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negl

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Fabrice Mouhartem 2018-06-12 16:54:21 +02:00
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5
chap-GS-background.tex
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@ -306,8 +306,7 @@ adversarially-controlled user.
attacks} if, for any $\ppt$ adversary $\adv$ involved in Experiment~$\Exp{\textrm{mis-id}}{\adv}(\lambda)$ attacks} if, for any $\ppt$ adversary $\adv$ involved in Experiment~$\Exp{\textrm{mis-id}}{\adv}(\lambda)$
described in Figure~\ref{exp:mis-id}, we have: described in Figure~\ref{exp:mis-id}, we have:
\[\advantage{\adv}{\mathrm{mis}\textrm{-}\mathrm{id}}(\lambda) \triangleq \[\advantage{\adv}{\mathrm{mis}\textrm{-}\mathrm{id}}(\lambda) \triangleq
\Proba{\,\Exp{\mathrm{mis}\textrm{-}\mathrm{id}}{\adv}(\lambda)=1} = \Proba{\,\Exp{\mathrm{mis}\textrm{-}\mathrm{id}}{\adv}(\lambda)=1} \leq \negl[\lambda].\]
\negl[\lambda].\]
\end{definition} \end{definition}
@ -364,7 +363,7 @@ The adversary eventually aims at framing an honest group member.
% %
A dynamic group signature scheme is secure against \emph{framing attacks} if, A dynamic group signature scheme is secure against \emph{framing attacks} if,
for any $\ppt$ adversary $\adv$ involved in the experiment~$\Exp{\mathrm{fra}}{\adv}(\lambda)$ described Figure~\ref{exp:frame}), it holds that for any $\ppt$ adversary $\adv$ involved in the experiment~$\Exp{\mathrm{fra}}{\adv}(\lambda)$ described Figure~\ref{exp:frame}), it holds that
\[ \advantage{\adv}{\mathrm{fra}}(\lambda)=\Proba{\Exp{\mathrm{fra}}{\adv}(\lambda)=1} \in \negl[\lambda]. \] \[ \advantage{\adv}{\mathrm{fra}}(\lambda)=\Proba{\Exp{\mathrm{fra}}{\adv}(\lambda)=1} \leq \negl[\lambda]. \]
% %
\end{definition} \end{definition}

2
chap-OT-LWE.tex
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@ -103,7 +103,7 @@ The distribution of outputs of the environment in the different settings is deno
\begin{definition} \begin{definition}
An AC-OT protocol is said to securely implement the functionality if for any real-world adversary $\adv$ and any real world environment $\mathcal E$, there exists an ideal-world simulator $\mathcal A'$ controlling the same parties in the ideal-world as $\adv$ does in the real-world, such that An AC-OT protocol is said to securely implement the functionality if for any real-world adversary $\adv$ and any real world environment $\mathcal E$, there exists an ideal-world simulator $\mathcal A'$ controlling the same parties in the ideal-world as $\adv$ does in the real-world, such that
\[ | \mathbf{Real}_{\mathcal E, \adv}(\lambda) - \mathbf{Ideal}_{\mathcal{E}, \adv}(\lambda) | = \negl(\lambda). \] \[ | \mathbf{Real}_{\mathcal E, \adv}(\lambda) - \mathbf{Ideal}_{\mathcal{E}, \adv}(\lambda) | \leq \negl(\lambda). \]
\end{definition} \end{definition}

2
chap-proofs.tex
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@ -123,7 +123,7 @@ an attack is successful if the probability that it succeed is noticeable.
\index{Probability!Negligible} \index{Probability!Noticeable} \index{Probability!Overwhelming} \index{Probability!Negligible} \index{Probability!Noticeable} \index{Probability!Overwhelming}
Let $f : \NN \to [0,1]$ be a function. The function $f$ is said to be \emph{negligible} if $f(n) = n^{-\omega(1)}_{}$, and this is written $f(n) = \negl[n]$.\\ Let $f : \NN \to [0,1]$ be a function. The function $f$ is said to be \emph{negligible} if $f(n) = n^{-\omega(1)}_{}$, and this is written $f(n) = \negl[n]$.\\
Non-negligible functions are also called \emph{noticeable} functions.\\ Non-negligible functions are also called \emph{noticeable} functions.\\
Finally, if $f = 1- \negl[n]$, $f$ is said to be \emph{overwhelming}. Finally, if $f = 1 - \negl[n]$, $f$ is said to be \emph{overwhelming}.
\end{definition} \end{definition}
Now, we have to define two more notions to be able to work on security proofs. Now, we have to define two more notions to be able to work on security proofs.