## CryptoDB

### Paper: Pairing-Friendly Elliptic Curves of Prime Order

Authors: Paulo S. L. M. Barreto Michael Naehrig URL: http://eprint.iacr.org/2005/133 Search ePrint Search Google 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 $\rho \sim q/(q-1)$.
##### BibTeX
@misc{eprint-2005-12469,
title={Pairing-Friendly Elliptic Curves of Prime Order},
booktitle={IACR Eprint archive},
keywords={public-key cryptography / elliptic curves, pairing-based cryptosystems},
url={http://eprint.iacr.org/2005/133},
note={Revised version presented at SAC'2005 and published in LNCS 3897, pp. 319--331, Springer, 2006. pbarreto@larc.usp.br 13207 received 8 May 2005, last revised 28 Feb 2006},
author={Paulo S. L. M. Barreto and Michael Naehrig},
year=2005
}