Constructing pairing-friendly genus 2 curves with split Jacobian

Abstract

Using genus 2 curves with simple but not absolutely simple Jacobians one can obtain pairing-based cryptosystems more efficient than for a generic genus 2 curve. We describe a new framework to construct pairing-friendly abelian surfaces, which are simple but not absolutely simple. The main contribution is the generalization of the notion of complete, complete with variable discriminant, and sparse families of elliptic curves introduced by Freeman, Scott and Teske [13]. We give algorithms to construct families of abelian surfaces of each type, which generalize the Brezing-Weng method. To realize these abelian surfaces as Jacobians we use curves of the form y2 = x5 + ax3 + bx or y2 = x6 + ax 3 + b, and apply the method of Freeman and Satoh [12]. As applications we give variable-discriminant families with best ρ-values. We also give some families with record ρ-value. © Springer-Verlag 2012.