Gold Functions and Switched Cube Functions Are Not 0-Extendable in Dimension $n > 5$

TL;DR

This paper proves that Gold functions and certain switched cube APN functions in dimensions greater than 5 cannot be extended to higher dimensions while maintaining maximum linearity, highlighting a fundamental limitation in their extendability.

Contribution

It establishes the non-extendability of Gold APN functions and similar quadratic APN functions to higher dimensions with maximum linearity, revealing a key structural property.

Findings

Gold APN functions in dimension >5 cannot be extended to maximum linearity in dimension +1

Switched cube APN functions of the form x↦x^3+μ(x) also cannot be extended similarly

The special case n ≤ 5 allows such extensions, unlike higher dimensions.

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 $n \leq 5$ 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.