On the Design of LIL Tests for (Pseudo) Random Generators and Some Experimental Results

TL;DR

This paper introduces new statistical testing techniques, including LIL-based tests, to improve the detection of deviations from true randomness in pseudorandom generators, addressing limitations of the NIST SP800-22 suite.

Contribution

It proposes novel LIL-based and statistical distance testing methods to enhance randomness testing and reduces Type II errors in existing NIST tests.

Findings

LIL-based tests effectively detect non-randomness in pseudorandom sequences.

Experimental results show low statistical distance (~0.07) from uniform distribution.

Generated 30TB of sequences to validate testing techniques.

Abstract

NIST SP800-22 (2010) proposes the state of art testing suite for (pseudo) random generators to detect deviations of a binary sequence from randomness. On the one hand, as a counter example to NIST SP800-22 test suite, it is easy to construct functions that are considered as GOOD pseudorandom generators by NIST SP800-22 test suite though the output of these functions are easily distinguishable from the uniform distribution. Thus these functions are not pseudorandom generators by definition. On the other hand, NIST SP800-22 does not cover some of the important laws for randomness. Two fundamental limit theorems about random binary strings are the central limit theorem and the law of the iterated logarithm (LIL). Several frequency related tests in NIST SP800-22 cover the central limit theorem while no NIST SP800-22 test covers LIL. This paper proposes techniques to address the above challenges that NIST SP800-22 testing suite faces. Firstly, we propose statistical distance based testing techniques for (pseudo) random generators to reduce the above mentioned Type II errors in NIST SP800-22 test suite. Secondly, we propose LIL based statistical testing techniques, calculate the probabilities, and carry out experimental tests on widely used pseudorandom generators by generating around 30TB of pseudorandom sequences. The experimental results show that for a sample size of 1000 sequences (2TB), the statistical distance between the generated sequences and the uniform distribution is around 0.07 (with $0$ for statistically indistinguishable and $1$ for completely distinguishable) and the root-mean-square deviation is around 0.005.