With $hash_n$, I mean a standard cryptographic hash like sha256, scaled up to have arbitrary length $n$ of its output with the same underlying principles. What is the time complexity class of the following problem?
Given an $n \in \mathbb{N}$ and data $d$ with $length(d) = n$, determine whether $\exists p : ((length(p) \le n) \wedge (hash_n(p) = d))$.
I find it really hard to find a class for it because it seems like a hard problem and it's definitely not $\in P$ (except if $P = NP$ or there is a vulnerability of the hash function (assume there isn't one not already known (so sha256 basically just printing the state of a state machine can be used, of course))) but you can't use it to solve any of the classical NP hard problems.
Determining the space complexity is of course trivial: It's only $\mathcal{O}(n)$.
This is of course just one example of a hash decision problem and it's meant to only be an example. I'm more interested in the general classification of hash decision problems.
Edit: You may assume a perfect hash function instead of a commonly used cryptographic hash function. E.g. if you're more comfortable with it, you don't want to exploit properties of cryptographic hash functions which are commonly used in practice, what you want to show is threatened by those properties, etc.
