Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions
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| Formato: | Preprint |
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2026
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| _version_ | 1866914439452688384 |
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| author | Çeşmelioğlu, Ayça Riera, Constanza Stănică, Pantelimon |
| author_facet | Çeşmelioğlu, Ayça Riera, Constanza Stănică, Pantelimon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01673 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions Çeşmelioğlu, Ayça Riera, Constanza Stănică, Pantelimon Number Theory Information Theory Combinatorics 94A60, 11T71, 06E30, 94C10 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. |
| title | Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions |
| topic | Number Theory Information Theory Combinatorics 94A60, 11T71, 06E30, 94C10 |
| url | https://arxiv.org/abs/2604.01673 |