Elliptic curves over finite fields GF(2^{n}) play a prominent role in
modern cryptography. Published quantum algorithms dealing with such curves build
on a short Weierstrass form in combination with affine or projective coordinates.
In this paper we show that changing the curve representation allows a substantial
reduction in the number of T-gates needed to implement the curve arithmetic. As a
tool, we present a quantum circuit for computing multiplicative inverses in
GF(2^{n}) in depth O(n log n) using a polynomial basis representation,
which may be of independent interest.