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.