Pseudorandom vector generation by the multiple-recursive matrix method

Abstract

Abstract.Pseudorandom vectors are of importance for parallelized simulation methods. In this paper we carry out an in-depth analysis of the multiplerecursive matrix method for the generation of uniform pseudorandom vectors which was introduced in an earlier paper of the author. We study, in particular, the periodicity properties, the lattice structure, and the behavior under the serial test for sequences of pseudorandom vectors generated by this method.

Improved Bounds in the Multiple-Recursive Matrix Method for Pseudorandom Number and Vector Generation

“…In this paper we show that if we vary the -splitting subspace or, in other words, if we pick a ''good'' -splitting subspace, then we can get results that are nearly best possible. This completes a task posed by the author in [10,Section 6]. We remark that a brief announcement (without proofs) of the results of the present paper is contained in [12].…”

“…with an explicitly given ms-dimensional lattice L. For s Ͼ k there is a figure of merit associated with the points in (15) by [10,Definition 3]. As in Section 2 we write…”

“…The multiple-recursive matrix method for the generation of (uniform) pseudorandom numbers and vectors was introduced by the author in [8] and studied in detail in [9,10]. It is a method based on linear vector recursions over finite fields which generalizes several classical methods, such as the multiplicative linear congruential method (with prime modulus), the multiple-recursive congruential method, and the GFSR method, and provides a unified framework for studying many linear methods for pseudorandom number and vector generation.…”

“…We obviously have per(u n ) ϭ per(z n ). [9,10] on the quality of pseudorandom numbers and vectors generated by the multiple-recursive matrix method operate in a setting where the -splitting subspace is fixed, and these results are far from optimal. In this paper we show that if we vary the -splitting subspace or, in other words, if we pick a ''good'' -splitting subspace, then we can get results that are nearly best possible.…”

Primitive polynomials, singer cycles and word-oriented linear feedback shift registers

“…After a preliminary version of this paper was prepared, we found that the seemingly new notion of a σ-LFSR can, in fact, be traced back to the work of Niederreiter (1993Niederreiter ( -1996 mainly in the context of pseudorandom number generation. Indeed, in a series of papers [12,13,14,15], Niederreiter has introduced the so called multiple recursive matrix method and the notion of recursive vector sequences. The latter are essentially the same as sequences generated by a σ-LFSR, modulo a natural isomorphism between the field F q m with q m elements and the vector space F m q of dimension m over The question of counting the number of primitive σ-LFSRs of a given order n over F q m is considered in [13, p. 11] under a different guise (cf.…”