key recovery in the ideal cipher model I'm reading up on the ideal cipher model and can't wrap my head around some basic calculation.

Let $E$ is an ideal cipher with $n$-bit key and $n$-bit block. For each key $K$, $E_K$ is random permutation chosen uniformly random from the $2^n!$ of them. My question is, given an ideal cipher, and $q$-queries to it: $(m_i, E_K(m_i))$ for $i\in[q]$, what does it leak about $K$?

For any given $q$ mappings $\{ m_i \to E_K(m_i) \}$, there are $2^{n-q}!$ permutations that respect the mappings. So $E_K$ could be any one of them. Does this mean those $q$ queries leak nothing about the key?
In the ideal cipher model, $E_K$ and $K$ are independent, so no amount of information about $E_K$ will tell you anything about $K$ itself.