Last update: Fri Mar 15 02:02:37 MDT 2019
@Article{Hildebrand:1993:RPF, author = "Martin Hildebrand", title = "Random Processes of the Form {$ X_{n + 1} = a_n X_n + b_n \pmod p $}", journal = j-ANN-PROBAB, volume = "21", number = "2", pages = "710--720", month = apr, year = "1993", CODEN = "APBYAE", ISSN = "0091-1798 (print), 2168-894X (electronic)", ISSN-L = "0091-1798", bibdate = "Sun Apr 20 10:44:17 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/annprobab1990.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.jstor.org/stable/2244672; http://projecteuclid.org/euclid.aop/1176989264", acknowledgement = ack-nhfb, fjournal = "Annals of Probability", journal-URL = "http://projecteuclid.org/all/euclid.aop", remark = "The author proposes a congruential generator whose multiplier and constant come from two separate independent random number streams. There is no discussion of the period of such a generator, but if $p$ is prime, the period should be the product of the periods of the three generators, and based on the effect of shuffling \cite{Bays:1976:IPR,Bays:1990:CIR}, any lattice structure should be well hidden. Hildebrand shows that convergence to a uniformly distributed sequence is rapid: under mild, and easily satisfied, restrictions on $ a_n $ and $ b_n $ only $ O((\log p)^2) $ steps are required. A test implementation of such a generator using two 16-bit generators from \cite{Kao:1996:EAP} with different prime moduli for the $ a_n $ and $ b_n $, and $ p = 2^{32} - 5 $, passes the Diehard Battery Test Suite.", }