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New pseudorandom sequences constructed by quadratic residues and Lehmer numbers

Author: Huaning Liu
Journal: Proc. Amer. Math. Soc. 135 (2007), 1309-1318
MSC (2000): Primary 11A07, 11K45
Published electronically: November 14, 2006
MathSciNet review: 2276639
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Abstract: Let $ p$ be an odd prime. Define

$\displaystyle 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.

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Additional Information

Huaning Liu
Affiliation: Department of Mathematics, Northwest University, Xi’an, Shaanxi, People’s Republic of China

Keywords: Pseudorandom, binary sequence, inverse
Received by editor(s): October 28, 2005
Received by editor(s) in revised form: December 23, 2005
Published electronically: November 14, 2006
Additional Notes: This work was supported by the NSF (10271093, 60472068) of P. R. China.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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