旋转对称向量布尔函数若干关键问题研究

Vectorial 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 vectorial functions. Rotation symmetric vectorial Boolean functions, used as.nonlinear components in cryptographic algorithms, can achieve a better trade-off between their security and efficiency, as they can reduce the memory needed for describing the functions. This project focuses on constructions and analyzation of cryptographically important rotation symmetric vectorial Boolean functions, such as: we evaluate two classes of quadratic rotation symmetric vectorial functions used in cryptosystems. We will characterize the differential uniformity of the rotation symmetric vectorial Boolean functions and construct several classes of such functions with low differential uniform. Together with computer program, we will explore new methods for construct rotation symmetric almost bent and vectorial bent functions. We shall develop an automatic algorithm for searching for rotation symmetric S-boxes with excellent performance, which can be applied to the design of hash functions and other symmetric encryption algorithms, especially to lightweight ones. Through the research of this project, we can not only evaluate the cryptographic significance of rotation symmetric vectorial Boolean functions but also provide lots of nonlinear functions to cryptographic algorithms with better security and efficiency.

作为密码算法的核心部件，向量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。旋转对称向量布尔函数是一类结构简单、运算速度快的密码函数，将其作为密码算法的非线性部件时可以兼顾算法的安全性和效率，因此特别适用于资源受限的环境中。本项目旨在研究密码学中旋转对称向量布尔函数，具体内容有：两类已知旋转对称向量函数密码学性质的研究；探讨旋转对称向量布尔函数差分特征，构造低差分均匀的旋转对称向量函数；给出旋转对称几乎bent函数和向量bent函数的构造方法；设计自动化搜索具有优良密码学性质的旋转对称S盒的算法。通过本项目的研究不但能够丰富和发展研究旋转对称向量函数的理论，而且还能为密码算法的设计提供安全性强、效率高的非线性函数。

安全性与效率是密码算法设计的核心要素。本项目首先研究了兼顾安全性和效率的旋转对称函数的密码学性质，计算了一类二次旋转对称向量函数的非线性度和差分均匀度，以及其逆置换。设计了搜索旋转对称S盒的最速下降法和搜索旋转对称布尔函数的遗传算法，得到了众多非线性度高、差分均匀度低的旋转对称函数。其次研究了非线性和差分性均最优的bent函数的递归构造法，证明了几类著名的构造法可由plateaued布尔函数和bent函数的复合来表示，应用Lagrange插值公式计算了其对偶函数。通过深入研究了布尔函数的复合理论，提出了对偶自同构bent函数的概念，从而使递归构造法有了理论遵循。最后还研究了对称布尔函数的渐进相关免疫性和周期性，证明了当代数次数给定时，变元个数充分大的相关免疫对称函数是不存在的。这极大地改进了前人的结果。研究了对称布尔函数的简约值向量和简约代数标准型向量的周期性，证明了在某些条件下二者具有相同的周期。