Miller–Rabin primality test

 

 

1 #include<iostream>    // The program is Goldbach's guess (want to output all combinations) 
2 #include<cmath>
 3 #include<cstdlib>
 4 #include<ctime>
 5 #include<cstdio>
 6  
7  using  namespace std;
 8  
9 typedef unsigned long  long ull;
 10 typedef unsigned long  long LL;
 11  
12 LL prime[ 6 ] = { 2 , 3 , 5 , 233 , 331 };
 13LL qmul(LL x, LL y, LL mod) { // Multiplication to prevent overflow, if p * p does not explode LL, it can be multiplied directly; O(1) multiplication or converted to binary addition
 14  // Fast multiplication modulo algorithm 
15  
16      return (x * y - ( long  long )(x / ( long  double )mod * y + 1e- 3 ) *mod + mod) % mod;
 17      /* 
18      LL ret = 0;
 19      while(y) {
 20          if(y & 1)
 21              ret = (ret + x) % mod;
 22          x = x * 2 % mod;
 23          y >>= 1;
 24      }
 25      return ret;
 26      */
27 }
28 
29 LL qpow(LL a, LL n, LL mod) {
30     LL ret = 1;
31     while(n) {
32         if(n & 1) ret = qmul(ret, a, mod);
33         a = qmul(a, a, mod);
34         n >>= 1;//n=n/2二进制乘除法
35     }
36     return ret;
37 }
38 
39 
40 bool Miller_Rabin(LL p) {
41     if(p < 2) return 0;
 42      if (p != 2 && p % 2 == 0 ) return  0 ;
 43      LL s = p - 1 ;
 44      while (! (s & 1 )) s >>= 1 ; // exclude even numbers 
45      for ( int i = 0 ; i < 5 ; ++ i) {
 46          if (p == prime[i]) return  1 ;
 47          LL t = s, m = qpow(prime[i], s, p);
 48          // Second detection theorem Carmichael number guarantees that the number is prime
49         //卡米歇尔数若一个数为合数当0<x<p,则方程x^p≡a(mod p)
50         //二次探测定理如果p是一个素数，0<x<p,则方程x^2≡1(mod p)的解为x=1,p-1
51         while(t != p - 1 && m != 1 && m != p - 1) {
52             m = qmul(m, m, p);
53             t <<= 1;
54         }
55         if(m != p - 1 && !(t & 1)) return 0;//不是奇数且m!=p-1
56     }
57     return 1;
58 }