- $S$ be a finite set
- $O$ be a random oracle from $S$ to $S$, such that $O$ is bijective
- $f$ be a random permutation of $S$
Is there any difference between $O$ and $f$?
Does it makes any difference if $S$ is not finite?
At some level, there is no essential difference. Certainly, there is no difference in the distribution on the random variable $O$ vs $f$.
However, there is a potential difference in how the terms are typically used.
Hopefully the set of parties who have access to $O$/$f$ should be made clear elsewhere (e.g., in the formalization of security), so there is no real danger of serious confusion, but this might serve as a hint to let readers know what to expect.