Issue No. 05 - May (1999 vol. 48)

ISSN: 0018-9340

pp: 522-527

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/12.769434

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. , pp. 522-527, May 1999, doi:10.1109/12.769434