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Amaury JOLY
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Recherche/BFT-ARBover/6_BFT_ARB/index.tex
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@@ -36,119 +36,128 @@ There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that :
\paragraph{Termination.} Every operation $\BFTAPPEND(x)$, $\BFTPROVE(x)$, and $\BFTREAD()$ invoked by a correct process always returns.
\paragraph{APPEND Validity.} The invocation of $\BFTAPPEND(x)$ by a correct process $p_i$ is valid iff $i \in \Pi_M$. Otherwise the operation is invalid.
% \paragraph{APPEND Validity.} The invocation of $\BFTAPPEND(x)$ by a correct process $p_i$ is valid iff $i \in \Pi_M$. Otherwise the operation is invalid.
\paragraph{PROVE Validity.} If the invocation of a $op = \BFTPROVE(x)$ by a correct process $p$ is not valid, then:
\begin{itemize}
\item $p \not\in \Pi_V$; \textbf{or}
\item At least $t+1$ valid $\BFTAPPEND(x)$ appears before $op$ in $\Seq$.
\end{itemize}
Otherwise, the operation is valid.
\paragraph{PROVE Validity.} The invocation of $op = \BFTPROVE(x)$ by a correct process is valid iff there exist a set of correct process $C$ such that $\forall c \in C$, $c$ invoke $op_2 = \BFTAPPEND(x)$ with $op_2 \prec op_1$ and $|C| \leq t$
\paragraph{PROVE Anti-Flickering.} If the invocation of a operation $op = \BFTPROVE(x)$ by a correct process $p \in \Pi_V$ is invalid, then any $\BFTPROVE(x)$ operation that appears after $op$ in $\Seq$ is invalid.
\paragraph{READ Validity.} The invocation of $op = \BFTREAD()$ by a process $p$ returns the list of valid invocations of $\BFTPROVE$ that appears before $op$ in $\Seq$ along with the names of the processes that invoked each operation.
\subsubsection{t-BFT-GE}
\paragraph{Local consistency.} For all correct process $p_i$ such that $p_i$ invoke an valid $\BFTPROVE(x)$ before a $P \gets \BFTREAD()$ operation in his local execution. Then the set of valid $\BFTPROVE(x)$ in $P$ must contain the previous valid $\BFTPROVE(x)$.
We consider a t-Byzantine Fault Tolerant Group Election Object (t-$\BFTGE[r]$) per round $r \in \mathcal{R}$ with the following properties.
\paragraph{Liveness} For all correct process $p_i$ such that $p_i$ invoke an invalid $\BFTPROVE(x)$ before a $P \gets \BFTREAD()$ operation in his local execution. Then the set of valid $\BFTPROVE(x)$ in $P$ must be the same for any $p_i$.
There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$.
% \subsubsection{t-BFT-GE}
\paragraph{Termination.} Every operation $\BFTVOTE(i, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$ invoked by a correct process always returns.
% We consider a t-Byzantine Fault Tolerant Group Election Object (t-$\BFTGE[r]$) per round $r \in \mathcal{R}$ with the following properties.
\paragraph{Stability.} If there exist at least $n-f$ invocations of $\BFTCOMMIT(r)$ by distincts processes and let call $\BFTCOMMIT(r)^\star$ the $(n-f)^{th}$ such invocation in the linearization of $\Seq$. Then any invocation of $\BFTRESULT(r)$ that appears after $\BFTCOMMIT(r)^\star$ in $\Seq$ returns the same set of processes $W_r$.
% There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$.
\paragraph{VOTE-Validity.} The invocation of $\BFTVOTE(j, r)$ by a correct process is not valid if $\BFTCOMMIT(r)^\star$ appears before in $\Seq$. Otherwise, the operation is valid.
% \paragraph{Termination.} Every operation $\BFTVOTE(i, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$ invoked by a correct process always returns.
\paragraph{Election.} If at least $f+1$ correct processes invoked a valid $\BFTVOTE(j, r)$ for the same process $j$ then $j$ will be enventually included in the set $W_r$ returned by $\BFTRESULT(r)$.
% \paragraph{Stability.} If there exist at least $n-f$ invocations of $\BFTCOMMIT(r)$ by distincts processes and let call $\BFTCOMMIT(r)^\star$ the $(n-f)^{th}$ such invocation in the linearization of $\Seq$. Then any invocation of $\BFTRESULT(r)$ that appears after $\BFTCOMMIT(r)^\star$ in $\Seq$ returns the same set of processes $W_r$.
% \paragraph{VOTE-Validity.} The invocation of $\BFTVOTE(j, r)$ by a correct process is not valid if $\BFTCOMMIT(r)^\star$ appears before in $\Seq$. Otherwise, the operation is valid.
% \paragraph{Election.} If at least $f+1$ correct processes invoked a valid $\BFTVOTE(j, r)$ for the same process $j$ then $j$ will be enventually included in the set $W_r$ returned by $\BFTRESULT(r)$.
\subsection{DL $\Rightarrow$ t-BFT-DL}
\begin{lemma}
For any fixed value $t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object.
For any fixed value $t$ such that $3t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object.
\end{lemma}
\begin{proof}
Fix $t < |M|$. Let
Fix $3t < |M|$. Let
\[
\mathcal{T} = \{\, T \subseteq M \mid |T| = t \,\}.
\mathcal{U} = \{\, U \subseteq M \mid |U| = |M| - t \,\}.
\]
For each $T \in \mathcal{T}$, we instantiate one DenyList object $DL_T$ whose authorization sets are
For each $U \in \mathcal{U}$, we instantiate one DenyList object $DL_U$ whose authorization sets are
\[
\Pi_M(DL_T) = S_T = M \setminus T
\Pi_M(DL_T) = S_T = U
\qquad\text{and}\qquad
\Pi_V(DL_T) = V.
\]
Let
\[
K = \{\, DL_T \mid T \in \mathcal{T} \,\},
K = \{\, DL_U \mid U \in \mathcal{U} \,\},
\qquad\text{so that}\qquad
|K| = \binom{|M|}{t}.
|U| = \binom{|M|}{|M| - t}.
\]
\begin{algorithmic}
\State $K \gets \{DL_T : T \subseteq M, |T|=t\}$
\begin{algorithmic}[1]
% \State $K \gets \{DL_T : T \subseteq M, |T|=t\}$
\Function{BFTAPPEND}{x}
\If{$p_i \notin M$}
\State \Return \textbf{false}
\EndIf
\For{\textbf{each } $DL_T \in K$}
\State $DL_T.\APPEND(x)$
\EndFor
\State \Return \textbf{true}
\EndFunction
\renewcommand{\algletter}{DL}
\begin{algorithm}[H]
\caption{\BFTAPPEND}
\Function{$\BFTAPPEND$}{x}
\For{\textbf{each } $DL_U \in K$ such that $p_i \in U$}
\State $DL_U.\APPEND(x)$
\EndFor
\EndFunction
\end{algorithm}
\Function{BFTPROVE}{x}
\If{$p_i \notin V$}
\State \Return $\bot$
\EndIf
\State $state \gets false$
\For{\textbf{each } $DL_T \in K$}
\State $state \gets state \textbf{ OR } DL_T.\PROVE(x)$
\EndFor
\State \Return $state$
\EndFunction
% \renewcommand{\algletter}{B}
\begin{algorithm}[H]
\caption{\BFTPROVE}
\Function{$\BFTPROVE$}{x}
\State $state \gets false$
\For{\textbf{each } $DL_U \in K$}
\State $state \gets state \textbf{ OR } DL_U.\PROVE(x)$
\EndFor
\State \Return $state$
\EndFunction
\end{algorithm}
\Function{BFTREAD}{}
\State $results \gets \emptyset$
\For{\textbf{each } $DL_T \in K$}
\State $results \gets results \cup DL_T.\READ()$
\EndFor
\State \Return $results$
\EndFunction
% \renewcommand{\algletter}{C}
\begin{algorithm}[H]
\caption{\BFTREAD}
\Function{$\BFTREAD$}{$\bot$}
\State $results \gets \emptyset$
\For{\textbf{each } $DL_U \in K$}
\State $results \gets results \cup DL_U.\READ()$
\EndFor
\State \Return $results$
\EndFunction
\end{algorithm}
\end{algorithmic}
\paragraph{BFT-APPEND Validity.} Let $A\subseteq M$ be the set of distinct issuers that invoked a valid $\BFTAPPEND(x)$ Suppose by contradiction that there exists $T\in\mathcal{T}$ with $A\cap S_T=\emptyset$. Since $S_T=M\setminus T$, this implies $A\subseteq T$, hence $|A|\le |T|=t$, contradicting $|A|\ge t+1$.
% \paragraph{BFT-APPEND Validity.} Let $A\subseteq M$ be the set of distinct issuers that invoked a valid $\BFTAPPEND(x)$ Suppose by contradiction that there exists $T\in\mathcal{T}$ with $A\cap S_T=\emptyset$. Since $S_T=M\setminus T$, this implies $A\subseteq T$, hence $|A|\le |T|=t$, contradicting $|A|\ge t+1$.
\paragraph{BFT-PROVE Validity.} Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i$. Let $A\subseteq M$ be the set of distinct issuers that invoked a valid $\BFTAPPEND(x)$ before $op$ in $\Seq$.
\smallskip
\noindent\textbf{Case (i): $i\notin V$.}
For every $T\in\mathcal{T}$, we configured $\Pi_V(DL_T)=V$, hence the induced operation $DL_T.\PROVE(x)$ is invalid by \textbf{PROVE Validity} of $\DL$. Therefore $op$ is invalid.
\smallskip
\noindent\textbf{Case (ii): $|A|\ge t+1$ and $i\in V$.} Fix any $T\in\mathcal{T}$. By BFT-APPEND Validity, $A\cap S_T\neq\emptyset$. Pick $j\in A\cap S_T$. Since $j\in S_T$, the call $\BFTAPPEND^{(j)}(x)$ triggers $DL_T.\APPEND(x)$, and because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, this induces a valid $DL_T.\APPEND(x)$ that appears before the induced $DL_T.\PROVE(x)$. By \textbf{PROVE Validity} of $\DL$, the induced $DL_T.\PROVE(x)$ is invalid. As this holds for every $T\in\mathcal{T}$, there is \emph{no} component $DL_T$ where $\PROVE(x)$ is valid, so $op$ is invalid.
\smallskip
\noindent\textbf{Case (iii): $|A|\le t$ and $i\in V$.}
By BFT-APPEND Validity, there exists $T^\star\in\mathcal{T}$ such that $A\cap S_{T^\star}=\emptyset$, i.e., $A\subseteq T^\star$. For any $j\in A$, we have $j\notin S_{T^\star}$, so $\BFTAPPEND^{(j)}(x)$ does \emph{not} call $DL_{T^\star}.\APPEND(x)$. Hence no valid $DL_{T^\star}.\APPEND(x)$ appears before the induced $DL_{T^\star}.\PROVE(x)$. Since also $i\in V=\Pi_V(DL_{T^\star})$, by \textbf{PROVE Validity} of $\DL$ the induced $DL_{T^\star}.\PROVE(x)$ is valid. Therefore, there exists a component with a valid $\PROVE(x)$, so by the lifting convention $op$ is valid.
\begin{itemize}
% \item \textbf{Case (i): $i\notin V$.}
% For every $T\in\mathcal{T}$, we configured $\Pi_V(DL_T)=V$, hence the induced operation $DL_T.\PROVE(x)$ is invalid by \textbf{PROVE Validity} of $\DL$. Therefore $op$ is invalid.
\item \textbf{Case (i): $|A|\ge t+1$.}
Fix any $U\in\mathcal{U}$. $A\cap U\neq\emptyset$. Pick $j\in A\cap U$. Since $j\in u$, the call $\BFTAPPEND^{(j)}(x)$ triggers $DL_U.\APPEND(x)$, and because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, this induces a valid $DL_U.\APPEND(x)$ that appears before the induced $DL_U.\PROVE(x)$. By \textbf{PROVE Validity} of $\DL$, the induced $DL_U.\PROVE(x)$ is invalid. As this holds for every $T\in\mathcal{T}$, there is \emph{no} component $DL_U$ where $\PROVE(x)$ is valid, so $op$ is invalid.
\item \textbf{Case (ii): $|A|\le t$.}
Fix any $U\in\mathcal{U}$, there exists $U^\star\in\mathcal{U}$ such that $A\cap U^\star=\emptyset$. For any $j\in A$, we have $j\notin U^\star$, so $\BFTAPPEND^{(j)}(x)$ does \emph{not} call $DL_{U^\star}.\APPEND(x)$. Hence no valid $DL_{U^\star}.\APPEND(x)$ appears before the induced $DL_{U^\star}.\PROVE(x)$. Since also $i\in V=\Pi_V(DL_{U^\star})$, by \textbf{PROVE Validity} of $\DL$ the induced $DL_{U^\star}.\PROVE(x)$ is valid. Therefore, there exists a component with a valid $\PROVE(x)$, so by the lifting convention $op$ is valid.
\end{itemize}
\smallskip
Combining the cases yields the claimed characterization of invalidity.
\paragraph{BFT-PROVE Anti-Flickering.} Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i\in V$ that is \emph{invalid} in $\Seq$.
By BFT-PROVE Validity, since $i\in V$, this implies that there exist at least $t+1$ \emph{distinct} processes in $M$ that invoked a \emph{valid} $\BFTAPPEND(x)$ before $op$ in $\Seq$. Let $A\subseteq M$ denote that set, with $|A|\ge t+1$.
By BFT-PROVE Validity, this implies that there exist at least $t+1$ \emph{distinct} processes in $M$ that invoked a \emph{valid} $\BFTAPPEND(x)$ before $op$ in $\Seq$. Let $A\subseteq M$ denote that set, with $|A|\ge t+1$.
Fix any $T\in\mathcal{T}$ and $DL_T$ with $\Pi_M(DL_T)=S_T=M\setminus T$. By BFT-APPEND Validity, we have $A\cap S_T\neq\emptyset$. Pick $j\in A\cap S_T$. Since $j\in S_T$, the call $\BFTAPPEND^{(j)}(x)$ triggers a call $DL_T.\APPEND(x)$. Moreover, because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, the induced $DL_T.\APPEND(x)$ appears before the induced $DL_T.\PROVE(x)$ of $op$ in the projection $\Seq_T$.
Fix any $U\in\mathcal{U}$. We have $A\cap U\neq\emptyset$. Pick $j\in A\cap U$. Since $j\in U$, the call $\BFTAPPEND^{(j)}(x)$ triggers a call $DL_U.\APPEND(x)$. Moreover, because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, the induced $DL_U.\APPEND(x)$ appears before the induced $DL_U.\PROVE(x)$ of $op$ in the projection $\Seq_U$.
Hence, in $\Seq_T$, there exists a \emph{valid} $DL_T.\APPEND(x)$ that appears before the $DL_T.\PROVE(x)$ induced by $op$. By \textbf{PROVE Validity} the base $\DL$ object, the induced $DL_T.\PROVE(x)$ is therefore \emph{invalid} in $\Seq_T$.
Hence, in $\Seq_U$, there exists a \emph{valid} $DL_U.\APPEND(x)$ that appears before the $DL_U.\PROVE(x)$ induced by $op$. By \textbf{PROVE Validity} the base $\DL$ object, the induced $DL_U.\PROVE(x)$ is therefore \emph{invalid} in $\Seq_U$.
Now let $op'=\BFTPROVE(x)$ be any invocation such that $op\prec op'$ in $\Seq$. Fix again any $T\in\mathcal{T}$. Hence, the $DL_T.\PROVE(x)$ induced by $op'$ appears after the $DL_T.\PROVE(x)$ induced by $op$ in $\Seq_T$. Since the induced $DL_T.\PROVE(x)$ of $op$ is invalid, by \textbf{PROVE Anti-Flickering} of $\DL$, \emph{every} subsequent $DL_T.\PROVE(x)$ in $\Seq_T$ is invalid.
Now let $op'=\BFTPROVE(x)$ be any invocation such that $op\prec op'$ in $\Seq$. Fix again any $U\in\mathcal{U}$. Hence, the $DL_U.\PROVE(x)$ induced by $op'$ appears after the $DL_U.\PROVE(x)$ induced by $op$ in $\Seq_U$. Since the induced $DL_U.\PROVE(x)$ of $op$ is invalid, by \textbf{PROVE Anti-Flickering} of $\DL$, \emph{every} subsequent $DL_U.\PROVE(x)$ in $\Seq_U$ is invalid.
As this holds for every $T\in\mathcal{T}$, there is no component $DL_T$ in which the induced $\PROVE(x)$ of $op'$ is valid.
As this holds for every $U\in\mathcal{U}$, there is no component $DL_U$ in which the induced $\PROVE(x)$ of $op'$ is valid.
\paragraph{Local consistency.}
\paragraph{Liveness.}
\end{proof}

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\documentclass{beamer}
\usetheme{Boadilla}
\usecolortheme{orchid}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage{stackengine}
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\usebeamercolor[fg]{footline}%
\hspace{1em}%
\insertframenumber/\inserttotalframenumber
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[shadow=true]
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\begin{frame}
\tableofcontents[sections=\value{section}]
\end{frame}
}
\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{calc}
\usetikzlibrary{arrows.meta}
\title{Amaury JOLY - Winter School GDR Cybersécurité}
\author{Amaury JOLY}
\institute{Université Aix-Marseille \\ Laboratoire d'Informatique et Systèmes (LIS)}
\date{Janvier 2026}
\begin{document}
\begin{frame}
\titlepage
\vspace{-1.2em}
\begin{center}
\includegraphics[height=1cm]{images/logoamu}\hspace{1cm}%
\includegraphics[height=1.5cm]{images/logolis}\hspace{1cm}
\includegraphics[height=0.9cm]{images/logodalgo}\hspace{1cm}%
\includegraphics[height=0.6cm]{images/logoparsec}
\end{center}
\end{frame}
\begin{frame}{Who am I?}
\begin{itemize}
\item \textbf{Amaury JOLY}
\item 3rd-year PhD candidate at the \textbf{Laboratory of Informatics and Systems (LIS)}
\item \textbf{Distributed Algorithms} team
\item Supervised by \textbf{Emmanuel GODARD} and \textbf{Corentin TRAVERS}
\item \textbf{CIFRE} PhD with the company \textbf{Parsec}
\end{itemize}
\end{frame}
%------------------------------------------------------------
\begin{frame}{Parsec: company \& product}
\begin{itemize}
\item \textbf{Parsec} develops an end-to-end encrypted file sharing platform
\item Client--server solution: users collaborate through shared workspaces
\item \textbf{End-to-end encryption}: the server only sees \emph{ciphertexts}
\item \textbf{Distributed PKI management among clients} (keys/identities handled at the edge)
\item \alert{one central server stores encrypted data and redistributes it to clients}
\end{itemize}
\end{frame}
%------------------------------------------------------------
\begin{frame}{My topic (high level)}
\begin{block}{Problem statement}
\textit{``Weak consistency in a zero-trust cloud for real-time collaborative applications.''}
\end{block}
\vspace{0.4em}
\begin{itemize}
\item Real-time collaboration: concurrent updates, low latency, intermittent connectivity
\item We want \textbf{weak consistency} (e.g., eventual / causal behaviors) with clear semantics under concurrency
\item \textbf{Zero-trust cloud} (Parsec-like setting):
\begin{itemize}
\item central server trusted for \textbf{availability} only
\item server can be \textbf{honest-but-curious} (observes metadata, stores/forwards ciphertexts)
\item \textbf{no trust assumptions on clients} (they may be compromised or mutually distrustful)
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Our Model}
\begin{block}{Assumptions}
\begin{itemize}
\item The system is highly connected
\item The system is not partitionable
\item The system is asynchronous (i.e., no assumption on message delays or relative process speeds)
\item Nodes can fail by \textbf{crash} (i.e., stop functioning)
\item Nodes can be \textbf{Byzantine} (i.e., arbitrary behavior)
\item The communication network is reliable but byzantine nodes can delay or reorder messages
\item There is a Reliable Broadcast abstraction available
\end{itemize}
\end{block}
\end{frame}
\begin{frame}
\frametitle{Consistency classes}
\begin{columns}
\column{0.5\textwidth}
\resizebox{\columnwidth}{!}{
\includegraphics{images/carte_criteres.png}
}
\column{0.5\textwidth}
One approach to define the consistency of an algorithm is to place the concurrent history it produces into a consistency class. \\
We can define 3 consistency classes:
\begin{itemize}
\item \textbf{State Locality} (LS)
\item \textbf{Validity} (V)
\item \textbf{Eventual Consistency} (EC)
\end{itemize}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{State Locality (LS)}
\begin{columns}
\column{0.4\textwidth}
\include{localiteetat_hc}
\column{0.6\textwidth}
\begin{block}{Definition}
For every process $p$, there exists a linearization containing all of $p$'s read operations. \\
\end{block}
\begin{math}
\begin{array}{ll}
e.g.: & \textcolor{blue}{C_{p_1} = \{r/(0,0), r/(0,2)^w, w(2)\}}, \\
& \textcolor{red}{C_{p_2} = \{r/(0,0), r/(0,1)^w, w(1)\}}, \\
& \textcolor{blue}{r/(0,0) \bullet w(2) \bullet r/(0,2)^w} \\
& \textcolor{red}{r/(0,0) \bullet w(1) \bullet r/(0,1)^w} \\
\end{array}
\end{math}
\end{columns}
\begin{flushright}
\begin{math}
LS = \left\{
\begin{array}{l}
\mathcal{T} \rightarrow \mathcal{P}(\mathcal{H}) \\
T \rightarrow \left\{
\begin{tabular}{lll}
$H \in \mathcal{H}:$ & \multicolumn{2}{l}{$\forall p \in \mathcal{P}_H, \exists C_p \subset E_H,$} \\
& & $\hat{Q}_{T,H} \subset C_p$ \\
& $\land$ & $lin(H[p \cap C_p / C_p]) \cap L(T) \neq \emptyset$ \\
\end{tabular}
\right. \\
\end{array}
\right.
\end{math}
\end{flushright}
\end{frame}
\begin{frame}
\frametitle{Validity (V)}
\begin{columns}
\column{0.4\textwidth}
\include{validite_hc}
\column{0.6\textwidth}
\begin{block}{Definition}
There exists a co-finite set of events such that for each of them, a linearization of all write operations justifies them. \\
\end{block}
\begin{math}
\begin{array}{ll}
e.g.: & E' = \{r/(2,1)^w, r/(1,2)^w\} \\
& w(2) \bullet w(1) \bullet \textcolor{red}{r/(2,1)^w} \\
& w(1) \bullet w(2) \bullet \textcolor{red}{r/(1,2)^w} \\
\end{array}
\end{math}
\end{columns}
\begin{flushright}
\begin{math}
V = \left\{
\begin{array}{l}
\mathcal{T} \rightarrow \mathcal{P}(\mathcal{H}) \\
T \rightarrow \left\{
\begin{array}{lll}
H \in \mathcal{H}: & \multicolumn{2}{l}{|U_{T,H}| = \infty} \\
& \lor & \exists E' \subset E_H, (|E_H \setminus E'| < \infty \\
& & \land \forall e \in E', lin(H[E_H / {e}]) \cap L(T) \neq \emptyset) \\
\end{array}
\right. \\
\end{array}
\right.
\end{math}
\end{flushright}
\end{frame}
\begin{frame}
\frametitle{Eventual Consistency (EC)}
\begin{columns}
\column{0.4\textwidth}
\include{convergence_hc}%
\column{0.5\textwidth}
\begin{block}{Definition}
There exists a co-finite set of events such that for each of them, a single linearization justifies them. \\
\end{block}
\begin{math}
\begin{array}{ll}
e.g.: & E' = \{r/(1,2)^w, r/(1,2)^w\} \\
& w(1) \bullet w(2) \bullet \textcolor{red}{r/(1,2)^w} \\
\end{array}
\end{math}
\end{columns}
\begin{flushright}
\begin{math}
EC = \left\{
\begin{array}{l}
\mathcal{T} \rightarrow \mathcal{P}(\mathcal{H}) \\
T \rightarrow \left\{
\begin{array}{lll}
H \in \mathcal{H}: & \multicolumn{2}{l}{|U_{T,H}| = \infty} \\
& \lor & \exists E' \subset E_H, |E_H \setminus E'| < \infty \\
& & \land \displaystyle\bigcap_{e \in E'} \delta_T^{-1}(\lambda(e)) \neq \emptyset \\
\end{array}
\right. \\
\end{array}
\right.
\end{math}
\end{flushright}
\end{frame}
\begin{frame}{Main Work}
\begin{itemize}
\item We designed a distributed algorithm that use a \textbf{byzantine-tolerant eventually consistent register} in our model to achieve Agreement with $t < n/3$ Byzantine nodes.
\item I'm working on a framework using this algorithm to build collaborative applications for the context of a zero-trust cloud.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Thank you for your attention!}
\vfill
\begin{center}
\includegraphics[height=1cm]{images/logoamu}\hspace{1cm}%
\vspace{1em}
\includegraphics[height=1.5cm]{images/logolis}\hspace{1cm}%
\vspace{1em}
\includegraphics[height=0.9cm]{images/logodalgo}\hspace{1cm}%
\vspace{1em}
\includegraphics[height=0.6cm]{images/logoparsec}
\end{center}
\end{frame}
\end{document}

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