Isogeny volcanoes

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.

Invited ANTS X paper, minor edits, 18 pages

References in corpus (2)

• Class polynomials for nonholomorphic modular functions
• Edwards curves and CM curves

Cited by in corpus (13)

• Computing images of Galois representations attached to elliptic curves
• Class polynomials for nonholomorphic modular functions
• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Torsion points on elliptic curves over number fields of small degree
• Adventures in Supersingularland
• Sato-Tate distributions of twists of the Fermat and the Klein quartics
• Extensions of CM elliptic curves and orbit counting on the projective line
• Correspondences without a Core
• Generalized class group actions on oriented elliptic curves with level structure
• CM points on Shimura curves via QM-equivariant isogeny volcanoes
• On self-correspondences on curves
• Ordinary Isogeny Graphs with Level Structure
• Elliptic curves over Hasse pairs