Isogeny volcanoes
This expository work provides insights for computational number theorists and elliptic curve cryptography practitioners, but it is incremental as it recounts existing theory and algorithms.
The paper reviews the theory of isogeny graphs of elliptic curves over finite fields and examines recent algorithms that achieve significant performance improvements by leveraging this theory.
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.