Identifying supersingular elliptic curves

TL;DR

This paper introduces a new, efficient algorithm for determining whether an elliptic curve over a field of positive characteristic is supersingular, improving upon previous methods in complexity and practicality.

Contribution

The authors develop a novel algorithm that exploits structural differences in isogeny graphs to efficiently identify supersingular elliptic curves, with significant complexity improvements.

Findings

The new algorithm operates in O(n^3 log^2 n) time and O(n) space.

It outperforms existing methods in practical computations.

The approach leverages structural differences in isogeny graphs.

Abstract

Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in F_p^2, determines the supersingularity of E in O(n^3 log^2 n) time and O(n) space, where n=O(log p). Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.