A study of 128-bit multipliers for congruential pseudorandom number generators
For the multiplicative congruential pseudorandom number generator \[ X_{n+1}=AX_n\text{ mod }2^{128}, \] the authors search for good multipliers of the form \(A=5^K\text{ mod }2^{128}\) with a prime number \(K\leq 2^{31}-1\). The paper is the continuation of the authors' paper [ibid. 103, No.~2-3, 103-130 (1997; reviewed above)]. For these generators with a period of \(2^{126}\), the naturally underlying lattice structure of pairs, triplets etc. is so fine that one cannot observe it. By using of parallel computing technique and mixing several flows of random numbers, this effect will even be amplified. The arbitrary postulate \(K>100000\) and five rather rigorously working number-theoretical tests, mainly from the literature (bitmap, crosstalk, lattice, lacunary and others) lead to a reduction of 2155 ``good'' primes \(K\). They remained unnamed and can be taken from a CPC library. The used software and its origin is mentioned in detail. It follow statistical tests as frequency, Kolmogorov-Smirnov, gap, birthday spacing and others. The test results are presented graphically and well correspond to what expected. Difficulties in computational converting are discussed in detail.
scientific article; zbMATH DE number 1437224