具有良好密码学性质的平衡函数和bent函数的研究

鉴于硬差错攻击在流密码上取得的成功和人们还不能完全认识bent函数的事实，本项目旨在研究密码函数抗硬差错攻击的能力和bent函数的设计与分析。具体内容包括：1）研究具有良好密码学性质的平衡函数抗硬差错攻击的能力，尝试构造具有高非线性度最优代数免疫度且能抵抗硬差错攻击的平衡函数；2）系统研究多项式bent函数和bent函数的"间接构造"方法，利用Walsh变换、迹函数以及指数和等知识，探讨bent函数的新构造方法；3）采用二阶差分法，分析"间接构造"所构造bent函数与"完全Maiorana- McFarland（M-M）类bent函数"之间的关系,探寻不属于"完全M-M类bent函数"的函数并研究它们的共同特征。本课题的研究成果将为对称密码和编码理论的设计提供新型资源，为分析密码函数提供新思路，为构造良好性质的密码函数提供新方法。本项目的研究内容符合我们的专业方向和长期积累。

The hard fault analysis is effective to stream ciphers and we do not fully understand bent functions, so the project studies the resistance of cryptographic functions against the hard fault analysis and the design and analysis of bent functions. The contents include the followings: 1) We shall investigate the resistance of balanced functions with good cryptographic properties against the hard fault analysis, and on this basis construct balanced functions which can be resistant to hard fault analysis and have high nonlinearities and an optimal algebraic immunity. 2) By digging into bent functions in polynomial forms and secondary constructions of bent functions, we will study new constructions of bent functions based on Walsh transform and the knowledges of trace functions and exponential sums. 3) The relationships between bent functions constructed by secondary constructions and the completed Maiorana-McFarland (M-M) class bent functions will be studied by using second-order differences. In addition, we shall find the functions which do not belong to the completed M-M class bent functions and study their same characters. The contributions of the project will provide novel design resource for symmetric ciphers and coding theory, good ideas for analysing cryptographic functions and new methods for constructing functions with good cryptographic properties. The contents of the project are in line with our professional direction and long-term accumulation of knowledge.