Basis Rigidity of the AES S-box and Generic Rigidity of Inversion under Affine Transformations
Abstract
The AES S-box is constructed from finite field inversion followed by a fixed affine transformation. Since inversion possesses intrinsic Frobenius symmetries among its coordinate realizations, we study how these basis symmetries are altered by outer affine transformations. We first develop a deterministic rigidity criterion for transformed inversion and apply it to the AES S-box. This shows that the linear part of the AES S-box affine transformation alone makes the transformed inversion map basis rigid. We then investigate the corresponding generic problem when the outer invertible linear transformation varies. The existence of a nontrivial linear stabilizer is reduced to a conjugacy problem for semilinear candidates arising from two sided linear equivalences of inversion, which we characterize in terms of relative norms and Frobenius orbits. We also determine the dimensions of the associated centralizer algebras exactly. These structural results imply that, for a uniformly chosen outer linear transformation, the probability that the linear stabilizer is nontrivial is bounded by 2−Ω(n2), with sharper finite dimensional bounds obtained from the exact conjugacy condition. Computational experiments independently verify the AES rigidity result, the conjugacy and centralizer formulas, and the finite dimensional estimates in small dimensions.
Related
- Stochastic Inversion of Multivariate Uniform-Distribution-Preserving Transformations
- Renormalization and rigidity of analytic non-linear interval exchange transformations
- Local centralizer rigidity for a non-generic diagonal map
- Common invariant measures for beta-transformations with rational affine relations
- Copula Transformations for Data-Consistent Inversion
BibTeX
@misc{zhang2026basis,
title = {{Basis Rigidity of the AES S-box and Generic Rigidity of Inversion under Affine Transformations}},
author = {Zheng Zhang and Na Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13644},
eprint = {2609.13644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13644},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Basis Rigidity of the AES S-box and Generic Rigidity of Inversion under Affine Transformations
AU - Zheng Zhang
AU - Na Zhang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13644
UR - https://arxiv.org/abs/2609.13644
AB - The AES S-box is constructed from finite field inversion followed by a fixed affine transformation. Since inversion possesses intrinsic Frobenius symmetries among its coordinate realizations, we study how these basis symmetries are altered by outer affine transformations. We first develop a deterministic rigidity criterion for transformed inversion and apply it to the AES S-box. This shows that the linear part of the AES S-box affine transformation alone makes the transformed inversion map basis rigid. We then investigate the corresponding generic problem when the outer invertible linear transformation varies. The existence of a nontrivial linear stabilizer is reduced to a conjugacy problem for semilinear candidates arising from two sided linear equivalences of inversion, which we characterize in terms of relative norms and Frobenius orbits. We also determine the dimensions of the associated centralizer algebras exactly. These structural results imply that, for a uniformly chosen outer linear transformation, the probability that the linear stabilizer is nontrivial is bounded by 2^−Ω(n^2), with sharper finite dimensional bounds obtained from the exact conjugacy condition. Computational experiments independently verify the AES rigidity result, the conjugacy and centralizer formulas, and the finite dimensional estimates in small dimensions.
ER -