Budaghyan and Carlet (2008) [4] constructed a family of almost perfect nonlinear (APN) hexanomials over a field with elements, and with terms of degrees , , , , , and , where and with . The construction requires a certain technical condition, which was verified empirically in a finite number of examples. Bracken, Tan, and Tan (2011) [1] proved that the condition holds when or . In this article, we prove that the construction of Budaghyan and Carlet produces APN polynomials for all relatively prime values of m and n.
More generally, if , Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are -to-one maps from to , provided the same technical condition holds. We prove that their construction produces polynomials with this property for all m and n.