Further analysis of bent functions from C and D which are provably outside or inside M
Further analysis of bent functions from C and D which are provably outside or inside M
复制标题
进一步分析 C 和 D 的弯曲函数，可证明它们位于 M 之外或之内
DOI：
10.1016/j.dam.2020.06.012
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发表时间：
2020
影响因子：
1.1
通讯作者：
Yongzhuang Wei
中科院分区：
文献类型：
--
作者：
Fengrong Zhang;Nastja Cepak;Enes Pasalic;Yongzhuang Wei
90年代初，Carlet(1994)引入了两类新的Bent函数，这两类函数都是从Maiorana-McFarland(M)类衍生出来的，并分别命名为C类和D类。除了D的一个子类，由Carlet表示为D-0，它被证明是在两个主要的(完备的)Bent函数的初等类之外，关于它们的有效构造知之甚少。更重要的是，这两个类都可以很容易地保留在基础M类中，这在Mandal等人中已经提到过。(2016)。假设可以指定属于这两个类(D-0除外)之一的Bent函数f，那么最重要的问题是确定f是否仍然包含在已知的初级类中，或者位于它们的完整版本之外。在这篇文章中，我们进一步阐述了对Zhang等人给出的一组充分条件的分析。(2017)关于C和D中可证明在M之外的Bent函数的规范，证明了这些与D类Bent函数有关的条件可以放宽，以便即使那些分量函数允许线性结构的排列仍然可以在设计中使用。文中还证明了形式为x(2r+1)的单项置换的逆对于n>；4绝不是二次的，这导致了在C中但在M之外的Bent函数的无穷大类。类似地，利用D和M之外的Bent函数的一组宽松的充分条件，我们识别出了这类Bent函数的一个显式无穷类。我们还推广了C和D中Bent函数的某些子类的包含性质，正如最初在Carlet(1994)，Mandal等人中讨论的那样。(2016)，最终属于已完成的M级。最值得注意的是，我们指定了D的其他一般和显式子类，用D-k表示，因为k是{1，...，n-2}的元素，其成员是可证明在完全M类之外的Bent函数。(C)2020爱思唯尔B.V.保留所有权利。
In early nineties Carlet (1994) introduced two new classes of bent functions, both derived from the Maiorana-McFarland (M) class, and named them C and D class, respectively. Apart from a subclass of D, denoted by D-0 by Carlet, which is provably outside two main (completed) primary classes of bent functions, little is known about their efficient constructions. More importantly, both classes may easily remain in the underlying M class which has already been remarked in Mandal et al. (2016). Assuming the possibility of specifying a bent function f that belongs to one of these two classes (apart from D-0), the most important issue is then to determine whether f is still contained in the known primary classes or lies outside their completed versions. In this article, we further elaborate on the analysis of the set of sufficient conditions given in Zhang et al. (2017) concerning the specification of bent functions in C and D which are provably outside M. It is shown that these conditions, related to bent functions in class D, can be relaxed so that even those permutations whose component functions admit linear structures still can be used in the design. It is also shown that monomial permutations of the form x(2r+1) have inverses which are never quadratic for n > 4, which gives rise to an infinite class of bent functions in C but outside M. Similarly, using a relaxed set of sufficient conditions for bent functions in D and outside M, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in C and D, as addressed initially in Carlet (1994), Mandal et al. (2016) that are ultimately within the completed M class. Most notably, we specify other generic and explicit subclasses of D, denoted by D-k for k is an element of {1, ..., n - 2}, whose members are bent functions provably outside the completed M class. (C) 2020 Elsevier B.V. All rights reserved.