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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2512.17603 |
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| _version_ | 1866908994736488448 |
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| author | Koo, Namhun Kwon, Soonhak Ko, Minwoo Kim, Byunguk |
| author_facet | Koo, Namhun Kwon, Soonhak Ko, Minwoo Kim, Byunguk |
| contents | Recently, several studies have shown that when $q\equiv3\pmod{4}$, for certain choices of $r$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\Fq$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \Fq$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \Fqmul$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17603 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic Koo, Namhun Kwon, Soonhak Ko, Minwoo Kim, Byunguk Information Theory Number Theory 94A60, 06E30 Recently, several studies have shown that when $q\equiv3\pmod{4}$, for certain choices of $r$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\Fq$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \Fq$ with $χ(x)=χ(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \Fqmul$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$. |
| title | Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic |
| topic | Information Theory Number Theory 94A60, 06E30 |
| url | https://arxiv.org/abs/2512.17603 |