发表机构
Fuzhou University Zhicheng College（福州大学至诚学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分类了K³中的二元线性两项Frobenius线性化算子，推导了其典范模型，将正交导数与纯σ二次APN映射关联，并给出两类构造实现及Gold表示，为相关关系提供扩域域标记。
AI 中文摘要
我们对有限域𝔽₂的有限扩域K上K³中的二元线性两项Frobenius线性化算子L(Y)=AY^σ+BY进行分类，其中σ是K的非平凡固定Frobenius自同构，其固定域为𝔽₂。在系数秩与二元核条件下，若A和B均具有K秩2，且L在𝔽₂上具有一维核，则对于该固定σ，可逆的K线性输入与输出变换可将L简化为典范模型(α,β,γ)↦(α^σ+α,β^σ,γ)。证明过程从两个系数核方向与二元核构造坐标框架，在这些坐标中，第一对偶输出行恰好是唯一非零迹伴随正规形，具有精确的K值归一化。对于纯σ二次几乎完美非线性（APN）映射，这通过π_F(X)^T F(X)=1识别正交导数；在奇扩域次数下，还得到置换行为及从射影平面到其对偶的双射。Gologlu与Kolsch的三射影构造、Li、Zhou、Li与Qu的三次范数扭曲构造是两种不同代数构造产生的实现：三射影情形还具有行列式分解与完整对偶框架，而范数扭曲实现表明纯映射的结论并非仅由算子定理得出。自然Gold表示具有系数秩对(3,3)，界定了秩二子类；该正规形还为已知的分量根与Walsh支撑关系提供精确扩域域标记。
英文摘要
We classify all admissible parameters of the G"ologlu--K"olsch triprojective construction of almost perfect nonlinear (APN) functions up to Carlet--Charpin--Zinoviev (CCZ) equivalence. The construction has three coefficients in $K=\mathbb{F}_{2^m}$ and a Frobenius exponent $k$ coprime to $m$. For every $m>1$, we give a necessary-and-sufficient criterion that includes changes of both the coefficients and the exponent. Each parameter choice determines one of the two irreducible cubic polynomials over $\mathbb{F}_2$. Two functions are equivalent precisely when their exponents and cubics agree, or when the exponents are negatives modulo $m$ and the cubics are reciprocal. When $7\nmid m$, every admissible member is equivalent to a Li--Kaleyski representative: the larger coefficient space adds no CCZ classes. When $7\mid m$, those representatives are inadmissible; we construct replacements and classify the full family. There are exactly $φ(m)$ CCZ classes, extending the previously known count for the older subfamilies to all admissible coefficients and all degrees. We also count the parameter triples and compute canonical representatives and explicit equivalence maps in deterministic polynomial time for every degree, including degrees divisible by seven. The proof combines a reduction to classical semilinear conjugacy with an intrinsic recovery of the scalar field from the polar bilinear map. The reduction from arbitrary EL equivalence is algebraic in every degree, including degrees three and six.
Comments17 pages
