快速幂
O(logk)

ll qmi(ll m, ll k, ll p)//求m^k%p,p>1e9用龟速乘
{
    ll res = 1;
    while (k)
    {
        if (k & 1)
        {
            res = res * m % p;
        }
        m = m * m % p;
        k >>= 1;
    }
    return res;
}

快速幂求逆元
O(logp)

ll a, p;//求a%p的乘法逆元,p为质数
cin >> a >> p;
if(a%p)
{
    cout << qmi(a, p-2, p) << endl;
}
else
{
    cout<<"impossible"<<endl;
}

矩阵快速幂
O(logn)

int n, m;//n为次数,m为模数

void mul(int c[], int a[], int b[][N])
{
    int temp[N] = {0};
    for (int i = 0; i < N; i++)
    {
        for (int j = 0; j < N; j++)
        {
            temp[i] = (temp[i] + (ll)a[j] * b[j][i]) % m;
        }
    }
    memcpy(c, temp, sizeof temp);
}

void mul(int c[][N], int a[][N], int b[][N])
{
    int temp[N][N] = {0};
    for (int i = 0; i < N; i++)
    {
        for (int j = 0; j < N; j++)
        {
            for (int k = 0; k < N; k++)
            {
                temp[i][j] = (temp[i][j] + (ll)a[i][k] * b[k][j]) % m;
            }
        }
    }
    memcpy(c, temp, sizeof temp);
}

int main()
{
    std::ios::sync_with_stdio(false), std::cin.tie(0), std::cout.tie(0);
    cin >> n >> m;
    int res[N] = {1, 1, 1}; //初始化
    int a[N][N] = {         //初始化
        {0, 1, 0},
        {1, 1, 1},
        {0, 0, 1}
    };
    n--;
    while (n)
    {
        if (n & 1)
        {
            mul(res, res, a);
        }
        mul(a, a, a);
        n >>= 1;
    }
    cout << res[2] << endl;//输出对应结果
    return 0;
}

龟速乘

ll qmul(ll a, ll b, ll p)
{
    ll sum = 0;
    while (b)
    {
        if (b & 1)
        {
            sum = (sum + a) % p;
        }
        a = (a + a) % p;
        b >>= 1;
    }
    return sum;
}

ll qmi(ll m, ll k, ll p) //求m^k%p
{
    ll res = 1;
    while (k)
    {
        if (k & 1)
        {
            res = qmul(res, m, p);
        }
        m = qmul(m, m, p);
        k >>= 1;
    }
    return res;
}