Adaptive chi-square test and its application to some cryptographic problems

We address the problem of testing the hypothesis H0 that the letters from some alphabet A = {a1, a2, . . . , ak}, are distributed uniformly (i.e. p(a1) = p(a2) = . . . = p(ak) = 1/k) against the alternative hypothesis H1 that the true distribution is not uniform, in case k is large. (It is typical for random number testing and some cryptographic problems where k = 210 ∼ 230 and more, see [2, 8, 6]). In such a case it is difficult to use the chi-square test because the sample size must be greater than k. We suggest the adaptive chi-square test which can be successfully applied for testing some kinds of H1 even in case when the sample size is much less than k. This statement is confirmed theoretically and experimentally. The theoretical proof is based on the consideration of one kind of the alternative hypothesis H1 where the suggested test rejects the null hypothesis when the sample size is O( √ k) (instead of const ·k for the usual chi-square test ). For experimental investigation of the suggested test we consider a problem of testing ciphered Russian texts. It turns out that the suggested test can distinguish the ciphered texts from random sequences basing on a sample which is much smaller than that required for the usual chi-square test.