\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{thmtools} \usepackage{enumitem} \usepackage{csquotes} \usepackage[hidelinks]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \usepackage[ruled, vlined, linesnumbered, algonl, titlenumbered]{algorithm2e} \usepackage{graphicx} \SetKwProg{Fn}{Function}{}{EndFunction} \SetKwFunction{Wait}{Wait Until} \SetKwProg{Upon}{Upon}{}{EndUpon} \SetKwComment{Comment}{}{} \usepackage{tikz} \graphicspath{{diagrams/out}} \usepackage{xspace} % \usepackage{plantuml} \usepackage[fr-FR]{datetime2} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \fancyfoot[L]{Compilé le \DTMnow} \fancyfoot[C]{\thepage} \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}{Definition} \theoremstyle{remark} \newtheorem{remark}{Remark} \newcommand{\RB}{\textsf{RB}\xspace} \newcommand{\ARB}{\textsf{ARB}\xspace} \newcommand{\DL}{\textsf{DL}} \newcommand{\append}{\ensuremath{\mathsf{append}}} \newcommand{\prove}{\ensuremath{\mathsf{prove}}} % \newcommand{\PROVEtrace}{\ensuremath{\mathsf{prove}}} \newcommand{\readop}{\ensuremath{\mathsf{read}}} % Backward compatibility aliases \newcommand{\APPEND}{\append} \newcommand{\PROVE}{\prove} \newcommand{\READ}{\readop} \newcommand{\BFTAPPEND}{\textsc{bft-append}} \newcommand{\BFTPROVE}{\textsc{bft-prove}} \newcommand{\BFTREAD}{\textsc{bft-read}} \newcommand{\ABbroadcast}{\textsc{abroadcast}} \newcommand{\ABdeliver}{\textsc{adeliver}} \newcommand{\validated}{\ensuremath{\textsc{validated}}} \newcommand{\rbcast}{\ensuremath{\mathsf{rbcast}}} \newcommand{\rbreceived}{\ensuremath{\mathsf{rreceived}}} \newcommand{\order}{\ensuremath{\mathsf{order}}} % Backward compatibility aliases \newcommand{\RBcast}{\rbcast} \newcommand{\rdeliver}{\ensuremath{\mathsf{rdeliver}}} \newcommand{\send}{\ensuremath{\mathsf{send}}} \newcommand{\receive}{\ensuremath{\mathsf{receive}}} % Local variables \newcommand{\unordered}{\ensuremath{\mathit{unordered}}} \newcommand{\ordered}{\ensuremath{\mathit{ordered}}} \newcommand{\delivered}{\ensuremath{\mathit{delivered}}} \newcommand{\prop}{\ensuremath{\mathit{prop}}} \newcommand{\winners}{\ensuremath{\mathit{winners}}} \newcommand{\done}{\ensuremath{\mathit{done}}} \newcommand{\res}{\ensuremath{\mathit{res}}} \newcommand{\flag}{\ensuremath{\mathit{flag}}} %% Used in BFT-DL implementation \newcommand{\state}{\ensuremath{\mathit{state}}} \newcommand{\results}{\ensuremath{\mathit{results}}} % Invariant/concept names (used in proofs) \newcommand{\Winners}{\mathsf{Winners}} \newcommand{\Messages}{\mathsf{Messages}} \newcommand{\received}{\ensuremath{\mathsf{received}}} \newcommand{\current}{\ensuremath{\mathsf{current}}} \newcommand{\Seq}{\mathsf{Seq}} \newcommand{\BFTDL}{\textsf{BFT\text{-}DL}} \crefname{theorem}{Theorem}{Theorems} \crefname{lemma}{Lemma}{Lemmas} \crefname{definition}{Definition}{Definitions} \crefname{algorithm}{Algorithm}{Algorithms} % Pour pouvoir referencer des lignes dans le pseudocode % \crefname{ALC@Line}{Lignes}{Lignes} % \Crefname{ALC@Line}{Ligne}{Lignes} \crefname{AlgoLine}{ligne}{lignes} \Crefname{AlgoLine}{Ligne}{Lignes} % Code exécuté par tout processus p_i \begin{document} \section{Model 1: Crash} We consider a static set $\Pi$ of $n$ processes with known identities, communicating by reliable point-to-point channels, in a complete graph. Messages are uniquely identifiable. At most $f$ processes can crash, with $n \geq f$, in the standard asynchronous crash-failure message-passing model~\cite{ChandraToueg96}. \paragraph{Synchrony.} The network is asynchronous. \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 $\READ()$. \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). For any operation $F \in O$,$F_i(...)$ denotes that the operation $F$ is invoked by process $p_i$. %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{Atomic Reliable Broadcast (ARB)} \input{3_ARB_Def/index.tex} \section{ARB using DL} \input{4_ARB_with_RB_DL/index.tex} \section{BFT-ARB over RB and DL} \input{5_BFT_ARB/index.tex} % \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} % \begin{algorithm}[H] % \caption{$\READ'() \rightarrow \mathcal{L}(\mathbb{R} \times \PROVEtrace(\mathbb{R}))$} % $j \gets$ the process invoking $\READ'()$\; % $\res \gets \emptyset$\; % \ForAll{$i \in \{1, \dots, k\}$}{ % $\res \gets \res \cup DL_i.\READ()$\; % } % \Return{$\res$}\; % \end{algorithm} % \begin{algorithm}[H] % \caption{$\APPEND'(\sigma) \rightarrow ()$} % $j \gets$ the process invoking $\APPEND'(\sigma)$\; % \ForAll{$M_i \in \{M_k \in M : j \in M_k\}$}{ % $DL_i.\APPEND(\sigma)$\; % } % \end{algorithm} % \begin{algorithm}[H] % \caption{$\PROVE'(\sigma) \rightarrow \{0, 1\}$} % $j \gets$ the process invoking $\PROVE'(\sigma)$\; % $\flag \gets false$\; % \ForAll{$i \in \{1, \dots, k\}$}{ % $\flag \gets \flag$ OR $DL_i.\PROVE(\sigma)$\; % } % \Return{$\flag$}\; % \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{frey:disc23} Davide Frey, Mathieu Gestin, and Michel Raynal. \newblock The synchronization power (consensus number) of access-control objects: The case of allowlist and denylist. \newblock {\em LIPIcs, DISC 2023}, 281:21:1--21:23, 2023. \newblock doi:10.4230/LIPIcs.DISC.2023.21. \bibitem{Bracha87} Gabriel Bracha. \newblock Asynchronous byzantine agreement protocols. \newblock {\em Information and Computation}, 75(2):130--143, 1987. \bibitem{Defago2004} Xavier Defago, Andre Schiper, and Peter Urban. \newblock Total order broadcast and multicast algorithms: Taxonomy and survey. \newblock {\em ACM Computing Surveys}, 36(4):372--421, 2004. \bibitem{ChandraToueg96} Tushar Deepak Chandra and Sam Toueg. \newblock Unreliable failure detectors for reliable distributed systems. \newblock {\em Journal of the ACM}, 43(2):225--267, 1996. \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}