Optimized quantum implementation of elliptic curve arithmetic over binary fields

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a ``quantum attack'' of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In \cite{PZ03}, Proos and Zalka show how to implement the extended Euclidean algorithm to compute inverses in the prime field $\GF(p)$. They employ a number of optimizations to achieve a running time of $O(n^2)$, and a space-requirement of $O(n)$ qubits, where $n$ is the number of bits in the binary representation of $p$ (there are some trade-offs that they make, sacrificing a few extra qubits to reduce running-time). In practice, elliptic curve cryptosystems often use curves over the binary field $\GF(2^m)$. In this paper, I show how to implement the extended Euclidean algorithm for polynomials to compute inverses in $\GF(2^m)$. Working under the assumption that qubits will be an `expensive' resource in realistic implementations, I optimize specifically to reduce the qubit space requirement, while keeping the running-time polynomial. The implementation here differs from that in $\cite{PZ03}$ for $\GF(p)$, and we are able to take advantage of some properties of the binary field $\GF(2^m)$. I also optimize the overall qubit space requirement for computing the group operation for elliptic curves over $\GF(2^m)$ by decomposing the group operation to make it ``piecewise reversible'' (similar to what is done in \cite{PZ03} for curves over $\GF(p)$).