Efficient quantum circuits for binary elliptic curve arithmetic: reducing T-gate complexity

Elliptic curves over finite fields F2n 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 F2n in depth O(n logn) using a polynomial basis representation, which may be of independent interest.