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

TL;DR

This paper introduces optimized quantum circuits for elliptic curve arithmetic over GF(2^n), significantly reducing T-gate complexity by changing curve representations and presenting an efficient inverse computation method.

Contribution

It demonstrates that alternative curve representations can lower T-gate counts and provides a new quantum circuit for multiplicative inverses in GF(2^n).

Findings

Reduced T-gate complexity for elliptic curve operations

Quantum circuit for GF(2^n) inversion with depth O(n log n)

Potential for more efficient quantum cryptography implementations

Abstract

Elliptic curves over finite fields GF(2^n) 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 GF(2^n) in depth O(n log n) using a polynomial basis representation, which may be of independent interest.