Fast Point Multiplication on Elliptic Curves through Isogenies

Abstract

Elliptic curve cryptosystems are usually implemented over fields of characteristic two or over (large) prime fields. For large prime fields, projective coordinates are more suitable as they reduce the computational workload in a point multiplication. In this case, choosing for parameter a the value −3 further reduces the workload. Over $\mathbb{F}_p$, not all elliptic curves can be rescaled through isomorphisms to the case a = −3. This paper suggests the use of the more general notion of isogenies to rescale the curve. As a side result, this also illustrates that selecting elliptic curves with a = −3 (as those recommended in most standards) is not restrictive.