paper

A note on the differential spectrum of the Ness-Helleseth function

arXiv:2409.03189

Abstract

Let $n ⩾ 3$ be an odd integer and $u$ an element in the finite field $\gf_{3^n}$. The Ness-Helleseth function is the binomial $f_{u} (x) = u x^{d_{1}} + x^{d_{2}}$ 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 $I ¨ (u + 1) = I ¨ (u - 1) = I ¨ (u)$ , is differentially $3$ -uniform when $I ¨ (u + 1) = I ¨ (u - 1) \neq = I ¨ (u)$ , and has differential uniformity at most 4 if $I ¨ (u + 1) \neq = I ¨ (u - 1)$ and $u\notin\gf_3$. Here $I ¨ (\cdot)$ 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 $I ¨ (u + 1) = I ¨ (u - 1)$ or $u\in\gf_3$. The remaining problem is the differential spectrum of $f_{u}$ with $I ¨ (u + 1) \neq = I ¨ (u - 1)$ 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.