Skip to main content
Log in

Abstract

Digit-serial architectures are best suited for systems requiring moderate sample rate and where area and power consumption are critical. This paper presents a new approach for designing digit-serial/parallel finite field multipliers. This approach combines both array-type and parallel multiplication algorithms, where the digit-level array-type algorithm minimizes the latency for one multiplication operation and the parallel architecture inside of each digit cell reduces both the cycle-time as well as the switching activities, hence power consumption. By appropriately constraining the feasible primitive polynomials, the mod p(x) operation involved in finite field multiplication can be performed in a more efficient way. As a result, the computation delay and energy consumption of one finite field multiplication using the proposed digit-serial/parallel architectures are significantly less than of those obtained by folding the parallel semi-systolic multipliers. Furthermore, their energy-delay products are reduced by a even larger percentage. Therefore, the proposed digit-serial/parallel architectures are attractive for both low-energy and high-performance applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R.E. Blahut, Theory and Practice of Error Control Codes, Addison Wesley, 1984.

  2. R.J. McEliece, The Theory of Information and Coding, Addison-Wesley, 1977.

  3. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977.

    MATH  Google Scholar 

  4. S. Lin and D.J. Costello, Error Control Coding:Fundamentals and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1983.

    Google Scholar 

  5. C.-S. Yeh, I.S. Reed, and T.K. Truong, “Systolic multipliers for finite fields GF(2m),” IEEE Trans. Comput., Vol. C-33, pp. 357- 360, 1984.

    Article  MathSciNet  Google Scholar 

  6. S. Bandyopadhyay and A. Sengupta, “Algorithms for multiplication in Galois Field for implementation using systolic arrays,” IEE Proceedings-E, Vol. 135, pp. 336-339, 1988.

    Google Scholar 

  7. C.L. Wang and J.L. Lin, “Systolic array implementation of multipliers for finite field GF(2m),” IEEE Trans. Circuits and Systems, Vol. 38, pp. 796-800, 1991.

    Article  Google Scholar 

  8. M.A. Hasan and V.K. Bhargava, “Bit-serial systolic divider and multiplier for finite field GF(2m),” IEEE Trans. Comput., Vol. 40, pp. 972-980, 1992.

    Article  MathSciNet  Google Scholar 

  9. S.W. Wei, “A systolic power-sum circuit for GF(2m),” IEEE Trans. Comput., Vol. 43, pp. 226-229, 1994.

    Article  Google Scholar 

  10. P.A. Scott, S.E. Tavares, and L.E. Peppard, “A Fast VLSI Multiplier for GF(2m),” IEEE Journal on Selected Areas in Communications, Vol. SAC-4, pp. 62-66, 1986.

    Article  Google Scholar 

  11. S.K. Jain and K.K. Parhi, “Low latency standard basis GF(2m) multiplier and squarer architectures,” Proc. IEEE ICASSP, Detroit (MI), pp. 2747-2750, 1995.

  12. L. Song and K.K. Parhi, “Efficient finite field serial/parallel multiplication,” Proc. of International Conf. on Application Specific Systems, Architectures and Processors, Chicago, pp. 72-82, 1996.

  13. S.K. Jain, L. Song, and K.K. Parhi, “Efficient semi-systolic architectures for finite field arithmetic,” IEEE Trans. on VLSI Systems, (to appear).

  14. E.R. Berlekamp, “Bit serial Reed-Solomon encoders,” IEEE Trans. inform. Theory, Vol. IT-28, pp. 869-874, 1982.

    Article  Google Scholar 

  15. S.T.J. Fenn, M. Benaissa, and D. Taylor, “GF(2m) multiplication and division over the dual basis,” IEEE Trans. Comput., Vol. 45, pp. 319-327, 1996.

    Article  MATH  Google Scholar 

  16. C.C. Wang et al., “VLSI architectures for computing multiplications and inverses in GF(2m),” IEEE Trans. Comput., Vol C-34, pp. 709-716, 1985.

    Article  Google Scholar 

  17. I.S. Hsu, T.K. Truong, L.J. Deutsch, and I.S. Reed, “A comparison of VLSI architecture of finite field multipliers using dual, normal, or standard bases,” IEEE Trans. Comput., Vol. 37, pp. 735-739, 1988.

    Article  Google Scholar 

  18. A.P. Chandrakasan and R.W. Brodersen, Low Power Digital CMOS Design, Kluwer Academic Publishers, 1995.

  19. L.B. Jackson, J.F. Kaiser, and H.S. McDonald, “An approach to implementation of digital filters,” IEEE Trans. on Audio Electroacoust, Vol. 16, pp. 413-421, 1968.

    Article  Google Scholar 

  20. P.B. Denyer and D. Renshaw, VLSI Signal Processing: A Bit-Serial Approach, Addison Wesley, 1986.

  21. K.K. Parhi, “A systematic approach for design of digit-serial signal processing architectures,” IEEE Trans. Circuits and Systems, Vol. 38, pp. 358-375, 1991.

    Article  Google Scholar 

  22. R.I. Hartley and K.K. Parhi, Digit-Serial Computation, Kluwer Academic Publishers, 1995.

  23. K.K. Parhi and D.G. Messerschmitt, “Static rate-optimal scheduling of iterative data-flow programs via optimum unfolding,” IEEE Trans. Comput., Vol. 40, pp. 178-195, 1991.

    Article  Google Scholar 

  24. K.K. Parhi, C.-Y. Wang, and A.P. Brown, “Synthesis of control circuits in folded pipelined DSP architectures,” IEEE Journal of Solid-State Circuits, Vol. 27, pp. 29-43, 1992.

    Article  Google Scholar 

  25. J.H. Satyanarayana and K.K. Parhi, “HEAT: Hierarchical energy analysis tool,” Proc. 33rd ACM/IEEE Design Automation Conf., Las Vegas, pp. 9-14, 1996.

  26. J.M. Rabaey and M. Pedram, Low Power Design Methodologies, Kluwer Academic Publishers, 1996.

  27. W.W. Peterson and E.J. Weldon, Error-Correcting Codes, The MIT Press, 1972.

  28. R.J. McEliece, Finite Fields for Computer Scientists and Engineers, Kluwer Academic Publishers, 1987.

  29. V.K. De, W. Lee, and K. Itoh, “Low voltage technologies and circuits: An industrial perspective,” International Symposium on Low Power Electronics and Design'97, Tutorial No. 1, 1997.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Song, L., Parhi, K.K. Low-Energy Digit-Serial/Parallel Finite Field Multipliers. The Journal of VLSI Signal Processing-Systems for Signal, Image, and Video Technology 19, 149–166 (1998). https://doi.org/10.1023/A:1008013818413

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008013818413

Keywords

Navigation