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Mathematics
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Computational Algebra, Coding Theory and Cryptography
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Computational Algebra, Coding Theory and Cryptography

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (1 March 2024) | Viewed by 5567

Special Issue Editor

Dr. Alla Levina

E-Mail Website
Guest Editor
Faculty of Computer Science and Technology, Saint Petersburg Electrotechnical University “LETI”, Professora Popova Str. 5, 100190 Saint-Petersburg, Russia
Interests: cryptography; information security; coding theory; wavelet transformation; computational algebra

Special Issue Information

Dear Colleagues,

In today’s world, information security, more precisely cryptography, plays a major role. We use it in our daily lives, even if we are not always aware of it. In addition, in the current era of artificial intelligence, its security comes to the foreground. This Special Issue will be focusing on cryptography, coding theory and other aspects of IS. In this Special Issue, we hope to gather the most recent research in the aforementioned areas. Topics include, but are not limited to, the following:

  • Algebraic codes in traditional communication channels;
  • Code-based cryptography;
  • Algebraic codes constructions;
  • Effective coding/decoding algorithms;
  • Cryptographic algorithms;
  • Cryptanalyses;
  • Side-channel attacks;
  • Security of AI.

Dr. Alla Levina
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • cryptography
  • coding theory
  • codes
  • cryptanalyses
  • AI security

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

Published Papers (4 papers)

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Research

11 pages, 251 KiB  
Article
A Method for Specifying Complete Signature Randomization and an Algebraic Algorithm Based on It
by Alexandr Moldovyan, Dmitriy Moldovyan, Nikolay Moldovyan and Alyona Kurysheva
Mathematics 2024, 12(13), 1970; https://doi.org/10.3390/math12131970 - 25 Jun 2024
Viewed by 1029
Abstract
To eliminate the limitations of signature randomization in known algebraic algorithms with a hidden group, the security of which is based on the computational complexity of solving large systems of power equations, a method for ensuring complete randomization is proposed. Based on this [...] Read more.
To eliminate the limitations of signature randomization in known algebraic algorithms with a hidden group, the security of which is based on the computational complexity of solving large systems of power equations, a method for ensuring complete randomization is proposed. Based on this method, a new algorithm of the indicated type was developed, using a four-dimensional finite non-commutative associative algebra as an algebraic basis. We obtained estimates of the security of algorithms to direct attacks as well as from attacks based on known signatures, which confirm the effectiveness of the proposed signature randomization method. Due to the relatively small size and signature of the public and private keys, the developed algorithm is of interest as a potential practical post-quantum digital signature scheme. Full article
(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography)
23 pages, 2123 KiB  
Article
Statistical Analysis of the Negative–Positive Transformation in Image Encryption
by Manuel Alejandro Cardona-López, Juan Carlos Chimal-Eguía, Víctor Manuel Silva-García and Rolando Flores-Carapia
Mathematics 2024, 12(6), 908; https://doi.org/10.3390/math12060908 - 20 Mar 2024
Viewed by 990
Abstract
The negative–positive transformation (NPT) is a widely employed technique for encrypting images on pixel blocks, commonly integrated into cryptosystems compatible with compression algorithms. The existing literature on NPT analysis can be categorized into two types: theoretical analyses with results that apply to any [...] Read more.
The negative–positive transformation (NPT) is a widely employed technique for encrypting images on pixel blocks, commonly integrated into cryptosystems compatible with compression algorithms. The existing literature on NPT analysis can be categorized into two types: theoretical analyses with results that apply to any image, primarily focused on compression compatibility, and numerical analyses that report empirical results from specific images, some without explaining the causes of the security results, while others are only related to the compression performance. Consequently, there is a significant gap in understanding the implications of applying the NPT for data protection. For that reason, this paper conducts a theoretical statistical analysis, presenting, demonstrating, and verifying six theorems to understand the security contributions of NPT. Two theorems examine the shape of the image histogram and the scatter plot of adjacent pixels after the NPT application. The subsequent four theorems explore the influence of NPT on the mean, variance, covariance, and correlation within each pixel block. The findings indicate that the NPT generates images with symmetrical histograms, the correlation of pixel blocks remains invariant, and distinct vertical and horizontal reflections manifest on the scatter plot. These theorems are verified by encrypting the Lena image with four pixel-block sizes. The histogram symmetry passed the goodness-of-fit test at a significance level of 5%, revealing consistent results. The correlation of pixel blocks remained unchanged, and the scatter plot exhibited an x-shaped pattern. Therefore, as the NPT alone does not achieve desirable encryption results, such as uniform histograms, scatter plots, and decreasing correlation, cryptosystems should complement it with additional techniques. Full article
(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography)
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16 pages, 623 KiB  
Article
An Optimized Point Multiplication Strategy in Elliptic Curve Cryptography for Resource-Constrained Devices
by Nawras H. Sabbry and Alla B. Levina
Mathematics 2024, 12(6), 881; https://doi.org/10.3390/math12060881 - 17 Mar 2024
Cited by 1 | Viewed by 1915
Abstract
Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution [...] Read more.
Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point. Full article
(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography)
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18 pages, 3030 KiB  
Article
An Efficient Audio Encryption Scheme Based on Elliptic Curve over Finite Fields
by Hafeez Ur Rehman, Mohammad Mazyad Hazzazi, Tariq Shah, Zaid Bassfar and Dawood Shah
Mathematics 2023, 11(18), 3824; https://doi.org/10.3390/math11183824 - 6 Sep 2023
Cited by 4 | Viewed by 1065
Abstract
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 cryptosystem for encrypting digital audio. As [...] Read more.
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 cryptosystem for encrypting digital audio. As a preliminary step, the system uses an EC of a particular type over a binary extension field to distort the digital audio pixel position. It reduces the inter-correlation between pixels in the original audio, making the system resistant to statistical attacks. In creating confusion in the data, an EC over a binary extension field is used to make a different number of substitution boxes (S-boxes). The suggested design employs a unique curve that relies on efficient EC arithmetic operations in the diffusion module. As a result, it generates high-quality pseudo-random numbers (PRNs) and achieves optimal diffusion in encrypted audio files with less processing work. Audio files of various sizes and kinds can all be encrypted using the provided algorithm. Moreover, the results show that this method effectively protects many kinds of audio recordings and is more resistant to statistical and differential attacks. Full article
(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography)
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