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 ) (p ≥ 2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen generalized Rijndael like ciphers. 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.

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Let (R, M) be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of R. Special attention is given to the set of atoms in M\M 2 and to the existence of atoms in M 2 . While our main interest is in local Cohen-Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight nonassociate atoms having an atom in M 2 .