rewrote multi-user proofs

thesis/sections/mu_security_of_eddsa/mu-gamez_implies_mu-uf-nma.tex

@@ -2,11 +2,10 @@
 
 This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signature scheme using the Random Oracle Model. The section starts by first providing an intuition of the proof followed by the detailed security proof.
 
-\paragraph{\underline{Introducing MU-\igame}} This game followed closely the definition of the \igame game. It again replaces the random oracle with the \ioracle oracle. The only difference to the \igame game is that the adversary gets access to $n$ public keys. The adversary again has to output a valid result for any commitment challenge pair generated by the \ioracle oracle for any of the public keys. The MU-\igame game is depicted in figure \ref{game:mu-igame}.
+\paragraph{\underline{Introducing MU-\igame}} This game followed closely the definition of the \igame game. It again replaces the random oracle with the \ioracle oracle. The only difference to the \igame game is that the adversary gets access to $N$ public keys. The adversary again has to output a valid result for any commitment challenge pair generated by the \ioracle oracle for any of the public keys. The MU-\igame game is depicted in figure \ref{game:mu-igame}.
 
-%TODO: Fix collision
 \begin{definition}[MU-\igame]
-	Let $n$ and $N$ be positive integers. For an adversary $\adversary{A}$ we define its advantage in the MU-\igame as following:
+	Let $n$ and $N$ be positive integers. For an adversary $\adversary{A}$, receiving $N$ public keys as input, we define its advantage in the MU-\igame as following:
 	
 	\[ \advantage{\adversary{A}}{\text{MU-\igame}}(\secparamter) \assign | \Pr[\text{MU-\igame}^{\adversary{A}} \Rightarrow 1] |. \]
 \end{definition}
@@ -42,7 +41,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 	\[ \advantage{\adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \advantage{\adversary{B}}{\text{MU-\igame}}(\secparamter). \]
 \end{theorem}
 
-\paragraph{\underline{Proof Overview}} Like the single-user setting the adversary has to query the random oracle to get the hash value $H(\encoded{R}|\encoded{A_i}|m)$. Again the programmability of the random oracle can be used to embed the challenge from \ioracle oracle into the answer of the random oracle. By embedding the challenge from the \ioracle oracle answer into the answer of the random oracle a valid forgery of the signature also becomes a valid solution for the MU-\igame game.
+\paragraph{\underline{Proof Overview}} Like the single-user setting the adversary has to query the random oracle to get the hash value $H(\encoded{R}|\encoded{A_i}|m)$. Again the programmability of the random oracle can be used to embed the challenge from \ioracle oracle into the answer of the random oracle. By embedding the challenge from the \ioracle oracle answer into the answer of the random oracle, a valid forgery of the signature also becomes a valid solution for the MU-\igame game.
 
 \paragraph{\underline{Formal Proof}}
 
@@ -73,7 +72,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 \end{figure}
 
 \begin{proof}
-	\item Let $G_0$ be defined in figure \ref{fig:mu-igame_implies_mu-uf-nma} and $G_0$ be MU-UF-NMA. By definition,
+	\item Let $G_0$ be defined in figure \ref{fig:mu-igame_implies_mu-uf-nma}. Then $G_0$ is the same as MU-UF-NMA with EdDSA. By definition,
 	
 	\[ \advantage{\text{EdDSA}, \adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \Pr[\text{MU-UF-NMA}^{\adversary{A}} \Rightarrow 1 ] = \Pr[G_0^{\adversary{A}} \Rightarrow 1]. \]
 	
@@ -107,7 +106,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 		\label{fig:adversary_mu-igame}
 	\end{figure}
 
-	\item To proof (\ref{eq:adv_mu-igame}), we define an adversary $\adversary{B}$ attacking MU-\igame that simulates $\adversary{A}$'s view in $G_0$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversary_mu-igame} is run in the \igame game and adversary $\adversary{B}$ simulates the random oracle $H$ for the adversary $\adversary{A}$. $H$ is simulated perfectly.
+	\item To proof (\ref{eq:adv_mu-igame}), we define an adversary $\adversary{B}$ attacking MU-\igame that simulates $\adversary{A}$'s view in $G_0$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversary_mu-igame} is run in the \igame game and adversary $\adversary{B}$ simulates the random oracle $H$ for the adversary $\adversary{A}$. $H$ is perfectly simulated because the \ioracle oracle also outputs a uniformly random 2b-bit bitstring. For this reason, $H$ returns a uniformly random 2b-bit bitstring for all queries, as expected.
 	
 	Finally, consider $\adversary{A}$'s output $(\m^*, \signature^* \assign (\encoded{R}, S))$. It is known that: