Scalable and Unified Hardware to Compute Montgomery Inverse in GF(p) and GF(2n)

  • Adnan Abdul-Aziz Gutub
  • Alexandre F. Tenca
  • Erkay Savaş
  • RCCetin K. Koç
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2523)


Computing the inverse of a number in finite fields GF(p) or GF(2n) is equally important for cryptographic applications. This paper proposes a novel scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2n) fields. We adjust and modify a GF(2n) Montgomery inverse algorithm to accommodate multi-bit shifting hardware, making it very similar to a previously proposed GF(p) algorithm. The architecture is intended to be scalable, which allows the hardware to compute the inverse of long precision numbers in a repetitive way. After implementing this unified design it was compared with other designs. The unified hardware was found to be eight times smaller than another reconfigurable design, with comparable performance. Even though the unified design consumes slightly more area and it is slightly slower than the scalable inverter implementations for GF(p) only, it is a practical solution whenever arithmetic in the two finite fields is needed.


  1. 1.
    E. Savas and C. K. Koç. The Montgomery Modular Inverse R3-Revisited. IEEE Trans. on Computers, 49(7): 763–766, July 2000.CrossRefGoogle Scholar
  2. 2.
    T. Kobayashi and H. Morita. Fast Modular Inversion Algorithm to Match Any Operation Unit. IEICE Trans. Fundamentals, E82-A(5):733–740, May 1999.Google Scholar
  3. 3.
    B. S. Kaliski. The Montgomery Inverse and its Applications. IEEE Trans. on Computers, 44(8):1064–1065, Aug. 1995.MATHCrossRefGoogle Scholar
  4. 4.
    E. Savas, A. F. Tenca, and C. K. Koç. A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2k). In Cryptographic Hardware and Embedded Systems, Lecture notes in Computer Science. Springer, Berlin, Germany, 2000.Google Scholar
  5. 5.
    I. Blake, G. Seroussi, and N. Smart. Elliptic Curves in Cryptography. Cambridge University Press: New York, 1999.MATHGoogle Scholar
  6. 6.
    M. D. Ercegovac, T. Lang, and J. H. Moreno. Introduction to Digital System. John Wiley & Sons, Inc., New York, 1999.Google Scholar
  7. 7.
    P. Montgomery. Modular Multiplication without Trail Division. Mathematics of Computation, 44(170): 519–521, April 1985.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    N. Takagi. Modular Inversion Hardware with a Redundant Binary Representation. IEICE Trans. on Information and Systems, E76-D(8): 863–869, Aug. 1993.Google Scholar
  9. 9.
    J.-H. Guo, and C.-L. Wang. Hardware-Efficient Systolic Architecture for Inversion and Division in GF(2m). IEE Proceedings: Computers and Digital Techniques, 145(4): 272–278, July 1998.Google Scholar
  10. 10.
    Choudhury, Pal, and Barua. Cellular Automata Based VLSI Architecture for Computing Multiplication and Inverses in GF(2m). Proceedings of the 7th IEEE International Conference on VLSI Design, Calcutta, India, January 5-8 1994.Google Scholar
  11. 12.
    M. A. Hasan. Efficient Computation of Multiplicative Inverses for Cryptographic Applications. Proceeding of the 15th IEEE Symposium on Computer Arithmetic, June 2001.Google Scholar
  12. 13.
    M. Feng. A VLSI Architecture for Fast Inversion in GF(2m). IEEE Trans. on Computers, 38(10):1383–1386, Oct. 1989.CrossRefGoogle Scholar
  13. 14.
    A. A.-A. Gutub, A. F. Tenca, and C. K. Koç. Scalable VLSI Architecture for GF(p) Montgomery Modular Inverse Computation. ISVLSI 2002: IEEE Computer Society Annual Symposium on VLSI, Pittsburgh, Pennsylvania, April 25-26 2002.Google Scholar
  14. 15.
    J. R. Michener and S. D. Mohan. Clothing the E-Emperor. IEEE Compute, 34(9):116–118, Sep. 2001.Google Scholar
  15. 16.
    J. Goodman and A. P. Chandrakasan. An Energy-Efficient Reconfigurable Public-Key Cryptogrphy Processor. IEEE Journal of Solid-State Circuits, 36(11):1808–1820, Nov. 2001.CrossRefGoogle Scholar
  16. 17.
    D. Knuth. The Art of Computer Programming R3-Seminumerical Algorithms, 2nd ed. Vol. 2, Reading, MA: Addison-Wesley, 1981.Google Scholar
  17. 18.
    A. A.-A. Gutub and A. F. Tenca. A Scalable VLSI Architecture for Montgomery Inversion in GF(p). Submitted for publication in March 2002 to IEEE Trans. on VLSI.Google Scholar
  18. 19.
    A. A.-A. Gutub, New Hardware Algorithms and Designs for Montgomery Modular Inverse Computation in Galois Fields GF(p) and GF(2n), Ph.D. thesis, Oregon State University, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Adnan Abdul-Aziz Gutub
    • 1
  • Alexandre F. Tenca
    • 1
  • Erkay Savaş
    • 1
  • RCCetin K. Koç
    • 1
  1. 1.Department of Electrical & Computer EngineeringOregon State UniversityCorvallis, OregonUSA

Personalised recommendations