On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic

• URL: http://arxiv.org/abs/2506.11486v1
• Date: Fri, 13 Jun 2025 06:23:32 GMT
• Title: On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic
• Authors: Namhun Koo, Soonhak Kwon,
• Abstract summary: We show that $F_r,pm1$ is locally-PN with boomerang uniformity $0$ when $pn equiv 3 pmod8$.<n>We also provide complete classifications of the differential and boomerang spectra of $F_r,pm1$.
• Score: 1.104960878651584
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + u\chi(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $\chi(x) = x^{\frac{p^n -1}{2}}$ 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, 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 $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$.

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