Maximally nonlinear functions over finite fields

IF 0.3 Q4 MATHEMATICS, APPLIED
V. G. Ryabov
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引用次数: 2

Abstract

Abstract An n-place function over a field Fq $ \mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is qn−1(q−1)−qn2−1 $ q^{n-1}(q - 1) - q^{\frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q>2 $ q \gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.
有限域上的最大非线性函数
域Fq$\mathbf上的n位函数{F}_q如果具有q个元素的$在所有q值n位函数中具有最大非线性,则称其为最大非线性。我们证明,即使n=2,一个函数也是最大非线性的,当且仅当它的非线性是qn−1(q−1)−qn2−1$q^{n-1}(q-1)-q ^{\frac n2-1}$;对于n=1,最大非线性的相应准则是q − 2.对于q>2$q\gt 2$,甚至n=2,我们描述了所有最大非线性二次函数的集合,并找到了它的基数。在这种情况下,所有最大非线性二次函数都是二次弯曲函数,并且它们的数量小于弯曲函数的减半数量。
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来源期刊
CiteScore
0.60
自引率
20.00%
发文量
29
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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