Prime Numbers

400 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
}
return x; // New random number.
Thanks to the shifts and “and” operations, this algorithm involves no explicit
multiplication or division. Furthermore, the generator fares well under some
established statistical tests [Wu 1997]. Of course, this generator can be
generalized, yet as with any machine generator, caution should be taken in
choosing the parameters; for example, the parameters c, M should be chosen
so that c is a primitive root for the prime M to achieve long period. We
should also add an important caution: Very recent experiments and analyses
have uncovered weaknesses in the generator of the type in Algorithm 8.2.3.
Whereas this kind of generator evidently does well on spectral tests, there
are certain bit-population statistics with respect to which such generators
are unsatisfactory [L’Ecuyer and Simard 1999]. Even so, there are still good
reasons to invoke such a generator, such as its very high speed, ease of
implementation, and good performance on some, albeit not all, statistical
tests.
Variants to these congruential generators abound. One interesting development
concerns generators with extremely long periods. A result along such
lines concerns random number generation via matrix–vector multiplication.
If T is a k × k matrix, and x a k-component vector, we may consider the
next vector in a generator’s iteration to be x = Tx, say,withsomerulefor
extracting bits or values from the current vector.
Theorem 8.2.4 (Golomb). For prime p, denote by Mk(p) the group of
nonsingular k × k matrices (mod p), andletx be a nonzero vector in Z k p.
Then the iterated sequence
x, Tx, T 2 x, ...
has period p k − 1 if and only if the order of T ∈ Mk(p) is p k − 1.
This elegant theorem can be applied in the same fashion as we have
constructed the previous iterative generators. However, as [Golomb 1982] and
[Marsaglia 1991] point out, there are much more efficient ways to provide
extreme periods. In this case it is appropriate for the key theorem to follow
the algorithm description, because of the iterative nature of the generator.
Algorithm 8.2.5 (Long-period random generator).
This algorithm assumes input integers b ≥ 2 and r>s>0 and produces an
iterative sequence of pseudorandom integers, each calculated from r previous
values and a running carry bit c. We start with a (vector) seed/carry entity v with
its first r components assumed in [0,b− 1], and last component c =0or 1.
1. [Procedure seed]
seed() {
Choose parameters b ≥ 2 and r>s>0;