  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.
