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Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2^{m})
August 2004 (vol. 53 no. 8)
pp. 945-959
ASCII Text
x 
 
Arash Reyhani-Masoleh, M. Anwar Hasan, "Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2^{m})," IEEE Transactions on Computers, vol. 53, no. 8, pp. 945-959, August, 2004.
 
 
BibTex
x
 
@article{ 10.1109/TC.2004.47,
author = {Arash Reyhani-Masoleh and M. Anwar Hasan},
title = {Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2^{m})},
journal ={IEEE Transactions on Computers},
volume = {53},
number = {8},
issn = {0018-9340},
year = {2004},
pages = {945-959},
doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2004.47},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Computers
TI - Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2^{m})
IS - 8
SN - 0018-9340
SP945
EP959
EPD - 945-959
A1 - Arash Reyhani-Masoleh,
A1 - M. Anwar Hasan,
PY - 2004
KW - Finite or Galois field
KW - Mastrovito multiplier
KW - all-one polynomial
KW - polynomial basis
KW - trinomial
KW - pentanomial and equally-spaced polynomial.
VL - 53
JA - IEEE Transactions on Computers
ER -
 

Abstract—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GF(2^{m}). In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix {\bf Q}. The main advantage of this new formulation is that it can be used with any field defining irreducible polynomial. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix {\bf Q}. To the best of our knowledge, this is the first time that these complexities are given in terms of {\bf Q}. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the well-known Mastrovito's multiplier, the proposed architecture has fewer routed signals. In this article, the proposed generalized architecture is further optimized for three special types of polynomials, namely, equally spaced polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic.

[1] G.B. Agnew, T. Beth, R.C. Mullin, and S.A. Vanstone, Arithmetic Operations in$GF(2^m)$ J. Cryptology, vol. 6, pp. 3-13, 1993.
[2] G.B. Agnew, R.C. Mullin, and S.A. Vanstone, An Implementation of Elliptic Curve Cryptosystems over$F_{2^{155}}$ IEEE J. Selected Areas in Comm., vol. 11, no. 5, pp. 804-813, June 1993.
[3] T.C. Bartee and D.I. Schneider, Computation with Finite Fields Information and Computers, vol. 6, pp. 79-98, Mar. 1963.
[4] E.R. Berlekamp, Algebraic Coding Theory. McGraw-Hill, 1968.
[5] R.E. Blahut, Fast Algorithms for Digital Signal Processing. Addison-Wesley, 1985.
[6] T.A. Gulliver, M. Serra, and V.K. Bhargava, The Generation of Primitive Polynomials in$GF(q)$with Independent Roots and Their Application for Power Residue Codes, VLSI Testing and Finite Field Multipliers Using Normal Bases Int'l J. Electronics, vol. 71, no. 4, pp. 559-576, 1991.
[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. 1161-1167, Oct. 1998.
[8] A. Halbutogullari and C.K. Koc, Mastrovito Multiplier for General Irreducible Polynomials IEEE Trans. Computers, vol. 49, no. 5, pp. 503-518, 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. 962-971, Aug. 1992.
[10] T. Itoh and S. Tsujii, Structure of Parallel Mutipliers for a Class of Fields$GF(2^m)$ Information and Computation, vol. 83, pp. 21-40, 1989.
[11] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications. Cambridge Univ. Press, 1994.
[12] E.D. Mastrovito, VLSI Designs for Multiplication over Finite Fields$GF(2^m)$ Proc. Sixth Symp. Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-6), pp. 297-309, July 1988.
[13] E.D. Mastrovito, VLSI Architectures for Computation in Galois Fields PhD thesis, Linkoping Univ., Linkoping, Sweden, 1991.
[14] A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields. Kluwer Academic, 1993.
[15] Nat'l Inst. of Standards and Tech nology, Digital Signature Standard, FIPS Publication 186-2, Jan. 2000.
[16] I.S. Reed and X. Chen, Error-Control Coding for Data Networks. Kluwer Academic, 1999.
[17] A. Reyhani-Masoleh and M.A. Hasan, A New Efficient Architecture of Mastrovito Multiplier over$GF(2^m)$ Proc. 20th Biennial Symp. Comm., pp. 59-63, May 2000.
[18] A. Reyhani-Masoleh and M.A. Hasan, Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over$GF(2^{m})$ Technical Report CORR 2003-19, Dept. of C&O, Univ. of Waterloo, Canada, July 2003.
[19] A. Reyhani-Masoleh and M.A. Hasan, On Low Complexity Bit Parallel Polynomial Basis Multipliers Proc. Cryptographic Hardware and Embedded Systems (CHES 2003), pp. 189-202, Sept. 2003.
[20] F. Rodriguez-Henriquez and C.K. Koc, Parallel Multipliers Based on Special Irreducible Pentanomials IEEE Trans. Computers, vol. 52, no. 12, pp. 1535-1542, Dec. 2003.
[21] P.A. Scott, S.J. Simmons, S.E. Tavares, and L.E. Peppard, Architectures for Exponentiation in$GF(2^m)$ IEEE J. Selected Areas in Comm., vol. 6, no. 3, pp. 578-586, Apr. 1988.
[22] G. Seroussi, Table of Low-Weight Binary Irreducible Polynomials HP Labs Tech. Report HPL-98-135, Aug. 1998.
[23] L. Song and K.K. Parhi, Low Complexity Modified Mastrovito Multipliers over Finite Fields$GF(2^M)$ Proc. IEEE Int'l Symp. Circuits and Systems (ISCAS-99), pp. 508-512, 1999.
[24] B. Sunar and Ç.K. Koç, Mastrovito Multiplier for All Trinomials IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[25] H. Wu, Bit-Parallel Finite Field Multiplier and Squarer Using Polynomial Basis IEEE Trans. Computers, vol. 51, no. 7, pp. 750-758, July 2002.
[26] H. Wu and M.A. Hasan, Efficient Exponentiation of a Primitive Root in$GF(2^m)$ IEEE Trans. Computers, vol. 46, no. 2, pp. 162-172, Feb. 1997.
[27] Y. Wu and M.I. Adham, Scan-Based BIST Fault Diagnosis IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 18, no. 2, pp. 203-211, Feb. 1999.
[28] T. Zhang and K.K. Parhi, Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials IEEE Trans. Computers, vol. 50, no. 7, pp. 734-748, July 2001.

Index Terms:
Finite or Galois field, Mastrovito multiplier, all-one polynomial, polynomial basis, trinomial, pentanomial and equally-spaced polynomial.
Citation:
Arash Reyhani-Masoleh, M. Anwar Hasan, "Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2^{m})," IEEE Transactions on Computers, vol. 53, no. 8, pp. 945-959, Aug. 2004, doi:10.1109/TC.2004.47
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