Recently, Beierle and Leander found two new sporadic quadratic APN permutations in dimension 9. Up to EA-equivalence, we present a single trivariate representation of those two permutations as $C_u \colon (\mathbb{F}_{2^m})^3 \rightarrow (\mathbb{F}_{2^m})^3, (x,y,z) \mapsto (x^3+uy^2z, y^3+uxz^2,z^3+ux^2y)$, where $m=3$ and $u \in \mathbb{F}_{2^3}\setminus\{0,1\}$ such that the two permutations correspond to different choices of $u$. We then analyze the differential uniformity and the nonlinearity of $C_u$ in a more general case. In particular, for $m \geq 3$ being a multiple of 3 and $u \in \mathbb{F}_{2^m}$ not being a 7-th power, we show that the differential uniformity of $C_u$ is bounded above by 8, and that the linearity of $C_u$ is bounded above by $8^{1+\lfloor \frac{m}{2} \rfloor}$. Based on numerical experiments, we conjecture that $C_u$ is not APN if $m$ is greater than $3$. We also analyze the CCZ-equivalence classes of the quadratic APN permutations in dimension 9 known so far and derive a lower bound on the number of their EA-equivalence classes. We further show that the two sporadic APN permutations share an interesting similarity with Gold APN permutations in odd dimension divisible by 3, namely that a permutation EA-inequivalent to those sporadic APN permutations and their inverses can be obtained by just applying EA transformations and inversion to the original permutations.
Linearly Self-Equivalent APN Permutations in Small Dimension
迄今为止，我们所知道的所有几乎完美非线性（APN）排列都承认一种特殊的线性自等价，即，在它们的CCZ等价类中存在一个置换$G$和两个线性置换$A$和$B$，使得$G \circ A = B \circ G$。在提供了一个调查的APN功能，重点是存在的自等价，我们寻找APN排列在6，7，8维，承认这样的线性自等价。在第六维中，我们能够进行穷举搜索，并获得只有一个这样的APN置换到CCZ-等价。在7维和8维中，我们对除了少数几类线性自等价之外的所有线性自等价进行了穷举搜索，并且我们没有找到任何新的APN置换。作为一个有趣的结果，在7维，我们得到的所有APN置换多项式的系数在$\mathbb{F}_2$必须是（直到CC Z-等价）单项函数。
Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially $4$-uniform functions
有限域上的置换多项式是基本对象，因为它们用于密码学、编码理论、组合设计和相关主题中的各种理论和实际应用中。这一系列的多项式构成了一个积极的研究领域，其中不断取得进展。特别是，构造有限域上具有良好微分性质（即低）的置换多项式的无限类仍然是一个令人兴奋的问题，尽管在这个方向上进行了多年的研究。本文给出了奇特征有限域上的低微分均匀幂置换。具体地说，它的目标是双重的关于幂函数$F(x)=x^{\frac{p^n+3}{2}}$定义在有限域$F_{p^n}$的顺序$p^n$，其中$p$是一个奇素数，$n$是一个正整数。首先是补充一些以前的结果发起Helleseth和桑德伯格在\cite{HS}解决开放的问题留下了二十多年的关于确定的差分谱的$F$时，$p^n\equiv3\pmod 4$和$p\neq 3$。二是确定其微分均匀性的精确值。我们的成果首先通过计算$F_{p^n}$上的一些指数和（这相当于计算一些相关曲线上的$F_{p^n}$ -有理点的数目），其次通过计算Helleseth，Rong和Sandberg在[“New families of almost perfect nonlinear power mapping，”IEEE Trans. Inform.理论，45卷。No.2，1999]，在测定$F$的微分光谱时自然出现。我们表明，在所考虑的情况下（$p^n\equiv3\pm…
Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension
Research Institute of Criminal Psychiatry / Sex Offender Medical Center, Tokyo, Japan（东京日本刑事精神病学研究所/性犯罪者医疗中心）；Department of Neuropsychiatry, Kyoto University, Kyoto, Japan（京都大学神经精神医学系）
Coefficient choice in Galois-ring lifts of APN permutations: sharp unit-direction spectra and one-term optimal lifts
Research Institute of Criminal Psychiatry / Sex Offender Medical Center（犯罪精神病学研究所/性犯罪者医疗中心）；Department of Neuropsychiatry, Kyoto University（京都大学神经精神医学系）
Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$ , $B$ , and an affine function $C$ satisfying $G = A \circ F \circ B + C$ . While the problem has a simple formulation, it is very difficult in practice to test whether two functions are EA-equivalent. This problem has two variants: { \em EA-partitioning} deals with partitioning a set of functions into disjoint EA-equivalence classes, and \emph{EA-recovery} is about recovering the tuple $(A,B,C)$ if it exists. In this paper, we prese…
