1084 lines
63 KiB
TeX
1084 lines
63 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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\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}\xspace}
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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{\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}{\mathsf{CANDIDATE}}
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\newcommand{\CLOSE}{\mathsf{CLOSE}}
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\newcommand{\READGE}{\mathsf{READGE}}
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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{\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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\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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% ------------------------------------------------------------------------------
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\section{Primitives}
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\subsection{Reliable Broadcast (RB)}
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\RB provides the following properties in the model.
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\begin{itemize}[leftmargin=*]
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\item \textbf{Integrity}: Every message received was previously sent. $\forall p_i:\ \RBreceived_i(m) \Rightarrow \exists p_j:\ \RBcast_j(m)$.
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\item \textbf{No-duplicates}: No message is received more than once at any process.
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\item \textbf{Validity}: If a correct process broadcasts $m$, every correct process eventually receives $m$.
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\end{itemize}
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\subsection{Group Election Object}
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We consider a Groupe Election object ($\GE[r]$) per round $r \in \mathcal{R}$ with the following properties.
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There are three operations: $\CANDIDATE(r), \CLOSE(r)$ and $\READGE(r)$ such that:
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\begin{itemize}
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\item \textbf{Termination.} A $\CANDIDATE(r), \CLOSE(r)$ or $\READGE(r)$ operation invoked by a correct process always returns.
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\item \textbf{Election.} If there exists at least one $\CLOSE(r)$ operation and let $\CLOSE(r)^\star$ denote the first $\CLOSE(r)$ in the linearization order. If some correct process $p$ invokes $\CANDIDATE(r)$ and the invocation of $\CANDIDATE(r)$ appears before $\CLOSE(r)^\star$ in the linearization order, then $\Winners_r \neq \emptyset$.
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\item \textbf{Prefix Inclusion.} If $\CLOSE(r)^\star$ exists, then there exists a set $\Winners_r \subseteq \Pi$ such that, for any process $p_j$: $p_j \in \Winners_r$ iff $p_j$ invokes $\CANDIDATE(r)$ and its $\CANDIDATE(r)$ operation is linearized before $\CLOSE(r)^\star$.
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\item \textbf{Stability.} If $\CLOSE(r)^\star$ exists, then every $\READGE(r)$ operation linearized after
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$\CLOSE(r)^\star$ returns exactly $\Winners_r$.
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\item \textbf{READ Validity.} The invocation of $op = \READGE(r)$ by a process $p$ returns the list of valid invocations of $\CANDIDATE(r)$ that appears before $op$ in the linearization order along with the names of the processes that invoked each operation.
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\end{itemize}
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\section{Group Election Object Consensus Number}
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\begin{definition}
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We assume a synchronous DenyList (\DL) object as in~\cite{frey:disc23} with the following properties.
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The DenyList object type supports three operations: $\APPEND$, $\PROVE$, and $\READ$. These operations appear as if executed in a sequence $\Seq$ such that:
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\begin{itemize}
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\item \textbf{Termination.} A $\PROVE$, an $\APPEND$, or a $\READ$ operation invoked by a correct process always returns.
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\item \textbf{APPEND Validity.} The invocation of $\APPEND(x)$ by a process $p$ is valid if:
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\begin{itemize}
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\item $p \in \Pi_M \subseteq \Pi$; \textbf{and}
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\item $x \in S$, where $S$ is a predefined set.
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\end{itemize}
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Otherwise, the operation is invalid.
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\item \textbf{PROVE Validity.} If the invocation of a $op = \PROVE(x)$ by a correct process $p$ is not valid, then:
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\begin{itemize}
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\item $p \not\in \Pi_V \subseteq \Pi$; \textbf{or}
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\item A valid $\APPEND(x)$ appears before $op$ in $\Seq$.
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\end{itemize}
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Otherwise, the operation is valid.
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\item \textbf{PROVE Anti-Flickering.} If the invocation of a operation $op = \PROVE(x)$ by a correct process $p \in \Pi_V$ is invalid, then any $\PROVE(x)$ operation that appears after $op$ in $\Seq$ is invalid.
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\item \textbf{READ Validity.} The invocation of $op = \READ()$ by a process $p \in \pi_V$ returns the list of valid invocations of $\PROVE$ that appears before $op$ in $\Seq$ along with the names of the processes that invoked each operation.
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item \textbf{Anonymity.} Let us assume the process $p$ invokes a $\PROVE(v)$ operation. If the process $p'$ invokes a $\READ()$ operation, then $p'$ cannot learn the value $v$ unless $p$ leaks additional information.
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\end{itemize}
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We assume that $\Pi_M = \Pi_V = \Pi$ (all processes can invoke \APPEND and \PROVE).
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\end{definition}
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\begin{lemma}[From DenyList to Group Election]\label{lem:dl-to-ge}
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For any fixed value $r \in S$, one DenyList object can be used to wait-free implement a Group Election object $\GE[r]$.
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\end{lemma}
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\begin{proof}
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We implement the operations of $\GE[r]$ using the operations of the
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DenyList as follows.
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\begin{algorithmic}
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\Function{CANDIDATE}{$r$}
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\State $\PROVE(r)$
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\EndFunction
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\vspace{1em}
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\Function{CLOSE}{$r$}
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\State $\APPEND(r)$
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\EndFunction
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\vspace{1em}
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\Function{READGE}{$r$}
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\State $P \gets \READ()$
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\State \Return $\{ j : (j,\PROVEtrace(r)) \in P \}$
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\EndFunction
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\end{algorithmic}
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\paragraph{Termination.} Termination follows from the Termination property of DenyList operations. Consider now a fixed sequential history $\mathsf{Seq}$ of the DenyList.
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\paragraph{Prefix Inclusion.} Let $\APPEND(r)^\star$ denote the first valid $\APPEND(r)$ in $\mathsf{Seq}$, if it exists. From the PROVE Validity and anti-flickering properties of the DenyList, a process $p_j$ has a valid $\PROVE(r)$ in $\mathsf{Seq}$ if and only if its $\PROVE(r)$ invocation appears before $\APPEND(r)^\star$ in $\mathsf{Seq}$.
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Hence, by construction, $p_j \in \Winners_r$ iff $p_j$ invokes $\CANDIDATE(r)$ and its $\CANDIDATE(r)$ is linearized before $\CLOSE(r)^\star$ where we identify $\CLOSE(r)^\star$ with $\APPEND(r)^\star$. This is exactly the Prefix inclusion property.
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\paragraph{Stability.} Moreover, after $\APPEND(r)^\star$, no new $\PROVE(r)$ can become valid (anti-flickering), so every subsequent $\READ^\star()$
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returns the same set of valid $\PROVE(r)$ invocations. Consequently,
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every $\READGE(r)$ linearized after $\CLOSE(r)^\star$ returns the same
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set $\Winners_r$, which proves Stability.
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\paragraph{Election.} Finally, if some process invokes $\CANDIDATE(r)$ before $\CLOSE(r)^\star$, its proof is valid and thus appears in the set returned by $\READ^\star()$. Hence $\Winners_r \neq \emptyset$, which proves Election.
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\paragraph{Validity.} Validity is immediate from the construction of $\Winners_r$: a process belongs to $\Winners_r$ only if it invoked $\PROVE(r)$, i.e., only if it invoked $\CANDIDATE(r)$.
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Thus the constructed object satisfies all properties of a Group Election object.
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\end{proof}
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\begin{lemma}[From Group Election to DenyList]\label{lem:ge-to-dl}
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Fix a value $r \in S$. A Group Election object $\GE[r]$ can be used to wait-free implement an DenyList.
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\end{lemma}
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\begin{proof}
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We implement the DenyList for value $r$ as follows.
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We use one Group Election object $\GE[r]$.
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The operations are implemented as follows.
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\begin{algorithmic}
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\Function{APPEND}{$r$}
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\State $\CLOSE(r)$
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\EndFunction
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\vspace{1em}
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\Function{PROVE}{$r$}
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\State $\CANDIDATE(r)$
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\State $W_r \gets \READGE(r)$
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\If{$p \in W_r$}
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\State \Return \texttt{True}
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\Else
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\State \Return \texttt{False}
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\EndIf
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\EndFunction
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\vspace{1em}
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\Function{READ}{$ $}
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\State $W \gets \bigcup_{\forall r \in S}\READGE(r)$
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\State \Return $\{(p,r) \mid p \in W_r\}$
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\EndFunction
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\end{algorithmic}
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\paragraph{Termination.} Termination follows from the Termination property of Group Election operations. Consider now a fixed sequential history $\mathsf{Seq}$ of the Group Election.
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\paragraph{APPEND Validity.} By construction, any process invoking $\APPEND(r)$ invokes $\CLOSE(r)$. By definition of Group Election, $\CLOSE(r)$ is always valid.
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\paragraph{PROVE Validity.} By definition $\Pi_V = \Pi$, so any process invoking $\PROVE(r)$ is in $\Pi_V$. So the case $p \not\in \Pi_V$ cannot happen. Now, if some process invokes $\APPEND(r)$ before the invocation of $\PROVE(r)$ in $\mathsf{Seq}$, then by the Prefix Inclusion property of Group Election, the set $\Winners_r$ is already fixed and any subsequent $\CANDIDATE(r)$ cannot be in $\Winners_r$. Hence the invocation of $\PROVE(r)$ is invalid. Conversely, if no $\APPEND(r)$ appears before $\PROVE(r)$ in $\mathsf{Seq}$, then by the Election property of Group Election, if some correct process invoked $\CANDIDATE(r)$ before any $\CLOSE(r)$, then $\Winners_r \neq \emptyset$ and hence the invoking process belongs to $\Winners_r$. Thus its invocation of $\PROVE(r)$ is valid. This proves PROVE Validity.
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\paragraph{PROVE Anti-Flickering.} If some invocation of $\PROVE(r)$ is invalid, then some $\APPEND(r)$ must have appeared before it in $\mathsf{Seq}$. By the Prefix Inclusion property of Group Election, the set $\Winners_r$ is fixed after the first $\CLOSE(r)$, so any subsequent $\CANDIDATE(r)$ cannot be in $\Winners_r$. Hence any subsequent invocation of $\PROVE(r)$ is also invalid. This proves PROVE Anti-Flickering.
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\paragraph{READ Validity.} Finally, by construction, the invocation of $\READ()$ returns the list of valid invocations of $\PROVE$ that appear before it in $\mathsf{Seq}$ along with the names of the processes that invoked each operation. This proves READ Validity.
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\end{proof}
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\begin{theorem}[Consensus number of Group Election]\label{thm:ge-consensus}
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Let $|\Pi_V| = k$. The Group Election object type with verifier set $\Pi_V$
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has consensus number $k$. In particular, when $\Pi_V = \Pi$, the consensus
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number of Group Election is $|\Pi|$.
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\end{theorem}
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\begin{proof}
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We first recall that, for a DenyList object with $|\Pi_V| = k$ (a
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$k$-DenyList), the consensus number is exactly $k$.
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\paragraph{Lower bound.}
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By Lemma~\ref{lem:dl-to-ge}, for any fixed value $r \in S$, one $k$-DenyList object can be used to wait-free implement a Group Election object $\GE[r]$ over the same set of processes. Since the k-DenyList has consensus number $k$, it follows that the Group Election type has consensus number at least $k$.
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\paragraph{Upper bound.}
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Conversely, by Lemma~\ref{lem:ge-to-dl}, one Group Election object
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can be used to wait-free implement a DenyList object restricted to value $r$.
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This restricted DenyList satisfies the same specification as the original k-DenyList on value $r$, and in particular it has consensus number $k$. Therefore, the consensus
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number of the Group Election type cannot exceed $k$.
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\medskip
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Combining the two bounds, we obtain that the consensus number of the Group
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Election object type is exactly $k = |\Pi_V|$. When we instantiate
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$\Pi_V = \Pi$, we get that the consensus number of Group Election is $|\Pi|$.
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\end{proof}
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% ------------------------------------------------------------------------------
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\section{Target Abstraction: Atomic Reliable Broadcast (ARB)}
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Processes export \ABbroadcast$(m)$ and \ABdeliver$(m)$. \ARB requires total order:
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\begin{equation*}
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\forall m_1,m_2,\ \forall p_i,p_j:\ \ \ABdeliver_i(m_1) < \ABdeliver_i(m_2) \Rightarrow \ABdeliver_j(m_1) < \ABdeliver_j(m_2),
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\end{equation*}
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plus Integrity/No-duplicates/Validity (inherited from \RB and the construction).
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\section{ARB over RB and DL}
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\begin{theorem}[RB + Group Election implements F-ARB]\label{thm:ge-to-farb}
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In an asynchronous message-passing system with crash failure. We can wait-free implement a FIFO-Atomic Reliable Broadcast from a Reliable Broadcast (RB) primitive and one Group Election object $\GE[r]$ per round $r \in \mathbb{N}$.
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\end{theorem}
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\begin{proof}
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By Theorem~\ref{thm:ge-consensus}, the Group Election object type with verifier set $\Pi_V$ has consensus number $|\Pi_V|$. In particular, when $\Pi_V = \Pi$, using one instance $\GE[r]$ per round $r$ we can implement wait-free consensus among all processes in~$\Pi$.
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It is well known that, in a crash-prone asynchronous message-passing system, consensus and atomic (total order) broadcast are equivalent (defago et al): given consensus, one can implement atomic broadcast by using an infinite sequence of consensus instances to decide the sequence of messages to deliver, and conversely atomic broadcast can be used to implement consensus by deciding a single value in the first
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position of the total order.
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In our setting, we already have a Reliable Broadcast (RB) primitive, which provides RB-Validity, RB-Agreement, and RB-Integrity for the dissemination of messages. Using the consensus power provided by the Group Election objects, we can therefore apply the standard reduction from consensus to atomic broadcast: each position (or \emph{slot}) in the global delivery sequence is chosen by a consensus instance, whose proposals are messages that have been RB-delivered but not yet ordered. This yields an atomic
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reliable broadcast (ARB) primitive.
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To obtain FIFO-Atomic Reliable Broadcast (F-ARB), it suffices to enforce per-sender FIFO order on top of ARB. This can be done in the usual way by tagging each message broadcast by a process $p_i$ with a local sequence number $s \in \mathbb{N}$, and by ensuring that only the message with the smallest pending sequence number for $p_i$ is ever proposed to a consensus instance. As a result, for every sender $p_i$, messages with tags $(p_i,s)$ and $(p_i,t)$ with $s < t$ are decided (and thus delivered) in this order at all processes.
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Hence, RB plus Group Election objects is sufficient to implement FIFO-Atomic Reliable Broadcast.
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\end{proof}
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% ------------------------------------------------------------------------------
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% \subsection{Algorithm}
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% % granularité diff commentaire de code et paragraphe pre algo
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% \begin{definition}[Closed round]\label{def:closed-round}
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% Given a \DL{} linearization $H$, a round $r\in\mathcal{R}$ is \emph{closed} in $H$ iff $H$ contains an operation $\APPEND(r)$.
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% Equivalently, there exists a time after which every $\PROVE(r)$ is invalid in $H$.
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% \end{definition}
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% \subsubsection{Variables}
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% Each process $p_i$ maintains:
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% %on met toutes les variables locales ici
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% \begin{algorithmic}
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% \State $\received \gets \emptyset$ \Comment{Messages received via \RB but not yet delivered}
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% \State $\delivered \gets \emptyset$ \Comment{Messages already delivered}
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% \State $\prop[r][j] \gets \bot,\ \forall r,j$ \Comment{Proposal from process $j$ for round $r$}
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% \State $\current \gets 0$
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% \end{algorithmic}
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% \paragraph{DenyList.} The \DL is initialized empty. We assume $\Pi_M = \Pi_V = \Pi$ (all processes can invoke \APPEND and \PROVE).
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% \subsubsection{Handlers and Procedures}
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% \renewcommand{\algletter}{A}
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% \begin{algorithm}[H]
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% \caption{\RB handler (at process $p_i$)}\label{alg:rb-handler}
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% \begin{algorithmic}[1]
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% \Function{RBreceived}{$S, r, j$}
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% % \State \textbf{on} $\RBreceived(S, r, j)$ \textbf{do}
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% \State $\received \leftarrow \received \cup \{S\}$
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% \State $\prop[r][j] \leftarrow S$ \Comment{Record sender $j$'s proposal $S$ for round $r$}
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% \EndFunction
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% \end{algorithmic}
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% \end{algorithm}
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% % \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.
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% \renewcommand{\algletter}{B}
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% \begin{algorithm}[H]
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% \caption{\ABbroadcast$(m)$ (at process $p_i$)}\label{alg:ab-bcast}
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% \begin{algorithmic}[1]
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% \Function{ABbroadcast}{$m$}
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% \State $P \leftarrow \READ()$ \Comment{Fetch latest \DL state to learn recent $\PROVE$ operations}
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% \State $r_{max} \leftarrow \max(\{ r' : \exists j,\ (j,\PROVE(r')) \in P \})$ \Comment{Pick current open round}
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% \State $S \leftarrow (\received \setminus \delivered) \cup \{m\}$ \Comment{Propose all pending messages plus the new $m$}
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% \vspace{1em}
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|
||
% \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{BFT-ARB over RB and DL}
|
||
|
||
|
||
\subsection{Model extension}
|
||
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 or be byzantine; at most $f = \frac{n}{2} - 1$ processes can be faulty.
|
||
|
||
\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{Byzantine behaviour}
|
||
A process exhibits Byzantine behavior if it deviates arbitrarily from the specified algorithm. This includes, but is not limited to, the following actions:
|
||
|
||
\begin{itemize}
|
||
\item Invoking primitives (\RBcast, \APPEND, \PROVE, etc.) with invalid or maliciously crafted inputs.
|
||
\item Colluding with other Byzantine processes to manipulate the system's state or violate its guarantees.
|
||
\item Delaying or accelerating message delivery to specific nodes to disrupt the expected timing of operations.
|
||
\item Withholding messages or responses to create inconsistencies in the system's state.
|
||
\end{itemize}
|
||
|
||
Byzantine processes are constrained by the following:
|
||
\begin{itemize}
|
||
\item They cannot forge valid cryptographic signatures or threshold shares without the corresponding private keys.
|
||
\item They cannot violate the termination, validity, or anti-flickering properties of the \DL{} object.
|
||
\item They cannot break the integrity, no-duplicates, or validity properties of the \RB{} primitive.
|
||
\end{itemize}
|
||
|
||
\paragraph{Notation.} Let $\Pi$ be the finite set of process identifiers and let $n \triangleq |\Pi|$. Two authorization subsets are $M \subseteq \Pi$ (processes allowed to issue \APPEND) and $V \subseteq \Pi$ (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 $j$.
|
||
|
||
% ------------------------------------------------------------------------------
|
||
\subsection{Primitives}
|
||
|
||
\subsubsection{BFT DenyList}
|
||
We consider a \DL object that satisfies the following properties despite the presence of up to $f$ byzantine processes:
|
||
\begin{itemize}
|
||
\item \textbf{Termination.} Every operation $\APPEND(x)$, $\PROVE(x)$, and $\READ()$ invoked by a correct process eventually returns.
|
||
\item \textbf{APPEND/PROVE/READ Validity.} The preconditions for invoking $\APPEND(x)$, $\PROVE(x)$, and $\READ()$ are respected (cf. \#2.2). The return values of these operations conform to the sequential specification of the \DL object.
|
||
\item \textbf{APPEND Justification.} For any element $x$, if an operation $\APPEND(x)$ invoked by a correct process completes successfully, then there exists at least one valid $\PROVE(x)$ operation that that precedes this $\APPEND(x)$ in the \DL linearization.
|
||
\item \textbf{PROVE Anti-Flickering.} Once an element $x$ has been appended to the \DL by any process, all subsequent invocations of $\PROVE(x)$ by any process return ``invalid''.
|
||
\end{itemize}
|
||
|
||
\subsubsection{t-out-of-n Threshold Random Number Generator}
|
||
|
||
We consider a function that with t out of n shares any process can reconstruct a deterministic random number. The function is defined as follows:
|
||
|
||
\begin{itemize}
|
||
\item \textbf{$t$-reconstruction.} Given any subset $S$ of at least $t$ valid shares derived from the same value $r$, there exists a unique value $\sigma$ consistent with all shares in~$S$, and $\sigma$ can be efficiently reconstructed from~$S$.
|
||
\item \textbf{$(t-1)$-non-reconstructibility.} Given any subset $S$ of at most $t-1$ valid shares derived from the same value $r$, there exist two distinct values $\sigma$ and $\sigma'$ that are both consistent with all shares in~$S$. In particular, no algorithm that only sees the shares in $S$ can always distinguish whether the underlying value is $\sigma$ or $\sigma'$. \item \textbf{Per-process non-equivocation.} For any process $p$ and value $r$, there is at most one valid share that $p$ can derive from $r$. Formally, if $\sigma$ and $\sigma'$ are two valid shares output by process $p$ from the same value $r$, then $\sigma = \sigma'$. In particular, a single process cannot emit two different valid shares for the same underlying value~$r$.
|
||
\end{itemize}
|
||
|
||
\paragraph{Interface.}
|
||
For algorithmic purposes, we model the $t$-out-of-$n$ threshold random number generator
|
||
as providing the following interface to each process $p \in \Pi$.
|
||
|
||
\begin{itemize}
|
||
\item{$\mathsf{SHARE}_{p_i}(r)$:} On input a value $r$, run locally by process $p_i$, returns a valid share $\sigma_r^i$. By per-process share uniqueness, for any fixed $p_i$ and $r$ the value $\sigma_r^i$ is uniquely determined.
|
||
|
||
\item{$\mathsf{COMBINE}(S)$:} On a set $S$ of at least $t$ pairs $(p_i,\sigma_r^i)$, returns the reconstructed value $\sigma_r$. By $t$-reconstruction, this value is well defined and independent of the particular set $S$ of valid shares of size at least $t$.
|
||
|
||
\item{$\mathsf{VERIFY}(r,\sigma_{r'})$:} On input a value $r$ and a candidate value $\sigma_{r'}$, returns \textsf{true} if and only if there exists a set $S$ of at least $t$ valid shares for $r$ such that $\mathsf{Combine}(r,S) = \sigma_{r'}$, and \textsf{false} otherwise. We say that $\sigma_{r'}$ is \emph{valid for $r$} if $\mathsf{Verify}(r,\sigma_{r'})=\textsf{true}$.
|
||
\end{itemize}
|
||
|
||
|
||
\subsection{Algorithm}
|
||
|
||
\subsubsection{Variables}
|
||
Each process $p_i$ maintains the following local variables:
|
||
\begin{algorithmic}
|
||
\State $\current \gets 0$
|
||
\State $\received \gets \emptyset$
|
||
\State $\delivered \gets \emptyset$
|
||
\State $\prop[r][j] \gets \bot, \forall r, j$
|
||
\State $X_r \gets \bot, \forall r$
|
||
\State $\resolved[r] \gets \bot, \forall r$
|
||
\end{algorithmic}
|
||
|
||
\renewcommand{\algletter}{D}
|
||
\begin{algorithm}[H]
|
||
\caption{\ABbroadcast}\label{alg:ab-cast}
|
||
\begin{algorithmic}[1]
|
||
\Function{ABcast}{$m$}
|
||
\State $S \gets (\received \setminus \delivered) \cup \{m\}$
|
||
\State $\RBcast(prop, S, r, i)$
|
||
\State \textbf{wait until} $|X_r| \geq f+1$
|
||
\State $\sigma_r \gets \COMBINE(X_r)$
|
||
\State $\PROVE(\sigma_r); \APPEND(\sigma_r);$
|
||
\State $\RBcast(submit, S, \sigma_r, r, i)$
|
||
\EndFunction
|
||
\end{algorithmic}
|
||
\end{algorithm}
|
||
|
||
\renewcommand{\algletter}{E}
|
||
\begin{algorithm}[H]
|
||
\caption{\ABdeliver}\label{alg:ab-deliver}
|
||
\begin{algorithmic}[1]
|
||
\Function{$\ABdeliver$}{}
|
||
\State $r \gets \current; \sigma_r \gets \resolved[r];$
|
||
\If{$\sigma_r == \bot$}
|
||
\State \Return $\bot$
|
||
\EndIf
|
||
\State $P \gets \READ()$
|
||
\If{$\forall j : (j,prove(\sigma_r)) \not\in P$}
|
||
\State \Return $\bot$
|
||
\EndIf
|
||
\State $\APPEND(\sigma_r); P \gets \READ();$
|
||
\State $W_r \gets \{j : (j, \PROVEtrace(\sigma_r)) \in P\}$
|
||
\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 $m \gets \ordered(M_r)[0]$
|
||
\State $\delivered \gets \delivered \cup \{m\};$
|
||
\If{$M_r \setminus \delivered = \emptyset$}
|
||
\State $\current \gets \current + 1;$
|
||
\EndIf
|
||
\State \Return $m$
|
||
\EndFunction
|
||
\end{algorithmic}
|
||
|
||
\end{algorithm}
|
||
|
||
\renewcommand{\algletter}{F}
|
||
\begin{algorithm}[H]
|
||
\caption{RBreceived handler}\label{alg:rb-handler}
|
||
\begin{algorithmic}[1]
|
||
\Function{RBrcvd}{$prop, S_j, r_j, j$}
|
||
\If{$r_j \geq r$}
|
||
\State $\prop[r_j][j] = S_j$
|
||
\State $\sigma^i_{r_j} \gets \SHARE(r_j)$
|
||
\State $send_j(r, \sigma^i_{r_j})$
|
||
\EndIf
|
||
\EndFunction
|
||
\end{algorithmic}
|
||
\end{algorithm}
|
||
|
||
|
||
\renewcommand{\algletter}{G}
|
||
\begin{algorithm}[H]
|
||
\caption{RBreceived handler}\label{alg:rb-handler-2}
|
||
\begin{algorithmic}[1]
|
||
\Function{RBrcvd}{$submit, S_j, \sigma_{r_j}, r_j, j$}
|
||
\If{$\VERIFY(r_j, \sigma_{r_j})$}
|
||
\State $\resolved[r_j] \gets \sigma_{r_j}$
|
||
\EndIf
|
||
\EndFunction
|
||
\end{algorithmic}
|
||
\end{algorithm}
|
||
|
||
|
||
\renewcommand{\algletter}{H}
|
||
\begin{algorithm}[H]
|
||
\caption{Share received handler}\label{alg:share-handler}
|
||
\begin{algorithmic}[1]
|
||
\Function{received}{$r_j, \sigma^j_{r_j}, j$}
|
||
\If{$r_j == r$}
|
||
\State $X_r \gets X_r \cup \sigma^j_{r}$
|
||
\EndIf
|
||
\EndFunction
|
||
\end{algorithmic}
|
||
\end{algorithm}
|
||
|
||
\subsection{Example execution}
|
||
|
||
\begin{figure}[H]
|
||
\centering
|
||
\input{diagrams/classic_seq.tex}
|
||
\caption{Expected Executions of P1 willing to send a message at round r}
|
||
\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}
|