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

Abstract

We describe a computational search for quadratic APN (Almost Perfect Nonlinear) functions over $\mathbb{F}_{2^8}$ within a structured algebraic subspace defined by a self-equivalence constraint. The search space is the 40-dimensional $\mathbb{F}_2$-linear subspace $V_A = \{F : F \circ A = A \circ F\}$ for a specific linear automorphism $A$ of order 5 (class index 22 in the taxonomy of Beierle, Brinkmann, and Leander); this subspace was previously reported to contain no APN functions under their recursive tree search method. We combine two phases: (1) random sampling inside $V_A$ via an explicit RREF parameterization to find APN center functions, and (2) Groebner basis computation in Magma over the Boolean polynomial ring to enumerate all APN functions in a 24-dimensional hyperplane through each center. From 428 hyperplane computations (covering 0.65% of the 65,536 total hyperplanes in $V_A$) we obtained 566 quadratic APN functions falling into six CCZ-equivalence classes under the ortho-derivative invariant. Four of these classes, comprising 500 functions, match no entry in the Beierle et al. 2025 database of 3,775,599 quadratic APN functions and no entry in the pre-2020 compilation of 12,921 instances. Two classes (66 functions) are identified as CCZ-equivalents of the Gold functions x^3 and x^9, confirming pipeline correctness. For quadratic APN functions, a signature mismatch rigorously certifies CCZ-inequivalence, by Yoshiara's theorem (CCZ = EA for quadratic APN) together with the ortho-derivative invariant; the absence of a signature match in the above databases therefore constitutes a rigorous proof of CCZ-inequivalence for the new classes. The complete dataset, source code, and verification scripts are publicly available.