The Discrete Fourier Transform Test (DFTT) is a randomness test in NIST SP800-22. However, to date, the theoretical reference distribution of the DFTT statistic has not been derived, which is problematic. We propose a new test using power spectrum variance as the test statistic whose reference distribution can be derived theoretically. Note that the purpose of both the DFTT and the proposed test is to detect periodic features. Experimental results demonstrate that the proposed test has stronger detection power than the DFTT and that it test can be used even for short sequences.

Linear complexity can be used to detect predictable nonrandom sequences, and hence it is included in the NIST randomness test suite. But, as shown in this paper, the NIST test suite cannot detect nonrandom sequences that are generated, for instance, by concatenating two different M-sequences with low linear complexity. This defect comes from the fact that the NIST linear complexity test uses deviation from the ideal value only in the last part of the whole linear complexity profile. In this paper, a new faithful linear complexity test is proposed, which uses deviations in all parts of the linear complexity profile and hence can detect even the above nonrandom sequences. An efficient formula is derived to compute the exact area distribution needed for the proposed test. Furthermore, a simple procedure is given to compute the proposed test statistic from linear complexity profile, which requires only O(M) time complexity for a sequence of length M.

Accurate values for occurrence probabilities of the template used in the overlapping template matching test included in NIST randomness test suite (NIST SP800-22) have been analyzed. The inaccurate values used in the NIST randomness test suite cause significant difference of pass rate. When the inaccurate values are used and significance level is set to 1%, the experimental mean value of pass rate, which is calculated by use of random number sequences taken from DES (Data Encryption Standard), is about 98.8%. In contrast, our new values derived from a set of recurrence formulas for the NIST randomness test suite give an empirical distribution of pass rate that meets the theoretical binomial distribution. Here, the experimental mean value of pass rate is about 99%, which corresponds to the significance level 1%.

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.