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Issue No.05 - May (2013 vol.62)
pp: 929-943
A. Cilardo , Dept. of Comput. Sci., Univ. of Naples Federico II, Naples, Italy
ABSTRACT
Numerous works have addressed efficient parallel GF(2m) multiplication based on polynomial basis or some of its variants. For those field degrees where neither irreducible trinomials nor Equally Spaced Polynomials (EPSs) exist, the best area/time performance has been achieved for special-type irreducible pentanomials, which however do not exist for all degrees. In other words, no multiplier architecture has been proposed so far achieving the best performance and, at the same time, being general enough to support any field degrees. In this paper, we propose a new representation, based on what we called Generalized Polynomial Bases (GPBs), covering polynomial bases and the so-called Shifted Polynomial Bases (SPBs) as special cases. In order to study the new representation, we introduce a novel formulation for polynomial basis and its variants, which is able to express concisely all implementation aspects of interest, i.e., gate count, subexpression sharing, and time delay. The methodology enabled by the new formulation is completely general and repetitive in its application, allowing the development of an ad-hoc software tool to derive proofs for area complexity and time delays automatically. As the central contribution of this paper, we introduce some new types of irreducible pentanomials and an associated GPB. Based on the above formulation, we prove that carefully chosen GPBs yield multiplier architectures matching, or even outperforming, the best special-type pentanomials from both the area and time point of view. Most importantly, the proposed GPB architectures require pentanomials existing for all degrees of practical interest. A list of suitable irreducible pentanomials for all degrees less than 1,000 is given in the appendix (Fig. 5 and Tables 4-11 are provided in a separate file containing the body of Appendix, which can be found on the Computer Society Digital Library at >http://doi.ieeecomputersociety.org/10.1109/TC.2012.63).
INDEX TERMS
digital arithmetic, computational complexity, computer architecture, computer society digital library, fast parallel GF(2m) polynomial multiplication, irreducible trinomials, equally spaced polynomials, EPS, special-type irreducible pentanomials, multiplier architecture, generalized polynomial bases, shifted polynomial bases, SPB, ad-hoc software tool, area complexity, time delays, GPB architectures, Polynomials, Logic gates, Delay, Delay effects, Computer architecture, Vectors, parallel $({GF}(2^m))$ multiplication, Polynomials, Logic gates, Delay, Delay effects, Computer architecture, Vectors, irreducible binary pentanomials, $({GF}(2^m))$ multiplication, polynomial basis, shifted polynomial basis
CITATION
A. Cilardo, "Fast Parallel GF(2^m) Polynomial Multiplication for All Degrees", IEEE Transactions on Computers, vol.62, no. 5, pp. 929-943, May 2013, doi:10.1109/TC.2012.63
REFERENCES
 [1] R. Lidl and H. Niederreiter, Finite Fields, second ed. Cambridge Univ. Press, 1997. [2] I. Blake, G. Seroussi, and N. Smart, Elliptic Curves in Cryptography. Cambridge Univ. Press, 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] T. Zhang and 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. [5] A. Reyhani-Masoleh and M.A. Hasan, "Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over ${GF}(2^m)$ ," IEEE Trans. Computers, vol. 53, no. 8, pp. 945-959, Aug. 2004. [6] G. Seroussi, "Table of Low-Weight Binary Irreducible Polynomials," Technical Report HPL-98- 135, Hewlett-Packard Laboratones, Aug. 1998. [7] F. Rodríguez-Henríquez and Ç.K. Koç, "Parallel Multipliers Based on Special Irreducible Pentanomials," IEEE Trans. Computers, vol. 52, no. 12, pp. 1535-1542, Dec. 2003. [8] J. Imaña, R. Hermida, and F. Tirado, "Low Complexity Bit-Parallel Multipliers Based on a Class of Irreducible Pentanomials," IEEE Trans. Very Large Scale Integration Systems, vol. 14, no. 12, pp. 1388-1393, Dec. 2006. [9] H. Fan and M.A. Hasan, "Fast Bit Parallel-Shifted Polynomial Basis Multipliers in ${GF}(2^n)$ ," IEEE Trans. Circuits and Systems I: Regular Papers, vol. 53, no. 12, pp. 2606-2615, Dec. 2006. [10] C. Negre, "Quadrinomial Modular Arithmetic Using Modified Polynomial Basis," Proc. Int'l Conf. Information Technology: Coding and Computing (ITCC '05), vol. 1, pp. 550-555, Apr. 2005. [11] H. Fan and Y. Dai, "Fast Bit Parallel ${GF}(2^m)$ Multiplier for All Trinomials," IEEE Trans. Computers, vol. 54, no. 4, pp. 485-490, Apr. 2005. [12] S.-M. Park, K.-Y. Chang, and D. Hong, "Efficient Bit-Parallel Multiplier for Irreducible Pentanomials Using a Shifted Polynomial Basis," IEEE Trans. Computers, vol. 55, no. 9, pp. 1211-1215, Sept. 2006. [13] A. Cilardo, "Efficient Bit-Parallel ${GF}(2^m)$ Multiplier for a Large Class of Irreducible Pentanomials," IEEE Trans. Computers, vol. 58, no. 7, pp. 1001-1008, July 2009. [14] E.D. Mastrovito, "VLSI Architectures for Multiplication over Finite Field ${\rm GF}(2^m)$ ," Proc. Sixth Int'l Conf. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes (AAECC-6), pp. 297-309, July 1988. [15] S.S. Erdem, T. Yanik, and Ç.K. Koç, "Polynomial Basis Multiplication over $GF(2^m)$ ," Acta Applicandae Mathematicae, vol. 93, no. 1, pp. 33-55, 2006. [16] A. Reyhani-Masoleh, "A New Bit-Serial Architecture for Field Multiplication Using Polynomial Bases," Proc. 10th Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '08), 10th Int'l Workshop, pp. 300-314, Aug. 2008. [17] A. Hariri and A. Reyhani-Masoleh, "Bit-Serial and Bit-Parallel Montgomery Multiplication and Squaring over $GF(2^m)$ ," IEEE Trans. Computers, vol. 58, no. 10, pp. 1332-1345, Oct. 2009. [18] Digital Signature Standard (DSS), Nat'l Inst. of Standards and Technology (NIST) Std. Fed. Information Processing Standards Publication 186-2, Feb. 2000. [19] A. Cilardo, N. Mazzocca, and A. Mazzeo, "Representation of Elements in ${F}_{2^m}$ Enabling Unified Field Arithmetic for Elliptic Curve Cryptography," Electronics Letters, vol. 41, no. 14, pp. 798-800, July 2005.
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