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<inf>n</inf>(x⁹) which is APN over F<inf>2</inf><inf>n</inf> for any positive integer n. The present paper is a continuation of this work. We give sufficient conditions on linear functions L<inf>1</inf> and L<inf>2</inf> from F<inf>2</inf><inf>n</inf> to itself such that the function L<inf>1</inf>(x³) + L<inf>2</inf>(x⁹) is APN over F<inf>2</inf><inf>n</inf>. 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<inf>n</inf>(a³x⁹ + a⁶x¹⁸) and x³ + a⁻¹ tr³<inf>n</inf>(a⁶x¹⁸ + a¹²x³⁶) over F<inf>2</inf><inf>n</inf> for any n divisible by 3 and a Є F^∗<inf>2</inf><inf>n</inf>. 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<inf>1</inf> and L<inf>2</inf> are satisfied.