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Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes

Authors Rodrigo S. V. Martins, Daniel Panario, Claudio Qureshi, Eric Schmutz

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Author Details

Rodrigo S. V. Martins
  • Universidade Tecnológica Federal do Paraná, Apucarana, PR 86812-460, Brazil
Daniel Panario
  • Carleton University, Colonel By Drive, Ottawa, ON K1S 5B6, Canada
Claudio Qureshi
  • Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
Eric Schmutz
  • Drexel University, Philadelphia, Pa 19104, USA

Funding

  • Panario, Daniel: This author was partially funded by NSERC of Canada.
  • Qureshi, Claudio: This author was funded by FAPESP under grants 2015/26420-1 and 2013/25977-7.

Cite As Get BibTex

Rodrigo S. V. Martins, Daniel Panario, Claudio Qureshi, and Eric Schmutz. Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.30

Abstract

Let f be a uniformly random element of the set of all mappings from [n] = {1, ..., n} to itself. Let T(f) and B(f) denote, respectively, the least common multiple and the product of the lengths of the cycles of f. Harris proved in 1973 that log T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for uniform random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) >= 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x^k + a over the integers modulo p, where k divides p - 1. We exhibit and discuss our numerical results on this heuristic.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
Keywords
  • random mappings with indegree restrictions
  • Brent-Pollard heuristic
  • periods of mappings

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References

  1. James Arney and Edward A. and Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103(2):269-294, 1982. Google Scholar
  2. R. Arratia, A. D. Barbour, and S. Tavaré. Limits of logarithmic combinatorial structures. Ann. Probab., 28(4):1620-1644, 2000. URL: http://dx.doi.org/10.1214/aop/1019160500.
  3. Richard Arratia, A. D. Barbour, and Simon Tavaré. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. URL: http://dx.doi.org/10.4171/000.
  4. A. D. Barbour and Simon Tavaré. A rate for the Erdős-Turán law. Combin. Probab. Comput., 3(2):167-176, 1994. URL: http://dx.doi.org/10.1017/S0963548300001097.
  5. Richard P. Brent and John M. Pollard. Factorization of the eighth Fermat number. Math. Comp., 36(154):627-630, 1981. URL: http://dx.doi.org/10.2307/2007666.
  6. Charles Burnette and Eric Schmutz. Periods of iterated rational functions. Int. J. Number Theory, 13(5):1301-1315, 2017. URL: http://dx.doi.org/10.1142/S1793042117500713.
  7. Michael Drmota and Michèle Soria. Images and preimages in random mappings. SIAM J. Discrete Math., 10(2):246-269, 1997. URL: http://dx.doi.org/10.1137/S0895480194268421.
  8. P. Erdős and P. Turán. On some problems of a statistical group-theory. III. Acta Math. Acad. Sci. Hungar., 18:309-320, 1967. URL: http://dx.doi.org/10.1007/BF02280290.
  9. Philippe Flajolet and Andrew M. Odlyzko. Random mapping statistics. In Advances in cryptology - EUROCRYPT '89 (Houthalen, 1989), volume 434 of Lecture Notes in Comput. Sci., pages 329-354. Springer, Berlin, 1990. URL: http://dx.doi.org/10.1007/3-540-46885-4_34.
  10. Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. URL: http://dx.doi.org/10.1017/CBO9780511801655.
  11. Jennie C. Hansen and Jerzy Jaworski. Random mappings with exchangeable in-degrees. Random Structures Algorithms, 33(1):105-126, 2008. URL: http://dx.doi.org/10.1002/rsa.20187.
  12. Bernard Harris. The asymptotic distribution of the order of elements in symmetric semigroups. J. Combinatorial Theory Ser. A, 15:66-74, 1973. URL: http://dx.doi.org/10.1016/0097-3165(73)90036-8.
  13. Valentin F. Kolchin. Random mappings. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York, 1986. Translated from the Russian, With a foreword by S. R. S. Varadhan. Google Scholar
  14. Rodrigo S. V. Martins and Daniel Panario. On the heuristic of approximating polynomials over finite fields by random mappings. Int. J. Number Theory, 12(7):1987-2016, 2016. URL: http://dx.doi.org/10.1142/S1793042116501219.
  15. J. M. Pollard. A Monte Carlo method for factorization. Nordisk Tidskr. Informationsbehandling (BIT), 15(3):331-334, 1975. URL: http://dx.doi.org/10.1007/BF01933667.
  16. Vijay K. Rohatgi and A. K. Md. Ehsanes Saleh. An introduction to probability and statistics. Wiley Series in Probability and Statistics: Texts and References Section. Wiley-Interscience, New York, second edition, 2001. URL: http://dx.doi.org/10.1002/9781118165676.
  17. H Rubin and R Sitgreaves. Probability distributions related to random transformations of a finite set. Technical Report SOL ONR 19A, Stanford, Stanford, 1954. Google Scholar
  18. Eric Schmutz. Period lengths for iterated functions. Combin. Probab. Comput., 20(2):289-298, 2011. URL: http://dx.doi.org/10.1017/S0963548310000337.
  19. Richard Stong. The average order of a permutation. Electron. J. Combin., 5:Research Paper 41, 6, 1998. Google Scholar
  20. Michael J. Wiener and Robert J. Zuccherato. Faster attacks on elliptic curve cryptosystems. In Selected areas in cryptography (Kingston, ON, 1998), volume 1556 of Lecture Notes in Comput. Sci., pages 190-200. Springer, Berlin, 1999. URL: http://dx.doi.org/10.1007/3-540-48892-8_15.
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