This Article 
 Bibliographic References 
 Add to: 
An Architecture for Computing Zech's Logarithms in GF(2m)
May 2000 (vol. 49 no. 5)
pp. 519-524

Abstract—In this paper, a new method for calculation of Zech's logarithm in $GF(2^m)$ is presented. For a given element, the logarithm is calculated by bit operations performed on its binary representation. No look-up tables are used. The proposed method makes feasible the implementation of an universal-type device for finite field arithmetic.

[1] R.E. Blahut, Theory and Practice of Error Control Codes. Owego, N.Y.: Addison-Wesley, 1983.
[2] T. ElGamal, A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms IEEE Trans. Information Theory, vol. 31, no. 4, pp. 469-472, 1985.
[3] S.T.J. Fenn, M. Benaissa, and D. Taylor, $GF(2^m)$Multiplication and Division over the Dual Basis IEEE Trans. Computers, vol. 45, no. 3, pp. 319-327, Mar. 1996.
[4] C. Paar, Some Remarks on Efficient Inversion in Finite Fields Proc. 1995 Int'l Symp. Information Theory, 1995.
[5] C. Paar, “A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields,” IEEE Trans. Computers, vol. 45, no. 7, pp. 846-861, July 1996.
[6] E.R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
[7] N.J. Sloane and F.J. MacWilliams, The Theory of Error-Correcting Codes. Amsterdam, New York: North Holland, 1977.
[8] K. Imamura, “A Method for Computing Addition Tables in$GF(p^m)$,” IEEE Trans. Information Theory, vol. 26, no. 3, pp. 367-369, May 1980.
[9] K. Huber, “Some Comments on Zech's Logarithms,” IEEE Trans. Information Theory, vol. 36, no. 4, pp. 946-950, July 1990.
[10] K. Huber, “Solving Equations in Finite Fields and Some Results Concerning the Structure of$GF(p^m)$,” IEEE Trans. Information Theory, vol. 38, no. 3, pp. 1,154-1,162, May 1992.
[11] K.Y. Siu and J. Bruck, “Neural Computation of Arithmetic Functions,” Proc. IEEE, vol. 78, no. 10, pp. 1,669-1,675, Oct. 1990.
[12] K.-Y. Siu, V.P. Roychowdhury, and T. Kailath, “Depth-Size Tradeoffs for Neural Computation,” IEEE Trans. Computers, vol. 40, no. 12, pp. 1,402-1,412, Dec. 1991.
[13] J. Bruck, T. Kailath, K.-Y. Siu, and T. Hofmeister, “Depth Efficient Neural Networks for Division and Related Problems,” IEEE Trans. Information Theory, vol. 39, no. 3, pp. 946-956, May 1993.
[14] T. Kailath, K.Y. Siu, and V. Roychowdhury, Discrete Neural Computation, A Theoretical Foundation. Englewood Cliffs, N.J.: Prentice Hall, 1995.

Index Terms:
Zech logarithms, finite fields, discrete neural networks.
Francisco M. Assis, C.e. Pedreira, "An Architecture for Computing Zech's Logarithms in GF(2m)," IEEE Transactions on Computers, vol. 49, no. 5, pp. 519-524, May 2000, doi:10.1109/12.859543
Usage of this product signifies your acceptance of the Terms of Use.