Statistical Tests for Checking Independence of Random Variables, Which Describe Sequences Generation in Cryptoalgorithms

Abstract

When a crypto-primitive, whose functions include the generation of a random/pseudo-random gamma, is admitted to operation, a necessary part of its quality checking is a sta-tistical testing of its output gamma and, often, intermediate gamma(s). Such requirement is applied, for example, to random/pseudo-random number generators (RNG/PRNG), stream ciphers, and block ciphers in different "stream" modes (such as OFB, CBC, etc). There exist widely used and well-known tools for checking the statistical properties of se-quences and generators, which are based on a set of statistical tests, like NIST STS set, Diehard, etc. At the same time, the other and very similar question, about independence of the se-quences (more precisely — independence of the corresponding random variables, that the considered sequences are their realizations) generated in such cryptoalgorithms, usually doesn’t attract enough attention. Nevertheless, it is also of great importance, because the dependence of the sequences can lead to predictability of the output gamma, which makes the cryptoprimitive vulnerable to statistical attacks. Therefore, there are no adequate and suitable tools for checking independence of different sequences, generated in the algo-rithms. In this work we deve¬loped and justified new set of three statistical tests for check-ing independence of random va¬riables, which realizations are internal or output sequences in encryption algorithms or RNG/PRNG. We also calculated reference values for limit sta-tistics for different parameters of sequences and different significance levels of tests. Re-sults of tests applications for independent and dependent random variables are given, which confirm correctness of proposed tests.