Skip to main content
SpringerLink
Log in

Random Mappings with Restricted Preimages

  • Conference paper
Progress in Cryptology – LATINCRYPT 2012 (LATINCRYPT 2012)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7533))

Abstract

In this paper we refer to finite endofunctions where each image has at most r preimages as r-mappings. Probabilistic analysis of these mappings provides heuristics for problems arising in cryptography which involve similar but more complicated classes of mappings. We give asymptotic probabilities and expectations related to the joint distribution of some pairs of mapping parameters for these mappings, extending work done previously for general, unrestricted mappings. We give an elementary derivation of the expected value of the maximum of tail length and cycle length of a random node in a random r-mapping, which is a useful component in the heuristic analysis of Pollard’s rho algorithm. All distributions considered are uniform.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldous, D.J.: Exchangeability and related topics. In: École D’été de Probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math., vol. 1117, pp. 1–198. Springer, Berlin (1985)

    Chapter  Google Scholar 

  2. Arratia, R., Tavaré, S.: Limit theorems for combinatorial structures via discrete process approximations. Random Structures Algorithms 3(3), 321–345 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baron, G., Drmota, M., Mutafchiev, L.: Predecessors in random mappings. Combin. Probab. Comput. 5, 317–335 (1995)

    MathSciNet  Google Scholar 

  4. DeLaurentis, J.M.: Components and Cycles of a Random Function. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 231–242. Springer, Heidelberg (1988)

    Google Scholar 

  5. Drmota, M., Gittenberger, B.: Strata of random mappings—a combinatorial approach. Stochastic Process. Appl. 82, 157–171 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hansen, J.C.: A functional central limit theorem for random mappings. Ann. Probab. 17, 317–332 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Harris, B.: Probability distributions related to random mappings. The Annals of Mathematical Statistics 31, 1045–1062 (1960)

    Article  MATH  Google Scholar 

  8. Kolchin, V.F.: A problem of the allocation of particles in cells and random mappings. Theory Probab. Appl. 21, 48–63 (1976)

    Article  MATH  Google Scholar 

  9. Kolchin, V.F.: Random mappings. Translation Series in Mathematics and Engineering. Optimization Software Inc. Publications Division, New York (1986); Translated from the Russian, with a foreword by Varadhan, S.R.S.

    Google Scholar 

  10. Mutafchiev, L.R.: Limit properties of components of random mappings. C. R. Acad. Bulgare Sci. 31(10), 1257–1260 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Mutafchiev, L.R.: The limit distribution of the number of nodes in low strata of a random mapping. Statist. Probab. Lett. 7(3), 247–251 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Flajolet, P., Odlyzko, A.M.: Random Mapping Statistics. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 329–354. Springer, Heidelberg (1990)

    Google Scholar 

  13. Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  14. Aldous, D., Pitman, J.: The asymptotic distribution of the diameter of a random mapping. C. R. Math. Acad. Sci. Paris 334(11), 1021–1024 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Drmota, M., Soria, M.: Marking in combinatorial constructions: generating functions and limiting distributions. Theoret. Comput. Sci. 144(1-2), 67–99 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hansen, J.C., Jaworski, J.: Local properties of random mappings with exchangeable in-degrees. Adv. in Appl. Probab. 40(1), 183–205 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hansen, J.C., Jaworski, J.: Random mappings with a given number of cyclical points. Ars Combin. 94, 341–359 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Pollard, J.M.: A Monte Carlo method for factorization. Nordisk Tidskr. Informationsbehandling (BIT) 15(3), 331–334 (1975)

    MathSciNet  MATH  Google Scholar 

  19. Arney, J., Bender, E.A.: Random mappings with constraints on coalescence and number of origins. Pacific J. Math. 103(2), 269–294 (1982)

    MathSciNet  MATH  Google Scholar 

  20. Gittenberger, B.: On the number of predecessors in constrained random mappings. Statist. Probab. Lett. 36(1), 29–34 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rubin, H., Sitgreaves, R.: Probability distributions related to random transformations on a finite set. Technical Report 19a, Applied Mathematics and Statistics Laboratory, Stanford (1954)

    Google Scholar 

  22. Stepanov, V.E.: Limit distributions of certain characteristics of random mappings. Teor. Verojatnost. i Primenen. 14, 639–653 (1969)

    MathSciNet  Google Scholar 

  23. Drmota, M., Soria, M.: Images and preimages in random mappings. SIAM J. Discrete Math. 10(2), 246–269 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216–240 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wilf, H.S.: The asymptotic behavior of the Stirling numbers of the first kind. J. Combin. Theory Ser. A 64(2), 344–349 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gourdon, X.: Largest component in random combinatorial structures. Discrete Math. 180, 185–209 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Brent, R.P.: An improved Monte Carlo factorization algorithm. BIT 20(2), 176–184 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  28. Knuth, D.E.: A Classical Mind: Essays in Honour of C.A.R. Hoare, pp. 247–258. Prentice Hall International (UK) Ltd., Hertfordshire (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Andrew MacFie & Daniel Panario

Authors
  1. Andrew MacFie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Panario
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Chile, Blanco Encalada 2120, Tercer Piso, Santiago, Chile

    Alejandro Hevia

  2. IBM Research - Zürich, Säumerstrasse 4, 8803, Rüschlikon, Switzerland

    Gregory Neven

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

MacFie, A., Panario, D. (2012). Random Mappings with Restricted Preimages. In: Hevia, A., Neven, G. (eds) Progress in Cryptology – LATINCRYPT 2012. LATINCRYPT 2012. Lecture Notes in Computer Science, vol 7533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33481-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33481-8_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33480-1

  • Online ISBN: 978-3-642-33481-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation