Pairing-Friendly Elliptic Curves of Prime Order

Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree k 6. More general methods produce curves over Fp 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 ρ ≡ log(p)/ log(r) ∼ 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 operations need no more than Fp4 arithmetic, and pairing values can be compressed to one third of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants D to minimize ρ; in particular, for embedding degree k = 2q where q is prime we show that the ability to handle log(D)/ log(r) ∼ (q − 3)/(q − 1) enables building curves with ρ ∼ q/...

• Type: Conference
• Year: 2005
• Where: SACRYPT
• Authors: Paulo S. L. M. Barreto, Michael Naehrig
• Keywords: Embedding, Methods Produce Curves, Pairing-friendly Curves, SACRYPT 2005