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 x(3) + tr(n) (x(9)) 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 L-1 and L-2 from F-2n to itself such that the function L-1(x(3)) + L-2(x(9)) 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 x(3) + a(-1) tr(n)(3) (a(3)x(9) + a(6)x(18)) and x(3) + a(-1) tr(n)(3) (a(6)x(18) + a(12)x(36)) over F-2n for any n divisible by 3 and a is an element of 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 L-1 and L-2 are satisfied.