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

We show that for any elliptic curve $E(\mathbb{F}_{q^n})$, if an adversary has access to a Static Diffie-Hellman Problem (Static DHP) oracle, then by making $O(q^{1-\frac{1}{n+1}})$ Static DHP oracle queries during an initial learning phase, for fixed $n>1$ and $q \rightarrow \infty$ the adversary can solve {\em any} further instance of the Static DHP in {\em heuristic} time $\tilde{O}(q^{1-\frac{1}{n+1}})$. Our proposal also solves the {\em Delayed Target DHP} as defined by Freeman, and naturally extends to provide algorithms for solving the {\em Delayed Target DLP}, the {\em One-More DHP} and {\em One-More DLP}, as studied by Koblitz and Menezes in the context of Jacobians of hyperelliptic curves of small genus. We also argue that for {\em any} group in which index calculus can be effectively applied, the above problems have a natural relationship, and will {\em always} be easier than the DLP. While practical only for very small $n$, our algorithm reduces the security provided by the elliptic curves defined over $\mathbb{F}_{p^2}$ and $\mathbb{F}_{p^4}$ proposed by Galbraith, Lin and Scott at EUROCRYPT 2009, should they be used in any protocol where a user can be made to act as a proxy Static DHP oracle, or if used in protocols whose security is related to any of the above problems.