2019/05/25 by Lijing Zheng, Jie Peng, Zheng, Lijing +7 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1905.10504
openalex publication_date 2019/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any positive integers n=2k and m such that m≥ k, in this paper we show the maximal number of bent components of any (n,m)-functions is equal to 2m-2m-k, and for those attaining the equality, their algebraic degree is at most k. It is easily seen that all (n,m)-functions of the form G(x)=(F(x),0) with F(x) being any vectorial bent (n,k)-function, have the maximum number of bent components. Those simple functions G are called trivial in this paper. We show that for a power (n,n)-function, it has such large number of bent components if and only if it is trivial under a mild condition. We also consider the (n,n)-function of the form Fi(x)=x^2ih(\rm Trne(x)), where h: \mathbbF2e → \mathbbF2e, and show that Fi has such large number if and only if e=k, and h is a permutation over \mathbbF2k. It proves that all the previously known nontrivial such functions are subclasses of the functions Fi. Based on the Maiorana-McFarland class, we present constructions of large numbers of (n,m)-functions with maximal number of bent components for any integer m in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is found that our constructions can also provide new plateaued vectorial functions.