A Taxonomy of Pairing-Friendly Elliptic Curves

Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairingbased cryptographic systems. Such “pairing-friendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature. We also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairing-friendly curves to choose to best satisfy a variety of performance and security requirements. Keywords. Elliptic curves, pairing-based cryptosystems, embedding degree, efficient implementation. 1 

• Type: Journal
• Year: 2010
• Where: JOC
• Authors: David Freeman, Michael Scott, Edlyn Teske
• Keywords: Constructions, Embedding, JOC 2010, Pairing-friendly Curves