Efficient quantum circuits for binary elliptic curve arithmetic: reducing $T$-gate complexity (pp0631-0644)
          Brittanney Amento, Martin Rotteler, and Rainer Steinwandt
          doi: https://doi.org/10.26421/QIC13.7-8-5
Abstracts: 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 log2 n) using a polynomial basis representation, which may be of independent interest.
Key words: quantum circuit; elliptic curve; finite field