Constructing Pairing-Friendly Genus 2 Curves with Split Jacobian

Genus 2 curves with simple but not absolutely simple jacobians can be used to construct pairing-based cryptosystems more efficient than for a generic genus 2 curve. We show that there is a full analogy between methods for constructing ordinary pairing-friendly elliptic curves and simple abelian varieties, which are iogenous over some extension to a product of elliptic curves. We extend the notion of complete, complete with variable discriminant, and sparse families introduced in by Freeman, Scott and Teske [11] for elliptic curves, and we generalize the Cocks-Pinch method and the Brezing-Weng method to construct families of each type. To realize abelian surfaces as jacobians we use of genus 2 curves of the form y = x + ax + bx or y = x + ax + b, and apply the method of Freeman and Satoh [10]. As applications we find some families of abelian surfaces with recorded ρ-value ρ = 2 for embedding degrees k = 3, 4, 6, 12, or ρ = 2.1 for k = 27, 54. We also give variable-discriminant families with best ρ-values.