Let $\gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over $\gf_{p^n}[x]$ with $u\in\gf_{p^n}\setminus\{0,\pm1\}$ was recently studied by Budaghyan and Pal in \cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^n\equiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=\pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{\pm1}$ in terms of the quadratic character sums.

翻译：令$\gf_{p^n}$表示包含$p^n$个元素的有限域，其中$n$为正整数，$p$为素数。Budaghyan与Pal近期在文献\cite{Budaghyan2024ArithmetizationorientedAP}中研究了定义在$\gf_{p^n}[x]$上、参数$u\in\gf_{p^n}\setminus\{0,\pm1\}$的函数$f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$，并证明当$p^n\equiv3~(模~4)$时其差分均匀度至多为$5$。本文针对$u=\pm1$的情形，研究$f_u$的差分均匀度与差分谱。我们首先给出任意密码函数差分谱的若干性质。进一步地，通过求解有限域上的若干方程组，我们借助二次特征和表达了$f_{\pm1}$的差分谱。