An application of the O’Nan-Scott theorem to the group generated by the round functions of an AES-like cipher

Abstract

Abstract. In a previous paper, we had proved that the permutation group generated by the round functions of an AES-like cipher is primitive. Here we apply the O'Nan Scott classification of primitive groups to prove that this group is the alternating group.

On the group generated by the round functions of translation based ciphers over arbitrary finite fields

“…In this paper we extend the results of [4] and [6] to translation based ciphers defined over an arbitrary finite field. The main point is the move from vector spaces defined over the field F 2 with two elements, to vector spaces defined over a field F p , where p is an arbitrary prime.…”

“…Moreover in [6], using the O'Nan-Scott classification of primitive groups, it was proved that if such a cipher satisfies some additional cryptographic assumptions, then the group is the alternating or the symmetric group.…”

The round functions of KASUMI generate the alternating group

“…[2,10,16]). For the DES [4], AES [22], and other ciphers, several results on the cyclic and group theoretic structure of their components have already been found (see [3,5,8,13,21,23,24]). …”

Inversion and subspaces of a finite field

“…In response to a question of Andrea Caranti, for use in [CDVS09], the author determined in [Mat07] the additive subgroups of a field which are closed with respect to inverting nonzero elements. The more general question with a division ring instead of a field was independently answered in [GGSZ06].…”