The Differential and Boomerang Properties of a Class of Binomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909326243790848 |
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| author | Mesnager, Sihem Wu, Huawei |
| author_facet | Mesnager, Sihem Wu, Huawei |
| contents | 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 $F_{2,u}(x)=x^2\big(1+uη(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $η$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Differential and Boomerang Properties of a Class of Binomials Mesnager, Sihem Wu, Huawei Number Theory Cryptography and Security Information Theory 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 $F_{2,u}(x)=x^2\big(1+uη(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $η$ 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. |
| title | The Differential and Boomerang Properties of a Class of Binomials |
| topic | Number Theory Cryptography and Security Information Theory |
| url | https://arxiv.org/abs/2409.14264 |