New pseudorandom sequences constructed by quadratic residues and Lehmer numbers

Abstract:

Let $p$ be an odd prime. Define \[ e_n = \begin {cases} (-1)^{n+\overline {n}}, & \text {if $n$ is a quadratic residue mod $p$},\\ (-1)^{n+\overline {n}+1}, & \text {if $n$ is a quadratic nonresidue mod $p$}, \end {cases} \] 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.