Constructions of complete permutations in multiplication

IF 1.2 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2025-02-14 DOI:10.1007/s10623-025-01593-0

Kangquan Li

引用次数: 0

Abstract

Complete permutations in addition over finite fields have attracted many scholars’ attention due to their wide applications in combinatorics, cryptography, sequences, and so on. In 2020, Tu et al. introduced the concept of the complete permutation in the sense of multiplication (CPM for short). In this paper, we further study the constructions and applications of CPMs. We mainly construct many classes of CPMs through three different approaches, i.e., index, self-inverse binomial, which is a new concept proposed in this paper, and linearized polynomial. Particularly, we provide a modular algorithm to produce all CPMs with a given index and determine all CPMs with index 3. Many infinite classes of complete self-inverse binomials are proposed, which explain most of the experimental results about complete self-inverse binomials over \({\mathbb {F}}_{2^n}\) with \(n\le 10\). Six classes of linearized CPMs are given by using standard arguments from fast symbolic computations and a general method is proposed by the AGW criterion. Finally, two applications of CPMs in cryptography are discussed.

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乘法中完全排列的构造

有限域上的完全置换由于在组合学、密码学、序列学等领域的广泛应用而引起了学者们的广泛关注。2020年，Tu等人引入了乘法意义上的完全置换（简称CPM）概念。在本文中，我们进一步研究了cpm的结构和应用。我们主要通过三种不同的方法构造许多类cpm，即指数、自逆二项（本文提出的一个新概念）和线性化多项式。特别是，我们提供了一个模块化算法来生成具有给定索引的所有cpm，并确定具有索引3的所有cpm。提出了许多无限类的完全自逆二项，它们解释了大多数关于完全自逆二项的实验结果 \({\mathbb {F}}_{2^n}\) 有 \(n\le 10\)。利用快速符号计算的标准参数给出了6类线性化cpm，并根据AGW准则提出了一种通用方法。最后，讨论了cpm在密码学中的两个应用。

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

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.

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