The construction of circulant matrices related to MDS matrices

IF 0.2 Q4 MATHEMATICS, APPLIED

Prikladnaya Diskretnaya Matematika Pub Date : 2022-01-01 DOI:10.17223/20710410/56/2

S. S. Malakhov, M. I. Rozhkov

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引用次数: 0

Abstract

The objective of this paper is to suggest a method of the construction of circulant matrices, which are appropriate for being MDS (Maximum Distance Separable) matrices utilising in cryptography. Thus, we focus on designing so-called bi-regular circulant matrices, and furthermore, impose additional restraints on matrices in order that they have the maximal number of some element occurrences and the minimal number of distinct elements. The reason to construct bi-regular matrices is that any MDS matrix is necessarily the bi-regular one, and two additional restraints on matrix elements grant that matrix-vector multiplication for the samples constructed may be performed efficiently. The results obtained include an upper bound on the number of some element occurrences for which the circulant matrix is bi-regular. Furthermore, necessary and sufficient conditions for the circulant matrix bi-regularity are derived. On the basis of these conditions, we developed an efficient bi-regularity verification procedure. Additionally, several bi-regular circulant matrix layouts of order up to 31 with the maximal number of some element occurrences are listed. In particular, it appeared that there are no layouts of order 32 with more than 5 occurrences of any element which yield a bi-regular matrix (and hence an MDS matrix).

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与MDS矩阵相关的循环矩阵的构造

本文的目的是提出一种循环矩阵的构造方法，这种循环矩阵适合于在密码学中使用的最大距离可分离矩阵。因此，我们专注于设计所谓的双正则循环矩阵，并进一步对矩阵施加额外的约束，以使它们具有某些元素出现的最大次数和不同元素的最小数量。构造双正则矩阵的原因是任何MDS矩阵都必须是双正则矩阵，并且对矩阵元素的两个附加约束使得所构造的样本的矩阵向量乘法可以有效地执行。所得到的结果包括循环矩阵为双正则的某些元素出现次数的上界。进一步给出了循环矩阵双正则性的充分必要条件。在这些条件的基础上，我们开发了一个有效的双规则验证程序。此外，还列出了几个双规则循环矩阵布局，其顺序最高为31，其中某些元素出现的最大次数。特别是，似乎没有32阶的布局，任何元素的出现次数超过5次，从而产生双正则矩阵(因此是MDS矩阵)。

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

Prikladnaya Diskretnaya Matematika MATHEMATICS, APPLIED-

CiteScore

0.60

自引率

50.00%

发文量

期刊介绍： The scientific journal Prikladnaya Diskretnaya Matematika has been issued since 2008. It was registered by Federal Control Service in the Sphere of Communications and Mass Media (Registration Witness PI № FS 77-33762 in October 16th, in 2008). Prikladnaya Diskretnaya Matematika has been selected for coverage in Clarivate Analytics products and services. It is indexed and abstracted in SCOPUS and WoS Core Collection (Emerging Sources Citation Index). The journal is a quarterly. All the papers to be published in it are obligatorily verified by one or two specialists. The publication in the journal is free of charge and may be in Russian or in English. The topics of the journal are the following: 1.theoretical foundations of applied discrete mathematics – algebraic structures, discrete functions, combinatorial analysis, number theory, mathematical logic, information theory, systems of equations over finite fields and rings; 2.mathematical methods in cryptography – synthesis of cryptosystems, methods for cryptanalysis, pseudorandom generators, appreciation of cryptosystem security, cryptographic protocols, mathematical methods in quantum cryptography; 3.mathematical methods in steganography – synthesis of steganosystems, methods for steganoanalysis, appreciation of steganosystem security; 4.mathematical foundations of computer security – mathematical models for computer system security, mathematical methods for the analysis of the computer system security, mathematical methods for the synthesis of protected computer systems;[...]