关于单值Bent函数的两个问题
Two Problems about Monomial Bent Functions
摘要：
在2008年，Langevin和Leander利用Stickelberger定理确定了三类单项bent函数的对偶函数：Dillon，Gold和Kasami。在他们的论文中，他们提出了一个非常强的条件，使得他们的方法有效，并证明了Gold指数和Kasami指数都满足这个条件。2018年，Pott { \em et al.}研究了具有最大弯曲分量数的向量函数的问题。他们发现一类二项式函数达到了上界。他们还提出了一个关于具有最大弯曲分量数的单项函数的公开问题。本文得到了关于Langevin和Leander条件的一个有趣结果，并解决了Pott { \em et al.}的公开问题.具体而言，我们表明：1）对于$\mathbb{F}_{2^{2k}}$上的单项Bent函数，如果其指数满足Langevin和Leander条件的第一部分，则它满足整函数条件; 2）$x^{2^k+1}$是$\mathbb{F}_{2^{2k}}$上唯一具有最大Bent分支数的单项函数。幸运的是，作为结果，我们还解决了2006年Ness和Helleseth的公开问题。
英文摘要：
In 2008, Langevin and Leander determined the dual function of three classes of monomial bent functions with the help of Stickelberger's theorem: Dillon, Gold and Kasami. In their paper, they proposed one very strong condition such that their method works, and showed that both Gold exponent and Kasami exponent satisfy this condition. In 2018, Pott {\em et al.} investigated the issue of vectorial functions with maximal number of bent components. They found one class of binomial functions which attains the upper bound. They also proposed an open problem regarding monomial function with maximal number of bent components. In this paper, we obtain an interesting result about the condition of Langevin and Leander, and solve the open problem of Pott {\em et al.}. Specifically, we show that: 1) for a monomial bent function over $\mathbb{F}_{2^{2k}}$, if the exponent satisfies the first part of the condition of Langevin and Leander, then it satisfies the entire condition; 2) $x^{2^k+1}$ is the only monomial function over $\mathbb{F}_{2^{2k}}$ which has maximal number of bent components. Fortunately, as a consequence, we also solve an open problem of Ness and Helleseth in 2006.
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