论文标题
在小维度中线性自我等效的APN排列
Linearly Self-Equivalent APN Permutations in Small Dimension
论文作者
论文摘要
迄今为止，我们知道的所有几乎完美的非线性（APN）排列都承认了一种特殊的线性自我等量，即，在其CCZ等值类中存在置换$ g $，并且两个线性排列$ a $ a $ a $ a $ a $ a $ a $ a $ a $ b $，因此$ g \ g \ g \ g \ circe circe circe a = b \ circc g $。在对已知的APN功能的调查以重点是自我等量的存在之后，我们在第6、7和8的apn排列中搜索了这种线性自我等量。在第六维度中，我们能够进行详尽的搜索，并获得只有一个apn置换量为CCZ等效性。在维度7和8中，我们对除几类线性自我等量外的所有类别进行了详尽的搜索，但我们没有找到任何新的APN置换。作为维度7中的一个有趣结果，我们获得了$ \ mathbb {f} _2 $中具有系数的所有APN置换多项式，必须是（最高ccz-querivalence）单函数。
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 $G \circ A = B \circ G$. 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 $\mathbb{F}_2$ must be (up to CCZ-equivalence) monomial functions.
