Prime Numbers

8.2 Random-number generation 399
Algorithm 8.2.2 (32-bit random-number generator (Knuth, Lewis)).
This algorithm provides seeding and random functions for a certain generator
known to have fairly good statistical behavior. We take M =2 32 as the generator
modulus, and will speedily effect operations modulo M by logical “and” (&)
with M −1. One first calls the seed() procedure, then calls random() successively
to get random numbers.
1. [Procedure seed]
seed() {
Choose starting seed x; // x is an integer in [0,M − 1].
return;
}
2. [Function random]
random() {
x = (1664525x + 1013904223) & (M − 1);
return x; // New random number.
}
Note that the “and” operation with M − 1 is simply the taking of the low 32
bits of the number involved. Along similar lines, the popular generator
xn+1 = (16807xn) modM31,
where M31 =231 − 1 is a Mersenne prime, has enjoyed a certain success in
passing many (but not all) experimental tests [Park and Miller 1988], [Press
et al. 1996].
An interesting optimization of certain congruential generators has been
forwarded in [Wu 1997]. The recursion is
xn+1 = (2 30 − 2 19
)xn mod M61,
where the fact of M61 being a Mersenne prime allows some rapid arithmetic.
Algorithm 8.2.3 (Fast, 61-bit random generator). This algorithm provides
seeding and random functions for the Wu generator, modulus M =2 61 − 1 and
multiplier c =2 30 − 2 19 . Though modular multiplications occur in principle, the
explicit operations below are relegated to addition/subtraction, left/right shifts
(>, respectively), and logical “and” (&) which acts as a certain mod
operation.
1. [Procedure seed]
seed() {
Choose starting seed x; // x is an integer in [1,M − 1].
return;
}
2. [Function random]
random() {
x =(x>>31) + ((x 42) − ((x