Harmonic analysis and a bentness-like notion in certain finite Abelian groups over some finite fields
This work addresses theoretical problems in harmonic analysis and cryptography for researchers in finite fields and coding theory, but it appears incremental as it builds on prior bent function definitions.
The authors developed a modular character theory and Fourier transform for specific finite Abelian groups using a Hermitian-like structure from degree two finite field extensions, and introduced a new notion of bent functions for finite field valued functions, proving it generalizes an existing concept and extends to a vectorial setting.
It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties. In particular we prove that this bentness notion is a consequence of that of Logachev, Salnikov and Yashchenko, introduced in "Bent functions on a finite Abelian group" (1997). In addition this new bentness notion is also generalized to a vectorial setting.