Edwards curves and CM curves

Abstract

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the links with Kubert parameterizations of X 0 (2) and X 0 (4). We classify CM curves that admit an Edwards or Montgomery form over a finite field, and justify the use of isogenous curves when needed.