International Association
for Cryptologic Research

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.