Abstract
The problem of constructing effective statistical tests for random number generators (RNG) is considered. Currently, there are hundreds of RNG statistical tests that are often combined into so-called batteries, each containing from a dozen to more than one hundred tests. When a battery test is used, it is applied to a sequence generated by the RNG, and the calculation time is determined by the length of the sequence and the number of tests. Generally speaking, the longer is the sequence, the smaller are the deviations from randomness that can be found by a specific test. Thus, when a battery is applied, on the one hand, the “better” are the tests in the battery, the more chances there are to reject a “bad” RNG. On the other hand, the larger is the battery, the less time it can spend on each test and, therefore, the shorter is the test sequence. In turn, this reduces the ability to find small deviations from randomness. To reduce this trade-off, we propose an adaptive way to use batteries (and other sets) of tests, which requires less time but, in a certain sense, preserves the power of the original battery. We call this method time-adaptive battery of tests. The suggested method is based on the theorem which describes asymptotic properties of the so-called p-values of tests. Namely, the theorem claims that, if the RNG can be modeled by a stationary ergodic source, the value goes to when n grows, where is the sequence, is the p-value of the most powerful test, and h is the limit Shannon entropy of the source.
Highlights
Randomness has many applications in cryptography, statistical sampling, computer modeling, and numerical Monte Carlo methods, as well as in games, gambling, and other fields
For practically used random number generators (RNG) and pseudo random number generators (PRNGs), this property is verified experimentally using statistical tests developed for this purpose
There are more than one hundred applicable statistical tests, as well as dozens RNGs based on different physical processes, and an even greater number of PRNGs based on different mathematical algorithms
Summary
Randomness has many applications in cryptography, statistical sampling, computer modeling, and numerical Monte Carlo methods, as well as in games, gambling, and other fields. Especially in cryptographic applications, this requirement is formulated as follows: an RNG must pass a so-called battery of statistical tests, that is, some fixed set of tests. Given a certain time budget, one can either use more tests and relatively short sequences generated by the RNG, or use fewer tests, but longer sequences and, in turn, this gives more chances to find deviations of randomness of the considered RNG. If our goal is to choose the most powerful test, a good strategy is to choose the test i for which the ratio − log( p−valuei )/li is maximum This recommendation is based on the following theorem: if an RNG can be modeled by a stationary ergodic source, the value −log π ( x1 x2 ...xn )/n goes to 1 − h, if n grows, where x1 x2 ... As far as we know, the proposed approach to testing RNGs is new, but the idea of finding the best test among many, testing the tests step by step in an increasing sequence, is widely used in algorithmic information theory, where the notion of random sequence is formally investigated and discussed [8,9,10]
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