Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions

Abstract

Generalized bent (gbent) functions from an $n$-variable Boolean space to ${\mathbb{Z}}_{2^k}$ are central in cryptography and sequence design. Instead of the usual binary decomposition, we introduce a $2^{\ell }$-adic representation, for $k=\ell r$, writing such functions as linear combinations of $r$ component functions valued in ${\mathbb{Z}}_{2^{\ell }}$. We prove a general result on overconstrained character sums over finite abelian groups: under a common-argument hypothesis, sequences with two-level Fourier magnitude spectra must be extremely sparse, with a conditional extension to multi-level spectra. As an application, we derive consequences for generalized plateaued functions under suitable assumptions. We then show that if $f:{\mathbb{F}}_2^n\to {\mathbb{Z}}_{2^k}$ is landscape, then under the $2^{\ell }$-adic decomposition every function in a certain affine space over ${\mathbb{Z}}_{2^{\ell }}$ is again landscape with the same Walsh magnitudes. This gives an unconditional necessity result, with no structural assumptions on $f$, together with a complete characterization using only a small subset of these maps. For generalized bent and generalized plateaued functions, sufficiency is also obtained from linear combinations of lower components under natural assumptions; a counterexample shows these assumptions are essential. Our method reduces verification for landscape functions from $2^{2^{k-1}}$ checks to fewer than $2^{k-\ell +1}+1$ conditions; for gbent functions this drops to a single basis function under the common-argument hypothesis, and for generalized plateaued functions, under additional assumptions, to $2^{k-\ell }$ checks. The $2^{\ell }$-adic framework also preserves key properties, including duality and differential uniformity.