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 \leqslant 6 Unknown control sequence '\leqslant'. More general methods produce curves over \mathbb FpFp 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 ¿ = 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 \mathbb Fp4Fp4 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 ¿; 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 ¿ ~ q/(q–1).