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There are several test suites referenced at csrc.nist.gov designed to test a (pseudo-)random byte sequence for randomness, and discover possible skews, correlations, patterns, or other hidden non-randomness in it.

Are there implementations of any of them or other similar test suites in Mathematica? Or some precompiled libraries (including Java or .NET modules) that can be invoked from Mathematica running on Windows?

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As far as I know there are no test suites that can be invoked directly from Mathematica. One can of course use the traditional ones such as the one you mention or the NIST or the Marsaglia's diehard tests. I implemented some "toy tests" in this Wolfram Demonstration to illustrate Mathematica's various built-in PRNGs, which fail or succeed the toy tests depending on various factors. There is also this paper that implemented the NIST tests in Mathematica. It is written in Spanish though, although the abstract is in English. Its content is in the context of Elementary Cellular Automata.

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This page has a download link for Ilja Gerhardt's updated version of Galbreath's notebook: https://gerhardt.ch/random.php

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