Advertisement

A Las Vegas Algorithm to Solve the Elliptic Curve Discrete Logarithm Problem

  • Ayan Mahalanobis
  • Vivek Mohan Mallick
  • Ansari Abdullah
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11356)

Abstract

In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the group of rational points on an elliptic curve and is thus not a generic algorithm. The algorithm that we describe has some similarities with the most powerful index-calculus algorithm for the discrete logarithm problem over a finite field. The algorithm has no restriction on the finite field over which the elliptic curve is defined.

Keyword

Elliptic curve discrete logarithm problem 

Notes

Acknowledgment

We are indebted to the anonymous referees for their careful reading of the manuscript and detail comments. Due to lack of time, we were not able to incorporate the new research directions suggested. However, those comments have certainly piqued our interest and we thank the referees for those.

References

  1. 1.
    Amadori, A., Pintore, F., Sala, M.: On the discrete logarithm problem for prime-field elliptic curve. Finite Fields Appl. 51, 168–182 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Lange, T.: Non-uniform cracks in the concrete: the power of free precomputation. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8270, pp. 321–340. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-42045-0_17CrossRefGoogle Scholar
  3. 3.
    Fulton, W.: Algebraic Curves (2008, self-published)Google Scholar
  4. 4.
    Galbraith, S., Gaudry, P.: Recent progress on the elliptic curve discrete logarithm problem. Des. Codes Crypt. 78, 51–78 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Galbraith, S.D., Gebregiyorgis, S.W.: Summation polynomial algorithms for elliptic curves in characteristic two. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 409–427. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13039-2_24CrossRefzbMATHGoogle Scholar
  6. 6.
    Gaudry, P.: Index calculus for abeian varieties of small dimension and the elliptic curve discrete logarithm problem. J. Symbolic Comput. 44, 1690–1702 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harris, J.: Algebraic Geometry. Springer, New York (1992).  https://doi.org/10.1007/978-1-4757-2189-8CrossRefzbMATHGoogle Scholar
  8. 8.
    Silverman, J.H., Pipher, J., Hoffstein, J.: An Introduction to Mathematical Cryptography. UTM. Springer, New York (2008).  https://doi.org/10.1007/978-0-387-77993-5CrossRefzbMATHGoogle Scholar
  9. 9.
    Jacobson, M.J., Koblitz, N., Silverman, J.H., Stein, A., Teske, E.: Analysis of the xedni calculus attack. Des. Codes Crypt. 20(1), 41–64 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Milne, J.S.: Elliptic Curves. BookSurge Publishers (2006)Google Scholar
  11. 11.
    Semaev, I.: Summation polynomials and the discrete logarithm problem on elliptic curves (2004). https://eprint.iacr.org/2004/031
  12. 12.
    Shoup, V.: NTL: a library for doing number theory (2016). http://www.shoup.net/ntl
  13. 13.
    Silverman, J.H.: The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Crypt. 20(1), 5–20 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wiener, M.J., Zuccherato, R.J.: Faster Attacks on Elliptic Curve Cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48892-8_15CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ayan Mahalanobis
    • 1
  • Vivek Mohan Mallick
    • 1
  • Ansari Abdullah
    • 2
  1. 1.IISER PunePuneIndia
  2. 2.Savitribai Phule Pune UniversityPuneIndia

Personalised recommendations