Determine the unresolved residue-direction behavior in the kernel direction

Determine the complete residue-level differential behavior of quadratic optimal APN permutations in the distinguished direction \rho, the nonzero element of the kernel of the linear part K of the derivative p'=\epsilon+K, after the differential rows in all directions outside \{0,\rho\} have been characterized.

Background

For a quadratic optimal APN permutation p of F_q, the kernel of the linear part of its derivative has the form {0,\rho}. Theorem \ref{thm:R} determines exact differential rows for coefficient lifts in unit directions whose residue is outside {0,\rho}, and in the top ideal directions.

The introduction identifies the residue direction \rho as the remaining unresolved direction. Proposition \ref{prop:W} supplies a compatibility obstruction for a particular two-word, two-layer linear-layer configuration, but it does not determine the full differential behavior in that direction.

References

A short residue-level statement about the direction $\rho$ left open in (R) is recorded in Proposition~\ref{prop:W}.

— Explicit reduced APN permutations: ramification and optimal Galois-ring lifts  (2610.21178 - Fukui, 15 Oct 2026) in Section 1, Introduction; developed further in Section 6, immediately before Proposition 6.1 (Proposition \ref{prop:W})