Abstract
In implementing an efficient pairing calculation, it is said that the lower bound of the number of iterations of Miller’s algorithm is log2 r/ϕ(k), where ϕ(·) is the Euler’s function. Ate pairing reduced the number of the loops of Miller’s algorithm of Tate pairing from \(\lfloor\log_2r\rfloor\) to \(\lfloor \log_2(t-1)\rfloor\). Recently, it is known to systematically prepare a pairing–friendly elliptic curve whose parameters are given by a polynomial of integer variable “χ”. For the curve, this paper gives integer variable χ –based Ate pairing that achieves the lower bound by reducing it to \(\lfloor\log_2\chi\rfloor\).
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
Akane, M., Kato, H., Okimoto, T., Nogami, Y., Morikawa, Y.: An Improvement of Miller’s Algorithm in Ate Pairing with Barreto–Naehrig Curve. In: Proc. of Computer Security Symposium 2007 (CSS 2007), pp. 489–494 (2007)
Akane, M., Kato, H., Okimoto, T., Nogami, Y., Morikawa, Y.: Efficient Parameters for Ate Pairing Computation with Barreto-Naehrig Curve. In: Proc. of Computer Security Symposium 2007 (CSS 2007), pp. 495–500 (2007)
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)
Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)
Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Mathematics and Its Applications. Chapman & Hall CRC (2005)
Devegili, A.J., Scott, M., Dahab, R.: Implementing Cryptographic Pairings over Barreto-Naehrig Curves. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 197–207. Springer, Heidelberg (2007)
Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves (preprint, 2006), http://math.berkeley.edu/~dfreeman/papers/taxonomy.pdf
Galbraith, S.D., Scott, M.: Exponentiation in pairing-friendly groups using homomorphisms. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS. Springer, Heidelberg (to appear, 2008)
GNU MP, http://gmplib.org/
Hess, F., Smart, N., Vercauteren, F.: The Eta Pairing Revisited. IEEE Trans. Information Theory, 4595–4602 (2006)
Itoh, T., Tsujii, S.: A Fast Algorithm for Computing Multiplicative Inverses in GF(2m) Using Normal Bases. Inf. and Comp. 78, 171–177 (1988)
Kato, H., Nogami, Y., Yoshida, T., Morikawa, Y.: Cyclic Vector Multiplication Algorithm Based on a Special Class of Gauss Period Normal Basis. ETRI Journal 29(6), 769–778 (2007), http://etrij.etri.re.kr/Cyber/servlet/BrowseAbstract?paperid=RP0702-0040
Knuth, D.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1981)
Lee, E., Lee, H., Park, C.: Efficient and Generalized Pairing Computation on Abelien Varieties, IACR ePrint archive, http://eprint.iacr.org/2008/040
Matsuda, S., Kanayama, N., Hess, F., Okamoto, E.: Optimised Versions of the Ate and Twisted Ate Pairings. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 302–312. Springer, Heidelberg (2007)
Nakanishi, T., Funabiki, N.: Verifier-Local Revocation Group Signature Schemes with Backward Unlinkability from Bilinear Maps. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 443–454. Springer, Heidelberg (2005)
Nogami, Y., Morikawa, Y.: A Fast Implementation of Elliptic Curve Cryptosystem with Prime Order Defined over \(F_{p^{8}}\). Memoirs of the Faculty of Engineering Okayama University 37(2), 73–88 (2003)
Sakemi, Y., Kato, H., Akane, M., Okimoto, T., Nogami, Y., Morikawa, Y.: An Improvement of Twisted Ate Pairing Using Integer Variable with Small Hamming Weight. In: The 2008 Symposium on Cryptography and Information Security (SCIS 2008), January 22-25 (2008)
Vercauteren, F.: Optimal Pairings, IACR ePrint archive, http://eprint.iacr.org/2008/096
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nogami, Y., Akane, M., Sakemi, Y., Kato, H., Morikawa, Y. (2008). Integer Variable χ–Based Ate Pairing. 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_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-85538-5_13
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)