Let $F$ be a finite field, let $f$ be a function from $F$ to $F$, and let $a$ be a nonzero element of $F$. The discrete derivative of $f$ in direction $a$ is $Δ_a f \colon F \to F$ with $(Δ_a f)(x)=f(x+a)-f(x)$. The differential spectrum of $f$ is the multiset of cardinalities of all the fibers of all the derivatives $Δ_a f$ as $a$ runs through $F^*$. An almost perfect nonlinear (APN) function is one for which the largest cardinality in its differential spectrum is $2$. Almost perfect nonlinear functions are of interest as cryptographic primitives. If $d$ is a positive integer, then the power function over $F$ with exponent $d$ is the function $f \colon F \to F$ with $f(x)=x^d$ for every $x \in F$. There is a small number of known infinite families of APN power functions. In this paper, we re-express the exponents for one such family in a more convenient form. This enables us not only to obtain the differential spectrum of each power function $f$ with an exponent in our family, but also to determine the elements that lie in an arbitrary fiber of the discrete derivative of $f$. This differential analysis, which is far more detailed than previous results, is achieved by composing the discrete derivative of $f$ with some permutations and a double covering of its domain to obtain a function whose fibers can more readily be analyzed.

翻译：设$F$为一个有限域，$f$为从$F$到$F$的函数，$a$为$F$中的非零元素。$f$在方向$a$上的离散导数为$Δ_a f \colon F \to F$，其中$(Δ_a f)(x)=f(x+a)-f(x)$。$f$的差分谱是当$a$遍历$F^*$时所有导数$Δ_a f$的所有纤维基数的多重集。若某函数差分谱中的最大基数为$2$，则称其为几乎完全非线性（APN）函数。几乎完全非线性函数作为密码学原语具有重要意义。若$d$为正整数，则$F$上指数为$d$的幂函数定义为$f \colon F \to F$，使得对所有$x \in F$满足$f(x)=x^d$。目前已知的APN幂函数无限族数量有限。本文以更便利的形式重新表达了其中一个无限族的指数表达式。这不仅使我们能够获得该族中每个幂函数$f$的差分谱，还能确定$f$的离散导数任意纤维中的元素。通过将$f$的离散导数与某些置换及其定义域的双重覆盖复合，我们得到了纤维结构更易分析的函数，从而实现了比以往结果更为精细的差分分析。