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Issue No.01 - Jan. (2013 vol.62)
pp: 193-199
J. Adikari , University of Waterloo, Waterloo
A. Barsoum , University of Waterloo, Waterloo
M.A. Hasan , University of Waterloo, Waterloo
A.H. Namin , University of Waterloo, Waterloo
C. Negre , Universite de Perpignan, Perpignan, and Universite Montpellier 2, Montpellier
ABSTRACT
In this paper, we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases. The schemes are based on a recently proposed method known as block recombination, which efficiently computes the sum of two products of Toeplitz matrices and vectors. Specifically, here we take advantage of some structural properties of the matrices and vectors involved in the formulation of field multiplication using optimal normal bases. This yields new space and time complexity results for corresponding bit parallel multipliers.
INDEX TERMS
Complexity theory, Computer architecture, Logic gates, Symmetric matrices, Delay, Matrix decomposition, Polynomials, block recombination, Binary field, optimal normal basis, Toeplitz matrix
CITATION
J. Adikari, A. Barsoum, M.A. Hasan, A.H. Namin, C. Negre, "Improved Area-Time Tradeoffs for Field Multiplication Using Optimal Normal Bases", IEEE Transactions on Computers, vol.62, no. 1, pp. 193-199, Jan. 2013, doi:10.1109/TC.2011.198
REFERENCES
 [1] D.J. Bernstein, “Batch Binary Edwards,” Proc. 29th Ann. Int'l Cryptology Conf. Advances in Cryptology (CRYPTO '09), pp. 317-336, 2009. [2] D.J. Bernstein and T. Lange, “Type-II Optimal Polynomial Bases,” Proc. Third Int'l Conf. Arithmetic of Finite Fields (WAIFI '10), pp. 41-61, 2010. [3] H. Fan and M.A. Hasan, “A New Approach to Sub-quadratic Space Complexity Parallel Multipliers for Extended Binary Fields,” IEEE Trans. Computers, vol. 56, no. 2, pp. 224-233, Feb. 2007. [4] H. Fan and M.A. Hasan, “Subquadratic Computational Complexity Schemes for Extended Binary Field Multiplication Using Optimal Normal Bases,” IEEE Trans. Computers, vol. 56, no. 10, pp. 1435-1437, Oct. 2007. [5] H. Fan, J. Sun, M. Gu, and K.-Y. Lam, “Overlap-Free Karatsuba-Ofman Polynomial Multiplication Algorithm,” IET Information Security, vol. 4, pp. 8-14, Mar. 2010. [6] P. Gallagher and C. Furlani, “FIPS PUB 186-3, Digital Signature Standard (DSS),” 2009. [7] S. Gao and H.W. Lenstra, “Optimal Normal Bases,” Design, Codes and Cryptography, vol. 2, pp. 315-323, Dec. 1992. [8] M.A. Hasan, N. Méloni, A.H. Namin, and C. Negre, “Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product,” IEEE Trans. Comp., To appear. [9] M.A. Hasan, M. 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. [10] C.K. Koc 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] M. Leone, “A New Low Complexity Parallel Multiplier for a Class of Finite Fields,” Proc. Third Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '01), pp. 160-170, 2001. [12] J.L. Massey and J.K. Omura, “Computational Method and Apparatus for Finite Field Arithmetic,” US patent number 4,587,627, Washington, D.C., 1984 [13] R.C. Mullin, S.A. Vanstone, and R.M. Wilson, “Optimal Normal Bases in GF($p^{n}$ ),” Discrete Applied Math., vol. 22, no. 2, pp. 149-161, 1989. [14] B. Sunar and Ç. K. Koç, “An Efficient Optimal Normal Basis Type II Multiplier,” IEEE Trans. Computers, vol. 50, no. 1, pp. 83-87, 2001. [15] J. von zur Gathen, A. Shokrollahi, and J. Shokrollahi, “Efficient Multiplication Using Type 2 Optimal Normal Bases,” Proc. First Int'l Workshop Arithmetic of Finite Fields (WAIFI '07), pp. 55-68, 2007.