This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Equally Spaced Polynomials, Dual Bases, and Multiplication in F2^n
May 2002 (vol. 51 no. 5)
pp. 588-591

A recently proposed multiplier for finite fields given by equally spaced polynomials is based on the transformation from the polynomial basis to its dual basis, combined with multiplication by a constant. We classify the constants that are optimal regarding the cost of this operation and investigate the cost of the inverse transformation.

[1] E.R. Berlekamp,"Bit-Serial Reed-Solomon Encoders," IEEE Trans. Information Theory, vol. 28, pp. 869-874, Nov. 1982.
[2] P. Gaudry, F. Hess, and N. Smart, “Constructive and Destructive Facets of Weil Descent on Elliptic Curves,” J. Cryptology, vol. 15, pp. 19-46, 2002.
[3] D. Gollmann, “Dual Bases and Bit-Serial Multiplication in${\rm F}_{q^n}$,” Theoretical Computer Science, vol. 226, pp. 45-59, Sept. 1999.
[4] M.A. Hasan and V.K. Bhargava,"Division and Bit-Serial Multiplication overGF(qm)," IEE Proc. E., vol. 139, pp. 230-236, May 1992.
[5] M. Hasan and V. Bhargava, “Low Complexity Architecure for Exponentiation in$GF(2^m)$,” Electronics Letters, vol. 28, pp. 1,984-1,986, Oct. 1992.
[6] M.A. Hasan and V.K. Bhargava, "Architecture for a Low Complexity Rate-Adaptive Reed-Solomon Encoder," IEEE Trans. Computers, vol. 44, no. 7, pp. 938-942, July 1995.
[7] M.A. Hasan, M. Wang, and V.K. Bhargava, Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields$GF(2^m)$ IEEE Trans. Computers, vol. 41, no. 8, pp. 962-971, Aug. 1992.
[8] J.J. Komo and W.J. Reid III, “Generation of Canonicalm-Sequences and Dual Bases,” IEEE Trans. Information Theory, vol. 42, no. 2, pp. 638-641, Mar. 1996.
[9] R. Lidl and H. Niederreiter,An Introduction to Finite Fields and Their Applications.Cambridge: Cambridge Univ. Press, 1986.
[10] M. Wang and I.F. Blake,"Bit-Serial Multiplication in Finite Fields," SIAM J. Discrete Maths., vol. 3, pp. 140-148, Feb. 1990.
[11] H. Wu, “Efficient Computations in Finite Fields with Cryptographic Significance,” PhD thesis, Dept. of Electrical and Computer Eng., Waterloo, Ontario, Canada, 1998.
[12] H. Wu and M.A. Hasan, "Low Complexity Bit-parallel Multipliers for a Class of Finite Fields," IEEE Trans. Computers, vol. 47, no. 8, pp. 883-887, Aug. 1998.

Index Terms:
Equally spaced polynomials, dual bases, trace, multipliers for {\rm F}_{2^n}
Citation:
D. Gollmann, "Equally Spaced Polynomials, Dual Bases, and Multiplication in F2^n," IEEE Transactions on Computers, vol. 51, no. 5, pp. 588-591, May 2002, doi:10.1109/TC.2002.1004597
Usage of this product signifies your acceptance of the Terms of Use.