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**2005/133**

**Title:**

Pairing-Friendly Elliptic Curves of Prime Order

**Authors:**
Paulo S. L. M. Barreto, 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).

**ePrint:**
https://eprint.iacr.org/2005/133

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