Addition law structure of elliptic curves

TL;DR

This paper investigates the algebraic structure of addition laws on elliptic curves, focusing on models with torsion symmetries to improve computational efficiency in cryptography.

Contribution

It provides a detailed analysis of the module structure of addition morphisms and explicit dimension formulas for sections and eigenspaces related to torsion groups.

Findings

Derived explicit dimension formulas for spaces of sections.

Analyzed the module structure of addition morphisms.

Applied results to models with parametrized torsion subgroups.

Abstract

The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more efficiently computable algorithms for the group law than the standard Weierstrass model. Examples of such models arise via symmetries induced by a rational torsion structure. We analyze the module structure of the space of sections of the addition morphisms, determine explicit dimension formulas for the spaces of sections and their eigenspaces under the action of torsion groups, and apply this to specific models of elliptic curves with parametrized torsion subgroups.