On Prime-Order Elliptic Curves with Embedding Degrees 3, 4 and 6

Bilinear pairings on elliptic curves have many cryptographic applications such as identity based encryption, one-round three-party key agreement protocols, and short signature schemes. The elliptic curves which are suitable for pairing-based cryptography are called pairing friendly curves. The prime-order pairing friendly curves with embedding degrees k=3,4 and 6 were characterized by Miyaji, Nakabayashi and Takano. We study this characterization of MNT curves in details. We present explicit algorithms to obtain suitable curve parameters and to construct the corresponding elliptic curves. We also give a heuristic lower bound for the expected number of isogeny classes of MNT curves. Moreover, the related theoretical findings are compared with our experimental results

On Finite fields for pairing based cryptography

Here, we improve our previous bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist. We also give some heuristic arguments which lead to a lower bound which in some cases is close to our upper bound.6 page(s