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Bentness and nonlinearity of functions on finite groups. (English) Zbl 1359.11092
Summary: Perfect nonlinear functions between two finite abelian groups were studied by C. Carlet and C. Ding [J. Complexity 20, No. 2–3, 205–244 (2004; Zbl 1053.94011)] and 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 O. A. Logachev et al. [Discrete Math. Appl. 7, 547–564 (1997; Zbl 0982.94012)]. 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.

MSC:
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
20C99 Representation theory of groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
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