International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Fast hashing to G2 on pairing friendly curves

Michael Scott
Naomi Benger
Manuel Charlemagne
Luis J. Dominguez Perez
Ezekiel J. Kachisa
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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 $E(\F_p)$ and $G_2$ is instantiated as a group of points with coordinates on some extension field, over a twisted curve $E'(\F_{p^d})$, 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.
  title={Fast hashing to G2 on pairing friendly curves},
  booktitle={IACR Eprint archive},
  keywords={implementation / Tate Pairing, Addition Chains},
  note={ 14231 received 18 Dec 2008},
  author={Michael Scott and Naomi Benger and Manuel Charlemagne and Luis J. Dominguez Perez and Ezekiel J. Kachisa},