Hidden subgroup problem

Setup
A group $$G$$ described by means of an encoding. A subgroup $$H$$ of $$G$$.

Conversion to be achieved
We are given a coset-separating function for $$H$$ in $$G$$, say $$f:G \to X$$.

We need to output a generating set for $$H$$ (in terms of encodings of the various elements).

Log-size assumption
We study the hidden subgroup proble under the log-size assumption, that is, we assume that:


 * The encoding of $$G$$ has logarithmic size with the multiplication and inversion implemented in polylogarithmic time
 * The set $$X$$ is also encoded using words whose length is at most logarithmic in the order of $$G$$
 * The time taken by this coset-separating function (which behaves like a black box) is also polylogarithmic in the size of the group

Under these assumptions, we want an algorithm that is polylogarithmic in the size of the group.