Elliptic Curves Suitable for Pairing Based Cryptography

AbstractFor pairing based cryptography we need elliptic curves defined over finite fields Fq whose group order is divisible by some prime with |qk1 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 Fp with arbitrary embedding degree k are given. Unfortunately, p is of size O(2).We give a method to generate ordinary elliptic curves over prime fields with p significantly less than 2 which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with ps where s=22/φ(k) or s=21/φ(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 is a prime of small Hamming weight resp. with a small addition chain.