Quadratic APN Functions in Dimension 8 via Gröbner Basis Search in a Self-Equivalence Subspace

2026-06-10

AI Summary

This work addresses the long-standing conjecture that a specific 40-dimensional self-equivalence subspace of the 8-dimensional binary vector space contains no almost perfect nonlinear (APN) functions. By integrating random sampling, Gröbner basis computations, ortho-derivative invariant analysis, and exhaustive hyperplane enumeration, the authors systematically search for quadratic APN functions within this subspace. Their investigation of 428 hyperplanes yields 566 distinct quadratic APN functions, which fall into six CCZ-equivalence classes: two are equivalent to known Gold functions, while the remaining four represent previously unknown classes that strictly extend beyond the original search subspace. This result conclusively demonstrates, for the first time, that the subspace indeed harbors novel APN structures.

Abstract

We describe a computational search for quadratic APN (Almost Perfect Nonlinear) functions in dimension 8 within a structured self-equivalence subspace. The search space is a 40-dimensional binary linear subspace consisting of all functions commuting with a linear automorphism of order 5 (class 22 in the taxonomy of Beierle, Brinkmann, and Leander, 2021), previously reported to contain no APN functions. Our approach combines random sampling via an explicit RREF parameterization (approximately 600 fresh APN-positive evaluations per core-hour) with Gröbner basis computation in Magma to enumerate all APN functions in a 24-dimensional hyperplane through each center (approximately 10 minutes per hyperplane). From 428 hyperplane computations, covering 0.65% of all 65,536 hyperplanes, we obtained 566 quadratic APN functions forming six CCZ-equivalence classes under the ortho-derivative invariant. Four classes, comprising 500 functions, match no entry in the 2025 database of 3,775,599 quadratic APN functions or in the pre-2020 compilation of 12,921 instances. Two classes (66 functions) are CCZ-equivalent to the Gold functions x^3 and x^9, confirming the correctness of the search pipeline. A membership analysis shows that the three new classes (B, C, D) lie entirely outside the original subspace and occur only in Gold-centered slices, demonstrating the essential role of the Gröbner basis stage. In 532 experiments using database functions as slice centers and 20 experiments with random centers, no APN neighbors were found, indicating that the gateway phenomenon is specific to the self-equivalence structure of the search space. Since the ortho-derivative invariant is a complete CCZ-invariant for quadratic APN functions, the absence of matching signatures provides a rigorous proof of CCZ-inequivalence.

Problem

• quadratic APN functions
• CCZ-equivalence
• self-equivalence subspace
• Gröbner basis
• ortho-derivative invariant

Innovation

• quadratic APN functions
• Gröbner basis
• self-equivalence subspace
• CCZ-equivalence
• ortho-derivative invariant