Ordinary pairing-friendly genus 2 hyperelliptic curves with absolutely simple Jacobians

Abstract

We present a method for producing pairing-friendly, simple, ordinary Jacobian varieties of genus 2 hyperelliptic curves defined over a prime field Fp. The proposed method heavily relies on the construction of a suitable p-Weil number and a corresponding quartic CM-field. Our Jacobians are absolutely simple and for this special class of Jacobians we give the first examples in the literature with ρ -values below 4, while previous results had in general ρ -values between 6 and 8. These examples derive from “families” of pairing-friendly Jacobians, which are basically polynomial representations of the Jacobian parameters.