Gold functions and switched cube functions are not 0-extendable in dimension n > 5

Abstract

In the independent works by Kalgin and Idrisova and by Beierle, Leander and Perrin, it was observed that the Gold APN functions over \(\mathbb {F}_{2^5}\) give rise to a quadratic APN function in dimension 6 having maximum possible linearity of \(2^5\) (that is, minimum possible nonlinearity \(2^4\)). In this article, we show that the case of \(n \le 5\) is quite special in the sense that Gold APN functions in dimension \(n>5\) cannot be extended to quadratic APN functions in dimension \(n+1\) having maximum possible linearity. In the second part of this work, we show that this is also the case for APN functions of the form \(x \mapsto x^3 + \mu (x)\) with \(\mu \) being a quadratic Boolean function.