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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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