Distortion maps for genus two curves
  Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12). 
Steven D. Galbraith, Jordi Pujol\`as, Christophe Ritzenthaler and   Benjamin Smith