有限域上的完全置换多项式与Bent-Negabent函数构造研究

Complete permutation polynomials are a special class of permutation polynomials, they have important applications in coding theory and cryptography. Due to the difficulty of characterizing the features of complete permutation polynomials over finite fields, the known constructions of complete permutation polynomials are few. This is a major barrier to the widespread applications of complete permutation polynomials. Bent-Negabent functions are the functions which have flat spectra with respect to two special kinds of generalized discrete Fourier transforms. However, there are only a few known constructions of Bent-Negabent functions due to their strict criterion. .The main research contents of this project are as follows. First, based on the analysis of our computer search results, complete permutation polynomials which are unknown before will be constructed by considering the properties of some partial exponential sums and the number of solutions of some special equations over finite fields. The mathematical tools such as finite fields, number theory and algebraic combinatorics will be used. Second, from the different viewpoints of algebraic combinatorics and finite fields, we will explore effective methods for constructing bent-negabent functions by using complete permutation polynomials over finite fileds, and then construct bent-negabent functions with high algebraic degree by using these methods. Theoretically, this project will extend the list of known complete permutation polynomials, and in practice, this project will play an important role in the study of certain topics in cryptography and coding theory.

有限域上的完全置换多项式是一类特殊的置换多项式，在编码理论和密码系统中有重要的应用。由于其形式多样，规律较难刻画，导致已发现的完全置换多项式还很少，进而限制了其在各领域中的应用。Bent-Negabent函数是在两种不同的广义离散傅立叶变换下都具有均匀频谱的布尔函数。然而，由于此类函数条件要求苛刻，目前其构造还很少。.本课题研究内容有两方面。首先，总结计算机搜索结果的内在规律，利用有限域、数论和代数组合论等数学工具，通过分析有限域上部分指数和的性质以及特定条件下方程的解数等问题，构造一系列前人未知的完全置换多项式。其次，从代数组合论和有限域等不同角度，探索利用有限域上的完全置换多项式构造Bent-Negabent函数的有效途径，构造高代数次数的Bent-Negabent函数。本课题将在理论上丰富完全置换多项式的结果，在应用上对密码学和编码理论的研究发挥重要作用。

本课题主要研究了两方面。首先，通过分析有限域上部分指数和的性质以及特定条件下方程的解数等问题，构造了一系列前人未知的置换多项式。其次，从代数组合论和有限域等不同角度，构造了一系列Bent-Negabent函数。取得的主要成果如下：1）提出一种新的构造置换多项式的方法，构造出了一系列前人未知的置换多项式；2）利用置换多项式及其逆函数的性质，构造了一系列有限域上的Negabent 函数和Bent-Negabent 函数；3）利用有限域中指数和的思想，解决了前人一个关于Negabent函数的公开问题；4）利用Bent函数和Bent_4函数之间的联系构造了几类具有至少两个均匀频谱的布尔函数。本课题将在理论上丰富置换多项式的结果，在应用上对密码学和编码理论的研究发挥重要作用。