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

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 $G{L}_n({\mathbb{F}}_2)$. 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.