4-差分置换与向量Bent函数的密码性质及构造

非线性密码函数是对称密码算法的重要组件，其密码性质直接决定了密码体制的安全性。低差分函数具有良好的抵抗差分攻击的能力，而Bent函数具有最优的抵抗线性攻击的能力。这两类重要密码函数的研究涉及代数与组合中的一些困难问题，具有一定的挑战性。本项目拟研究4-差分置换和向量Bent函数的密码性质及构造方面的几个关键问题，具体包括：（1）研究向量布尔函数的差分均匀度与代数次数、非线性度等其他密码指标的内在联系；（2）给出几种构造4-差分置换的新方法，构造兼具高代数次数与已知最高非线性度的4-差分置换；（3）研究具有最大Bent分支数的向量布尔函数的密码性质及构造；（4）给出几种构造向量Bent 函数的新方法，等等。这些都是近几年密码函数基础理论研究方面受到国内外同行高度关注的前沿热点问题。本项目的研究成果将对密码函数在密码、编码与组合中的应用产生积极影响。

Nonlinear cryptographic functions are an important component of the symmetric key algorithms. The cryptographic properties of the functions play a crucial role in the security of symmetric key algorithms. Differentially low uniform functions have good ability to resist the differential attack, and Bent functions possess the best ability to resist the linear attack. The research on these two important classes of cryptographic functions relates to the study on some difficult problems in algebra and combinatorics, hence is full of challenging. This project intends to investigate some important problems relating to the cryptographic properties and constructions of differentially 4-uniform permutations and vectorial Bent functions, including: (1) to study the relations between differential uniformity and other cryptographic properties including algebraic degree and nonlinearity; (2) to propose new methods to construct differentially 4-uniform permutations, and to construct differentially 4-uniform permutations with high algebraic degree and the best known nonlinearity; (3) to study the cryptographic properties and constructions of vectorial Boolean fuctions with the maximum number of Bent components; (4) to propose new methods to construct vectorial Bent functions, and so on. These problems are basic and from the frontiers of cryptographic function research and have attracted much attention from many cryptanalysts so far. This project will make important contributions to the applications of cryptographic functions in cryptography, coding theory and combinatorics.