remove RB for crash algorithms + some syntaxes fix in BFT algo
This commit is contained in:
Amaury JOLY
@@ -48,7 +48,7 @@
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\newcommand{\DL}{\textsf{DL}}
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\newcommand{\append}{\ensuremath{\mathsf{append}}}
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\newcommand{\prove}{\ensuremath{\mathsf{prove}}}
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\newcommand{\PROVEtrace}{\ensuremath{\mathsf{prove}}}
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% \newcommand{\PROVEtrace}{\ensuremath{\mathsf{prove}}}
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\newcommand{\readop}{\ensuremath{\mathsf{read}}}
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% Backward compatibility aliases
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@@ -65,7 +65,7 @@
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\newcommand{\validated}{\ensuremath{\textsc{validated}}}
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\newcommand{\rbcast}{\ensuremath{\mathsf{rbcast}}}
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\newcommand{\rbreceived}{\ensuremath{\mathsf{rreceived}}}
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% \newcommand{\ordered}{\ensuremath{\mathsf{order}}}
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\newcommand{\order}{\ensuremath{\mathsf{order}}}
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% Backward compatibility aliases
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\newcommand{\RBcast}{\rbcast}
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@@ -117,7 +117,7 @@ We consider a static set $\Pi$ of $n$ processes with known identities, communica
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\paragraph{Synchrony.} The network is asynchronous.
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\paragraph{Communication.} Processes can exchange through a Reliable Broadcast ($\RB$) primitive (defined below) which is invoked with the functions $\RBcast(m)$ and $m = \rbreceived()$. There exists a shared object called DenyList ($\DL$) (defined below) that is interfaced with a set $O$ of operations. There exist three types of these operations: $\APPEND(x)$, $\PROVE(x)$ and $\READ()$.
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\paragraph{Communication.} Processes communicate through reliable, error-free point-to-point channels. Messages sent by a correct process to another correct process are eventually delivered without loss or corruption. There exists a shared object called DenyList ($\DL$) (defined below) that is interfaced with a set $O$ of operations. There exist three types of these operations: $\APPEND(x)$, $\PROVE(x)$ and $\READ()$.
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\paragraph{Notation.} For any indice $x$ we defined by $\Pi_x$ a subset of $\Pi$. We consider two subsets $\Pi_M$ and $\Pi_V$ two authorization subsets. Indices $i \in \Pi$ 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).
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For any operation $F \in O$,$F_i(...)$ denotes that the operation $F$ is invoked by process $p_i$.
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@@ -132,7 +132,7 @@ For any operation $F \in O$,$F_i(...)$ denotes that the operation $F$ is invoked
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\input{3_ARB_Def/index.tex}
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\section{ARB over RB and DL}
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\section{ARB using DL}
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\input{4_ARB_with_RB_DL/index.tex}
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@@ -143,156 +143,156 @@ For any operation $F \in O$,$F_i(...)$ denotes that the operation $F$ is invoked
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\section{Implementation of BFT-DenyList and Threshold Cryptography}
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% \section{Implementation of BFT-DenyList and Threshold Cryptography}
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\subsection{DenyList}
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% \subsection{DenyList}
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\paragraph{BFT-DenyList}
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In our algorithm we use multiple DenyList as follows:
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% \paragraph{BFT-DenyList}
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% In our algorithm we use multiple DenyList as follows:
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\begin{itemize}
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\item Let $\mathcal{DL} = \{DL_1, \dots, DL_k\}$ be the set of DenyList used by the algorithm.
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\item We set $k = \binom{n}{f}$.
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\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 $|M_i| = n-f$.
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\item Let $\mathcal{M} = \{M_1, \dots, M_k\}$. We require that the $M_i$ are pairwise distinct:
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\[
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\forall i,j \in \{1,\dots,k\},\ i \neq j \implies M_i \neq M_j.
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\]
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\end{itemize}
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% \begin{itemize}
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% \item Let $\mathcal{DL} = \{DL_1, \dots, DL_k\}$ be the set of DenyList used by the algorithm.
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% \item We set $k = \binom{n}{f}$.
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% \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 $|M_i| = n-f$.
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% \item Let $\mathcal{M} = \{M_1, \dots, M_k\}$. We require that the $M_i$ are pairwise distinct:
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% \[
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% \forall i,j \in \{1,\dots,k\},\ i \neq j \implies M_i \neq M_j.
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% \]
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% \end{itemize}
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\begin{lemma}
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$\exists M_i \in M : \forall p \in M_i$ $p$ is correct.
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\end{lemma}
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% \begin{lemma}
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% $\exists M_i \in M : \forall p \in M_i$ $p$ is correct.
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% \end{lemma}
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\begin{proof}
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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.
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\end{proof}
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% \begin{proof}
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% 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.
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% \end{proof}
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\begin{lemma}
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$\forall M_i \in M, \exists p \in M_i$ such that $p$ is correct.
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\end{lemma}
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% \begin{lemma}
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% $\forall M_i \in M, \exists p \in M_i$ such that $p$ is correct.
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% \end{lemma}
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\begin{proof}
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$\forall i \in \{1, \dots, k\}, |M_i| = n-f$ with $n \geq 2f+1$. We can say that $|M_i| \geq 2f+1-f = f+1 > f$
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\end{proof}
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% \begin{proof}
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% $\forall i \in \{1, \dots, k\}, |M_i| = n-f$ with $n \geq 2f+1$. We can say that $|M_i| \geq 2f+1-f = f+1 > f$
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% \end{proof}
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Each process can invoke the following functions :
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% Each process can invoke the following functions :
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\begin{itemize}
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\item $\READ' : () \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$
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\item $\APPEND' : \mathbb{R} \rightarrow ()$
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\item $\PROVE' : \mathbb{R} \rightarrow \{0, 1\}$
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\end{itemize}
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% \begin{itemize}
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% \item $\READ' : () \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$
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% \item $\APPEND' : \mathbb{R} \rightarrow ()$
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% \item $\PROVE' : \mathbb{R} \rightarrow \{0, 1\}$
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% \end{itemize}
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Such that :
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% Such that :
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% % \begin{algorithm}[H]
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% % \caption{$\READ'() \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$}
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% % \begin{algorithmic}
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% % \Function{READ'}{}
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% % \State $j \gets$ the process invoking $\READ'()$
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% % \State $res \gets \emptyset$
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% % \ForAll{$i \in \{1, \dots, k\}$}
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% % \State $res \gets res \cup DL_i.\READ()$
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% % \EndFor
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% % \State \Return $res$
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% % \EndFunction
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% % \end{algorithmic}
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% % \end{algorithm}
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% % \begin{algorithm}[H]
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% % \caption{$\APPEND'(\sigma) \rightarrow ()$}
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% % \begin{algorithmic}
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% % \Function{APPEND'}{$\sigma$}
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% % \State $j \gets$ the process invoking $\APPEND'(\sigma)$
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% % \ForAll{$M_i \in \{M_k \in M : j \in M_k\}$}
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% % \State $DL_i.\APPEND(\sigma)$
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% % \EndFor
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% % \EndFunction
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% % \end{algorithmic}
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% % \end{algorithm}
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% % \begin{algorithm}[H]
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% % \caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$}
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% % \begin{algorithmic}
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% % \Function{PROVE'}{$\sigma$}
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% % \State $j \gets$ the process invoking $\PROVE'(\sigma)$
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% % \State $flag \gets false$
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% % \ForAll{$i \in \{1, \dots, k\}$}
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% % \State $flag \gets flag$ OR $DL_i.\PROVE(\sigma)$
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% % \EndFor
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% % \State \Return $flag$
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% % \EndFunction
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% % \end{algorithmic}
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% % \end{algorithm}
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% \begin{algorithm}[H]
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% \caption{$\READ'() \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$}
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% \begin{algorithmic}
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% \Function{READ'}{}
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% \State $j \gets$ the process invoking $\READ'()$
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% \State $res \gets \emptyset$
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% \ForAll{$i \in \{1, \dots, k\}$}
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% \State $res \gets res \cup DL_i.\READ()$
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% \EndFor
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% \State \Return $res$
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% \EndFunction
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% \end{algorithmic}
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% \end{algorithm}
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% $j \gets$ the process invoking $\READ'()$\;
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% $\res \gets \emptyset$\;
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% \ForAll{$i \in \{1, \dots, k\}$}{
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% $\res \gets \res \cup DL_i.\READ()$\;
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% }
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% \Return{$\res$}\;
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% \end{algorithm}
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% \begin{algorithm}[H]
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% \caption{$\APPEND'(\sigma) \rightarrow ()$}
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% \begin{algorithmic}
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% \Function{APPEND'}{$\sigma$}
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% \State $j \gets$ the process invoking $\APPEND'(\sigma)$
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% \ForAll{$M_i \in \{M_k \in M : j \in M_k\}$}
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% \State $DL_i.\APPEND(\sigma)$
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% \EndFor
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% \EndFunction
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% \end{algorithmic}
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% \end{algorithm}
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% \begin{algorithm}[H]
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% \caption{$\APPEND'(\sigma) \rightarrow ()$}
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% $j \gets$ the process invoking $\APPEND'(\sigma)$\;
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% \ForAll{$M_i \in \{M_k \in M : j \in M_k\}$}{
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% $DL_i.\APPEND(\sigma)$\;
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% }
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% \end{algorithm}
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% \begin{algorithm}[H]
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% \caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$}
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% \begin{algorithmic}
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% \Function{PROVE'}{$\sigma$}
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% \State $j \gets$ the process invoking $\PROVE'(\sigma)$
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% \State $flag \gets false$
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% \ForAll{$i \in \{1, \dots, k\}$}
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% \State $flag \gets flag$ OR $DL_i.\PROVE(\sigma)$
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% \EndFor
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% \State \Return $flag$
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% \EndFunction
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% \end{algorithmic}
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% \end{algorithm}
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% \begin{algorithm}[H]
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% \caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$}
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% $j \gets$ the process invoking $\PROVE'(\sigma)$\;
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% $\flag \gets false$\;
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% \ForAll{$i \in \{1, \dots, k\}$}{
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% $\flag \gets \flag$ OR $DL_i.\PROVE(\sigma)$\;
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% }
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% \Return{$\flag$}\;
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% \end{algorithm}
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\begin{algorithm}[H]
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\caption{$\READ'() \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$}
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$j \gets$ the process invoking $\READ'()$\;
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$\res \gets \emptyset$\;
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\ForAll{$i \in \{1, \dots, k\}$}{
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$\res \gets \res \cup DL_i.\READ()$\;
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}
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\Return{$\res$}\;
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\end{algorithm}
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% \subsection{Threshold Cryptography}
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\begin{algorithm}[H]
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\caption{$\APPEND'(\sigma) \rightarrow ()$}
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$j \gets$ the process invoking $\APPEND'(\sigma)$\;
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\ForAll{$M_i \in \{M_k \in M : j \in M_k\}$}{
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$DL_i.\APPEND(\sigma)$\;
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}
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\end{algorithm}
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% We are using the Boneh-Lynn-Shacham scheme as cryptography primitive to our threshold signature scheme.
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% With :
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\begin{algorithm}[H]
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\caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$}
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$j \gets$ the process invoking $\PROVE'(\sigma)$\;
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$\flag \gets false$\;
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\ForAll{$i \in \{1, \dots, k\}$}{
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$\flag \gets \flag$ OR $DL_i.\PROVE(\sigma)$\;
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}
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\Return{$\flag$}\;
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\end{algorithm}
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% \begin{itemize}
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% \item $G : \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R} $
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% \item $S : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R} $
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% \item $V : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\} $
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% \end{itemize}
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\subsection{Threshold Cryptography}
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% Such that :
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We are using the Boneh-Lynn-Shacham scheme as cryptography primitive to our threshold signature scheme.
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With :
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% \begin{itemize}
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% \item $G(x) \rightarrow (pk, sk)$ : where $x$ is a random value such that $\nexists x_1, x_2: x_1 \neq x_2, G(x_1) = G(x_2)$
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% \item $S(sk, m) \rightarrow \sigma_m$
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% \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 $k = 0$
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% \end{itemize}
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\begin{itemize}
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\item $G : \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R} $
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\item $S : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R} $
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\item $V : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\} $
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\end{itemize}
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% \paragraph{threshold Scheme}
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Such that :
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% In our algorithm we are only using the following functions :
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\begin{itemize}
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\item $G(x) \rightarrow (pk, sk)$ : where $x$ is a random value such that $\nexists x_1, x_2: x_1 \neq x_2, G(x_1) = G(x_2)$
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\item $S(sk, m) \rightarrow \sigma_m$
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\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 $k = 0$
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\end{itemize}
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% \begin{itemize}
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% \item $G' : \mathbb{R} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{R} \times (\mathbb{R} \times \mathbb{R})^n$ : with $n \triangleq |\Pi|$
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% \item $S' : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R}$
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% \item $C' : \mathbb{R}^n \times \mathcal{R} \times \mathbb{R} \times \mathbb{R}^t \rightarrow \{\mathbb{R}, \bot\}$ : with $t \leq n$
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% \item $V' : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\}$
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% \end{itemize}
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\paragraph{threshold Scheme}
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% Such that :
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In our algorithm we are only using the following functions :
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\begin{itemize}
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\item $G' : \mathbb{R} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{R} \times (\mathbb{R} \times \mathbb{R})^n$ : with $n \triangleq |\Pi|$
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\item $S' : \mathbb{R} \times \mathcal{R} \rightarrow \mathbb{R}$
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\item $C' : \mathbb{R}^n \times \mathcal{R} \times \mathbb{R} \times \mathbb{R}^t \rightarrow \{\mathbb{R}, \bot\}$ : with $t \leq n$
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\item $V' : \mathbb{R} \times \mathcal{R} \times \mathbb{R} \rightarrow \{0, 1\}$
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\end{itemize}
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Such that :
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\begin{itemize}
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\item $G'(x, n, t) \rightarrow (pk, pk_1, sk_1, \dots, pk_n, sk_n)$ : let define $pkc = {pk_1, \dots, pk_n}$
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\item $S'(sk_i, m) \rightarrow \sigma_m^i$
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\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 $\sigma = \bot$.
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\item $V'(pk, m_1, \sigma_{m_2}) \rightarrow V(pk, m_1, \sigma_{m_2})$
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\end{itemize}
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% \begin{itemize}
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% \item $G'(x, n, t) \rightarrow (pk, pk_1, sk_1, \dots, pk_n, sk_n)$ : let define $pkc = {pk_1, \dots, pk_n}$
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% \item $S'(sk_i, m) \rightarrow \sigma_m^i$
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% \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 $\sigma = \bot$.
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% \item $V'(pk, m_1, \sigma_{m_2}) \rightarrow V(pk, m_1, \sigma_{m_2})$
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% \end{itemize}
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\bibliographystyle{plain}
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