The geometry of efficient arithmetic on elliptic curves

TL;DR

This paper explores the geometric structure of elliptic curve arithmetic, using line bundles and global sections to develop more efficient algorithms for addition and scalar multiplication, especially for curves with torsion points.

Contribution

It introduces a geometric framework that reduces elliptic curve operations to linear algebra, improving computational complexity for key operations.

Findings

Enhanced algorithms for doubling and tripling elliptic curve points

Reduced complexity in elliptic curve arithmetic using geometric methods

Effective use of torsion points to optimize computations

Abstract

The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripling, by considering families of elliptic curves admiting a $2$-torsion or $3$-torsion point.