Mathematical objects in this paper are group codes. In the first part of the paper we present a survey with some of the main results about group codes, mainly the existence of group codes that are not abelian group codes, the minimal length and the minimal dimension of such codes and the existence of a non-abelian group code that has better parameters than any abelian group code. In particular, in a previous paper [1], we have shown that the minimal dimension of a group code that is not abelian group code is 4. However, all known examples of group codes of dimension 4 that are non-abelian group codes are constructed using groups that are not p-groups. We do not know if such codes exist for the case of p-groups, but in the second part of this paper we prove that, under some restrictions on the base field, all four-dimensional G-codes for an arbitrary finite p-group G are abelian.

In this paper, we present some results concerning orthogonality of l-generalized derivations and r-generalized derivations on ideals in semiprime rings, we also study the connections between orthogonality and some properties of the composition of a l-generalized derivation and a r-generalized derivation. These results are related to results of Brešar and Vukman (Rad Mat 5(2):237–246, 1989), that extend a theorem by Posner (Proc Am Math Soc 8:1093–1100, 1957) about products of derivations on prime rings.

Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dimF⁡(C)≤3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when charF>2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4.

Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of modern cryptology. However, security practitioners are not always familiar with probability theory, and thus fail to foresee the impact of (seemingly small) deviations from the theoretical description of a scheme at the implementation level. On the other hand, many cryptographic scenarios involve mutually distrusting parties, which need however to cooperate towards a joint goal. In order to attain assurance of the good behavior of one party, interactive validation methods (also known as interactive proof systems) are employed. Randomness is at the core of such methods, which most oftenwill only provide relative assurance, in the sense that they will establish correctness in a probabilistic way. In this paper we will briefly discuss the role of probability theory within modern cryptology, reviewing probabilistic proof systems as a powerful tool towards efficient protocol design, and provable security, as an invaluable framework for deriving formal security proofs.

We prove that there exist non-Abelian group codes over an arbitrary finite field.

Herstein (J Algebra 14:561–571, 1970) proved that given a semiprime 2-torsion free ring R and an inner derivation \(d_{t}\), if \(d_{t}^{2}(U)=0\) for a Lie ideal U of R then \(d_{t}(U)=0\). Carini (Rend Circ Mat Palermo 34:122–126, 1985) extended this result for an arbitrary derivation d, proving that \(d^{2}(U)=0\) implies \(d(U)\subseteq Z(R)\). The aim of this paper is to extend the results mentioned above for right (resp. left) generalized derivations. Precisely, we prove that if R admits a right generalized derivation F associated with a derivation d such that \(F^{2}(U) = (0)\), then \(d^{3}(U)= (0)\) and \((d^{2}(U))^{2}= (0)\). Furthermore, if F is also a left generalized derivation on U, then \(d(U)=F(U)=(0)\), and \(d(R), F(R)\subseteq C_{R}(U)\). On the other hand, if (F, d), (G, g) are, respectively, right and left generalized derivations that satisfy \(F(u)v=uG(v)\) for all \(u, v \in U\), then \(d(U), g(U)\subseteq C_{R}(U)\).

We generalize the notion of reverse derivation by introducing generalized reverse derivations. We define an l-generalized reverse derivation (r-generalized reverse derivation) as an additive mapping F : R → R, satisfying F(xy) = F(y)x + yd(x) (F(xy) = d(y)x + yF(x)) for all x, y ∈ R, where d is a reverse derivation of R. We study the relationship between generalized reverse derivations and generalized derivations on an ideal in a semiprime ring. We prove that if F is an l-generalized reverse (or r-generalized) derivation on a semiprime ring R, then R has a nonzero central ideal.

It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring F5S4. The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F-3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F-2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F-2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.

Let G be a finite group and F a field. We show that all G-codes over F are abelian if the order of G is less than 24, but for F = ℤ5 and G = S4 there exist non-abelian G-codes over F, answering to an open problem posed in [J. J. Bernal, Á. del Río and J. J. Simón, An intrinsical description of group codes, Des. Codes Cryptogr.51(3) (2009) 289–300]. This problem is related to the decomposability of a group as the product of two abelian subgroups. We consider this problem in the case of p-groups, finding the minimal order for which all p-groups of such order are decomposable. Finally, we study if the fact that all G-codes are abelian remains true when the base field is changed.

In this paper we study the hardness of some discrete logarithmlike problems defined in linear recurring sequences over finitefields from a point of view as general as possible. Theintractability of these problems plays a key role in the securityof the class of public key cryptographic constructions based onlinear recurring sequences. We define new discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems for any nontrivial linear recurring sequence in any finite field whose minimal polynomial is irreducible. Then, we prove that these problems are polynomially equivalent to the discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems in the subgroup generated by any root of the minimal polynomial of thesequence.

Reed–Solomon codes and Reed–Muller codes are represented as ideals of the group ring S = QH of an elementary Abelian p-group H over a finite field Q = \( {\mathbb{F}_q} \) of characteristic p. Such representations for these codes are already known. Our technique differs from the previously used method in the following. There, the codes in question were represented as kernels of some homomorphisms; in other words, these were defined by some kind of parity-check relations. Here, we explicitly specify generators for the ideals presenting the codes. In this case Reed–Muller codes are obtained by applying the trace function to some sums of one-dimensional subspaces of Q S in a fixed set of q such subspaces, whose sums also present Reed–Solomon codes.

Let G be a finite group and F be a field. Any linear code over F that is permutation equivalent to some code defined by an ideal of the group ring FG will be called a G-code. The theory of these “abstract” group codes was developed in 2009. A code is called Abelian if it is an A-code for some Abelian group A. Some conditions were given that all G-codes for some group G are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all G-codes over any field are Abelian if |G| < 128 and |G| ∉ {24, 48, 54, 60, 64, 72, 96, 108, 120}, but for F = \( {\mathbb{F}_5} \) and G = S4 there exist non-Abelian G-codes over F. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p(n), for some prime number p, such that its top-factor (S) over bar = S/pS semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to ((S) over bar, .) , the so-called Teichmuller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop (S) over bar* is right (or left) primitive.

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]q-codes over Fq for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed–Solomon codes as ideals of the group ring FqG where G is an elementary abelian group of order q.

Algebraic methods play an important role in coding theory. For instance, there are many connections between codes and groups. In this paper we will present two results that show different applications of algebraic methods in coding theory. One of them refers to the classical context and another one to the quantum error correcting theory. These results can be found in [5] and [13] respectively, where proofs and more details can be found.

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S , S * = S \ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = Z/nZ with n = 4 or n = p(r) for some odd prime p . Without associativity, S * is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.

том 16 ВЫПУСК 1 * 2004 УДК 519.7 Групповые коды и их неассоциативные обобщения © 2004 г.С. Гонсалес, Е. Коусело, В. Т. Марков, А