一些商环上的可逆矩阵：识别、生成和分析

IF 0.3 Q4 MATHEMATICS, APPLIED

Discrete Mathematics and Applications Pub Date : 2022-08-01 DOI:10.1515/dma-2022-0022

V. Vysotskaya, L. Vysotsky

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引用次数: 3

摘要

摘要研究了二元域上商环上的矩阵模一元多项式。得到了给定大小的所有可逆矩阵的分数的下界。提出并分析了一种计算这些商环上矩阵行列式的有效算法和生成随机可逆矩阵(在所有可逆矩阵的集合上均匀分布)的算法。考虑并分析了后一种算法对形式为xr−1的商环模多项式的有效版本。这些方法可以在基于准循环码(如LEDAcrypt)的密码方案的密钥生成中找到实际应用。

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Invertible matrices over some quotient rings: identification, generation, and analysis

Abstract We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.

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来源期刊

Discrete Mathematics and Applications MATHEMATICS, APPLIED-

CiteScore

0.60

自引率

20.00%

发文量

期刊介绍： The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.