Algebraic properties of generalized Rijndael-like ciphers

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(p^k) (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 [39], 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 ≥ 2) is equal to the symmetric group or the alternating group on the state space.