Bent函数和plateaued函数的构造和分类研究

在密码和编码领域，bent函数和plateaued函数起着重要的作用。然而，目前所知的bent（或plateaued）函数仅占它们总数的很小一部分。针对目前密码函数构造和分析方法相对较少这一问题，本项目从函数的“构造”和“分类”两个方面展开研究。借助“复合构造”和“间接构造”等方法来设计bent函数和plateaued函数，通过对抽象构造方法的具体化来获得构造函数的新方法。给出构造plateaued函数的“逆向构造”，进而获得新的plateaued函数和“谱不相交函数集”。在此基础上，提出构造多输出bent（或plateaued）函数（尤其是Almost Bent 函数）的新方法。结合所构造函数的特性，研究函数Carlet-Charpin-Zinoviev(CCZ)等价性的分析方法。最终给出不属于已知函数“完全类”的新密码函数。研究成果将为对称密码核心部件的设计与分析提供新思路及实用工具。

Bent and plateaued functions play very important roles in cryptography and coding theory. The design methods of bent and plateaued functions have been addressed in several works in the last few decades, but the number of known bent (or plateaued) functions seems negligible with respect to the total number of bent (or plateaued) functions. We shall focus on the constructions and classifications of cryptographic functions. The composite representation of Boolean functions and secondary constructions of bent functions will be firstly investigated. We shall provide several generic methods for constructing bent and plateaued functions by using the general constructions. A reversed design method that specifies plateaued functions in the Walsh spectral domain will be presented. We shall utilize this method to construct new plateaued functions and a large set of disjoint spectra functions. Further, on this basis, the constructions of multi-output bent or plateaued functions (in particular, Almost Bent) will be proposed. On the other hand, according to the properties of the constructed functions, we shall study the methods of analyzing Carlet-Charpin-Zinoviev (CCZ) equivalence between two cryptographic functions. In this project, we try to present several new functions which are not included in the completed classes of the known primary classes. These results will provide some new theoretical and practical analysis tools for the design and analysis of symmetric ciphers.