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NTT_any_mod.cpp
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NTT_any_mod.cpp
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//
// 任意 mod の畳み込み
//
// verified:
// Yosupo Library Checker - Convolution (Mod 1,000,000,007)
// https://judge.yosupo.jp/problem/convolution_mod_1000000007
//
#include <bits/stdc++.h>
using namespace std;
// modint
template<int MOD> struct Fp {
// inner value
long long val;
// constructor
constexpr Fp() : val(0) { }
constexpr Fp(long long v) : val(v % MOD) {
if (val < 0) val += MOD;
}
constexpr long long get() const { return val; }
constexpr int get_mod() const { return MOD; }
// arithmetic operators
constexpr Fp operator + () const { return Fp(*this); }
constexpr Fp operator - () const { return Fp(0) - Fp(*this); }
constexpr Fp operator + (const Fp &r) const { return Fp(*this) += r; }
constexpr Fp operator - (const Fp &r) const { return Fp(*this) -= r; }
constexpr Fp operator * (const Fp &r) const { return Fp(*this) *= r; }
constexpr Fp operator / (const Fp &r) const { return Fp(*this) /= r; }
constexpr Fp& operator += (const Fp &r) {
val += r.val;
if (val >= MOD) val -= MOD;
return *this;
}
constexpr Fp& operator -= (const Fp &r) {
val -= r.val;
if (val < 0) val += MOD;
return *this;
}
constexpr Fp& operator *= (const Fp &r) {
val = val * r.val % MOD;
return *this;
}
constexpr Fp& operator /= (const Fp &r) {
long long a = r.val, b = MOD, u = 1, v = 0;
while (b) {
long long t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
val = val * u % MOD;
if (val < 0) val += MOD;
return *this;
}
constexpr Fp pow(long long n) const {
Fp res(1), mul(*this);
while (n > 0) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
constexpr Fp inv() const {
Fp res(1), div(*this);
return res / div;
}
// other operators
constexpr bool operator == (const Fp &r) const {
return this->val == r.val;
}
constexpr bool operator != (const Fp &r) const {
return this->val != r.val;
}
constexpr Fp& operator ++ () {
++val;
if (val >= MOD) val -= MOD;
return *this;
}
constexpr Fp& operator -- () {
if (val == 0) val += MOD;
--val;
return *this;
}
constexpr Fp operator ++ (int) const {
Fp res = *this;
++*this;
return res;
}
constexpr Fp operator -- (int) const {
Fp res = *this;
--*this;
return res;
}
friend constexpr istream& operator >> (istream &is, Fp<MOD> &x) {
is >> x.val;
x.val %= MOD;
if (x.val < 0) x.val += MOD;
return is;
}
friend constexpr ostream& operator << (ostream &os, const Fp<MOD> &x) {
return os << x.val;
}
friend constexpr Fp<MOD> pow(const Fp<MOD> &r, long long n) {
return r.pow(n);
}
friend constexpr Fp<MOD> inv(const Fp<MOD> &r) {
return r.inv();
}
};
namespace NTT {
long long modpow(long long a, long long n, int mod) {
long long res = 1;
while (n > 0) {
if (n & 1) res = res * a % mod;
a = a * a % mod;
n >>= 1;
}
return res;
}
long long modinv(long long a, int mod) {
long long b = mod, u = 1, v = 0;
while (b) {
long long t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
u %= mod;
if (u < 0) u += mod;
return u;
}
int calc_primitive_root(int mod) {
if (mod == 2) return 1;
if (mod == 167772161) return 3;
if (mod == 469762049) return 3;
if (mod == 754974721) return 11;
if (mod == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
long long x = (mod - 1) / 2;
while (x % 2 == 0) x /= 2;
for (long long i = 3; i * i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) x /= i;
}
}
if (x > 1) divs[cnt++] = x;
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (modpow(g, (mod - 1) / divs[i], mod) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
int get_fft_size(int N, int M) {
int size_a = 1, size_b = 1;
while (size_a < N) size_a <<= 1;
while (size_b < M) size_b <<= 1;
return max(size_a, size_b) << 1;
}
// number-theoretic transform
template<class mint> void trans(vector<mint> &v, bool inv = false) {
if (v.empty()) return;
int N = (int)v.size();
int MOD = v[0].get_mod();
int PR = calc_primitive_root(MOD);
static bool first = true;
static vector<long long> vbw(30), vibw(30);
if (first) {
first = false;
for (int k = 0; k < 30; ++k) {
vbw[k] = modpow(PR, (MOD - 1) >> (k + 1), MOD);
vibw[k] = modinv(vbw[k], MOD);
}
}
for (int i = 0, j = 1; j < N - 1; j++) {
for (int k = N >> 1; k > (i ^= k); k >>= 1);
if (i > j) swap(v[i], v[j]);
}
for (int k = 0, t = 2; t <= N; ++k, t <<= 1) {
long long bw = vbw[k];
if (inv) bw = vibw[k];
for (int i = 0; i < N; i += t) {
mint w = 1;
for (int j = 0; j < t/2; ++j) {
int j1 = i + j, j2 = i + j + t/2;
mint c1 = v[j1], c2 = v[j2] * w;
v[j1] = c1 + c2;
v[j2] = c1 - c2;
w *= bw;
}
}
}
if (inv) {
long long invN = modinv(N, MOD);
for (int i = 0; i < N; ++i) v[i] = v[i] * invN;
}
}
// for garner
static constexpr int MOD0 = 754974721;
static constexpr int MOD1 = 167772161;
static constexpr int MOD2 = 469762049;
using mint0 = Fp<MOD0>;
using mint1 = Fp<MOD1>;
using mint2 = Fp<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
// small case (T = mint, long long)
template<class T> vector<T> naive_mul(const vector<T> &A, const vector<T> &B) {
if (A.empty() || B.empty()) return {};
int N = (int)A.size(), M = (int)B.size();
vector<T> res(N + M - 1);
for (int i = 0; i < N; ++i)
for (int j = 0; j < M; ++j)
res[i + j] += A[i] * B[j];
return res;
}
// mul by convolution
template<class mint> vector<mint> mul(const vector<mint> &A, const vector<mint> &B) {
if (A.empty() || B.empty()) return {};
int N = (int)A.size(), M = (int)B.size();
if (min(N, M) < 30) return naive_mul(A, B);
int MOD = A[0].get_mod();
int size_fft = get_fft_size(N, M);
if (MOD == 998244353) {
vector<mint> a(size_fft), b(size_fft), c(size_fft);
for (int i = 0; i < N; ++i) a[i] = A[i];
for (int i = 0; i < M; ++i) b[i] = B[i];
trans(a), trans(b);
vector<mint> res(size_fft);
for (int i = 0; i < size_fft; ++i) res[i] = a[i] * b[i];
trans(res, true);
res.resize(N + M - 1);
return res;
}
vector<mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0);
vector<mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0);
vector<mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0);
for (int i = 0; i < N; ++i)
a0[i] = A[i].val, a1[i] = A[i].val, a2[i] = A[i].val;
for (int i = 0; i < M; ++i)
b0[i] = B[i].val, b1[i] = B[i].val, b2[i] = B[i].val;
trans(a0), trans(a1), trans(a2), trans(b0), trans(b1), trans(b2);
for (int i = 0; i < size_fft; ++i) {
c0[i] = a0[i] * b0[i];
c1[i] = a1[i] * b1[i];
c2[i] = a2[i] * b2[i];
}
trans(c0, true), trans(c1, true), trans(c2, true);
mint mod0 = MOD0, mod01 = mod0 * MOD1;
vector<mint> res(N + M - 1);
for (int i = 0; i < N + M - 1; ++i) {
int y0 = c0[i].val;
int y1 = (imod0 * (c1[i] - y0)).val;
int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val;
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
};
//------------------------------//
// Examples
//------------------------------//
void Yosupo_Convolution_mod_1000000007() {
const int MOD = 1000000007;
using mint = Fp<MOD>;
int N, M;
cin >> N >> M;
vector<mint> a(N), b(M);
for (int i = 0; i < N; ++i) cin >> a[i];
for (int i = 0; i < M; ++i) cin >> b[i];
vector<mint> res = NTT::mul(a, b);
for (auto v : res) cout << v << " ";
cout << endl;
}
int main() {
Yosupo_Convolution_mod_1000000007();
}