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.
Categories: math.NT