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

Computer Science > Cryptography and Security [Submitted on 12 Aug 2026] Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps Jingchuan Ma, Yanhua Liu, Qiaoyun Huang We classify binary-linear two-term Frobenius-linearized operators L(Y)=A Y σ +BY on K 3 , where K is a finite extension of F 2 and σ is a fixed nontrivial Frobenius automorphism of K with fixed field 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 F 2 , then invertible K -linear input and output changes reduce L , for this fixed σ , to the canonical model (α,β,γ)↦( 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. Comments: 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 Subjects: Cryptography and Security (cs.CR) MSC classes: 94A60, 11T71 Cite as: arXiv:2608.11939 [cs.CR]   (or arXiv:2608.11939v1 [cs.CR] for this version)   https://doi.org/10.48550/arXiv.2608.11939 Focus to learn more Submission history From: Jingchuan Ma [view email] [v1] Wed, 12 Aug 2026 11:27:30 UTC (109 KB) Access Paper: HTML (experimental) view license Ancillary files (details): EXPECTED_OUTPUT.txt MANIFEST.sha256 README.md reproduce_proposition18.py src/__init__.py (3 additional files not shown) Current browse context: cs.CR < prev   |   next > new | recent | 2026-08 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)