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Prime Modulus


When the modulus m is prime, (usually very close to a power of 2 such as a Mersenne prime, tex2html_wrap_inline1495 , 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 tex2html_wrap_inline1503 where tex2html_wrap_inline1505 is the jth 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 tex2html_wrap_inline1505 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 gif. 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 jth integer relatively prime to m-1, and (2) does the good inter-stream correlation also ensure good intra-stream independence via the spectral test?