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A. ReyhaniMasoleh, M.A. Hasan, "A New Construction of MasseyOmura Parallel Multiplier over GF(2^{m})," IEEE Transactions on Computers, vol. 51, no. 5, pp. 511520, May, 2002.  
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@article{ 10.1109/TC.2002.1004590, author = {A. ReyhaniMasoleh and M.A. Hasan}, title = {A New Construction of MasseyOmura Parallel Multiplier over GF(2^{m})}, journal ={IEEE Transactions on Computers}, volume = {51}, number = {5}, issn = {00189340}, year = {2002}, pages = {511520}, doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2002.1004590}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  A New Construction of MasseyOmura Parallel Multiplier over GF(2^{m}) IS  5 SN  00189340 SP511 EP520 EPD  511520 A1  A. ReyhaniMasoleh, A1  M.A. Hasan, PY  2002 KW  Finite field KW  MasseyOmura multiplier KW  allone polynomial KW  optimal normal bases VL  51 JA  IEEE Transactions on Computers ER   
The MasseyOmura multiplier of GF(2^{m}) uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible allone polynomials, the parallel MasseyOmura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multipliers contain redundancy in that special class of finite fields, but it also has redundancy in fields GF(2^{m}) defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original MasseyOmura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible allone polynomials, the multiplier's circuit complexity matches the best result available in the open literature.
[1] G.B. Agnew, T. Beth, R.C. Mullin, and S.A. Vanstone, “Arithmetic Operations in$GF(2^m)$,” J. Cryptology, vol. 6, pp. 313, 1993.
[2] D.W. Ash, I.F. Blake, and S.A. Vanstone, “Low Complexity Normal Bases,” Discrete Applied Math., vol. 25, pp. 191210, 1989.
[3] G. Drolet, “A New Representation of Elements of Finite Fields$\big. {\rm GF}(2^m)\bigr.$Yielding Small Complexity Arithmetic Circuits,” IEEE Trans. Computers, vol. 47, no. 9, pp. 938946, Sept. 1998.
[4] M. Elia, M. Leone, and C. Visentin, “Low Complexity BitParallel Multipliers for$GF(2^m)$with Generator Polynomial$x^m+x^k+1$,” Electronics Letters, vol. 35, no. 7, pp. 551552, Apr. 1999.
[5] 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. 319327, Mar. 1996.
[6] S. Gao and H.W. Lenstra Jr., Optimal Normal Bases Designs, Codes and Cryptography, vol. 2, pp. 315323, 1992.
[7] J.H. Guo and C.L. Wang, Systolic Array Implementation of Euclid's Algorithm for Inversion and Division in$GF(2^m)$ IEEE Trans. Computers, vol. 47, no. 10, pp. 11611167, Oct. 1998.
[8] A. Halbutogullari and C.K. Koc, Mastrovito Multiplier for General Irreducible Polynomials IEEE Trans. Computers, vol. 49, no. 5, pp. 503518, May 2000.
[9] 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. 962971, Aug. 1992.
[10] M.A. Hasan, M.Z. Wang, and V.K. Bhargava, “A Modified MasseyOmura Parallel Multiplier for a Class of Finite Fields,” IEEE Trans. Computers, vol. 42, no. 10, pp. 12781280, Oct. 1993.
[11] I.S. Hsu,T.K. Truong,L.J. Deutsch, and I.S. Reed,"A Comparison of VLSI Architectures of Finite Field Multipliers Using Dual, Normal or Standard Bases," IEEE Trans. Computers, vol. 37, no. 6, pp. 735737, June 1988.
[12] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Finite Fields$GF(2^m)$,” Information and Computation, vol. 83, pp. 2140, 1989.
[13] Ç.K. Koç and B. Sunar, LowComplexity BitParallel Canonical and Normal Basis Multipliers for a Class of Finite Fields IEEE Trans. Computers, vol. 47, no. 3, pp. 353356, Mar. 1998.
[14] R. Lidl and H. Niederreiter,An Introduction to Finite Fields and Their Applications.Cambridge: Cambridge Univ. Press, 1986.
[15] J.L. Massey and J.K. Omura, Computational Method and Apparatus for Finite Field Arithmetic, US Patent No. 4,587,627, to OMNET Assoc., Sunnyvale CA, Washington, D.C.: Patent and Trademark Office, 1986.
[16] E.D. Mastrovito,"VLSI Design for Multiplication over Finite Fields," LNCS357, Proc. AAECC6, pp. 297309,Rome, July 1988, SpringerVerlag.
[17] E.D. Mastrovito, “VLSI Architectures for Computation in Galois Fields,” PhD thesis, Linkoping Univ., Linkoping, Sweden, 1991.
[18] A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer Academic, 1993.
[19] R.C. Mullin,I.M. Onyszchuk,S.A. Vanstone, and R.M. Wilson,"Optimal Normal Bases inGF(pn)," Discrete Applied Maths., pp. 142169, 1988/89.
[20] 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. 846861, July 1996.
[21] C. Paar, P. Fleischmann, and P. Roelse, “Efficient Multiplier Architectures for Galois Fields,” IEEE Trans. Computers, vol. 47, no. 2, pp. 162170, Feb. 1998.
[22] A. ReyhaniMasoleh and M.A. Hasan, “A Reduced Redundancy MasseyOmura Parallel Multiplier over$GF(2^m)$,” Proc. 20th Biennial Symp. Comm., pp. 308312, May 2000.
[23] B. Sunar and Ç.K. Koç, Mastrovito Multiplier for All Trinomials IEEE Trans. Computers, vol. 48, no. 5, pp. 522527, May 1999.
[24] C.C. Wang,T.K. Truong,H.M. Shao,L.J. Deutsch,J.K. Omura, and I.S. Reed,"VLSI Architectures for Computing Multiplications and Inverses inGF(2m)," IEEE Trans. Computers, vol. 34, no. 8, pp. 709716, Aug. 1985.
[25] H. Wu and M.A. Hasan, Efficient Exponentiation of a Primitive Root in$GF(2^m)$ IEEE Trans. Computers, vol. 46, no. 2, pp. 162172, Feb. 1997.
[26] H. Wu and M.A. Hasan, "Low Complexity Bitparallel Multipliers for a Class of Finite Fields," IEEE Trans. Computers, vol. 47, no. 8, pp. 883887, Aug. 1998.
[27] H. Wu, M.A. Hasan, and I.F. Blake, New LowComplexity BitParallel Finite Field Multipliers Using Weakly Dual Bases IEEE Trans. Computers, vol. 47, no. 11, pp. 12231233, Nov. 1998.