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Non abelian bent functions. (English) Zbl 1282.11165
Summary: Perfect nonlinear functions from a finite group $$G$$ to another one $$H$$ are those functions $$f: G\to H$$ such that for all nonzero $$\alpha \in G$$, the derivative $$d_{a}f: x \mapsto f(ax) f(x)^{-1}$$ is balanced. In the case where both $$G$$ and $$H$$ are abelian groups, $$f: G \to H$$ is perfect nonlinear if, and only if, $$f$$ is bent, i.e., for all nonprincipal characters $$\chi$$ of $$H$$, the (discrete) Fourier transform of $$\chi \deg f$$ has a constant magnitude equal to $$|G|$$. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where $$G$$ and/or $$H$$ are (finite) non-abelian groups. Thus we extend the concept of bent functions to the framework of non-abelian groups.

##### MSC:
 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 06E30 Boolean functions 20D99 Abstract finite groups 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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