an:06008465
Zbl 1282.11165
Poinsot, Laurent
Non abelian bent functions
EN
Cryptogr. Commun. 4, No. 1, 1-23 (2012).
00295734
2012
j
11T71 06E30 20D99 43A30
bent functions; perfect nonlinearity; finite non Abelian groups; Fourier transform; linear representations
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.