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Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials
July 2001 (vol. 50 no. 7)
pp. 734-749

Abstract—This paper considers the design of bit-parallel dedicated finite field multipliers using standard basis. An explicit algorithm is proposed for efficient construction of Mastrovito product matrix, based on which we present a systematic design of Mastrovito multiplier applicable to $GF(2^m)$ generated by an arbitrary irreducible polynomial. This design effectively exploits the spatial correlation of elements in Mastrovito product matrix to reduce the complexity. Using a similar methodology, we propose a systematic design of modified Mastrovito multiplier, which is suitable for $GF(2^m)$ generated by high-Hamming weight irreducible polynomials. For both original and modified Mastrovito multipliers, the developed multiplier architectures are highly modular, which is desirable for VLSI hardware implementation. Applying the proposed algorithm and design approach, we study the Mastrovito multipliers for several special irreducible polynomials, such as trinomial and equally-spaced-polynomial, and the obtained complexity results match the best known results. Moreover, we have discovered several new special irreducible polynomials which also lead to low-complexity Mastrovito multipliers.

[1] S.A. Vanstone and P.C. Oorschot, An Introduction to Error Correcting Codes with Applications. Kluwer, 1989.
[2] A.J. Menezes, Handbook of Applied Cryptography. CRC, 1997.
[3] A.J. Menezes, Applications of Finite Fields. Kluwer Academic, 1994.
[4] R. Lidl and H. Niederreiter,An Introduction to Finite Fields and Their Applications.Cambridge: Cambridge Univ. Press, 1986.
[5] D. Jungnickel, Finite Fields: Structure and Arithmetics. Wissenschaftsverlag, 1993.
[6] E.D. Mastrovito,"VLSI Design for Multiplication over Finite Fields," LNCS-357, Proc. AAECC-6, pp. 297-309,Rome, July 1988, Springer-Verlag.
[7] K.K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation. John Wiley&Sons, 1999.
[8] B. Sunar and Ç.K. Koç, Mastrovito Multiplier for All Trinomials IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[9] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Finite Fields$GF(2^m)$,” Information and Computation, vol. 83, pp. 21-40, 1989.
[10] 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.
[11] 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.
[12] H. Wu and M.A. Hasan, "Low Complexity Bit-parallel Multipliers for a Class of Finite Fields," IEEE Trans. Computers, vol. 47, no. 8, pp. 883-887, Aug. 1998.
[13] A. Halbutogullari and Ç.K. Koç, “Mastrovito Multiplier for General Irreducible Polynomials,” Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 498-507, 1999.
[14] A. Halbutogullari and C.K. Koc, Mastrovito Multiplier for General Irreducible Polynomials IEEE Trans. Computers, vol. 49, no. 5, pp. 503-518, May 2000.
[15] 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.
[16] T. Zhang and K.K. Parhi, “Systematic Design Approach of Mastrovito Multipliers over${\rm GF}(2^m)$,” Proc. IEEE Workshop Signal Processing Systems (SiPS), pp. 507-516, Oct. 2000.
[17] G. Golub and C. Van Loan, Matrix Computations, third ed. Baltimore: Johns Hopkins Univ. Press, 1996.

Index Terms:
Finite (or Galois) field, standard basis, multiplication, irreducible polynomials, complexity, VLSI architecture, Toeplitz matrix.
Citation:
Tong Zhang, Keshab K. Parhi, "Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials," IEEE Transactions on Computers, vol. 50, no. 7, pp. 734-749, July 2001, doi:10.1109/12.936239
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