A RANDOMNESS TEST FOR PSEUDO-RANDOM NUMBER GENERATORS AND CHAOS GENRATORS

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(1) We have introduced a new serial correlation test which is applied to a binary sequence with the probability ration into which a real-valued sequence is transformed using the threshold function;This test is desined to determine whether or not for each threshold value the 2nd order autocorrelation functions of a binary sequences is delta function. Furthermore, we have pointed out the usefulness of its higher-order statistics.(2) In order to assess randomness of real-valued sequences generated by the well-known generators, for any threshold value we have compared each binary sequence with a ratio to Bernoulli trials via the classical chi-square goodness-of-fit test of three kinds of tests. We have concluded that each of linear congruenial sequences and chaotic sequences generated by both the Tchebycheff map of high degree and some high iterate of the tent map is superior to the Lowis ＆ Payne sequence. Because randomness of the latter is remarably influenced by its initialization methods, despite its arbitrarily long period.(3) Discussing the design of piecewise-linear Markov maps generating white chaos, we have concluded that a map whose incident matrix is of rank 1 is realizable and shown several maps whose the incident matrix is of rank 2.(4) In order to compensate empirical test results, we have introduced a new theoretical test for random unmber generators by utilizing its Perron-Frobenius oparator and cofirmed its availability by the agreement of the theoretical test results with the empirical one for the Tchebycheff map.