Abstract
Most implementations of pairing-based cryptography are using pairing-friendly curves with an embedding degree k ≤ 12. They have security levels of up to 128 bits. In this paper, we consider a family of pairing-friendly curves with embedding degree k = 24, which have an enhanced security level of 192 bits. We also describe an efficient implementation of Tate and Ate pairings using field arithmetic in \(F_{q^{24}}\); this includes a careful selection of the parameters with small hamming weight and a novel approach to final exponentiation, which reduces the number of computations required.
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References
Hess, F., Smart, N.P., Vercauteren, F.: The Eta Pairing Revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)
Miller, V.S.: The Weil pairing and its efficient calculation. Journal of Cryptography 17(4), 235–261 (2004)
Freeman, D., Scott, M., Teske, E.: A Taxonomy of Pairing-Friendly Elliptic Curves. Journal of Cryptology 23, 224–280 (2010)
Scott, M.: A note on twists for pairing friendly curves (2005), ftp://ftp.coomputing.dcu.ie/pub/resources/crypto/twists.pdf
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)
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)
Granger, R., Page, D., Smart, N.P.: High Security Pairing-Based Cryptography Revisited. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 480–494. Springer, Heidelberg (2006)
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)
Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Designs, Codes and Cryptography 37(1), 133–141 (2005)
Freeman, D.: Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 452–465. Springer, Heidelberg (2006)
Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory 39, 1639–1646 (1993)
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)
Montgomery, P.L.: Modular multiplication without trial division. Mathematics of Computation 44(170), 519–521 (1985)
CNSS Policy, no. 15, fact sheet no. 1, National Policy on the Use of the Advanced Encryption Standard(AES) to Protect National Security Systems and National Security Information. NIST (2003)
Cocks, C., Pinch, R.G.E.: Identity-based cryptosystems based on the Weil pairing (2001) (unpublished manuscript)
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Kim, I.T., Park, C., Hwang, S.O., Park, CM. (2011). Implementation of Bilinear Pairings over Elliptic Curves with Embedding Degree 24. In: Kim, Th., et al. Multimedia, Computer Graphics and Broadcasting. MulGraB 2011. Communications in Computer and Information Science, vol 262. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27204-2_5
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DOI: https://doi.org/10.1007/978-3-642-27204-2_5
Publisher Name: Springer, Berlin, Heidelberg
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