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

01/25/2022

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by   Christof Beierle, et al.

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In the independent works by Kalgin and Idrisova and by Beierle, Leander and Perrin, it was observed that the Gold APN functions over 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 ≤ 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 ↦ x^3 + μ(x) with μ being a quadratic Boolean function.