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Representation Redundancy and Structural Complexity in Finite-Field Inversion

2026-09-07 12:00 Science 🔥 40.2 heat score
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On September 7, 2026, arXiv cs.LG published a research paper titled “Representation Redundancy and Structural Complexity in Finite-Field Inversion”. This study focuses on the representation redundancy phenomenon present in the inverse operation of finite fields and the structural complexity challenges it poses. It aims to analyze the relevant mechanisms and explore optimization pathways.

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A arXiv cs.LG en 2026-09-07 12:00

Representation Redundancy and Structural Complexity in Finite-Field Inversion

研究针对有限域 \(\mathbb F_{2^n}\) 上的求逆运算，证明两个有序基诱导相同坐标求逆映射当且仅当它们属于同一 Galois 轨道。由于每个轨道大小为 \(n\)，有序基与不同求逆映射的对应关系为 \(n\)-对一。分析三种布尔形式的求逆：参考形式代数次数为 \(n-1\)、联合 ANF 跳跃为 1；混合表示形式代数次数为 \(2(n-1)\)、联合 ANF 跳跃为 2；完整原始形式代数次数至多为 \(3(n-1)\)、联合 ANF 跳跃至少为 \(n\)。穷举计算验证了理论结果与界限，多层感知机实验显示学习难度顺序一致，但 Galois 轨道冗余在测试条件下仅提供有限的泛化收益。结果表明，当表示作为输入暴露时，表示间的精确冗余可与布尔结构及学习行为的改变共存。

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