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I have written my own random number generator as a C library and I tested it on NIST Statistical Test Suite.

Now I would like to get a certification to have a formal proof that my C library generates random numbers randomly enough according SP800-22.

On a NIST web pages I'm not able to find some list of independent laboratory that could take my library, test it, and provide me certification.

Where can I find a list of companies/independent laboratories that can certify my RNG library according NIST SP800-22?

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This certification is typically done through FIPS certification.

The list of labs certified by NIST to perform FIPS certifications is here

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