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Symmetry
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The Advances in Algebraic Coding Theory
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The Advances in Algebraic Coding Theory

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

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 10732

Special Issue Editors

Dr. Nadir Murru

E-Mail Website
Guest Editor
Department of Mathematics, University of Trento, via Calepina, 14-38122 Trento, Italy
Interests: matematica; algebra
Dr. Alessio Meneghetti

E-Mail Website
Guest Editor
Department of Mathematics, University of Trento, Via Calepina, 14-38122 Trento, Italy
Interests: cryptography; coding theory; finite fields; combinatorics; discrete mathematics; computational algebra; blockchain technology
Special Issues, Collections and Topics in MDPI journals
Dr. Danilo Bazzanella

E-Mail Website
Guest Editor
Department of Mathematics, Politecnico of Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy
Interests: cryptography; number theory
Dr. Stefano Barbero

E-Mail Website
Guest Editor
Department of Mathematical Sciences, Politecnico of Torino, Corso Duca degli Abruzzi, 24 10129 Torino, Italy
Interests: number theory; cryptography

Special Issue Information

Dear Colleagues,

Coding Theory covers several topics and is a widely studied multidisciplinary subject, involving techniques from computer science, engineering, information theory and mathematics.

From both a theoretical and a computational point of view, algebra has established itself as one of the main reference areas in researching codes, their properties, encoding and decoding algorithms and other related aspects of Coding Theory. The continuous development and increasing relevance of digital data utilization makes Coding Theory a research field of primary interest that needs constant study and updates for keeping up with the technological demands of modern society. In particular, it is essential that messages reach the destination correctly, without errors that can occur due the presence of noise in the communication channel. Therefore, error correction codes are necessary in order to obtain efficient methods for detecting and correcting as many errors as possible.

The aim of the present Special Issue is to encourage the study of algebraic topics related to coding theory, as well as the development of new techniques for detecting and correcting errors, the improvement and analysis of existing error correction codes and also their applications in cryptography. We are soliciting contributions covering a broad range of these topics, including (though not limited to) the following:

- Families of codes and their properties (e.g., general linear and nonlinear codes, cyclic codes, AG codes, LCD codes);

- Analysis of codes in any metric (e.g., Hamming metric, rank metric);

- Optimality (e.g., bounds on parameters, asymptotic behavior of families of algebraic codes);

- Algebraic and computational aspects of the theory underlying codes (e.g., finite fields, cyclotomic polynomials, complexity issues);

- Encoding and decoding procedures;

- Code-based post-quantum cryptography;

- Algebraic construction of quantum codes.

Dr. Nadir Murru
Dr. Alessio Meneghetti
Dr. Danilo Bazzanella
Dr. Stefano Barbero
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

  • error correction codes
  • optimality
  • decoding algorithms
  • finite fields
  • post-quantum cryptography
  • quantum codes

Published Papers (5 papers)

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Research

11 pages, 306 KiB  
Article
(β,γ)-Skew QC Codes with Derivation over a Semi-Local Ring
by Mohammad Ashraf, Amal S. Alali, Mohd Asim and Ghulam Mohammad
Symmetry 2023, 15(1), 225; https://doi.org/10.3390/sym15010225 - 13 Jan 2023
Viewed by 1255
Abstract
In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication [...] Read more.
In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so that S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;β,γ] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software. Full article
(This article belongs to the Special Issue The Advances in Algebraic Coding Theory)
21 pages, 3542 KiB  
Article
Security Enhanced Symmetric Key Encryption Employing an Integer Code for the Erasure Channel
by Miodrag J. Mihaljević, Aleksandar Radonjić, Lianhai Wang and Shujiang Xu
Symmetry 2022, 14(8), 1709; https://doi.org/10.3390/sym14081709 - 17 Aug 2022
Cited by 1 | Viewed by 1303
Abstract
An instance of the framework for cryptographic security enhancement of symmetric-key encryption employing a dedicated error correction encoding is addressed. The main components of the proposal are: (i) a dedicated error correction coding and (ii) the use of a dedicated simulator of the [...] Read more.
An instance of the framework for cryptographic security enhancement of symmetric-key encryption employing a dedicated error correction encoding is addressed. The main components of the proposal are: (i) a dedicated error correction coding and (ii) the use of a dedicated simulator of the noisy channel. The proposed error correction coding is designed for the binary erasure channel where at most one bit is erased in each codeword byte. The proposed encryption has been evaluated in the traditional scenario where we consider the advantage of an attacker to correctly decide to which of two known messages the given ciphertext corresponds. The evaluation shows that the proposed encryption provides a reduction of the considered attacker’s advantage in comparison with the initial encryption setting. The implementation complexity of the proposed encryption is considered, and it implies a suitable trade-off between increased security and increased implementation complexity. Full article
(This article belongs to the Special Issue The Advances in Algebraic Coding Theory)
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18 pages, 374 KiB  
Article
Rotational Cryptanalysis on ChaCha Stream Cipher
by Stefano Barbero, Danilo Bazzanella and Emanuele Bellini
Symmetry 2022, 14(6), 1087; https://doi.org/10.3390/sym14061087 - 25 May 2022
Cited by 3 | Viewed by 2122
Abstract
In this paper we consider the ChaCha20 stream cipher in the related-key scenario and we study how to obtain rotational-XOR pairs with nonzero probability after the application of the first quarter round. The ChaCha20 input can be viewed as a 4×4 [...] Read more.
In this paper we consider the ChaCha20 stream cipher in the related-key scenario and we study how to obtain rotational-XOR pairs with nonzero probability after the application of the first quarter round. The ChaCha20 input can be viewed as a 4×4 matrix of 32-bit words, where the first row of the matrix is fixed to a constant value, the second two rows represent the key, and the fourth some initialization values. Under some reasonable independence assumptions and a suitable selection of the input, we show that the aforementioned probability is about 2−251.7857, a value greater than 2−256, which is the one expected from a random permutation. We also investigate the existence of constants, different from the ones used in the first row of the ChaCha20 input, for which the rotational-XOR probability increases, representing a potential weakness in variants of the ChaCha20 stream cipher. So far, to our knowledge, this is the first analysis of the ChaCha20 stream cipher from a rotational-XOR perspective. Full article
(This article belongs to the Special Issue The Advances in Algebraic Coding Theory)
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17 pages, 392 KiB  
Article
Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials
by Emanuele Bellini, Massimiliano Sala and Ilaria Simonetti
Symmetry 2022, 14(2), 213; https://doi.org/10.3390/sym14020213 - 22 Jan 2022
Cited by 1 | Viewed by 2208
Abstract
We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves [...] Read more.
We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O(n2n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexity of established algorithms, such as those based on the fast Walsh transform, is reached. Full article
(This article belongs to the Special Issue The Advances in Algebraic Coding Theory)
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20 pages, 586 KiB  
Article
Perfect Reconciliation in Quantum Key Distribution with Order-Two Frames
by Luis Adrián Lizama-Pérez and José Mauricio López-Romero
Symmetry 2021, 13(9), 1672; https://doi.org/10.3390/sym13091672 - 10 Sep 2021
Cited by 2 | Viewed by 1736
Abstract
We present an error reconciliation method for Quantum Key Distribution (QKD) that corrects 100% of errors generated in regular binary frames transmitted over a noisy quantum channel regardless of the quantum channel error rate. In a previous investigation, we introduced a novel distillation [...] Read more.
We present an error reconciliation method for Quantum Key Distribution (QKD) that corrects 100% of errors generated in regular binary frames transmitted over a noisy quantum channel regardless of the quantum channel error rate. In a previous investigation, we introduced a novel distillation QKD algorithm whose secret key rate descends linearly with respect to the channel error rate. Now, as the main achievement of this work, we demonstrate an improved algorithm capable of retaining almost all the secret information enclosed in the regular binary frames. Remarkably, this technique increases quadratically the secret key rate as a function of the double matching detection events and doubly quadratically in the number of the quantum pulses. Furthermore, this reconciliation method opens up the opportunity to use less attenuated quantum pulses, would allow greater QKD distances at drastically increased secret key rate. Since our method can be implemented as a software update, we hope that quantum key distribution technology would be fast deployed over global data networks in the quantum era. Full article
(This article belongs to the Special Issue The Advances in Algebraic Coding Theory)
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