Tyler Yip, UnixWeenie(tm), asks:

What characteristics of the multiplier and modulator provide large periods?

The standard reference is D. Knuth, The Art of Computer Programming (as I recall), Volume II, the chapter on random numbers. Knuth gives conditions involving a, k and m for n = ( n*a + k ) mod m to have a long (or maximal) period. For the less mathematically-inclined, a pleasing and quick way to eliminate weak pseudorandom number generators is to use the generator to pick row and column pixel positions on a graphics screen. Turn on "randomly" selected pixels. If the screen does not get completely filled in a visibly random way then the generator is weak. A particularly weak generator will turn on 1% or so of the pixels then nothing further happens because it has entered a cycle. A weak generator may fill the screen with parallel lines. Writing a program to test generators in this way is a useful, easy and amusing task and is left as an exercise for the reader. Someone may be inclined to reply that this test does not show that a generator is cryptographically strong, to which the answer is: True, but it certainly eliminates the ones that aren't, and it's fun to watch the pixel display for different generators. Well, on second thought, maybe some generators that are not crypto- graphically bullet-proof might pass this test. But if some generator does not, you can throw it away immediately.