Abstract
In this paper, we present explicit formulae for Miller’s algorithm to compute the Tate pairing on generalized Hessian curves using projective coordinates. Firstly, we propose the geometric interpretation of the group law and construct Miller function on generalized Hessian curves. The computation of Tate pairing using these functions in Miller’s algorithm costs 10m in addition steps and 5m+6s in doubling steps. Finally, we present the parallel algorithm for computing Tate pairing on generalized Hessian curves.
Keywords
- Elliptic curves
- Hessian curves
- Tate pairing
- Miller algorithm
- Cryptography
Supported in part by the National Natural Science Foundation of China No. 11101002.
This is a preview of subscription content, access via your 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
Arène, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster computation of the Tate pairing, Arxiv preprint, arXiv:0904.0854 (2009)
Barreto, P.S.L.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient Algorithms for Pairing-Based Cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–369. Springer, Heidelberg (2002)
Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008)
Blake, I.F., Seroussi, G., Smart, N.P.: Advances in elliptic curve cryptography. Cambridge Univ. Pr. (2005)
Chatterjee, S., Sarkar, P., Barua, R.: Efficient Computation of Tate Pairing in Projective Coordinate over General Characteristic Fields. In: Park, C.-S., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 168–181. Springer, Heidelberg (2005)
Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Mathematics 7(4), 385–434 (1986)
Das, M.P.L., Sarkar, P.: Pairing Computation on Twisted Edwards Form Elliptic Curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 192–210. Springer, Heidelberg (2008)
Edwards, H.M.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44(3), 393–422 (2007)
Farashahi, R.R., Joye, M.: Efficient Arithmetic on Hessian Curves. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 243–260. Springer, Heidelberg (2010)
Gu, H., Gu, D., Xie, W.: Efficient Pairing Computation on Elliptic Curves in Hessian Form. In: Rhee, K.-H., Nyang, D. (eds.) ICISC 2010. LNCS, vol. 6829, pp. 169–176. Springer, Heidelberg (2011)
Ionica, S., Joux, A.: Another Approach to Pairing Computation in Edwards Coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008)
Joye, M., Quisquater, J.-J.: Hessian Elliptic Curves and Side-Channel Attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)
Ostrowski, A.: Über dirichletsche reihen und algebraische differentialgleichungen. Mathematische Zeitschrift 8(3), 241–298 (1920)
Silverman, J.H.: The arithmetic of elliptic curves, vol. 106. Springer (2009)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, L., Zhang, F. (2012). Tate Pairing Computation on Generalized Hessian Curves. In: Lee, D.H., Yung, M. (eds) Information Security Applications. WISA 2012. Lecture Notes in Computer Science, vol 7690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35416-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-35416-8_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35415-1
Online ISBN: 978-3-642-35416-8
eBook Packages: Computer ScienceComputer Science (R0)
