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% Line-number prefix configuration (A/B/C)
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\newcommand{\RB}{\textsf{RB}\xspace}
\newcommand{\res}{\mathsf{res}}
\newcommand{\ARB}{\textsf{ARB}\xspace}
\newcommand{\DL}{\textsf{DL}\xspace}
\newcommand{\APPEND}{\textsf{APPEND}}
\newcommand{\PROVE}{\textsf{PROVE}}
\newcommand{\PROVEtrace}{\textsf{prove}}
\newcommand{\READ}{\textsf{READ}}
\newcommand{\ABbroadcast}{\textsf{AB-broadcast}}
\newcommand{\ABdeliver}{\textsf{AB-deliver}}
\newcommand{\RBcast}{\textsf{RB-cast}}
\newcommand{\RBreceived}{\textsf{RB-received}}
\newcommand{\ordered}{\textsf{ordered}}
\newcommand{\Winners}{\mathsf{Winners}}
\newcommand{\Messages}{\mathsf{Messages}}
\newcommand{\ABlisten}{\textsf{AB-listen}}

\newcommand{\CANDIDATE}{\mathsf{CANDIDATE}}
\newcommand{\CLOSE}{\mathsf{CLOSE}}
\newcommand{\READGE}{\mathsf{READGE}}

\newcommand{\SHARE}{\mathsf{SHARE}}
\newcommand{\COMBINE}{\mathsf{COMBINE}}
\newcommand{\VERIFY}{\mathsf{VERIFY}}

\newcommand{\RETRIEVE}{\mathsf{RETRIEVE}}
\newcommand{\SUBMIT}{\mathsf{SUBMIT}}

\newcommand{\delivered}{\mathsf{delivered}}
\newcommand{\received}{\mathsf{received}}
\newcommand{\prop}{\mathsf{prop}}
\newcommand{\resolved}{\mathsf{resolved}}
\newcommand{\current}{\mathsf{current}}

\newcommand{\Seq}{\mathsf{Seq}}
\newcommand{\GE}{\mathsf{GE}}
\newcommand{\BFTDL}{\mathsf{BFT\text{-}DL}}

\crefname{theorem}{Theorem}{Theorems}
\crefname{lemma}{Lemma}{Lemmas}
\crefname{definition}{Definition}{Definitions}
\crefname{algorithm}{Algorithm}{Algorithms}
% Code exécuté par tout processus p_i

\begin{document}

\section{Model}
We consider a static set of $n$ processes with known identities, communicating by reliable point-to-point channels, in a complete graph. Messages are uniquely identifiable. 

\paragraph{Synchrony.} The network is asynchronous. Processes may crash; at most $f$ crashes occur.

\paragraph{Communication.} Processes can exchange through a Reliable Broadcast (\RB) primitive (defined below) which's invoked with the functions \RBcast$(m)$ and \RBreceived$(m)$. There exists a shared object called DenyList (\DL) (defined below) that is interfaced with the functions \APPEND$(x)$, \PROVE$(x)$ and \READ$()$. 

\paragraph{Notation.} Let $\Pi$ be the finite set of process identifiers and let $n \triangleq |\Pi|$. Two authorization subsets are $\Pi_M \subseteq \Pi$ (processes allowed to issue \APPEND) and $\Pi_V \subseteq \Pi$ (processes allowed to issue \PROVE). Indices $i,j \in \Pi$ refer to processes, and $p_i$ denotes the process with identifier $i$. Let $\mathcal{M}$ denote the universe of uniquely identifiable messages, with $m \in \mathcal{M}$. Let $\mathcal{R} \subseteq \mathbb{N}$ be the set of round identifiers; we write $r \in \mathcal{R}$ for a round. We use the precedence relation $\prec$ for the \DL{} linearization: $x \prec y$ means that operation $x$ appears strictly before $y$ in the linearized history of \DL. For any finite set $A \subseteq \mathcal{M}$, \ordered$(A)$ returns a deterministic total order over $A$ (e.g., lexicographic order on $(\textit{senderId},\textit{messageId})$ or on message hashes). For any round $r \in \mathcal{R}$, define $\Winners_r \triangleq \{\, j \in \Pi \mid (j,\PROVEtrace(r)) \prec \APPEND(r) \,\}$, i.e., the set of processes whose $\PROVE(r)$ appears before the first $\APPEND(r)$ in the \DL{} linearization.
We denoted by $\PROVE^{(j)}(r)$ or $\APPEND^{(j)}(r)$ the operation $\PROVE(r)$ or $\APPEND(r)$ invoked by process $j$.

% ------------------------------------------------------------------------------
\section{Primitives}

\input{2_Primitives/index.tex}


\section{Group Election Object Consensus Number}

\input{3_GEO_CS/index.tex}

% ------------------------------------------------------------------------------
\section{Target Abstraction: Atomic Reliable Broadcast (ARB)}

\input{4_ARB_Def/index.tex}


\section{ARB over RB and DL}

\input{5_ARB_with_RB_DL/index.tex}
% ------------------------------------------------------------------------------

\section{BFT-ARB over RB and DL}

\input{6_BFT_ARB/index.tex}

\section{Example of implementation of ARB with DL and RB}

We present below an example of implementation of Atomic Reliable Broadcast (\ARB) using a Reliable Broadcast (\RB) primitive and a DenyList (\DL) object according to the model and notations defined in Section 2.

\subsection{Algorithm}
% granularité diff commentaire de code et paragraphe pre algo

\begin{definition}[Closed round]\label{def:closed-round}
  Given a \DL{} linearization $H$, a round $r\in\mathcal{R}$ is \emph{closed} in $H$ iff $H$ contains an operation $\APPEND(r)$.
  Equivalently, there exists a time after which every $\PROVE(r)$ is invalid in $H$.
\end{definition}

\subsubsection{Variables}
Each process $p_i$ maintains:

%on met toutes les variables locales ici
\begin{algorithmic}
  \State $\received \gets \emptyset$ \Comment{Messages received via \RB but not yet delivered}
  \State $\delivered \gets \emptyset$ \Comment{Messages already delivered}
  \State $\prop[r][j] \gets \bot,\ \forall r,j$ \Comment{Proposal from process $j$ for round $r$}
  \State $\current \gets 0$
\end{algorithmic}

\paragraph{DenyList.} The \DL is initialized empty. We assume $\Pi_M = \Pi_V = \Pi$ (all processes can invoke \APPEND and \PROVE).

\subsubsection{Handlers and Procedures}

\renewcommand{\algletter}{A}
\begin{algorithm}[H]
  \caption{\RB handler (at process $p_i$)}\label{alg:rb-handler}
  \begin{algorithmic}[1]
    \Function{RBreceived}{$S, r, j$}
    % \State \textbf{on} $\RBreceived(S, r, j)$ \textbf{do}
      \State $\received \leftarrow \received \cup \{S\}$
      \State $\prop[r][j] \leftarrow S$ \Comment{Record sender $j$'s proposal $S$ for round $r$}
    \EndFunction
  \end{algorithmic}
\end{algorithm}

% \paragraph{} An \ABbroadcast$(m)$ chooses the next open round from the \DL view, proposes all pending messages together with the new $m$, disseminates this proposal via \RB, then executes $\PROVE(r)$ followed by $\APPEND(r)$ to freeze the winners of the round. The loop polls \DL until (i) some winner’s proposal includes $m$ in a \emph{closed} round and (ii) all winners' proposals for closed rounds are known locally, ensuring eventual inclusion and delivery.

\renewcommand{\algletter}{B}
\begin{algorithm}[H]
  \caption{\ABbroadcast$(m)$ (at process $p_i$)}\label{alg:ab-bcast}
  \begin{algorithmic}[1]
    \Function{ABbroadcast}{$m$} 
      \State $P \leftarrow \READ()$ \Comment{Fetch latest \DL state to learn recent $\PROVE$ operations}
      \State $r_{max} \leftarrow \max(\{ r' : \exists j,\ (j,\PROVE(r')) \in P \})$ \Comment{Pick current open round}
      \State $S \leftarrow (\received \setminus \delivered) \cup \{m\}$ \Comment{Propose all pending messages plus the new $m$}

      \vspace{1em}

      \For{\textbf{each}\ $r \in \{r_{max}, r_{max}+1, \cdots \}$} 
        \State $\RBcast(S, r, i)$; $\PROVE(r)$; $\APPEND(r)$;
        \State $P \leftarrow \READ()$ \Comment{Refresh local view of \DL}
        \If{($\big((i, \PROVEtrace(r)) \in P\ \vee\ (\exists j, r': (j, \PROVEtrace(r')) \in P \wedge \ m \in \prop[r'][j]))$)} 
          \State \textbf{break} \Comment{Exit loop once $m$ is included in some closed round}
        \EndIf
      \EndFor
    \EndFunction
  \end{algorithmic}
\end{algorithm}

\renewcommand{\algletter}{C}
\begin{algorithm}[H]
  \caption{\ABdeliver() at process $p_i$}\label{alg:delivery}
  \begin{algorithmic}[1]
    \Function{ABdeliver}{}
      \State $r \gets \current$
      \State $P \gets \READ()$ 
      \If{$\forall j : (j, \PROVEtrace(r)) \not\in P$} 
        \State \Return $\bot$
      \EndIf
      \State $\APPEND(r)$; $P \gets \READ()$
      \State $W_r \gets \{ j : (j, \PROVEtrace(r)) \in P \}$
      \If{$\exists j \in W_r,\ \prop[r][j] = \bot$} 
        \State \Return $\bot$
      \EndIf
      \State $M_r \gets \bigcup_{j \in W_r} \prop[r][j]$
      \State $m \gets \ordered(M_r \setminus \delivered)[0]$ \Comment{Set $m$ as the smaller message not already delivered}
      \State $\delivered \leftarrow \delivered \cup \{m\}$
      \If{$M_r \setminus \delivered = \emptyset$} \Comment{Check if all messages from round $r$ have been delivered}
        \State $\current \leftarrow \current + 1$
      \EndIf
      \State \textbf{return} $m$
    \EndFunction
  \end{algorithmic}
\end{algorithm}

% ------------------------------------------------------------------------------
\subsection{Correctness}

\begin{lemma}[Stable round closure]\label{lem:closure-stable}
If a round $r$ is closed, then there exists a linearization point $t_0$ of $\APPEND(r)$ in the \DL, and from that point on, no $\PROVE(r)$ can be valid.
Once closed, a round never becomes open again.
\end{lemma}

\begin{proof}
  By \Cref{def:closed-round}, some $\APPEND(r)$ occurs in the linearization $H$. \\
  $H$ is a total order of operations, the set of $\APPEND(r)$ operations is totally ordered, and hence there exists a smallest $\APPEND(r)$ in $H$. We denote this operation $\APPEND^{(\star)}(r)$ and $t_0$ its linearization point. \\
  By the validity property of \DL, a $\PROVE(r)$ is valid iff $\PROVE(r) \prec \APPEND^{(\star)}(r)$. Thus, after $t_0$, no $\PROVE(r)$ can be valid. \\
  $H$ is a immutable grow-only history, and hence once closed, a round never becomes open again. \\
  Hence there exists a linearization point $t_0$ of $\APPEND(r)$ in the \DL, and from that point on, no $\PROVE(r)$ can be valid and the closure is stable. 
\end{proof}

\begin{definition}[First APPEND]\label{def:first-append}
  Given a \DL{} linearization $H$, for any closed round $r\in\mathcal{R}$, we denote by $\APPEND^{(\star)}(r)$ the earliest $\APPEND(r)$ in $H$.  
\end{definition}

\begin{lemma}[Across rounds]\label{lem:across}
If there exists a $r$ such that $r$ is closed, $\forall r'$ such that $r' < r$, r' is also closed.
\end{lemma}

\begin{proof}
  \emph{Base.} For a closed round $k=0$, the set $\{k' \in \mathcal{R},\ k' < k\}$ is empty, hence the lemma is true.

  \emph{Induction.} Assume the lemma is true for round $k\geq 0$, we prove it for round $k+1$.
  
  \smallskip
  Assume $k+1$ is closed and let $\APPEND^{(\star)}(k+1)$ be the earliest $\APPEND(k+1)$ in the DL linearization $H$.
  By Lemma 1, after an $\APPEND(k)$ is in $H$, any later $\PROVE(k)$ is rejected also, a process’s program order is preserved in $H$.
  
  There are two possibilities for where $\APPEND^{(\star)}(k+1)$ is invoked.

  \begin{itemize}
    \item \textbf{Case (B6) :}
    Some process $p^\star$ executes the loop (lines B5–B11) and invokes $\APPEND^{(\star)}(k+1)$ at line B6.
    Immediately before line B6, line B5 sets $r\leftarrow r+1$, so the previous loop iteration (if any) targeted $k$. We consider two sub-cases.
    
    \begin{itemize}
      \item \emph{(i) $p^\star$ is not in its first loop iteration.}
      In the previous iteration, $p^\star$ executed $\PROVE^{(\star)}(k)$ at B6, followed (in program order) by $\APPEND^{(\star)}(k)$. 
      If round $k$ wasn't closed when $p^\star$ execute $\PROVE^{(\star)}(k)$ at B9, then the condition at B8 would be true hence the tuple $(p^\star, \PROVEtrace(k))$ should be visible in P which implies that $p^\star$ should leave the loop at round $k$, contradicting the assumption that $p^\star$ is now executing another iteration.
      Since the tuple is not visible, the $\PROVE^{(\star)}(k)$ was rejected by the DL which implies by definition an $\APPEND(k)$ already exists in $H$. Thus in this case $k$ is closed.
    
      \item \emph{(ii) $p^\star$ is in its first loop iteration.}
      To compute the value $r_{max}$, $p^\star$ must have observed one or many $(\_ , \PROVEtrace(k+1))$ in $P$ at B2/B3, issued by some processes (possibly different from $p^\star$). Let's call $p_1$ the issuer of the first $\PROVE^{(1)}(k+1)$ in the linearization $H$. \\
      When $p_1$ executed $P \gets \READ()$ at B2 and compute $r_{max}$ at B3, he observed no tuple $(\_,\PROVEtrace(k+1))$ in $P$ because he's the issuer of the first one. So when $p_1$ executed the loop (B5–B11), he run it for the round $k$, didn't seen any $(1,\PROVEtrace(k))$ in $P$ at B8, and then executed the first $\PROVE^{(1)}(k+1)$ at B6 in a second iteration. \\
      If round $k$ wasn't closed when $p_1$ execute $\PROVE^{(1)}(k)$ at B6, then the condition at B8 should be true which implies that $p_1$ sould leave the loop at round $k$, contradicting the assumption that $p_1$ is now executing $\PROVE^{(1)}(r+1)$. In this case $k$ is closed.
    \end{itemize}
    
    \item \textbf{Case (C8) :} 
    Some process invokes $\APPEND(k+1)$ at C8. 
    Line C8 is guarded by the presence of $\PROVE(\textit{next})$ in $P$ with $\textit{next}=k+1$ (C5). 
    Moreover, the local pointer $\textit{next}$ grow by increment of 1 and only advances after finishing the current round (C17), so if a process can reach $\textit{next}=k+1$ it implies that he has completed round $k$, which includes closing $k$ at C8 when $\PROVE(k)$ is observed. 
    Hence $\APPEND^\star(k+1)$ implies a prior $\APPEND(k)$ in $H$, so $k$ is closed.
  \end{itemize}
  
  \smallskip
  In all cases, $k+1$ closed implie $k$ closed. By induction on $k$, if the lemme is true for a closed $k$ then it is true for a closed $k+1$.
  Therefore, the lemma is true for all closed rounds $r$.
\end{proof}

\begin{definition}[Winner Invariant]\label{def:winner-invariant}
  For any closed round $r$, define
  \[
    \Winners_r \triangleq \{ j : \PROVE^{(j)}(r) \prec \APPEND^\star(r) \}
  \] 
  as the unique set of winners of round $r$.
\end{definition}

\begin{lemma}[Invariant view of closure]\label{lem:closure-view}
  For any closed round $r$, all correct processes eventually observe the same set of valid tuples $(\_,\PROVEtrace(r))$ in their \DL view.
\end{lemma}
  
\begin{proof}
  Let's take a closed round $r$. By \Cref{def:first-append}, there exists a unique earliest $\APPEND(r)$ in the DL linearization, denoted $\APPEND^\star(r)$.

  Consider any correct process $p$ that invokes $\READ()$ after $\APPEND^\star(r)$ in the DL linearization. Since $\APPEND^\star(r)$ invalidates all subsequent $\PROVE(r)$, the set of valid tuples $(\_,\PROVEtrace(r))$ observed by any correct process after $\APPEND^\star(r)$ is fixed and identical across all correct processes.

  Therefore, for any closed round $r$, all correct processes eventually observe the same set of valid tuples $(\_,\PROVEtrace(r))$ in their \DL view.
\end{proof}

\begin{lemma}[Well-defined winners]\label{lem:winners}
  For any correct process and round $r$, if the process computes $W_r$ at line C9, then :
  \begin{itemize}
    \item $\Winners_r$ is defined;
    \item the computed $W_r$ is exactly $\Winners_r$.
  \end{itemize}
\end{lemma}

\begin{proof}
  Let take a correct process $p_i$ that reach line C9 to compute $W_r$. \\
  By program order, $p_i$ must have executed $\APPEND^{(i)}(r)$ at C8 before, which implies by \Cref{def:closed-round} that round $r$ is closed. So by \Cref{def:winner-invariant}, $\Winners_r$ is defined. \\
  By \Cref{lem:closure-view}, all correct processes eventually observe the same set of valid tuples $(\_,\PROVEtrace(r))$ in their \DL view. Hence, when $p_i$ executes the $\READ()$ at C8 after the $\APPEND^{(i)}(r)$, it observes a set $P$ that includes all valid tuples $(\_,\PROVEtrace(r))$ such that
  \[
    W_r = \{ j : (j,\PROVEtrace(r)) \in P \} = \{j : \PROVE^{(j)}(r) \prec \APPEND^\star(r) \} = \Winners_r
  \]
\end{proof}

\begin{lemma}[No APPEND without PROVE]\label{lem:append-prove}
If some process invokes $\APPEND(r)$, then at least a process must have previously invoked $\PROVE(r)$.
\end{lemma}

\begin{proof}[Proof]
  Consider the round $r$ such that some process invokes $\APPEND(r)$. There are two possible cases
  
  \begin{itemize}
    \item \textbf{Case (B6) :}
    There exists a process $p^\star$ who's the issuer of the earliest $\APPEND^{(\star)}(r)$ in the DL linearization $H$. By program order, $p^\star$ must have previously invoked $\PROVE^{(\star)}(r)$ before invoking $\APPEND^{(\star)}(r)$ at B6. In this case, there is at least one $\PROVE(r)$ valid in $H$ issued by a correct process before $\APPEND^{(\star)}(r)$.
    
    \item \textbf{Case (C8) :}
    There exist a process $p^\star$ invokes $\APPEND^{(\star)}(r)$ at C8. Line C8 is guarded by the condition at C5, which ensures that $p$ observed some $(\_,\PROVEtrace(r))$ in $P$. In this case, there is at least one $\PROVE(r)$ valid in $H$ issued by some process before $\APPEND^{(\star)}(r)$.
  \end{itemize}
    
  In both cases, if some process invokes $\APPEND(r)$, then some process must have previously invoked $\PROVE(r)$.
\end{proof}

\begin{lemma}[No empty winners]\label{lem:nonempty}
  Let $r$ be a round, if $\Winners_r$ is defined, then $\Winners_r \neq \emptyset$.
\end{lemma}

\begin{proof}[Proof]
  If $\Winners_r$ is defined, then by \Cref{def:winner-invariant}, round $r$ is closed. By \Cref{def:closed-round}, some $\APPEND(r)$ occurs in the DL linearization. \\
  By \Cref{lem:append-prove}, at least a process must have invoked a valid $\PROVE(r)$ before $\APPEND^{(\star)}(r)$. Hence, there exists at least one $j$ such that $\{j: \PROVE^{(j)}(r) \prec \APPEND^\star(r)\}$, so $\Winners_r \neq \emptyset$.
\end{proof}

\begin{lemma}[Winners must propose]\label{lem:winners-propose}
  For any closed round $r$, $\forall j \in \Winners_r$, process $j$ must have invoked a $\RBcast(S^{(j)}, r, j)$
\end{lemma}

\begin{proof}[Proof]
  Fix a closed round $r$. By \Cref{def:winner-invariant}, for any $j \in \Winners_r$, there exist a valid $\PROVE^{(j)}(r)$ such that $\PROVE^{(j)}(r) \prec \APPEND^\star(r)$ in the DL linearization. By program order, if $j$ invoked a valid $\PROVE^{(j)}(r)$ at line B6 he must have invoked $\RBcast(S^{(j)}, r, j)$ directly before.
\end{proof}

\begin{definition}[Messages invariant]\label{def:messages-invariant}
  For any closed round $r$ and any correct process $p_i$ such that $\nexists j \in \Winners_r : prop^{[i)}[r][j] = \bot$, define
  \[
    \Messages_r \triangleq \bigcup_{j\in\Winners_r} \prop^{(i)}[r][j]
  \]
  as the unique set of messages proposed by the winners of round $r$.
\end{definition}

\begin{lemma}[Non-empty winners proposal]\label{lem:winner-propose-nonbot}
  For any closed round $r$, $\forall j \in \Winners_r$, for any correct process $p_i$, eventually $\prop^{(i)}[r][j] \neq \bot$.  
\end{lemma}

\begin{proof}[Proof]
  Fix a closed round $r$. By \Cref{def:winner-invariant}, for any $j \in \Winners_r$, there exist a valid $\PROVE^{(j)}(r)$ such that $\PROVE^{(j)}(r) \prec \APPEND^\star(r)$ in the DL linearization. By \Cref{lem:winners-propose}, $j$ must have invoked $\RBcast(S^{(j)}, r, j)$.

  Let take a process $p_i$, by \RB \emph{Validity}, every correct process eventually receives $j$'s \RB message for round $r$, which sets $\prop[r][j]$ to a non-$\bot$ value at line A3.
\end{proof}

\begin{lemma}[Eventual proposal closure]\label{lem:eventual-closure}
  If a correct process $p_i$ define $M_r$ at line C13, then for every $j \in \Winners_r$, $\prop^{(i)}[r][j] \neq \bot$. 
\end{lemma}

\begin{proof}[Proof]
  Let take a correct process $p_i$ that computes $M_r$ at line C13. By \Cref{lem:winners}, $p_i$ computes the unique winner set $\Winners_r$.

  By \Cref{lem:nonempty}, $\Winners_r \neq \emptyset$. The instruction at line C13 where $p_i$ computes $M_r$ is guarded by the condition at C10, which ensures that $p_i$ has received all \RB messages from every winner $j \in \Winners_r$. Hence, when $p_i$ computes $M_r = \bigcup_{j\in\Winners_r} \prop^{(i)}[r][j]$, we have $\prop^{(i)}[r][j] \neq \bot$ for all $j \in \Winners_r$.
\end{proof}

\begin{lemma}[Unique proposal per sender per round]\label{lem:unique-proposal}
  For any round $r$ and any process $p_i$, $p_i$ invokes at most one $\RBcast(S, r, i)$.
\end{lemma}

\begin{proof}[Proof]
  By program order, any process $p_i$ invokes $\RBcast(S, r, i)$ at line B6 must be in the loop B5–B11. No matter the number of iterations of the loop, line B5 always uses the current value of $r$ which is incremented by 1 at each turn. Hence, in any execution, any process $p_i$ invokes $\RBcast(S, r, j)$ at most once for any round $r$.
\end{proof}

\begin{lemma}[Proposal convergence]\label{lem:convergence}
  For any round $r$, for any correct processes $p_i$ that define $M_r$ at line C13, we have
  \[
    M_r^{(i)} = \Messages_r
  \]
\end{lemma}

\begin{proof}[Proof]
  Let take a correct process $p_i$ that define $M_r$ at line C13. That implies that $p_i$ has defined $W_r$ at line C9. It implies that, by \Cref{lem:winners}, $r$ is closed and $W_r = \Winners_r$. \\
  By \Cref{lem:eventual-closure}, for every $j \in \Winners_r$, $\prop^{(i)}[r][j] \neq \bot$. By \Cref{lem:unique-proposal}, each winner $j$ invokes at most one $\RBcast(S^{(j)}, r, j)$, so $\prop^{(i)}[r][j] = S^{(j)}$ is uniquely defined. Hence, when $p_i$ computes
  \[
    M_r^{(i)} = \bigcup_{j\in\Winners_r} \prop^{(i)}[r][j] = \bigcup_{j\in\Winners_r} S^{(j)} = \Messages_r.
  \]
\end{proof}

\begin{lemma}[Inclusion]\label{lem:inclusion}
  If some correct process invokes $\ABbroadcast(m)$, then there exist a round $r$ and a process $j\in\Winners_r$ such that $p_j$ invokes
  \[
  \RBcast(S, r, j)\quad\text{with}\quad m\in S.
  \]
\end{lemma}

\begin{proof}
  Fix a correct process $p_i$ that invokes $\ABbroadcast(m)$ and eventually exits the loop (B5–B11) at some round $r$. There are two possible cases.

  \begin{itemize}
    \item \textbf{Case 1:} $p_i$ exits the loop because $(i, \PROVEtrace(r)) \in P$. In this case, by \Cref{def:winner-invariant}, $p_i$ is a winner in round $r$. By program order, $p_i$ must have invoked $\RBcast(S, r, i)$ at B6 before invoking $\PROVE^{(i)}(r)$ at B7. By line B4, $m \in S$. Hence there exist a closed round $r$ and a correct process $j=i\in\Winners_r$ such that $j$ invokes $\RBcast(S, r, j)$ with $m\in S$.
    
    \item \textbf{Case 2:} $p_i$ exits the loop because $\exists j, r': (j, \PROVEtrace(r')) \in P \wedge m \in \prop[r'][j]$. In this case, by \Cref{lem:winners-propose} and \Cref{lem:unique-proposal} $j$ must have invoked a unique $\RBcast(S, r', j)$. Which set $\prop^{(i)}[r'][j] = S$ with $m \in S$.   
  \end{itemize}
    
  In both cases, if some correct process invokes $\ABbroadcast(m)$, then there exist a round $r$ and a correct process $j\in\Winners_r$ such that $j$ invokes
  \[
    \RBcast(S, r, j)\quad\text{with}\quad m\in S.
  \]
\end{proof}

\begin{lemma}[Broadcast Termination]\label{lem:bcast-termination}
  If a correct process invokes $\ABbroadcast(m)$, then he eventually exit the function and returns.
\end{lemma}

\begin{proof}[Proof]
  Let a correct process $p_i$ that invokes $\ABbroadcast(m)$. The lemma is true if $\exists r_1$ such that $r_1 \geq r_{max}$ and if $(i, \PROVEtrace(r_1)) \in P$; or if $\exists r_2$ such that $r_2 \geq r_{max}$ and if $\exists j: (j, \PROVEtrace(r_2)) \in P \wedge m \in \prop[r_2][j]$ (like guarded at B8).

  Let admit that there exists no round $r_1$ such that $p_i$ invokes a valid $\PROVE(r_1)$. In this case $p_i$ invokes infinitely many $\RBcast(S, \_, i)$ at B6 with $m \in S$ (line B4).\\
  The assumption that no $\PROVE(r_1)$ invoked by $p$ is valid implies by \DL \emph{Validity} that for every round $r' \geq r_{max}$, there exists at least one $\APPEND(r')$ in the DL linearization, hence by \Cref{lem:nonempty} for every possible round $r'$ there at least a winner. Because there is an infinite number of rounds, and a finite number of processes, there exists at least a correct process $p_k$ that invokes infinitely many valid $\PROVE(r')$ and by extension infinitely many $\ABbroadcast(\_)$. By \RB \emph{Validity}, $p_k$ eventually receives $p_i$ 's \RB messages. Let call $t_0$ the time when $p_k$ receives $p_i$ 's \RB message. \\
  At $t_0$, $p_k$ execute \Cref{alg:rb-handler} and do $\received \leftarrow \received \cup \{S\}$ with $m \in S$ (line A2). 
  For the first invocation of $\ABbroadcast(\_)$ by $p_k$ after the execution of \Cref{alg:rb-handler}, $p_k$ must include $m$ in his proposal $S$ at line B4 (because $m$ is pending in $j$'s $\received \setminus \delivered$ set). There exists a minimum round $r_2$ such that $p_k \in \Winners_{r_2}$ after $t_0$. By \Cref{lem:winner-propose-nonbot}, eventually $\prop^{(i)}[r_2][k] \neq \bot$. By \Cref{lem:unique-proposal}, $\prop^{(i)}[r_2][k]$ is uniquely defined as the set $S$ proposed by $p_k$ at B6, which by \Cref{lem:inclusion} includes $m$. Hence eventually $m \in \prop^{(i)}[r_2][k]$ and $k \in \Winners_{r_2}$. 

  So if $p_i$ is a winner he cover the condition $(i, \PROVEtrace(r_1)) \in P$. And we show that if the first condition is never satisfied, the second one will eventually be satisfied. Hence either the first or the second condition will eventually be satisfied, and $p_i$ eventually exits the loop and returns from $\ABbroadcast(m)$.
\end{proof}

\begin{lemma}[Validity]\label{lem:validity}
  If a correct process $p$ invokes $\ABbroadcast(m)$, then every correct process that invokes a infinitely often times $\ABdeliver()$ eventually delivers $m$.
\end{lemma}

\begin{proof}[Proof]
  Let $p_i$ a correct process that invokes $\ABbroadcast(m)$ and $p_q$ a correct process that infinitely invokes $\ABdeliver()$. By \Cref{lem:inclusion}, there exist a closed round $r$ and a correct process $j\in\Winners_r$ such that $p_j$ invokes
  \[
    \RBcast(S, r, j)\quad\text{with}\quad m\in S.
  \]

  By \Cref{lem:eventual-closure}, when $p_q$ computes $M_r$ at line C13, $\prop[r][j]$ is non-$\bot$ because $j \in \Winners_r$. By \Cref{lem:unique-proposal}, $p_j$ invokes at most one $\RBcast(S, r, j)$, so $\prop[r][j]$ is uniquely defined. Hence, when $p_q$ computes
  \[
    M_r = \bigcup_{k\in\Winners_r} \prop[r][k],
  \]
  we have $m \in \prop[r][j] = S$, so $m \in M_r$. By \Cref{lem:convergence}, $M_r$ is invariant so each computation of $M_r$ by any correct process that defines it includes $m$. At each invocation of $\ABdeliver()$ which deliver $m'$, $m'$ is add to $\delivered$ until $M_r \subseteq \delivered$. Once this append we're assured that there exist an invocation of $\ABdeliver()$ which return $m$. Hence $m$ is well delivered.
\end{proof}

\begin{lemma}[No duplication]\label{lem:no-duplication}
  No correct process delivers the same message more than once.
\end{lemma}

\begin{proof}
  Let consider two invokations of $\ABdeliver()$ made by the same correct process which returns $m$. Let call these two invocations respectively $\ABdeliver^{(A)}()$ and $\ABdeliver^{(B)}()$.

  When $\ABdeliver^{(A)}()$ occur, by program order and because it reach line C19 to return $m$, the process must have add $m$ to $\delivered$. Hence when $\ABdeliver^{(B)}()$ reach line C14 to extract the next message to deliver, it can't be $m$ because $m \not\in (M_r \setminus \{..., m, ...\})$. So a $\ABdeliver^{(B)}()$ which deliver $m$ can't occur.
\end{proof}

\begin{lemma}[Total order]\label{lem:total-order}
  For any two messages $m_1$ and $m_2$ delivered by correct processes, if a correct process $p_i$ delivers $m_1$ before $m_2$, then any correct process $p_j$ that delivers both $m_1$ and $m_2$ delivers $m_1$ before $m_2$.  
\end{lemma}

\begin{proof}
  Consider any correct process that delivers both $m_1$ and $m_2$. By \Cref{lem:validity}, there exist closed rounds $r'_1$ and $r'_2$ and correct processes $k_1 \in \Winners_{r'_1}$ and $k_2 \in \Winners_{r'_2}$ such that
  \[
    \RBcast(S_1, r'_1, k_1)\quad\text{with}\quad m_1\in S_1,
  \]
  \[
    \RBcast(S_2, r'_2, k_2)\quad\text{with}\quad m_2\in S_2.
  \]

  Let consider three cases :
  \begin{itemize}
    \item \textbf{Case 1:} $r_1 < r_2$. By program order, any correct process must have waited to append in $\delivered$ every messages in $M_{r_1}$ (which contains $m_1$) to increment $\current$ and eventually set $\current = r_2$ to compute $M_{r_2}$ and then invoke the $\ABdeliver()$ that returns $m_2$. Hence, for any correct process that delivers both $m_1$ and $m_2$, it delivers $m_1$ before $m_2$.
    
    \item \textbf{Case 2:} $r_1 = r_2$. By \Cref{lem:convergence}, any correct process that computes $M_{r_1}$ at line C13 computes the same set of messages $\Messages_{r_1}$. By line C14 the messages are pull in a deterministic order defined by $\ordered(\_)$. Hence, for any correct process that delivers both $m_1$ and $m_2$, it delivers $m_1$ and $m_2$ in the deterministic order defined by $\ordered(\_)$.
  \end{itemize}

  In all possible cases, any correct process that delivers both $m_1$ and $m_2$ delivers $m_1$ and $m_2$ in the same order.
\end{proof}

\begin{lemma}[Fifo Order]\label{lem:fifo-order}
  For any two messages $m_1$ and $m_2$ broadcast by the same correct process $p_i$, if $p_i$ invokes $\ABbroadcast(m_1)$ before $\ABbroadcast(m_2)$, then any correct process $p_j$ that delivers both $m_1$ and $m_2$ delivers $m_1$ before $m_2$.
\end{lemma}

\begin{proof}
  Let take two messages $m_1$ and $m_2$ broadcast by the same correct process $p_i$, with $p_i$ invoking $\ABbroadcast(m_1)$ before $\ABbroadcast(m_2)$. By \Cref{lem:validity}, there exist closed rounds $r_1$ and $r_2$ and correct processes $k_1 \in \Winners_{r_1}$ and $k_2 \in \Winners_{r_2}$ such that
  \[
    \RBcast(S_1, r_1, k_1)\quad\text{with}\quad m_1\in S_1,
  \]
  \[
    \RBcast(S_2, r_2, k_2)\quad\text{with}\quad m_2\in S_2.
  \]

  By program order, $p_i$ must have invoked $\RBcast(S_1, r_1, i)$ before $\RBcast(S_2, r_2, i)$. By \Cref{lem:unique-proposal}, any process invokes at most one $\RBcast(S, r, i)$ per round, hence $r_1 < r_2$. By \Cref{lem:total-order}, any correct process that delivers both $m_1$ and $m_2$ delivers them in a deterministic order.
  
  In all possible cases, any correct process that delivers both $m_1$ and $m_2$ delivers $m_1$ before $m_2$.
\end{proof}

\begin{theorem}[FIFO-\ARB]
Under the assumed \DL synchrony and \RB reliability, the algorithm implements FIFO Atomic Reliable Broadcast.
\end{theorem}

\begin{proof}
  We show that the algorithm satisfies the properties of FIFO Atomic Reliable Broadcast under the assumed \DL synchrony and \RB reliability.

  First, by \Cref{lem:bcast-termination}, if a correct process invokes \ABbroadcast$(m)$, then it eventually returns from this invocation.
  Moreover, \Cref{lem:validity} states that if a correct process invokes \ABbroadcast$(m)$, then every correct process that invokes \ABdeliver() infinitely often eventually delivers $m$.
  This gives the usual Validity property of \ARB.

  Concerning Integrity and No-duplicates, the construction only ever delivers messages that have been obtained from the underlying \RB primitive.
  By the Integrity property of \RB, every such message was previously \RBcast by some process, so no spurious messages are delivered.
  In addition, \Cref{lem:no-duplication} states that no correct process delivers the same message more than once.
  Together, these arguments yield the Integrity and No-duplicates properties required by \ARB.

  For the ordering guarantees, \Cref{lem:total-order} shows that for any two messages $m_1$ and $m_2$ delivered by correct processes, every correct process that delivers both $m_1$ and $m_2$ delivers them in the same order.
  Hence all correct processes share a common total order on delivered messages.
  Furthermore, \Cref{lem:fifo-order} states that for any two messages $m_1$ and $m_2$ broadcast by the same correct process, any correct process that delivers both messages delivers $m_1$ before $m_2$ whenever $m_1$ was broadcast before $m_2$.
  Thus the global total order extends the per-sender FIFO order of \ABbroadcast.

  All the above lemmas are proved under the assumptions that \DL satisfies the required synchrony properties and that the underlying primitive is a Reliable Broadcast (\RB) with Integrity, No-duplicates and Validity.
  Therefore, under these assumptions, the algorithm satisfies Validity, Integrity/No-duplicates, total order and per-sender FIFO order, and hence implements FIFO Atomic Reliable Broadcast, as claimed.
\end{proof}

\subsection{Reciprocity}
% ------------------------------------------------------------------------------

So far, we assumed the existence of a synchronous DenyList (\DL) object and
showed how to upgrade a Reliable Broadcast (\RB) primitive into FIFO Atomic
Reliable Broadcast (\ARB). We now briefly argue that, conversely, an \ARB{}
primitive is strong enough to implement a synchronous \DL object (ignoring the
anonymity property).

\paragraph{DenyList as a deterministic state machine.}
Without anonymity, the \DL specification defines a
deterministic abstract object: given a sequence $\Seq$ of operations
$\APPEND(x)$, $\PROVE(x)$, and $\READ()$, the resulting sequence of return
values and the evolution of the abstract state (set of appended elements,
history of operations) are uniquely determined.

\paragraph{State machine replication over \ARB.}
Assume a system that exports a FIFO-\ARB primitive with the guarantees that if a correct process invokes $\ABbroadcast(m)$, then every correct process eventually $\ABdeliver(m)$ and the invocation eventually returns.
Following the classical \emph{state machine replication} approach
such as described in Schneider~\cite{Schneider90}, we can implement a fault-tolerant service by ensuring the following properties:
\begin{quote}
  \textbf{Agreement.} Every nonfaulty state machine replica receives every request. \\
  \textbf{Order.} Every nonfaulty state machine replica processes the requests it receives in
  the same relative order.
\end{quote}

Which are cover by our FIFO-\ARB specification.

\paragraph{Correctness.}


\begin{theorem}[From \ARB to synchronous \DL]\label{thm:arb-to-dl}
  In an asynchronous message-passing system with crash failures, assume a
  FIFO Atomic Reliable Broadcast primitive with Integrity, No-duplicates,
  Validity, and the liveness of $\ABbroadcast$. Then, ignoring anonymity, there
  exists an implementation of a synchronous DenyList object that satisfies the
  Termination, Validity, and Anti-flickering properties.
\end{theorem}

\begin{proof}
  Because the \DL object is deterministic, all correct processes see the same
  sequence of operations and compute the same sequence of states and return
  values. We obtain:

  \begin{itemize}[leftmargin=*]
    \item \textbf{Termination.} The liveness of \ARB ensures that each
          $\ABbroadcast$ invocation by a correct process eventually returns, and
          the corresponding operation is eventually delivered and applied at all
          correct processes. Thus every $\APPEND$, $\PROVE$, and $\READ$ operation invoked by a correct process
          eventually returns.
    \item \textbf{APPEND/PROVE/READ Validity.} The local code that forms
          \ABbroadcast requests can achieve the same preconditions as in the
          abstract \DL specification (e.g., $p\in\Pi_M$, $x\in S$ for
          $\APPEND(x)$). Once an operation is delivered, its effect and return
          value are exactly those of the sequential \DL specification applied in
          the common order.
    \item \textbf{PROVE Anti-Flickering.} In the sequential \DL specification,
          once an element $x$ has been appended, all subsequent $\PROVE(x)$ are
          invalid forever. Since all replicas apply operations in the same order,
          this property holds in every execution of the replicated implementation:
          after the first linearization point of $\APPEND(x)$, no later
          $\PROVE(x)$ can return ``valid'' at any correct process.
  \end{itemize}

  Formally, we can describe the \DL object with the state machine approach for
  crash-fault, asynchronous message-passing systems with a total order broadcast
  layer~\cite{Schneider90}.
\end{proof}

\subsubsection{Example executions}

\begin{figure}[H]
  \centering
  \resizebox{0.4\textwidth}{!}{
    \input{diagrams/nonBFT_behaviour.tex}
  }
  \caption{Example execution of the ARB algorithm in a non-BFT setting}
\end{figure}


\begin{figure}
  \centering
  \resizebox{0.4\textwidth}{!}{
    \input{diagrams/BFT_behaviour.tex}
  }
  \caption{Example execution of the ARB algorithm with a byzantine process}
\end{figure}


\section{Implementation of BFT-DenyList and Threshold Cryptography}

\subsection{DenyList}

\paragraph{BFT-DenyList}
In our algorithm we use multiple DenyList as follows:

\begin{itemize}
  \item Let $\mathcal{DL} = \{DL_1, \dots, DL_k\}$ be the set of DenyList used by the algorithm.
  \item We set $k = \binom{n}{f}$.
  \item For each $i \in \{1,\dots,k\}$, let $M_i$ be the set of moderators associated with $DL_i$ according to the DenyList definition, so that $|M_i| = n-f$.
  \item Let $\mathcal{M} = \{M_1, \dots, M_k\}$. We require that the $M_i$ are pairwise distinct:
        \[
          \forall i,j \in \{1,\dots,k\},\ i \neq j \implies M_i \neq M_j.
        \]
\end{itemize}


\begin{lemma}
  $\exists M_i \in M :  \forall p \in M_i$ $p$ is correct.
\end{lemma}

\begin{proof}
  Let consider the set $F$ of faulty processes, with $|F| = f$. We can construct the set $M_i = \Pi \setminus F$ such that $|M_i| = n - |F| = n - f$. By construction, $\forall p \in M_i$ $p$ is correct.
\end{proof}

\begin{lemma}
  $\forall M_i \in M, \exists p \in M_i$ such that $p$ is correct.  
\end{lemma}

\begin{proof}
  $\forall i \in \{1, \dots, k\}, |M_i| = n-f$ with $n \geq 2f+1$. We can say that $|M_i| \geq 2f+1-f = f+1 > f$
\end{proof}

Each process can invoke the following functions :

\begin{itemize}
  \item $\READ' : () \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$
  \item $\APPEND' : \mathbb{R} \rightarrow ()$
  \item $\PROVE' : \mathbb{R} \rightarrow \{0, 1\}$
\end{itemize}

Such that :

\begin{algorithm}[H]
  \caption{$\READ'() \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$}
  \begin{algorithmic}
    \Function{READ'}{}
    \State $j \gets$ the process invoking $\READ'()$
    \State $res \gets \emptyset$
    \ForAll{$i \in \{1, \dots, k\}$}
      \State $res \gets res \cup DL_i.\READ()$
    \EndFor
    \State \Return $res$
    \EndFunction
  \end{algorithmic}
\end{algorithm}

\begin{algorithm}[H]
  \caption{$\APPEND'(\sigma) \rightarrow ()$}
  \begin{algorithmic}
    \Function{APPEND'}{$\sigma$}
    \State $j \gets$ the process invoking $\APPEND'(\sigma)$
    \ForAll{$M_i \in \{M_k \in  M : j \in M_k\}$}
      \State $DL_i.\APPEND(\sigma)$
    \EndFor
    \EndFunction
  \end{algorithmic}
\end{algorithm}

\begin{algorithm}[H]
  \caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$}
  \begin{algorithmic}
    \Function{PROVE'}{$\sigma$}
    \State $j \gets$ the process invoking $\PROVE'(\sigma)$
    \State $flag \gets false$
    \ForAll{$i \in \{1, \dots, k\}$}
      \State $flag \gets flag$ OR $DL_i.\PROVE(\sigma)$
    \EndFor
    \State \Return $flag$
    \EndFunction
  \end{algorithmic}
\end{algorithm}

\subsection{Threshold Cryptography}

We are using the Boneh-Lynn-Shacham scheme as cryptography primitive to our threshold signature scheme.
With :

\begin{itemize}
  \item $G : \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R} $
  \item $S : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R} $
  \item $V : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\} $
\end{itemize}

Such that :

\begin{itemize}
  \item $G(x) \rightarrow (pk, sk)$ : where $x$ is a random value such that $\nexists x_1, x_2: x_1 \neq x_2, G(x_1) = G(x_2)$
  \item $S(sk, m) \rightarrow \sigma_m$
  \item $V(pk, m_1, \sigma_{m_2}) \rightarrow k$ : with $k = 1$ iff $m_1 == m_2$ and $\exists x \in \mathbb{R}$ such that $G(x) \rightarrow (pk, sk)$; otherwise $k = 0$
\end{itemize}

\paragraph{threshold Scheme}

In our algorithm we are only using the following functions :

\begin{itemize}
  \item $G' : \mathbb{R} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{R} \times (\mathbb{R} \times \mathbb{R})^n$ : with $n \triangleq |\Pi|$
  \item $S' : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R}$
  \item $C' : \mathbb{R}^n \times \mathcal{R} \times \mathbb{R} \times \mathbb{R}^t \rightarrow \{\mathbb{R}, \bot\}$ : with $t \leq n$
  \item $V' : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\}$
\end{itemize}

Such that :

\begin{itemize}
  \item $G'(x, n, t) \rightarrow (pk, pk_1, sk_1, \dots, pk_n, sk_n)$ : let define $pkc = {pk_1, \dots, pk_n}$
  \item $S'(sk_i, m) \rightarrow \sigma_m^i$
  \item $C'(pkc, m_1, J, \{\sigma_{m_2}^j\}_{j \in J}) \rightarrow \sigma$ : with $J \subseteq \Pi$; and $\sigma = \sigma_{m_1}$ iff $|J| \geq t, \forall j \in J: V(pk_j, m_1, \sigma_{m_2}^j) == 1$; otherwise $\sigma = \bot$.
  \item $V'(pk, m_1, \sigma_{m_2}) \rightarrow V(pk, m_1, \sigma_{m_2})$
\end{itemize}


\bibliographystyle{plain}
\begin{thebibliography}{9}
% (left intentionally blank)
\bibitem{Schneider90}
Fred B.~Schneider.
\newblock Implementing fault-tolerant services using the state machine
  approach: a tutorial.
\newblock {\em ACM Computing Surveys}, 22(4):299--319, 1990.
\end{thebibliography}

\end{document}
