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 $2^5$. In this note, 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.
翻译:在Kalgin和Idrissova以及Beierle、Leander和Perrin的独立著作中,人们注意到,金APN的功能大于$mathbb{F ⁇ 2 ⁇ 5},在6维中产生了四边式APN的功能,其最大可能线性为2 ⁇ 5美元。在本说明中,我们表明,$\leq 5美元的情况非常特殊,因为金APN的维度为$ >5美元,不能扩大到维度为$+1美元的四边式APN的功能,在最大线性上不能扩大到$+1美元。