This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product
February 2012 (vol. 61 no. 2)
pp. 151-163
M. Anwar Hasan, University of Waterloo, Waterloo
Nicolas Méloni, Université de Toulon-Var, Toulon
Ashkan Hosseinzadeh Namin, University of Waterloo, Waterloo
Christophe Negre, University of Waterloo, Waterloo and Université de Perpignan, Perpignan
In this paper, we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity. The method is based on decomposing the building blocks of the Fan-Hasan subquadratic Toeplitz matrix-vector multiplier. We reduce the space complexity of their architecture by recombining the building blocks. In comparison to other similar schemes available in the literature, our proposal presents a better space complexity while having the same time complexity. We also show that block recombination can be used for efficient implementation of the GHASH function of Galois Counter Mode (GCM).

[1] J.-C. Bajard, L. Imbert, and G.A. Jullien, "Parallel Montgomery Multiplication in GF($2^{{\rm k}}$ ) Using Trinomial Residue Arithmetic," Proc. IEEE Symp. Computer Arithmetic, pp. 164-171, 2005.
[2] D.J. Bernstein, "Batch Binary Edwards," CRYPTO '09: Proc. Conf. Advances in Cryptology, pp. 317-336, 2009.
[3] A.E. Cohen and K.K. Parhi, "Implementation of Scalable Elliptic Curve Cryptosystem Crypto-Accelerators for GF($2^m$ )," Proc. 13th Asilomar Conf. Signals, Systems, and Computers, pp. 471-477, 2004.
[4] H. Fan and M.A. Hasan, "A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields," IEEE Trans. Computers, vol. 56, no. 2, pp. 224-233, Sept. 2007.
[5] H. Fan, J. Sun, M. Gu, and K.-Y. Lam, "Overlap-Free Karatsuba-Ofman Polynomial Multiplication Algorithms," Information Security, IET, vol. 4, pp. 8-14, Mar. 2010.
[6] M.A. Hasan and V.K. Bhargava, "Division and Bit-Serial Multiplication over GF($q^m$ )," Proc. IEEE Computers and Digital Techniques, pp. 230-236, may 1992.
[7] T. Itoh and S. Tsujii, "Structure of Parallel Multipliers for a Class of Fields ${\rm GF}(2^m)$ ," Information and Computation, vol. 83, pp. 21-40, 1989.
[8] A. Karatsuba and Y. Ofman, "Multiplication of Multidigit Numbers on Automata," Soviet Physics-Doklady, vol. 7, no. 7, pp. 595-596, 1963.
[9] N. Koblitz, "Elliptic Curve Cryptosystems," Math. Computation, vol. 48, pp. 203-209, 1987.
[10] E. Mastrovito, "VLSI Designs for Multiplication over Finite Fields GF($2^m$ )," Proc. Sixth Int'l Conf. Applied Algebra, Algebraic Algorithm, and Error-Correcting Codes (AAECC-6), pp. 297-309, 1988.
[11] D.A. McGrew and J. Viega, "The Security and Performance of the Galois/Counter Mode (GCM) of Operation," Proc. INDOCRYPT, pp. 343-355, 2004.
[12] V. Miller, "Use of Elliptic Curves in Cryptography," CRYPTO '85: Proc. Advances in Cryptology, pp. 417-426, 1986.
[13] 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. 856-861, July 1996.
[14] A. Reyhani-Masoleh and M.A. Hasan, "A New Construction of Massey-Omura Parallel Multiplier over GF($2^{m}$ )," IEEE Trans. Computers, vol. 51, no. 5, pp. 511-520, May 2002.
[15] F. Rodriguez-Henriquez and C.K. Koç, "Parallel Multipliers Based on Special Irreducible Pentanomials," IEEE Trans. Computers, vol. 52, no. 12, pp. 1535-1542, Dec. 2003.
[16] A. Satoh, "High-Speed Parallel Hardware Architecture for Galois Counter Mode," Proc. IEEE Symp. Circuits and Systems (ISCAS), pp. 1863-1866, 2007.
[17] B. Sunar, "A Generalized Method for Constructing Subquadratic Complexity GF($2^k$ ) Multipliers," IEEE Trans. Computers, vol. 53, no. 9, pp. 1097-1105, Sept. 2004.
[18] B. Sunar and C. Koc, "Mastrovito Multiplier for All Trinomials," IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[19] S. Winograd, Arithmetic Complexity of Computations. Soc. For Industrial & Applied Math., 1980.

Index Terms:
Binary field, subquadratic space complexity multiplier, Toeplitz matrix, block recombination.
Citation:
M. Anwar Hasan, Nicolas Méloni, Ashkan Hosseinzadeh Namin, Christophe Negre, "Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product," IEEE Transactions on Computers, vol. 61, no. 2, pp. 151-163, Feb. 2012, doi:10.1109/TC.2010.276
Usage of this product signifies your acceptance of the Terms of Use.