Summary: Assume that \(p\) is an odd prime, and \(f(x)\in \mathbb F_p[x]\). For \((a, p)=1\), denote the multiplicative inverse of \(a\) by \(\overline{a}\) with \(a\overline{a}\equiv 1 \bmod p\) and \(1\leq \overline{a}\leq p-1\). Let \(E_{p-1}=\{e_1,\ldots, e_{p-1}\}\) be defined by \[ e_n=\begin{cases} +1,&\text{if }0\leq R_p(f(n)+\overline{n})<p/2,\\-1,&\text{if }p/2\leq R_p(f(n)+\overline{n})<p,\end{cases} \] where \(R_p(n)\) denotes the least non-negative residue of \(n\) modulo \(p\). In this paper, we study the collision and avalanche effect of \(E_{p-1}\) by using the methods in analytic number theory.