A Matrix Decomposition Method for Optimal Normal Basis Multiplication

We introduce a matrix decomposition method and prove that multiplication in GF
$(2^k)$
with a Type 1 optimal normal basis for can be performed using
$k^2-1$
XOR gates irrespective of the choice of the irreducible polynomial generating the field. The previous results achieved this bound only with special irreducible polynomials. Furthermore, the decomposition method performs the multiplication operation using
$1.5k(k-1)$
XOR gates for Type 2a and 2b optimal normal bases, which matches previous bounds.