For pairing based cryptography we need elliptic curves defined over finite fields F q whose group order is divisible by some prime with | q k − 1 where k is relatively small. In Barreto et al. and Dupont et al. 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( 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 p ≈ s where s = 2 − 2/ϕ(k) or s = 2 − 1/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.
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