On the Decisional Diffie-Hellman Problem in genus 2

by Jordi Pujolas Boix

RHUL-MA-2009-13

Abstract:

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.