Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1113–1129 | Cite as

On the elliptic curve endomorphism generator

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Abstract

For an elliptic curve \({E}\) over a finite field we define the point sequence \((P_n)\) recursively by \(P_n=\vartheta (P_{n-1})=\vartheta ^n(P_0)\) with an endomorphism \(\vartheta \in {{\mathrm{End}}}({E})\) and with some initial point \(P_0\) on \({E}\). We study the distribution and the linear complexity of sequences obtained from \((P_n)\).

Keywords

Elliptic curves Complex multiplication Character sums Linear complexity Power generator 

Mathematics Subject Classification

11T23 65C10 14H52 94A55 94A60 

Notes

Acknowledgements

The author would like to thank Arne Winterhof and Igor Shparlinski for helpful comments. The author is partially supported by the Austrian Science Fund FWF Project F5511-N26 which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

References

  1. 1.
    Beelen P.H.T., Doumen J.M.: Pseudorandom Sequences from Elliptic Curves, Finite Fields with Applications to Coding Theory, Cryptography and Related Areas (Oaxaca, 2001), pp. 37–52. Springer, Berlin (2002).CrossRefMATHGoogle Scholar
  2. 2.
    Bisson G.: Computing endomorphism rings of elliptic curves under the GRH. J. Math. Cryptol. 5(2), 101–113 (2011).MathSciNetMATHGoogle Scholar
  3. 3.
    Bisson G., Sutherland A.V.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. J. Number Theory 131, 815–831 (2011).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cohen H.: A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993).CrossRefGoogle Scholar
  5. 5.
    Cox D.A.: Primes of the Form \(X^2 + nY^2\): Fermat, Class Field Theory, and Complex Multiplication. Wiley, New York (1989).Google Scholar
  6. 6.
    El-Mahassni E., Shparlinski I.E.: On the uniformity of distribution of congruential generators over elliptic curves. In: Proc. Intern. Conf. on Sequences and their Applications, Bergen, 2001, pp. 257–264. Springer, London (2002).Google Scholar
  7. 7.
    El-Mahassni E., Shparlinski I.E.: On the distribution of the elliptic curve power generator. In: Finite Fields and Applications, Contemporary Mathematics, vol. 461, pp. 111–118. American Mathematical Society, Providence, RI (2008).Google Scholar
  8. 8.
    Freeman D., Scott M., Teske E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptol. 23, 224–280 (2010).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Friedlander J.B., Shparlinski I.E.: On the distribution of the power generator. Math. Comput. 70(236), 1575–1589 (2001).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hallgren S.: Linear congruential generators over elliptic curves, pp. 1–10, Preprint CS-94-143, Dept. of Comp. Sci., Cornegie Mellon Univ. (1994).Google Scholar
  11. 11.
    Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008).MATHGoogle Scholar
  12. 12.
    Hess F., Shparlinski I.E.: On the linear complexity and multidimensional distribution of congruential generators over elliptic curves. Des. Codes Cryptogr. 35, 111–117 (2005).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kohel D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California at Berkeley (1996).Google Scholar
  14. 14.
    Kohel D., Shparlinski I.E.: Exponential sums and group generators for elliptic curves over finite fields. In: Proc. Algorithmic Number Theory Symposium, Leiden, LNCS vol. 1838, pp. 395–404. Springer, Berlin (2000).Google Scholar
  15. 15.
    Lange T., Shparlinski I.E.: Certain exponential sums and random walks on elliptic curves. Can. J. Math. 57, 338–350 (2005).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lange T., Shparlinski I.E.: Collisions in fast generation of ideal classes and points on hyperelliptic and elliptic curves. Appl. Algebra Eng. Commun. Comput. 15, 329–337 (2005).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lange T., Shparlinski I.E.: Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1(1), 1–11 (2007).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Luca F., Shparlinski I.E.: Discriminants of complex multiplication fields of elliptic curves over finite fields. Can. Math. Bull. 50(3), 409–417 (2007).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Marcus D.A.: Number Fields, Universitext. Springer, New York (1977).CrossRefGoogle Scholar
  20. 20.
    Meidl W., Winterhof A.: Linear complexity of sequences and multisequences. In: Mullen G., Panario D. (eds.) Handbook of Finite Fields, pp. 324–336. Chapman & Hall (2013).Google Scholar
  21. 21.
    Mérai L.: On the elliptic curve power generator. Unif. Distrib. Theory 9(2), 59–65 (2014).MathSciNetMATHGoogle Scholar
  22. 22.
    Mérai L.: On Pseudorandom Properties of Certain Sequences of Points on Elliptic Curve, Lecture Notes in Computer Science, vol. 10064. Springer, Berlin (2017).Google Scholar
  23. 23.
    Mérai L., Winterhof A.: On the linear complexity profile of some sequences derived from elliptic curves. Des. Codes Cryptogr. 81(2), 259–267 (2016).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Niederreiter H.: Linear complexity and related complexity measures for sequences. In: Progress in Cryptology—INDOCRYPT 2003, Lecture Notes in Computer Science, vol. 2904, pp. 1–17. Springer, Berlin (2003).Google Scholar
  25. 25.
    Schoof R.: Counting points on elliptic curves over finite fields. J. Thor. Nombres Bordeaux. 7(1), 219–254 (1995).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Shparlinski I.: On the linear complexity of the power generator. Des. Codes Cryptogr. 23(1), 5–10 (2001).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Shparlinski I.E.: Pseudorandom number generators from elliptic curves. In: Recent Trends in Cryptography, Contemporary Mathematics, vol. 477, pp. 121–141. American Mathematical Society, Providence, RI (2009).Google Scholar
  28. 28.
    Smart N.P.: Elliptic curve cryptosystems over small fields of odd characteristic. J. Cryptol. 12, 141–151 (1999).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Solinas J.A.: Efficient arithmetic on Koblitz curves. Des. Codes Cryptogr. 19(2–3), 195–249 (2000).MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Silverman J.H.: The Arithmetic of Elliptic Curves. Springer, Berlin (1995).Google Scholar
  31. 31.
    Winterhof A.: Linear complexity and related complexity measures. In: Selected Topics in Information and Coding Theory, pp. 3–40. World Scientific, Singapore (2010).Google Scholar
  32. 32.
    Winterhof A.: Some estimates for character sums and applications. Des. Codes Cryptogr. 22(2), 123–131 (2001).MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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