On a construction of quadratic APN functions

In a recent paper, the authors introduced a method for constructing new quadratic APN functions from known ones. Applying this method, they obtained the function x³ + tr_n(x⁹) which is APN over F_2n for any positive integer n. The present paper is a continuation of this work. We give sufficient conditions on linear functions L₁ and L₂ from F_2n to itself such that the function L₁(x³) + L₂(x⁹) is APN over F_2n. We show that this can lead to many new cases of APN functions. In particular, we get two families of APN functions x³ + a^-1 tr³ _n(a³x⁹ + a⁶x¹⁸) and x³ + a^-1 tr³ _n(a⁶x¹⁸ + a¹²x³⁶) over F_2n for any n divisible by 3 and a ¿ F_2n*. We prove that for n = 9, these families are pairwise different and differ from all previously known families of APN functions, up to the most general equivalence notion, the CCZ-equivalence. We also investigate further sufficient conditions under which the conditions on the linear functions L₁ and L₂ are satisfied.