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I'm studying different tests for uniform random number generators. I've read about the diehard tests and TestU01. These tests test different properties of a uniform random number generator. The birthday test for instance, tests the uniformity of a random number generator, while the permutation test checks if values are correlated (if I understand it correctly).

Question: Is it possible to assign different tests for RNGs into different categories? And if so; which categories are there? Are there just uniformity and iid-ness?

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L'Ecuyer and Simard's paper at the TestU01 site divides TestU01's tests into categories that are more fine-grained than the categories into which Knuth divides them in The Art of Computer Programming 3rd ed., volume 2, chapter 3. (Knuth divides them into three categories: empirical, theoretical, and spectral tests, which are both empirical and theoretical.)

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