Is there any more asymetric-time permutation than x^3 (mod p)?
One approach is suggested by the ideas introduced by this paper.  I'll spell them out for you:


The verifier selects two large safe primes $p, q$ (that is, primes where $(p-1)/2$ and $(q-1)/2$ are also prime), the value $n = p \times q$, an integer $z$, and a quadratic residue ($\bmod n$) $a$.
The challenge (the "hard direction") is, given the values $n, z$ and $a$, compute $a^{2^{z}} \bmod n$
The verification (the "easy direction") is, given $a^{2^z} \bmod n$, recover $a$ (which is moderately easy to do, effectively an RSA operation, if you know the factorization of $n$).


This is a trapdoor function, but one which the verifier knows the trap door (and doesn't need to rely on anyone else).  This works fine if we have a unique verifier (as in a challenge-response system); it's less useful if the verification logic needs to be public.

And, the function is a permutation, if we restrict ourselves to the quadratic residues modulo $n$.

The point of this construction is that the asymmetry ratio can be as large as desired; it's effectively $O(z / \log n)$, as $z$ is an arbitrary integer, we can set this to whatever we feel like.