Abstract
An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman. In this paper, we give other explicit constructions of pairing-friendly hyperelliptic curves with ordinary Jacobians based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type y 2 = x 5 + ax. We present two methods in this paper. One is an analogue of the Cocks-Pinch method and the other is a cyclotomic method. By using these methods, we construct a pairing-friendly hyperelliptic curve y 2 = x 5 + ax over a finite prime field \({\mathbb F}_p\) whose Jacobian is ordinary and simple over \({\mathbb F}_p\) with a prescribed embedding degree. Moreover, the analogue of the Cocks-Pinch produces curves with ρ ≈ 4 and the cyclotomic method produces curves with 3 ≤ ρ ≤ 4.
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
Preview
Unable to display preview. Download preview PDF.
References
Balasubramanian, R., Koblitz, N.: The improbability that an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm. J. Cryptology 11(2), 141–145 (1998)
Barreto, P.S.L.M., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)
Blake, I.-F., Seroussi, G., Smart, N.-P.: Advances in Elliptic Curve Cryptography. Cambridge University Press, Cambridge (2005)
Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM Journal of Computing 32(3), 586–615 (2003)
Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Design, Codes and Cryptography 37, 133–141 (2005)
Cardona, G., Nart, E.: Zeta Function and Cryptographic Exponent of Supersingular Curves of Genus 2. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 132–151. Springer, Heidelberg (2007)
Cocks, C., Pinch, R.G.E.: Identity-based cryptosystems based on the Weil pairing (unpublished manuscript, 2001)
Freeman, D.: Constructing pairing-friendly genus 2 curves with ordinary Jacobians. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 152–176. Springer, Heidelberg (2007)
Freeman, D.: A generalized Brezing-Weng method for constructing pairing-friendly ordinary abelian varieties. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 146–163. Springer, Heidelberg (2008)
Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves, Cryptology ePrint Archive, Report 2006/372 (2006), http://eprint.iacr.org/
Freeman, D., Stevenhagen, P., Streng, M.: Abelian varieties with prescribed embedding degree. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 60–73. Springer, Heidelberg (2008)
Furukawa, E., Kawazoe, M., Takahashi, T.: Counting Points for Hyperelliptic Curves of Type y 2 = x 5 + ax over Finite Prime Fields. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 26–41. Springer, Heidelberg (2004)
Galbraith, S., McKee, J., Valença, P.: Ordinary abelian varieties having small embedding degree. Finite Fields and Their Applications 13, 800–814 (2007)
Haneda, M., Kawazoe, M., Takahashi, T.: Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of Type y 2 = x 2k + 1 + ax. In: Gaires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 539–550. Springer, Heidelberg (2005)
Joux, A.: A one round protocol for tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS-IV 2000. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000); Full version: Journal of Cryptology 17, 263–276 (2004)
Kawazoe, M., Sakaeyama, R., Takahashi, T.: Pairing-friendly Hyperelliptic Curves of type y 2 = x 5 + ax. In: 2008 Symposium on Cryptography and Information Security (SCIS 2008), Miyazaki, Japan (2008)
Kawazoe, M., Takahashi, T.: Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type y 2 = x 5 + ax, Extended version of the present paper, Cryptology ePrint Archive, Report 2008/026 (2008), http://eprint.iacr.org/ .
Miyaji, A., Nakabayashi, M., Takano, S.: New explicit conditions of elliptic curve traces for FR-reduction. IEICE Transactions on Fundamentals E84-A(5), 1234–1243 (2001)
Rubin, K., Silverberg, A.: Supersingular abelian varieties in cryptology. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 336–353. Springer, Heidelberg (2002)
Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystem based on pairing. In: 2000 Symposium on Cryptography and Information Security (SCIS 2000), Okinawa, Japan (2000)
Scott, M., Barreto, P.S.L.M.: Generating more MNT elliptic curves. Designs, Codes and Cryptography 38, 209–217 (2006)
Wolfram Research, Inc., Mathematica, Version 6.0, Champaign, IL (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kawazoe, M., Takahashi, T. (2008). Pairing-Friendly Hyperelliptic Curves with Ordinary Jacobians of Type y 2 = x 5 + ax . In: Galbraith, S.D., Paterson, K.G. (eds) Pairing-Based Cryptography – Pairing 2008. Pairing 2008. Lecture Notes in Computer Science, vol 5209. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85538-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-85538-5_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85503-3
Online ISBN: 978-3-540-85538-5
eBook Packages: Computer ScienceComputer Science (R0)