From b08241b07f03865dd09bf14f6392a182a2220fc7 Mon Sep 17 00:00:00 2001 From: Aaron Kaiser Date: Thu, 2 Mar 2023 11:02:35 +0100 Subject: [PATCH] clarified notation --- thesis/Abschlussarbeit.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/thesis/Abschlussarbeit.tex b/thesis/Abschlussarbeit.tex index 9c7aea4..3ebe4bf 100644 --- a/thesis/Abschlussarbeit.tex +++ b/thesis/Abschlussarbeit.tex @@ -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}