Algebraic properties of generalized Rijndael-like ciphers

Abstract.

We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF (pk) (p≥2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [Discrete Appl. Math. 156 (2008), 3139–3149], 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 (2k) 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 (pk) (p≥2) is equal to the symmetric group or the alternating group on the state space.