Paper 2008/530

Fast hashing to G2 on pairing friendly curves

Michael Scott, Naomi Benger, Manuel Charlemagne, Luis J. Dominguez Perez, and Ezekiel J. Kachisa

Abstract

When using pairing-friendly ordinary elliptic curves over prime fields to implement identity-based protocols, there is often a need to hash identities to points on one or both of the two elliptic curve groups of prime order $r$ involved in the pairing. Of these $G_1$ is a group of points on the base field $E({\text{F}}_p)$ and $G_2$ is instantiated as a group of points with coordinates on some extension field, over a twisted curve $E^{\prime }({\text{F}}_{p^d})$, where $d$ divides the embedding degree $k$. While hashing to $G_1$ is relatively easy, hashing to $G_2$ has been less considered, and is regarded as likely to be more expensive as it appears to require a multiplication by a large cofactor. In this paper we introduce a fast method for this cofactor multiplication on $G_2$ which exploits an efficiently computable homomorphism.