New pseudorandom sequences constructed by quadratic residues and Lehmer numbers

Let $p$ be an odd prime. Define

$e_n=\left\{\begin{array}{ll}\displaystyle (-1)^{n+\overline{n}}, ... ...e{n}+1}, & \hbox{if $n$ is a quadratic nonresidue mod $p$}, \end{array}\right.$

where $\overline{n}$ is the multiplicative inverse of $n$ modulo $p$ such that $1\leq \overline{n}\leq p-1$. This paper shows that the sequence $\{e_n\}$ is a ``good" pseudorandom sequence, by using the properties of exponential sums, character sums, Kloosterman sums and mean value theorems of Dirichlet $L$-functions.