The Differential and Boomerang Properties of a Class of Binomials
<|reference_start|>The Differential and Boomerang Properties of a Class of Binomials: 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.<|reference_end|>
@article{mesnager2024the,
    title={The Differential and Boomerang Properties of a Class of Binomials},
    author={Sihem Mesnager and Huawei Wu},
    journal={arXiv preprint arXiv:2409.14264},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.14264},
    primaryClass={math.NT cs.CR cs.IT math.IT}
}