Estimates of the mean size of the subset image under composition of random mappings
-
Andrey M. Zubkov
and
Aleksandr A. Serov
Abstract
Let XN be a set of N elements and F1, F2,… be a sequence of random independent equiprobable mappings XN → N. 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.
Note: Originally published in Diskretnaya Matematika (2018) 30, N∘2, 27–36 (in Russian).
Communicated by Anatolij Dvurečenskij
References
[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).Search in 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.Search in Google Scholar
[3] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P. Random allocations, V. H. Winston & Sons, Washington, 1978, 262.Search in Google Scholar
[4] Borovkov A. A. Probability Theory, New York: Gordon & Breach, 1998, 474.Search in Google Scholar
[5] Hellman M.E., “A cryptanalytic time–memory trade-off”, IEEE Trans. Inf. Theory, 1980, 401–406.10.1109/TIT.1980.1056220Search in Google Scholar
[6] Flajolet P., Odlyzko A.M., “Random mapping statistics”, Advances in Cryptology –- Eurocrypt’89, Lect. Notes Comput. Sci., 1990 434, 329–354.10.1007/3-540-46885-4_34Search in Google Scholar
[7] Oechslin P., “Making a faster cryptanalytic time-memory trade-off”, Lect. Notes Comput. Sci, 2729, 2003, 617–630.10.1007/978-3-540-45146-4_36Search in Google Scholar
[8] Hong J. , “The cost of false alarms in Hellman and rainbow tradeoffs”, Designs, Codes and Cryptography, 57:3, 2010, 293–327.10.1007/s10623-010-9368-xSearch in Google Scholar
[9] Hong J., Moon S., “A comparison of cryptanalytic tradeoff algorithms”, J. Cryptology, 26 2013, 559–637.10.1007/s00145-012-9128-3Search in Google 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.Search in 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 , 179–185.10.1515/dma-2015-0017Search in Google Scholar
[12] Serov A..A. , “Images of a finite set under iterations of two random dependent mappings”, Discrete Math. Appl., 2016, 26 :3 , 175–181.10.1515/dma-2016-0015Search in Google 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 , 131–138.10.1515/dma-2018-0013Search in Google 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).Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A subcritical decomposable branching process in a mixed environment
- Limit theorems for bounded branching processes
- On coincidences of tuples in a q-ary tree with random labels of vertices
- A generalization of Shannon function
- Reduced critical Bellman–Harris branching processes for small populations
- Estimates of the mean size of the subset image under composition of random mappings
- Necessary conditions for power commuting in finite-dimensional algebras over a field
Articles in the same Issue
- Frontmatter
- A subcritical decomposable branching process in a mixed environment
- Limit theorems for bounded branching processes
- On coincidences of tuples in a q-ary tree with random labels of vertices
- A generalization of Shannon function
- Reduced critical Bellman–Harris branching processes for small populations
- Estimates of the mean size of the subset image under composition of random mappings
- Necessary conditions for power commuting in finite-dimensional algebras over a field
