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Mastrovito Multiplier for All Trinomials
May 1999 (vol. 48 no. 5)
pp. 522-527

Abstract—An efficient algorithm for the multiplication in $GF(2^m)$ was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial $x^m+x+1$ was given as $m^2-1$ XOR and $m^2$ AND gates. In this paper, we describe an architecture based on a new formulation of the multiplication matrix and show that the Mastrovito multiplier for the generating trinomial $x^m+x^n+1$, where $m \not=2n$, also requires $m^2-1$ XOR and $m^2$ AND gates. However, $m^2-m/2$ XOR gates are sufficient when the generating trinomial is of the form $x^m+x^{m/2}+1$ for an even $m$. We also calculate the time complexity of the proposed Mastrovito multiplier and give design examples for the irreducible trinomials $x^7+x^4+1$ and $x^6+x^3+1$.

[1] G. Golub and C. Van Loan, Matrix Computations, third ed. Baltimore: Johns Hopkins Univ. Press, 1996.
[2] J. Guajardo and C. Paar, “Efficient Algorithms for Elliptic Curve Cryptosystems,” Advances in Cryptology—CRYPTO 97, B.S. Kaliski, ed., pp. 342-356, 1997.
[3] 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.
[4] R. Lidl and H. Niederreiter,An Introduction to Finite Fields and Their Applications.Cambridge: Cambridge Univ. Press, 1986.
[5] E.D. Mastrovito,"VLSI Design for Multiplication over Finite Fields," LNCS-357, Proc. AAECC-6, pp. 297-309,Rome, July 1988, Springer-Verlag.
[6] E.D. Mastrovito, “VLSI Architectures for Computation in Galois Fields,” PhD thesis, Linköping Univ., Dept. of Electrical Eng., Linköping, Sweden, 1991.
[7] Applications of Finite Fields, A.J. Menezes, ed. Boston: Kluwer Academic, 1993.
[8] A.J. Menezes, Elliptic Curve Public Key Cryptosystems. Boston: Kluwer Academic, 1993.
[9] C. Paar, “Efficient VLSI Architectures for Bit Parallel Computation in Galois Fields,” PhD thesis, Universität GH Essen, VDI Verlag, 1994.
[10] 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. 846-861, July 1996.
[11] C. Paar, private communication, 1997.

Index Terms:
Finite fields, multiplication, standard basis, irreducible trinomial.
Citation:
B. Sunar, Ç.k. Koç, "Mastrovito Multiplier for All Trinomials," IEEE Transactions on Computers, vol. 48, no. 5, pp. 522-527, May 1999, doi:10.1109/12.769434
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