# Revisiting the DFT Test in the NIST SP 800-22 Randomness Test Suite

### Hiroki Okada, Kazuhide Fukushima

#### 2023

#### Abstract

The National Institute of Standards and Technology (NIST) released SP 800-22, which is a test suite for evaluating pseudorandom number generators for cryptographic applications. The discrete Fourier transform (DFT) test, which is one of the tests in NIST SP 800-22, was constructed to detect some periodic features of input sequences. There was a crucial problem in the construction of the DFT test: its reference distribution of the test statistic was not derived mathematically; instead, it was numerically estimated. Thus, the DFT test was constructed under the assumption that the pseudorandom number generator (PRNG) used for the estimation generated “truly” random numbers, which is a circular reasoning. Recently, Iwasaki (Iwasaki, 2020) performed a novel analysis to theoretically derive the correct reference distribution (without numerical estimation). However, Iwasaki’s analysis relied on some heuristic assumptions. In this paper, we present theoretical evidence for one of the assumptions. Let x0,··· , xn−1 be an n-bit input sequence. Its Fourier coefficients are defined as F0,...,Fn−1. Iwasaki assumed that Σn2 −1j=0|Fj|2 = n2/2. We use a quantitative analysis to show that this holds when n is sufficiently large. We also verify that our analysis is sufficiently accurate with numerical experiments.

Download#### Paper Citation

#### in Harvard Style

Okada H. and Fukushima K. (2023). **Revisiting the DFT Test in the NIST SP 800-22 Randomness Test Suite**. In *Proceedings of the 9th International Conference on Information Systems Security and Privacy - Volume 1: ICISSP,* ISBN 978-989-758-624-8, pages 366-372. DOI: 10.5220/0011626300003405

#### in Bibtex Style

@conference{icissp23,

author={Hiroki Okada and Kazuhide Fukushima},

title={Revisiting the DFT Test in the NIST SP 800-22 Randomness Test Suite},

booktitle={Proceedings of the 9th International Conference on Information Systems Security and Privacy - Volume 1: ICISSP,},

year={2023},

pages={366-372},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0011626300003405},

isbn={978-989-758-624-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 9th International Conference on Information Systems Security and Privacy - Volume 1: ICISSP,

TI - Revisiting the DFT Test in the NIST SP 800-22 Randomness Test Suite

SN - 978-989-758-624-8

AU - Okada H.

AU - Fukushima K.

PY - 2023

SP - 366

EP - 372

DO - 10.5220/0011626300003405