On constructions and properties of (n, m)-functions with maximal number of bent components

• Published in: Designs, codes, and cryptography codes, and cryptography, 2020-10, Vol.88 (10), p.2171-2186
• Main Authors: Zheng, Lijing, Peng, Jie, Kan, Haibin, Li, Yanjun, Luo, Juan
• Format: Article
• Language: English
• Subjects: Bivariate analysis; Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Permutations
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-020-00770-7

For any positive integers n = 2 k and m such that m ≥ k , in this paper we show that the maximal number of bent components of any ( n ,  m )-function is equal to 2 m - 2 m - 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 maximal 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 the maximal number of bent components if and only if it is trivial. We also consider the ( n ,  n )-function of the form F ( x ) = x h ( Tr e n ( x ) ) , where h : F 2 e → F 2 e , and show that F has the maximal number of bent components if and only if e = k , and h is a permutation over F 2 e . It essentially shows that all previously known nontrivial functions with maximal number of bent components are subclasses of the class described by F . 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 spectra and Walsh spectra of the constructed functions. It turns out that our constructions can also provide new plateaued vectorial functions.