diff --git a/exam/ex.tex b/exam/ex.tex index 93f3ce5..3324f88 100644 --- a/exam/ex.tex +++ b/exam/ex.tex @@ -290,7 +290,20 @@ at a soundness error of $1/2^{260}$, which is well below the required $1/2^{192}$. - \item \TODO + \item A simulator $\mathcal{S}$ is built as follows: + \begin{itemize} + \item $\mathcal{S}$ starts $\mathcal{V}^*$ with $G_i$ and + $i\in\{0,\dots,2^{130}-1\}$. + \item $\mathcal{S}$ makes a guess $\mathsf{ch}^*$ and calculates + $G'\leftarrow\psi(\phi_{ch^*}^{-1}(G_{ch^*}))$. + \item $\mathcal{S}$ gets a challenge $\mathsf{ch}$ from $\mathcal{V}^*$. + If $\mathsf{ch}=\mathsf{ch}^*$, $\mathcal{S}$ outputs + $(G',\mathsf{ch}^*,\phi_{ch^*}^{-1}\psi)$. If + $\mathsf{ch}\neq\mathsf{ch}^*$, $\mathcal{S}$ rewinds $\mathcal{V}^*$ + and goes to step 2. + \end{itemize} + The simular $\mathcal{S}$ is expected probabilistic polynomial-time with + $2^{130}n$ time and the protocol is zero-knowledge. \item For completeness see 5e.