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Symmetry
Special Issues
Asymmetric and Symmetric Study on Number Theory and Cryptography
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Asymmetric and Symmetric Study on Number Theory and Cryptography

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 9438

Special Issue Editors

Prof. Dr. Ákos Pintér

E-Mail Website
Guest Editor
Institute of Mathematics, Faculty of Sciences and Technology, University of Debrecen, Debrecen, Hungary
Interests: polynomials; diophantine number theory
Special Issues, Collections and Topics in MDPI journals
Dr. Andrea Huszti

E-Mail Website
Guest Editor
Faculty of Informatics, University of Debrecen, Debrecen, Hungary
Interests: cryptographic protocols

Special Issue Information

Dear Colleagues,

The Number Theory is one of the oldest mathematical subjects. Mathematics is the queen of sciences and number theory is the queen of mathematics as stated by Gauss. We will focus on Diophantine Number Theory including Diophantine equations recurrence sequences, and number-theoretical applications of certain special polynomials. The cryptographic protocols or more generally cybersecurity and cryptography are state-of-the-art areas of the scientific and real-life world. Our subjects are relatives and most of the cases of the number of theoretical problems provide a scientific background for cryptography. We emphasize that the symmetry and asymmetry properties play an important role in both subjects and one can think of symmetric and asymmetric encryption or symmetric polynomials respectively. Our Special Issue guarantees an up-to-date forum for cryptographic and number-theoretical scientific communities.

Prof. Dr. Ákos Pintér
Dr. Andrea Huszti
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • number theory
  • Diophantine equations
  • recurrence sequences
  • special polynomials
  • cryptographic primitives and protocols
  • applied cryptography

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (6 papers)

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Research

34 pages, 389 KiB  
Article
On Generalized Fibospinomials: Generalized Fibonacci Polynomial Spinors
by Ece Gülşah Çolak, Nazmiye Gönül Bilgin and Yüksel Soykan
Symmetry 2024, 16(6), 694; https://doi.org/10.3390/sym16060694 - 5 Jun 2024
Viewed by 999
Abstract
Spinors are important objects in physics, which have found their place more and more after the discovery that particles have an intrinsic angular momentum shape and Cartan’s mathematical expression of this situation. Recent studies using special number sequences have also revealed a new [...] Read more.
Spinors are important objects in physics, which have found their place more and more after the discovery that particles have an intrinsic angular momentum shape and Cartan’s mathematical expression of this situation. Recent studies using special number sequences have also revealed a new approach to the use of spinors in mathematics and have provided a different perspective for spinor research that can be used as a source for future physics studies. The purpose of this work is to expand the generalized Fibonacci quaternion polynomials to the generalized Fibonacci polynomial spinors by associating spinors with quaternions, and to introduce and investigate a new polynomial sequence that can be used to benefit from the potential advantages of spinors in physical applications, and thus, to provide mathematical arguments, such as new polynomials, for studies using spinors and quaternions in quantum mechanics. Starting from this point of view, in this paper we introduce and investigate a new family of sequences called generalized Fibospinomials (or generalized Fibonacci polynomial spinors or Horadam polynomial spinors). Being particular cases, we use (r,s)-Fibonacci and (r,s)-Lucas polynomial spinors. We present Binet’s formulas, generating functions and the summation formulas for these polynomials. In addition, we obtain some special identities of these new sequences and matrices related to these polynomials. The importance of this study is that generalized Fibospinomials are currently the most generalized sequence in the literature when moving from Fibonacci quaternions to spinor structure, and that a wide variety of new spinor sequences can be obtained from this particular polynomial sequence. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
10 pages, 299 KiB  
Article
Bi-Unitary Superperfect Polynomials over 𝔽2 with at Most Two Irreducible Factors
by Haissam Chehade, Domoo Miari and Yousuf Alkhezi
Symmetry 2023, 15(12), 2134; https://doi.org/10.3390/sym15122134 - 30 Nov 2023
Viewed by 845
Abstract
A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. [...] Read more.
A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. gcdu(B,A/B)=1), where gcdu(B,A/B) denotes the greatest common unitary divisor of B and A/B. We denote by σ**(A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F2 if the sum of all bi-unitary monic divisors of σ**(A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F2. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
19 pages, 424 KiB  
Article
Secure Registration Protocol for the Internet of Drones Using Blockchain and Physical Unclonable Function Technology
by Norbert Oláh, Botond Molnár and Andrea Huszti
Symmetry 2023, 15(10), 1886; https://doi.org/10.3390/sym15101886 - 7 Oct 2023
Cited by 4 | Viewed by 1914
Abstract
Unmanned aerial vehicles (UAVs) have become increasingly popular in recent years and are applied in various fields, from commercial and scientific to military and humanitarian operations. However, their usage presents many challenges, including limited resources, scalability issues, insecure communication, and inefficient solutions. We [...] Read more.
Unmanned aerial vehicles (UAVs) have become increasingly popular in recent years and are applied in various fields, from commercial and scientific to military and humanitarian operations. However, their usage presents many challenges, including limited resources, scalability issues, insecure communication, and inefficient solutions. We developed a secure and scalable registration protocol to address these issues using LoRa technology. Our solution involves the usage of the physical unclonable function (PUF) and blockchain technology for key exchange. PUF also ensures security against physical tampering, and blockchain is applied to share the symmetric key among the base stations. After the registration, the later communication messages are encrypted with AES-GCM to provide authentication and confidentiality between the parties. We conducted a security analysis of the registration protocol using the ProVerif tool, and our solution meets the security requirements, including the mutual authentication of entities, key freshness, key secrecy and also key confirmation properties. Besides the Proverif-based analysis, an informal security analysis is also provided that shows that the registration is protected against a variety of well-known active and passive security attacks. As drone resources are limited, we also prepared a proof of concept to test our solution under real-life conditions, focusing on efficiency and lightweight operations. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
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14 pages, 4128 KiB  
Article
Effective Modified Fractional Reduced Differential Transform Method for Solving Multi-Term Time-Fractional Wave-Diffusion Equations
by Adel Al-rabtah and Salah Abuasad
Symmetry 2023, 15(9), 1721; https://doi.org/10.3390/sym15091721 - 7 Sep 2023
Cited by 3 | Viewed by 1301
Abstract
In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very [...] Read more.
In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
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26 pages, 401 KiB  
Article
Results on Minkowski-Type Inequalities for Weighted Fractional Integral Operators
by Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Artion Kashuri and Nejmeddine Chorfi
Symmetry 2023, 15(8), 1522; https://doi.org/10.3390/sym15081522 - 2 Aug 2023
Cited by 3 | Viewed by 1765
Abstract
This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful [...] Read more.
This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
20 pages, 971 KiB  
Article
New Numerical Results on Existence of Volterra–Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type
by Hawsar HamaRashid, Hari Mohan Srivastava, Mudhafar Hama, Pshtiwan Othman Mohammed, Eman Al-Sarairah and Musawa Yahya Almusawa
Symmetry 2023, 15(6), 1144; https://doi.org/10.3390/sym15061144 - 24 May 2023
Cited by 10 | Viewed by 1878
Abstract
Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that [...] Read more.
Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach. Full article
(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)
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