Abstract
Elliptic curve cryptography on smart cards is vulnerable under a particular Side Channel Attack: the existence of zero-value points (ZVP). One approach to face this drawback relies on changing the curve for an isogenous one, until a resistant curve is found. This paper focuses on an alternative strategy: exploiting the properties of a recently introduced form of elliptic curves, Edwards curves. We show that these curves achieve conditions for being resistant to ZVP-attacks. Hence, using Edwards curves is a good countermeasure to avoid these attacks.
Similar content being viewed by others
Explore related subjects
Discover the latest articles and news from researchers in related subjects, suggested using machine learning.References
Akishita, T., Takagi, T.: Zero-value point attacks on elliptic curve cryptosystem. In: Information Security, ISC 2003, LNCS 2851, pp. 218–233 (2003)
Akishita, T., Takagi, T.: On the optimal parameter choice for elliptic curve cryptosystems using isogeny. In: Public Key Cryptography, PKC 2004, LNCS 2947, pp. 346–359 (2004)
Avanzi, R., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of elliptic and hyperelliptic curve cryptography. Discret. Math. Appl. Chapman & Hall/CRC (2006)
Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: ECM using Edwards curves, Cryptology ePrint Archive, Report 2008/016, 2008. Accessed 1 Apr 2011
Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: ASIACRYPT 2007. LNCS 4833, pp. 29–50, Springer (2007)
Bernstein, D.J., Lange, T.: Inverted Edwards Coordinates. In: 17 Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS 4851, pp. 20–27, Springer (2007)
Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Progress in cryptology–AFRICACRYPT 2008. LNCS 5023, pp. 389–405, Springer (2008)
Baldwin, B., Byrne, A., McGuire, G., Moloney, R., Marnane, W.P.: A Hardware Analysis of Twisted Edwards Curves for an Elliptic Curve Cryptosystem. In: Reconfigurable Computing: Architectures, Tools and Applications, ARC 2009, LNCS 5453, pp. 355–361, Springer (2009)
Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. London Math. Soc. Lecture Note Ser. vol. 265, Cambridge University Press, Cambridge (1999)
Das, M.P., & Sarkar, P.: Pairing computation on twisted Edwards form elliptic curves. In: Pairing-Based Cryptography, Pairing 2008. LNCS 5209, pp. 192–210. Springer (2008)
Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc. 44, 393–422 (2007)
Fan, J., Gierlichs, B., Vercauteren, F.: To infinity and beyond: combined attack on ECC using points of low order CHES 2011. LNCS 6917, 143–159 (2011)
Goubin, L.: A refined power-analysis attack on elliptic curve cryptosystems. In: Public Key Cryptography, PKC 2003, LNCS 2567, pp. 199–211 (2003)
Hisil, H., Wong, K.K., Carter, G., Dawson, E.: Twisted Edwards curves revisited. ASIACRYPT 2008, LNCS 5350, 326–346. Springer (2008)
Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48(177), 203–209 (1987)
Miller, V.: Use of elliptic curves in cryptography. In: CRYPTO 85, LNCS 218, pp. 417–426. Springer, Berlin (1986)
Miret, J., Sadornil, D., Tena, J., Tomàs, R., Valls, M.: On avoiding ZVP-attacks using isogeny volcanoes. In: Workshop on Information Security Applications, WISA 2008, LNCS 5379, pp. 266–277. Springer (2009)
Morain, F.: Edwards curves and CM curves. arXiv:0904.2243, 2009. Accessed 13 Sept 2011
Standards for Efficient Cryptography Group (SECG). SEC 2: Recommended Elliptic Curve Domain Parameters, Version 1.0, 2000. http://www.secg.org/secg_docs.htm. Accessed 13 Sept 2013
Smart, N.: An analysis of Goubin’s refined power analysis attack. In: Workshop on Cryptographic Hardware and Embedded Systems, CHES 2003, LNCS 2779, pp. 281–290 (2003)
Acknowledgments
This work has been partially supported by grants MTM2010-21580-C02-01/02 and MTM2010-16051 from Spanish Ministerio de Economía y Competitividad.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Martínez, S., Sadornil, D., Tena, J. et al. On Edwards curves and ZVP-attacks. AAECC 24, 507–517 (2013). https://doi.org/10.1007/s00200-013-0211-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-013-0211-2