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Diskretnaya Matematika, 2004, Volume 16, Issue 2, Pages 54–78
DOI: https://doi.org/10.4213/dm152
(Mi dm152)
 

Spectral properties of a linear congruential generator in special cases

A. S. Rybakov
References:
Abstract: In this paper for the linear congruent generator
$$ z_{N+1}=G(z_N),\qquad N=1,2,\dots, $$
where $G(x)=\lambda x+c \pmod W$, $W=p^F$, $p$ is a prime number, we find a non-trivial lower bound for the least non-zero wave number $e_L(\lambda)$, the fundamental characteristic introduced in the spectral test to check for randomness on the base of analysis of the frequence of occurrences of $L$-tuples $(t_1,\ldots,t_L)$ in the sequence $(z_N)$.
The lower bound obtained is of the form $W^{1/L-\delta}$, where $\delta$ is some variable explicitly depending on parameters which determine the factor $\lambda$. Under an appropriate choice of the parameters, $\delta$ can be made as small as desired. The factor $1/L$ cannot be changed for a greater one. Such bounds are necessary in studying classes of multipliers that pass the spectral test.
Received: 26.02.2003
English version:
Discrete Mathematics and Applications, 2004, Volume 14, Issue 3, Pages 231–255
DOI: https://doi.org/10.1515/1569392031905584
Bibliographic databases:
UDC: 519.7
Language: Russian
Citation: A. S. Rybakov, “Spectral properties of a linear congruential generator in special cases”, Diskr. Mat., 16:2 (2004), 54–78; Discrete Math. Appl., 14:3 (2004), 231–255
Citation in format AMSBIB
\Bibitem{Ryb04}
\by A.~S.~Rybakov
\paper Spectral properties of a linear congruential generator in special cases
\jour Diskr. Mat.
\yr 2004
\vol 16
\issue 2
\pages 54--78
\mathnet{http://mi.mathnet.ru/dm152}
\crossref{https://doi.org/10.4213/dm152}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2084569}
\zmath{https://zbmath.org/?q=an:1121.11051}
\transl
\jour Discrete Math. Appl.
\yr 2004
\vol 14
\issue 3
\pages 231--255
\crossref{https://doi.org/10.1515/1569392031905584}
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  • https://doi.org/10.4213/dm152
  • https://www.mathnet.ru/eng/dm/v16/i2/p54
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