When the modulus *m* is prime, (usually very close to a power of 2
such as a Mersenne prime, , so that the operation of taking
the modulus will be faster) a method based on using the multiplier,
*a*, as the parameter to generate many sequences has been proposed.
We start with a reference value of *a* and choose the multiplier for
the *jth* stream as where is the
*j*th integer relatively prime to *m*-1. This is closely related to
the leapfrog method method discussed earlier. Conditions on *a* and
efficient algorithms for computing can be found in a recent
work of one of the authors [16].

The scheme given above can be justified based on exponential sums,
which is explained in section . Two
important open questions remain: (1) is it more efficient overall to
choose *m* to be amenable to fast modular multiplication or fast
calculation of the *j*th integer relatively prime to *m*-1, and (2)
does the good inter-stream correlation also ensure good intra-stream
independence via the spectral test?

The National Center for Supercomputing Applications

University of Illinois at Urbana-Champaign

ashoks@ncsa.uiuc.edu