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?