Translation based block cipher

Question

The paper "On the group generated by the round functions of translation based ciphers over arbitrary finite fields" states:

Definition 3.3. A block cipher $C = \{ \tau_k | k ∈ K \}$ over $F_q$ is called translation based (tb) if:

1. each $\tau_k$ is the composition of $l$ round functions $\tau_{k,h}$ , for $k ∈ K$, and $h = 1, . . . , l$, where in turn each round function $\tau_{k,h}$ can be written as a composition $\gamma_h\lambda_h\sigma_{φ(k,h)}$ of three permutations of $V$ , where:

   • $\gamma_h$ is a bricklayer transformation not depending on $k$ and $0\gamma_h = 0$
   • $\lambda_h$ is a linear permutation not depending on $k$,
   • $φ : K × \{1, . . ., l\}→ V$ is the key scheduling function, so that $φ (k, h)$ is the $h$-th round key, given the master key $k$;

2. for at least one round index $h_0$ we have that

   • $\lambda_{h_0}$ is a proper mixing layer, and
   • the map $K → V$ given by $k → φ (k, h_0)$ is surjective, that is, every element of $V$ occurs as an $h_0$-th round key

How do I check if this map is surjective? Is AES a translation based block cipher?

Answer 1

Well, you probably cannot check this without a somewhat explicit specification of $\varphi.$ In general, if the key scheduling function has a simple structure, and if $|V|=|K|$ this becomes easier; In that case, if $\varphi(h_0,\cdot)$ is a permutation you are done.