Exact raw ANF degree and leap for general field dimension

Prove, for every field dimension n beyond the computed cases n=3 and n=4, that the complete raw inversion map has algebraic degree 3(n-1) and joint ANF leap n.

Background

The paper proves an upper bound of 3(n-1) for the algebraic degree of the complete raw inversion map and a lower bound of n for its joint ANF leap. Exhaustive computations for n=3 and n=4 attain both bounds.

The authors explicitly state that these two finite cases do not establish equality for arbitrary n. Determining whether both bounds are sharp in all dimensions is consequently left unresolved.

References

These computations suggest that

\deg(f_{\mathrm{raw})=3(n-1), \qquad L_{\mathrm{ANF}{\mathrm{joint}(f_{\mathrm{raw})=n

may hold for general $n$, although the cases $n=3$ and $n=4$ do not prove either equality.

— Representation Redundancy and Structural Complexity in Finite-Field Inversion  (2609.04583 - Zhang et al., 4 Sep 2026) in Section 4, subsection “Experiment 0: Exact ANF computation”

Establishing an intrinsic notion of ANF complexity for the restricted domain is left for future work.

— Representation Redundancy and Structural Complexity in Finite-Field Inversion  (2609.04583 - Zhang et al., 4 Sep 2026) in Section 6, subsection “ANF structure and learning difficulty”

The equality of the coordinate-level profiles in the complete raw formulation is an observed property of the $n=3$ and $n=4$ computations. We do not claim that this equality holds for arbitrary $n$ or for every choice of reference basis.

— Representation Redundancy and Structural Complexity in Finite-Field Inversion  (2609.04583 - Zhang et al., 4 Sep 2026) in Appendix, subsection “Coordinate-level ANF statistics”