On Second-Order Derivatives of Boolean Functions and Cubic APN Permutations in Even Dimension

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-08 DOI:10.1007/s00009-024-02660-x

Augustine Musukwa, Massimiliano Sala, Irene Villa, Marco Zaninelli

引用次数: 0

Abstract

The big APN problem is one of the most important challenges in the theory of Boolean functions, i.e. finding a new APN permutation in even dimension. Among this class of functions, those with the lowest possible degree are cubic. Yet, none has been found so far. In this paper, we introduce new parameters for Boolean functions and for vectorial Boolean functions, mostly derived from the behavior of their second-order derivatives. These parameters are invariant under extended affine equivalence, and they are particularly relevant for small-degree functions. They allow studying bent, semi-bent and APN functions of degrees two and three. In particular, they allow tackling the big APN problem for cubic permutations. Notably, we focus on the case of dimension 8, providing some computational results.

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论偶数维布尔函数的二阶导数和立方 APN 排列

大 APN 问题是布尔函数理论中最重要的挑战之一，即在偶数维中寻找新的 APN 变换。在这一类函数中，度数最低的是立方函数。然而，迄今为止还没有发现。在本文中，我们为布尔函数和矢量布尔函数引入了新的参数，这些参数主要来自其二阶导数的行为。这些参数在扩展仿射等价条件下是不变的，它们与小度函数尤其相关。它们允许研究二阶和三阶的弯曲、半弯曲和 APN 函数。特别是，它们可以解决立方排列的大 APN 问题。值得注意的是，我们重点研究了维数为 8 的情况，并提供了一些计算结果。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.