an:06477345
Zbl 1359.11092
Xu, Bangteng
Bentness and nonlinearity of functions on finite groups
EN
Des. Codes Cryptography 76, No. 3, 409-430 (2015).
0925-1022 1573-7586
2015
j
11T71 20C99 43A30 94A55
bent functions; perfect nonlinear functions; Fourier transforms; class functions; representations; characters
Summary: Perfect nonlinear functions between two finite abelian groups were studied by \textit{C. Carlet} and \textit{C. Ding} [J. Complexity 20, No. 2--3, 205--244 (2004; Zbl 1053.94011)] and \textit{A. Pott} [Discrete Appl. Math. 138, No. 1--2, 177--193 (2004; Zbl 1035.05023)], which can be regarded as a generalization of bent functions on finite abelian groups studied by \textit{O. A. Logachev} et al. [Discrete Math. Appl. 7, 547--564 (1997; Zbl 0982.94012)]. \textit{L. Poinsot} [Multidimensional bent functions. GESTS Int. Trans. Comput. Sci. Eng. 18, No. 1, 185--195 (2005); J. Discrete Math. Sci. Cryptography 9, No. 2, 349--364 (2006; Zbl 1105.43002), Cryptogr. Commun. 4, No. 1, 1--23 (2012; Zbl 1282.11165)] extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.
1053.94011; 1035.05023; 0982.94012; 1105.43002; 1282.11165