**Pairing-Friendly Elliptic Curves of Prime Order**

*Paulo S. L. M. Barreto and Michael Naehrig*

**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 $\rho \sim q/(q-1)$.

**Category / Keywords: **public-key cryptography / elliptic curves, pairing-based cryptosystems

**Publication Info: **Revised version presented at SAC'2005 and published in LNCS 3897, pp. 319--331, Springer, 2006.

**Date: **received 8 May 2005, last revised 28 Feb 2006

**Contact author: **pbarreto at larc usp br

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **The new section 3 deals with implementation issues, and suggest that the proposed family of curves are in some aspects more efficient than previously known curves of prime order.

**Version: **20060228:184025 (All versions of this report)

**Short URL: **ia.cr/2005/133

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