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On a problem of D.H. Lehmer and pseudorandom binary sequences

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Abstract

Let p be an odd prime, and f(x), g(x) ∈ \( \mathbb{F}_p \)[x]. Define

$$ e'_n = \left\{ \begin{gathered} + 1,if\overline {f(n)} \equiv R_p (g(n))(\bmod 2), \hfill \\ - 1,if\overline {f(n)} \not \equiv R_p (g(n))(\bmod 2), \hfill \\ \end{gathered} \right. $$

where \( \bar x \) is the inverse of x modulo p with \( \bar x \) ∈ {1, ..., p − 1}, and R p (x) denotes the unique r ∈ {0, 1, ..., p − 1} with x ≡ r(mod p). This paper shows that the sequences {e′ n } is a “good” pseudorandom binary sequences, and give a generalization on a problem of D.H. Lehmer.

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Authors and Affiliations

  1. Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China

    Huaning Liu

  2. Department of Mathematics, Shangluo University, Shangluo, Shaanxi, P.R. China

    Cundian Yang

Authors
  1. Huaning Liu
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  2. Cundian Yang
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Correspondence to Huaning Liu.

Additional information

Supported by the National Natural Science Foundation of China under Grant No. 60472068 and No. 10671155; Natural Science Foundation of Shaanxi province of China under Grant No. 2006A04; and the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant No. 06JK168.

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Liu, H., Yang, C. On a problem of D.H. Lehmer and pseudorandom binary sequences. Bull Braz Math Soc, New Series 39, 387–399 (2008). https://doi.org/10.1007/s00574-008-0012-6

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  • DOI: https://doi.org/10.1007/s00574-008-0012-6

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