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

Computer Science > Information Theory [Submitted on 22 May 2026] On APN Exponents and the Differential and Boomerang Properties of Binomials in Characteristic 3 Namhun Koo, Soonhak Kwon, Minwoo Ko, Byunguk Kim Recent studies on binomials of the form F_r(x) = x^r(1 + \chi(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. Subjects: Information Theory (cs.IT); Cryptography and Security (cs.CR) MSC classes: 94A60, 06E30 Cite as: arXiv:2605.23224 [cs.IT]   (or arXiv:2605.23224v1 [cs.IT] for this version)   https://doi.org/10.48550/arXiv.2605.23224 Focus to learn more Submission history From: Namhun Koo [view email] [v1] Fri, 22 May 2026 04:33:54 UTC (23 KB) Access Paper: HTML (experimental) view license Current browse context: cs.IT < prev   |   next > new | recent | 2026-05 Change to browse by: cs cs.CR math math.IT References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)