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.

• The paper introduces a novel 2^ℓ-adic framework that exponentially reduces verification checks for generalized Boolean functions.
• It establishes unconditional sparsity results for overconstrained character sums, ensuring practical validation of cryptographic properties.
• The study integrates algebraic, combinatorial, and spectral methods to enhance cryptographic S-box design and maintain structural integrity.

Overconstrained Character Sums and $2^\ell$-adic Decomposition of Generalized Bent, Plateaued, and Landscape Functions

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Introduction

This paper systematically develops a $2^\ell$-adic structural framework for generalized Boolean functions mapping from $n$-dimensional vector spaces over $\mathbb{F}_2$ to the ring $\mathbb{Z}_{2^k}$. The primary focus is on efficient algebraic/spectral verification and decomposition of generalized bent (gbent), plateaued (gplateaued), and landscape functions. The authors introduce new unconditional results on the sparsity of overconstrained character sums and leverage these to provide hierarchical characterizations of generalized Boolean functions, leading to practically viable checks for cryptographic suitability.

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Structural Decomposition: Beyond Binary Components

Classically, gbent functions are examined through binary decompositions where the function is viewed as a linear combination of Boolean components. Mesnager et al.'s result [mtqwwf] provides a characterization, but demands verification of exponentially many ($2^{2^{k-1}}$) Boolean conditions. This paper generalizes such decompositions to $2^\ell$-adic representations, for $k = \ell r$, treating $f$ as a combination of $r$ components over $2^\ell$0. This reformulation is not only algebraically natural for composite $2^\ell$1, but also aligns with hardware and algorithmic constraints in cryptographic implementations.

The $2^\ell$2-adic structure reveals previously obscured spectral properties and enables efficient partitioning of the domain according to lower-level digits. Hierarchical dependencies in the function reflect Galois-theoretic and combinatorial structures, providing a new lens to analyze duality, derivative behavior, and differential uniformity.

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Overconstrained Character Sums: Sparsity and Structure

The paper establishes foundational results on character sums whose Fourier transforms exhibit two-level or multi-level spectral magnitudes. The main result is that, under the common-argument hypothesis, such sequences must be extremely sparse—often supported on at most two cosets of a subgroup. Specifically, for two-level spectral constraints, the support collapses to a single coset unless an algebraic exceptional case occurs. This sparsity result (Theorem 3.5) is unconditional and essential for subsequent decomposition theorems.

Generalizations to multi-level spectra are provided, contingent upon technical conditions regarding sumset growth and functional calculus in the group algebra. The authors show that merely imposing two-level magnitude constraints does not guarantee sparsity unless phase alignment (common-argument) is enforced, emphasizing the necessity of spectral structure in cryptographic applications.

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Landscape, gbent, and gplateaued Function Characterization

The paper introduces a partition-based framework for decomposing generalized Boolean functions. By analyzing the $2^\ell$3-adic expansion and partitioning $2^\ell$4 by lower-level digits, the authors derive a spectral decomposition:

$2^\ell$5

where $2^\ell$6 encapsulates the contribution from each partition cell. Crucially, the independence of the basis elements over $2^\ell$7 ensures that, for gbent functions, exactly one coefficient per input $2^\ell$8 is nonzero—a spectral sparsity phenomenon.

Necessity and Sufficiency — Main Results:

• Unconditional necessity: If $2^\ell$9 is landscape, all affine combinations of its $n$0-adic components generate landscape functions with identical Walsh spectra.
• Sufficiency (conditional): If the nonzero coefficients satisfy a common-argument hypothesis and certain sumset stabilizer conditions, gbent or gplateaued status can be checked via a small subset of the affine-generated family, reducing verification from $n$1 binary checks to $n$2 or fewer.
• For gbents, under common-argument, verification drops to a single basis function. Exact characterization of the spectral conditions is provided.

Counterexamples are given, demonstrating that sufficiency fails without the common-argument or structural sparsity hypotheses, and that affine subfamily verification is not generally adequate.

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Implications for Cryptographic Practice: Efficiency and Structure Preservation

The $n$3-adic decomposition provides exponential computational savings for verifying cryptographic properties—a notable practical advance given the infeasibility of brute-force checks for large $n$4. The decomposition is shown to preserve or interact predictably with key cryptographic properties:

• Duality: The lower digits of the dual function correspond exactly to the unique nonzero partition cell of the original function.
• Maiorana–McFarland structure: Decomposition is compatible with the M-M class, with detailed formulas accounting for carries and digit expansion.
• Quadraticity and Derivatives: Algebraic degree and derivative properties map cleanly through the decomposition.
• Differential Uniformity: Bounds for the differential spectrum are given, and the decomposition enables tight control over uniformity properties.

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Empirical Analysis: S-boxes and Boolean Optimization

The paper provides computational evidence that standard cipher S-boxes (PRESENT, GIFT, PRINCE, SKINNY) do not satisfy the landscape, gbent, or gplateaued spectral constraints when viewed as functions to $n$5. The Walsh spectra contain non-integer and high-degree algebraic values, reinforcing that Boolean optimization in S-box design does not guarantee the hierarchical spectral properties derived from the $n$6-adic framework. This supports the necessity for the new design and verification approaches advocated by the authors.

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Theoretical and Practical Implications, Open Directions

The results have deep theoretical ramifications:

• They connect additive combinatorics, cyclotomic field extensions, and Fourier analysis to the structural theory of generalized Boolean functions, suggesting further algebraic number theory generalizations.
• The $n$7-adic methodology may be adapted to non-binary or $n$8-adic settings, enabling broader applications.
• Identification of minimal structural conditions linking affine subfamily checks to full characterization remains an open problem.

Practically, the results pave the way for algorithmic S-box and cryptographic primitive design optimized for spectral constraints at the modular/composite level, rather than purely Boolean criteria.

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Conclusion

The $n$9-adic decomposition provides a rigorous framework for analyzing and constructing cryptographically valuable generalized Boolean functions, enabling a hierarchy of algebraic, spectral, and combinatorial criteria for gbent, plateaued, and landscape classes. The sparsity and structural characterizations, grounded in unconditional combinatorial results, permit exponentially faster verification, preserve cryptographic properties, and reveal hierarchies inaccessible via classical binary decomposition. The interplay between algebraic structure and spectral constraints is established as central in both theoretical analysis and cryptographic design. The results open significant avenues for further exploration in algebraic combinatorics, number theory, and practical cryptography.