The relation between Proportion test and Uniformity test in NIST SP800-22

NIST SP800-22 statistical test tool is one of the randomness test suites. It consists of 15 kinds of tests, and each test outputs the p-value for a sequence. Each test outputs multiple p-values when we test plural sequences. SP800-22 suggests to perform additional hypothesis tests for the multiple p-values, called the proportion test and the uniformity test, and judge the randomness of the tested sequences. To make a rational criterion for the two tests, the relation between the proportion test and the uniformity test is essential. In this paper, we derive the probability that both the proportion and uniformity tests simultaneously reject the null hypothesis. In the derivation process, we use a numerical method, but it is not a stochastic algorithm. Thus, we can ensure the properness of the derived value of the probability.