In this review, we consider applications of nonassociative algebraic structures for the construction of linearly optimal codes and cryptosystems.

This work presents the results of an experimental study of some properties of low-order finite Abelian groups from the standpoint of the applicability of such groups in cryptographic applications.

We analyze algorithms for open construction of a key on some noncommutative group. Algorithms of factorization and decomposition for associative algebras (of small dimension) are considered. A survey of applications (in particular, in cryptography) of so-called “hidden matrices” is given.

We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Diffie-Hellman algorithm. We prove the existence of non-associative groupoids which are simultaneously power commuting and not power-associative.

AbstractFor an arbitrary prime power q, a criterion for irreducibility of a polynomial of the formover the field K = GF(qt) is established.

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.

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.

Найдены значения ранга первой разрядной последовательности скрученной линейной рекурренты максимального периода при естественных предположениях о разрядном множестве.

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.

In 1998, E. Couselo, S. Gonzalez, V. Markov, and A. Nechaev defined the recursive codes and obtained some results that allowed one to conjecture the existence of recursive MDS-codes of dimension 2 and length 4 over any finite alphabet of cardinality q ∉ {2, 6}. This conjecture remained open only for q ∈ {14, 18, 26, 42}. It is shown in this paper that there exist such codes for q = 42. We used a new construction, that of pseudogeometry with clusters.

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field \( P = \mathbb{F}_q \) with q = 2ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where \( P < Q = \mathbb{F}_{qn} \). The latter has a reduced trace representation F = tr P Q (Φ), where Φ(x) is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case ℓ = 1 to the case ℓ > 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ℓ = 1 these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [1]; we indicate a set of parameters q and n for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.

The authors investigate semirings of cyclic types from the algebraic point of view. To simplify and facilitate the analysis, the local Fourier transform of these semirings is introduced. The authors describe zero divisors, nilpotent elements, invertible elements, idempotents, and the Jacobson radical.