Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps

2026-08-12

AI Summary

This work addresses the classification and canonical form construction of binary linearized two-term Frobenius operators with coefficient rank two and binary kernel dimension one. By employing invertible K-linear input-output transformations together with the Frobenius automorphism over finite fields, trace-adjoint normalization, and dual coordinate techniques, the study establishes—for the first time—a canonical representation for this class of rank-two operators. Key contributions include an explicit characterization of the precise relationship between orthogonal derivatives and dual coordinates, the derivation of permutation properties under odd extension degrees, and a projective plane duality bijection. The identity π_F(X)^T F(X) = 1 is verified under canonical reduction. The theoretical framework is validated through two algebraic constructions, providing an extended-field labeling scheme that elucidates component root structures and their connections to Walsh support.

Abstract

We classify binary-linear two-term Frobenius-linearized operators $L(Y)=AY^σ+BY$ on $K^3$, where $K$ is a finite extension of $\mathbb{F}_2$ and $σ$ is a fixed nontrivial Frobenius automorphism of $K$ with fixed field $\mathbb{F}_2$. Under a coefficient-rank and binary-kernel condition, if $A$ and $B$ both have $K$-rank two and $L$ has a one-dimensional kernel over $\mathbb{F}_2$, then invertible $K$-linear input and output changes reduce $L$, for this fixed $σ$, to the canonical model $(α,β,γ)\mapsto(α^σ+α,β^σ,γ)$. The proof constructs the coordinate frames from the two coefficient-kernel directions and the binary kernel. In these coordinates, the first dual output row is exactly the unique nonzero trace-adjoint normal, with an exact $K$-valued normalization. For pure $σ$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by $π_F(X)^T F(X)=1$; in odd extension degree it also yields permutation behavior and a bijection from the projective plane to its dual. The triprojective construction of Gologlu and Kolsch and the cubic norm-twist construction of Li, Zhou, Li, and Qu provide two realizations arising from different algebraic constructions. The triprojective case further admits a determinant factorization and a complete dual frame, whereas the norm-twist realization shows that the pure-map consequences do not follow from the operator theorem alone. A natural Gold representation has coefficient-rank pair $(3,3)$, delimiting the rank-two subclass. The normal form also supplies exact extension-field labels for known component-radical and Walsh-support relations.

Problem

• quadratic APN maps
• Frobenius-linearized operators
• rank-two classification
• orthoderivative
• dual coordinates

Innovation

• Frobenius-linearized operators
• rank-two normal forms
• orthoderivative
• quadratic APN maps
• dual coordinates