Choosing Coordinate Forms for Solving ECDLP Using Shor's Algorithm

TL;DR

This paper investigates the impact of coordinate choice on the efficiency of Shor's quantum algorithm for solving the elliptic curve discrete logarithm problem, highlighting limitations of projective coordinates.

Contribution

It demonstrates that projective coordinates do not offer the same benefits as affine coordinates in quantum resource optimization for ECDLP using Shor's algorithm.

Findings

Projective coordinates lack uniqueness without modular division.

Affine coordinates are more advantageous for quantum resource use.

Projective coordinates do not improve Shor's algorithm efficiency.

Abstract

Shor's algorithm is well-known for its capability to address the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. The enhancement of its quantum resources continues to be a crucial focus of research. Nevertheless, the application of projective coordinates for quantum resource optimization remains an unresolved issue, mainly because the representation of projective coordinates lacks uniqueness without employing modular division operations. Our study reveals that projective coordinates do not provide the same advantages as affine coordinates when utilizing Shor's method to tackle the ECDLP.