On existence of Budaghyan-Carlet APN hexanomials

Budaghyan and Carlet [4] constructed a family of almost perfect nonlinear (APN) hexanomials over a field with r2 elements, and with terms of degrees r + 1, s + 1, rs+ 1, rs+ r, rs+ s, and r + s, where r = 2m and s = 2n with GCD(m,n) = 1. The construction requires a certain technical condition, which was verified empirically in a finite number of examples. Bracken, Tan, and Tan [1] proved the condition holds when m ≡ 2 or 4 (mod 6). In this article, we prove that the construction of Budaghyan and Carlet produces APN polynomials for all values of m and n. More generally, if GCD(m,n) = k ≥ 1, Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are 2k-to-one maps from Fr2 to Fr2 , provided the same technical condition holds. We prove their construction produces polynomials with this property for all m and n.