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The answer is yes; see Chapter 21 of Galbraith's book. Suppose we have your Fixed-Inverse-DH oracle $O(\cdot)$, and given $g^a$ and $g^b$ we want to find $g^{ab}$. We do this in two steps. First, we use $O$ to compute $g^{a^2}$ from $g^{a}$—this is a related problem called the Square-DH problem. Then we use the quarter-squares identity to compute $g^{ab}$. ...

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