Computing Canonical Heights on Elliptic Curves in Quasi-Linear Time

TL;DR

This paper presents a fast algorithm for computing the canonical height of points on elliptic curves over rationals, significantly improving efficiency by avoiding integer factorization.

Contribution

It introduces a novel quasi-linear time algorithm for the non-archimedean component of the height, eliminating the need for integer factorization.

Findings

Algorithm computes heights efficiently in quasi-linear time.

No integer factorization needed for the non-archimedean term.

Applicable to elliptic curves over the rationals.

Abstract

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.