Newest papers

Linear Self-Symmetries of Inverse-Based S-boxes: Generic Rigidity and Applications

Zheng Zhang, Na Zhang · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.29837 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

フィニットフィールドインバージョンは、シンメトリック暗号化のためのSボックスの設計における基本的な非線形構造ブロックです。逆に基づくSボックスでは、フィニットフィールドインバージョンはしばしば、逆機能のアルジェビアシンメトリーを保存または破壊する可能性のある線形またはアフィン層と組み合わされています。私たちは、逆機能のアルジェビアシンメトリーをどのくらい頻繁に構築するかを研究します。外部の反転式線形変換が \(GL_n(\mathbb F_2)\)を超える可能性の上下限と上限を取得し、これらは、逆転の両側の線形等価の分類を使用して、我々は、

Machine translation (Meta M2M100); the English original follows.

Abstract

Finite-field inversion is a basic nonlinear building block in the design of S-boxes for symmetric cryptography. In inverse-based S-boxes, finite-field inversion is often combined with linear or affine layers that may preserve or destroy algebraic symmetries of the inverse function. We study how often nontrivial linear self-symmetries remain as the outer invertible linear transformation varies over \(GL_n(\mathbb F_2)\). Using the classification of two-sided linear equivalences of inversion, we reduce the problem to semilinear conjugacy and centralizer estimates. We obtain matching upper and lower bounds for the probability that a nontrivial linear self-symmetry remains, and these bounds determine its leading quadratic exponent. This exponent is governed by the smallest prime divisor of the extension degree. Thus the decay rate depends on the arithmetic of the field extension and not only on its dimension. As cryptographic applications, we provide a practical rigidity criterion, prove that the full AES S-box has no nontrivial linear self-symmetry, and illustrate the broader relevance of the framework through ARIA and Vision.

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Topics: Mathematics, Invertible matrix, Affine transformation, Linear map, Conjugacy class, Centralizer and normalizer

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