素数判定Miller_Rabin 算法详解

素数判定Miller_Rabin 算法详解 

例如：

Goldbach

#include
using namespace std;
unsigned long long n;
const int times = 5;
int number = 0;

unsigned long long Random( unsigned long long n )         //生成[ 0 , n ]的随机数
{
    return ((double)rand( ) / RAND_MAX*n + 0.5);
}

unsigned long long q_mul( unsigned long long a, unsigned long long b, unsigned long long mod ) //快速计算 (a*b) % mod
{
    unsigned long long ans = 0;
    while(b)
    {
        if(b & 1)
        {
            b--;
            ans =(ans+ a)%mod;
        }
        b /= 2;
        a = (a + a) % mod;

    }
    return ans;
}

unsigned long long q_pow( unsigned long long a, unsigned long long b, unsigned long long mod ) //快速计算 (a^b) % mod
{
    unsigned long long ans = 1;
    while(b)
    {
        if(b & 1)
        {
            ans = q_mul( ans, a, mod );
        }
        b /= 2;
        a = q_mul( a, a, mod );
    }
    return ans;
}

bool witness( unsigned long long a, unsigned long long n )//miller_rabin算法的精华
{//用检验算子a来检验n是不是素数
    unsigned long long tem = n - 1;
    int j = 0;
    while(tem % 2 == 0)
    {
        tem /= 2;
        j++;
    }
    //将n-1拆分为a^r * s

    unsigned long long x = q_pow( a, tem, n ); //得到a^r mod n
    if(x == 1 || x == n - 1) return true;   //余数为1则为素数
    while(j--) //否则试验条件2看是否有满足的 j
    {
        x = q_mul( x, x, n );
        if(x == n - 1) return true;
    }
    return false;
}

bool miller_rabin( unsigned long long n )  //检验n是否是素数
{

    if(n == 2)
        return true;
    if(n < 2 || n % 2 == 0)
        return false;               //如果是2则是素数，如果<2或者是>2的偶数则不是素数

    for(int i = 1; i <= times; i++)  //做times次随机检验
    {
        unsigned long long a = Random( n - 2 ) + 1; //得到随机检验算子 a
        if(!witness( a, n ))                        //用a检验n是否是素数
            return false;
    }
    return true;
}
int main(){
	int T;
	scanf("%d",&T);
	while(T--){
		int a,b,c;
		char ch;
		cin>>n;
		if(n==2){
			cout<<1<<" "<<1<