  We provide two sufficient conditions to guarantee that the round functions of
a translation based cipher generate a primitive group. Furthermore, under the
same hypotheses, and assuming that a round of the cipher is strongly proper and
consists of m-bit S-Boxes, with m = 3; 4 or 5, we prove that such a group is
the alternating group. As an immediate consequence, we deduce that the round
functions of some lightweight translation based ciphers, such as the PRESENT
cipher, generate the alternating group.
