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On the lattice structure of a nonlinear generator with modulus \(2^{\alpha}\). (English) Zbl 0702.65006

Let \(Z_ m=\{0,1,...,m-1\}\), \(Z^*_{2^{\alpha}}=\{1,3,...,2^{\alpha}-1\}\) for m,\(\alpha\in N\). Define a nonlinear generator by \(x_{n+1}\equiv ax_ n^{-1}+b(mod 2^{\alpha})\) (1), \(n\geq 0\), where \(x^{-1}\) is the unique element in \(Z^*_{2^{\alpha}}\) with x \(x^{-1}\equiv 1 mod(2^{\alpha})\) for \(x\in Z^*_{2^{\alpha}}\) and \(x_ 0\in Z^*_{2^{\alpha}}\). Assumptions such that (1) has maximal period length \(2^{\alpha -1}\) are given. It is proved that the set of consecutive pseudorandom numbers forms a superposition of shifted lattices. The lattice bases are determined.
Reviewer: W.Grecksch

MSC:

65C10 Random number generation in numerical analysis
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References:

[1] Eichenauer, J.; Lehn, J., On the structure of quadratic congruential sequences, Manuscripta Math., 58, 129-140 (1987) · Zbl 0598.65004
[2] Eichenauer, J.; Lehn, J.; Topuzoǧlu, A., A nonlinear congruential pseudorandom number generator with power of two modulus, Math. Comp., 51, 757-759 (1988) · Zbl 0701.65008
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