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 (p k )${\mathrm {GF}(p^k)}$ ( p≥2${p\ge 2}$ ) 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 (2 k )${\mathrm {GF}(2^k)}$ 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 )${\mathrm {GF}(p^k)}$ ( p≥2${p\ge 2}$ ) is equal to the symmetric group or the alternating group on the state space.