Abstract
The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation (Gaudry); for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors (Diem, Kochinke). Recently, Sarkar and Singh have proposed a sieving technique, inspired by an earlier work of Joux and Vitse, to speed up the relation search in the hyperelliptic case. We give a new description of this technique, and show that this new formulation applies naturally to the non-hyperelliptic case with or without large prime variations. In particular, we obtain a speed-up by a factor approximately 3 for the relation search in Diem and Kochinke’s methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The terminology of index calculus stems from the context of integer factorization. In our setting, “large primes” are arbitrary elements, and involve no notion of size.
- 2.
This is only true asymptotically. For actual instances of the DLP many other factors have to be taken into account, and large prime variations are not always appropriate.
- 3.
More fundamentally, large prime variations are interesting for the asymptotic complexity analysis, but are not always well-suited in practice ; other methods such as the Gaussian structured elimination [9] can be more efficient.
References
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993)
Diem, C.: An index calculus algorithm for plane curves of small degree. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 543–557. Springer, Heidelberg (2006)
Diem, C., Kochinke, S.: Computing discrete logarithms with special linear systems (2013). http://www.math.uni-leipzig.de/diem/preprints/dlp-linear-systems.pdf
Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000)
Gaudry, P., Hess, F., Smart, N.P.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptol. 15(1), 19–46 (2002)
Gaudry, P., Thomé, E., Thériault, N., Diem, C.: A double large prime variation for small genus hyperelliptic index calculus. Math. Comput. 76(257), 475–492 (2007)
Joux, A., Vitse, V.: Cover and decomposition index calculus on elliptic curves made practical. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 9–26. Springer, Heidelberg (2012)
Laine, K., Lauter, K.: Time-memory trade-offs for index calculus in genus 3. J. Math. Cryptol. 9(2), 95–114 (2015)
LaMacchia, B.A., Odlyzko, A.M.: Computation of discrete logarithms in prime fields. Des. Codes Crypt. 1(1), 47–62 (1991)
Sarkar, P., Singh, S.: A new method for decomposition in the Jacobian of small genus hyperelliptic curves. Cryptology ePrint Archive, Report 2014/815 (2014)
Thériault, N.: Index calculus attack for hyperelliptic curves of small genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003)
Acknowlegdements
We would like to thank the anonymous referees for their useful comments during the elaboration of the article.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Vitse, V., Wallet, A. (2015). Improved Sieving on Algebraic Curves. In: Lauter, K., Rodríguez-Henríquez, F. (eds) Progress in Cryptology -- LATINCRYPT 2015. LATINCRYPT 2015. Lecture Notes in Computer Science(), vol 9230. Springer, Cham. https://doi.org/10.1007/978-3-319-22174-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-22174-8_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22173-1
Online ISBN: 978-3-319-22174-8
eBook Packages: Computer ScienceComputer Science (R0)