A High Speed Coprocessor for Elliptic Curve Scalar Multiplications over \(\mathbb{F}_p\)

  • Nicolas Guillermin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6225)

Abstract

We present a new hardware architecture to compute scalar multiplications in the group of rational points of elliptic curves defined over a prime field. We have made an implementation on Altera FPGA family for some elliptic curves defined over randomly chosen ground fields offering classic cryptographic security level. Our implementations show that our architecture is the fastest among the public designs to compute scalar multiplication for elliptic curves defined over a general prime ground field. Our design is based upon the Residue Number System, guaranteeing carry-free arithmetic and easy parallelism. It is SPA resistant and DPA capable.

Keywords

elliptic curve high speed RNS prime field FPGA 

References

  1. 1.
    Bajard, J.-C., Didier, L.-S., Kornerup, P.: An rns montgomery modular multiplication algorithm. IEEE Transactions on Computers 47(7), 766–776 (1998)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bajard, J.-C., Imbert, L., Liardet, P.-Y., Teglia, Y.: Leak resistant arithmetic. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 116–145. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Chen, L., Yanpu, C., Zhengzhong, B.: An implementation of fast algorithm for elliptic curve cryptosystem over GF(p). Journal of Electronics (China) 21(4), 346–352 (2004)CrossRefGoogle Scholar
  4. 4.
    Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Edwards, H.: A normal form for elliptic curves. Bull. Amer. Math. Soc. 44 (2007)Google Scholar
  6. 6.
    Fouque, P.-A., Valette, F.: The doubling attack – why upwards is better than downwards. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 269–280. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Güneysu, T., Paar, C.: Ultra high performance ecc over nist primes on commercial fpgas. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 62–78. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Jarvinen, K.U., Skytta, J.O.: High-speed elliptic curve cryptography accelerator for koblitz curves. In: Annual IEEE Symposium on Field-Programmable Custom Computing Machines, pp. 109–118 (2008)Google Scholar
  9. 9.
    Joye, M., Sung-Min-Yen: The montgomery powering ladder. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 291–302. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Kawamura, S., Koike, M., Sano, F., Shimbo, A.: Cox-rower architecture for fast parallel montgomery multiplication. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 523–538. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Mentens, N.: Secure and Efficient Coprocessor Design for Cryptographic Applications on FPGAs. PhD thesis, Ruhr-University Bochum (2007)Google Scholar
  12. 12.
    Messerges, T.S., Dabbish, E.A., Sloan, R.H.: Power analysis attacks of modular exponentiation in smartcards. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 144–157. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    de Dormale, G.M., Quisquater, J.-J.: High-speed hardware implementations of elliptic curve cryptography: A survey. J. Syst. Archit. 53(2-3), 72–84 (2007)CrossRefGoogle Scholar
  14. 14.
    Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  15. 15.
    Montgomery, P.L.: Modular multiplication without trial division. Mathematics of Computation 44, 519–521 (1985)MATHMathSciNetGoogle Scholar
  16. 16.
    Nozaki, H., Motoyama, M., Shimbo, A., Kawamura, S.-i.: Implementation of rsa algorithm based on rns montgomery multiplication. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 364–376. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    National Institute of Science and Technology. The digital signature standard. Technical report, http://csrc.nist.gov/publications/fips/archive/fips186-2/fips186-2.pdf
  18. 18.
    White Paper. Stratix vs. virtex-ii pro fpga performance analysis. Technical report, http://www.altera.com/literature/wp/wpstxvrtxII.pdf
  19. 19.
    Posch, K.C., Posch, R.: Modulo reduction in residue number systems. IEEE Trans. Parallel Distrib. Syst. 6(5), 449–454 (1995)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Ecegovac, M., Duquesne, S., Bajard, J.C.: Combining leak-resistant arithmetic for elliptic curves define over \(\mathbb{F}_p\) and rns representationGoogle Scholar
  21. 21.
    Sakiyama, K., Mentens, N., Batina, L., Preneel, B., Verbauwhede, I.: Reconfigurable modular arithmetic logic unit for high-performance public-key cryptosystems. In: Bertels, K., Cardoso, J.M.P., Vassiliadis, S. (eds.) ARC 2006. LNCS, vol. 3985, pp. 347–357. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Satoh, A., Takano, K.: A scalable dual-field elliptic curve cryptographic processor. IEEE Transactions on Computers 52, 449–460 (2003)CrossRefGoogle Scholar
  23. 23.
    Schinianakis, D.M., Fournaris, A.P., Michail, H.E., Kakarountas, A.P., Stouraitis, T.: An rns implementation of an fpelliptic curve point multiplier. Trans. Cir. Sys. Part I 56(6), 1202–1213 (2009)CrossRefGoogle Scholar
  24. 24.
    Shenoy, P.P., Kumaresan, R.: Fast base extension using a redundant modulus in rns. IEEE Trans. Comput. 38(2), 292–297 (1989)MATHCrossRefGoogle Scholar
  25. 25.
    Szerwinski, R., Gayneysu, T.: Exploiting the power of GPUs for asymmetric cryptography. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 79–99. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nicolas Guillermin
    • 1
    • 2
  1. 1.DGA Information SuperiorityBruzFrance
  2. 2.IRMARUniversité Rennes 1France

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