论文标题
黄金功能和开关立方体功能在尺寸$ n> 5 $中不可延长0-
Gold Functions and Switched Cube Functions Are Not 0-Extendable in Dimension $n > 5$
论文作者
论文摘要
在Kalgin和Idrisova的独立作品以及Beierle，Leander和Perrin的独立作品中，观察到，金APN在$ \ Mathbb {f} _ {2^5} $上的功能引起了二次APN在维度6中的二次APN函数，其最大可能的线性性能为$ 2^5 $（这是$ 2^5 $）（这是最低限度的，最低限度的$ 2^4 $ 2^4 $ 2^4 $ 2^44.4 $ 2^44.4 $ 2^4.4 $ 2^4 $ 2^4 $ 2^4 $ 2^4 $ 2^4 $ 2^4 $ 2^4 $ 2^4 $ 2^4。在本文中，我们表明$ n \ leq 5 $的情况是非常特别的，因为黄金APN功能$ n> 5 $不能扩展到二次APN函数$ n+1 $具有最大可能的线性性。在这项工作的第二部分中，我们表明，$ x \ mapsto x^3 +μ（x）$的APN函数也是如此，$μ$是二次布尔函数。
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.
