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.