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

(1)引入了一种新的序列相关检验方法，该方法应用于二值序列，其概率比是用阈值函数将实值序列变换为二值序列；该测试旨在确定对于每个阈值，二值序列的二阶自相关函数是否为δ函数。此外，我们还指出了其高阶统计量的有用性。(2)为了评估由知名生成器生成的实值序列的随机性，对于任意阈值，我们通过三种检验的经典卡方拟合优度检验将每个二值序列与伯努利试验的比率进行比较。我们得出结论，由高阶Tchebycheff映射和一些高迭代的tent映射生成的线性同余序列和混沌序列都优于Lowis & Payne序列。因为后者的随机性受到其初始化方法的显著影响，尽管其周期任意长。(3)讨论了产生白混沌的分段线性马尔可夫映射的设计，得出了一个事件矩阵为秩1的映射是可实现的，并给出了几个事件矩阵为秩2的映射。(4)为了弥补实证检验结果，我们引入了一种新的随机数生成器理论检验，利用其Perron-Frobenius算子，并通过理论检验结果与Tchebycheff映射的实证检验结果的一致性，证实了其有效性。

(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.