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A multiple recursive nonlinear congruential pseudo random number generator. (English) Zbl 0609.65005
A nonlinear multiple recursive congruential pseudo-random number generator with prime modulus p is introduced. Let $$x_ n$$, $$n\geq 0$$, be the sequence generated by a usual linear $$(r+1)$$-step recursive congruential generator with prime modulus p and denote by N(n), $$n\geq 0$$, the sequence of non-negative integers with $$x_{N(n)}\not\equiv 0$$ (mod p). The nonlinear generator is defined by $$z_ n\equiv x_{N(n+1)}\cdot x^{-1}_{N(n)}\quad (mod p),$$ $$n\geq 0$$, where $$x^{-1}_{N(n)}$$ denotes the inverse element of $$x_{N(n)}$$ in the Galois field $$GF(p).$$ A condition is given which ensures that the generated sequence is purely periodic with period length $$p^ r$$ and all $$(p-1)^ r$$ r-tuples $$(y_ 1,...,y_ r)$$ with $$1\leq y_ 1,...,y_ r<p$$ are generated once per period when r-tuples of consecutive numbers of the generated sequence are formed. For $$r=1$$ this generator coincides with a nonlinear generator introduced by the first and the third author in an earlier paper.

##### MSC:
 65C10 Random number generation in numerical analysis
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##### References:
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