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).Comment: 16 page

On the Decisional Diffie-Hellman Problem in Genus 2

We investigate the Decisional Diffie-Hellman problem in the Jacobian variety of supersingular curves of genus two over finite fields. A solution to this problem is useful in Public Key Cryptography, for example in Digital Signatures and Identity-Based Cryptography. The existence of a non-degenerate, bilinear pairing reduces the solution to DDH to the existence of sufficiently many distortion maps. These maps are found in the endomorphism ring of the Jacobian variety. We show examples of supersingular curves over finite fields of both even and odd characteristics such that the endomorphism algebra is 16-dimensional over the rationals, and we solve DDH in some cases