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首页> 外文期刊>IEEE Transactions on Information Theory >Constructing Bent Functions Outside the Maiorana–McFarland Class Using a General Form of Rothaus
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Constructing Bent Functions Outside the Maiorana–McFarland Class Using a General Form of Rothaus

机译：使用Rothaus的一般形式构造Maiorana–McFarland类之外的Bent函数

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摘要

In the mid 1960s, Rothaus proposed the so-called “most general form” of constructing new bent functions by using three (initial) bent functions whose sum is again bent. In this paper, we utilize a special case of Rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an n/2 -dimensional subspace. This simplification allows us to treat the induced bent conditions more easily, also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so-called non-normal bent functions. Affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class (stemming from the complete Desarguesian spread). It is also shown that when the initial bent functions belong to the class D , then, under certain conditions, the constructed functions provably do not belong to the completed Maiorana–McFarland class. We conjecture that our method potentially generates an infinite class of non-normal bent functions (all tested ten-variable functions are non-normal but unfortunately they are weakly normal) though there are no efficient computational tools for confirming this.
机译：在1960年代中期，Rothaus提出了所谓的“最一般形式”，即通过使用三个（初始）弯曲函数求和，它们的总和再次被弯曲来构造新的弯曲函数。在本文中，当这三个弯曲函数中的两个因适当选择的n / 2维子空间特征函数而不同时，我们利用了Rothaus结构的特殊情况。这种简化使我们能够更轻松地处理诱发的弯曲条件，这还意味着可以在部分扩展类别中指定初始函数，最显着的是可以识别所谓的非正常弯曲函数的几种情况。然后使用部分扩展类别（从完整的Desarguesian扩展中提取）中的初始弯曲函数的适当选择来识别此类中的仿射不等价弯曲函数。还表明，当初始弯曲函数属于D类时，则在某些条件下，构造函数可证明不属于完整的Maiorana–McFarland类。我们推测，尽管没有有效的计算工具来确认这一点，但我们的方法可能会生成无限类的非正态弯曲函数（所有测试的十个变量函数均非正态，但不幸的是它们是弱正态的）。

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