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 $x^6+x^3+1$.