A note on the differential spectrum of the Ness-Helleseth function
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910590409113600 |
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| author | Ren, Ketong Xiong, Maosheng Yan, Haode |
| author_facet | Ren, Ketong Xiong, Maosheng Yan, Haode |
| contents | Let $n\geqslant3$ be an odd integer and $u$ an element in the finite field $\gf_{3^n}$. The Ness-Helleseth function is the binomial $f_u(x)=ux^{d_1}+x^{d_2}$ over $\gf_{3^n}$, where $d_1=\frac{3^n-1}{2}-1$ and $d_2=3^n-2$. In 2007, Ness and Helleseth showed that $f_u$ is an APN function when $χ(u+1)=χ(u-1)=χ(u)$, is differentially $3$-uniform when $χ(u+1)=χ(u-1)\neqχ(u)$, and has differential uniformity at most 4 if $ χ(u+1)\neqχ(u-1)$ and $u\notin\gf_3$. Here $χ(\cdot)$ denotes the quadratic character on $\gf_{3^n}$. Recently, Xia et al. determined the differential uniformity of $f_u$ for all $u$ and computed the differential spectrum of $f_u$ for $u$ satisfying $χ(u+1)=χ(u-1)$ or $u\in\gf_3$. The remaining problem is the differential spectrum of $f_u$ with $χ(u+1)\neqχ(u-1)$ and $u\notin\gf_3$. In this paper, we fill in the gap. By studying differential equations arising from the Ness-Helleseth function $f_u$ more carefully, we express the differential spectrum of $f_u$ for such $u$ in terms of two quadratic character sums. This complements the previous work of Xia et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the differential spectrum of the Ness-Helleseth function Ren, Ketong Xiong, Maosheng Yan, Haode Cryptography and Security Information Theory Let $n\geqslant3$ be an odd integer and $u$ an element in the finite field $\gf_{3^n}$. The Ness-Helleseth function is the binomial $f_u(x)=ux^{d_1}+x^{d_2}$ over $\gf_{3^n}$, where $d_1=\frac{3^n-1}{2}-1$ and $d_2=3^n-2$. In 2007, Ness and Helleseth showed that $f_u$ is an APN function when $χ(u+1)=χ(u-1)=χ(u)$, is differentially $3$-uniform when $χ(u+1)=χ(u-1)\neqχ(u)$, and has differential uniformity at most 4 if $ χ(u+1)\neqχ(u-1)$ and $u\notin\gf_3$. Here $χ(\cdot)$ denotes the quadratic character on $\gf_{3^n}$. Recently, Xia et al. determined the differential uniformity of $f_u$ for all $u$ and computed the differential spectrum of $f_u$ for $u$ satisfying $χ(u+1)=χ(u-1)$ or $u\in\gf_3$. The remaining problem is the differential spectrum of $f_u$ with $χ(u+1)\neqχ(u-1)$ and $u\notin\gf_3$. In this paper, we fill in the gap. By studying differential equations arising from the Ness-Helleseth function $f_u$ more carefully, we express the differential spectrum of $f_u$ for such $u$ in terms of two quadratic character sums. This complements the previous work of Xia et al. |
| title | A note on the differential spectrum of the Ness-Helleseth function |
| topic | Cryptography and Security Information Theory |
| url | https://arxiv.org/abs/2409.03189 |