发表机构
Towson University（陶森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二元扩域𝔽_{2ⁿ}上求逆运算的表示冗余，证明有序基与求逆映射为n对1对应，分析三种布尔形式的代数特性，实验验证学习难度排序及轨道冗余的有限泛化收益。
AI 中文摘要
数学运算的表示选择会同时影响其代数形式和实际学习难度。我们针对二元扩域𝔽_{2ⁿ}上的求逆运算研究该现象，其中域元素以不同的有序𝔽₂基表示。我们证明，两个有序基诱导出相同的坐标求逆映射当且仅当它们属于同一个伽罗瓦轨道。由于每个轨道大小为n，有序基与不同求逆映射之间的对应关系恰好是n对1。随后我们分析了求逆的三种布尔形式：参考形式的代数次数为n-1，联合代数正规型（ANF）跳跃为1；混合表示形式的次数为2(n-1)，联合ANF跳跃为2；完全原始形式的次数至多为3(n-1)，联合ANF跳跃至少为n。穷尽计算在考虑的案例中与理论结果和边界一致。对多层感知机的受控实验显示学习难度存在相同的排序，而在测试条件下，伽罗瓦轨道冗余仅提供有限的泛化收益。这些结果表明，当表示作为输入的一部分被呈现时，精确的表示冗余可与布尔结构变化及学习行为共存。
英文摘要
The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over \(\mathbb F_{2^n}\), with field elements expressed in varying ordered \(\mathbb F_2\)-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size \(n\), the correspondence between ordered bases and distinct inversion maps is exactly \(n\)-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree \(n-1\) and joint ANF leap \(1\), the mixed representation formulation has degree \(2(n-1)\) and joint ANF leap \(2\), and the complete raw formulation has degree at most \(3(n-1)\) and joint ANF leap at least \(n\). Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.
