On the Static Diffie-Hellman Problem on Elliptic Curves over Extension Fields

Recent work by Koblitz and Menezes has highlighted the existence, in some cases, of apparent separations between the hardness of breaking discrete logarithms in a particular group, and the hardness of solving in that group problems to which the security of certain cryptosystems are provably related. We consider one such problem in the context of elliptic curves over extension fields, and report potential weaknesses of the Galbraith-Lin-Scott curves from EUROCRYPT 2009, as well as two very different practical attacks on the Oakley Key Determination Protocol curves.