How to create a 'hashing' function that maps up to 1 to N numbers uniquely to numbers 1 to N

Question

I would like to use or create a hashing algorithm that takes K inputs from 1 to N and maps them uniquely to different numbers on 1 to N. K can be 1 to N. Ideally I would like the hashing alg to be able to be salted, but I'm not sure that is possible if some of these numbers must be prime.

Someone in #algorithms on libera suggested using $g^i\bmod p$ where $p$ is a prime and $g$ is a generator of the $p$ prime ring, but I never took an abstract algebra class, so this is a bit over my head. I have this:

uint64_t primeHash(uint64_t const& i, uint64_t const& p)
{
    uint64_t g = 4294967311; // first prime after sqrt(p_max)                                                                                                                                
    for (unsigned int _ = 0; _ < i; ++_)
        g *= g;
                                                                                                                                                                    
    return g % p;
}

and I am thinking I would set p to N, which would be p = 100, but p is then not prime and the first 3 numbers are not unique:

i: 1 gen: 128849019105         g^i mod p: 5
i: 2 gen: 57982058546625       g^i mod p: 25
i: 3 gen: 5870683425282890625  g^i mod p: 25

I am all mixed up. Where am I going wrong with my understanding? How do I bound my outputs to 1 to N (here 100) ?

Answer 1

Look for “linear congruential random number generator”. If you take a random number generator with range 0 <= r < n that’s basically what you want. And it’s quite fast.