We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a...
For the most extensive range of tasks, such as real-time data processing in intelligent transport systems, etc., advanced computer-based techniques are required. They include field-programmable gate arrays (FPGAs). This paper proposes a method of pre...
Irreducible polynomials are widely used in modern cryptography; however, algorithms for finding such polynomials remain quite complex and require significant computational resources. In this study, a new approach to finding irreducible equations over...
Both the act of keeping information secret and the research on how to achieve it are included in the broad category of cryptography. When people refer to “information security,” they are referring to the study and use of methods that make...
The rapid development of artificial intelligence in recent years has been driven primarily by engineering advances, while the philosophical foundations of the concept of “intelligence” itself have been reconsidered much more slowly. This...
Substitution boxes (S-Boxes) function as essential nonlinear elements in contemporary cryptographic systems, offering robust protection against cryptanalytic attacks. This study presents a novel technique for generating compact 8-bit S-Boxes based on...
Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, our research investigates the relationship between c...
The Internet of Things (IoT), which is characteristic of the current industrial revolutions, is the connection of physical devices through different protocols and sensors to share information. Even though the IoT provides revolutionary opportunities,...
Attacks on the Internet of Things (IoT) are not highly considered during the design and implementation. The prioritization is making profits and supplying services to clients. Most cryptographic algorithms that are commonly used on the IoT are vulner...
In substitution–permutation network as a cryptosystem, substitution boxes play the role of the only nonlinear part. It would be easy for adversaries to compromise the security of the system without them. 8-bit S-boxes are the most used cryptogr...
A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of expe...
This paper focuses on a block cipher adaptation of the Galois Extension Fields (GEF) combination technique for PRNGs and targets application in the Internet of Things (IoT) space, an area where the combination technique was concluded as a quality str...
Radiation induced transient faults like single event upsets (SEU) and multiple event upsets (MEU) in memories are well researched. As a result of the technology scaling, it is observed that the logic blocks are also vulnerable to malfunctioning when...
This paper proposes a spatial analog of the compass rose, constructed using finite fields and the discrete logarithm operation. The basic idea is to match the geometric elements of regular and semiregular polyhedra with elements of Galois fields (GF)...
Mosaic ornaments and periodic geometric patterns are deeply rooted in cultural heritage and contemporary design, where symmetry plays a fundamental role in both aesthetic and cognitive perception. This study develops an algebraic method for generatin...
A novel approach to constructing an algorithm for computing discrete logarithms, which holds significant interest for advancing cryptographic methods and the applied use of multivalued logic, is proposed. The method is based on the algebraic delta fu...
Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension
It is shown that the analysis of rotating circular images containing n = 2p−1 pixels, the state of which is described by variables of binary logic, and p is an integer, is expedient to carry out using digital spectra obtained using the Fourier&...
The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more gener...
It is shown that it is reasonable to use Galois fields, including those obtained by algebraic extensions, to describe the position of a point in a discrete Cartesian coordinate system in many cases. This approach is applicable to any problem in which...
We construct Galois coordinates that connect general three-dimensional shape representations with a layerwise spectral description of convolutional networks. Two spirals on an auxiliary ellipsoid form a closed contour; for q ≡ 3 (mod̳...
Random linear network coding (RLNC) can greatly aid data transmission in lossy wireless networks. However, RLNC requires computationally complex matrix multiplications and inversions in finite fields (Galois fields). These computations are highly dem...
This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of
We provide three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is based on Galois rings. The main achievement in this work is to obtain optimal impers...
The method of non-standard algebraic extensions based on the use of additional formal solutions of the reduced equations is extended to the case corresponding to three-dimensional space. This method differs from the classical one in that it leads to...
The role of substitution boxes is very important in block ciphers. Substitution boxes are utilized to create confusion in the cryptosystem. However, to create both confusion and diffusion in any cryptosystem p-boxes and chaos base substitution boxes...
This paper presents fine-grained Field Programmable Gate Arrays (FPGA) architectures for the Advanced Encryption Standard (AES) MixColumns and InvMixColumns transformations, targeting improved performance and resource utilization. The proposed method...
A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose res...
Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on q...
Binary field (
Data security is one of the biggest concerns in the modern world due to advancements in technology, and cryptography ensures that the privacy, integrity, and authenticity of such information are safeguarded in today’s digitally connected world....
The well known Nakano–Nishijima–Gell-Mann (NNG) formula relates certain quantum numbers of elementary particles to their charge number. This equation, which phenomenologically introduces the quantum numbers
Secret sharing is a data security technique that divides secret information into multiple parts, embeds these parts into various shares, and distributes these shares to different participants. The original secret information can be retrieved only whe...
In this paper, the theoretical lower-bound on the success probability of blind reconstruction of Bose–Chaudhuri–Hocquenghem (BCH) codes is derived. In particular, the blind reconstruction method of BCH codes based on the consecutive roots...
This paper introduces a novel method for selecting parameters of finite fields formed by 2 × 2 matrices over a finite field of integers modulo a prime p. The method aims to simultaneously determine both the field parameters and primitive elemen...
Bose–Chaudhuri–Hocquenghem (BCH) codes are broadly used to correct errors in flash memory systems and digital communications. These codes are cyclic block codes and have their arithmetic fixed over the splitting field of their generator p...
Many stream ciphers employ linear feedback shift registers (LFSRs) to generate pseudorandom sequences. Many recent LFSRs are defined in
It is demonstrated that neuromorphic materials designed for computational tasks can be effectively implemented by drawing an analogy with trigger-based systems built upon classical binary elements. Among the most promising approaches in this context...
Statistical analysis shows that the most common errors in the transmission of information consist of single errors and transposition errors. Error detection and correction methods are often desired, particularly when the accuracy of information is of...
The proliferation of Internet of Things (IoT) devices has introduced significant security challenges, including weak authentication, insufficient data protection, and firmware vulnerabilities. To address these issues, we propose a linguistic secret s...
Data security and privacy have become essential due to the increasingly advanced interconnectivity in today’s world, hence the reliance on cryptography. This paper introduces a new algorithm that uses a novel hybrid Tent–May chaotic map t...
Many data processing applications involve binary matrices for storing digital information. At present, there are limited results in the literature about algorithms for inverting large binary matrices. This paper contributes the following three result...
Wireless Sensor Network (WSN) technology has constrained computing resources that require efficient and low-cost cryptographic hardware to provide security services, particularly when dealing with large modular polynomial multiplication in cryptograp...
Elliptic curve (EC) based cryptographic systems are more trustworthy than the currently used cryptographic approaches since they require less computational work while providing good security. This paper shows how to use an EC to make a good cryptosys...
This paper proposes a spatial analog of the compass rose, interpreted as a discrete analog of cylindrical coordinates and considered as a basis for direction-based command filtering in Unmanned Aerial Vehicle (UAV) groups. The initial formulation is...
Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are int...
Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusin...
An involution of an exceptional geometry of type
According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous...
The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over
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