On the period length of pseudorandom vector sequences generated by matrix generators. (English) Zbl 0657.65008

Let m be a positive integer, let \({\mathbb{Z}}_ m=\{0,1,...,m-1\}\), and let A be an \(r\times r\) matrix with elements in \({\mathbb{Z}}_ m\) which is nonsingular (mod m). The following linear recursive congruential matrix generator for generating r-dimensional pseudorandom vectors is considered \((*)\quad \bar x_{n+1}\equiv A\bar x_ n (mod m)\), \(\bar x_{n+1}\in {\mathbb{Z}}^ r_ m\), \(n\geq 0\), with \(\bar x_ 0\in {\mathbb{Z}}^ r_ m\). The sequence \(\{z_ n:\) \(n\geq 0\}\) is purely periodic and let \(\lambda(A,\bar x_ 0,m)\) denote its period length. In this paper the case \(m=p^{\alpha}\), \(\alpha\geq 2\), is considered with p a prime number. It is shown that for \(p\geq 3\) and \(r\geq 2\) there exist matrix generators (*) with \(\lambda (A,\bar x_ 0,p^{\alpha})=(p^ r- 1)p_{\alpha -1}\) for any \(\bar x_ 0\in {\mathbb{Z}}^ r_{p^{\alpha}}\) with \(\bar x_ 0\equiv \bar O (mod p)\).
Reviewer: R.Theodorescu


65C10 Random number generation in numerical analysis
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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