Algebraic properties of generalized Rijndael-like ciphers

TL;DR

This paper investigates the algebraic structure of generalized Rijndael-like ciphers over finite fields, providing conditions for their permutation groups to be symmetric or alternating, which informs their cryptographic strength and potential for multiple encryption.

Contribution

It extends previous work by characterizing the permutation groups generated by Rijndael-like functions over arbitrary finite fields, not just GF(2^k), and discusses implications for cipher security.

Findings

Conditions for the group to be symmetric or alternating

Implications for multiple encryption security

Extension of algebraic analysis to GF(p^k) with p ≥ 2

Abstract

We provide conditions under which the set of Rijndael functions considered as permutations of the state space and based on operations of the finite field $\GF (p^k)$ ($p\geq 2$ a prime number) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the AES (Rijndael block cipher with specific block sizes) in case AES became practically insecure. In Sparr and Wernsdorf (2008), R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field $\GF (2^k)$ is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field $\GF (p^k)$ ($p\geq 2$) is equal to the symmetric group or the alternating group on the state space.