On circulant involutory and orthogonal MDS matrices over finite commutative rings

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2024-04-27 DOI:10.1007/s00200-024-00656-4

Shakir Ali, Atif Ahmad Khan, Bhupendra Singh

引用次数: 0

Abstract

Let \(k>1\) be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), proved the non existence of \(2^k \times 2^k\) orthogonal circulant MDS matrices and involutory circulant MDS matrices over finite fields of characteristic 2. The main aim of this paper is to prove the non-existence of orthogonal circulant MDS matrices of order \(2^k\times 2^k\) and involutory circulant MDS matrices of order k over finite commutative rings of characteristic 2. Precisely, we prove that any circulant orthogonal matrix of order \(2^k\) over finite commutative rings of characteristic 2 with identity is not a MDS matrix. Moreover, some related results are also discussed. Finally, we provide some examples to prove that the assumed restrictions on our main results are not superfluous.

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关于有限交换环上的循环非正交矩阵和正交 MDS 矩阵

让 \(k>1\) 是一个固定整数。在 Gupta 和 Ray (Cryptography and Communications 7: 257-287, 2015)一文中，证明了在特征 2 的有限域上\(2^k \times 2^k\) 正交环形 MDS 矩阵和非正交环形 MDS 矩阵的不存在性。本文的主要目的是证明在特征 2 的有限交换环上，阶为 \(2^k\times 2^k\) 的正交循环 MDS 矩阵和阶为 k 的非法定循环 MDS 矩阵的不存在性。确切地说，我们证明了在特征 2 的有限交换环上任何阶为 \(2^k\) 的环状正交矩阵都不是 MDS 矩阵。此外，我们还讨论了一些相关结果。最后，我们提供了一些例子来证明我们对主要结果的假定限制并非多余。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.