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.
• I guess that makes sense. Even if one figured out $E_K$ completely, there is still no leakage about $K$. Aug 5 '19 at 2:00