All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation G in their CCZ-equivalence class and two linear permutations A and B, such that G °A = B °G. After providing a survey on the known APN functions with a focus on the existence of self-equivalences, we search for APN permutations in dimension 6, 7, and 8 that admit such a linear self-equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equivalence. In dimensions 7 and 8, we performed an exhaustive search for all but a few classes of linear self-equivalences and we did not find any new APN permutation. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in \mathbb F2 must be (up to CCZ-equivalence) monomial functions.

中文翻译：

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小维线性自等价 APN 排列


迄今为止我们所知道的所有几乎完美的非线性（APN）排列都承认一种特殊的线性自等价性，即在它们的CCZ等价类中存在一个排列G和两个线性排列A和B，使得G°A = B°G。在对已知的 APN 函数进行了调查并重点关注自等价性的存在之后，我们在第 6、7 和 8 维中搜索允许这种线性自等价性的 APN 排列。在第六维度中，我们能够进行详尽的搜索并发现只有一种这样的 APN 排列达到 CCZ 等价。在维度 7 和 8 中，我们对除几类线性自等价类之外的所有类别进行了详尽的搜索，但没有发现任何新的 APN 排列。作为第 7 维中的一个有趣结果，我们得到所有系数为 \mathbb F2 的 APN 置换多项式必须是（直到 CCZ 等价）单项式函数。