Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware

  • Kris Gaj
  • Soonhak Kwon
  • Patrick Baier
  • Paul Kohlbrenner
  • Hoang Le
  • Mohammed Khaleeluddin
  • Ramakrishna Bachimanchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4249)

Abstract

A novel portable hardware architecture for the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proof-of-concept design by Pelzl, Šimka, et al. has been performed, and a substantial improvement has been demonstrated in terms of both the execution time and the area-time product. The ECM architecture has been ported across three different families of FPGA devices in order to select the family with the best performance to cost ratio. A timing comparison with a highly optimized software implementation, GMP-ECM, has been performed. Our results indicate that low-cost families of FPGAs, such as Xilinx Spartan 3, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio.

Keywords

Cipher-breaking factoring ECM FPGA 

References

  1. 1.
    Pollard, J.M.: Factoring with cubic integers. Lecture Notes in Mathematics, vol. 1554, pp. 4–10. Springer, Heidelberg (1993)Google Scholar
  2. 2.
    Lenstra, A.K., Lenstra, H.W.: The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)MATHCrossRefGoogle Scholar
  3. 3.
    Bahr, F., Boehm, M., Franke, J., Kleinjung, T.: Factorization of RSA-200, http://crypto-world.com/announcements/rsa200.txt
  4. 4.
    Zimmermann, P.: 20 years of ECM (preprint, 2005), http://www.loria.fr/~zimmerma/papers/ecm-submitted.pdf
  5. 5.
    Fougeron, J., Fousse, L., Kruppa, A., Newman, D., Zimmermann, P.: GMP-ECM (2005), http://www.komite.net/laurent/soft/ecm/ecm-6.0.1.html
  6. 6.
    Šimka, M., Pelzl, J., Kleinjung, T., Franke, J., Priplata, C., Stahlke, C., Drutarovsky, M., Fischer, V., Paar, C.: Hardware factorization based elliptic curve method. In: IEEE Symposium on Field-Programmable Custom Computing Machines - FCCM 2005, Napa, CA, USA (2005)Google Scholar
  7. 7.
    Pelzl, J., Šimka, M., Kleinjung, T., Franke, J., Priplata, C., Stahlke, C., Drutarovsky, M., Fischer, V., Paar, C.: Area-time efficient hardware architecture for factoring integers with the elliptic curve method. IEEE Proceedings on Information Security 152(1), 67–78 (2005)CrossRefGoogle Scholar
  8. 8.
    Hankerson, D., Menezes, A.J., Vanstone, S.A.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)MATHGoogle Scholar
  9. 9.
    Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Brent, R.P.: Some integer factorization algorithms using elliptic curves. Australian Computer Science Communications 8, 149–163 (1986)Google Scholar
  11. 11.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Montgomery, P.L.: An FFT extension of the elliptic curve method of factorization., Ph.D. Thesis, UCLA (1992)Google Scholar
  13. 13.
    Montgomery, P.L.: Modular multiplication without trivial division. Mathematics of Computation 44, 519–521 (1985)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    McIvor, C., McLoone, M., McCanny, J., Daly, A., Marnane, W.: Fast Montgomery modular multiplication and RSA cryptographic processor architectures. In: Proc. 37th IEEE Computer Society Asilomar Conference on Signals, Systems and Computers, Monterey, USA, November 2003, pp. 379–384 (2003)Google Scholar
  15. 15.
    Kumar, S., Paar, C., Pelzl, J., Pfeiffer, G., Rupp, A., Schimmler, M.: How to break DES for 8,980 Euro. In: 2nd Workshop on Special-purpose Hardware for Attacking Cryptographic Systems - SHARCS 2006, Cologne, Germany, April 3-4 (2006)Google Scholar
  16. 16.
    Franke, J., Kleinjung, T., Paar, C., Pelzl, J., Priplata, C., Stahlke, C.: SHARK: A realizable special hardware sieving device for factoring 1024-bit integers. In: Rao, J.R., Sunar, B. (eds.) CHES 2005. LNCS, vol. 3659, pp. 119–130. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Geiselmann, W., Januszewski, F., Koepfer, H., Pelzl, J., Steinwandt, R.: A simpler sieving device: Combining ECM and TWIRL, Cryptology ePrint Archive, http://eprint.iacr.org/2006/109
  18. 18.
    SRC Computers, Inc., http://www.srccomp.com
  19. 19.
    Silverman, R.D., Wagstaff, S.S.: A practical analysis of the elliptic curve factoring algorithm. Mathematics of Computation 61(203), 462–465 (1993)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kris Gaj
    • 1
  • Soonhak Kwon
    • 2
  • Patrick Baier
    • 1
  • Paul Kohlbrenner
    • 1
  • Hoang Le
    • 1
  • Mohammed Khaleeluddin
    • 1
  • Ramakrishna Bachimanchi
    • 1
  1. 1.Dept. of Electrical and Computer EngineeringGeorge Mason UniversityFairfaxUSA
  2. 2.Inst. of Basic ScienceSungkyunkwan UniversitySuwonKorea

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