  We consider a quantum polynomial-time algorithm which solves the discrete
logarithm problem for points on elliptic curves over $GF(2^m)$. We improve over
earlier algorithms by constructing an efficient circuit for multiplying
elements of binary finite fields and by representing elliptic curve points
using a technique based on projective coordinates. The depth of our proposed
implementation, executable in the Linear Nearest Neighbor (LNN) architecture,
is $O(m^2)$, which is an improvement over the previous bound of $O(m^3)$
derived assuming no architectural restrictions.
