Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves

TL;DR

This paper introduces new heuristic algorithms for computing class and modular polynomials modulo a prime by leveraging recent advances in supersingular elliptic curves, challenging the traditional focus on ordinary curves.

Contribution

The paper presents novel heuristic algorithms that efficiently compute class and modular polynomials using supersingular elliptic curves, inspired by recent cryptographic developments.

Findings

Algorithms outperform previous methods based on ordinary curves

Supersingular curves can be used efficiently for polynomial computations

Advances in isogeny-based cryptography enable practical supersingular computations

Abstract

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, the recent advances in the study of supersingular curves through the Deuring correspondence motivated by isogeny-based cryptography has provided all the tools to perform the necessary tasks efficiently.