Isogeny volcanoes

TL;DR

This paper explores the structure of isogeny graphs of elliptic curves over finite fields, explaining their theoretical properties and discussing recent algorithms that leverage this structure for improved computational efficiency.

Contribution

It provides an exposition of the theory of isogeny volcanoes and analyzes new algorithms that significantly enhance performance in elliptic curve computations.

Findings

Enhanced algorithms exploiting isogeny volcano structures

Substantial performance improvements in elliptic curve computations

Comprehensive theoretical overview of isogeny graphs

Abstract

The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This expository paper recounts the theory behind these graphs and examines several recently developed algorithms that realize substantial (often dramatic) performance gains by exploiting this theory.