In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + uχ(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $χ(x) = x^{\frac{p^n -1}{2}}$ is the quadratic character in $\mathbb{F}_{p^n}$. We show that $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n \equiv 3 \pmod{8}$. To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity $0$, and the first such example over odd characteristic fields with $p > 3$. Moreover, we show that $F_{r,\pm1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n \equiv 7 \pmod{8}$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$.

翻译：本文研究了有限域 $\mathbb{F}_{p^n}$ 上一类二项式 $F_{r,u}(x) = x^r(1 + uχ(x))$ 的差分与回旋镖性质，其中 $r = \frac{p^n+1}{4}$，$p^n \equiv 3 \pmod{4}$，且 $χ(x) = x^{\frac{p^n -1}{2}}$ 是 $\mathbb{F}_{p^n}$ 上的二次特征。我们证明当 $p^n \equiv 3 \pmod{8}$ 时，$F_{r,\pm1}$ 是局部-PN 函数，且其回旋镖均匀度为 $0$。据我们所知，这是已知的第二个具有零回旋镖均匀度的非-PN 函数类，也是 $p > 3$ 的奇特征域上的首个此类实例。此外，我们证明当 $p^n \equiv 7 \pmod{8}$ 时，$F_{r,\pm1}$ 是局部-APN 函数，且其回旋镖均匀度至多为 $2$。我们还完整给出了 $F_{r,\pm1}$ 的差分谱与回旋镖谱分类。进一步地，我们系统研究了当 $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$ 时 $F_{r,u}$ 的差分均匀性。