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Journal of Automation and Information Sciences
SJR: 0.275 SNIP: 0.59 CiteScore™: 0.8

ISSN Imprimir: 1064-2315
ISSN On-line: 2163-9337

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Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i10.30
pages 31-45

Statistical Analysis of Local Sections of Bits Sequences

Vladimir I. Masol
Kiev National Taras Shevchenko University, Kiev
Svetlana V. Popereshnyak
Kiev National Taras Shevchenko University, Kiev


Consideration is given to the joint distribution of the number of 2-chains and the number of 3-chains of the fixed form of the random (0, 1)-sequence which allowed one to carry out the statistical analysis of local sections of this sequence. Two theorems are formulated and proved. For the number of s-chains of the form tt*, t1t*, t0t* t1t1*, t11t1* (t1t1*, t1t, t0t, ttt, tt*t), which appeared in a random bit sequence of length n, n > 0 Theorems 1, 2 established explicit expressions of the joint distributions of such events:
{η(tt*) = k1, η(t1t*) + η(t0t*) = k2 }, {η(t1t1*) = k1, η(t1tt1*) = k2 }, {η(tt*) = k1
η(t1t*) = k2,η(t0t*) = k3}, ({η(t1t1*) = k1, η(t1t) + η(t0t) = k2}, {η(t1t1*) = k1,
η(ttt) = k2}, {η(t1t1*) = k1, η(tt*t) = k2}, {η(t1t1*) = k1, η(ttt) = k2, η(tt*t) = k3}),
where η(t1t2... ts) is the number of s-chains of the form t1t2... ts in the initial n-dimensional (0, 1)-sequence; k1, k2 and k3 are suitable nonnegative integers. One of the main assumptions of each theorem is that zeros and ones in a bit sequence are independent identically distributed random variables. The proofs of formulas for distributions of these events are based on counting the number of corresponding conductive events provided that (0, 1)-sequence contains a fixed number of zeros and ones. As examples of the use of explicit expressions of joint distributions we presented tables that contain the values of probabilities of the above events for the random (0, 1)-sequence of the length n, n = 20, and some values of parameters k1, k2 and k3 under assumptions that zeroes and unities appear equally possible. For illustrative purpose some of the tables are presented by bubble chart. The established formulas may be of interest for tasks on testing local sections formed at the output of pseudorandom number generators. Also they may be suitable for some tasks of information protection from unauthorized access as well as in other areas where it becomes necessary to analyze bit sequences.


  1. Gaidyshev I.P., AtteStat data analysis software [in Russian], Rukovodstvo polzovatelya. Versiya 13, 2012 .

  2. Rukhin A., Soto J., Nechvatal J., Smid M., Barker E., Leigh S., Levenson M., Vangel M., Banks D., Heckert A., Dray J., Vo S., A statistical test suite for random and pseudorandom number generators for cryptographic applications, National Institute of Standards and Technology. Special Publication 800-22 revision 1a, 2010 .

  3. Masol V.I., On distribution of some statistics of (0, 1)-vector, Issledovanie operatsiy i ASU, Vyp. 29, 1987, 23-27 .

  4. Hu Y., Polk T., Yang J., Zhao Y., Liu S., Spot-tracking lens: A zoomable user interface for animated bubble charts, IEEE Pacific Visualization Symposium (PacificVis), 2016, 16-23 .

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