A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m)

  • Erkay Savaš
  • Alexandre F. Tenca
  • Çetin K. Koç
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1965)

Abstract

We describe a scalable and unified architecture for a Montgomery multiplication module which operates in both types of finite fields GF(p) and GF(2m). The unified architecture requires only slightly more area than that of the multiplier architecture for the field GF(p). The multiplier is scalable, which means that a fixed-area multiplication module can handle operands of any size, and also, the wordsize can be selected based on the area and performance requirements. We utilize the concurrency in the Montgomery multiplication operation by employing a pipelining design methodology. The upper limit on the precision of the scalable and unified Montgomery multiplier is dictated only by the available memory to store the operands and internal results, and the module is capable of performing infinite-precision Montgomery multiplication in both types of finite fields.

Keywords

Prime fields binary extension fields multiplication Montgomery multiplication scalability hardware implementation 

References

  1. 1.
    G. B. Agnew, R. C. Mullin, and S. A. Vanstone. An implementation of elliptic curve cryptosystems over F2155. IEEE Journal on Selected Areas in Communications, 11(5):804–813, June 1993.CrossRefGoogle Scholar
  2. 2.
    A. Bernal and A. Guyot. Design of a modular multiplier based on Montgomery’s algorithm. In 13th Conference on Design of Circuits and Integrated Systems, pages 680–685, Madrid, Spain, November 17-20 1998.Google Scholar
  3. 3.
    W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, 22:644–654, November 1976.Google Scholar
  4. 4.
    S. E. Eldridge and C. D. Walter. Hardware implementation of Montgomery’s modular multiplication algorithm. IEEE Transactions on Computers, 42(6):693–699, June 1993.CrossRefGoogle Scholar
  5. 5.
    Steve Furber. ARM System Architecture. Addison-Wesley, Reading, MA, 1997.Google Scholar
  6. 6.
    B. S. Kaliski Jr. The Montgomery inverse and its applications. IEEE Transactions on Computers, 44(8):1064–1065, August 1995.MATHCrossRefGoogle Scholar
  7. 7.
    N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, 48(177):203–209, January 1987.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ç. K. Koç. High-Speed RSA Implementation. Technical Report TR 201, RSA Laboratories, 73 pages, November 1994.Google Scholar
  9. 9.
    Ç. K. Koç and T. Acar. Montgomery multiplication in GF(2k). Designs, Codes and Cryptography, 14(1):57–69, April 1998.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ç. K. Koç, T. Acar, and B. S. Kaliski Jr. Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro, 16(3):26–33, June 1996.CrossRefGoogle Scholar
  11. 11.
    P. Kornerup. High-radix modular multiplication for cryptosystems. In E. Swartzlander, M. J. Irwin, and G. Jullien, editors, Proceedings, 11th Symposium on Computer Arithmetic, pages 277–283, Windsor, Ontario, June 29-July 2 1993. IEEE Computer Society Press, Los AlamitoGoogle Scholar
  12. 12.
    A. J. Menezes. Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Boston, MA, 1993.MATHGoogle Scholar
  13. 13.
    P. L. Montgomery. Modular multiplication without trial division. Mathematics of Computation, 44(170):519–521, April 1985.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Naccache and D. M’Raïhi. Cryptographic smart cards. IEEE Micro, 16(3):14–24, June 1996.Google Scholar
  15. 15.
    National Institute for Standards and Technology. Digital Signature Standard (DSS). FIPS PUB 186-2, January 2000.Google Scholar
  16. 16.
    H. Orup. Simplifying quotient determination in high-radix modular multiplication. In S. Knowles and W. H. McAllister, editors, Proceedings, 12th Symposium on Computer Arithmetic, pages 193–199, Bath, England, July 19-21 1995. IEEE Computer Society Press, Los Alamitos, CA.Google Scholar
  17. 17.
    J.-J. Quisquater and C. Couvreur. Fast decipherment algorithm for RSA publickey cryptosystem. Electronics Letters, 18(21):905–907, October 1982.CrossRefGoogle Scholar
  18. 18.
    A. Royo, J. Moran, and J. C. Lopez. Design and implementation of a coprocessor for cryptography applications. In European Design and Test Conference, pages 213–217, Paris, France, March 17-20 1997.Google Scholar
  19. 19.
    E. Sava§ and Ç. K. Koç. The Montgomery modular inverse-revisited. IEEE Transactions on Computers, 49(8), July 2000. To appear.Google Scholar
  20. 20.
    A. F. Tenca and Ç. K. Koç. A scalable architecture for Montgomery multiplication. In Ç. K. Koç and C. Paar, editors, Cryptographic Hardware and Embedded Systems, Lecture Notes in Computer Science, No. 1717, pages 94–108. Springer, Berlin, Germany, 1999.CrossRefGoogle Scholar
  21. 21.
    C. D. Walter. Space/Time trade-offs for higher radix modular multiplication using repeated addition. IEEE Transactions on Computers, 46(2):139–141, February 1997.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Erkay Savaš
    • 1
  • Alexandre F. Tenca
    • 1
  • Çetin K. Koç
    • 1
  1. 1.Electrical & Computer EngineeringOregon State UniversityCorvallis

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