Algebraic Theory Research Articles

Role of Linear Diophantine Equations in RSA Encryption

Abstract. Diophantine equations are mathematical equations that include only integer solutions and two or more unknown variables. To illustrate, the most elementary form of Diophantine equations are linear Diophantine equations when two different one-degree monomials added up to a constant value. Such equations were given the name of the extraordinary Greek mathematician Diophantus who made great contribution to algebra and number theory. In addition, they help in defining concepts in algebraic geometry and contribute to the development of algorithms for cryptography. They work in the same way as public keys do in cryptocurrencies within blockchain technology to curb cyber threats and scams targeting public systems. They serve as a means of securing transactions on a computer network when using cryptocurrencies such as bitcoins. These encryption techniques also permit cryptographers and mathematicians to create new ideas in relation to these number systems, thus making it possible for further cryptographic techniques to be put into place.

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

Boundary algebras of the Kitaev quantum double model

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.

Cluster Algebras and Its Applications

This workshop focused on recent developments in cluster algebras and their applications as well as interactions with other areas of mathematics. In addition to new advances in the theory of cluster algebras themselves, it included applications to knot theory and geometry as well as interactions with representation theory and categorification, Grassmannians, combinatorics, geometric surfaces models and Lie theory.

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Using Schur Rings to Produce GRRs for Dihedral Groups

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following: Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn. We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

Intersections of Algebraic Geometry and Coding Theory: A Study of Error-Correcting Codes

Intersections of Algebraic Geometry and Coding Theory: A Study of Error-Correcting Codes

About Supports of Local Cohomology Modules

The article provides the relation between the theory of local cohomology modules, and vanishing results, and also about the theory of support of such modules. Here, we put results about the theory, and also we provide a relation of local cohomology in the theory of commutative algebra and homological algebra.

Spectral Bipartition via Gap Cut on DNA Sequences

Deoxyribonucleic Acid (DNA) and graph partitioning are two distinct fields of study which can be linked in the structure of biological networks. Graph partitioning has been extensively studied but not its application in the biological field. This research explored on the application of spectral graph partitioning in DNA splicing, aiming to simulate the cleavage of DNA by performing spectral bipartition on DNA sequences in a DNA splicing system. This research incorporates Fiedler theory and algebraic graph theory, which are commonly utilized in network analysis and the analysis of graph connectivity. Some DNA sequences of even length are selected and expressed in graphical representations. The adjacency matrix, Laplacian matrix, and degree matrix are computed from the graphs, as well as the Fiedler value and Fiedler vector associated with the graphs. Gap cut is used as a method of spectral bipartition which produces two partitions of DNA sequence of unequal lengths. The generalizations of gap cut on DNA sequences of even length are provided as lemmas and theorem.

Fixed-time consensus and finite-time flocking of second-order nonlinear multi-agent systems under false data injection attacks

This paper focuses on addressing consensus and flocking issues in second-order nonlinear multi-agent systems (MASs) under the false data injection attacks. Novel adaptive control protocols are proposed to effectively mitigate the impact of these attacks by updating adaptive parameters. Based on algebraic graph theory, matrix theory and the Lyapunov approach, various sufficient conditions are established, which show the fixed-time consensus and finite-time flocking can be achieved for the MASs. Moreover, some upper bounds for the settling time are derived in terms of the parameters of the model. Finally, three simulation examples are presented to validate the efficacy of our proposed theoretical results.

Advance in Applied Cryptography Theory: Survey and New Results. Part 1. Key Cryptography

In the current paper, consisting from two parts, are presented both results already published before (but hard for access) and new once. Actuality of this work is firstly in a fact that recently has been obtained a number of new results in area of applied cryptography that are needed both in a clarification and be put into practice. This is namely the main goal of the current paper. The setting problem in the first paper part concerns to a complexity of symmetric cipher breaking while in the second part of the paper is discussed, so called, keyless cryptography, namely: wiretap channel concept, execution of communication channels which allow to provide information security without of key exchange procedure between legal correspondences. In the part widely used methods of applied mathematics, namely: algebra, number, probability and information theories. Computer simulation also used there. A novelty of the first part of work consists in the following: first of all it is clarified the sense of a key lifetime limitation for different symmetric cipher modes, secondly, it is explained an approach of cipher breaking by the use of quantum computers, finely, the key authentication for the Diffie ‒ Hellman protocol based on the mobile device pairing technology is investigated in detail. In the second part of the current paper has been presented a vulnerability of Dean ‒ Goldsmith cryptosystem under some extension of attacks. The main results of this paper are: estimation of the key lifetime of single key for symmetric cipher in CBC mode, clarifying of Grover’s algorithm breaking of symmetric ciphers by brute force attack, development of a method for authentication of Diffie ‒ Hellman values based on pre-distributed sequences, selection of ciphers which allow to execute with Shamir’s protocol without any key sharing in advance, breaking of Dean ‒ Goldsmith protocol under some conclusions, proof the fact regarding of a possible breakability of the key sharing protocol over noiseless communication channels. Practical application of paper results consists in the fact of stimulation the correct choice of ciphers and their parameters in order to provide their resistance to different attacks and more attention to algorithms of keyless cryptography.

Symmetry and Representation Theory of Lie Groups and Lie Algebras

Symmetry and Representation Theory of Lie Groups and Lie Algebras

Fixed-Time Distributed Event-Triggered Cooperative Guidance Methods for Multiple Vehicles with Limited Communications to Achieve Simultaneous Arrival

Aiming at the salvo-attack problem of multiple missiles, a distributed cooperative guidance law based on the event-triggered mechanism is proposed, which enables missiles with large differences in spatial location and velocity to achieve simultaneous attacks with only a few dozen information exchanges. It effectively reduces the generation of control commands and communication frequency, thereby reducing channel load and improving communication efficiency and reliability. Compared to traditional periodic sampling communication, the number of communications has been reduced by over 90%. The guidance process is divided into two stages. The first stage is the cooperative guidance stage, where missiles achieve consensus of the time-to-go estimates through information exchange. In this stage, each missile is designed with an event-triggered function based on its own state error, and the missile only updates and transmits its information in the communication network when the error meets the set threshold, effectively reducing the occupancy rate of missile-borne resources during the cooperation process. The second stage is the independent guidance stage, where missiles can hit the target simultaneously while keeping the communication network silent. This is achieved by ensuring that the time-to-go estimates of missiles can represent the real time-to-go after achieving consensus. By the design of the two-stage guidance law and the replacement of the event-triggered function, the cooperative guidance system can be ensured to remain stable in scenarios where the leader missile is present and destroyed, and this excludes Zeno behavior. The stability of the cooperative guidance law is rigorously proved by algebraic graph theory, matrix theory, and the Lyapunov method. Finally, the numerical simulation results demonstrate the validity of the algorithm and the correctness of the stability analysis.

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP1. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2<c, the generic c-nilpotent Lie algebra over Fp is strictly NSOP4 and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.