On the Computational Complexity of ECDLP for Elliptic Curves in Various Forms Using Index Calculus

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10779)

Abstract

The security of elliptic curve cryptography is closely related to the computational complexity of the elliptic curve discrete logarithm problem (ECDLP). Today, the best practical attacks against ECDLP are exponential-time, generic discrete logarithm algorithms such as Pollard’s rho method. Recently, there is a line of research on index calculus for ECDLP started by Semaev, Gaudry, and Diem. Under certain heuristic assumptions, such algorithms could lead to subexponential attacks to ECDLP in some cases. In this paper, we investigate the computational complexity of ECDLP for elliptic curves in various forms—including Hessian, Montgomery, (twisted) Edwards, and Weierstrass using index calculus. The research question we would like to answer is: Using index calculus, is there any significant difference in the computational complexity of ECDLP for elliptic curves in various forms? We will provide some empirical evidence and insights showing an affirmative answer in this paper.

Keywords

Security evaluation ECDLP Index calculus Summation polynomial Point decomposition problem 

Notes

Acknowledgments

This work is partially supported by JSPS KAKENHI Grant (C)(JP15K00183) and (JP15K00189) and Japan Science and Technology Agency, CREST and Infrastructure Development for Promoting International S&T Cooperation and Project for Establishing a Nationwide Practical Education Network for IT Human Resources Development, Education Network for Practical Information Technologies.

References

  1. 1.
    Bailey, D.V., Paar, C.: Optimal extension fields for fast arithmetic in public-key algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0055748 CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J.: Curve25519: new Diffie-Hellman speed records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 207–228. Springer, Heidelberg (2006).  https://doi.org/10.1007/11745853_14 CrossRefGoogle Scholar
  3. 3.
    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. IACR Cryptology ePrint Archive, 2008:13 (2008)Google Scholar
  4. 4.
    Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. IACR Cryptology ePrint Archive, 2007:286 (2007)Google Scholar
  5. 5.
    Diem, C.: On the discrete logarithm problem in class groups of curves. Math. Comput. 80(273), 443–475 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Faugère, J., Gaudry, P., Huot, L., Renault, G.: Using symmetries in the index calculus for elliptic curves discrete logarithm. J. Cryptol. 27(4), 595–635 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Faugère, J.-C., Perret, L., Petit, C., Renault, G.: Improving the complexity of index calculus algorithms in elliptic curves over binary fields. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 27–44. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_4 CrossRefGoogle Scholar
  8. 8.
    Galbraith, S.D., Gaudry, P.: Recent progress on the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 78(1), 51–72 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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_24 Google Scholar
  10. 10.
    Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. J. Symb. Comput. 44(12), 1690–1702 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Huang, Y.-J., Petit, C., Shinohara, N., Takagi, T.: Improvement of Faugère et al.’s method to solve ECDLP. In: Sakiyama, K., Terada, M. (eds.) IWSEC 2013. LNCS, vol. 8231, pp. 115–132. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-41383-4_8 CrossRefGoogle Scholar
  12. 12.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987). http://links.jstor.org/sici?sici=0025-5718(198701)48:177<243:STPAEC>2.0.CO;2-3
  13. 13.
    Petit, C., Quisquater, J.: On polynomial systems arising from a Weil descent. IACR Cryptology ePrint Archive 2012:146 (2012)Google Scholar
  14. 14.
    Pollard, J.M.: Monte Carlo methods for index computation mod \(p\). Math. Comput. 32, 918–924 (1978)MathSciNetMATHGoogle Scholar
  15. 15.
    Semaev, I.A.: Summation polynomials and the discrete logarithm problem on elliptic curves. IACR Cryptology ePrint Archive 2004:31 (2004)Google Scholar
  16. 16.
    Smart, N.P.: The Hessian form of an elliptic curve. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 118–125. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44709-1_11 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of EngineeringOsaka UniversitySuitaJapan

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