Constructing Pairing-Friendly Genus 2 Curves with Ordinary Jacobians

Abstract

We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm is modeled on the Cocks-Pinch method for constructing pairing-friendly elliptic curves [5], and works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields ${{\mathbb F}}_q$ with q ≈ r ⁴. We also provide an algorithm for constructing genus 2 curves over prime fields ${{\mathbb F}}_q$ with ordinary Jacobians J having the property that $J[r] \subset J({{\mathbb F}}_{q})$ or $J[r] \subset J({{\mathbb F}}_{q^k})$ for any even k.