发表机构
Towson University（托森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究AES S-box所基于的有限域逆运算在仿射变换下的基刚性，提出确定性判据证明其线性部分使映射基刚性，并推广至通用情形，给出稳定子非平凡概率的指数界及精确中心化子维数，实验验证了理论结果。
AI 中文摘要
AES S-box由有限域逆运算后接一个固定的仿射变换构成。由于逆运算在其坐标实现中具有内在的Frobenius对称性，我们研究这些基对称性如何被外部仿射变换所改变。我们首先为变换后的逆运算建立了一个确定性的刚性判据，并将其应用于AES S-box。这表明AES S-box仿射变换的线性部分单独就使得变换后的逆映射具有基刚性。随后，我们研究了当外部可逆线性变换变化时相应的通用问题。非平凡线性稳定子的存在性被归结为逆运算的双侧线性等价所产生的半线性候选者的共轭问题，我们通过相对范数和Frobenius轨道对其进行了刻画。我们还精确确定了相关中心化子代数的维数。这些结构结果表明，对于均匀选择的外部线性变换，线性稳定子为非平凡的概率以$2^{-\Omega(n^2)}$为界，并且通过精确的共轭条件获得了更精确的有限维界。计算实验独立验证了AES刚性结果、共轭与中心化子公式以及小维度下的有限维估计。
英文摘要
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.
