This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Bit-Parallel Finite Field Multipliers for Irreducible Trinomials
May 2006 (vol. 55 no. 5)
pp. 520-533
A new formulation for the canonical basis multiplication in the finite fields GF(2^m) based on the use of a triangular basis and on the decomposition of a product matrix is presented. From this algorithm, a new method for multiplication (named transpositional) applicable to general irreducible polynomials is deduced. The transpositional method is based on the computation of 1-cycles and 2--cycles given by a permutation defined by the coordinate of the product to be computed and by the cardinality of the field GF(2^m). The obtained cycles define groups corresponding to subexpressions that can be shared among the different product coordinates. This new multiplication method is applied to five types of irreducible trinomials. These polynomials have been widely studied due to their low-complexity implementations. The theoretical complexity analysis of the corresponding bit-parallel multipliers shows that the space complexities of our multipliers match the best results known to date for similar canonical GF(2^m) multipliers. The most important new result is the reduction, in two of the five studied trinomials, of the time complexity with respect to the best known results.

[1] 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.
[2] A. Halbutogullari and Ç.K. Koç, “Mastrovito Multiplier for General Irreducible Polynomials,” Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 498-507, 1999.
[3] A. Halbutogullari and Ç.K. Koç, “Mastrovito Multiplier for General Irreducible Polynomials,” IEEE Trans. Computers, vol. 49, no. 5, pp. 503-518, May 2000.
[4] M.A. Hasan, “Double-Basis Multiplicative Inversion over $GF(2^m)$ ,” IEEE Trans. Computers, vol. 47, no. 9, pp. 960-970, Sept. 1998.
[5] M.A. Hasan and V.K. Bhargava, “Architecture for a Low Complexity Rate-Adaptive Reed-Solomon Encoder,” IEEE Trans. Computers, vol. 44, no. 6, pp. 938-942, June 1995.
[6] M.A. Hasan, M.Z. 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.
[7] M.A. Hasan, M.Z. Wang, and V.K. Bhargava, “A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields,” IEEE Trans. Computers, vol. 42, no. 10, pp. 1278-1280, Oct. 1993.
[8] J.L. Imaña and J.M. Sánchez, “A New Reconfigurable-Oriented Method for Canonical Basis Multiplication Over a Class of Finite Fields $GF(2^m)$ ,” Proc. 13th Int'l Conf. Field Programmable Logic and Applications, pp. 1127-1130, 2003.
[9] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Finite Fields $GF(2^m)$ ,” Information and Computation, vol. 83, pp. 21-40, 1989.
[10] Ç.K. Koç and B. Sunar, “Low-Complexity Bit-Parallel Canonical and Normal Basis Multipliers for a Class of Finite Fields,” IEEE Trans. Computers, vol. 47, no. 3, pp. 353-356, Mar. 1998.
[11] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications. New York: Cambridge Univ. Press, 1994.
[12] E.D. Mastrovito, “VLSI Architectures for Multiplication over Finite Fields $GF(2^m)$ ,” Proc. Sixth Int'l Conf. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes (AAECC-6), pp. 297-309, July 1988.
[13] Applications of Finite Fields, A.J. Menezes, ed. Boston: Kluwer Academic, 1993.
[14] J. Omura and J. Massey, “Computational Method and Apparatus for Finite Field Arithmetic,” US Patent Number 4,587,627, May 1986.
[15] K.K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation. John Wiley & Sons, 1999.
[16] A. Reyhani-Masoleh and M.A. Hasan, “On Low Complexity Bit Parallel Polynomial Basis Multipliers,” Proc. Workshop on Cryptographic Hardware and Embedded Systems (CHES 2003), pp. 189-202, 2003.
[17] B. Sunar and Ç.K. Koç, “Mastrovito Multiplier for All Trinomials,” IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[18] H. Wu, “Bit-Parallel Finite Field Multiplier and Squarer Using Polynomial Basis,” IEEE Trans. Computers, vol. 51, no. 7, pp. 750-758, July 2002.
[19] 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.
[20] T. Zhang and K.K. Parhi, “Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials,” IEEE Trans. Computers, vol. 50, no. 7, pp. 734-749, July 2001.

Index Terms:
Finite (or Galois) fields, multiplication, canonical basis, irreducible trinomials, complexity, triangular basis, matrix decomposition, permutation, cycles, transpositions.
Citation:
Jos? Luis Ima?, Juan Manuel S?nchez, Francisco Tirado, "Bit-Parallel Finite Field Multipliers for Irreducible Trinomials," IEEE Transactions on Computers, vol. 55, no. 5, pp. 520-533, May 2006, doi:10.1109/TC.2006.69
Usage of this product signifies your acceptance of the Terms of Use.