International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Pairing-Friendly Elliptic Curves of Prime Order

Paulo S. L. M. Barreto
Michael Naehrig
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Abstract: Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree $k \leqslant 6$. More general methods produce curves over $\F_p$ where the bit length of $p$ is often twice as large as that of the order $r$ of the subgroup with embedding degree $k$; the best published results achieve $\rho \equiv \log(p)/\log(r) \sim 5/4$. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic curves of prime order and embedding degree $k = 12$. The new curves lead to very efficient implementation: non-pairing cryptosystem operations only need $\F_p$ and $\F_{p^2}$ arithmetic, and pairing values can be compressed to one \emph{sixth} of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants $D$ to minimize $\rho$; in particular, for embedding degree $k = 2q$ where $q$ is prime we show that the ability to handle $\log(D)/\log(r) \sim (q-3)/(q-1)$ enables building curves with $\rho \sim q/(q-1)$.
  title={Pairing-Friendly Elliptic Curves of Prime Order},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / elliptic curves, pairing-based cryptosystems},
  note={Revised version presented at SAC'2005 and published in LNCS 3897, pp. 319--331, Springer, 2006. 13207 received 8 May 2005, last revised 28 Feb 2006},
  author={Paulo S. L. M. Barreto and Michael Naehrig},