The answer remains open. A witness must be a bijection \(F:\mathbb F_{2^8}\to\mathbb F_{2^8}\) satisfying \[ \max_{a\ne0,\,b}\#\{x:F(x+a)+F(x)=b\}=2. \] The complete search space contains \(256!\) lookup tables, so every published computation uses structural restrictions or selected CCZ-equivalence classes.

Beierle, Brinkmann, and Leander searched APN permutations admitting a nontrivial linear self-equivalence. Their dimension-eight search exhausted all except a few self-equivalence classes and found no witness. Beierle, Langevin, Leander, Polujan, and Rasoolzadeh later generated 3,775,599 inequivalent quadratic APN functions in dimension eight. None of those functions is CCZ-equivalent to a permutation.

A June 2026 preprint by Kuznetsov found four further quadratic APN CCZ-classes absent from the 2025 database. Its displayed functions are nonpermutations, and the paper leaves open whether any of the four CCZ-classes contains a permutation. This latest result preserves the unrestricted open status.

For \(\lambda\ne0\), write the Boolean component as \[ F_\lambda(x)=\lambda\cdot F(x). \] Bijectivity makes all 255 such components balanced. Calderini, Sala, and Villa prove that an APN permutation in even dimension has no partially bent component. In particular, no component can be quadratic, so the vectorial algebraic degree is at least three.

Musukwa, Sala, Villa, and Zaninelli prove a derivative restriction valid for every APN permutation: no nonzero component has an identically zero derivative in a nonzero direction, and each component has at most one direction whose derivative is identically one.

Their dimension-eight classification gives a sharper test for a cubic candidate. Every nonzero component must have degree three. For their invariant \(\mathcal M\), each component satisfies either \(\mathcal M=192\) or \(\mathcal M\ge288\). If \[ \Lambda=\{\lambda\ne0:\mathcal M(F_\lambda)\le256\}, \] then \[ 85\le |\Lambda|\le252. \] Up to EA-equivalence, every component indexed by \(\Lambda\) belongs to the finite list in their Table 5. This restriction prunes cubic searches while leaving higher-degree candidates and the remaining cubic combinations unresolved.

The audit followed three computational programs. The 2021 self-equivalence search treats permutations with a nontrivial linear self-equivalence after CCZ transformation. It exhausted all except a few dimension-eight automorphism classes and found no new APN permutation. This is a structural search rather than a classification of every permutation.

The 2025 quadratic search generated 3,775,599 inequivalent APN functions in eight variables and tested their CCZ classes for permutation representatives. No class passed. The authors estimate roughly six million quadratic APN classes in total, so their database is explicitly incomplete.

The June 2026 Gröbner-basis search covered 428 of 65,536 normalized slices around one order-five self-equivalence subspace. It found four classes absent from the 2025 database. The source states that its representatives are nonpermutations and leaves CCZ-equivalent permutation testing for future work. Together these searches rule out power permutations and several million quadratic CCZ classes. They do not cover arbitrary cubic or higher-degree permutations.

The program represents \(\mathbb F_{2^8}\) with the AES polynomial \[ x^8+x^4+x^3+x+1 \] and computes every derivative value directly. It checks all 128 exponents \(d\) with \(\gcd(d,255)=1\), exactly the exponents for which \(x^d\) permutes \(\mathbb F_{256}\). The minimum differential uniformity is four. It is attained for \[ d\in\{127,191,223,239,247,251,253,254\}. \] This exhausts the monomial-permutation family and does not address general lookup tables.

Two controls expose the even-dimension obstruction. Over \(\mathbb F_{2^7}\), \(x^3\) is both a permutation and APN. Over \(\mathbb F_{2^8}\), the same Gold map remains APN but takes each nonzero cube value three times. The inverse permutation \(x^{254}\) on \(\mathbb F_{256}\) has differential uniformity four. The canonical report has SHA-256 digest `7a35e41b9243f97ac244c69402b8bbf0a8ad747db186329e0d509ab923d4c9a8`.