A Scalable Architecture for Montgomery Nultiplication

  • Alexandre F. Tenca
  • Çetin K. Koç
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1717)

Abstract

This paper describes the methodology and design of a scalable Montgomery multiplication module. There is no limitation on the maximum number of bits manipulated by the multiplier, and the selection of the word-size is made according to the available area and/or desired performance. We describe the general view of the new architecture, analyze hardware organization for its parallel computation, and discuss design tradeoffs which are useful to identify the best hardware configuration.

References

  1. 1.
    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
  2. 2.
    W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, 22:644–654, November 1976.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    T. Hamano, N. Takagi, S. Yajima, and F. P Preparata. O(n)-Depth circuit algorithm for modular exponentiation. In S. Knowles and W. H. McAllister, editors, Proceedings, 12th Symposium on Computer Arithmetic, pages 188–192, Bath, England, July 19–21 1995. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
  5. 5.
    Ç. K. KoÇ and T. Acar. Fast software exponentiation in GF(2k). In T. Lang, J.-M. Muller, and N. Takagi, editors, Proceedings, 13th Symposium on Computer Arithmetic, pages 225–231, Asilomar, California, July 6–9, 1997. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
  6. 6.
    Ç. K. KoÇ and T. Acar. Montgomery multiplication in GF(2k). Designs, Codes and Cryptography, 14(1):57–69, April 1998.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    P. Kornerup. High-radix modular multiplication for cryptosystems. In E. Swartz lander, Jr., M. J. Irwin, and G. Jullien, editors, Proceedings, 11th Symposium on Computer Arithmetic, pages 277–283, Windsor, Ontario, June 29-July 2 1993. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
  9. 9.
    A. J. Menezes. Elliptic Curve Public Key Cryptosystems. Boston, MA: Kluwer Academic Publishers, 1993.MATHGoogle Scholar
  10. 10.
    P. L. Montgomery. Modular multiplication without trial division. Mathematics of Computation, 44(170):519–521, April 1985.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
  12. 12.
    R. L. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2):120–126, February 1978.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    A. F. Tenca. Variable Long-Precision Arithmetic (VLPA) for Reconfigurable Co-processor Architectures. PhD thesis, Department of Computer Science, University of California at Los Angeles, March 1998.Google Scholar
  15. 15.
    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 1999

Authors and Affiliations

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

Personalised recommendations