On the classification of non-exceptional APN functions

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

Applicable Algebra in Engineering Communication and Computing Pub Date : 2024-01-18 DOI:10.1007/s00200-023-00642-2

Nurdagül Anbar, Tekgül Kalaycı, Nihal Yurdakul

引用次数: 0

Abstract

An almost perfect non-linear (APN) function over \(\mathbb {F}_{2^n}\) is called exceptional APN if it remains APN over infinitely many extensions of \(\mathbb {F}_{2^n}\). Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions.

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关于非特殊 APN 功能的分类

如果一个在 \(\mathbb {F}_{2^n}\) 上的近乎完美非线性（APN）函数在 \(\mathbb {F}_{2^n}\) 的无限多个扩展上仍然保持 APN，那么这个函数就被称为例外 APN。在过去的几十年里，异常 APN 函数引起了许多研究者的关注。赫尔南多（Hernando）和麦奎尔（McGuire）已经完成了异常 APN 单项式的分类，而奥布里（Aubry）、麦奎尔（McGuire）和罗迪耶（Rodier）则猜想，直到等价为止，唯一的异常 APN 函数是戈尔德（Gold）和卡萨米-韦尔奇（Kasami-Welch）单项式函数。从那时起，人们开始对非特殊 APN 多项式进行分类，并取得了许多部分成果。在本文中，为了对 APN 函数的例外性质进行分类，我们引入了一种新方法，该方法使用了有限域上曲线的技术。然后，我们将该方法与爱森斯坦不可重复性准则和库默尔定理结合起来，得到了新的非特殊 APN 函数。

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.