Generalized class polynomials

TL;DR

This paper introduces a generalized class polynomial that can produce smaller coefficients than traditional modular functions, surpassing known bounds and achieving significant size reductions for elliptic curve computations.

Contribution

It presents a new class polynomial generalization that exceeds previous size reduction bounds, with examples surpassing all known modular functions.

Findings

Achieves reduction factors matching Weber's functions

Surpasses known bounds for an infinite family of discriminants

Provides examples with at least double the reduction of existing functions

Abstract

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, that reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Br\"oker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Br\"oker-Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.