Constructing pairing-friendly genus 2 curves over prime fields with ordinary Jacobians

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 works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields Fq with q ≈ r. We also provide an algorithm for constructing genus 2 curves over prime fields Fq with ordinary Jacobians J having the property that J [r] ⊂ J(Fq) or J [r] ⊂ J(Fqk ) for any even k.