Further results on the $(-1)$-differential uniformity of some functions over finite fields with odd characteristic

Abstract

Functions with low differential uniformity have wide applications in cryptography. In this paper, by using the quadratic character of ${\mathbb {F}}_{p^n}^*$, we further investigate the $(-1)$-differential uniformity of these functions in odd characteristic: (1) $f_1(x)=x^d$, where $d=-\frac{p^n-1}{2}+p^k+1$, n and k are two positive integers satisfying $\frac{n}{\gcd (n,k)}$ is odd; (2) $f_2(x)=(x^{p^m}-x)^{\frac{p^n-1}{2}+1}+x+x^{p^m}$, where $n=3m$; (3) $f_3(x)=(x^{3^m}-x)^{\frac{3^n-1}{2}+1}+(x^{3^m}-x)^{\frac{3^n-1}{2}+3^m}+x$, where $n=3m$. The results show that the upper bounds on the $(-1)$-differential uniformity of the power function $f_1(x)$ are derived. Furthermore, we determine the $(-1)$-differential uniformity of two classes of permutation polynomials $f_2(x)$ and $f_3(x)$ over ${\mathbb {F}}_{p^n}$ and ${\mathbb {F}}_{3^n}$, respectively. Specifically, a class of permutation polynomial $f_3(x)$ that is of P$_{-1}$N or AP$_{-1}$N function over ${\mathbb {F}}_{3^n}$ is obtained.