The description of the sponge function on Crypto.Stackexchange contains the following text (source):
The cryptographic sponge is a construction scheme for hash functions (and other symmetric primitives) based on an unkeyed permutation.
But is it possible to make use of a keyed permutation $f_k(x)$ to construct the sponge function? All constructions described in the “One-way compression function” article on Wikipedia are based on the keyed permutation function. And section “Sponge construction” of this article contains the following information:
The sponge construction can be used to build one-way compression functions.
But the article “Sponge function” does not mention any presence of the key in the used permutation function $f(x)$. Does this imply that we can fix any public key $K$ and always use $f_K(x)$ as $f(x)$? If yes, then how does the length of $K$ impact the security of the $n$-bit output, assuming that $n$ will be equal to the capacity divided by 2, as in SHA-3 (but if $K$ is public, then it seems that its length should not impact the security at all!)? If no, then how to apply a pseudo-random permutation (that is, a keyed permutation function) $f_k(x)$ to construct a sponge function, and how will the length of $k$ impact the security of the $n$-bit output?