Elliptic curves suitable for pairing based cryptography
For pairing based cryptography we need elliptic curves defined over finite fields F-q whose group order is divisible by some prime l with l vertical bar 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 F-p with arbitrary embedding degree k are given. Unfortunately, p is of size O(l(2)). We give a method to generate ordinary elliptic curves over prime fields with p significantly less than l(2) which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with p approximate to l(s) where s = 2- 2/phi(k) or s = 2- 1/phi(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 l is a prime of small Hamming weight resp. with a small addition chain.
elliptic curves pairing based cryptagraphy