Quick Answer: Statistical randomness tests analyze a sequence of numbers and check whether they exhibit properties expected of truly random sequences. The industry standard is the NIST SP 800-22 test suite (15 tests). Passing all 15 NIST tests is required for any CSPRNG used in government and security applications.
What Statistical Tests for Randomness Check
- Frequency (Monobit) Test: Are 0s and 1s approximately equal in number?
- Block Frequency Test: Is each block of the sequence balanced?
- Runs Test: Are there too many or too few runs (consecutive identical values)?
- Longest Run Test: Is the longest run of 1s within expected limits?
- Binary Matrix Rank Test: Are there the right number of linearly independent rows?
- Spectral Test (DFT): Are there any periodic patterns in the sequence?
- Approximate Entropy Test: Is the entropy consistent with randomness?
- Cumulative Sums Test: Does the sequence show expected random walk behavior?
The NIST SP 800-22 Test Suite
The National Institute of Standards and Technology (NIST) publishes SP 800-22: A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications. It contains 15 statistical tests. Any CSPRNG used in US government applications must pass this suite. Many commercial and open-source implementations are tested against NIST standards.
The Diehard and TestU01 Suites
The Diehard test battery (by George Marsaglia) was the classic comprehensive randomness test suite before NIST. TestU01 (by Pierre L'Ecuyer) is the most comprehensive modern test suite for PRNGs, containing hundreds of tests in its "Crush" and "BigCrush" batteries. The Mersenne Twister passes all tests in small batteries but fails some BigCrush tests — irrelevant for practical use but noted for completeness.
No Test Suite Can Prove True Randomness
Statistical tests can detect detectable non-randomness — but they cannot prove that a sequence is truly random. A perfect PRNG (like a high-quality CSPRNG) will pass all tests. The tests confirm statistical properties expected of random sequences, not that the generator is physically random.