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

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

Problem Statement
The paper investigates the modification of the Gold almost perfect nonlinear (APN) function, defined as $F(x) = x^3$ over the finite field $\mathbb{F}_{2^n}$ (where $n$ is even), via a scalar trace-product switching mechanism. Specifically, the study analyzes functions of the form:
$$F_{n,\theta}(x) = x^3 + \theta Q_n(x)$$
where $Q_n(x) = \text{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(x) \cdot \text{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\lambda x)$ is a normalized quadratic Boolean form, $\theta \in \mathbb{F}_{2^n}$ is a coefficient, and $\lambda$ is a primitive element of $\mathbb{F}_4$. The central problem is to determine for which even dimensions $n$ and which nonzero coefficients $\theta$ the resulting function $F_{n,\theta}$ retains the APN property. While previous work (e.g., Arshad's dissertation) provided raw lists of admissible coefficients for small dimensions ($n=6, 8$), a unified theoretical description and a proof of nonexistence for larger dimensions were lacking.

Methodology
The author employs a combination of algebraic geometry, character sum estimates, and computational verification:

1. Low-Rank Derivative Framework: The paper establishes a "Low-Rank Fixed-Point Criterion" (Theorem 3.1). This generalizes existing hyperplane-injectivity criteria to arbitrary rank-$r$ updates. It reduces the APN verification of an update $F + U \circ q$ to checking whether a linear map $U$ avoids precomputed "forbidden sets" derived from the derivative images of the base function. This allows for efficient membership testing ($2^r - 1$ tests) rather than exhaustive APN checks.
2. Section-Product Criterion: For the specific scalar trace-product family, the author derives a geometric criterion (Theorem 4.2). They show that $F_{n,\theta}$ is APN if and only if $\theta$ is not representable as a "section product" $c^3 + d^3$ where $c, d$ satisfy specific relative trace conditions ($T(c)=1, T(d)=0$). This transforms the problem into counting solutions to the equation $c^3 + d^3 = \theta$ under trace constraints.
3. Analytic and Finite Bridges:
   • For small dimensions ($n=4, 6, 8$), the author uses direct algebraic manipulation and Frobenius-orbit certificates to classify admissible coefficients.
   • For $n \ge 10$, the paper utilizes additive-character estimates on the projective Fermat cubic curve $X^3 + Y^3 = \theta Z^3$. By bounding the number of affine points using Weil bounds and Artin–Schreier conductor analysis, they prove that the number of solutions $h_n(\theta)$ is strictly positive for all $\theta \neq 0$ when $n \ge 14$.
   • For the intermediate cases $n=10$ and $n=12$, exact finite computations (XOR-convolution of trace fibres) serve as "bridges" to confirm nonexistence.
4. Projective Classification (Dimension 8): In dimension 8, the author extends the analysis to rank-two combinations of projective trace forms. They utilize the projective line $\mathbb{P}^1(\mathbb{F}_4)$ to parametrize the extensions, employing a binary trace selector to identify valid coefficient pairs.

Key Contributions and Results

• Dimension Rigidity Theorem: The paper proves a strict "dimension ladder" for this switching mechanism:
  • $n=4$: Admissible coefficients are the nonzero elements of the trace-zero hyperplane ($\text{Tr}(\theta)=0$). These maps are Extended-Affine (EA) equivalent to the original Gold function.
  • $n=6$: Admissible coefficients are exactly the six elements of multiplicative order 9 in $\mathbb{F}_{64}$.
  • $n=8$: Admissible coefficients are the nonzero elements of the subfield $\mathbb{F}_4^*$.
  • $n \ge 10$ (even): No nonzero coefficient $\theta$ yields an APN function. The set of admissible coefficients is empty.
• Intrinsic Characterization: The paper provides the first intrinsic, basis-independent descriptions for the coefficient lists previously found in Arshad's dissertation for $n=6$ and $n=8$, replacing raw field coordinates with algebraic properties (order 9 and subfield membership, respectively).
• Projective Rank-Two Extensions ($n=8$): The author classifies normalized rank-two extensions in dimension 8. They demonstrate that the parameter space is $\mathbb{P}^1(\mathbb{F}_4)$, where each of the four non-base points supports exactly two accepted coefficient pairs. These eight marked switchings form exactly two EA classes (and consequently two CCZ classes), organized into two Frobenius cycles.
• Computational Tool: A reusable, center-independent fixed-point criterion is provided, which reduces APN testing for rank-$r$ updates to $2^r - 1$ incidence tests against precomputed forbidden sets.

Significance and Claims
The paper claims that its primary contribution is the uniform nonexistence result for dimensions $n \ge 10$ and the intrinsic geometric description of the admissible sets in lower dimensions.

• Methodological Value: The "dimension rigidity" result removes an entire class of switching mechanisms from the search space for new APN functions in larger even dimensions, streamlining future cryptographic research.
• Structural Insight: The results describe the "local organization" of known global APN classes around the Gold center. The paper clarifies that while the resulting global classes are known, the specific mechanism of trace-product switching is highly constrained, functioning only in dimensions 4, 6, and 8.
• Reproducibility: The work emphasizes rigorous reproducibility, providing exact JSON certificates, Python scripts, and SHA-256 checksums for all finite computations (including the 12-row and 33-row orbit certificates and the $n=10, 12$ bridges). The author explicitly states that the computer-assisted steps are auxiliary to the analytic proofs for $n \ge 14$ and the algebraic derivations for $n < 10$.

The paper does not claim to discover new global APN classes but rather to map the precise boundaries and structural properties of how the Gold function can be locally modified via trace-product switching.