Weak Boneh-Boyen Signature in Composite order group In the paper Ring Signatures of Sub-linear Size Without Random Oracles, authors have remarked that the scheme is instantiable in composite order setting, too. I am attaching the following reference below. Can anyone please help me in understanding the argument I have highlighted?
1) To perform verification in the order $p$ subgroup (if the order is $n=pq$ for $p,q$ prime, $a^q$ maps an element $a$ into the order $p$ subgroup - and this clearly also holds in the target group). 

2) The argumentation in the unforgeability proof is that switching from $h$
of order $n$ in the common reference string of the GS proof to $h$ of order $q$, then under the subgroup decision
assumption the probability of a forgery to happen only changes negligibly. So their idea is that
playing the original unforgeability game with the adversary (where $h$ is of order $n$) and a forgery happens,
then playing the modified game with $h$ of order $q$ does not change the probability of a forgery
to happen. But now as in the modified game the commitments are perfectly binding and the
proofs are perfectly sound (the adversary cannot forge a proof), a forgery output by the adversary implies a forgery of a BB signature (which we know can
only happen with negligible probability). 

Now, as the simulator who interacts with the adversary knows the the factorization of $n$ (trapdoor of the GS common reference string), it would not yield a valid reduction 
if the security of the BB signatures would be considered in the entire group and would require that the factorization of $n$ is unknown (as the simulator knows the factorization, it could then compute a forgery on it's own and thus the reduction would be meaningless). However, if one assumes that forging BB signatures in the order $p$ subgroup is hard even if knowing the factorization of $n$, then the reduction can use the forgery output by the adversary in the unforgeability game to break the unforgeability of the BB signature scheme.

As the proof in the paper is very sketchy, this does not show up anywhere else in the paper.