The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT.

A structure theorem is proved for finite groups with the property that, for some integer m with m ½ 2, every proper quotient group can be generated by m elements but the group itself cannot.1991 Mathematics subject classification (Amer. Math. Soc.): 20D20.

Abstract. In a previous paper, we had proved that the permutation group generated by the round functions of an AES-like cipher is primitive. Here we apply the O'Nan Scott classification of primitive groups to prove that this group is the alternating group.

We define a translation based cipher over an arbitrary finite field, and study the permutation group generated by the round functions of such a cipher. We show that under certain cryptographic assumptions this group is primitive. Moreover, a minor strengthening of our assumptions allows us to prove that such a group is the symmetric or the alternating group; this improves upon a previous result for the case of characteristic two.

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.2000 Mathematics subject classification: primary 20B05, 20P05.

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups.When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated by a couple of examples that might be of independent interest.

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.

No abstract