Building test batteries based on analysing random number generator tests within the framework of algorithmic information theory

TL;DR

The paper addresses how to compare randomness tests for RNGs beyond traditional statistics by framing test power with algorithmic information theory and Hausdorff dimension. It develops tests based on universal codes for memory-structured processes and proves ordering relations among them, while demonstrating that dictionary-based compression tests (e.g., LZ) add nontrivial detection power not captured by entropy-based tests. A constructive two-faced-process argument and an explicit LZ-based test show that some sequences can be non-random for LZ while remaining random for others, justifying the inclusion of LZ/grammar-based compressors in test batteries. The practical guidance recommends mixing tests across varying word lengths and including dictionary-based compressors to enhance the ability to detect non-randomness beyond stationary ergodic models.

Abstract

The problem of testing random number generators is considered and it is shown that an approach based on algorithmic information theory allows us to compare the power of different tests in some cases where the available methods of mathematical statistics do not distinguish between the tests. In particular, it is shown that tests based on data compression methods using dictionaries should be included in the test batteries.