Andrea Caranti

On some block ciphers and imprimitive groups

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

Abelian Hopf Galois structures on prime-power Galois field extensions

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m where m + 1 < p, then every regular abelian subgroup of the holomorph of N is isomorphic to N . The proof utilizes a connection, observed in [CDVS06], between regular abelian subgroups of the holomorph of N and nilpotent ring structures on (N, +). Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let L|K be a Galois extension of fields with abelian Galois group G. If also L|K is HHopf Galois where the K-Hopf algebra H has associated group N with N as above, then N is isomorphic to G.

An application of the O'Nan-Scott theorem to the group generated by the round functions of an AES-like cipher

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.

Finite p-groups of exponent p2 in which each element commutes with its endomorphic images

Various authors have studied the groups G in which each element commutes with all of its images under endomorphisms of G. Such groups are called E-groups. They can be defined equivalently as the groups G such that the near-ring generated by the endomorphisms of G in the near-ring of maps on G is actually a ring (see [7, Sect. 3; 3, Theorem 4.4.31; see also the remark following Lemma 3.1 below). Abelian groups are obviously E-groups, and since in an E-group each element commutes with all of its conjugates, every E-group is nilpotent of class at most 3 (see [S, 111.6.4 and 111.6.51). Various examples of finite nonabelian E-groups have already appeared in the literature [2, 8, 11. All the groups G appearing in these papers are finite p-groups of exponent p* and class 2, and satisfy the following conditions:

On the group generated by the round functions of translation based ciphers over arbitrary finite fields

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.

The group generated by the round functions of a GOST-like cipher

We define a cipher that is an extension of GOST, and study the permutation group generated by its round functions. We show that, under minimal assumptions on the components of the cipher, this group is the alternating group on the plaintext space. This we do by first showing that the group is primitive, and then applying the O’Nan-Scott classification of primitive groups.

Automorphism groups ofp-groups of class 2 and exponentp2: a classification on 4 generators

SuntoSi classificano i gruppi di automorfismi, modulo gli automorfismi centrali, di certi p-gruppi 4-generati di classe 2ed esponente p2.

The Round Functions of Cryptosystem PGM Generate the Symmetric Group

S. S. Magliveras et al. have described symmetric and public key cryptosystems based on logarithmic signatures (also known as group bases) for finite permutation groups.In this paper we show that if G is a nontrivial finite group which is not cyclic of order a prime, or the square of a prime, then the round (or encryption) functions of these systems, that are the permutations of G induced by the exact-transversal logarithmic signatures (also known as transversal group bases), generate the full symmetric group on G.This answers a question of S. S. Magliveras, D. R. Stinson and Tran van Trung.

A simple construction for a class of

We exhibit a simple construction, based on elementary linear algebra, for a class of examples of finite

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