Recent studies on binomials of the form $F_r(x) = x^r(1 + χ(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in characteristic $3$, where instances of extremely low boomerang uniformity-namely $0$ or $1$-seem to arise more frequently than in other characteristics. First, we provide a systematic analysis of Almost Perfect Nonlinear (APN) power functions in characteristic $3$. We present an explicit parametrization of APN exponents arising from the construction of Zha and Wang and demonstrate through numerical results for $n \le 13$ that this generalized framework accounts for several previously known and sporadic APN instances. Building on this classification, we identify and rigorously prove two classes of binomials $F_r$ that are locally-PN and possess the minimum possible boomerang uniformity of $0$. These classes involve exponents derived from the aforementioned APN construction and the differentially 4-uniform exponent $r = 2 \cdot 3^{\frac{n-1}{2}} + 1$. Furthermore, we analyze the binomial $F_r$ with $r = 3^n - 3$, proving that it is locally-APN with boomerang uniformity $1$ when $n\ge 5$ is odd, and completely determine its boomerang spectrum through the evaluation of character sums. Our results clarify and extend existing studies on the cryptographic properties of binomials, providing a systematic characterization of several classes of binomials with very low boomerang uniformity in characteristic $3$.

翻译：近期对形如 $F_r(x) = x^r(1 + χ(x))$ 的 $\mathbb{F}_{p^n}$ 上二项式的研究表明，此类函数可呈现极低的回笼均匀度。本文聚焦特征 $3$ 下该二项式的特殊行为，其中回笼均匀度极低（即 $0$ 或 $1$）的实例似乎比其他特征更为常见。首先，我们对特征 $3$ 下的几乎完美非线性（APN）幂函数进行系统分析。给出由Zha与Wang构造导出的APN指数的显式参数化，并通过 $n \le 13$ 的数值结果显示，这一广义框架涵盖了多个先前已知及零星的APN实例。基于该分类，我们识别并严格证明了两类局部PN且具有最小可能回笼均匀度 $0$ 的二项式 $F_r$。这些类涉及源自前述APN构造的指数以及差分为4-均匀的指数 $r = 2 \cdot 3^{\frac{n-1}{2}} + 1$。此外，我们分析了指数 $r = 3^n - 3$ 的二项式 $F_r$，证明当 $n\ge 5$ 为奇数时其为局部APN且回笼均匀度为 $1$，并通过特征和求值完全确定其回笼谱。我们的结果阐明并扩展了现有关于二项式密码学性质的研究，系统刻画了特征 $3$ 下若干类具有极低回笼均匀度的二项式。