The Distribution of the Spectrum for the Discrete Fourier Transform Test Included in SP800-22

Kenji HAMANO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A   No.1   pp.67-73
Publication Date: 2005/01/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.1.67
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Symmetric Key Cryptography
discrete Fourier transform,  random number,  statistical test,  SP800-22,  NIST,  central limit theorem,  

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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 to the value of 1.7308 , 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.