I need an efficient, invertible random permutation of $n$-bit strings, defined by a $128$-bit key. $n$ can be any positive integer from $6$ to $128$. I have no requirements on the security of the permutation; only that it is random (i.e. that outputs for different keys are uncorrelated).
Many standard algorithms exist for $n$ of $32$, $64$, and $128$ bits. I wondered if the Sponge Construction of Keccak could be used to transform such permutations into $n$-bit permutations. I understand the sponge construction generates a function of variable input length and arbitrary output length. However, it is not clear that if I use it to generate a function to turn one $n$-bit string into another $n$-bit string, that that function will encode a random permutation. Does it?