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 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 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 F_p⁴-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/(q − 1)$.