高非线性函数及其相关组合对象与编码研究

有限阿贝尔群上具有高非线性度或低差分均匀度的函数在密码学中被统称为高非线性函数。这类函数在密码、编码和组合数学中具有重要应用，可用于构造分组密码中的S-盒、具有优美结构的组合对象、具有优良相关性的序列编码和具有优异纠错能力的线性码。本项目拟基于有限域、指数和、组合设计、交织技术等理论和工具，对高非线性函数及其相关组合对象与编码进行深入研究：1. 研究高非线性函数的性质，设计新型高非线性函数；2. 研究高非线性函数的像集与原像集，构造新的组合对象；3. 利用高非线性函数构造编码，得到达到理论界的最优序列编码和纠错编码。这些课题的研究具有重要的理论意义和应用价值。

Highly nonlinear functions over Abelian group with high nonlinearity or low differential uniformity have important applications in cryptography, coding theory and combinatorics. It turns out that they can be used to construct S-box for block cipher, to obtain combinatorial objects with nice structures，to construct sequences with optimal correlation and error-correcting codes with good performance. The main objective of this project is to give a well-rounded treatment of highly nonlinear functions with applications in combinatorics and coding theory based on the theory of finite fields, exponential sums, combinatorial design， and interleaving technique. To achieve this goal, we will first study the properties and constructions for highly nonlinear functions. Secondly, we will study the combinatorial objects corresponding to these new proposed functions. Finally, we will employ these functions to obtain sequences and error-correcting codes whose parameters are optimal in the sense they meet some bounds. Study on these subjects has both important theoretical significance and application value.