Representation Redundancy and Structural Complexity in Finite-Field Inversion
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 ha...
| Authors | Zheng Zhang, Na Zhang |
|---|---|
| Primary category | cs.LG |
| Published | 4 September 2026 |
| Last updated at source | 4 September 2026 |
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