hdu 5895(矩阵快速幂+欧拉函数)

Posted 2020-08-11 AC菜鸟机

题目链接:http://acm.hdu.edu.cn/showproblem.php?pid=5895

f(n)=f(n-2)+2*f(n-1)

f(n)*f(n-1)=f(n-2)*f(n-1)+2*f(n-1)*f(n-1);

2*f(n-1)*f(n-1)=f(n)*f(n-1)-f(n-2)*f(n-1);

累加可得 g(n) = f(n)*f(n+1)/2

 

然后这个公式:A^x % m = A^(x%phi(m)+phi(m)) % m (x >= phi(m))

 

反正比赛没做出来.

#include <bits/stdc++.h>
#define LL long long
using namespace std;
struct Maxtri{
    LL v[2][2];
    Maxtri(){memset(v,0,sizeof(v));}
}ori;
LL n, y, x, s, mod ;
Maxtri mult(Maxtri a,Maxtri b){
    Maxtri temp;
    for(int i=0;i<2;i++){
        for(int j=0;j<2;j++){
            for(int k=0;k<2;k++){
                temp.v[i][j] = (temp.v[i][j]+(a.v[i][k]*b.v[k][j])%mod)%mod;
            }
        }
    }
    return temp;
}
LL pow_mod(Maxtri a,LL n){
    if(n==0) return 0;
    if(n==1) return 1;
    if(n==2) return 2;
    n-=2;
    Maxtri ans;
    for(int i=0;i<2;i++){
        ans.v[i][i] = 1;
    }
    while(n){
        if(n&1) ans = mult(ans,a);
        a = mult(a,a);
        n>>=1;
    }
    return (ans.v[0][0]*2+ans.v[0][1])%mod;
}
LL pow_mod1(LL a,LL n,LL mod){
    LL ans = 1;
    while(n){
        if(n&1) ans = ans*a%mod;
        a = a*a%mod;
        n>>=1;
    }
    return ans;
}
LL Phi(LL x)
{
    LL ans = x;
    for(LL i=2LL; i*i<=x; i++)
    {
        if(x % i == 0)
        {
            ans -= ans/i;
            while(x % i == 0)
                x /= i;
        }
    }
    if(x > 1)
        ans -= ans/x;
    return ans;
}
int main(){
    ori.v[0][0] = 2,ori.v[0][1] = 1;
    ori.v[1][0] = 1,ori.v[1][1] = 0;
    int tcase;
    scanf("%d",&tcase);
    while(tcase--){
        scanf("%lld%lld%lld%lld",&n, &y, &x, &s);
        s++;
        LL phi = 2*Phi(s);
        mod = 2*phi;
        LL fn = pow_mod(ori,n*y);
        LL fn1 = pow_mod(ori,n*y+1);
        LL ans = ((fn*fn1)%mod/2);
        ans+=phi;
        printf("%lld\n",pow_mod1(x,ans,s)%s);
    }
    return 0;
}