# Randomness Evaluation with the Discrete Fourier Transform Test Based on Exact Analysis of the Reference Distribution

## TL;DR

This paper derives a precise mathematical reference distribution for the DFT test in NIST SP 800-22, improving its reliability and sensitivity for evaluating randomness in cryptographic applications.

## Contribution

It provides an exact chi-squared distribution for the power spectrum in the DFT test, replacing the previous numerical estimation method.

## Key findings

- The proposed test is more reliable than the existing DFT test.
- The new test is more sensitive in detecting non-random sequences.
- Experimental results confirm the improved performance of the proposed method.

## Abstract

In this paper, we study the problems in the discrete Fourier transform (DFT) test included in NIST SP 800-22 released by the National Institute of Standards and Technology (NIST), which is a collection of tests for evaluating both physical and pseudo-random number generators for cryptographic applications. The most crucial problem in the DFT test is that its reference distribution of the test statistic is not derived mathematically but rather numerically estimated, the DFT test for randomness is based on a pseudo-random number generator (PRNG). Therefore, the present DFT test should not be used unless the reference distribution is mathematically derived. Here, we prove that a power spectrum, which is a component of the test statistic, follows a chi-squared distribution with 2 degrees of freedom. Based on this fact, we propose a test whose reference distribution of the test statistic is mathematically derived. Furthermore, the results of testing non-random sequences and several PRNGs showed that the proposed test is more reliable and definitely more sensitive than the present DFT test.