There are 2 different definitions of special soundness in the literature:
(1) can be found in Damgard:
We say that a Sigma-protocol $\Pi$ satisfies special soundness, if there exists a PPT extractor $\mathcal{E}$, such that given any pair of accepting transcripts $(com,ch_1,resp_1),(com,ch_2,resp_2)$ with $ch_1\neq ch_2$, $\mathcal{E}$ can recover $sk$.
(2) can be found in Katz: Digital Signatures:
$\Pi$ satisfies special soundness, if the following is negligible in $\lambda$ for all PPT adversaries $\mathcal{A}$:
\begin{align} \operatorname{Pr} \left[ \begin{array}{c} (pk,sk) \gets \mathrm{keygen}(\lambda) \\ (com,ch_1,resp_1,ch_2,resp_2) \gets\mathcal{A}(pk) \end{array} : \begin{array}{c} ch_1\neq ch_2\\ \land\\ (com,ch_1,resp_1),(com,ch_2,resp_2) \\ \text{are both accepting transcripts.} \end{array} \right] \end{align}
I believe (1) is strictly stronger than (2). Is that correct?
