密码学中完全置换多项式的研究

分组密码算法的安全性往往取决于其唯一的非线性部件S盒，人们通常将安全性能良好S盒的设计问题转化为具有良好性质多输出布尔函数的构造及分析等数学问题。偶特征有限域上的完全置换多项式是一类具有良好位独立性和雪崩特性的多输出布尔函数，可用于设计S盒，且其应用涉及密码、编码、组合学等多个领域。完全置换多项式的研究已成为当前密码编码等领域的研究热点。本项目将综合运用代数学、组合学以及计算机代数等相关工具来研究具有密码学应用背景的完全置换多项式。研究内容主要包括：（1）刻画完全置换多项式的圈结构、非线性度等密码学性质；（2）构造具有特殊形式、特殊圈结构或具有良好密码学性质的完全置换多项式；（3）探讨基于完全置换多项式的高非线性函数的设计问题。本项目的研究可为密码系统提供更多可供选择的优秀密码函数，为分组密码S盒的设计提供新的思路和技术，具有重要的理论和实际意义。

The security of block cipher algorithms often depends on their unique nonlinear component S-boxes. People usually transform the design of S-boxes with good security performance into the construction and analysis of multi-output Boolean functions with good properties. Complete permutation polynomials (CPPs) over finite fields of even characteristic are a class of multi-output Boolean functions with good bit independence and avalanche characteristics, which can be used in the design of S-boxes and have applications in the fields of cryptography, coding theory, combinatorics, etc. The research of CPPs has become a hot topic in the fields of cryptography, coding theory, etc. Applying the tools of algebra, combinatorics and computer algebra, etc., this project will focus on the construction and analysis of CPPs with cryptography application background. The main contents of this project are as follows: (1) we will characterize the cryptographic properties of CPPs, such as cycle structure and nonlinearity; (2) we will discuss the constructions of CPPs with special forms, special cycle structures or good cryptographic properties; (3) based on CPPs, the design of cryptographic functions with high nonlinearity will also be discussed. The research of this project can provide more cryptographic functions with good properties for the cryptographic systems, and provide new ideas and techniques for the design of S-boxes of block ciphers. It has important theoretical and practical significance.