What is the adversary advantage for soundness of a sigma protocol? Does a sigma protocol have an adversary advantage of zero for soundness? Or is this affected by the probability distribution of the indistinguishability of the simulator? Will in that case using the Fiat-Shamir heuristic affect the adversary advantage?
No, the advantage of an adversary against soundness in a $\Sigma$-protocol is not 0. In fact, a $\Sigma$-protocol satisfies honest-verifier zero-knowledge, meaning that one can always &quot;cheat&quot; by running the zero-knowledge simulator against an honest verifier, provided that we manage to guess the challenge in advance. This means that any $\Sigma$-protocol will have soundness error at least $1/|S|$, where $S$ is the set of all possible challenges. Most $\Sigma$-protocols from the literature actually achieve this bound exactly.