274 lines
10 KiB
TeX
274 lines
10 KiB
TeX
\documentclass[11pt]{article}
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\usepackage[margin=1in]{geometry}
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage{lmodern}
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\usepackage{microtype}
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\usepackage{amsmath,amssymb,amsthm,mathtools}
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\usepackage{thmtools}
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\usepackage{enumitem}
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\usepackage{csquotes}
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\usepackage[hidelinks]{hyperref}
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\usepackage[nameinlink,noabbrev]{cleveref}
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\usepackage{algorithm}
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\usepackage{algpseudocode}
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\usepackage{graphicx}
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% Line-number prefix configuration (A/B/C)
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\renewcommand{\thealgorithm}{\Alph{algorithm}} % Float labels: Algorithm A, B, C
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\newcommand{\algletter}{}
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\algrenewcommand\alglinenumber[1]{\scriptsize\textbf{\algletter}#1}
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\algnewcommand\algorithmicupon{\textbf{upon}}
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% \algnewcommand\algorithmicdo{\textbf{do}}
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\algdef{SE}[UPON]{Upon}{EndUpon}[1]{%
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\algorithmicupon\ #1\ \algorithmicdo%
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}{\textbf{end upon}}
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\usepackage{tikz}
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\graphicspath{{diagrams/out}}
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\usepackage{xspace}
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% \usepackage{plantuml}
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\usepackage[fr-FR]{datetime2}
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\usepackage{fancyhdr}
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\pagestyle{fancy}
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\fancyhf{}
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\fancyfoot[L]{Compilé le \DTMnow}
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\fancyfoot[C]{\thepage}
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\renewcommand{\headrulewidth}{0pt}
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\renewcommand{\footrulewidth}{0pt}
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\theoremstyle{plain}
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\newtheorem{theorem}{Theorem}
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\theoremstyle{definition}
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\newtheorem{definition}{Definition}
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\theoremstyle{remark}
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\newtheorem{remark}{Remark}
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\newcommand{\RB}{\textsf{RB}\xspace}
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\newcommand{\res}{\mathsf{res}}
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\newcommand{\ARB}{\textsf{ARB}\xspace}
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\newcommand{\DL}{\textsf{DL}}
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\newcommand{\APPEND}{\textsf{APPEND}}
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\newcommand{\PROVE}{\textsf{PROVE}}
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\newcommand{\PROVEtrace}{\textsf{prove}}
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\newcommand{\READ}{\textsf{READ}}
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\newcommand{\BFTAPPEND}{\textsf{BFT\text{-}APPEND}}
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\newcommand{\BFTPROVE}{\textsf{BFT\text{-}PROVE}}
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\newcommand{\BFTREAD}{\textsf{BFT\text{-}READ}}
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\newcommand{\ABbroadcast}{\textsf{AB-broadcast}}
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\newcommand{\ABdeliver}{\textsf{AB-deliver}}
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\newcommand{\RBcast}{\textsf{RB-cast}}
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\newcommand{\RBreceived}{\textsf{RB-received}}
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\newcommand{\ordered}{\textsf{ordered}}
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\newcommand{\Winners}{\mathsf{Winners}}
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\newcommand{\Messages}{\mathsf{Messages}}
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\newcommand{\ABlisten}{\textsf{AB-listen}}
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\newcommand{\CANDIDATE}{\textsf{VOTE}}
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\newcommand{\CLOSE}{\textsf{COMMIT}}
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\newcommand{\READGE}{\textsf{RESULT}}
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\newcommand{\SHARE}{\mathsf{SHARE}}
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\newcommand{\COMBINE}{\mathsf{COMBINE}}
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\newcommand{\VERIFY}{\mathsf{VERIFY}}
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\newcommand{\RETRIEVE}{\mathsf{RETRIEVE}}
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\newcommand{\SUBMIT}{\mathsf{SUBMIT}}
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\newcommand{\delivered}{\mathsf{delivered}}
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\newcommand{\received}{\mathsf{received}}
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\newcommand{\prop}{\mathsf{prop}}
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\newcommand{\resolved}{\mathsf{resolved}}
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\newcommand{\current}{\mathsf{current}}
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\newcommand{\Seq}{\mathsf{Seq}}
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\newcommand{\GE}{\mathsf{GE}}
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\newcommand{\BFTDL}{\textsf{BFT\text{-}DL}}
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\newcommand{\BFTGE}{\textsf{BFT\text{-}GE}}
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\newcommand{\BFTVOTE}{\textsf{BFT\text{-}VOTE}}
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\newcommand{\BFTCOMMIT}{\textsf{BFT\text{-}COMMIT}}
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\newcommand{\BFTRESULT}{\textsf{BFT\text{-}RESULT}}
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\crefname{theorem}{Theorem}{Theorems}
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\crefname{lemma}{Lemma}{Lemmas}
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\crefname{definition}{Definition}{Definitions}
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\crefname{algorithm}{Algorithm}{Algorithms}
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% Code exécuté par tout processus p_i
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\begin{document}
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\section{Model}
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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.
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\paragraph{Synchrony.} The network is asynchronous. Processes may crash; at most $f$ crashes occur.
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\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$()$.
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\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.
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We denoted by $\PROVE^{(j)}(r)$ or $\APPEND^{(j)}(r)$ the operation $\PROVE(r)$ or $\APPEND(r)$ invoked by process $j$.
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\section{Primitives}
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\input{2_Primitives/index.tex}
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\section{Target Abstraction: Atomic Reliable Broadcast (ARB)}
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\input{3_ARB_Def/index.tex}
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\section{ARB over RB and DL}
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\input{4_ARB_with_RB_DL/index.tex}
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\section{BFT-ARB over RB and DL}
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\input{5_BFT_ARB/index.tex}
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\section{Implementation of BFT-DenyList and Threshold Cryptography}
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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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\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{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{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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\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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\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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\subsection{Threshold Cryptography}
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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 : \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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Such that :
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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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\paragraph{threshold Scheme}
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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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\bibliographystyle{plain}
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\begin{thebibliography}{9}
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% (left intentionally blank)
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\bibitem{Schneider90}
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Fred B.~Schneider.
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\newblock Implementing fault-tolerant services using the state machine
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approach: a tutorial.
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\newblock {\em ACM Computing Surveys}, 22(4):299--319, 1990.
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\end{thebibliography}
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\end{document}
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