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Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields

  • S. Ugolini
Article

Abstract

In this paper we study the dynamics of rational maps induced by endomorphisms of ordinary elliptic curves defined over finite fields. For any such map we describe the cycle and the tree structures of the associated graphs.

Keywords

Dynamical systems Finite fields Arithmetic dynamics Endomorphisms of elliptic curves 

Mathematics Subject Classification

37P55 37P05 11G20 

Notes

Acknowledgements

The author is grateful to the anonymous Reviewer for the careful reading of the manuscript and the valuable comments and suggestions.

References

  1. 1.
    W.-S. Chou, I. Shparlinski, On the cycle structure of repeated exponentiation modulo a prime. J. Number Theory 107(2), 345–356 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    O. Colón-Reyes, A.S. Jarrah, R. Laubenbacher, B. Sturmfels, Monomial dynamical systems over finite fields. Complex Syst. 16(4), 333–342 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Elia, J.C. Interlando, R. Rosenbaum, On the structure of residue rings of prime ideals in algebraic number fields—PART I: unramified primes. Int. Math. Forum 5(56), 2795–2808 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Elia, J.C. Interlando, R. Rosenbaum, On the structure of residue rings of prime ideals in algebraic number fields—PART II: ramified primes. Int. Math. Forum 6(12), 565–589 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    T. Gassert, Chebyshev action on finite fields. Discrete Math. 315, 83–94 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H.W. Lenstra, Complex multiplication structure of elliptic curves. J. Number Theory 56, 227–241 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. Lübeck, Conway Polynomials for Finite Fields, http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html. Online; Accessed 25 Jan 2018
  8. 8.
    C. Qureshi, D. Panario, Rédei actions on finite fields and multiplication map in cyclic group. SIAM J. Discrete Math. 29(3), 1486–1503 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    T. Rogers, The graph of the square mapping on the prime fields. Discrete Math. 148, 317–324 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Sha, S. Hu, Monomial dynamical systems of dimension one over finite fields. Acta Arith. 148(4), 309–331 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    The GAP Group, GAP—Groups, Algorithms and Programming, http://www.gap-system.org
  12. 12.
    The Sage Developers, Sage Mathematics Software (Version 6.1.1) (2014), http://www.sagemath.org
  13. 13.
    S. Ugolini, Graphs associated with the map \(x \mapsto x + x^{-1}\) in finite fields of characteristic two, in Theory and Applications of Finite Fields, Contemp. Math., vol. 579, Amer. Math. Soc., Providence, RI (2012)Google Scholar
  14. 14.
    S. Ugolini, Graphs associated with the map \({X} \mapsto X + {X}^{-1}\) in finite fields of characteristic three and five. J. Number Theory 133, 1207–1228 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. Ugolini, On the iterations of certain maps \({X} \mapsto K \cdot ({X} + {X}^{-1})\) over finite fields of odd characteristic. J. Number Theory 142, 274–297 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    T. Vasiga, J. Shallit, On the iteration of certain quadratic maps over \({G} F (p)\). Discrete Math. 277, 219–240 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Z.-X. Wan, Finite Fields and Galois Rings (World Scientific Publishing, Singapore, 2012)zbMATHGoogle Scholar
  18. 18.
    C. Wittmann, Group structure of elliptic curves over finite fields. J. Number Theory 88(2), 335–344 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Università degli Studi di TrentoTrentoItaly

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