On APN Exponents and the Differential and Boomerang Properties of Binomials in Characteristic 3

Abstract

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 $3$.