EFFICIENT QUANTUM CIRCUITS FOR BINARY ELLIPTIC CURVE ARITHMETIC: REDUCING T-GATE COMPLEXITY
Elliptic curves over finite fields F-2n 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 F-2n in depth O(n log(2) n) using a polynomial basis representation, which may be of independent interest.
quantum circuit elliptic curve finite field