二次剩余

参考资料。

类似fft，自定义一种复数，使得第二维的平方等于w。

然后就可以进行SAO操作了。

注意sgn要返回long long

例题：Timus Online Judge 1132

#include <cstdio>
#include <algorithm>
#include <climits>

typedef long long LL;
const int N = 100010;

LL w, MO;

inline int rd() {
    return (rand() * (1ll << 16) + rand()) % LONG_MAX;
}

struct cp {
    LL x, y;
    cp(LL X = 0, LL Y = 0) {
        x = X;
        y = Y;
    }
    inline cp operator *(const cp &W) const {
        return cp(x * W.x % MO + y * W.y % MO * w % MO, x * W.y % MO + y * W.x % MO);
    }
    inline cp operator %(const LL &W) const {
        return cp(x % W, y % W);
    }
};

template <class T> inline T qpow(T a, LL b) {
    T ans(1);
    while(b) {
        if(b & 1) ans = ans * a % MO;
        a = a * a % MO;
        b = b >> 1;
    }
    return ans;
}

inline int sgn(LL x) {
    return qpow(x, (MO - 1) >> 1);
}

inline void solve() {
    LL n;
    scanf("%lld%lld", &n, &MO);
    n %= MO;
    if(MO == 2) {
        printf("%d
", n);
        return;
    }
    if(sgn(n) == MO - 1) {
        printf("No root
");
        return;
    }
    LL a;
    while(1) {
        a = rd() % MO;
        w = (a * a % MO - n + MO) % MO;
        if(sgn(w) == MO - 1) break;
    }
    cp ans(a, 1);
    ans = qpow(ans, (MO + 1) >> 1);
    LL b = ans.x, c = MO - b;
    if(b > c) std::swap(b, c);
    printf("%lld %lld 
", b, c);
    return;
}

int main() {
    srand(19384);
    int T;
    scanf("%d", &T);
    while(T--) solve();
    return 0;
}