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

(1)提出了一种新的序列相关性检验方法，该检验方法适用于二值序列的概率比为实值序列的阈值变换后的序列，该检验用于判断二值序列的二阶自相关函数对于每个阈值是否为Delta函数。为了评估由知名生成元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来比较每个二进制序列与Bernoulli试验的比率。我们得出结论：由帐篷映射的高次切比雪夫映射和某些高次迭代产生的线性同余序列和混沌序列均优于Lowis&Payne序列。(3)讨论了产生白色混沌的分段线性马尔可夫映射的设计，得到了一个关联矩阵为1的映射是可实现的，并给出了多个关联矩阵为2的映射。(4)为了补偿经验检验结果，我们利用Perron-Frobenius算子对随机数生成器引入了一个新的理论检验，并通过理论检验结果与切比雪夫映射的经验检验结果的一致性来证实其有效性。

(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.