[洛谷P4245]【模板】任意模数NTT

题目大意：给你两个多项式 $f(x)$ 和 $g(x)$ 以及一个模数 $p(p\leqslant10^9)$ ，求 $f*g\pmod p$

任意模数 $NTT$ ，最大的数为 $p^2\times\max\{n,m\}\leqslant10^{23}$ ，所以一般选 $3$ 个模数即可，求出这三个模数下的答案，然后中国剩余定理即可。

假设这一位的答案是 $x$ ，三个模数分别为 $A,B,C$ ，那么：

$x\equiv x_1\pmod{A}\\ x\equiv x_2\pmod{B}\\ x\equiv x_3\pmod{C}$

先把前两个合并：

$x_1+k_1A=x_2+k_2B\\ x_1+k_1A\equiv x_2\pmod{B}\\ k_1\equiv \frac{x_2-x_1}A\pmod{B}$

于是求出了 $k_1$ ，也就求出了 $x\equiv x_1+k_1A\pmod{AB}$ ，记 $x_4=x_1+k_1A$

$x_4+k_4AB=x_3+k_3C\\ x_4+k_4AB\equiv x_3\pmod{C}\\ k_4\equiv \dfrac{x_3-x_4}{AB}\pmod{C}$

求出了 $k_4$ ， $x\equiv x_4+k_4AB\pmod{ABC}$ ，因为 $x<ABC$ ，所以 $x=x_4+k_4AB$

$Wn$ 数组开小，中国剩余定理写错

C++ Code：

#include <algorithm>
#include <cstdio>
#include <cstring>
int mod;
namespace Math {
    inline int pw(int base, int p, const int mod) {
        static int res;
        for (res = 1; p; p >>= 1, base = static_cast<long long> (base) * base % mod) if (p & 1) res = static_cast<long long> (res) * base % mod;
        return res;
    }
    inline int inv(int x, const int mod) { return pw(x, mod - 2, mod); }
}
 
const int mod1 = 998244353, mod2 = 1004535809, mod3 = 469762049, G = 3;
const long long mod_1_2 = static_cast<long long> (mod1) * mod2;
const int inv_1 = Math::inv(mod1, mod2), inv_2 = Math::inv(mod_1_2 % mod3, mod3);
struct Int {
    int A, B, C;
    explicit inline Int() { }
    explicit inline Int(int __num) : A(__num), B(__num), C(__num) { }
    explicit inline Int(int __A, int __B, int __C) : A(__A), B(__B), C(__C) { }
    static inline Int reduce(const Int &x) {
        return Int(x.A + (x.A >> 31 & mod1), x.B + (x.B >> 31 & mod2), x.C + (x.C >> 31 & mod3));
    }
    inline friend Int operator + (const Int &lhs, const Int &rhs) {
        return reduce(Int(lhs.A + rhs.A - mod1, lhs.B + rhs.B - mod2, lhs.C + rhs.C - mod3));
    }
    inline friend Int operator - (const Int &lhs, const Int &rhs) {
        return reduce(Int(lhs.A - rhs.A, lhs.B - rhs.B, lhs.C - rhs.C));
    }
    inline friend Int operator * (const Int &lhs, const Int &rhs) {
        return Int(static_cast<long long> (lhs.A) * rhs.A % mod1, static_cast<long long> (lhs.B) * rhs.B % mod2, static_cast<long long> (lhs.C) * rhs.C % mod3);
    }
    inline int get() {
        long long x = static_cast<long long> (B - A + mod2) % mod2 * inv_1 % mod2 * mod1 + A;
        return (static_cast<long long> (C - x % mod3 + mod3) % mod3 * inv_2 % mod3 * (mod_1_2 % mod) % mod + x) % mod;
    }
} ;
 
#define maxn 131072
 
namespace Poly {
#define N (maxn << 1)
    int lim, s, rev[N];
    Int Wn[N | 1];
    inline void init(int n) {
        s = -1, lim = 1; while (lim < n) lim <<= 1, ++s;
        for (register int i = 1; i < lim; ++i) rev[i] = rev[i >> 1] >> 1 | (i & 1) << s;
        const Int t(Math::pw(G, (mod1 - 1) / lim, mod1), Math::pw(G, (mod2 - 1) / lim, mod2), Math::pw(G, (mod3 - 1) / lim, mod3));
        *Wn = Int(1); for (register Int *i = Wn; i != Wn + lim; ++i) *(i + 1) = *i * t;
    }
    inline void NTT(Int *A, const int op = 1) {
        for (register int i = 1; i < lim; ++i) if (i < rev[i]) std::swap(A[i], A[rev[i]]);
        for (register int mid = 1; mid < lim; mid <<= 1) {
            const int t = lim / mid >> 1;
            for (register int i = 0; i < lim; i += mid << 1) {
                for (register int j = 0; j < mid; ++j) {
                    const Int W = op ? Wn[t * j] : Wn[lim - t * j];
                    const Int X = A[i + j], Y = A[i + j + mid] * W;
                    A[i + j] = X + Y, A[i + j + mid] = X - Y;
                }
            }
        }
        if (!op) {
            const Int ilim(Math::inv(lim, mod1), Math::inv(lim, mod2), Math::inv(lim, mod3));
            for (register Int *i = A; i != A + lim; ++i) *i = (*i) * ilim;
        }
    }
#undef N
}
 
int n, m;
Int A[maxn << 1], B[maxn << 1];
int main() {
    scanf("%d%d%d", &n, &m, &mod); ++n, ++m;
    for (int i = 0, x; i < n; ++i) scanf("%d", &x), A[i] = Int(x % mod);
    for (int i = 0, x; i < m; ++i) scanf("%d", &x), B[i] = Int(x % mod);
    Poly::init(n + m);
    Poly::NTT(A), Poly::NTT(B);
    for (int i = 0; i < Poly::lim; ++i) A[i] = A[i] * B[i];
    Poly::NTT(A, 0);
    for (int i = 0; i < n + m - 1; ++i) {
        printf("%d", A[i].get());
        putchar(i == n + m - 2 ? '\n' : ' ');
    }
    return 0;
}