On the Walsh spectrum of a family of quadratic APN functions with five terms

Abstract

Recently, a family of quadratic APN functions was demonstrated by Bracken et al. to exist over $\mathbb{F}_{2^{2k} }$ with k even and 3 ∤ k. This family of APN functions was firstly proposed by Budaghyan et al. and they exist provided the existence of a quadratic polynomial of the type $x^{2^s + 1} + cx^{2^s } + c^{2^k } x + 1$ with no zeros in $\mathbb{F}_{2^{2k} }$. Bracken et al. constructed such polynomials when k is even and 3 ∤ k. In this paper, we show that such polynomials exist for all even integers k. As a result, the APN functions over $\mathbb{F}_{2^{2k} }$ exist for all even k. Furthermore, the Walsh spectra of these APN functions is shown to be the same as the one of the Gold APN functions. This gives a positive answer to one conjecture.