  Finite fields of the form GF(2^m) play an important role in coding theory and
cryptography. We show that the choice of how to represent the elements of these
fields can have a significant impact on the resource requirements for quantum
arithmetic. In particular, we show how the use of Gaussian normal basis
representations and of `ghost-bit basis' representations can be used to
implement inverters with a quantum circuit of depth O(m log(m)). To the best of
our knowledge, this is the first construction with subquadratic depth reported
in the literature. Our quantum circuit for the computation of multiplicative
inverses is based on the Itoh-Tsujii algorithm which exploits that in normal
basis representation squaring corresponds to a permutation of the coefficients.
We give resource estimates for the resulting quantum circuit for inversion over
binary fields GF(2^m) based on an elementary gate set that is useful for
fault-tolerant implementation.
