On finite field arithmetic in characteristic 2

Abstract We are interested in extending normal bases of F 2 n / F 2 to bases of F 2 n d / F 2 which allow fast arithmetic in F 2 n d . This question has been studied by Thomson and Weir in 2018 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.