On vectorial functions with maximal number of bent components

IF 1.4 2区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS
Xianhong Xie, Yi Ouyang, Honggang Hu
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Abstract

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of \(F(x)=x^{2^e}h(\textrm{Tr}_{2^{2m}/2^m}(x))\), where \(e\ge 0\) and h(x) is a permutation over \({\mathbb {F}}_{2^m}\). If h(x) is monomial, the nonlinearity of F(x) is shown to be at most \( 2^{2\,m-1}-2^{\lfloor \frac{3\,m}{2}\rfloor }\) and some non-plateaued and plateaued functions attaining the upper bound are found. If h(x) is linear, the exact nonlinearity of F(x) is determined. Secondly, we give a construction of vectorial functions with maximal number of bent components from known ones, thus obtain two new classes from the Niho class and the Maiorana-McFarland class. Our construction gives a quadratic vectorial function that is not equivalent to the known functions of the form xh(x), and also contains vectorial functions outside the completed Maiorana-McFarland class. Finally, we show that the vectorial function \(F: {\mathbb {F}}_{2^{2m}}\rightarrow {\mathbb {F}}_{2^{2m}}\), \(x\mapsto x^{2^m+1}+x^{2^i+1}\) has maximal number of bent components if and only if \(i=0\).

关于具有最大弯曲分量数的向量函数
本文研究具有最大弯曲分量数的向量函数。我们首先研究了\(F(x)=x^{2^e}h(\textrm{Tr}_{2^{2m}/2^m}(x))\)的Walsh变换和非线性,其中\(e\ge 0\)和h(x)是\({\mathbb {F}}_{2^m}\)上的一个置换。如果h(x)是单项式,则F(x)的非线性不超过\( 2^{2\,m-1}-2^{\lfloor \frac{3\,m}{2}\rfloor }\),并找到了一些达到上界的非平稳函数和平稳函数。如果h(x)是线性的,F(x)的非线性是确定的。其次,我们从已知的向量函数中构造出弯曲分量数目最大的向量函数,从而从Niho类和Maiorana-McFarland类中得到两个新的类。我们的构造给出了一个二次向量函数,它不等同于已知形式的xh(x)函数,并且还包含了已完成的Maiorana-McFarland类之外的向量函数。最后,我们证明了向量函数\(F: {\mathbb {F}}_{2^{2m}}\rightarrow {\mathbb {F}}_{2^{2m}}\), \(x\mapsto x^{2^m+1}+x^{2^i+1}\)具有最大的弯曲分量当且仅当\(i=0\)。
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来源期刊
Designs, Codes and Cryptography
Designs, Codes and Cryptography 工程技术-计算机:理论方法
CiteScore
2.80
自引率
12.50%
发文量
157
审稿时长
16.5 months
期刊介绍: Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.
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