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Issue No.05 - May (1999 vol.48)
pp: 522-527
ABSTRACT
<p><b>Abstract</b>—An efficient algorithm for the multiplication in <tmath>\$GF(2^m)\$</tmath> was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial <tmath>\$x^m+x+1\$</tmath> was given as <tmath>\$m^2-1\$</tmath> XOR and <tmath>\$m^2\$</tmath> 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 <tmath>\$x^m+x^n+1\$</tmath>, where <tmath>\$m \not=2n\$</tmath>, also requires <tmath>\$m^2-1\$</tmath> XOR and <tmath>\$m^2\$</tmath> AND gates. However, <tmath>\$m^2-m/2\$</tmath> XOR gates are sufficient when the generating trinomial is of the form <tmath>\$x^m+x^{m/2}+1\$</tmath> for an even <tmath>\$m\$</tmath>. We also calculate the time complexity of the proposed Mastrovito multiplier and give design examples for the irreducible trinomials <tmath>\$x^7+x^4+1\$</tmath> and <tmath>\$x^6+x^3+1\$</tmath>.</p>
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