Permutation for which an inverse is a hard problem Is it possible to construct (even if we don't know how) a permutation for which an inverse must exist but is difficult to find (brute force required)? The one way permutation contains no hidden information (keys) or state - it's public.

Or can it be proven that such a construct is not possible?

In case I'm not specific enough in what I'm asking, such a construct would allow for a sponge construction with no hidden state because no inverse permutation for it can be found.
It is obviously possible; one simple example is the function $f(x) = g^x - 1 \bmod p$, where $p$ is a large prime, and $g$ generates the entire group $\mathbb{Z}_p^*$; this is permutation over the range $[0, p-1)$, and is difficult to invert if the discrete log problem is hard.

What may be more difficult is to design an efficient hard-to-invert permutation; most efficient permutations are designed as a composition of trivially invertible permutations (and hence the result can be easily inverted).