Abstract
At ASIACRYPT 2012, Petit and Quisquater suggested that there may be a subexponential-time index-calculus type algorithm for the Elliptic Curve Discrete Logarithm Problem (ECDLP) in characteristic two fields. This algorithm uses Semaev polynomials and Weil Descent to create a system of polynomial equations that subsequently is to be solved with Gröbner basis methods. Its analysis is based on heuristic assumptions on the performance of Gröbner basis methods in this particular setting. While the subexponential behaviour would manifest itself only far beyond the cryptographically interesting range, this result, if correct, would still be extremely remarkable. We examined some aspects of the work by Petit and Quisquater experimentally.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bettale, L., Faugère, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Crypt. 3, 177–197 (2009)
Collins, G.: The calculation of multivariate polynomial resultants. Journal of the Association of Computing Machinery 18, 515–522 (1971)
Diem, C.: On the discrete logarithm problem in elliptic curves. Compositio Mathematica 147(1), 75–104 (2011)
Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139, 61–88 (1999)
Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 75–83. ACM Press (2002)
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)
Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. J. Symbolic Computation 44(12), 1690–1702 (2009)
Bettale, L.: Hybrid approach for solving multivariate polynomial systems over finite fields, http://www-polsys.lip6.fr/~bettale/hybrid
Petit, C., Quisquater, J.-J.: On polynomial systems arising from a Weil descent. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 451–466. Springer, Heidelberg (2012)
Semaev, I.: Summation polynomials and the discrete logarithm problem on elliptic curves, Cryptology ePrint Archive Report 2004/031 (2004), http://eprint.iacr.org/2004/031/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Shantz, M., Teske, E. (2013). Solving the Elliptic Curve Discrete Logarithm Problem Using Semaev Polynomials, Weil Descent and Gröbner Basis Methods – An Experimental Study. In: Fischlin, M., Katzenbeisser, S. (eds) Number Theory and Cryptography. Lecture Notes in Computer Science, vol 8260. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-42001-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-42001-6_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-42000-9
Online ISBN: 978-3-642-42001-6
eBook Packages: Computer ScienceComputer Science (R0)