Chaotic Random Number Generators with Random Cycle Lengths

A known cycle length has hitherto been considered an indispensable requirement for pseudorandom number generators. This requirement has restricted random number generators to suboptimal designs with known deficiencies. The present article shows that the requirement for a known cycle length can be avoided by including a self-test facility. The distribution of cycle lengths is analyzed theoretically and experimentally. As an example, a class of chaotic random number generators based on bit-rotation and addition are analyzed theoretically and experimentally. These generators, suitable for Monte Carlo applications, have good randomness, long cycle lengths, and higher speed than other generators of similar quality. Introduction The literature on random number generators has often emphasized that random number generators must be supported by theoretical analysis (Knuth 1998). In fact, most treatises on random number generators have focused mainly on very simple generators, such as linear congruential generators, in order to make theoretical analysis possible (e.g. Knuth 1998, Niederreiter 1992, Fishman 1996). Unfortunately, such simple generators are known to have serious defects (Entacher 1998). The sequence produced by a pseudo-random number generator is deterministic, and hence not absolutely random. It always has some kind of structure. The difference between good and bad generators is that the good ones have a better hidden structure, i.e. a structure that is more difficult to detect by statistical tests and less likely to interfere with specific applications (Couture and L’Ecuyer 1998). This criterion appears to be in direct conflict with the need for mathematical tractability. The best random number generators are likely to be the ones that are most difficult to analyze theoretically. The idea that mathematical intractability may in fact be a desired quality has so far only been explored in cryptographic applications (e.g. Blum et al 1986), not in Monte Carlo applications. Mathematicians have done admirable efforts to analyze complicated random number generators, but this doesn’t solve the fundamental dilemma between mathematical tractability and randomness. Thus, we are left with the paradox that it may be impossible to know which type of generator is best. Two characteristics of random number generators need to be analyzed: randomness and cycle length. Randomness may be tested either by theoretical analysis or by statistical tests. Both methods are equally valid in the sense that a particular defect may be detected by either method. Some defects are most easily detected by theoretical analysis; other defects are easier to detect experimentally. Thus, it is recommended that a generator be subjected to both types of testing. The discipline of designing random number generators has reached a state where good generators pass all experimental tests. Attempts at improvement have therefore in recent years relied increasingly on theoretical testing. It is often required that random number generators have very long cycle lengths (L'Ecuyer 1999). Theoretical analysis is the only way to find the exact cycle length in case the cycle is too long to measure experimentally. However, experimental tests can assure that the cycle is longer than the sequence of random numbers needed for a particular application. As is demonstrated below, such a test can be performed "on the fly" in a very efficient way if the state transition function is invertible. Cycle lengths in random maps A pseudo random number generator is based on the sequence S s s f s n n ∈ = − ), ( 1 (1) where the state transition function f maps the finite set S into itself. The number of possible states is the cardinality of S: