Abstract
We provide software implementation timings for pairings over composite-order and prime-order elliptic curves. Composite orders must be large enough to be infeasible to factor. In the literature, protocols use orders which are product of 2 up to 5 large prime numbers. Our contribution is three-fold. First, we extend the results of Lenstra concerning the RSA modulus sizes to multi-prime modulus, for various security levels. We then implement a Tate pairing over a composite order supersingular curve and an optimal ate pairing over a prime-order Barreto-Naehrig curve, both at the 128-bit security level. Thirdly we use our implementation timings to deduce the total cost of the homomorphic encryption scheme of Boneh, Goh and Nissim and its translation by Freeman in the prime-order setting. We also compare the efficiency of the unbounded Hierarchical Identity Based Encryption protocol of Lewko and Waters and its translation by Lewko in the prime order setting. Our results strengthen the previously observed inefficiency of composite-order bilinear groups and advocate the use of prime-order group whenever possible in protocol design.
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Acar, T., Lauter, K., Naehrig, M., Shumow, D.: Affine pairings on ARM. In: Abdalla, M., Lange, T. (eds.) Pairing 2012. LNCS, vol. 7708, pp. 203–209. Springer, Heidelberg (2013)
Aranha, D.F., Karabina, K., Longa, P., Gebotys, C.H., López, J.: Faster explicit formulas for computing pairings over ordinary curves. In: Paterson (ed.) [22], pp. 48–68
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)
Bernstein, D.J., Lange, T.: Computing small discrete logarithms faster. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 317–338. Springer, Heidelberg (2012)
Beuchat, J.-L., González-Díaz, J.E., Mitsunari, S., Okamoto, E., Rodríguez-Henríquez, F., Teruya, T.: High-speed software implementation of the optimal ate pairing over Barreto-Naehrig curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 21–39. Springer, Heidelberg (2010)
Boneh, D., Boyen, X., Goh, E.-J.: Hierarchical identity based encryption with constant size ciphertext. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 440–456. Springer, Heidelberg (2005)
Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005)
Boneh, D., Rubin, K., Silverberg, A.: Finding composite order ordinary elliptic curves using the Cocks-Pinch method. Journal of Number Theory 131(5), 832–841 (2011)
Devegili, A.J., Héigeartaigh, C.O., Scott, M., Dahab, R.: Multiplication and squaring on pairing-friendly fields. Cryptology ePrint Archive, Report 2006/471 (2006)
Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptology 23(2), 224–280 (2010)
Freeman, D.M.: Converting pairing-based cryptosystems from composite-order groups to prime-order groups. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 44–61. Springer, Heidelberg (2010)
Granger, R., Scott, M.: Faster squaring in the cyclotomic subgroup of sixth degree extensions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 209–223. Springer, Heidelberg (2010)
Herrmann, M.: Improved cryptanalysis of the multi-prime φ - hiding assumption. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 92–99. Springer, Heidelberg (2011)
Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. Cambridge University Press (2005)
Kiltz, E., O’Neill, A., Smith, A.: Instantiability of RSA-OAEP under chosen-plaintext attack. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 295–313. Springer, Heidelberg (2010)
Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)
Lenstra, A.K.: Unbelievable security: Matching AES security using public key systems. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 67–86. Springer, Heidelberg (2001)
Lenstra, A.K., Lenstra, H.W.J. (eds.): The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)
Lewko, A.: Tools for simulating features of composite order bilinear groups in the prime order setting. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 318–335. Springer, Heidelberg (2012)
Lewko, A.B., Waters, B.: Unbounded HIBE and attribute-based encryption. In: Paterson (ed.) [22], pp. 547–567
Naehrig, M., Niederhagen, R., Schwabe, P.: New software speed records for cryptographic pairings. In: Abdalla, M., Barreto, P.S.L.M. (eds.) LATINCRYPT 2010. LNCS, vol. 6212, pp. 109–123. Springer, Heidelberg (2010)
Paterson, K.G. (ed.): EUROCRYPT 2011. LNCS, vol. 6632. Springer, Heidelberg (2011)
Seo, J.H.: On the (im)possibility of projecting property in prime-order setting. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 61–79. Springer, Heidelberg (2012)
Vercauteren, F.: Optimal pairings. IEEE Transactions on Information Theory 56(1), 455–461 (2010)
Waters, B.: Dual system encryption: Realizing fully secure IBE and HIBE under simple assumptions. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 619–636. Springer, Heidelberg (2009)
Zhao, C., Zhang, F., Xie, D.: Faster computation of self-pairings. IEEE Transactions on Information Theory 58(5), 3266–3272 (2012)
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Guillevic, A. (2013). Comparing the Pairing Efficiency over Composite-Order and Prime-Order Elliptic Curves. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds) Applied Cryptography and Network Security. ACNS 2013. Lecture Notes in Computer Science, vol 7954. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38980-1_22
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DOI: https://doi.org/10.1007/978-3-642-38980-1_22
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