A new characterization of group action-based perfect nonlinearity

The left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X.