Abstract
This paper presents an efficient implementation of optimal-ate pairing over BN curves. It exploits the highly optimized IP cores available in modern FPGAs to speed up pairing computation. The pipelined datapaths for \(\mathbb{F}_{p}\)-operations and suitable memory cores help to reduce the overall clock cycle count more than 50%. The final design, on a Virtex-6 FPGA, computes an optimal-ate pairing having 126-bit security in 0.375 ms which is a 32% speedup from state of the art result.
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Aranha, D.F., Beuchat, J.L., Detrey, J., Estibals, N.: Optimal Eta pairing on supersingular genus-2 binary hyperelliptic curves. Cryptology ePrint Archive, Report 2010/559, http://eprint.iacr.org/
Aranha, D.F., Karabina, K., Longa, P., Gebotys, C.H., López, J.: Faster Explicit Formulas for Computing Pairings over Ordinary Curves. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 48–68. Springer, Heidelberg (2011)
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)
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)
Beuchat, J.L., Detrey, J., Estibals, N., Okamoto, E., Rodríguez-Henríquez, F.: Fast architectures for the η T pairing over small-characteristic supersingular elliptic curves. IEEE Transactions on Computers 60(2) (2011)
Cheung, R.C.C., Duquesne, S., Fan, J., Guillermin, N., Verbauwhede, I., Yao, G.X.: FPGA Implementation of Pairings Using Residue Number System and Lazy Reduction. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 421–441. Springer, Heidelberg (2011)
Costello, C., Lange, T., Naehrig, M.: Faster Pairing Computations on Curves with High-Degree Twists. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 224–242. Springer, Heidelberg (2010)
Devegili, A., ÓhÉigeartaigh, C., Scott, M., Dahab, R.: Multiplication and squaring on pairing-friendly fields. Cryptology ePrint, Report 2006/471 (2006)
Duquesne, S., Guillermin, N.: A FPGA pairing implementation using the residue number system. Cryptology ePrint Archive, Report 2011/176 (2011), http://eprint.iacr.org/
Estibals, N.: Compact Hardware for Computing the Tate Pairing over 128-Bit-Security Supersingular Curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 397–416. Springer, Heidelberg (2010)
Fan, J., Vercauteren, F., Verbauwhede, I.: Faster \(\mathbb{F}_p\)-Arithmetic for Cryptographic Pairings on Barreto-Naehrig Curves. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 240–253. Springer, Heidelberg (2009)
Fan, J., Vercauteren, F., Verbauwhede, I.: Efficient Hardware Implementation of \(\mathbb{F}_{p}\)-arithmetic for Pairing-Friendly Curves. IEEE Trasaction on Computers (2011), http://dx.doi.org/10.1109/TC.2011.78
Ghosh, S., Mukhopadhyay, D., Roychowdhury, D.: Petrel: power and timing attack resistant elliptic curve scalar multiplier based on programmable arithmetic unit. IEEE Trans. on Circuits and Systems I 58(11), 1798–1812 (2011)
Ghosh, S., Roychowdhury, D., Das, A.: High Speed Cryptoprocessor for η T Pairing on 128-bit Secure Supersingular Elliptic Curves over Characteristic Two Fields. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 442–458. Springer, Heidelberg (2011)
Ghosh, S., Mukhopadhyay, D., Roychowdhury, D.: High Speed Flexible Pairing Cryptoprocessor on FPGA Platform. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 450–466. 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)
Hankerson, D., Menezes, A., Scott, M.: Software implementation of pairings. Cryptology and Info. Security Series, ch. 12, pp. 188–206. IOS Press (2009)
IEEE: P1363.3: Standard for Identity-Based Cryptographic Techniques using Pairings (2006), http://grouper.ieee.org/groups/1363/IBC/submissions/
Kammler, D., Zhang, D., Schwabe, P., Scharwaechter, H., Langenberg, M., Auras, D., Ascheid, G., Mathar, R.: Designing an ASIP for Cryptographic Pairings over Barreto-Naehrig Curves. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 254–271. Springer, Heidelberg (2009)
Montgomery, P.: Speeding the Pollard and Elliptic Curve Methods of Factorization. Mathematics of Computation 48, 243–264 (1987)
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)
Pereira, G.C.C.F., Simplício Jr., M.A., Naehrig, M., Barreto, P.S.L.M.: A Family of Implementation-Friendly BN Elliptic Curves. Journal of Systems and Software 84(8), 1319–1326 (2011)
Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., Kachisa, E.J.: On the Final Exponentiation for Calculating Pairings on Ordinary Elliptic Curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)
Vercauteren, F.: Optimal pairings. IEEE Transactions on Information Theory 56(1), 455–461 (2010)
Xilinx. LogiCORE IP Block Generator (2010), http://www.xilinx.com
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Ghosh, S., Verbauwhede, I., Roychowdhury, D. (2013). Core Based Architecture to Speed Up Optimal Ate Pairing on FPGA Platform. In: Abdalla, M., Lange, T. (eds) Pairing-Based Cryptography – Pairing 2012. Pairing 2012. Lecture Notes in Computer Science, vol 7708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36334-4_9
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DOI: https://doi.org/10.1007/978-3-642-36334-4_9
Publisher Name: Springer, Berlin, Heidelberg
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