Generalization of a Class of APN Binomials to Gold-Like Functions

Abstract

In 2008 Budaghyan, Carlet and Leander generalized a known instance of an APN function over the finite field $\mathbb {F}_{2^{12}}$ and constructed two new infinite families of APN binomials over the finite field $\mathbb {F}_{2^n}$, one for n divisible by 3, and one for n divisible by 4. By relaxing conditions, the family of APN binomials for n divisible by 3 was generalized to a family of differentially $2^t$-uniform functions in 2012 by Bracken, Tan and Tan; in this sense, the binomials behave in the same way as the Gold functions. In this paper, we show that when relaxing conditions on the APN binomials for n divisible by 4, they also behave in the same way as the Gold function $x^{2^s+1}$ (with s and n not necessarily coprime). As a counterexample, we also show that a family of APN quadrinomials obtained as a generalization of a known APN instance over $\mathbb {F}_{2^{10}}$ cannot be generalized to functions with $2^t$-to-1 derivatives by relaxing conditions in a similar way.