  The algebraic structure of the group generated by the encryption functions of
a block cipher depends on the key schedule algorithm used for generating the
round keys. For such a reason, in general, studying this group does not appear
to be an easy task. Previous works, focusing on the algebraic properties of
groups associated to a cipher, have studied the group generated by the round
functions of the cipher considering independent round keys. In this paper, we
want to study the more realistic group generated by the encryption functions,
where the key schedule satisfies certain requirements. In this contest, we are
able to identify sufficient conditions that permit to guarantee the primitivity
of this group and the security of the cipher with respect to the
partition-based trapdoor. This type of trapdoor has been recently introduced by
Bannier et al. (2016) and it is a generalization of that introduced by Paterson
in 1999.
