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Verify/LC_multivariate_convolution_cyclic.test.cpp
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- Last update: 2024-04-26 03:32:16+09:00
- Problem: https://judge.yosupo.jp/problem/multivariate_convolution_cyclic
Depends on
- Arbitrary Mod Convolution
(Convolution/arbitrary.hpp)
- Multivarate Convolution Cyclic
(Convolution/multivariatecyclic.hpp)
- Number Theoretic Transform
(Convolution/ntt.hpp)
- Formal Power Series (Arbitrary mod)
(FPS/arbitraryfps.hpp)
- Multipoint Evaluation on Geometric Sequence
(FPS/multievalgeom.hpp)
- Dynamic Modint
(Math/dynamic.hpp)
- Fast Division
(Math/fastdiv.hpp)
- Miller-Rabin
(Math/miller.hpp)
- Modint
(Math/modint.hpp)
- Pollard-Rho
(Math/pollard.hpp)
- Primitive Function
(Math/primitive.hpp)
- Template/template.hpp
- Fast IO
(Utility/fastio.hpp)
- Random
(Utility/random.hpp)
Code
#define PROBLEM \
"https://judge.yosupo.jp/problem/multivariate_convolution_cyclic"
#include "Template/template.hpp"
#include "Utility/fastio.hpp"
#include "Math/dynamic.hpp"
#include "Convolution/arbitrary.hpp"
#include "FPS/arbitraryfps.hpp"
template <>
vector<Fp> Poly<Fp>::mult(const vector<Fp> &a, const vector<Fp> &b) const {
return ArbitraryMult<Fp>(a, b);
}
#include "Convolution/multivariatecyclic.hpp"
int main() {
int p, k;
read(p, k);
Fp::set_mod(p);
vector<int> a(k);
read(a);
int n = 1;
for (auto &x : a)
n *= x;
vector<Fp> f(n), g(n);
rep(i, 0, n) read(f[i].v);
rep(i, 0, n) read(g[i].v);
auto ret = MultivariateCyclic(f, g, a);
rep(i, 0, n) print(ret[i].v);
return 0;
}#line 1 "Verify/LC_multivariate_convolution_cyclic.test.cpp"
#define PROBLEM \
"https://judge.yosupo.jp/problem/multivariate_convolution_cyclic"
#line 1 "Template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b-1); i >= (int)(a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
const int inf = 0x3fffffff;
const ll INF = 0x1fffffffffffffff;
template <typename T> inline bool chmax(T &a, T b) {
if (a < b) {
a = b;
return 1;
}
return 0;
}
template <typename T> inline bool chmin(T &a, T b) {
if (a > b) {
a = b;
return 1;
}
return 0;
}
template <typename T, typename U> T ceil(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U> T floor(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T> int popcnt(T x) {
return __builtin_popcountll(x);
}
template <typename T> int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <typename T> int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#ifdef LOCAL
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define show(...) true
#endif
template <typename T> void _show(int i, T name) {
cerr << '\n';
}
template <typename T1, typename T2, typename... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
for (; a[i] != ',' && a[i] != '\0'; i++)
cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << "P(" << p.first << ", " << p.second << ")";
return os;
}
template <typename T, template <class> class C>
ostream &operator<<(ostream &os, const C<T> &v) {
os << "[";
for (auto d : v)
os << d << ", ";
os << "]";
return os;
}
#line 2 "Utility/fastio.hpp"
#include <unistd.h>
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
char num[10000][4];
constexpr Pre() : num() {
for (int i = 0; i < 10000; i++) {
int n = i;
for (int j = 3; j >= 0; j--) {
num[i][j] = n % 10 | '0';
n /= 10;
}
}
}
} constexpr pre;
inline void load() {
memmove(ibuf, ibuf + pil, pir - pil);
pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
pil = 0;
if (pir < SZ)
ibuf[pir++] = '\n';
}
inline void flush() {
fwrite(obuf, 1, por, stdout);
por = 0;
}
void rd(char &c) {
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
}
void rd(string &x) {
x.clear();
char c;
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
do {
x += c;
if (pil == pir)
load();
c = ibuf[pil++];
} while (!isspace(c));
}
template <typename T> void rd_real(T &x) {
string s;
rd(s);
x = stod(s);
}
template <typename T> void rd_integer(T &x) {
if (pil + 100 > pir)
load();
char c;
do
c = ibuf[pil++];
while (c < '-');
bool minus = 0;
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (c == '-') {
minus = 1, c = ibuf[pil++];
}
}
x = 0;
while ('0' <= c) {
x = x * 10 + (c & 15), c = ibuf[pil++];
}
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (minus)
x = -x;
}
}
void rd(int &x) {
rd_integer(x);
}
void rd(ll &x) {
rd_integer(x);
}
void rd(i128 &x) {
rd_integer(x);
}
void rd(uint &x) {
rd_integer(x);
}
void rd(ull &x) {
rd_integer(x);
}
void rd(u128 &x) {
rd_integer(x);
}
void rd(double &x) {
rd_real(x);
}
void rd(long double &x) {
rd_real(x);
}
template <class T, class U> void rd(pair<T, U> &p) {
return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T> void rd_tuple(T &t) {
if constexpr (N < std::tuple_size<T>::value) {
auto &x = std::get<N>(t);
rd(x);
rd_tuple<N + 1>(t);
}
}
template <class... T> void rd(tuple<T...> &tpl) {
rd_tuple(tpl);
}
template <size_t N = 0, typename T> void rd(array<T, N> &x) {
for (auto &d : x)
rd(d);
}
template <class T> void rd(vector<T> &x) {
for (auto &d : x)
rd(d);
}
void read() {}
template <class H, class... T> void read(H &h, T &...t) {
rd(h), read(t...);
}
void wt(const char c) {
if (por == SZ)
flush();
obuf[por++] = c;
}
void wt(const string s) {
for (char c : s)
wt(c);
}
void wt(const char *s) {
size_t len = strlen(s);
for (size_t i = 0; i < len; i++)
wt(s[i]);
}
template <typename T> void wt_integer(T x) {
if (por > SZ - 100)
flush();
if (x < 0) {
obuf[por++] = '-', x = -x;
}
int outi;
for (outi = 96; x >= 10000; outi -= 4) {
memcpy(out + outi, pre.num[x % 10000], 4);
x /= 10000;
}
if (x >= 1000) {
memcpy(obuf + por, pre.num[x], 4);
por += 4;
} else if (x >= 100) {
memcpy(obuf + por, pre.num[x] + 1, 3);
por += 3;
} else if (x >= 10) {
int q = (x * 103) >> 10;
obuf[por] = q | '0';
obuf[por + 1] = (x - q * 10) | '0';
por += 2;
} else
obuf[por++] = x | '0';
memcpy(obuf + por, out + outi + 4, 96 - outi);
por += 96 - outi;
}
template <typename T> void wt_real(T x) {
ostringstream oss;
oss << fixed << setprecision(15) << double(x);
string s = oss.str();
wt(s);
}
void wt(int x) {
wt_integer(x);
}
void wt(ll x) {
wt_integer(x);
}
void wt(i128 x) {
wt_integer(x);
}
void wt(uint x) {
wt_integer(x);
}
void wt(ull x) {
wt_integer(x);
}
void wt(u128 x) {
wt_integer(x);
}
void wt(double x) {
wt_real(x);
}
void wt(long double x) {
wt_real(x);
}
template <class T, class U> void wt(const pair<T, U> val) {
wt(val.first);
wt(' ');
wt(val.second);
}
template <size_t N = 0, typename T> void wt_tuple(const T t) {
if constexpr (N < std::tuple_size<T>::value) {
if constexpr (N > 0) {
wt(' ');
}
const auto x = std::get<N>(t);
wt(x);
wt_tuple<N + 1>(t);
}
}
template <class... T> void wt(tuple<T...> tpl) {
wt_tuple(tpl);
}
template <class T, size_t S> void wt(const array<T, S> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
template <class T> void wt(const vector<T> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
void print() {
wt('\n');
}
template <class Head, class... Tail> void print(Head &&head, Tail &&...tail) {
wt(head);
if (sizeof...(Tail))
wt(' ');
print(forward<Tail>(tail)...);
}
void __attribute__((destructor)) _d() {
flush();
}
} // namespace fastio
using fastio::flush;
using fastio::print;
using fastio::read;
inline void first(bool i = true) {
print(i ? "first" : "second");
}
inline void Alice(bool i = true) {
print(i ? "Alice" : "Bob");
}
inline void yes(bool i = true) {
print(i ? "yes" : "no");
}
inline void Yes(bool i = true) {
print(i ? "Yes" : "No");
}
inline void No() {
print("No");
}
inline void YES(bool i = true) {
print(i ? "YES" : "NO");
}
inline void NO() {
print("NO");
}
inline void Yay(bool i = true) {
print(i ? "Yay!" : ":(");
}
inline void Possible(bool i = true) {
print(i ? "Possible" : "Impossible");
}
inline void POSSIBLE(bool i = true) {
print(i ? "POSSIBLE" : "IMPOSSIBLE");
}
/**
* @brief Fast IO
*/
#line 6 "Verify/LC_multivariate_convolution_cyclic.test.cpp"
#line 2 "Math/fastdiv.hpp"
struct FastDiv{
using u64=uint64_t;
using u128=__uint128_t;
constexpr FastDiv():m(),s(),x(){}
constexpr FastDiv(int _m)
:m(_m),s(__lg(m-1)),x(((u128(1)<<(s+64))+m-1)/m){}
constexpr int get(){return m;}
constexpr friend u64 operator/(u64 n,const FastDiv& d){
return (u128(n)*d.x>>d.s)>>64;
}
constexpr friend int operator%(u64 n,const FastDiv& d){
return n-n/d*d.m;
}
constexpr pair<u64,int> divmod(u64 n)const{
u64 q=n/(*this);
return {q,n-q*m};
}
int m,s; u64 x;
};
/**
* @brief Fast Division
*/
#line 3 "Math/dynamic.hpp"
struct Fp{
using u64=uint64_t;
int v;
static int get_mod(){return _getmod();}
static void set_mod(int _m){bar=FastDiv(_m);}
Fp inv() const{
int tmp,a=v,b=get_mod(),x=1,y=0;
while(b){
tmp=a/b,a-=tmp*b;
swap(a,b);
x-=tmp*y;
swap(x,y);
}
if(x<0){x+=get_mod();}
return x;
}
Fp():v(0){}
Fp(ll x){
v=x%get_mod();
if(v<0)v+=get_mod();
}
Fp operator-()const{return Fp()-*this;}
Fp pow(ll t){
assert(t>=0);
Fp res=1,b=*this;
while(t){
if(t&1)res*=b;
b*=b;
t>>=1;
}
return res;
}
Fp& operator+=(const Fp& x){
v+=x.v;
if(v>=get_mod())v-=get_mod();
return *this;
}
Fp& operator-=(const Fp& x){
v+=get_mod()-x.v;
if(v>=get_mod())v-=get_mod();
return *this;
}
Fp& operator*=(const Fp& x){
v=(u64(v)*x.v)%bar;
return *this;
}
Fp& operator/=(const Fp& x){
(*this)*=x.inv();
return *this;
}
Fp operator+(const Fp& x)const{return Fp(*this)+=x;}
Fp operator-(const Fp& x)const{return Fp(*this)-=x;}
Fp operator*(const Fp& x)const{return Fp(*this)*=x;}
Fp operator/(const Fp& x)const{return Fp(*this)/=x;}
bool operator==(const Fp& x)const{return v==x.v;}
bool operator!=(const Fp& x)const{return v!=x.v;}
private:
static FastDiv bar;
static int _getmod(){return bar.get();}
};
FastDiv Fp::bar(998244353);
/**
* @brief Dynamic Modint
*/
#line 2 "Convolution/ntt.hpp"
template <typename T> struct NTT {
static constexpr int rank2 = __builtin_ctzll(T::get_mod() - 1);
std::array<T, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<T, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<T, std::max(0, rank2 - 2 + 1)> rate2;
std::array<T, std::max(0, rank2 - 2 + 1)> irate2;
std::array<T, std::max(0, rank2 - 3 + 1)> rate3;
std::array<T, std::max(0, rank2 - 3 + 1)> irate3;
NTT() {
T g = 2;
while (g.pow((T::get_mod() - 1) >> 1) == 1) {
g += 1;
}
root[rank2] = g.pow((T::get_mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
void ntt(std::vector<T> &a, bool type = 0) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
if (type) {
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
T irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(T::get_mod() + l.v - r.v) *
irot.v;
;
}
if (s + 1 != (1 << (len - 1)))
irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
T irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
T irot2 = irot * irot;
T irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].v;
auto a1 = 1ULL * a[i + offset + 1 * p].v;
auto a2 = 1ULL * a[i + offset + 2 * p].v;
auto a3 = 1ULL * a[i + offset + 3 * p].v;
auto a2na3iimag =
1ULL * T((T::get_mod() + a2 - a3) * iimag.v).v;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (T::get_mod() - a1) + a2na3iimag) *
irot.v;
a[i + offset + 2 * p] =
(a0 + a1 + (T::get_mod() - a2) +
(T::get_mod() - a3)) *
irot2.v;
a[i + offset + 3 * p] =
(a0 + (T::get_mod() - a1) +
(T::get_mod() - a2na3iimag)) *
irot3.v;
}
if (s + 1 != (1 << (len - 2)))
irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
}
len -= 2;
}
}
T e = T(n).inv();
for (auto &x : a)
x *= e;
} else {
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
T rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len))
rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
T rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
T rot2 = rot * rot;
T rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * T::get_mod() * T::get_mod();
auto a0 = 1ULL * a[i + offset].v;
auto a1 = 1ULL * a[i + offset + p].v * rot.v;
auto a2 = 1ULL * a[i + offset + 2 * p].v * rot2.v;
auto a3 = 1ULL * a[i + offset + 3 * p].v * rot3.v;
auto a1na3imag =
1ULL * T(a1 + mod2 - a3).v * imag.v;
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] =
a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] =
a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len))
rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
}
len += 2;
}
}
}
}
vector<T> mult(const vector<T> &a, const vector<T> &b) {
if (a.empty() or b.empty())
return vector<T>();
int as = a.size(), bs = b.size();
int n = as + bs - 1;
if (as <= 30 or bs <= 30) {
if (as > 30)
return mult(b, a);
vector<T> res(n);
rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
return res;
}
int m = 1;
while (m < n)
m <<= 1;
vector<T> res(m);
rep(i, 0, as) res[i] = a[i];
ntt(res);
if (a == b)
rep(i, 0, m) res[i] *= res[i];
else {
vector<T> c(m);
rep(i, 0, bs) c[i] = b[i];
ntt(c);
rep(i, 0, m) res[i] *= c[i];
}
ntt(res, 1);
res.resize(n);
return res;
}
};
/**
* @brief Number Theoretic Transform
*/
#line 2 "Math/modint.hpp"
template <unsigned mod = 1000000007> struct fp {
unsigned v;
static constexpr int get_mod() {
return mod;
}
constexpr unsigned inv() const {
assert(v != 0);
int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, p -= t * q;
tmp = x, x = y, y = tmp;
tmp = p, p = q, q = tmp;
}
if (p < 0)
p += mod;
return p;
}
constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
fp operator-() const {
return fp() - *this;
}
fp pow(ull t) {
fp res = 1, b = *this;
while (t) {
if (t & 1)
res *= b;
b *= b;
t >>= 1;
}
return res;
}
fp &operator+=(const fp &x) {
if ((v += x.v) >= mod)
v -= mod;
return *this;
}
fp &operator-=(const fp &x) {
if ((v += mod - x.v) >= mod)
v -= mod;
return *this;
}
fp &operator*=(const fp &x) {
v = ull(v) * x.v % mod;
return *this;
}
fp &operator/=(const fp &x) {
v = ull(v) * x.inv() % mod;
return *this;
}
fp operator+(const fp &x) const {
return fp(*this) += x;
}
fp operator-(const fp &x) const {
return fp(*this) -= x;
}
fp operator*(const fp &x) const {
return fp(*this) *= x;
}
fp operator/(const fp &x) const {
return fp(*this) /= x;
}
bool operator==(const fp &x) const {
return v == x.v;
}
bool operator!=(const fp &x) const {
return v != x.v;
}
friend istream &operator>>(istream &is, fp &x) {
return is >> x.v;
}
friend ostream &operator<<(ostream &os, const fp &x) {
return os << x.v;
}
};
template <unsigned mod> void rd(fp<mod> &x) {
fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
fastio::wt(x.v);
}
template <typename T> T Inv(ll n) {
static const int md = T::get_mod();
static vector<T> buf({0, 1});
assert(n > 0);
n %= md;
while (SZ(buf) <= n) {
int k = SZ(buf), q = (md + k - 1) / k;
buf.push_back(buf[k * q - md] * q);
}
return buf[n];
}
template <typename T> T Fact(ll n, bool inv = 0) {
static const int md = T::get_mod();
static vector<T> buf({1, 1}), ibuf({1, 1});
assert(n >= 0 and n < md);
while (SZ(buf) <= n) {
buf.push_back(buf.back() * SZ(buf));
ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
}
return inv ? ibuf[n] : buf[n];
}
template <typename T> T nPr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nHr(int n, int r, bool inv = 0) {
return nCr<T>(n + r - 1, r, inv);
}
/**
* @brief Modint
*/
#line 4 "Convolution/arbitrary.hpp"
using M1 = fp<1045430273>;
using M2 = fp<1051721729>;
using M3 = fp<1053818881>;
NTT<M1> N1;
NTT<M2> N2;
NTT<M3> N3;
constexpr int r_12 = M2(M1::get_mod()).inv();
constexpr int r_13 = M3(M1::get_mod()).inv();
constexpr int r_23 = M3(M2::get_mod()).inv();
constexpr int r_1323 = M3(ll(r_13) * r_23).v;
constexpr ll w1 = M1::get_mod();
constexpr ll w2 = ll(w1) * M2::get_mod();
template <typename T>
vector<T> ArbitraryMult(const vector<int> &a, const vector<int> &b) {
if (a.empty() or b.empty())
return vector<T>();
int n = a.size() + b.size() - 1;
vector<T> res(n);
if (min(a.size(), b.size()) <= 60) {
rep(i, 0, a.size()) rep(j, 0, b.size()) res[i + j] += T(a[i]) * b[j];
return res;
}
vector<int> vals[3];
vector<M1> a1(ALL(a)), b1(ALL(b)), c1 = N1.mult(a1, b1);
vector<M2> a2(ALL(a)), b2(ALL(b)), c2 = N2.mult(a2, b2);
vector<M3> a3(ALL(a)), b3(ALL(b)), c3 = N3.mult(a3, b3);
for (M1 x : c1)
vals[0].push_back(x.v);
for (M2 x : c2)
vals[1].push_back(x.v);
for (M3 x : c3)
vals[2].push_back(x.v);
rep(i, 0, n) {
ll p = vals[0][i];
ll q = (vals[1][i] + M2::get_mod() - p) * r_12 % M2::get_mod();
ll r = ((vals[2][i] + M3::get_mod() - p) * r_1323 +
(M3::get_mod() - q) * r_23) %
M3::get_mod();
res[i] = (T(r) * w2 + q * w1 + p);
}
return res;
}
template <typename T>
vector<T> ArbitraryMult(const vector<T> &a, const vector<T> &b) {
vector<int> A, B;
for (auto &x : a)
A.push_back(x.v);
for (auto &x : b)
B.push_back(x.v);
return ArbitraryMult<T>(A, B);
}
/**
* @brief Arbitrary Mod Convolution
*/
#line 2 "FPS/arbitraryfps.hpp"
template <typename T> struct Poly : vector<T> {
Poly(int n = 0) {
this->assign(n, T());
}
Poly(const initializer_list<T> f) : vector<T>::vector(f) {}
Poly(const vector<T> &f) {
this->assign(ALL(f));
}
int deg() const {
return this->size() - 1;
}
T eval(const T &x) {
T res;
for (int i = this->size() - 1; i >= 0; i--)
res *= x, res += this->at(i);
return res;
}
Poly rev() const {
Poly res = *this;
reverse(ALL(res));
return res;
}
void shrink() {
while (!this->empty() and this->back() == 0)
this->pop_back();
}
Poly operator>>(ll sz) const {
if ((int)this->size() <= sz)
return {};
Poly ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
Poly operator<<(ll sz) const {
Poly ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
Poly inv() const {
assert(this->front() != 0);
const int n = this->size();
Poly res(1);
res.front() = T(1) / this->front();
for (int k = 1; k < n; k <<= 1) {
Poly g = res, h = *this;
h.resize(k * 2);
res.resize(k * 2);
g = (g.square() * h);
g.resize(k * 2);
rep(i, k, min(k * 2, n)) res[i] -= g[i];
}
res.resize(n);
return res;
}
Poly square() const {
return Poly(mult(*this, *this));
}
Poly operator-() const {
return Poly() - *this;
}
Poly operator+(const Poly &g) const {
return Poly(*this) += g;
}
Poly operator+(const T &g) const {
return Poly(*this) += g;
}
Poly operator-(const Poly &g) const {
return Poly(*this) -= g;
}
Poly operator-(const T &g) const {
return Poly(*this) -= g;
}
Poly operator*(const Poly &g) const {
return Poly(*this) *= g;
}
Poly operator*(const T &g) const {
return Poly(*this) *= g;
}
Poly operator/(const Poly &g) const {
return Poly(*this) /= g;
}
Poly operator%(const Poly &g) const {
return Poly(*this) %= g;
}
pair<Poly, Poly> divmod(const Poly &g) const {
Poly q = *this / g, r = *this - g * q;
r.shrink();
return {q, r};
}
Poly &operator+=(const Poly &g) {
if (g.size() > this->size())
this->resize(g.size());
rep(i, 0, g.size()) {
(*this)[i] += g[i];
}
return *this;
}
Poly &operator+=(const T &g) {
if (this->empty())
this->push_back(0);
(*this)[0] += g;
return *this;
}
Poly &operator-=(const Poly &g) {
if (g.size() > this->size())
this->resize(g.size());
rep(i, 0, g.size()) {
(*this)[i] -= g[i];
}
return *this;
}
Poly &operator-=(const T &g) {
if (this->empty())
this->push_back(0);
(*this)[0] -= g;
return *this;
}
Poly &operator*=(const Poly &g) {
*this = mult(*this, g);
return *this;
}
Poly &operator*=(const T &g) {
rep(i, 0, this->size())(*this)[i] *= g;
return *this;
}
Poly &operator/=(const Poly &g) {
if (g.size() > this->size()) {
this->clear();
return *this;
}
Poly g2 = g;
reverse(ALL(*this));
reverse(ALL(g2));
int n = this->size() - g2.size() + 1;
this->resize(n);
g2.resize(n);
*this *= g2.inv();
this->resize(n);
reverse(ALL(*this));
shrink();
return *this;
}
Poly &operator%=(const Poly &g) {
*this -= *this / g * g;
shrink();
return *this;
}
Poly diff() const {
Poly res(this->size() - 1);
rep(i, 0, res.size()) res[i] = (*this)[i + 1] * (i + 1);
return res;
}
Poly inte() const {
Poly res(this->size() + 1);
for (int i = res.size() - 1; i; i--)
res[i] = (*this)[i - 1] / i;
return res;
}
Poly log() const {
assert(this->front() == 1);
const int n = this->size();
Poly res = diff() * inv();
res = res.inte();
res.resize(n);
return res;
}
Poly exp() const {
assert(this->front() == 0);
const int n = this->size();
Poly res(1), g(1);
res.front() = g.front() = 1;
for (int k = 1; k < n; k <<= 1) {
g = (g + g - g.square() * res);
g.resize(k);
Poly q = *this;
q.resize(k);
q = q.diff();
Poly w = (q + g * (res.diff() - res * q)), t = *this;
w.resize(k * 2 - 1);
t.resize(k * 2);
res = (res + res * (t - w.inte()));
res.resize(k * 2);
}
res.resize(n);
return res;
}
Poly shift(const int &c) const {
const int n = this->size();
Poly res = *this, g(n);
g[0] = 1;
rep(i, 1, n) g[i] = g[i - 1] * c / i;
vector<T> fact(n, 1);
rep(i, 0, n) {
if (i)
fact[i] = fact[i - 1] * i;
res[i] *= fact[i];
}
res = res.rev();
res *= g;
res.resize(n);
res = res.rev();
rep(i, 0, n) res[i] /= fact[i];
return res;
}
Poly pow(ll t) {
if (t == 0) {
Poly res(this->size());
res[0] = 1;
return res;
}
int n = this->size(), k = 0;
while (k < n and (*this)[k] == 0)
k++;
Poly res(n);
if (__int128_t(t) * k >= n)
return res;
n -= t * k;
Poly g(n);
T c = (*this)[k], ic = c.inv();
rep(i, 0, n) g[i] = (*this)[i + k] * ic;
g = g.log();
for (auto &x : g)
x *= t;
g = g.exp();
c = c.pow(t);
rep(i, 0, n) res[i + t * k] = g[i] * c;
return res;
}
vector<T> mult(const vector<T> &a, const vector<T> &b) const;
};
/**
* @brief Formal Power Series (Arbitrary mod)
*/
#line 10 "Verify/LC_multivariate_convolution_cyclic.test.cpp"
template <>
vector<Fp> Poly<Fp>::mult(const vector<Fp> &a, const vector<Fp> &b) const {
return ArbitraryMult<Fp>(a, b);
}
#line 2 "Math/miller.hpp"
struct m64 {
using i64 = int64_t;
using u64 = uint64_t;
using u128 = __uint128_t;
static u64 mod;
static u64 r;
static u64 n2;
static u64 get_r() {
u64 ret = mod;
rep(_,0,5) ret *= 2 - mod * ret;
return ret;
}
static void set_mod(u64 m) {
assert(m < (1LL << 62));
assert((m & 1) == 1);
mod = m;
n2 = -u128(m) % m;
r = get_r();
assert(r * mod == 1);
}
static u64 get_mod() { return mod; }
u64 a;
m64() : a(0) {}
m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){};
static u64 reduce(const u128 &b) {
return (b + u128(u64(b) * u64(-r)) * mod) >> 64;
}
u64 get() const {
u64 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
m64 &operator*=(const m64 &b) {
a = reduce(u128(a) * b.a);
return *this;
}
m64 operator*(const m64 &b) const { return m64(*this) *= b; }
bool operator==(const m64 &b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
bool operator!=(const m64 &b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
m64 pow(u128 n) const {
m64 ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
};
typename m64::u64 m64::mod, m64::r, m64::n2;
bool Miller(ll n){
if(n<2 or (n&1)==0)return (n==2);
m64::set_mod(n);
ll d=n-1; while((d&1)==0)d>>=1;
vector<ll> seeds;
if(n<(1<<30))seeds={2, 7, 61};
else seeds={2, 325, 9375, 28178, 450775, 9780504};
for(auto& x:seeds){
if(n<=x)break;
ll t=d;
m64 y=m64(x).pow(t);
while(t!=n-1 and y!=1 and y!=n-1){
y*=y;
t<<=1;
}
if(y!=n-1 and (t&1)==0)return 0;
} return 1;
}
/**
* @brief Miller-Rabin
*/
#line 2 "Utility/random.hpp"
namespace Random {
mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count());
using u64 = unsigned long long;
u64 get() {
return randgen();
}
template <typename T> T get(T L) { // [0,L]
return get() % (L + 1);
}
template <typename T> T get(T L, T R) { // [L,R]
return get(R - L) + L;
}
double uniform() {
return double(get(1000000000)) / 1000000000;
}
string str(int n) {
string ret;
rep(i, 0, n) ret += get('a', 'z');
return ret;
}
template <typename Iter> void shuffle(Iter first, Iter last) {
if (first == last)
return;
int len = 1;
for (auto it = first + 1; it != last; it++) {
len++;
int j = get(0, len - 1);
if (j != len - 1)
iter_swap(it, first + j);
}
}
template <typename T> vector<T> select(int n, T L, T R) { // [L,R]
if (n * 2 >= R - L + 1) {
vector<T> ret(R - L + 1);
iota(ALL(ret), L);
shuffle(ALL(ret));
ret.resize(n);
return ret;
} else {
unordered_set<T> used;
vector<T> ret;
while (SZ(used) < n) {
T x = get(L, R);
if (!used.count(x)) {
used.insert(x);
ret.push_back(x);
}
}
return ret;
}
}
void relabel(int n, vector<pair<int, int>> &es) {
shuffle(ALL(es));
vector<int> ord(n);
iota(ALL(ord), 0);
shuffle(ALL(ord));
for (auto &[u, v] : es)
u = ord[u], v = ord[v];
}
template <bool directed, bool simple> vector<pair<int, int>> genGraph(int n) {
vector<pair<int, int>> cand, es;
rep(u, 0, n) rep(v, 0, n) {
if (simple and u == v)
continue;
if (!directed and u > v)
continue;
cand.push_back({u, v});
}
int m = get(SZ(cand));
vector<int> ord;
if (simple)
ord = select(m, 0, SZ(cand) - 1);
else {
rep(_, 0, m) ord.push_back(get(SZ(cand) - 1));
}
for (auto &i : ord)
es.push_back(cand[i]);
relabel(n, es);
return es;
}
vector<pair<int, int>> genTree(int n) {
vector<pair<int, int>> es;
rep(i, 1, n) es.push_back({get(i - 1), i});
relabel(n, es);
return es;
}
}; // namespace Random
/**
* @brief Random
*/
#line 4 "Math/pollard.hpp"
vector<ll> Pollard(ll n) {
if (n <= 1)
return {};
if (Miller(n))
return {n};
if ((n & 1) == 0) {
vector<ll> v = Pollard(n >> 1);
v.push_back(2);
return v;
}
for (ll x = 2, y = 2, d;;) {
ll c = Random::get(2LL, n - 1);
do {
x = (__int128_t(x) * x + c) % n;
y = (__int128_t(y) * y + c) % n;
y = (__int128_t(y) * y + c) % n;
d = __gcd(x - y + n, n);
} while (d == 1);
if (d < n) {
vector<ll> lb = Pollard(d), rb = Pollard(n / d);
lb.insert(lb.end(), ALL(rb));
return lb;
}
}
}
/**
* @brief Pollard-Rho
*/
#line 4 "Math/primitive.hpp"
ll mpow(ll a, ll t, ll m) {
ll res = 1;
FastDiv im(m);
while (t) {
if (t & 1)
res = __int128_t(res) * a % im;
a = __int128_t(a) * a % im;
t >>= 1;
}
return res;
}
ll minv(ll a, ll m) {
ll b = m, u = 1, v = 0;
while (b) {
ll t = a / b;
a -= t * b;
swap(a, b);
u -= t * v;
swap(u, v);
}
u = (u % m + m) % m;
return u;
}
ll getPrimitiveRoot(ll p) {
vector<ll> ps = Pollard(p - 1);
sort(ALL(ps));
rep(x, 1, inf) {
for (auto &q : ps) {
if (mpow(x, (p - 1) / q, p) == 1)
goto fail;
}
return x;
fail:;
}
assert(0);
}
ll extgcd(ll a, ll b, ll &p, ll &q) {
if (b == 0) {
p = 1;
q = 0;
return a;
}
ll d = extgcd(b, a % b, q, p);
q -= a / b * p;
return d;
}
pair<ll, ll> crt(const vector<ll> &vs, const vector<ll> &ms) {
ll V = vs[0], M = ms[0];
rep(i, 1, vs.size()) {
ll p, q, v = vs[i], m = ms[i];
if (M < m)
swap(M, m), swap(V, v);
ll d = extgcd(M, m, p, q);
if ((v - V) % d != 0)
return {0, -1};
ll md = m / d, tmp = (v - V) / d % md * p % md;
V += M * tmp;
M *= md;
}
V = (V % M + M) % M;
return {V, M};
}
ll ModLog(ll a, ll b, ll p) {
ll g = 1;
for (ll t = p; t; t >>= 1)
g = g * a % p;
g = __gcd(g, p);
ll t = 1, c = 0;
for (; t % g; c++) {
if (t == b)
return c;
t = t * a % p;
}
if (b % g)
return -1;
t /= g, b /= g;
ll n = p / g, h = 0, gs = 1;
for (; h * h < n; h++)
gs = gs * a % n;
unordered_map<ll, ll> bs;
for (ll s = 0, e = b; s < h; bs[e] = ++s)
e = e * a % n;
for (ll s = 0, e = t; s < n;) {
e = e * gs % n, s += h;
if (bs.count(e)) {
return c + s - bs[e];
}
}
return -1;
}
ll mod_root(ll k, ll a, ll m) {
if (a == 0)
return k ? 0 : -1;
if (m == 2)
return a & 1;
k %= m - 1;
ll g = gcd(k, m - 1);
if (mpow(a, (m - 1) / g, m) != 1)
return -1;
a = mpow(a, minv(k / g, (m - 1) / g), m);
FastDiv im(m);
auto _subroot = [&](ll p, int e, ll a) -> ll { // x^(p^e)==a(mod m)
ll q = m - 1;
int s = 0;
while (q % p == 0) {
q /= p;
s++;
}
int d = s - e;
ll pe = mpow(p, e, m),
res = mpow(a, ((pe - 1) * minv(q, pe) % pe * q + 1) / pe, m), c = 1;
while (mpow(c, (m - 1) / p, m) == 1)
c++;
c = mpow(c, q, m);
map<ll, ll> mp;
ll v = 1, block = sqrt(d * p) + 1,
bs = mpow(c, mpow(p, s - 1, m - 1) * block % (m - 1), m);
rep(i, 0, block + 1) mp[v] = i, v = v * bs % im;
ll gs = minv(mpow(c, mpow(p, s - 1, m - 1), m), m);
rep(i, 0, d) {
ll err = a * minv(mpow(res, pe, m), m) % im;
ll pos = mpow(err, mpow(p, d - 1 - i, m - 1), m);
rep(j, 0, block + 1) {
if (mp.count(pos)) {
res = res *
mpow(c,
(block * mp[pos] + j) * mpow(p, i, m - 1) %
(m - 1),
m) %
im;
break;
}
pos = pos * gs % im;
}
}
return res;
};
for (ll d = 2; d * d <= g; d++)
if (g % d == 0) {
int sz = 0;
while (g % d == 0) {
g /= d;
sz++;
}
a = _subroot(d, sz, a);
}
if (g > 1)
a = _subroot(g, 1, a);
return a;
}
ull floor_root(ull a, ull k) {
if (a <= 1 or k == 1)
return a;
if (k >= 64)
return 1;
if (k == 2)
return sqrtl(a);
constexpr ull LIM = -1;
if (a == LIM)
a--;
auto mul = [&](ull &x, const ull &y) {
if (x <= LIM / y)
x *= y;
else
x = LIM;
};
auto pw = [&](ull x, ull t) -> ull {
ull y = 1;
while (t) {
if (t & 1)
mul(y, x);
mul(x, x);
t >>= 1;
}
return y;
};
ull ret = (k == 3 ? cbrt(a) - 1 : pow(a, nextafter(1 / double(k), 0)));
while (pw(ret + 1, k) <= a)
ret++;
return ret;
}
/**
* @brief Primitive Function
*/
#line 2 "FPS/multievalgeom.hpp"
template<typename T>vector<T> MultievalGeomSeq(vector<T>& f,T a,T w,int m){
int n=f.size();
vector<T> ret(m);
if(w==0){
T base=1;
rep(i,0,n)ret[0]+=base*f[i],base*=a;
rep(i,1,m)ret[i]=f[0];
return ret;
}
vector<T> tri(n+m-1),itri(n+m-1);
tri[0]=itri[0]=1;
T iw=w.inv(),pww=1,pwiw=1;
for(int i=1;i<n+m-1;i++,pww*=w,pwiw*=iw){
tri[i]=tri[i-1]*pww;
itri[i]=itri[i-1]*pwiw;
}
Poly<T> y(n),v(n+m-1);
T pwa=1;
for(int i=0;i<n;i++,pwa*=a){
y[i]=f[i]*itri[i]*pwa;
}
rep(i,0,n+m-1)v[i]=tri[i];
reverse(ALL(y));
y*=v;
rep(i,0,m)ret[i]=y[n-1+i]*itri[i];
return ret;
}
/**
* @brief Multipoint Evaluation on Geometric Sequence
*/
#line 4 "Convolution/multivariatecyclic.hpp"
template<typename T>vector<T> MultivariateCyclic
(vector<T> f,vector<T> g,vector<int>& a){
int MO=T::get_mod();
int k=a.size(),n=1;
for(auto& x:a)n*=x;
T pr=getPrimitiveRoot(MO),ipr=T(pr).inv();
int offset=1;
rep(x,0,k){
assert((MO-1)%a[x]==0);
T w=pr.pow((MO-1)/a[x]);
rep(i,0,n)if(i%(offset*a[x])<offset){
vector<T> na(a[x]),nb(a[x]);
rep(j,0,a[x]){
na[j]=f[i+offset*j];
nb[j]=g[i+offset*j];
}
na=MultievalGeomSeq(na,T(1),w,a[x]);
nb=MultievalGeomSeq(nb,T(1),w,a[x]);
rep(j,0,a[x]){
f[i+offset*j]=na[j];
g[i+offset*j]=nb[j];
}
}
offset*=a[x];
}
rep(i,0,n)f[i]*=g[i];
offset=1;
rep(x,0,k){
T iw=ipr.pow((MO-1)/a[x]);
rep(i,0,n)if(i%(offset*a[x])<offset){
vector<T> na(a[x]);
rep(j,0,a[x])na[j]=f[i+offset*j];
na=MultievalGeomSeq(na,T(1),iw,a[x]);
rep(j,0,a[x])f[i+offset*j]=na[j];
}
offset*=a[x];
}
T ninv=T(n).inv();
rep(i,0,n)f[i]*=ninv;
return f;
}
/**
* @brief Multivarate Convolution Cyclic
*/
#line 15 "Verify/LC_multivariate_convolution_cyclic.test.cpp"
int main() {
int p, k;
read(p, k);
Fp::set_mod(p);
vector<int> a(k);
read(a);
int n = 1;
for (auto &x : a)
n *= x;
vector<Fp> f(n), g(n);
rep(i, 0, n) read(f[i].v);
rep(i, 0, n) read(g[i].v);
auto ret = MultivariateCyclic(f, g, a);
rep(i, 0, n) print(ret[i].v);
return 0;
}