_Maybe for arbitrary n_

An idea for arbitrary n by extending the n==1 case:

In the Dual target game, instead of adversary sending back (H,G). He sends back n copies of H and n copies of G. As soon as he zeroes out anyone of those, he is forced to solve the DL of our random elements.

This is assuming _sets_ means multisets. 

If _sets_ means no duplicates, then he can send {H, 2H, 3H, ... nH}, {G, 2G, 3G, ..., nG}

Then if he zeroes out any of those terms, we can find the DL of G wrt H, since we know the DL of G wrt all of the terms in the set