Say we have an ideal 16-bit PRP. Since it appears that a permutation with a small domain can be used to turn it into a PRF, which can then be used in a Feistel network. On the surface, it makes it look like this means a trivial random permutation implemented using nothing more than a lookup table could then be used to create an ideal cipher with an arbitrary block and key size, which is obviously not the case. There must be something I am missing.
The way I see it:
A small, random permutation can be implemented in practice using a lookup table.
A random permutation can be used to implement a keyed PRP, e.g. $C = \pi(P \oplus K) \oplus K$.
A PRP with a small domain can be used to construct a secure PRF.
A secure PRF in a Feistel network can be used to create a secure block cipher.
A random permutation with a small domain can be implemented easily using a lookup table. Were I any more naïve, I would have proclaimed that I had just discovered an information theoretic secure cipher that fits the ideal cipher model, but obviously that is not the case. Why is this?
This is not a cryptosystem review question in disguise. I am simply curious to know the reason why an ideal but small PRP cannot be used to construct a cipher that fits the ideal cipher model.