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Marsaglia’s lattice test and non-linear congruential pseudo-random number generators. (English) Zbl 0653.65006
A recursive congruential non-additive generator of the form $$(1)\quad x_{n+1}\equiv f(x_ n)(mod p),$$ $$x_{n+1}\in {\mathbb{Z}}_ p$$, $$n\geq 0$$, is considered, where p is a large prime number, $${\mathbb{Z}}_ p=\{0,1,...,p-1\}$$, $$x_ 0\in {\mathbb{Z}}_ p$$, and f: $${\mathbb{Z}}_ p\to {\mathbb{Z}}_ p$$ is a function such that (1) has maximal period length. The sequences of integers $$\{x_ i:$$ $$i\geq 0\}$$ generated by (1) are divided into vectors of $$d\geq 2$$ consecutive numbers: $$v^ d_ i=(x_ i,...,x_{i+d-1})^ T\in {\mathbb{Z}}^ d_ p$$ and let $$w^ d_ i\equiv v_ i^ d-v^ d_ 0(mod p),$$ $$i\geq 0$$. For $$d\leq 3$$, it is shown that $$V^ d={\mathbb{Z}}^ d_ p,$$ where $$V^ d=\{v\in {\mathbb{Z}}^ d_ p| \quad v\equiv \sum^{p-1}_{i=1}z_ iw^ d_ i(mod p);\quad z_ 1,...,z_{p-1}\in {\mathbb{Z}}_ p\}.$$ In other words, (1) passes G. Marsaglia’s lattice test [Applications of number theory to numerical analysis, 249-285 (1972; Zbl 0266.65007)]. For $$d\geq 4$$ there are generators (1) which fail this test. It is also shown that the generators of a class of nonlinear generators introduced by the first and the third author [Stat. Hefte 27, 315-326 (1986; Zbl 0607.65001)] pass Marsaglia’s lattice test for $$d\leq (p-1)/2$$.
Reviewer: R.Theodorescu

##### MSC:
 65C10 Random number generation in numerical analysis
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##### References:
  Beyer WA, Roof RB, Williamson D (1971) The lattice structure of multiplicative pseudo-random vectors. Math Comp 25:345–363 · Zbl 0269.65003  Eichenauer J, Lehn J (1986) A non-linear congruential pseudo random number generator. Statistical Papers 27:315–326 · Zbl 0607.65001  Marsaglia G (1972) The structure of linear congruential sequences. In: Zaremba SK (ed) Applications of number theory to numerical analysis, pp 249–285 · Zbl 0266.65007
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