The Differential and Boomerang Properties of a Class of Binomials

• URL: http://arxiv.org/abs/2409.14264v2
• Date: Wed, 25 Sep 2024 18:10:48 GMT
• Title: The Differential and Boomerang Properties of a Class of Binomials
• Authors: Sihem Mesnager, Huawei Wu,
• Abstract summary: We study the differential and boomerang properties of the function $F_2,u(x)=x2big (1+ueta(x)big)$ over $mathbbF_q$. We disproving a conjecture proposed in citebudaghyan2024arithmetization which states that there exist infinitely many $q$ and $u$ such that $F_2,u$ is an APN function.
• Score: 28.489574654566677
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $q$ be an odd prime power with $q\equiv 3\ ({\rm{mod}}\ 4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $\eta$ is the quadratic character of $\mathbb{F}_{q}$. We determine the differential uniformity of $F_{2,u}$ for any $u\in\mathbb{F}_{q}^*$ and determine the differential spectra and boomerang uniformity of the locally-APN functions $F_{2,\pm 1}$, thereby disproving a conjecture proposed in \cite{budaghyan2024arithmetization} which states that there exist infinitely many $q$ and $u$ such that $F_{2,u}$ is an APN function.

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