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

TL;DR

This paper investigates the permutation group generated by round functions in translation-based ciphers over finite fields, proving it is primitive or even symmetric/alternating under certain assumptions, extending previous results.

Contribution

It establishes conditions under which the group generated by cipher round functions is primitive or symmetric/alternating, generalizing prior work to arbitrary finite fields.

Findings

Group is primitive under certain assumptions

Group is symmetric or alternating with strengthened assumptions

Extends previous results beyond characteristic two

Abstract

We define a translation based cipher over an arbitrary finite field, and study the permutation group generated by the round functions of such a cipher. We show that under certain cryptographic assumptions this group is primitive. Moreover, a minor strengthening of our assumptions allows us to prove that such a group is the symmetric or the alternating group; this improves upon a previous result for the case of characteristic two.