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The practical purpose of this research is to propose a candidate for post-quantum signature standard that is free of significant drawback of the finalists of the NIST world competition, which consists in the large size of the signature and the public key. The practical purpose is to propose a fundamentally new method for development of algebraic digital signature algorithms.

The proposed method is distinguished by the use of two different finite commutative associative algebras as a single algebraic support of the digital signature scheme and setting two different verification equation for a single signature. A single public key is computed as the first and the second public keys, elements of which are computed exponentiating two different generators of cyclic groups in each of the algebras.

Additionally, a scalar multiplication by a private integer is performed as final step of calculation of every element of the public key. The same powers and the same scalar values are used to compute the first and the second public keys by the same mathematic formulas. Due to such design, the said generators are kept in secret, providing resistance to quantum attacks. Two new finite commutative associative algebras, multiplicative group of which possesses four-dimensional cyclicity, have been proposed as a suitable algebraic support.

The introduced method is novel and includes new techniques for designing algebraic signature schemes that resist quantum attacks. On its base, a new practical post-quantum signature scheme with relatively small size of signature and public key is developed.

The authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.

The method is theoretical.

The authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.

The paper is original.

The purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.

The methods used are from the topic multiplicative ideal theory from commutative ring theory.

Let S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.

Motivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

The aim of this paper is to investigate the relationship between the ring structure of the twisted partial skew generalized power series ring RG,≤;Θ and the corresponding structure of its zero-divisor graph Γ̅RG,≤;Θ.

The authors first introduce the history and motivation of this paper. Secondly, the authors give a brief exposition of twisted partial skew generalized power series ring, in addition to presenting some properties of such structure, for instance, a-rigid ring, a-compatible ring and (G,a)-McCoy ring. Finally, the study’s main results are stated and proved.

The authors establish the relation between the diameter and girth of the zero-divisor graph of twisted partial skew generalized power series ring RG,≤;Θ and the zero-divisor graph of the ground ring R. The authors also provide counterexamples to demonstrate that some conditions of the results are not redundant. As well the authors indicate that some conditions of recent results can be omitted.

The results of the twisted partial skew generalized power series ring embrace a wide range of results of classical ring theoretic extensions, including Laurent (skew Laurent) polynomial ring, Laurent (skew Laurent) power series ring and group (skew group) ring and of course their partial skew versions.

The purpose of this paper is to determine the structure of nilpotent (n+6)-dimensional n-Lie algebras of class 2 when n≥4.

By dividing a nilpotent (n+6)-dimensional n-Lie algebra of class 2 by a central element, the authors arrive to a nilpotent (n+5) dimensional n-Lie algebra of class 2. Given that the authors have the structure of nilpotent (n+5)-dimensional n-Lie algebras of class 2, the authors have access to the structure of the desired algebras.

In this paper, for each n≥4, the authors have found 24 nilpotent (n+6) dimensional n-Lie algebras of class 2. Of these, 15 are non-split algebras and the nine remaining algebras are written as direct additions of n-Lie algebras of low-dimension and abelian n-Lie algebras.

This classification of n-Lie algebras provides a complete understanding of these algebras that are used in algebraic studies.

In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.

We introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.

We develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.

We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC. In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.

Let ℱ be a field of zero characteristic, let Nn(ℱ) denote the algebra of n×n strictly upper triangular matrices with entries in ℱ, and let f:Nn(ℱ)→Nn(ℱ) be a nonlinear Jordan centralizer of Nn(ℱ),that is, a map satisfying that f(XY+YX)=Xf(Y)+f(Y)X, for all X, Y∈Nn(ℱ). We prove that f(X)=λX+η(X) where λ∈ℱ and η is a map from Nn(ℱ) into its center 𝒵(Nn(ℱ)) satisfying that η(XY+YX)=0 for every X,Yin Nn(F).

In this paper, we study the different kinds of the primeness on the class of near-rings and we give new characterizations for them. For that purpose, we introduce new concepts called set-divisors, ideal-divisors, etc. and we give equivalent statements for 3-primeness which make 3-primeness looks like the forms of the other kinds of primeness. Also, we introduce a new different kind of primeness in near-rings called K-primeness which lies between 3-primeness and e-primeness. After that, we study different kinds of prime ideals in near-rings and find a connection between them and new concepts called set-attractors, ideal-attractors, etc. to make new characterizations for them. Also, we introduce a new different kind of prime ideals in near-rings called K-prime ideals.