Constructions of Bent—Negabent Functions and Their Relation to the Completed Maiorana—McFarland Class
Constructions of Bent—Negabent Functions and Their Relation to the Completed Maiorana—McFarland Class
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DOI：
10.1109/tit.2015.2393879
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发表时间：
2015-01
影响因子：
2.5
通讯作者：
Fengrong Zhang;Yongzhuang Wei;E. Pasalic
中科院分区：
文献类型：
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作者：
Fengrong Zhang;Yongzhuang Wei;E. Pasalic
通过最近提出的一系列构造方法，构造不属于完备的Maiorana-McFarland类的Bent-Negabent函数的问题隐含地出现了。这些方法试图优化Bent-Negabent函数的代数度，但所有构造的Bent-Negabent函数都属于完备的Maiorana-McFarland类。在本文中，我们利用Carlet于2004年提出的间接和结构来构造非可证明包含在这类函数中的Bent-Negabent函数，这是在这一方向上的首次重要尝试。为了实现这一点，我们首先提供了一类Bent函数，它具有某些不属于这一类的理想性质，并证明了该类成员的存在性。然后，将这些函数嵌入到间接和构造的框架中，我们能够给出Bent-Negabent函数不包含在完备的Maiorana-McFarland类中的充分条件。
The problem of constructing bent-negabent functions that do not belong to the completed Maiorana-McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent-negabent functions, but all of the constructed bent-negabent functions belong to the completed Maiorana-McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent-negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent-negabent functions not being contained in the completed Maiorana-McFarland class.