
unsigned z = a[offset + j + 3 * wd].val;
unsigned iy_m_iz = (ull)(M::get_mod() + y - z) * fftroot.root[2].val % M::get_mod();
a[offset + j] = M(w + x + y + z);
a[offset + j + wd] = M((ull)zeta.val * (2 * M::get_mod() + w - x - iy_m_iz));
a[offset + j + 2 * wd] = M((ull)zeta2.val * (2 * M::get_mod() + w + x - y - z));
a[offset + j + 3 * wd] = M((ull)zeta3.val * (M::get_mod() + w - x + iy_m_iz));
}
zeta *= fftroot.irate3[__builtin_ctz(~i)];
}
}
}
for (int i = 0; i < n; ++i) {
a[i] *= fftroot.inv2[l];
}
}
template <typename M>
void fft(std::vector<M> &a) {
fft(a.data(), (int)a.size());
}
template <typename M>
void ifft(std::vector<M> &a) {
ifft(a.data(), (int)a.size());
}
template <typename M>
std::vector<M> convolve_naive(const std::vector<M> &a,
const std::vector<M> &b) {
int n = (int)a.size();
int m = (int)b.size();
std::vector<M> c(n + m - 1);
if (n < m) {
for (int j = 0; j < m; ++j) {
for (int i = 0; i < n; ++i) {
c[i + j] += a[i] * b[j];
}
}
} else {
for (int i = 0; i < n; ++i) {
for (int j = 0; j < m; ++j) {
c[i + j] += a[i] * b[j];
}
}
}
return c;
}
template <typename M>
std::vector<M> convolve_fft(std::vector<M> a, std::vector<M> b) {
int n = (int)a.size() + (int)b.size() - 1;
int m = 1;
while (m < n) {
m <<= 1;
}
bool shr = false;
M last;
if (n >= 3 && n == m / 2 + 1) {
shr = true;
last = a.back() * b.back();
m /= 2;
while ((int)a.size() > m) {
a[(int)a.size() - 1 - m] += a.back();
a.pop_back();
}
while ((int)b.size() > m) {
b[(int)b.size() - 1 - m] += b.back();
b.pop_back();
}
}
a.resize(m);
b.resize(m);
fft(a);
fft(b);
for (int i = 0; i < m; ++i) {
a[i] *= b[i];
}
ifft(a);
a.resize(n);
if (shr) {
a[0] -= last;
a[n - 1] = last;
}
return a;
}
template <typename M>
std::vector<M> convolve(const std::vector<M> &a, const std::vector<M> &b) {
if (a.empty() || b.empty()) {
return std::vector<M>(0);
}
if (std::min(a.size(), b.size()) <= 60) {
return convolve_naive(a, b);
} else {
return convolve_fft(a, b);
}
}
#line 4 "convolution/mul_mod_p_conv.hpp"
template <typename T>
std::vector<T> mul_mod_p_convolution(const std::vector<T> &a,
const std::vector<T> &b, int g = -1) {
int p = (int)a.size();
if (g == -1) {
g = find_primitive_root(p);
}
std::vector<int> pw(p - 1);
pw[0] = 1;
for (int i = 1; i < p - 1; ++i) {
pw[i] = (long long)pw[i - 1] * g % p;
}
std::vector<T> at(p - 1), bt(p - 1);
for (int i = 0; i < p - 1; ++i) {
at[i] = a[pw[i]];
bt[i] = b[pw[i]];
}
std::vector<T> ct = convolve(at, bt);
std::vector<T> c(p, T(0));
for (int i = 0; i < p; ++i) {
c[0] += a[i] * b[0];
}
for (int i = 1; i < p; ++i) {
c[0] += a[0] * b[i];
}
for (int i = 0; i < (int)ct.size(); ++i) {
c[pw[i % (p - 1)]] += ct[i];
}
return c;
}
#line 1 "template/fastio.hpp"
#include <cstdio>
#include <cstring>
#line 5 "template/fastio.hpp"
// unable to read INT_MIN (int), LLONG_MIN (long long)
class Reader {
FILE *fp;
static constexpr int BUF = 1 << 18;
char buf[BUF];
char *pl, *pr;
void reread() {
int wd = pr - pl;
std::memcpy(buf, pl, wd);
pl = buf;
pr = buf + wd;
pr += std::fread(pr, 1, BUF - wd, fp);
}
char skip() {
char ch = *pl++;
while (ch <= ' ') {
ch = *pl++;
}
return ch;
}
template <typename T>
void read_unsigned(T &x) {
if (pr - pl < 64) {
reread();
}
x = 0;
char ch = skip();
while ('0' <= ch) {
x = 10 * x + (0xf & ch);
ch = *pl++;
}
}
template <typename T>
void read_signed(T &x) {
if (pr - pl < 64) {
reread();
}
x = 0;
bool neg = false;
char ch = skip();
if (ch == '-') {
ch = *pl++;
neg = true;
}
while ('0' <= ch) {
x = 10 * x + (0xf & ch);
ch = *pl++;
}
if (neg) {
x = -x;
}
}
void read_single(int &x) { read_signed(x); }
void read_single(unsigned &x) { read_unsigned(x); }
void read_single(long &x) { read_signed(x); }
void read_single(unsigned long &x) { read_signed(x); }
void read_single(long long &x) { read_signed(x); }
void read_single(unsigned long long &x) { read_unsigned(x); }
public:
Reader(FILE *fp) : fp(fp), pl(buf), pr(buf) { reread(); }
void read() {}
template <typename Head, typename... Tail>
void read(Head &head, Tail &...tail) {
read_single(head);
read(tail...);
}
};
struct NumberToString {
char buf[10000][4];
constexpr NumberToString() : buf() {
for (int i = 0; i < 10000; ++i) {
int n = i;
for (int j = 3; j >= 0; --j) {
buf[i][j] = '0' + n % 10;
n /= 10;
}
}
}
} constexpr number_to_string_precalc;
class Writer {
FILE *fp;
static constexpr int BUF = 1 << 18;
char buf[BUF];
char *ptr;
void write_u32(unsigned x) {
if ((buf + BUF - ptr) < 32) {
flush();
}
static char sml[12];
int t = 8;
while (x >= 10000) {
unsigned n = x % 10000;
x /= 10000;
std::memcpy(sml + t, number_to_string_precalc.buf[n], 4);
t -= 4;
}
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, sml + (t + 4), 8 - t);
ptr += 8 - t;
}
void write_u64(unsigned long long x) {
if ((buf + BUF - ptr) < 32) {
flush();
}
if (x >= 10000000000000000) {
unsigned long long z = x % 100000000;
x /= 100000000;
unsigned long long y = x % 100000000;
x /= 100000000;
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, number_to_string_precalc.buf[y / 10000], 4);
std::memcpy(ptr + 4, number_to_string_precalc.buf[y % 10000], 4);
std::memcpy(ptr + 8, number_to_string_precalc.buf[z / 10000], 4);
std::memcpy(ptr + 12, number_to_string_precalc.buf[z % 10000], 4);
ptr += 16;
} else {
static char sml[12];
int t = 8;
while (x >= 10000) {
unsigned long long n = x % 10000;
x /= 10000;
std::memcpy(sml + t, number_to_string_precalc.buf[n], 4);
t -= 4;
}
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, sml + (t + 4), 8 - t);
ptr += 8 - t;
}
}
void write_char(char c) {
if (ptr == buf + BUF) {
flush();
}
*ptr++ = c;
}
template <typename T>
void write_unsigned(T x) {
if constexpr (std::is_same_v<T, unsigned long long> ||
std::is_same_v<T, unsigned long>) {
write_u64(x);
} else {
write_u32(x);
}
}
template <typename T>
void write_signed(T x) {
std::make_unsigned_t<T> y = x;
if (x < 0) {
write_char('-');
y = -y;
}
write_unsigned(y);
}
void write_string(const std::string &s) {
for (char c : s) {
write_char(c);
}
}
void write_single(int x) { write_signed(x); }
void write_single(unsigned x) { write_unsigned(x); }
void write_single(long x) { write_signed(x); }
void write_single(unsigned long x) { write_unsigned(x); }
void write_single(long long x) { write_signed(x); }
void write_single(unsigned long long x) { write_unsigned(x); }
void write_single(char c) { write_char(c); }
void write_single(const std::string &s) { write_string(s); }
public:
Writer(FILE *fp) : fp(fp), ptr(buf) {}
~Writer() { flush(); }
void flush() {
std::fwrite(buf, 1, ptr - buf, fp);
ptr = buf;
}
void write() {}
template <typename Head, typename... Tail>
void write(Head &&head, Tail &&...tail) {
write_single(head);
if (sizeof...(Tail)) {
write_char(' ');
}
write(std::forward<Tail>(tail)...);
}
template <typename... T>
void writeln(T &&...t) {
write(std::forward<T>(t)...);
write_char('\n');
}
};
Reader rd(stdin);
Writer wr(stdout);
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
#define OVERRIDE(a, b, c, d, ...) d
#define REP2(i, n) for (i32 i = 0; i < (i32)(n); ++i)
#define REP3(i, m, n) for (i32 i = (i32)(m); i < (i32)(n); ++i)
#define REP(...) OVERRIDE(__VA_ARGS__, REP3, REP2)(__VA_ARGS__)
#define PER2(i, n) for (i32 i = (i32)(n)-1; i >= 0; --i)
#define PER3(i, m, n) for (i32 i = (i32)(n)-1; i >= (i32)(m); --i)
#define PER(...) OVERRIDE(__VA_ARGS__, PER3, PER2)(__VA_ARGS__)
#define ALL(x) begin(x), end(x)
#define LEN(x) (i32)(x.size())
using namespace std;
using u32 = unsigned int;
using u64 = unsigned long long;
using i32 = signed int;
using i64 = signed long long;
using f64 = double;
using f80 = long double;
using pi = pair<i32, i32>;
using pl = pair<i64, i64>;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = V<V<T>>;
template <typename T>
using VVV = V<V<V<T>>>;
template <typename T>
using VVVV = V<V<V<V<T>>>>;
template <typename T>
using PQR = priority_queue<T, V<T>, greater<T>>;
template <typename T>
bool chmin(T &x, const T &y) {
if (x > y) {
x = y;
return true;
}
return false;
}
template <typename T>
bool chmax(T &x, const T &y) {
if (x < y) {
x = y;
return true;
}
return false;
}
template <typename T>
i32 lob(const V<T> &arr, const T &v) {
return (i32)(lower_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
i32 upb(const V<T> &arr, const T &v) {
return (i32)(upper_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
V<i32> argsort(const V<T> &arr) {
V<i32> ret(arr.size());
iota(ALL(ret), 0);
sort(ALL(ret), [&](i32 i, i32 j) -> bool {
if (arr[i] == arr[j]) {
return i < j;
} else {
return arr[i] < arr[j];
}
});
return ret;
}
#ifdef INT128
using u128 = __uint128_t;
using i128 = __int128_t;
#endif
[[maybe_unused]] constexpr i32 INF = 1000000100;
[[maybe_unused]] constexpr i64 INF64 = 3000000000000000100;
struct SetUpIO {
SetUpIO() {
#ifdef FAST_IO
ios::sync_with_stdio(false);
cin.tie(nullptr);
#endif
cout << fixed << setprecision(15);
}
} set_up_io;
void scan(char &x) { cin >> x; }
void scan(u32 &x) { cin >> x; }
void scan(u64 &x) { cin >> x; }
void scan(i32 &x) { cin >> x; }
void scan(i64 &x) { cin >> x; }
void scan(f64 &x) { cin >> x; }
void scan(string &x) { cin >> x; }
template <typename T>
void scan(V<T> &x) {
for (T &ele : x) {
scan(ele);
}
}
void read() {}
template <typename Head, typename... Tail>
void read(Head &head, Tail &...tail) {
scan(head);
read(tail...);
}
#define CHAR(...) \
char __VA_ARGS__; \
read(__VA_ARGS__);
#define U32(...) \
u32 __VA_ARGS__; \
read(__VA_ARGS__);
#define U64(...) \
u64 __VA_ARGS__; \
read(__VA_ARGS__);
#define I32(...) \
i32 __VA_ARGS__; \
read(__VA_ARGS__);
#define I64(...) \
i64 __VA_ARGS__; \
read(__VA_ARGS__);
#define F64(...) \
f64 __VA_ARGS__; \
read(__VA_ARGS__);
#define STR(...) \
string __VA_ARGS__; \
read(__VA_ARGS__);
#define VEC(type, name, size) \
V<type> name(size); \
read(name);
#define VVEC(type, name, size1, size2) \
VV<type> name(size1, V<type>(size2)); \
read(name);
#line 5 "convolution/test/mul_modp_convolution.test.cpp"
void solve() {
using M = ModInt<998244353>;
i32 p;
rd.read(p);
V<M> a(p), b(p);
REP(i, p) {
rd.read(a[i].val);
}
REP(i, p) {
rd.read(b[i].val);
}
V<M> c = mul_mod_p_convolution(a, b);
REP(i, p) {
wr.write(c[i].val);
if (i != p - 1) {
wr.write(' ');
}
}
wr.writeln();
}
int main() {
i32 t = 1;
// cin >> t;
while (t--) {
solve();
}
}