On Division Polynomial PIT and Supersingularity

TL;DR

This paper introduces new algorithms for testing the supersingularity of elliptic curves over finite fields, leveraging Polynomial Identity Testing for division polynomials and high-order points.

Contribution

It presents novel algorithms based on PIT for division polynomials, improving supersingularity testing methods for elliptic curves.

Findings

Efficient supersingularity testing algorithms proposed

Algorithms utilize high-order points on elliptic curves

Potential improvements in elliptic curve cryptography applications

Abstract

For an elliptic curve $E$ over a finite field $\F_q$, where $q$ is a prime power, we propose new algorithms for testing the supersingularity of $E$. Our algorithms are based on the Polynomial Identity Testing (PIT) problem for the $p$-th division polynomial of $E$. In particular, an efficient algorithm using points of high order on $E$ is given.