Quadratic residues and related permutations

Abstract: Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\ &A_2: \overline{g^2},\ \overline{g^4},\ \cdots,\ \overline{g^{p-1}}, \end{align*} where $g\in\Z$ is a primitive root modulo $p$ and $1\le a_1<a_2<\cdots<a_{(p-1)/2}\le p-1$ are all quadratic residues modulo $p$. Obviously $A_i$ is a permutation of $A_j$ and we call this permutation $\sigma_{i,j}$. Sun obtained the sign of $\sigma_{0,1}$ when $p\equiv 3\pmod4$. In this paper we give the sign of $\sigma_{0,1}$ and determine the sign $\sigma_{0,2}$ when $p\equiv 1\pmod 4$.