A quantum circuit to find discrete logarithms on ordinary binary elliptic curves in depth O(log^2 n)

TL;DR

This paper presents a quantum circuit for solving discrete logarithms on binary elliptic curves with depth O(log^2 n), significantly improving the efficiency of quantum algorithms for this problem.

Contribution

It introduces quantum circuits for GF(2^n) multiplication and inversion with depths O(log n) and O(log^2 n), enabling a quadratic-depth quantum algorithm for discrete logarithms.

Findings

Quantum circuit depth for discrete logs is O(log^2 n)

GF(2^n) multiplication implemented in depth O(log n)

GF(2^n) inversion implemented in depth O(log^2 n)

Abstract

Improving over an earlier construction by Kaye and Zalka, Maslov et al. describe an implementation of Shor's algorithm which can solve the discrete logarithm problem on binary elliptic curves in quadratic depth O(n^2). In this paper we show that discrete logarithms on such curves can be found with a quantum circuit of depth O(log^2 n). As technical tools we introduce quantum circuits for GF(2^n) multiplication in depth O(log n) and for GF(2^n) inversion in depth O(log^2 n).