Issues on Computer Search for Large Order Multiple Recursive Generators

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 p k - 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 (p k - 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].