Resolution of the exceptional APN conjecture in the Gold degree case

Abstract

A function \(f: {\mathbb {F}}_q \rightarrow {\mathbb {F}}_q\), is called an almost perfect nonlinear (APN) if \(f(X+a)-f(X) =b\) has at most 2 solutions for every \(b,a \in {\mathbb {F}}_q\), with a nonzero. Furthermore, it is called an exceptional APN if it is an APN on infinitely many extensions of \({\mathbb {F}}_q\). These problems are equivalent to finding rational points on the corresponding variety \({\mathcal {X}}_f:=\phi _f(X,Y,Z)=0\). The Lang–Weil, Deligne, and Ghorpade–Lachaud bounds help solve these problems when \(\phi _f\) contains an absolutely irreducible factor in the defining field. The exceptional monomial APN functions had been classified up to CCZ equivalence by Hernando and McGuire (J Algebra 343:78–92, 2011), proving the conjecture of Janwa, Wilson, and McGuire (JMW) (1993, 1995). The main tools used were the computation and classification of the singularities of \({\mathcal {X}}_f\) and the algorithm of JMW for the absolute irreducibility testing using Bezout’s Theorem. Aubry et al. (2010) conjectured that the only exceptional APN functions of odd degree up to CCZ equivalence are the Gold \((2^k+1)\) and the Kasami-Welch \((2^{2k}-2^k+1)\) monomial functions. Here, we settle the first case (Theorem 20). We also prove a part of a conjecture on exceptional crooked functions. One of the main tools in our proofs is our new absolute irreducibility criterion (Theorem 9).