Streamlined advantage style

thesis/sections/security_of_eddsa/dlog'_implies_gamez.tex

@@ -32,7 +32,7 @@ For an adversary $\adversary{A}$ we define its advantage in the \sdlog game as f
 	\label{theorem:advgamez}
 	Let $\adversary{A}$ be an adversary against \igame with $\group{G}$ being a cyclic group of prime order $L$, making at most $\oraclequeries$ oracle queries. Then
 	
-	\[ \advantage{\igame,\adversary{A}}{\group{G}} \leq \advantage{\sdlog,\adversary{B}}{\group{G}} - \frac{\oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \].
+	\[ \advantage{\group{G},\adversary{A}}{\igame}(k) \leq \advantage{\group{G},\adversary{B}}{\sdlog}(k) - \frac{\oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \].
 \end{theorem}
 
 \paragraph{\underline{Proof Overview}}
@@ -78,7 +78,7 @@ The adversary has to call the \ioracle oracle with a commitment $\groupelement{R
 	\item \paragraph{\underline{$G_0$:}} Let $G_0$ be \igame. By definition,
 	
 	% TODO: Hier Sicherheitsparameter?
-	\[ \advantage{\group{G}}{\igame}(\adversary{A}) = \Pr[\igame^{\adversary{A}} \Rightarrow 1] = \Pr[G_0^{\adversary{A}} \Rightarrow 1] \].
+	\[ \advantage{\group{G},\adversary{A}}{\igame}(k) = \Pr[\igame^{\adversary{A}} \Rightarrow 1] = \Pr[G_0^{\adversary{A}} \Rightarrow 1] \].
 	
 	\item \paragraph{\underline{$G_1$:}} Game $G_1$ is exactly the same as $G_0$ with the only change being the bad flag being set inside an if condition. The bad flag is set if $2^c \ch_i = -r_2$. This represents cases where not all solutions from the adversary $\adversary{A}$ can be used to calculate the discrete logarithm of $\groupelement{A}$. This is just a conceptual change since the behavior of the game does not change whether the flag is set or not. Hence,
 	
@@ -92,7 +92,7 @@ The adversary has to call the \ioracle oracle with a commitment $\groupelement{R
 	\item Finally, Game $G_2$ is well prepared to show that there exists an adversary $\adversary{B}$ satisfying
 	
 	\begin{align}
-		\Pr[G_2^{\adversary{A}} \Rightarrow 1] = \advantage{\sdlog}{\adversary{B}} \label{eq:advbsdlog}
+		\Pr[G_2^{\adversary{A}} \Rightarrow 1] = \advantage{\group{G},\adversary{B}}{\sdlog}(k) \label{eq:advbsdlog}
 	\end{align}
 	
 	\begin{figure}