Efficient arithmetic of finite fields is important in implementing cryptosystems, error-correcting codes and computer algebra systems.
Normal bases offer considerable advantages.
Optimal normal bases in finite fields were introduced at the University of Waterloo by Mullin et al., and are used in practical hardware implementation of public-key cryptosystems. In the same paper, Mullin et al. constructed two families of optimal normal bases and, based on a computer experiment, they conjectured no more exist. This conjecture had remained open for several years before H. W. Lenstra, Jr. proved it for finite fields over . Lenstra's method, however, is not applicable to other fields. Together with Lenstra, we proved the conjecture holds even for any finite Galois extension of an arbitrary field and the proof for the general case is again different but simpler.