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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) ?

Drew
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  • The function $f(i)= i \bmod N +1 $ satisfies your requirements. Also your exponentiation function is slow prone to overflows. You can implement it in time $\approx \log i$ and keep the numbers small by using the relations $g^i \bmod p = 1$ if $i=0$, $g^i \bmod p =(g^{i/2} \bmod p)^2 \bmod p$ if $i \ge 2$ and $i$ is even, and $g^i \bmod p = \big( g ( g^{(i-1)/2} \bmod p)^2 \big) \bmod p$ if $i$ is odd. – Steven Nov 23 '23 at 12:46
  • https://cs.stackexchange.com/q/55807/755, https://cs.stackexchange.com/q/41124/755, https://cs.stackexchange.com/q/162415/755, https://cs.stackexchange.com/q/81664/755 – D.W. Nov 24 '23 at 23:57

1 Answers1

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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.

gnasher729
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  • from the wikipedia article on them it looks like I have to choose a modulus that is prime to get the full period. So, if I want to generate numbers from 1 to 99, I can't use modulus 100 to get a period of m-1 and would have to use either 97 or 101 and will either not generate 97, 98, 99, or I will generate 100 when I don't want to. Is there a way to get a period that is uniform on 1 to N if N+1 isn't prime? – Drew Nov 25 '23 at 11:14