Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Discrete Mathematics and Applications

Editor-in-Chief: Zubkov, Andrei

6 Issues per year

CiteScore 2016: 0.16

SCImago Journal Rank (SJR) 2016: 0.231
Source Normalized Impact per Paper (SNIP) 2016: 0.552

See all formats and pricing
More options …
Volume 28, Issue 5


Estimates of the mean size of the subset image under composition of random mappings

Andrey M. Zubkov / Aleksandr A. Serov
Published Online: 2018-10-26 | DOI: https://doi.org/10.1515/dma-2018-0029


Let XN be a set of N elements and F1, F2,… be a sequence of random independent equiprobable mappings XNN. For a subset S0 ⊂ XN, |S0|=m, we consider a sequence of its images St=Ft(…F2(F1(S0))…), t=1,2… An approach to the exact recurrent computation of distribution of |St| is described. Two-sided inequalities forM{|St|||S0|=m} such that the difference between the upper and lower bounds is o(m)for m, t, N → ∞, mt=o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

Keywords: compositions of random mappings; time-memory tradeoff method

Note: Originally published in Diskretnaya Matematika (2018) 30, N2, 27–36 (in Russian).


  • [1]

    Bonferroni C. E. “Teoria statistica delle classi e calcolo delle probabilità ”, Pubbl. d. R. Ist. Super. di Sci. Econom. e Commerciali di Firenze, 1936, No3–62 (in Italian).Google Scholar

  • [2]

    Gantmacher F. R. The Theory of Matrices. vol. 1 and vol. 2, Chelsea Publishing Company, New York, 1959, vol. 1: x+374 pp. vol. 2: x+277 pp.Google Scholar

  • [3]

    Kolchin V. F., Sevastyanov B. A., Chistyakov V. P. Random allocations, V. H. Winston & Sons, Washington, 1978, 262.Google Scholar

  • [4]

    Borovkov A. A. Probability Theory, New York: Gordon & Breach, 1998, 474.Google Scholar

  • [5]

    Hellman M.E., “A cryptanalytic time–memory trade-off”, IEEE Trans. Inf. Theory, 1980, 401–406.Google Scholar

  • [6]

    Flajolet P., Odlyzko A.M., “Random mapping statistics”, Advances in Cryptology –- Eurocrypt’89, Lect. Notes Comput. Sci., 1990 434, 329–354.Google Scholar

  • [7]

    Oechslin P., “Making a faster cryptanalytic time-memory trade-off”, Lect. Notes Comput. Sci, 2729, 2003, 617–630.CrossrefGoogle Scholar

  • [8]

    Hong J. , “The cost of false alarms in Hellman and rainbow tradeoffs”, Designs, Codes and Cryptography, 57:3, 2010, 293–327.Web of ScienceCrossrefGoogle Scholar

  • [9]

    Hong J., Moon S., “A comparison of cryptanalytic tradeoff algorithms”, J. Cryptology, 26 2013, 559–637.Web of ScienceCrossrefGoogle Scholar

  • [10]

    Pilshchikov D.V., “Estimation of the characteristics of time-memory-data tradeoff methods via generating functions of the number of particles and the total number of particles in the Galton-Watson process”, Matematicheskie Voprosy Kriptografii, 5:2, 2014, 103–108.Google Scholar

  • [11]

    Zubkov A.M., Serov A.A. , “Images of subset of finite set under iterations of random mappings”, Discrete Math. Appl., 2015, 25 :3 , 179–185.Web of ScienceGoogle Scholar

  • [12]

    Serov A..A. , “Images of a finite set under iterations of two random dependent mappings”, Discrete Math. Appl., 2016, 26 :3 , 175–181.Web of ScienceGoogle Scholar

  • [13]

    Zubkov A.M., Serov A.A. , “Limit theorem for the size of an image of subset under compositions of random mappings”, Discrete Math. Appl., 2018, 28 :2 , 131–138.CrossrefWeb of ScienceGoogle Scholar

  • [14]

    Zubkov A.M., Mironkin V.~O. , “The distribution of the length of the aperiodicity segment in the graph of a k-fold iteration of a random equiprobable mapping”, Mathematical Aspects of Cryptography,, 8 :4 , 63–74 (in Russian).Google Scholar

About the article

Received: 2018-03-28

Published Online: 2018-10-26

Published in Print: 2018-10-25

Communicated by Anatolij Dvurečenskij

Citation Information: Discrete Mathematics and Applications, Volume 28, Issue 5, Pages 331–338, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265, DOI: https://doi.org/10.1515/dma-2018-0029.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in