The Distribution of the Spectrum for the Discrete Fourier Transform Test Included in SP800-22
Abstract
In this paper, the problem in the distribution of the test statistic of the Discrete Fourier Transform (DFT) test included in SP800-22 released by the National Institute of Standards and Technology (NIST), which causes a very high rate of rejection compared with the significance level, is considered on the basis of the distribution of the spectrum. The statistic of the DFT test, which was supposed to follow the standard normal distribution N(0, 1) according to the central limit theorem, seems to follow the normal distribution N(0.691, 0.5) approximately. The author derived the distribution function of the spectrum, and changed the threshold value from the default value of √3n to the value of 1.7308 ... × n, where n is the length of a random number sequence. By this modification, the test statistic becomes to follow the normal distribution N(0, 0.5) approximately. The disagreement between this variance (= 0.5) and that of the standard normal distribution (= 1) can be considered to originate in the dependence of the spectrum. The evidences of the dependence are shown.
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The NIST reference distribution for N1 is normal with μ = 0.95·n/2 and σ2 = 0.95·0.05·n/4, as experimentally found by many authors.
“The reference distribution for N 1 indicated by NIST in [2] is a normal distribution with µ = N 0 = 0 . 95 · n/ 2 and σ 2 = 0 . 95 · 0 . 05 · n/ 4 . This has been experimentally found independently by many authors [14], [15].”
From On Statistical Tests for Randomness Included in the NIST SP800-22 Test Suite and Based on the Binomial Distribution (2012) · cites this paper as [15]
For the Spectral Test, the paper proposed a refinement of the reference distribution for p-value computation, and with this refinement the known issues of these tests are strongly reduced.
“Furthermore, for the Spectral Test, we have proposed a refinement of the reference distribution for the p -value computation. With this refinement, the issues of these tests that are known and addressed in many other papers [14], [15] are strongly reduced.”
From On Statistical Tests for Randomness Included in the NIST SP800-22 Test Suite and Based on the Binomial Distribution (2012) · cites this paper as [15]
Cites
Bassham, L. E., Rukhin, A. L., Soto, J., Nechvatal, J. R., Smid, M. E., Barker, E. B., Leigh, S. D., Levenson, M., Vangel, M., Banks, D. L., et al. (2010) · NIST Special Publication 800-22 Revision 1a · doi:10.6028/nist.sp.800-22r1a · Cited by 3,617 · see citing works
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Pareschi, F., Rovatti, R., & Setti, G. (2012) · IEEE Transactions on Information Forensics and Security · doi:10.1109/tifs.2012.2185227 · Cited by 189 · see citing works
Correction of Overlapping Template Matching Test Included in NIST Randomness Test Suite
HAMANO, K., & KANEKO, T. (2007) · IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences · doi:10.1093/ietfec/e90-a.9.1788 · Cited by 32 · see citing works
Self-citationError Analysis of NIST SP 800-22 Test Suite
Chen, D., Chen, H., Fan, L., & Luo, K. (2023) · IEEE Transactions on Information Forensics and Security · doi:10.1109/tifs.2023.3287391 · Cited by 14 · see citing works
- Authors
- HAMANO, K.
- Corresponding author
- HAMANO, K.
- Venue
- IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
- Published
- 2005-01-01
- Volume
- E88-A
- Issue
- 1
- Pages
- 67-73
- ISSN
- 0916-8508 (ISSN-L)
- 1745-1337
- OpenAlex
- W2077823812
- Cited by
- 34
- References
- 1
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 1 · 2023: 2 · 2021: 3 · 2020: 3 · 2019: 2 · 2018: 2 · 2017: 2 · 2016: 5 · 2015: 1 · 2014: 1 · 2013: 2 · 2012: 1
- FWCI
- 1.38
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Keywords
- NIST test suite, random number generators, randomness testing, test statistic distribution, rejection rate
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