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Equidistribution properties of nonlinear congruential pseudorandom numbers. (English) Zbl 0787.65003
Let $$p\geq 5$$ be a prime and identify $$\mathbb{Z}_ p:= \{0,1,\dots,p-1\}$$ with the finite field of order $$p$$. Let $$\gamma\in \mathbb{Z}_ p\backslash\{0\}$$, $$g: \mathbb{Z}\to\mathbb{Z}_ p$$ be a monic permutation polynomial of $$\mathbb{Z}_ p$$ with degree $$s$$ as a polynomial over $$\mathbb{Z}_ p$$, where $$3\leq s\leq p-2$$. Define a sequence of elements of $$\mathbb{Z}_ p$$: $$(y_ n)_{n\geq 0}$$ by $$y_ n\equiv \gamma g(n) (\text{mod } p)$$, $$n\geq 0$$, and let $$x_ n= y_ n/p$$ $$(n\geq 0)$$. The author proves that the discrepancy $$D_ N$$ of the sequence of nonlinear congruential pseudorandom numbers $$\{x_ 0,x_ 1,\dots,x_{N-1}\}$$ $$(1\leq N<p)$$ satisfies $D_ N<(s-1) {p^{1/2}\over N}\left({4\over \pi^ 2}\log p+ 0.38+ {0.608\over p}+ {0.116\over p^ 2}\right)^ 2+ {1\over p},$ and also shows that this upper bound for $$D_ N$$ is best possible up to the logarithmic factor. This estimate slightly improves the result of H. Niederreiter [Monatsh. Math. 106, No. 2, 149-159 (1988; Zbl 0652.65007)].

##### MSC:
 65C10 Random number generation in numerical analysis 11K45 Pseudo-random numbers; Monte Carlo methods 11K38 Irregularities of distribution, discrepancy
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