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In this work, we classify all templates according to their periods and for each template we evaluate the exact probabilities using generating functions. Afterwards, we propose a new family of statistical randomness tests, that is periodic template tests, for a collection of binary sequences. We apply these tests to the outputs of AES, SHA-3, SHA-2 family, SHA-1 and MD5 and the binary expansion of pi and root 2 and biased non-random data to test the power of new tests. Moreover, we give the probabilities for all templates for the overlapping template matching test in the NIST test suite. Afterwards, we analyse the power of templates and compare the periodic template tests with NIST overlapping template test. (C) 2019 Elsevier B.V. All rights reserved.

In this article a family of statistical randomness tests for binary strings are introduced, based on Golomb's pseudorandomness postulate R-2 on the number of runs. The basic idea is to construct recursive formulae with computationally tenable probability distribution functions. The technique is illustrated on testing strings of 2(7), 2(8), 2(10) and 2(12) bits. Furthermore, the expected value of the number of runs with a specific length is obtained. Finally the tests are applied to several collections of strings arising from different pseudorandom number generators.

In this work, we define a family of nine statistical randomness tests for collections of short binary strings, by making use of random walk statistics. For a binary sequence of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{n}$$\end{document}, we consider the probability of intersecting the line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y=t}$$\end{document} exactly at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{k}$$\end{document} distinct points. Although there are some explicit formulas for these probability values in the literature, those applicable to short sequences are not feasible for computations involving sequences of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{256}$$\end{document} bits or more. On the other hand, approximation techniques, or asymptotic approaches, that should be used only when testing long sequences, are not useful for testing sequences of length between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{256}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{4096}$$\end{document}. The recursive formulas, derived in this paper, made it possible to obtain exact values of the corresponding probability distribution functions. Using these formulas, we provide the necessary figures for testing collections of strings of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{2}<^>{\varvec{7}}, \ \varvec{2}<^>{\varvec{8}}, \ \varvec{2}<^>{\varvec{10}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{2}<^>{\varvec{12}}$$\end{document} bits. Finally, we apply these nine tests to various collections of strings obtained from different pseudorandom number generators as well as to biased sequences to assess whether the proposed tests can effectively detect non-random data.

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