An $n$-round Feistel network is a key-ed permutation defined by $$ {\sf Fstl}^{(m)}_{k_1,\dots,k_m}(L,R) := {\sf Fstl}^{(m-1)}_{k_1,\dots,k_{m-1}}\big( R, L\oplus F_{k_m}(R) \big)\;, $$ with the convention that ${\sf Fstl}^{(0)}:={\sf Id}$. It is well-known that, if the underlying $F_{k_i}$ is a pseudo-random function, then ${\sf Fstl}^{(3)}_{k_1,k_2,k_3}$ is a pseudo-random permutation (with forward query only), but what would happen if adversaries are given oracle access to $F_k$?
First of all, this immediately gives away inverse queries of ${\sf Fstl}^{(n)}_{k_1,\dots,k_m}$, so likely more rounds are needed to obtain meaningful security.
Formally, consider the key-less variant $$ {\sf Fstl}^{(m)}(L,R) := {\sf Fstl}^{(m-1)}\big( R, L\oplus H(m,R) \big)\;, $$ where now $H:[m]\times\{0,1\}^n\to\{0,1\}^n$ is a publicly accessible random oracle. Would the following be true for any sufficiently large $m$?
True for sufficiently large $m$ (or never)? For every adversary ${\cal A}^{{\sf Fstl}^{(m)}, ({\sf Fstl}^{(m)})^{-1}, H}$ given at most ${\sf poly}(n)$ many queries to every oracle, there is a stateful simulator ${\sf Sim}^{P, P^{-1}}$ that is given oracle access to a random permutation $P\in{\sf Sym}(\{0,1\}^{2n})$ and its inverse $P^{-1}$ for at most ${\sf poly}(n)$ many queries, such that $$ \left|\;\Pr\Big[1\gets{\cal A}^{{\sf Fstl}^{(m)}, ({\sf Fstl}^{(m)})^{-1}, H}\Big] - \Pr\Big[1\gets{\cal A}^{P,P^{-1},{\sf Sim}^{P,P^{-1}}}\Big]\;\right|\leq{\sf negl}(n)\;, $$ is negligibly small.
