This Article 
 Bibliographic References 
 Add to: 
Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields GF(2/sup m/)
August 1992 (vol. 41 no. 8)
pp. 962-971

Structures for parallel multipliers of a class of fields GF(2/sup m/) based on irreducible all one polynomials (AOP) and equally spaced polynomials (ESP) are presented. The structures are simple and modular, which is important for hardware realization. Relationships between an irreducible AOP and the corresponding irreducible ESPs have been exploited to construct ESP-based multipliers of large fields by a regular expansion of the basic modules of the AOP-based multiplier of a small field. Some features of the structures also enable a fast implementation of squaring and multiplication algorithms and therefore make fast exponentiation and inversion possible. It is shown that, if for a certain degree, an irreducible AOP as well as an irreducible ESP exist, then from the complexity point of view, it is advantageous to use the ESP-based parallel multiplier.

[1] R. Lidl and H. Niederreiter,Introduction to Finite Fields and Their Applications. Cambridge, MA: Cambridge Univ., 1986.
[2] C. C. Wang, T. K. Truong, H. M. Shao, L. J. Deutsch, J. K. Omura, and I. S. Reed, "VLSI architecture for computing multiplications and inverses in GF(2m),"IEEE Trans. Comput., vol. C-34, pp. 709-716, Aug. 1985.
[3] T. Itoh and S. Tsujii, "Structure of parallel multipliers for a class of fields GF(2m),"Inform. Comp., vol. 83, pp. 21-40, 1989.
[4] P. K. S. Wah and M. Z. Wang, "Realization and application of the Massey-Omura lock," inProc. International Zurich Seminar, Switzerland, 1984.
[5] C. E. Stroud and A. E. Barbour, "Design for testability and test generation for static redundancy system level fault-tolerant circuits," inProc. IEEE Int. Test Conf., Washington DC, Aug. 29-31, 1989, pp. 812-818.
[6] Y. Sugiyama, "An algorithm for solving discrete-time Wiener-Hopf equations based on Euclid's algorithm,"IEEE Trans. Inform. Theory, vol. IT-32, pp. 394-409, May 1986.
[7] M. A. Hasan and V. K. Bhargava, "Division and bit-serial multiplication over GF(qm),"IEE Proc., part E, vol. 139, no. 3, pp. 230-236, May 1992.
[8] T. Itoh. "A fast algorithm for computing multiplicative inverses in GF(2m),"Inform. Comp., vol. 78, pp. 171-177, Sept. 1988.
[9] M. Z. Wang, I. F. Blake, and V. K. Bhargava, "Normal bases and irreducible polynomials in the finite field GF(22r),"Disc. Appl. Math., 1990, submitted for publication.
[10] T. Itoh. "Characterization for a family of infinitely many irreducible equally spaced polynomials,"Inform. Process. Lett., vol. 37, pp. 272-277, Mar. 1991.
[11] D. Shanks,Solved and Unsolved Problems in Number Theory, vol. 1, Washington, DC, 1962.
[12] A. Pincin, "A new algorithm for multiplication in finite fields,"IEEE Trans. Comput., vol. C-38, pp. 1045-1049, July 1989.

Index Terms:
squaring algorithms; parallel multipliers; finite fields; irreducible all one polynomials; equally spaced polynomials; multiplication algorithms; exponentiation; inversion; complexity; computational complexity; digital arithmetic; multiplying circuits; number theory.
M.A. Hasan, M. Wang, V.K. Bhargava, "Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields GF(2/sup m/)," IEEE Transactions on Computers, vol. 41, no. 8, pp. 962-971, Aug. 1992, doi:10.1109/12.156539
Usage of this product signifies your acceptance of the Terms of Use.