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On Two-to-One Mappings Over Finite Fields

机译：关于有限域上的二对一映射

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Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of x(r) h(x((q-1)/d)), those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.
机译：有限域上的二对一（2-to-1）映射在对称密码学中扮演重要角色。特别是，它们允许设计APN功能，弯曲功能和半弯曲功能。在本文中，我们提供了对在有限域上定义的二对一映射的系统研究。我们通过沃尔什变换来表征这种映射。我们还提出了几种构造，包括类似AGW的准则，具有x（r）h（x（（q-1）/ d））形式的构造，来自置换多项式，来自线性转换器和来自APN函数的构造。然后，我们介绍古典多项式类别中的2对1多项式映射：线性多项式和单项式，低阶多项式，Dickson多项式和Muller-Cohen-Matthews多项式等。最后，我们展示了2对1映射在弯曲布尔和矢量弯曲函数，半弯曲函数，平面函数和置换多项式的构造的有限域。在所有这些方面，我们将回顾已知的内容并提供几个新结果。

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