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 $\Delta_a f \colon F \to F$ with $(\Delta_a f)(x)=f(x+a)-f(x)$. The differential spectrum of $f$ is the multiset of cardinalities of all the fibers of all the derivatives $\Delta_a f$ as $a$ runs through $F^*$. The function $f$ is almost perfect nonlinear (APN) if the largest cardinality in the differential spectrum is $2$. Almost perfect nonlinear functions are of interest as cryptographic primitives. If $d$ is a positive integer, 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 to give the differential spectrum and, even more, to determine the sizes of individual fibers of derivatives.

翻译：设$F$为有限域，$f$为从$F$到$F$的函数，$a$为$F$中的非零元素。$f$沿方向$a$的离散导数为$\Delta_a f \colon F \to F$，满足$(\Delta_a f)(x)=f(x+a)-f(x)$。$f$的差分谱是当$a$遍历$F^*$时所有导数$\Delta_a f$所有纤维基数的多重集。若差分谱中的最大基数为$2$，则函数$f$为几乎完美非线性(APN)函数。几乎完美非线性函数作为密码原语具有重要意义。若$d$为正整数，则$F$上指数为$d$的幂函数为$f \colon F \to F$，满足对于每个$x \in F$有$f(x)=x^d$。目前已知的APN幂函数无穷族数量极少。本文重新表述了其中一个无穷族指数的更便捷形式，从而能够给出其差分谱，并进一步确定各导数单个纤维的规模。