k阶拟Bent函数在密码设计和通信中的应用

王育民、何大可提出了布尔函数关于线性函数的r阶相关度E(r)的概念来刻划布尔函数抵抗相关攻击的能力，本文以极小化所有非零相关度E(r)为主要目的，利用k阶拟Bent函数的特殊性质，给出了一类基于k阶拟Bent函数的“最佳”非线性组合设计的实现，构造了一类平衡的，具有高阶相关免疫性，而且非零相关度一致地小的非退化的布尔函数，并比较了它与基于部分Bent函数的“最佳”非线性组合设计的优劣。最后我们又利用k阶拟Bent函数构造了一类Bent互补函数族和Bent侣，Bent互补函数族和Bent侣在最佳信号设计方面意义重大，这也表明k阶拟Bent函数在密码设计和通信领域都有比较广的应用前景。

The application of k-order quasi-Bent functions in cryptology and communication fields

Wang Yumin and He Dake proposed the definition of the rth Correlation measure E(r) to decide whether the nonlinear combining functions can avoid correlation attack. In this paper, the best design of nonlinear combining functions aiming to minimize all the nonzero rth Correlation measure E(r) is introduced. Based on the k-order quasi-Bent functions, a class of Boolean functions with higher order correlation-immunity is presented whose nonzero rth correlation measures are same. Moreover they are also nondegenerate. The cryptographic properties of this class of nonlinear combining functions are better than the function that is designed on the basis of partially-Bent functions. In the end of this paper, the method is given to construct the families of Bent complimentary functions, which shows that the k-order quasi-Bent functions are of much importance in the cryptology and communication field.