【算法编程】基于Miller-Rabin的大素数测试

基本原理：

费尔马小定理:如果p是一个素数,且0<a<p,则a^(p-1)%p=1.
       利用费尔马小定理,对于给定的整数n,可以设计素数判定算法,通过计算d=a^(n-1)%n来判断n的素性,当d!=1时,n肯定不是素数,当d=1时,n  很可能是素数.

二次探测定理:如果p是一个素数,且0<x<p,则方程x^2%p=1的解为:x=1或x=p-1.
       利用二次探测定理,可以再利用费尔马小定理计算a^(n-1)%n的过程中增加对整数n的二次探测,一旦发现违背二次探测条件,即得出n不是素数的结论.
    
    如果n是素数,则(n-1)必是偶数,因此可令(n-1)=m*(2^q),其中m是正奇数(若n是偶数,则上面的m*(2^q)一定可以分解成一个正奇数乘以2的k次方的形式 ),q是非负整数,考察下面的测试:
    序列:
         a^m%n; a^(2m)%n; a^(4m)%n; ……;a^(m*2^q)%n

Miller-Rabin素性测试伪代码描述：

1、找出整数k,q,其中k>0,q是奇数，使（n-1=2kq）。

2、随机选取整数a,1<a<n-1。

3、Ifaq mod n=1, printf("该数可能是素数!\n");

4、Forj=0 to k-1 , if a^(2^j*q) mod n = n – 1, printf("该数可能是素数!\n");如果步骤3、4都不成立，则printf("该数肯定不是素数!\n")

5、当该数可能是素数时，随机选取整数a,1<a<n-1。若多次都表明可能是素数，则我们有理由相信该数是素数。


具体代码实现： 

1.、BigInt.h文件

#ifndef _BIGNUM_H_#define _BIGNUM_H_#define SIZE                128  //一个大整数用个字节保存，最多表示位大整数#define SIZE_2                   2* SIZEtypedef unsigned char       UCHAR;typedef unsigned short      USHORT;UCHAR atox(char ch);  //将一个十六进制的字符(4位)转位数字，转换失败返回xff typedef struct BigNum  //大整数结构{UCHAR data[SIZE];  //空间为(SIZE * sizeof(UCHAR))，就是SIZE个字节}BigNum;BigNum Init(char *str);  //初始化大整数，str为十六进制字符串 int GetBitFront(BigNum bignum); //有多少bit (前面的0不算)int GetBitEnd(BigNum bignum);  //有多少0(即只算末尾的0个数) BigNum MovBitLetf(BigNum bignum, int n);//向左移n位BigNum MovBitRight(BigNum bignum, int n);  //右移n位 int Cmp(BigNum bignum_a, BigNum bignum_b);  //大整数比较大小，>返回1，<返回-1，==返回0BigNum Mod(BigNum bignum_a, BigNum bignum_b);  //大整数模运算BigNum Sub(BigNum bignum_a, BigNum bignum_b);  //大整数减法void print2(BigNum bignum); //以二进制打印BigNum Mul(BigNum bignum_a, BigNum bignum_b);  //大整数乘法BigNum Div(BigNum bignum_a, BigNum bignum_b);  //大整数除法BigNum Add(BigNum bignum_a, BigNum bignum_b);  //大整数加法void print10(BigNum bignum);//以十进制打印int b2d(BigNum bignum);  //二进制到转十进制BigNum modMDyn(BigNum a, BigNum power, BigNum mod); //求大整数幂的模BigNum d2b(int num); //十进制转二进制int checkprime(BigNum n,BigNum a);#endif

2. BigInt.c文件：

 #include <stdio.h>#include<stdlib.h>#include <string.h>#include "BigInt.h"#include "math.h"#include<time.h>void print2(BigNum bignum)//以二进制打印{  if(GetBitFront(bignum)==0)        printf("0\n");else{        for(int i=SIZE-GetBitFront(bignum);i<SIZE;i++)        {        printf("%c",bignum.data[i]);        }         printf("\n");}} BigNum Init(char *str)  //高位在0{int j=0;BigNum bignum;for(inti=SIZE-int(strlen(str));i<SIZE;i++){        bignum.data[i]=str[j];        j++;}for(i=SIZE-int(strlen(str))-1;i>=0;i--)        bignum.data[i]='0';return bignum;} int GetBitFront(BigNum bignum)  //有多少bit(前面的0不算){int BitOfBigNum = SIZE;int i=0;while ((bignum.data[i] == '0')&& (BitOfBigNum > 0)){        i++;        BitOfBigNum--;}return BitOfBigNum;} int GetBitEnd(BigNum bignum)  //有多少0(即只算末尾的0个数){int BitOfBigNum = SIZE;int num=0;while ((bignum.data[BitOfBigNum -1] == '0') && (BitOfBigNum > 0)){        num++;        BitOfBigNum--;}return num;} BigNum MovBitLetf(BigNum bignum, int n)//向左移n位{int bignum_len =GetBitFront(bignum);for (int i =SIZE- bignum_len; i<SIZE; i++){             if (i - n < 0)        {               printf("ok\n");               continue;        }        bignum.data[i - n] =bignum.data[i];} for (i = SIZE- n; i <SIZE; i++){        bignum.data[i] ='0';}return bignum;} BigNum MovBitRight(BigNum bignum, int n) //右移n位{int bignum_len =GetBitFront(bignum);for (int i = SIZE - 1; i >=SIZE-bignum_len; i--){        if (i<0)        {               continue;        }        bignum.data[i] =bignum.data[i-n];}for (i =0; i <SIZE-bignum_len;i++){        bignum.data[i] = '0';}return bignum;} int Cmp(BigNum bignum_a, BigNum bignum_b)  //大整数比较大小，>返回1，<返回-1，==返回0{int bignum_a_len =GetBitFront(bignum_a);int bignum_b_len =GetBitFront(bignum_b);if(bignum_a_len>bignum_b_len)return 1;if(bignum_a_len<bignum_b_len)return -1;if(bignum_a_len=bignum_b_len){        int max = bignum_a_len;        for (int i =SIZE-max; i<SIZE; i++)        {               if (bignum_a.data[i]> bignum_b.data[i])               {                      return 1;               }               if (bignum_a.data[i]< bignum_b.data[i])               {                      return -1;               }        }}return 0;} BigNum Sub(BigNum bignum_a, BigNum bignum_b)  //大整数减法{BigNum bignum_c;int temp=0;int temp1=0;int carry = 0;int i;int j=0;for (i = SIZE-1; i >=0; i--){        temp = bignum_a.data[i] -bignum_b.data[i] -carry;        temp1=temp;        if(temp==-1)               temp1=1;        if(temp==-2)               temp1=0;             bignum_c.data[i] =temp1+48;        if(temp<0)        carry=1;        else        carry=0;        j++;}return bignum_c;} BigNum Mod(BigNum bignum_a, BigNum bignum_b)  //大整数模运算{BigNum bignum_c =Init("0");BigNum B;B = bignum_b;int bignum_a_len;int bignum_b_len;int bignum_c_len;if (Cmp(bignum_b, bignum_c) == 0){        printf("错误！除数为\n");        return bignum_c;}bignum_a_len =GetBitFront(bignum_a);bignum_b_len =GetBitFront(bignum_b);bignum_c_len = bignum_a_len -bignum_b_len; while (bignum_c_len >= 0){          B = MovBitLetf(bignum_b,bignum_c_len);        int m=0;        m=Cmp(bignum_a, B);        while (Cmp(bignum_a, B) !=-1)//大于等于        {               bignum_a =Sub(bignum_a, B);        }        bignum_c_len--; }     return bignum_a;} BigNum Mul(BigNum bignum_a, BigNum bignum_b)  //大整数乘法{BigNum bignum_c =Init("0");BigNum bignum=Init("0");int wei=0;wei=GetBitFront(bignum_a)+GetBitFront(bignum_b)-1; int carry[SIZE_2];int carry1[SIZE_2];int mod[SIZE_2];for(int k=0;k<=SIZE_2;k++){        carry[k]=0;        carry1[k]=0;        mod[k]=0;} int i=0;int j=0;for(i=SIZE-1;i>=0;i--){        for(j=SIZE-1;j>=0;j--)          carry[i+j+1]=(bignum_a.data[i]-48)*(bignum_b.data[j]-48)+carry[i+j+1];}        for(k=SIZE_2-1;k>=0;k--)        {          if(k==SIZE_2-1)                 carry1[k]=carry[k];            else                 carry1[k]=carry1[k+1]/2+carry[k];        }                 wei=GetBitFront(bignum_a)+GetBitFront(bignum_b)-1;        bignum=d2b(carry1[SIZE_2-wei]);        for(i=SIZE-1,j=SIZE_2-wei;i>=0&&j>=0;i--,j--)              carry1[j]=bignum.data[i]-48;        for(k=0;k<SIZE_2;k++)        {          if(carry1[k]!=0)                 break;                 }        for(i=SIZE-1,j=SIZE_2-1;j>=k;i--,j--)        {               bignum_c.data[i]=carry1[j]%2+48;        } return bignum_c;} BigNum Div(BigNum bignum_a, BigNum bignum_b)  //大整数除法{BigNum bignum_c =Init("0");BigNum B;int bignum_a_len;int bignum_b_len;int bignum_c_len;if (Cmp(bignum_b, bignum_c) == 0){        printf("错误！除数为\n");        return bignum_c;}bignum_a_len =GetBitFront(bignum_a);bignum_b_len = GetBitFront(bignum_b);bignum_c_len = bignum_a_len -bignum_b_len;while (bignum_c_len >= 0){        B = MovBitLetf(bignum_b,bignum_c_len);        while (Cmp(bignum_a, B) !=-1)        {               bignum_a =Sub(bignum_a, B);               bignum_c.data[SIZE-1-bignum_c_len]++;        }        bignum_c_len--;}return bignum_c;} BigNum Add(BigNum bignum_a, BigNum bignum_b)  //大整数加法{BigNum bignum_c;int temp;int carry = 0;int i;for (i = SIZE-1; i>=0; i--){        temp = bignum_a.data[i]-48+ bignum_b.data[i]-48 + carry;               if(temp==2)        {               temp=0;            carry=1;        }        else if(temp==3)        {               temp=1;               carry=1;        }        else carry=0;        bignum_c.data[i] = temp+48;}return bignum_c;} int b2d(BigNum bignum) //二进制转十进制{int n=0;int j=0;int result=0;n=GetBitFront(bignum);    for(int i=SIZE-1;i>=0;i--){        result=result+(bignum.data[i]-48)*pow(2,j);        j++;}return result;} void print10(BigNum bignum)  //打印十进制大整数{int temp[SIZE];int i = 0;int j;BigNum c;while (Cmp(bignum,Init("0")) == 1){        c=Mod(bignum,Init("1010"));        temp[i] = b2d(c);        bignum = Div(bignum,Init("1010"));         i++;}for (j = i - 1; j >= 0; j--){        printf("%d",temp[j]);}printf("\n");}BigNum modMDyn(BigNum a, BigNum power, BigNum mod) //求大整数幂的模 {    BigNum temp;   BigNum result;   BigNum t1;   temp=Mod(a,mod);   result=Init("1");   for(inti=SIZE-1;i>=SIZE-GetBitFront(power);i--)   {               if(power.data[i]=='1')               {                           t1=Mul(result,temp);                   result=Mod(Mul(result,temp),mod);               }                    temp=Mod(Mul(temp,temp),mod);   }    return result; } BigNum d2b(int num) //十进制转二进制{BigNum bignum;bignum=Init("0");int a=0;int b=0;int i=1;while(num>0){        a=num%2;        num=num/2;        bignum.data[SIZE-i]=a+48;        i++;}    return bignum;}int checkprime(BigNum n,BigNum a){BigNum k;BigNum q;//     BigNum a;BigNum n1;//n1=n-1BigNum num1;//num1为常数1BigNum num2;//num2为常数2BigNum k2;//2^kBigNum k22;    int k1=0; //末尾0的个数num1=Init("1");num2=Init("10");k=Init("0");q=Init("0");n1=Init("0");k22=Init("10");//     a=Init("1010");//选择的数n1=Sub(n,num1);k1=GetBitEnd(n1);k=d2b(k1);q=MovBitRight(n1,k1);k2=Div(n1,q);if(Cmp(modMDyn(a,q,n),num1)==0){//    print2(n);//    printf("该数可能是素数!\n");        return 1;}    for(int i=0;i<b2d(k);i++){        k22=MovBitLetf(num1,i);        if(Cmp(modMDyn(a,Mul(k22,q),n),n1)==0)        {//           print2(n);//           printf("该数可能是素数!\n");               return 1;        }}print10(n);    printf("该数肯定不是素数!\n");      return 0;}void main()//主函数的内容可以根据你自己的需求编写！{BigNum n;//n为要判断的素数BigNum k;BigNum q;BigNum a;BigNum n1;//n1=n-1BigNum num1;//num1为常数1BigNum num2;//num2为常数2BigNum k2;//2^kBigNum k22;    int k1=0; //末尾0的个数int flag=0;int aa=10;int i=0;num1=Init("1");num2=Init("10");k=Init("0");q=Init("0");n1=Init("0");k22=Init("10");a=Init("1010");//选择的数n=Init("1111111111111111111111111111111111111111111111111111111111111111");//最大的64bit数 a可以设128bit内的值//     n=Init("1111"); srand(time(NULL));n1=Sub(n,Mul(num2,num1));for(int kk=0;kk<30;kk++)//这里可以自己设置循环次数{n=Sub(n,Mul(num2,num1));//n每次-2print10(n);printf("第%d次：\n",kk);flag=checkprime(n,a);while(flag==1&&i<10){        i++;        print10(n);        aa=rand()%10+5;//注意!n必须大于a        a=d2b(aa);        flag=checkprime(n,a);        printf("ok%d\n",aa);}if(i==10){        i=0;        print10(n);        printf("该数肯定是素数!\n");}printf("\n");}      } 

运行结果如下：

原文：http://blog.csdn.net/tengweitw/article/details/23952839

作者：nineheadedbird