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A Generalized Method for Constructing Subquadratic Complexity GF(2^k) Multipliers
September 2004 (vol. 53 no. 9)
pp. 1097-1105
Berk Sunar, IEEE Computer Society
We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms, the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n=d^k point multiplier with area that is subquadratic and delay that is logarithmic in the bit-length n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method, we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm.

[1] R.E. Blahut, Fast Algorithms for Digital Signal Processing. Reading, Mass.: Addison-Wesley, 1984.
[2] R.E. Blahut, Theory and Practice of Error Control Codes. Reading, Mass.: Addison-Wesley, 1983.
[3] Reed-Solomon Codes and Their Applications, S.B. Wicker and V.K.Bhargava, eds. IEEE Press, 1994.
[4] E.R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
[5] W.W. Peterson and E.J. Weldon Jr., Error-Correcting Codes. Cambrdige, Mass.: MIT Press, 1972.
[6] S.W. Golomb, Shift Register Sequences. San Francisco: Holden-Day, 1967.
[7] R.J. McEliece, Finite Fields for Computer Scientists and Engineers, second ed. Kluwer Academic, 1989.
[8] I.F. Blake, X.H. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghgoobin, Applications of Finite Fields. Kluwer Academic, 1993.
[9] W. Geiselmann and D. Gollmann, Self-Dual Bases in$F_{q^n}$ Designs, Codes, and Cryptographym vol. 3, pp. 333-345, 1993.
[10] 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.
[11] S.T.J. Fenn, M. Benaissa, and D. Taylor, Finite Field Inversion over the Dual Basis IEEE Trans. Very Large Scale Integration (VLSI) Systems, vol. 4, no. 1, pp. 134-136, Mar. 1996.
[12] E.D. Mastrovito, VLSI Architectures for Computation in Galois Fields PhD thesis, Dept. of Electrical Eng., Linköping Univ., Sweden, 1991.
[13] E.D. Mastrovito, VLSI Design for Multiplication over Finite Fields$GF(2^m)$ Proc. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-6), pp. 297-309, Mar. 1989.
[14] 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.
[15] C. Paar, Efficient VLSI Architectures for Bit-Parallel Computation in Galois Fields PhD thesis (English translation), Inst. for Experimental Math., Univ. of Essen, Germany, June 1994.
[16] 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.
[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] B. Sunar and Ç.K. Koç, An Efficient Optimal Normal Basis Type II Multiplier IEEE Trans. Computers, vol. 50, no. 1, pp. 83-87, Jan. 2001.
[19] A. Karatsuba and Y. Ofman, Multiplication of Multidigit Numbers on Automata Soviet Physics-Doklady (English translation), vol. 7, no. 7, pp. 595-596, 1963.
[20] S. Winograd, Some Bilinear Forms Whose Multiplicative Complexity Depends on the Field of Constants Math. Systems Theory, vol. 10, pp. 169-180, 1977.
[21] S. Winograd, Arithmetic Complexity of Computations. SIAM, 1980.
[22] M. Jacobson, A.J. Menezes, and A. Stein, Solving Elliptic Curve Discrete Logarithm Problems Using Weil Descent CACR Technical Report CORR2001-31, Univ. of Waterloo, May 2001.

Index Terms:
Bit-parallel multipliers, finite fields, Winograd convolution.
Berk Sunar, "A Generalized Method for Constructing Subquadratic Complexity GF(2^k) Multipliers," IEEE Transactions on Computers, vol. 53, no. 9, pp. 1097-1105, Sept. 2004, doi:10.1109/TC.2004.52
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