Question

I am implementing Wikipedia’s Miller-Rabin algorithm but don’t seem to be getting even vaguely apt results. 7, 11, 19, 23 etc. are reported composite. Infact, when k>12, even 5 is shown composite. I have read the maths behind Miller-Rabin but don’t quite understand it very well and am relying on the algorithm blindly. Any cues on where I’m going wrong?

Here’s my code:

#include<stdio.h>
#include<math.h>

int modpow(int b, int e, int m) {
    long result = 1;

    while (e > 0) {
        if ((e & 1) == 1) {
            result = (result * b) % m;
        }
        b = (b * b) % m;
        e >>= 1;
    }

    return result;
}

int isPrime(long n,int k){
        int a,s,d,r,i,x,loop;
        if(n<2)return 0;
        if(n==2||n==3)return 1;
        if(n%2==0)return 0;

        d=n-1;
        s=0;
        while(d&1==0){
                d>>=1;
                s++;
        }


        for(i=0;i<k;i++){
                loop=0;
                a=(int)(rand()*(n-1))+1;
                x=modpow(a,d,n);
                if(x==1 || x==n-1){
                        continue;

                }
                for(r=1;r<=s;r++){
                        x=modpow(x,2,n);
                        if(x==1)return 0;
                        if(x==n-1){
                                loop=1;
                                break;
                        }
                }
                if(!loop)return 0;

        }
        return 1; 

}

int main(){
        int i,k;
        scanf("%d",&k);
        for(i=5;i<100;i+=2){
                printf("%d : %d\n",i,isPrime(i,k));
        }
        return 0;
}

Answer 1

If the base is not coprime to the candidate, the strong Fermat check always returns “not a probable prime”.

With your mistake

a=(int)(rand()*(n-1))+1;

for a prime p, the base is not coprime to p (a multiple of p), if and only if the result of rand() has the form

k*p + 1

For small primes, that is practically guaranteed to happen even with few iterations.

The base should lie between 2 ans n/2 (choosing bases larger than n/2 is not necessary since a is a witness for compositeness if and only if n - a is one), so you want something like

a = rand() % (n/2 - 2) + 2;

if you don’t mind the modulo bias in the random number generation that favours small remainders, or

a = rand() /(RAND_MAX + 1.0) * (n/2 - 2) + 2;

if you want to distribute the bias over the entire possible range.