Elliptic Curves Suitable for Pairing Based Cryptography

Abstract

For pairing based cryptography we need elliptic curves defined over finite fields $\mathbb{F}_{q}$ whose group order is divisible by some prime $\ell$ with $\ell | q^{k-1}$ where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields $\mathbb{F}_{p}$ with arbitrary embedding degree k are given. Unfortunately, p is of size $O(\ell^{2})$.

We give a method to generate ordinary elliptic curves over prime fields with p significantly less than $\ell^{2}$ which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with $p \approx \ell^{s}$ where $s = 2 - 2/\varphi(k)$ or $s = 2 - 1/\varphi(k)$ depending on k. For special values of k even better results are obtained.

We present several examples. In particular, we found some curves where $\ell$ is a prime of small Hamming weight resp. with a small addition chain.