Abstract
Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis \({\widehat{G}}\), consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on \({\widehat{G}}\), and obtain characterizations of a bent function by its Fourier transforms (as functions on \({\widehat{G}}\)). For a function f from G to another finite group, we define a dual function \({\widetilde{f}}\) on \({\widehat{G}}\), and characterize the nonlinearity of f by its dual function \({\widetilde{f}}\). Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.
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The authors would like to thank the referees; their useful comments have improved the paper.
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Communicated by A. Pott.
The first author is supported by NSFC Grant 11271005.
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Fan, Y., Xu, B. Fourier transforms and bent functions on finite groups. Des. Codes Cryptogr. 86, 2091–2113 (2018). https://doi.org/10.1007/s10623-017-0439-0
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DOI: https://doi.org/10.1007/s10623-017-0439-0