A Random Number Generator Test Suite for the C++ ... - ETH Zürich

5. Tests for Studying Random DataIn this section we present different tests to study the behavior of random number generators.We can distinguish two different sorts of tests, statistical tests and physical test 1 .The only difference is the motivation to do the test. In the first case, we want to know thebehavior of some statistical properties, in the second case, we simulate a physical system.(Strictly speaking there are some more tests like “visual tests” or “theoretical test”. But wedo not look at them because of lack of automatism). Each of these tests checks a specialproperty of the generated numbers against the theoretically expected behavior. These testsare not my invention, I only collected them and add examples of usage to it. A reference tothe source (not source code) is mentioned with each test.Table 5.1 lists many known random number generator tests and its occurrence in oftencited test-benches. It is impossible to list all tests, there are an infinite number of them, sowe mention the most popular ones.A more interesting table for testers is table 5.2. It shows all available 2 tests in the test suiteand their class names. (The name of the header file is the concatenation of class name and.h).5.1. Equidistribution testIn this test we check if the generated numbers are equally distributed. See [16].The N measured random values in the interval ¢ α;β£ must be divided in k classesI 1 I 2 ¢¡¢¡¢¡£k . The classes contains N N 1¤N ¡¢¡¢¡¤ 2¤k¥values.For each class, the expected number is calculated with the assumption that all valuesappear with the same probability k Np¥INCheck the probability with the χ 2 test for the classes and use the KS test to check thewhole data.α¨ §β1 A nice description of physical tests is given in [29]Passing several tests does not prove the randomness of any sequence, however. This is dueto the fact that proving randomness requires that the sequence fulfils an actual definition forrandomness. An unfortunate fact is, however, that there is no unique definition for randomness.[...] Therefore, passing many tests is never a sufficient condition for the use of any pseudo randomnumber generator in all applications. In other words, in addition to standard tests, efficientapplication specific tests of randomness are also needed. This need is emphasized by recentsimulations, in which some physical models combined with special algorithms have been foundwhich are very sensitive to the quality of random numbers.2 I hope that by the time this paper is published the list will already be updated with further implementations18