How can I convert numbers into prime numbers?

Question

I'm working with one-way accumulators, but I'm not knowledgable in cryptography. Is there an easy peasy way to hash numbers (or whatever) into prime numbers? Obviously I'd like it to be collision resistant and all that, but this project is huge, I'm alone and I'm confident someone else in the field will pick up from there. Also, primality tests are linear in n or what? Thanks

Answer 1

Thank you for your answers but I think I found a better method.

1. Take the hash of your input $h(x)$, preferably with random oracle approximation
2. Sample the interval $[2^kh(x), 2^k(h(x)+1)]$ and pick only primes, for each of them
3. Hash it with an universal hashing function $f$ until you find that $f(p)=h(x)$
4. Write to memory: $H(x)=p$

Done!

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notes:

1. The random oracle makes collisions infeasible

2. Universal hashing gives high density of primes with high probability (there is a theorem out there but the principle is that, for a given prime, I have multiple hash outputs, so the probability that one of them is my input increases. See: Gennaro et al. - Secure hash-and-sign signatures without the Random Oracle, lemma 2)

Answer 2

Misiec´s conjecture:

Let

$$x= \frac{1}{n \cdot n^{\frac{1}{2} + n \cdot 2 \cdot i}}$$

if $n$ is a prime number then the $\sin(x)=x$, and if not the number is not a prime number.

Even though the conjecture does not hold true the $x$ value obtained turns a non-prime number into a complex prime number.