2025/11/02 by Bartoli, Daniele, Grimaldi, Giovanni Giuseppe, Stanica, Pantelimon
Computer Science · #11G20 #11T06 #12E20 #14Q10 #Algebraic Geometry (math.AG) #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2511.01003
openalex publication_date 2025/11/02 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
We systematically analyze a class of hexanomial functions over finite fields of characteristic 2 proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over \mathbbFq2, where q=2n, of the form F(x)=x(Ax2+Bxq+Cx2q)+x2(Dxq+Ex2q)+x3q, we derive necessary conditions on the coefficients A,B,C,D,E for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over \mathbbF22 and \mathbbF24, and random sampling over \mathbbF26 and \mathbbF28, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For q∈\2,4\, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family.