Added Proofs for lax parsing

thesis/sections/mu_security_of_eddsa/omdl'_implies_mu-gamez.tex

@@ -136,7 +136,7 @@ This section shows that \somdl implies MU-\igame using the Algebraic Group Model
 		\label{fig:adversary_omdl'}
 	\end{figure}
 	
-	To prove (\ref{eq:adv_omdl'}), we define an adversary $\adversary{B}$ attacking \sdlog that simulates $\adversary{A}$'s view in $G_2$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversary_omdl'} is run in the \sdlog game and adversary $\adversary{B}$ simulates \ioracle for adversary $\adversary{A}$. \ioracle is simulated perfectly.
+	To prove (\ref{eq:adv_omdl'}), we define an adversary $\adversary{B}$ attacking \somdl that simulates $\adversary{A}$'s view in $G_2$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversary_omdl'} is run in the \somdl game and adversary $\adversary{B}$ simulates \ioracle for adversary $\adversary{A}$. \ioracle is simulated perfectly.
 	
 	Finally, consider $\adversary{A}$'s output $s^*$. It is known that $\groupelement{R^*} = 2^c s^* \groupelement{B} - 2^c \ch^* \groupelement{A_i}$ for one of the public keys and one tuple $(R^*, \ch^*)$ generated by the \ioracle oracle. Using the \textit{DL} oracle we can get the discrete logarithms of all public keys but the one for which $s^*$ is a valid solution in the MU-\igame game. This way the \textit{DL} oracle gets called exactly $N-1$ times which is smaller than $N$ which is required by the \somdl game. Together with the representation of $R^*$ provided during the \ioracle oracle call and the discrete logarithms of the public keys we are able to generate a representation of $R^*$ looking like $\groupelement{R^*} = r_b \groupelement{B} + r_i \groupelement{A_i}$. By equating both equations we get: