Gold函数和交换立方函数在$n > 5$维上不可0-可扩
Gold Functions and Switched Cube Functions Are Not 0-Extendable in Dimension $n > 5$
摘要：
在Kalgin和Idrisova以及Beierle、Leander和Perrin的独立工作中，观察到在$\mathbb{F}_{2^5}$上的Gold APN函数产生在维度6中具有最大可能线性度$2^5$（即，最小可能非线性度$2^4$）的二次APN函数。在这篇文章中，我们表明，$n \leq 5$的情况下是非常特殊的意义上说，黄金APN功能的尺寸$n>5$不能扩展到二次APN功能的尺寸$n+1$具有最大可能的线性。在这项工作的第二部分，我们表明，这也是APN功能的形式$x \mapsto x^3 + μ(x)$与$μ$是一个二次布尔函数的情况。
英文摘要：
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 $x \mapsto x^3 + μ(x)$ with $μ$ being a quadratic Boolean function.
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