Honest Verifier Zero-Knowledge Game for Sigma Protocols I am looking for how an adversary to special HVZK would work. In Boneh and Shoup's book (BonehShoup) they have Attack Game 20.4 for special cHVZK.
Here, the adversary produces a pair (x,y) (witness and statement) and sends it to their challenger, who either uses P and V or Sim to produce the transcript.

Can I modify this game in the following ways?


Can the adversary generate the challenge?
Can I have the adversary ask on polynomially many challenges instead of just one? Why is he limited to a single output here?
This question requires clarifications regarding the definition of special computational Honest-Verifier-Zero-Knowledge (Special cHVZK) as introduced in the linked material by Boneh and Shoup:


  
  Can the adversary generate the challenge c of the transcript in Special cHVZK?
  


No, because this is meant to be a definition for a version of honest-verifier zero-knowledge, therefore challenges must be uniformly random and cannot be chosen by the adversary.


  
  Can the adversary ask the transcript for polynomially many challenge instead of just one?
  


Well, that is the definition. If you allow the adversary to ask transcripts of polynomially many challenges, then it is easy to see that you will end up with an equivalent definition. So in theory, you could allow the adversary to ask the transcript of polynomially-many challenges.