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DX-k, proposed by Deng and Xu [2003], is a special class of multiple Recursive Generators (MRGs) where all nonzero coefficients of the k-th order recurrence are equal. In particular, a DX-k generator requires only up to four nonzero coefficients in its recurrence equation, hence is very efficient in computation. However, a random number generator with few nonzero coefficients has a drawback that, when the k-dimensional state vector is close to the zero vector, the subsequent numbers generated may stay within a neighborhood of zero for quite many of them before they can break away from this near-zero land, a property apparently not desirable in the sense of randomness. Consequently, two generated sequences using the same DX generator with nearly identical initial state vectors may not depart from each other quickly enough. To avoid the above potential problem, we consider MRGs with very few zero coefficients. To make such generators efficient and portable, we propose selecting the same nonzero value for all coefficients (with at most one exception) in the recurrence equation. With this feature, the proposed generators can be implemented efficiently via a higher-order recurrence of few zero coefficients. Note that the new class of generators is an opposite of the DX generators in terms of the number of nonzero coefficients. Several such generators with the maximum period have been found via computer search and presented in the paper for ready implementation. [ABSTRACT FROM AUTHOR]

Multiple Recursive Generators (MRGs) have become the most popular random number generators recently. They compute the next value iteratively from the previous k values using a k-th order recurrence equation which, in turn, corresponds to a k-th degree primitive polynomial under a prime modulus p. In general, when k and p are large, checking if a k-th degree polynomial is primitive under a prime modulus p is known to be a hard problem. A common approach is to check the conditions given in Alanen and Knuth [1964] and Knuth [1998]. However, as mentioned in Deng [2004], this approach has two obvious problems: (a) it requires the complete factorization of pk - 1, which can be difficult; (b) it does not provide any early exit strategy for non-primitive polynomials. To avoid (a), one can consider a prime order k and prime modulus p such that (pk - 1)/(p - 1) is also a prime number as considered in L'Ecuyer [1999] and Deng [2004]. To avoid (b), one can use a more efficient iterative irreducibility test proposed in Deng [2004]. In this paper, we survey several leading probabilistic and deterministic methods for the problems of primality testing and irreducibility testing. To test primality of a large number, it is known that probabilistic methods are much faster than deterministic methods. On the other hand, a probabilistic algorithm in fact has a very tiny probability of, say, 10-200 to commit a false positive error in the test result. Moreover, even when such an unlikely event had happened, for a speci.c choice of k and p, it can be argued that such an error has a negligible e.ect on the successful search of a primitive polynomial. We perform a computer search for large-order DX generators proposed in Deng and Xu [2003] and present many such generators in the paper for ready implementation. An extensive empirical study shows that these large-order DX generators have passed the stringent Crush battery of the TestU01 package. [ABSTRACT FROM AUTHOR]