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The serial test for a nonlinear pseudorandom number generator

Authors:	Takashi Kato Li-Ming Wu Niro Yanagihara

Institution:	Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Chiba City, 263 Japan Li-Ming Wu ; Department of Mathematics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Chiba City, 263 Japan Niro Yanagihara ; Department of Mathematics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Chiba City, 263 Japan

Abstract:	Let $M = 2^{w},$ and $G_{M} = \{1,3,...,M-1 \}.$ A sequence $\{y_{n} \}, y_{n} \in G_{M},$ is obtained by the formula $y_{n+1} \equiv a{\overline{y}_{n}} + b + cy_{n} \mathrm{mod} M.$ The sequence $\{x_{n} \}, x_{n}=y_{n}/M,$ is a sequence of pseudorandom numbers of the maximal period length if and only if $a+c \equiv 1$ (mod 4), $b \equiv 2$ (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.

Keywords:	Pseudorandom number generator the inversive congruential method power of two modulus discrepancy $k$-dimensional serial test Kloostermann sum

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