Finished OMDL proof in GGM

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

@@ -11,7 +11,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 	\[ \advantage{\adversary{A}}{\text{MU-\igame}}(\secparamter) \assign | \Pr[\text{MU-\igame}^{\adversary{A}} \Rightarrow 1] |. \]
 \end{definition}
 
-\begin{figure}
+\begin{figure}[h]
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@@ -46,7 +46,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 
 \paragraph{\underline{Formal Proof}}
 
-\begin{figure}
+\begin{figure}[h]
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@@ -83,7 +83,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
 		\Pr[G_0^{\adversary{A}} \Rightarrow 1] = \advantage{\group{G}, \adversary{B}}{\text{MU-\igame}}(\secparamter). \label{eq:adv_mu-igame}
 	\end{align}
 
-	\begin{figure}
+	\begin{figure}[h]
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thesis/sections/mu_security_of_eddsa/mu-uf-nma_implies_mu-suf-cma.tex

@@ -15,7 +15,7 @@ Again the programmability of the random oracle together with the \simalg algorit
 
 \paragraph{\underline{Formal Proof}}
 
-\begin{figure}
+\begin{figure}[h]
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@@ -108,7 +108,7 @@ Again the programmability of the random oracle together with the \simalg algorit
 		\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter). \label{eq:adv_mu-uf-nma}
 	\end{align}
 
-	\begin{figure}
+	\begin{figure}[h]
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@@ -174,7 +174,7 @@ This section shows that MU-UF-NMA security of EdDSA implies the MU-EUF-CMA secur
 		\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter). \label{eq:adv2_mu-uf-nma}
 	\end{align}
 	
-	\begin{figure}
+	\begin{figure}[h]
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thesis/sections/mu_security_of_eddsa/omdl'_implies_mu-gamez.tex

@@ -5,12 +5,13 @@ This section shows that \somdl implies MU-\igame using the Algebraic Group Model
 \paragraph{\underline{Introducing \somdl}} Similar to \sdlog being a variant of the discrete logarithm problem the \somdl is a variant of the one-more discrete logarithm problem which represents the special distribution of secret keys resulting from the key generation algorithm of the EdDSA signature scheme. The only difference to the original one-more discrete logarithm game as introduced in \cite{JC:BNPS03} is that the secret scalars are chosen from the set $\{2^{n-1}, 2^{n-1} + 8, ..., 2^{n} - 8\}$ which represents all valid secret scalars regarding the key generation algorithm. A lower bound on the hardness of the \somdl problem is further analyzed in section \ref{sec:somdl}. The \somdl game is depicted in figure \ref{fig:somdl}.
 
 \begin{definition}[\somdl]
+	\label{def:somdl}
 	Let $n$ and $N$ be positive integer. For an adversary $\adversary{A}$ we define its advantage in the \somdl game as following:
 	
 	\[ \advantage{\adversary{A}}{\text{\somdl}}(\secparamter) \assign | \Pr[\text{\somdl}^{\adversary{A}} \Rightarrow 1] |. \]
 \end{definition}
 
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+\begin{figure}[h]
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 	\vspace{1mm}
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@@ -47,8 +48,7 @@ This section shows that \somdl implies MU-\igame using the Algebraic Group Model
 
 \paragraph{\underline{Formal Proof}}
 
-% TODO: clarify encoding of c
-\begin{figure}
+\begin{figure}[h]
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 	\large
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@@ -101,7 +101,7 @@ This section shows that \somdl implies MU-\igame using the Algebraic Group Model
 		\Pr[G_2^{\adversary{A}} \Rightarrow 1] = \advantage{\group{G},\adversary{B}}{\somdl}(\secparamter). \label{eq:adv_omdl'}
 	\end{align}
 	
-	\begin{figure}
+	\begin{figure}[h]
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 		\vspace{1mm}