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Multidimensional Fourier transforms and nonlinear functions on finite groups. (English) Zbl 1294.11216
Summary: In this paper we study characterizations of perfect nonlinear functions between arbitrary finite groups. We need to introduce multidimensional Fourier transforms and multidimensional bent functions on arbitrary finite groups, and discuss their properties. Then by using multidimensional bent functions, we obtain a characterization of a perfect nonlinear function between two arbitrary finite groups, and improve the results of L. Poinsot [Cryptogr. Commun. 4, No. 1, 1–23 (2012; Zbl 1282.11165)]. Our main result is the characterization of a perfect nonlinear function by the related difference family. As a direct consequence, a perfect nonlinear function determines a partitioned 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.
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