Mathematical (Arithmetic) representation of XOR

Question

I have spent the last 5 hours searching for an answer. Even though I have found many answers they have not helped in any way.

What I am basically looking for is a mathem

Answer 1

I do realize that this is sort of an old topic, but the question is worth answering and yes, this is possible using an algorithm. And rather than go into great detail about how it works, I'll just demonstrate with a simple example (written in C):

    #include 
    #include 
    #include    
    #include 

    typedef unsigned long
            number;

    number XOR(number a, number b)
    {
            number
                    result = 0,
            /*
                    The following calculation just gives us the highest power of 
                    two (and thus the most significant bit) for this data type.
            */          
                    power = pow(2, (sizeof(number) * 8) - 1);
    /*
            Loop until no more bits are left to test...
    */                
            while(power != 0)
            {
                    result *= 2;
            /*
                    The != comparison works just like the XOR operation.
            */                
                    if((power > a) != (power > b))
                            result += 1;
                    a %= power;
                    b %= power;
                    power /= 2;
            }
            return result;
    }

    int main()
    {
            srand(time(0));
            for(;;)
            {
                    number
                            a = rand(), 
                            b = rand();
                    printf("a = %lu\n", a);
                    printf("b = %lu\n", b);
                    printf("a ^ b     = %lu\n", a ^ b);
                    printf("XOR(a, b) = %lu\n", XOR(a, b));
                    getchar();
            }
    }