CCZ-Equivalence and Enumeration of Triprojective APN Functions
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.