On existence of Budaghyan–Carlet APN hexanomials

Abstract

Budaghyan and Carlet (2008) [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 (2011) [1] proved that the condition holds when m≡2 or 4(mod6). 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 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 that their construction produces polynomials with this property for all m and n.