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

2026-04-02

AI Summary

This study investigates the structural characterization and efficient verification of generalized bent, plateaued, and landscape functions mapping from Boolean spaces to $\mathbb{Z}_{2^k}$. It introduces a novel approach based on $2^\ell$-adic decomposition, representing the target function as a linear combination of $\mathbb{Z}_{2^\ell}$-valued component functions. By integrating character sums over finite abelian groups with Walsh spectral analysis, the work establishes necessary conditions and a complete characterization of landscape functions without imposing structural assumptions. This method substantially reduces verification complexity: the number of required checks for landscape functions drops from $2^{2^{k-1}}$ to fewer than $2^{k-\ell+1}+1$. Under shared parameters, verifying gbent functions necessitates checking only a single base function, while plateaued functions require at most $2^{k-\ell}$ verifications, all while preserving duality and differential uniformity properties.

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.

Problem

• generalized bent functions
• plateaued functions
• landscape functions
• character sums
• finite abelian groups

Innovation

• 2^ell-adic decomposition
• overconstrained character sums
• generalized bent functions
• landscape functions
• Walsh spectrum