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

Abstract

We classify binary-linear two-term Frobenius-linearized operators $L(Y)=A{Y}^{\sigma }+BY$ on $K^3$, where $K$ is a finite extension of ${\mathbb{F}}_2$ and $\sigma$ 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 $\sigma$, to the canonical model $(\alpha ,\beta ,\gamma )\mapsto ({\alpha }^{\sigma }+\alpha ,{\beta }^{\sigma },\gamma )$. 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 $\sigma$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by ${\pi }_F(X{)}^{T}F(X)=1$; 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.

13 pages. Ancillary files contain exact code and regression tests for the finite n=9 computation in Proposition 18. Submitted to IEEE Transactions on Information Theory