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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 ≤ 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/(q − 1)$.

Details

Publication typeInproceedings
Published inSelected Areas in Cryptography - SAC 2005
Pages319–331
Volume3897
SeriesLecture Notes in Computer Science
PublisherSpringer

Previous versions

Paulo S. L. M. Barreto and Michael Naehrig. Pairing-friendly elliptic curves of prime order, Springer, 2006.

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