Generalized class polynomials
Marc Houben, Marco Streng

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…
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Taxonomy
TopicsCoding theory and cryptography · Mathematical functions and polynomials
