有限域上函数最大非线性的判据

IF 0.3 Q4 MATHEMATICS, APPLIED

Discrete Mathematics and Applications Pub Date : 2023-03-01 DOI:10.1515/dma-2023-0012

V. G. Ryabov

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引用次数: 1

摘要

具有q个元素的域上的n位函数如果在所有这些函数中具有最大的非线性，则称为最大非线性。得到了最大非线性的判据和必要条件，这意味着，对于偶数n，最大非线性函数是bent函数，而对于q > 2，已知的bent函数族不是最大非线性的。对于任意有限域，得到了函数到所有仿射映射的Hamming距离与函数非平凡特征的傅立叶谱之间的关系。

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Criteria for maximal nonlinearity of a function over a finite field

Abstract An n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.

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来源期刊

Discrete Mathematics and Applications MATHEMATICS, APPLIED-

CiteScore

0.60

自引率

20.00%

发文量

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.