On proving it is hard to compute $g^{rb}$ with knowledge of $r$, given $g, g^a, g^{ab}$ I am trying to prove the following


  Given $g, g^a, g^{ab}$ it is hard to compute $r, g^r, g^{rb}$, for some arbitrarily chosen value of $r$ 


where $g ∈ \mathbb{G}, \mathbb{G}$ is a cyclic group of prime order $p$ and $r∈ \mathbb{Z}^*_p -\{0\} $ and $r \neq a$.

I don't have a clear (far from elegant) proof that, if there exists an algorithm $\mathcal{A}$ that can solve this, then we can solve *some*  hard problem..

The best sketch I can think of is reducing this to a the Okamato's Conference Key Sharing Scheme. However, again, I don't have a clear proof idea here...

Also, I argue, the adversary can submit the following,$(g^a)^t, (g^{ab})^t$,  $t∈_R Z$, however in that case, the adversary has no clue about the value of the exponent
Here, I cranked out the black box optimum to detect any subgroup of $S_{n}$, still exponential but much faster than brute force over all $n!$ permutations.

Your problem is around $O(n  \log  rb)$ with repeated doubling if you have $g$ given to you as a permutation in $S_{n}$.