On proving it is hard to compute $g^{rb}$ with knowledge of $r$, given $g, g^a, g^{ab}$
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 $S_{n}$.