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

*Boris Ryabko*

**Abstract: **We address the problem of testing the hypothesis H_0 that the
letters from some alphabet A= {a_1,a_2,..., a_k }, are
distributed uniformly
against the alternative hypothesis H_1 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= 2^{10} - 2^{30} and more). 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 H_1 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 H_1 where the suggested test rejects the null hypothesis when the sample size is O( \sqrt{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.

**Category / Keywords: **secret-key cryptography / adaptive testing, random number testing, block cipher testing

**Date: **received 8 Mar 2002

**Contact author: **ryabko at neic nsk su

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20020308:201111 (All versions of this report)

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