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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

Korean

Finite-field inversion은 대칭 암호화를 위한 S-box의 설계에서 기본적인 비선형 빌딩 블록입니다. Inverter-based S-box에서, finite-field inversion은 종종 반대 기능의 알제리적 대칭을 보존하거나 파괴 할 수있는 선형 또는 친화적 계층과 결합됩니다. 우리는 외부 반대형 선형 변환이 \(GL_n(\mathbb F_2)\) 이상으로 변하는 경우 얼마나 자주 nontrivial 선형 자기 대칭이 남아있는지 연구합니다. 반대의 양면적 선형 동등성의 분류를 사용하여, 우리는 문제를 semilinear conjugation 및 centralizer 추정으로 줄입니다. 우리는 nontrivial 선형 자기 대칭이 남아있는 확률에 대한 상위 및 낮은 경

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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