HDU 2814 斐波那契循环节 欧拉降幂

一看就是欧拉降幂，问题是怎么求$fib(a^b)$，C给的那么小显然还是要找循环节。数据范围出的很那啥..unsigned long long注意用防爆的乘法

 

/** @Date    : 2017-09-25 15:17:05
  * @FileName: HDU 2814 斐波那契循环节 欧拉降幂.cpp
  * @Platform: Windows
  * @Author  : Lweleth (SoungEarlf@gmail.com)
  * @Link    : https://github.com/
  * @Version : $Id$
  */
#include <bits/stdc++.h>
#define LL unsigned long long
#define PII pair<int ,int>
#define MP(x, y) make_pair((x),(y))
#define fi first
#define se second
#define PB(x) push_back((x))
#define MMG(x) memset((x), -1,sizeof(x))
#define MMF(x) memset((x),0,sizeof(x))
#define MMI(x) memset((x), INF, sizeof(x))
using namespace std;

const int INF = 0x3f3f3f3f;
const double eps = 1e-8;


LL mul(LL a, LL b, LL m) {
    LL ans = 0;
    while (b) {
        if (b & 1) {
            ans = (ans + a) % m;
            b--;
        }
        b >>= 1;
        a = (a + a) % m;
    }
    return ans;
}
/*    RERERERERE 精度
LL mul(LL x, LL y, LL mod)
{
    return (x * y - (LL)(x / (long double)mod * y + 1e-3) * mod + mod) % mod;
}*/
/*
struct Matrix
{
    LL m[M][M];
};

Matrix A;
Matrix I = {1, 0, 0, 1};

Matrix multi(Matrix a, Matrix b, LL MOD)
{
    Matrix c;
    for(int i = 0; i < M; i++)
    {
        for(int j = 0; j < M; j++)
        {
            c.m[i][j] = 0;
            for(int k = 0; k < M; k++)
                c.m[i][j] = (c.m[i][j] % MOD + (a.m[i][k] % MOD) * (b.m[k][j] % MOD) % MOD) % MOD;
            c.m[i][j] %= MOD;
        }
    }
    return c;
}

Matrix power(Matrix a, LL k, LL MOD)
{
    Matrix ans = I, p = a;
    while(k)
    {
        if(k & 1)
        {
            ans = multi(ans, p, MOD);
            k--;
        }
        k >>= 1;
        p = multi(p, p, MOD);
    }
    return ans;
}

LL gcd(LL a, LL b)
{
    return b ? gcd(b, a % b) : a;
}

const int N = 400005;
const int NN = 5005;

LL num[NN], pri[NN];
LL fac[NN];
int cnt, c;

bool prime[N];
int p[N];
int k;

void isprime()
{
    k = 0;
    memset(prime, true, sizeof(prime));
    for(int i = 2; i < N; i++)
    {
        if(prime[i])
        {
            p[k++] = i;
            for(int j = i + i; j < N; j += i)
                prime[j] = false;
        }
    }
}

LL fpow(LL a, LL b, LL m)
{
    LL ans = 1;
    a %= m;
    while(b)
    {
        if(b & 1)
        {
            ans = mul(ans,  a , m);
            b--;
        }
        b >>= 1;
        a = mul(a , a , m);
    }
    return ans;
}

LL legendre(LL a, LL p)
{
    if(fpow(a, (p - 1) >> 1, p) == 1)
        return 1;
    else return -1;
}

void Solve(LL n, LL pri[], LL num[])
{
    cnt = 0;
    LL t = (LL)sqrt(1.0 * n);
    for(int i = 0; p[i] <= t; i++)
    {
        if(n % p[i] == 0)
        {
            int a = 0;
            pri[cnt] = p[i];
            while(n % p[i] == 0)
            {
                a++;
                n /= p[i];
            }
            num[cnt] = a;
            cnt++;
        }
    }
    if(n > 1)
    {
        pri[cnt] = n;
        num[cnt] = 1;
        cnt++;
    }
}

void Work(LL n)
{
    c = 0;
    LL t = (LL)sqrt(1.0 * n);
    for(int i = 1; i <= t; i++)
    {
        if(n % i == 0)
        {
            if(i * i == n) fac[c++] = i;
            else
            {
                fac[c++] = i;
                fac[c++] = n / i;
            }
        }
    }
}

LL get_loop(LL n)
{
    Solve(n, pri, num);
    LL ans = 1;
    for(int i = 0; i < cnt; i++)
    {
        LL record = 1;
        if(pri[i] == 2)
            record = 3;
        else if(pri[i] == 3)
            record = 8;
        else if(pri[i] == 5)
            record = 20;
        else
        {
            if(legendre(5, pri[i]) == 1)
                Work(pri[i] - 1);
            else
                Work(2 * (pri[i] + 1));
            sort(fac, fac + c);
            for(int k = 0; k < c; k++)
            {
                Matrix a = power(A, fac[k] - 1, pri[i]);
                LL x = (a.m[0][0] % pri[i] + a.m[0][1] % pri[i]) % pri[i];
                LL y = (a.m[1][0] % pri[i] + a.m[1][1] % pri[i]) % pri[i];
                if(x == 1 && y == 0)
                {
                    record = fac[k];
                    break;
                }
            }
        }
        for(int k = 1; k < num[i]; k++)
            record *= pri[i];
        ans = ans / gcd(ans, record) * record;
    }
    return ans;
}

LL fib[5005];

void Init()
{
    A.m[0][0] = 1;
    A.m[0][1] = 1;
    A.m[1][0] = 1;
    A.m[1][1] = 0;
    fib[0] = 0;
    fib[1] = 1;
    for(int i = 2; i < 5005; i++)
        fib[i] = fib[i - 1] + fib[i - 2];
}*///会爆ULL C比较小300 不如直接求
LL fib[5500];

LL get_loop(LL n)
{
    LL lp = -1;
    fib[0] = 0, fib[1] = 1;
    for(int i = 2; i < 2010; i++)
    {
        fib[i] = (fib[i - 1] % n + fib[i - 2] % n) % n;
        if(fib[i] == 1 && fib[i - 1] == 0)
            return lp = i - 1;
    }
    return lp;
}
LL fpow(LL a, LL n, LL mod)
{
    LL res = 1;
    a %= mod;
    while(n)
    {
        if(n & 1)
            res = mul(res, a, mod);
        a = mul(a, a, mod);
        n >>= 1;
    }
    return res;
}

int get_phi(int x)
{
    int ans = x;
    for (int i = 2; i * i <= x; i++)
    {
        if (x % i == 0)
        {
            while (x % i == 0)
                x /= i;
            ans = ans - ans / i;
        }
    }
    if (x > 1)
        ans = ans - ans / x;
    return ans;
}

int main()
{
    LL n;
    //Init();
    //isprime();
    int T;
    cin >> T;
    int icas = 0;
    while(T--)
    {
        LL a, b, n, C;
        scanf("%llu%llu%llu%llu", &a, &b, &n, &C);
        if(C == 1)
        {
            printf("Case %d: 0\n", ++icas);
            continue;
        }
        int lp1 = get_loop(C);
        LL e1 = fpow(a, b, lp1);
        LL ans1 = fib[e1] % C;

        LL phi = get_phi(C);
        int lp2 = get_loop(phi);
        LL e2 = fpow(a, b, lp2);
        LL ans2 = fpow(fib[e2] % phi, n - 1, phi) + phi;

        LL ans = fpow(ans1, ans2, C);
        printf("Case %d: %llu\n", ++icas, ans);

    }
    return 0;
}