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 $F$ 为有限域，$f$ 是从 $F$ 到 $F$ 的函数，$a$ 为 $F$ 中非零元素。$f$ 在方向 $a$ 上的离散导数为 $Δ_a f \colon F \to F$，满足 $(Δ_a f)(x)=f(x+a)-f(x)$。$f$ 的差分谱是当 $a$ 取遍 $F^*$ 时所有导数 $Δ_a f$ 的纤维基数所构成的多重集。几乎完美非线性（APN）函数是指其差分谱中最大基数为 $2$ 的函数。几乎完美非线性函数作为密码学原语具有重要研究价值。若 $d$ 为正整数，则 $F$ 上指数为 $d$ 的幂函数定义为 $f \colon F \to F$，满足对所有 $x \in F$ 有 $f(x)=x^d$。目前已知的 APN 幂函数无穷族仅有少数几种。本文将该无穷族中指数重新表示为更便捷的形式。这一重表达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 $f$ 离散导数任意纤维中的元素。这种比以往结果更详尽的差分分析，是通过将 $f$ 的离散导数与若干置换及其定义域的双重覆盖进行复合，从而得到更易分析其纤维结构的函数来实现的。