O. A. Logachev, Sergey N. Fedorov, V. V. Yashchenko
{"title":"On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions","authors":"O. A. Logachev, Sergey N. Fedorov, V. V. Yashchenko","doi":"10.1515/dma-2022-0016","DOIUrl":null,"url":null,"abstract":"Abstract Let G be the extension of a general affine group by the group of affine functions. We study the action of G on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of G. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dma-2022-0016","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let G be the extension of a general affine group by the group of affine functions. We study the action of G on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of G. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.
期刊介绍：
The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.
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