旋转对称布尔函数的关键问题研究

Boolean functions play a central role in cryptographic algorithms since the security and efficiency of the algorithms are mainly determined by the cryptographic properties and the algebraic structure of the involved Boolean functions. Rotation symmetric Boolean function, used as a nonlinear component in cryptographic algorithms, can promote their security and efficiency simultaneously as they can reduce the memory needed for describing the function. This project focuses on constructing cryptographically important rotation symmetric Boolean functions and analyzing their cryptographic properties, such as: constructions of rotation symmetric bent functions, constructions of idempotent bent functions (rotation symmetric bent functions defined over finite field), and the asymptotic balancedness of symmetric Boolean functions with fixed algebraic degree. Through the research of this project, we can not only evaluate the cryptographic significance of rotation symmetric Boolean functions but also provide many nonlinear functions to cryptographic algorithms with better security and efficiency. ..

作为密码算法的核心部件, 布尔函数的密码学性质和代数结构在很大程度上决定着算法的安全性与效率。旋转对称布尔函数是一类结构简单、运算速度快的布尔函数，将其作为密码算法的非线性部件时可以大幅度地提高算法的安全性和效率。本项目研究密码学中旋转对称布尔函数的构造与分析，具体内容有：旋转对称 bent函数的构造、幂等bent函数（定义在有限域上的旋转对称bent函数）的构造和代数次数固定的对称布尔函数的渐进平衡性。通过本项目的研究不但能够丰富研究旋转对称布尔函数的理论方法，而且还能够为密码算法的设计提供安全性强、效率高的非线性函数。

布尔函数作为密码算法的核心部件, 其密码学性质和代数结构在很大程度上决定着算法的安全性与效率。旋转对称布尔函数是一类结构简单、运算速度快的布尔函数，将其作为密码算法的非线性部件时可以大幅度地提高算法的安全性和效率。本项目研究了密码学中旋转对称布尔函数的构造与分析，构造了旋转对称 bent函数、幂等bent函数、生成了性质优良的旋转对称S盒；证明了长达十年的Canteaut和Videau提出的关于对称布尔函数的渐进平衡性猜想。该项目的实施为密码算法的设计提供了理论基础和技术支持。