  We are interested in extending normal bases of
$\mathbf{F}_{\!2^n}/\mathbf{F}_{\!2}$ to bases of
$\mathbf{F}_{\!2^{nd}}/\mathbf{F}_{\!2}$ which allow fast arithmetic in
$\mathbf{F}_{\!2^{nd}}$. This question has been recently studied by Thomson and
Weir in case $d$ is equal to $2$. We construct efficient extended bases in case
$d$ is equal to $3$ and $4$. We also give conditions under which Thomson-Weir
construction can be combined with ours.
