  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.
