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?