Toward resolution of the exceptional APN conjecture in the Kasami–Welch degree case

An almost perfect nonlinear (APN) function is a function [Formula: see text] satisfying the property that its directional derivative at every nonzero point is two to one. APN functions arise in many areas of mathematics. For example, in cryptography, the APN function minimizes the probability of differential cryptanalysis’s success. Also, some of the curves and surfaces defined by the corresponding nonlinear functions have many rational points and have applications to Algebraic-Geometric (AG) codes [H. Janwa and R. Wilson, Hyperplane sections of Fermat varieties in [Formula: see text] in [Formula: see text] and some applications to cyclic codes, in 14th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-correcting Codes[Formula: see text]San Juan, PR, 1993[Formula: see text] , (Springer, Berlin, 1993), pp. 180–194; H. Janwa and R. Wilson, Rational points on the Klein quartic and words of weight [Formula: see text] in the binary cyclic codes generated by [Formula: see text], to appear]. An APN function is considered an exceptional APN if it is an APN over infinitely many extensions of [Formula: see text]. An equivalent algebraic geometric formulation in terms of rational points on a corresponding algebraic curve was derived in [H. Janwa and R. Wilson, Hyperplane sections of Fermat varieties in [Formula: see text] in [Formula: see text] and some applications to cyclic codes, in 14th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-correcting Codes[Formula: see text]San Juan, PR, 1993[Formula: see text] (Springer, Berlin, 1993), pp. 180–194]. They gave nonexistence results using the Weil estimate on the number of rational points on absolutely irreducible polynomials. In [H. Janwa and R. Wilson, Hyperplane sections of Fermat varieties in [Formula: see text] in [Formula: see text] and some applications to cyclic codes, in 14th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-correcting Codes[Formula: see text]San Juan, PR, 1993[Formula: see text] (Springer, Berlin, 1993), pp. 180–194; H. Janwa and R. Wilson, Rational points on the Klein quartic and words of weight [Formula: see text] in the binary cyclic codes generated by [Formula: see text], to appear] it was conjectured that monomial exceptional APN functions necessarily have the Gold ([Formula: see text]) or Kasami–Welch ([Formula: see text]) degree exponents. A breakthrough was obtained in [H. Janwa and R. Wilson, Hyperplane sections of Fermat varieties in [Formula: see text] in [Formula: see text] and some applications to cyclic codes, in 14th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-correcting Codes[Formula: see text]San Juan, PR, 1993[Formula: see text] (Springer, Berlin, 1993), pp. 180–194; H. Janwa and R. Wilson, Rational points on the Klein quartic and words of weight [Formula: see text] in the binary cyclic codes generated by [Formula: see text], to appear] using an absolute irreducibility criterion using Bezout’s theorem and intersection multiplicities. After substantial progress in [H. Janwa and R. Wilson, Hyperplane sections of Fermat varieties in [Formula: see text] in [Formula: see text] and some applications to cyclic codes, in 14th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-correcting Codes[Formula: see text]San Juan, PR, 1993[Formula: see text] (Springer, Berlin, 1993), pp. 180–194; H. Janwa and R. Wilson, Rational points on the Klein quartic and words of weight [Formula: see text] in the binary cyclic codes generated by [Formula: see text], to appear], a final resolution of this conjecture was obtained in [F. Hernando and G. McGuire, Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions, J. Algebra 343 (2011) 78–92]. Aubry et al. [A few more functions that are not APN infinitely often, in Finite Fields: Theory and Applications, Contemporary Mathematics, Vol. 518 (American Mathematical Society, Providence, RI, 2010), pp. 23–31] conjectured that up to CCZ equivalence, the only exceptional APN functions are the Gold and Kasami–Welch monomials. Using the solution to the monomial case of the conjecture, they established that the odd degrees are necessarily the Gold or Kasami–Welch exponents. The Weil estimate was replaced by Lang–Weil and Ghorpade–Lachaud estimates. Since then, several authors, including us, have made substantial progress (see [M. Delgado, H. Janwa and C. Agrinsoni, Some new techniques and progress towards the resolution of the conjecture of exceptional APN functions and absolutely irreducibility of a class of polynomials, Des. Codes Cryptogr. 91 (2023) 2481–2495] for citations to the latest published results). We recently resolved this conjecture when the degree exponent is Gold [C. Agrinsoni, H. Janwa and M. Delgado, Resolution of the Exceptional APN conjecture in the gold degree case, Des. Codes Cryptogr. (2025) 1–20, https://doi.org/10.1007/s10623-025-01607-x] . Substantial progress has been made toward resolving this conjecture in the Kasami–Welch degree case by Delgado and Janwa [On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in [Formula: see text] and the conjecture on exceptional APN functions: the Kasami-Welch degree case, preprint (2016), arXiv:1612.05997; Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: the Gold case, Adv. Math. Commun. 11(2) (2017) 389–396] and Ferard [A infinite class of Kasami functions that are not APN infinitely often, in Arithmetic, Geometry, Cryptography, and Coding Theory, Contemporary Mathematics, Vol. 686 (American Mathematical Society, Providence, RI, 2017), pp. 45–63] when the degree of the second term is odd. In this paper, we give a resolution of the exceptional APN conjecture in the Kasami–Welch degree case when the second term has odd degree. Additionally, we make some progress in the remaining case of resolution of the odd degree conjecture. Finally, we also provide some results on absolute irreducibility testing that are of independent interest.