Added Proofs for lax parsing

thesis/sections/security_notions.tex

@@ -105,6 +105,39 @@ Strong Existential Unforgeability against Chosen Message Attack (\cma) is a secu
 	\label{game:cma}
 \end{figure}
 
+\subsubsection{EUF-CMA}
+
+Existential Unforgability against Chosen Message Attack (EUF-CMA) is similar to the SUF-CMA security notion. The adversary also gets access to an \Osign oracle which generates arbitrary signatures. Unlike in the SUF-CMA game the adversary wins the EUF-CMA game only if he forges a valid signature for a message that was not queries by the \Osign oracle. The security game is depicted in figure \ref{game:euf-cma}.
+
+\begin{definition}[\cma]
+	Let $SIG = (\keygen, \sign, \verify)$ be a digital signature scheme. $SIG$ is EUF-CMA secure if for all ppt adversaries $\adversary{A}$ the $\advantage{SIG,\adversary{A}}{\text{EUF-CMA}}(\secparamter)$ is negligible in $\secparamter$.
+	
+	\[ \advantage{SIG,\adversary{A}}{\text{EUF-CMA}}(\secparamter) \assign \prone{\text{EUF-CMA}^{\adversary{A}}} \leq \epsilon \]
+\end{definition}
+
+\begin{figure}[h]
+	\hrule
+	\begin{multicols}{2}
+		\normalsize
+		\begin{algorithmic}[1] 
+			\Statex \underline{\game $\text{\cma}$}
+			\State $(\pubkey, \privkey) \randomassign \keygen(1^\secparamter)$
+			\State $(\m^*, \signature^*) \randomassign \adversary{A}^{\sign(\inp)}(\pubkey)$
+			\State \Return $\verify(\pubkey, \m^*, \signature^*) \test 1 \wedge \m^* \notin M$
+		\end{algorithmic}
+		\columnbreak
+		\begin{algorithmic}[1] 
+			\Statex \underline{\oracle \Osign($\m \in \messagespace$)}
+			\State $\signature \randomassign \sign(\privkey, \m)$
+			\State $M \assign M \cup \{\m\}$
+			\State \Return $\signature$
+		\end{algorithmic}
+	\end{multicols}
+	\hrule
+	\caption{EUF-CMA Security Game}
+	\label{game:euf-cma}
+\end{figure}
+
 \subsubsection{UF-NMA}
 
 Unforgeability against No Message Attack (UF-NMA) is a security notion for digital signature schemes. The difference to the \cma game is that the adversary does not get access to an \Osign oracle, which provides it with valid signatures for arbitrary messages. Like in the \cma setting the adversary is tasked to provide a valid signature for an arbitrary message. The game is depicted in figure \ref{game:uf-nma}.
@@ -162,6 +195,8 @@ MU-SUF-CMA is the multi-user variant of the SUF-CMA security notion. Instead of
 	\label{game:mu-suf-cma}
 \end{figure}
 
+\subsubsection{MU-EUF-CMA}
+
 \subsubsection{MU-UF-NMA}
 
 MU-UF-NMA  is the multi-user variant of the UF-NMA security notion. Instead of one public key the adversary gets access to $n$ public keys and has to forge a signature for any of the public keys. Unlike the MU-SUF-CMA the adversary does not get access to a signing oracle. The game is depicted in figure \ref{game:mu-uf-nma}.
@@ -185,4 +220,6 @@ MU-UF-NMA  is the multi-user variant of the UF-NMA security notion. Instead of o
 	\hrule
 	\caption{MU-UF-NMA Security Game}
 	\label{game:mu-uf-nma}
-\end{figure}
+\end{figure}
+
+\subsection{Security Assumptions}