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 $$\mathbb {F}_{2^5}$$ F 2 5 give rise to a quadratic APN function in dimension 6 having maximum possible linearity of $$2^5$$ 2 5 (that is, minimum possible nonlinearity $$2^4$$ 2 4 ). In this article, we show that the case of $$n \le 5$$ n ≤ 5 is quite special in the sense that Gold APN functions in dimension $$n>5$$ n > 5 cannot be extended to quadratic APN functions in dimension $$n+1$$ 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 $$x \mapsto x^3 + \mu (x)$$ x ↦ x 3 + μ ( x ) with $$\mu $$ μ being a quadratic Boolean function.