布尔函数的局部安全准则研究

The stream cipher based on permutation filter is applicable to homomorphic encryption in cloud service. Boolean functions used in this kind of stream ciphers should satisfy local cryptographic criteria to ensure security, and have low algebraic degree to decrease the noise of homomorphic evaluation. Up to now, there are only a few known results on local cryptographic criteria of Boolean functions. Constructing Boolean functions satisfying several local cryptographic criteria and with relatively low algebraic degree becomes a key problem in the design of homomorphic-friendly stream ciphers. The research objective of this project is to study local cryptographic criteria of Boolean functions. The main research contents include the following three aspects. 1. We will study the relationships among balancedness, algebraic degree, nonlinearity, and algebraic immunity of Boolean functions when the variables are restricted to the vectors with constant Hamming weight. 2. We will construct balanced Boolean functions such that all the sub-functions with variables restricted to the vectors of constant Hamming weight satisfy weightwise (almost) balancedness, high weightwise nonlinearity, and high weightwise algebraic immunity simultaneously. 3. We will explore the methods of decreasing algebraic degree without weakening the local cryptographic criteria of Boolean functions. This project will enrich the theoretical foundations of Boolean functions, and extend the application of Boolean functions in homomorphic encryption.

基于置换滤波的流密码可以适用于同态加密在云服务应用环境中的需求，其关键安全部件布尔函数应满足局部安全准则以保证系统的安全性，并且具有较低的代数次数以保证函数电路同态计算噪声较小。目前，关于布尔函数局部安全准则的相关研究较少，构造代数次数相对较低并且同时满足多项局部安全准则的布尔函数成为了面向同态加密的新型流密码设计中亟待解决的问题。本项目拟对布尔函数局部安全准则的关键问题进行深入研究：1.研究布尔函数自变量限制在等重向量集上的平衡性、代数次数、非线性度和代数免疫度之间的关系；2.构造平衡的布尔函数使其满足当自变量限制在各个等重集上时，子函数均满足局部（几乎）平衡性、局部高非线性度和局部高代数免疫度；3.研究在基本不改变局部安全准则的条件下降低布尔函数代数次数的方法。本项目的研究将丰富布尔函数的理论基础，扩展布尔函数在同态加密中的应用。

基于置换滤波的流密码FLIP适用于云服务下的同态加密，其关键部件布尔函数应满足局部安全准则和较低的代数次数以保证系统的安全性和同态计算乘法深度。构造代数次数较低并满足多项局部安全准则的布尔函数成为了流密码FLIP设计中亟待解决的问题。本项目对布尔函数局部安全准则的关键问题进行了深入的研究，给出了布尔函数局部安全准则之间的关系，构造了自变量限制在等重集上满足局部（几乎）平衡性和高非线性的布尔函数，并得到了降低代数次数的方法，同时，给出了适用于同态加密的布尔函数的相关应用。项目成果包括学术论文11篇，受理专利4项，培养硕士研究生6人。项目取得的成果对研究适用于同态加密的布尔函数具有重要的学术价值和理论意义，并在基于同态加密的密态数据融合、可信计算理论等领域具有重要的应用价值。