Fourier Transforms and Bent Functions on Finite Abelian Group-Acted Sets
This work addresses a theoretical extension in cryptography and coding theory for researchers studying nonlinear functions, but it is incremental as it builds directly on prior generalizations.
The paper tackles the problem of generalizing bent functions and perfect nonlinearity from finite abelian groups to sets acted upon by such groups, by introducing a dual set and Fourier transforms to characterize these functions uniformly, resulting in a framework that recovers known results and provides new criteria for bentness based on distance from linear functions.
Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot et al. [6,7] via Fourier transforms of functions on $G$. In this paper we introduce the so-called $G$-dual set $\widehat X$ of $X$, which plays the role similar to the dual group $\widehat G$ of $G$, and the Fourier transforms of functions on $X$, a generalization of the Fourier transforms of functions on finite abelian groups. Then we characterize the bent functions on $X$ in terms of their own Fourier transforms on $\widehat X$. Bent (perfect nonlinear) functions on finite abelian groups and $G$-bent ($G$-perfect nonlinear) functions on $X$ are treated in a uniform way in this paper, and many known results in [4,2,6,7] are obtained as direct consequences. Furthermore, we will prove that the bentness of a function on $X$ can be determined by its distance from the set of $G$-linear functions. In order to explain the main results clearly, examples are also presented.