利用 $$P_\tau$ 属性设计弯曲函数,可证明其在已完成的马约拉纳-麦克法兰类之外

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Enes Pasalic, Amar Bapić, Fengrong Zhang, Yongzhuang Wei
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引用次数: 0

摘要

在本文中,我们利用所谓的 \(P_\tau \) 属性构造出了弯曲函数的某些实例,这些实例可以证明是在完整的 Maiorana-McFarland (\({\mathcal{M}\mathcal{M}}^\#\)类之外。这也部分回答了 Kan 等人提出的一个开放问题(IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022)。我们证明,这种设计框架(使用 \(P_\tau \) 属性)可以提供已知弯曲函数类之外的弯曲函数实例、包括\({\mathcal{M}\mathcal{M}}^\#)、\({{mathcal {C}}},{{\mathcal {D}}}\)和\({{mathcal {D}}}_0\) 类,其中后三个类是卡雷特在九十年代初引入的。我们提供了两种识别这类实例的通用方法,其中最值得注意的是,其中一种方法使用的是可能包含线性结构的排列组合。我们首次提出了一系列充分条件,用于计算形式为+ G_1(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))G_2(Tr_1^m(\beta _{k+1}z),\ldots ,Tr_1^m(\beta _{\tau }z))+ G_3(Tr_1^m(\alpha _1y),\ldots 、({\mathcal{M}\mathcal{M}}^\#\)的外部是弯曲的,而没有一个强有力的假设,即 permutation \(\pi \)的成分不承认线性结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Using $$P_\tau $$ property for designing bent functions provably outside the completed Maiorana–McFarland class

In this article, we identify certain instances of bent functions, constructed using the so-called \(P_\tau \) property, that are provably outside the completed Maiorana–McFarland (\({\mathcal{M}\mathcal{M}}^\#\)) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the \(P_\tau \) property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes \({\mathcal{M}\mathcal{M}}^\#\), \({{\mathcal {C}}},{{\mathcal {D}}}\) and \({{\mathcal {D}}}_0\), where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form \(h(y,z)=Tr(y\pi (z)) + G_1(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))G_2(Tr_1^m(\beta _{k+1}z),\ldots ,Tr_1^m(\beta _{\tau }z))+ G_3(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))\) to be bent and outside \({\mathcal{M}\mathcal{M}}^\#\) is specified without a strong assumption that the components of the permutation \(\pi \) do not admit linear structures.

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来源期刊
Accounts of Chemical Research
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.
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