# Is there any block cipher which is also a pseudo-random-permutation over the key

As block ciphers are defined as a pseudo-random-permutation over the data (keyed with the key), I was wondering, if there are also constructions for which key and data can be switched and the cipher is a permutation over the key space for a fixed (data) input?

So the question is, if the output space of $$E_k(a)$$ for all possible $$k$$ covers the entire space of $$\{0,1\}^n$$

More formally:

$$E_k$$ is a block cipher with key size equal to block size: $$\{ 0, 1 \}^n \times \{ 0, 1 \}^n \rightarrow \{ 0, 1 \}^n$$

and $$\exists a \forall k_1, k_2: E_{k_1}(a) = E_{k_2}(a) \Rightarrow k_1 = k_2$$

Or more generally: with $$f$$ a function $$\{ 0, 1 \}^n \rightarrow \{ 0, 1 \}^n$$

$$\forall k_1, k_2: E_{k_1}(f(k_1)) = E_{k_2}(f(k_2)) \Rightarrow k_1 = k_2$$

Or is this even true for every block cipher?