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clarified notation

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This commit is contained in:
Aaron Kaiser
2023-03-02 11:02:35 +01:00
parent 91ec89c040
commit b08241b07f
1 changed files with 5 additions and 5 deletions
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10
thesis/Abschlussarbeit.tex
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@@ -368,7 +368,7 @@ Assuming that $r_2 + 2^c c$ is invertible in $\field{L}$ (not equal to $0$) we c
\State $\groupelement{R}_i = r_1 \groupelement{B} + r_2 \groupelement{A}$
\State $c_i \randomsample \{0,1\}^{2b}$
\BeginBox[draw=blue]
\State \textbf{If} $2^c c_i = -r_2$ \textbf{then}
\State \textbf{If} $2^c c_i \equiv -r_2 \pmod L$ \textbf{then}
\State \text{ } $bad \assign true$
\BeginBox[draw=red,dashed]
\State \text{ } $abort$
@@ -397,7 +397,7 @@ Game $G_0$ is defined in Figure \ref{fig:igamewithabort} by ignoring all boxes.
% TODO: hard bezüglich ggen
% TODO: min entropy von {0,1}^{2b} mod L?
\[ \advantage{\igame}{\adversary{A}} \leq \advantage{\sdlog}{\adversary{B}} - \frac{\oraclequeries}{2^{\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \]
\[ \advantage{\igame}{\adversary{A}} \leq \advantage{\sdlog}{\adversary{B}} - \frac{\oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \]
\end{theorem}
\begin{proof}
@@ -410,9 +410,9 @@ Game $G_0$ is defined in Figure \ref{fig:igamewithabort} by ignoring all boxes.
\[ \Pr[G_0^{\adversary{A}} \Rightarrow 1] = \Pr[G_1^{\adversary{A}} \Rightarrow 1] \]
% TODO: wählen von
\item \paragraph{\underline{$G_2:$}} Game $G_2$ aborts if the flag bad is set. For each individual \ioracle query the bad flag is set with probability at most $\frac{1}{2^{\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}$, since $c$ is chosen from $\{0,1\}^{2b}$ uniformly at random and then reduced modulo $L$. By the Union bound over all $\oraclequeries$ queries we obtain $\Pr[bad] = \frac{\oraclequeries}{2^{\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}$. Since $G_1$ and $G_2$ are identical-until-bad games, we have
\item \paragraph{\underline{$G_2:$}} Game $G_2$ aborts if the flag bad is set. For each individual \ioracle query the bad flag is set with probability at most $\frac{1}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}$. $-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})$ being the min entropy of $c$ since $c$ is chosen from $\{0,1\}^{2b}$ uniformly at random and then reduced modulo $L$ in the check during the if condition. By the Union bound over all $\oraclequeries$ queries we obtain $\Pr[bad] = \frac{\oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}$. Since $G_1$ and $G_2$ are identical-until-bad games, we have
\[ |\Pr[G_1^{\adversary{A}} \Rightarrow 1] - \Pr[G_2^{\adversary{A}} \Rightarrow 1]| \leq \Pr[bad] \leq \frac{\oraclequeries}{2^{\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \]
\[ |\Pr[G_1^{\adversary{A}} \Rightarrow 1] - \Pr[G_2^{\adversary{A}} \Rightarrow 1]| \leq \Pr[bad] \leq \frac{\oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \]
\item Finally, Game $G_2$ is well prepared to show that there exists an adversary $\adversary{B}$ satisfying
@@ -436,7 +436,7 @@ Game $G_0$ is defined in Figure \ref{fig:igamewithabort} by ignoring all boxes.
\Procedure{\ioracle}{$\agmgroupelement{R_i}{r_i} \in \group{G}$}
\State $\groupelement{R}_i = r_1 \groupelement{B} + r_2 \groupelement{A}$
\State $c_i \randomsample \{0,1\}^{2b}$
\State \textbf{If} $2^c c_i = -r_2$ \textbf{then}
\State \textbf{If} $2^c c_i \equiv -r_2 \pmod L$ \textbf{then}
\State \text{ } $bad \assign true$
\State \text{ } $abort$
\State \textbf{endIf}