Quasi-quadratic elliptic curve point counting using rigid cohomology

TL;DR

This paper introduces a deterministic, quasi-quadratic time algorithm for counting points on nonsupersingular elliptic curves over finite fields, utilizing rigid cohomology and deformation techniques, with promising implementation results.

Contribution

It presents a novel deterministic algorithm that improves upon existing methods by combining rigid cohomology with deformation, achieving quasi-quadratic complexity.

Findings

Algorithm works efficiently in small odd characteristic

Implementation yields very good practical results

Advances point counting methods for elliptic curves

Abstract

We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the canonical lift of E, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.