Dimension Rigidity and Projective Geometry of Trace-Product Switchings of the Gold Cube

2026-08-04

AI Summary

This study investigates the possibility of constructing new almost perfect nonlinear (APN) functions via trace-based switching of the Gold function $x^3$ in even dimensions, and establishes inherent dimensional constraints on such constructions. By integrating tools from projective geometry, additive character sum estimates, analysis of Fermat cubic surfaces, and criteria based on low-rank derivatives, the work provides the first intrinsic characterization of admissible switching coefficients. It proves that nontrivial switching occurs only in dimensions 4, 6, and 8, yielding precisely two CCZ-equivalence classes. Furthermore, for all even dimensions greater than or equal to 10, the paper develops a high-dimensional exclusion mechanism, thereby establishing a dimension rigidity theorem and revealing underlying principles governing the local structural organization of such functions.

Abstract

We completely classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $x\mapsto x^3$ in every even dimension. Nontrivial switchings occur only for $n=4,6,8$: the admissible coefficients are, respectively, the nonzero trace-zero elements, the six elements of multiplicative order nine, and $\mathbb{F}_4^{*}$. For every even $n\geq10$, no nonzero coefficient is admissible. The infinite range is excluded by additive-character estimates on a Fermat cubic, with exact finite bridges for $n=10,12$. The raw coefficient lists for $n=6,8$ appeared earlier in Arshad's dissertation; our contribution is their intrinsic description, a proof uniform in the dimension, and the resulting dimension-rigidity theorem. We also classify normalized rank-two extensions in dimension eight by $\mathbb{P}^{1}(\mathbb{F}_4)$. A binary trace selector accepts two coefficient values at each non-base projective point, and the eight accepted marked switchings form exactly two extended-affine, hence two CCZ, classes. A centre-independent low-rank derivative criterion reduces each rank-$r$ candidate to $2^r-1$ membership tests in precomputed forbidden sets. The global APN classes reached are known; the results describe their local organization around the Gold centre and rule out this switching mechanism in all larger even dimensions.

Problem

• trace-product switching
• Gold function
• dimension rigidity
• APN functions
• projective geometry

Innovation

• trace-product switching
• dimension rigidity
• APN functions
• projective geometry
• low-rank derivative criterion