\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{algorithm} \usepackage{algpseudocode} \usepackage{graphicx} % Line-number prefix configuration (A/B/C) \renewcommand{\thealgorithm}{\Alph{algorithm}} % Float labels: Algorithm A, B, C \newcommand{\algletter}{} \algrenewcommand\alglinenumber[1]{\scriptsize\textbf{\algletter}#1} \algnewcommand\algorithmicupon{\textbf{upon}} % \algnewcommand\algorithmicdo{\textbf{do}} \algdef{SE}[UPON]{Upon}{EndUpon}[1]{% \algorithmicupon\ #1\ \algorithmicdo% }{\textbf{end upon}} \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{\res}{\mathsf{res}} \newcommand{\ARB}{\textsf{ARB}\xspace} \newcommand{\DL}{\textsf{DL}} \newcommand{\APPEND}{\textsf{APPEND}} \newcommand{\PROVE}{\textsf{PROVE}} \newcommand{\PROVEtrace}{\textsf{prove}} \newcommand{\READ}{\textsf{READ}} \newcommand{\BFTAPPEND}{\textsf{BFT\text{-}APPEND}} \newcommand{\BFTPROVE}{\textsf{BFT\text{-}PROVE}} \newcommand{\BFTREAD}{\textsf{BFT\text{-}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}{\textsf{VOTE}} \newcommand{\CLOSE}{\textsf{COMMIT}} \newcommand{\READGE}{\textsf{RESULT}} \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}{\textsf{BFT\text{-}DL}} \newcommand{\BFTGE}{\textsf{BFT\text{-}GE}} \newcommand{\BFTVOTE}{\textsf{BFT\text{-}VOTE}} \newcommand{\BFTCOMMIT}{\textsf{BFT\text{-}COMMIT}} \newcommand{\BFTRESULT}{\textsf{BFT\text{-}RESULT}} \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{Target Abstraction: Atomic Reliable Broadcast (ARB)} \input{3_ARB_Def/index.tex} \section{ARB over RB and 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} \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}