论偶数维布尔函数的二阶导数和立方 APN 排列

IF 1.1 3区数学 Q1 MATHEMATICS

Mediterranean Journal of Mathematics Pub Date : 2024-05-08 DOI:10.1007/s00009-024-02660-x

Augustine Musukwa, Massimiliano Sala, Irene Villa, Marco Zaninelli

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引用次数: 0

摘要

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

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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来源期刊

Mediterranean Journal of Mathematics 数学-数学

CiteScore

1.80

自引率

0.00%

发文量

261

审稿时长

6-12 weeks

期刊介绍： The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003. The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience. In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.