281 lines
15 KiB
TeX
281 lines
15 KiB
TeX
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\subsection{Model extension}
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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 or be byzantine; at most $f = \frac{n}{2} - 1$ processes can be faulty.
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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{Byzantine behaviour}
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A process exhibits Byzantine behavior if it deviates arbitrarily from the specified algorithm. This includes, but is not limited to, the following actions:
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\begin{itemize}
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\item Invoking primitives (\RBcast, \APPEND, \PROVE, etc.) with invalid or maliciously crafted inputs.
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\item Colluding with other Byzantine processes to manipulate the system's state or violate its guarantees.
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\item Delaying or accelerating message delivery to specific nodes to disrupt the expected timing of operations.
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\item Withholding messages or responses to create inconsistencies in the system's state.
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\end{itemize}
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Byzantine processes are constrained by the following:
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\begin{itemize}
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\item They cannot forge valid cryptographic signatures or threshold shares without the corresponding private keys.
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\item They cannot violate the termination, validity, or anti-flickering properties of the \DL{} object.
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\item They cannot break the integrity, no-duplicates, or validity properties of the \RB{} primitive.
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\end{itemize}
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\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.
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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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\subsection{Primitives}
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\subsubsection{t-BFT-DL}
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We consider a t-Byzantine Fault Tolerant DenyList (t-$\BFTDL$) with the following properties.
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There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that :
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\paragraph{Termination.} Every operation $\BFTAPPEND(x)$, $\BFTPROVE(x)$, and $\BFTREAD()$ invoked by a correct process always returns.
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% \paragraph{APPEND Validity.} The invocation of $\BFTAPPEND(x)$ by a correct process $p_i$ is valid iff $i \in \Pi_M$. Otherwise the operation is invalid.
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\paragraph{PROVE Validity.} The invocation of $op = \BFTPROVE(x)$ by a correct process is valid iff there exist a set of correct process $C$ such that $\forall c \in C$, $c$ invoke $op_2 = \BFTAPPEND(x)$ with $op_2 \prec op_1$ and $|C| \leq t$
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\paragraph{PROVE Anti-Flickering.} If the invocation of a operation $op = \BFTPROVE(x)$ by a correct process $p \in \Pi_V$ is invalid, then any $\BFTPROVE(x)$ operation that appears after $op$ in $\Seq$ is invalid.
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\paragraph{READ Validity.} The invocation of $op = \BFTREAD()$ by a process $p$ returns the list of valid invocations of $\BFTPROVE$ that appears before $op$ in $\Seq$ along with the names of the processes that invoked each operation.
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\paragraph{Local consistency.} For all correct process $p_i$ such that $p_i$ invoke an valid $\BFTPROVE(x)$ before a $P \gets \BFTREAD()$ operation in his local execution. Then the set of valid $\BFTPROVE(x)$ in $P$ must contain the previous valid $\BFTPROVE(x)$.
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\paragraph{Liveness} For all correct process $p_i$ such that $p_i$ invoke an invalid $\BFTPROVE(x)$ before a $P \gets \BFTREAD()$ operation in his local execution. Then the set of valid $\BFTPROVE(x)$ in $P$ must be the same for any $p_i$.
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% \subsubsection{t-BFT-GE}
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% We consider a t-Byzantine Fault Tolerant Group Election Object (t-$\BFTGE[r]$) per round $r \in \mathcal{R}$ with the following properties.
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% There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$.
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% \paragraph{Termination.} Every operation $\BFTVOTE(i, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$ invoked by a correct process always returns.
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% \paragraph{Stability.} If there exist at least $n-f$ invocations of $\BFTCOMMIT(r)$ by distincts processes and let call $\BFTCOMMIT(r)^\star$ the $(n-f)^{th}$ such invocation in the linearization of $\Seq$. Then any invocation of $\BFTRESULT(r)$ that appears after $\BFTCOMMIT(r)^\star$ in $\Seq$ returns the same set of processes $W_r$.
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% \paragraph{VOTE-Validity.} The invocation of $\BFTVOTE(j, r)$ by a correct process is not valid if $\BFTCOMMIT(r)^\star$ appears before in $\Seq$. Otherwise, the operation is valid.
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% \paragraph{Election.} If at least $f+1$ correct processes invoked a valid $\BFTVOTE(j, r)$ for the same process $j$ then $j$ will be enventually included in the set $W_r$ returned by $\BFTRESULT(r)$.
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\subsection{DL $\Rightarrow$ t-BFT-DL}
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\begin{lemma}
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For any fixed value $t$ such that $3t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object.
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\end{lemma}
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\begin{proof}
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Fix $3t < |M|$. Let
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\[
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\mathcal{U} = \{\, U \subseteq M \mid |U| = |M| - t \,\}.
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\]
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For each $U \in \mathcal{U}$, we instantiate one DenyList object $DL_U$ whose authorization sets are
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\[
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\Pi_M(DL_T) = S_T = U
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\qquad\text{and}\qquad
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\Pi_V(DL_T) = V.
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\]
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Let
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\[
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K = \{\, DL_U \mid U \in \mathcal{U} \,\},
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\qquad\text{so that}\qquad
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|U| = \binom{|M|}{|M| - t}.
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\]
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\begin{algorithmic}[1]
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% \State $K \gets \{DL_T : T \subseteq M, |T|=t\}$
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\renewcommand{\algletter}{DL}
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\begin{algorithm}[H]
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\caption{\BFTAPPEND}
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\Function{$\BFTAPPEND$}{x}
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\For{\textbf{each } $DL_U \in K$ such that $p_i \in U$}
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\State $DL_U.\APPEND(x)$
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\EndFor
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\EndFunction
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\end{algorithm}
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% \renewcommand{\algletter}{B}
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\begin{algorithm}[H]
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\caption{\BFTPROVE}
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\Function{$\BFTPROVE$}{x}
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\State $state \gets false$
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\For{\textbf{each } $DL_U \in K$}
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\State $state \gets state \textbf{ OR } DL_U.\PROVE(x)$
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\EndFor
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\State \Return $state$
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\EndFunction
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\end{algorithm}
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% \renewcommand{\algletter}{C}
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\begin{algorithm}[H]
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\caption{\BFTREAD}
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\Function{$\BFTREAD$}{$\bot$}
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\State $results \gets \emptyset$
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\For{\textbf{each } $DL_U \in K$}
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\State $results \gets results \cup DL_U.\READ()$
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\EndFor
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\State \Return $results$
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\EndFunction
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\end{algorithm}
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\end{algorithmic}
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% \paragraph{BFT-APPEND Validity.} Let $A\subseteq M$ be the set of distinct issuers that invoked a valid $\BFTAPPEND(x)$ Suppose by contradiction that there exists $T\in\mathcal{T}$ with $A\cap S_T=\emptyset$. Since $S_T=M\setminus T$, this implies $A\subseteq T$, hence $|A|\le |T|=t$, contradicting $|A|\ge t+1$.
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\paragraph{BFT-PROVE Validity.} Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i$. Let $A\subseteq M$ be the set of distinct issuers that invoked a valid $\BFTAPPEND(x)$ before $op$ in $\Seq$.
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\begin{itemize}
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% \item \textbf{Case (i): $i\notin V$.}
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% For every $T\in\mathcal{T}$, we configured $\Pi_V(DL_T)=V$, hence the induced operation $DL_T.\PROVE(x)$ is invalid by \textbf{PROVE Validity} of $\DL$. Therefore $op$ is invalid.
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\item \textbf{Case (i): $|A|\ge t+1$.}
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Fix any $U\in\mathcal{U}$. $A\cap U\neq\emptyset$. Pick $j\in A\cap U$. Since $j\in u$, the call $\BFTAPPEND^{(j)}(x)$ triggers $DL_U.\APPEND(x)$, and because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, this induces a valid $DL_U.\APPEND(x)$ that appears before the induced $DL_U.\PROVE(x)$. By \textbf{PROVE Validity} of $\DL$, the induced $DL_U.\PROVE(x)$ is invalid. As this holds for every $T\in\mathcal{T}$, there is \emph{no} component $DL_U$ where $\PROVE(x)$ is valid, so $op$ is invalid.
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\item \textbf{Case (ii): $|A|\le t$.}
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Fix any $U\in\mathcal{U}$, there exists $U^\star\in\mathcal{U}$ such that $A\cap U^\star=\emptyset$. For any $j\in A$, we have $j\notin U^\star$, so $\BFTAPPEND^{(j)}(x)$ does \emph{not} call $DL_{U^\star}.\APPEND(x)$. Hence no valid $DL_{U^\star}.\APPEND(x)$ appears before the induced $DL_{U^\star}.\PROVE(x)$. Since also $i\in V=\Pi_V(DL_{U^\star})$, by \textbf{PROVE Validity} of $\DL$ the induced $DL_{U^\star}.\PROVE(x)$ is valid. Therefore, there exists a component with a valid $\PROVE(x)$, so by the lifting convention $op$ is valid.
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\end{itemize}
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\smallskip
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Combining the cases yields the claimed characterization of invalidity.
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\paragraph{BFT-PROVE Anti-Flickering.} Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i\in V$ that is \emph{invalid} in $\Seq$.
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By BFT-PROVE Validity, this implies that there exist at least $t+1$ \emph{distinct} processes in $M$ that invoked a \emph{valid} $\BFTAPPEND(x)$ before $op$ in $\Seq$. Let $A\subseteq M$ denote that set, with $|A|\ge t+1$.
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Fix any $U\in\mathcal{U}$. We have $A\cap U\neq\emptyset$. Pick $j\in A\cap U$. Since $j\in U$, the call $\BFTAPPEND^{(j)}(x)$ triggers a call $DL_U.\APPEND(x)$. Moreover, because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, the induced $DL_U.\APPEND(x)$ appears before the induced $DL_U.\PROVE(x)$ of $op$ in the projection $\Seq_U$.
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Hence, in $\Seq_U$, there exists a \emph{valid} $DL_U.\APPEND(x)$ that appears before the $DL_U.\PROVE(x)$ induced by $op$. By \textbf{PROVE Validity} the base $\DL$ object, the induced $DL_U.\PROVE(x)$ is therefore \emph{invalid} in $\Seq_U$.
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Now let $op'=\BFTPROVE(x)$ be any invocation such that $op\prec op'$ in $\Seq$. Fix again any $U\in\mathcal{U}$. Hence, the $DL_U.\PROVE(x)$ induced by $op'$ appears after the $DL_U.\PROVE(x)$ induced by $op$ in $\Seq_U$. Since the induced $DL_U.\PROVE(x)$ of $op$ is invalid, by \textbf{PROVE Anti-Flickering} of $\DL$, \emph{every} subsequent $DL_U.\PROVE(x)$ in $\Seq_U$ is invalid.
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As this holds for every $U\in\mathcal{U}$, there is no component $DL_U$ in which the induced $\PROVE(x)$ of $op'$ is valid.
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\paragraph{Local consistency.}
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\paragraph{Liveness.}
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\end{proof}
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\subsection{t-BFT-DL $\Rightarrow$ t-BFT-GE}
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\begin{lemma}
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For any fixed value $r \in S$, multiple BFT-DenyList Object can be used to implement a BFT-Group Election Object.
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\end{lemma}
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\begin{proof}
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\begin{algorithmic}
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\State $Y[i]$ \Comment{Is a set of $n$ $\BFTDL$ with $\Pi_M = \Pi_V = \Pi$}
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\vspace{1em}
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\Function{BFTVOTE}{j, r}
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\EndFunction
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\vspace{1em}
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\Function{BFTCOMMIT}{r}
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\EndFunction
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\vspace{1em}
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\Function{BFTRESULT}{r}
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\State $Z \gets \emptyset$
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\For{\textbf{each } $j \in \Pi$}
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\If{$|\{(\_, \PROVEtrace(\_, r)) \in Y[j].\BFTREAD(r)\}| \geq n-f$}
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\State $P \gets \BFTREAD()$
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\State \Return $\{j : (j, \PROVEtrace), \}$
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\EndIf
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\EndFor
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\EndFunction
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\vspace{1em}
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\end{algorithmic}
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\end{proof}
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\subsection{Algorithm}
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\subsubsection{Variables}
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Each process $p_i$ maintains the following local variables:
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\begin{algorithmic}
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\State $\current \gets 0$
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\State $\received \gets \emptyset$
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\State $\delivered \gets \emptyset$
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\State $\prop[r][j] \gets \bot, \forall r, j$
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% \State $X_r \gets \bot, \forall r$
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\State $W_r \gets \bot, \forall r$
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\State $\resolved[r] \gets \bot, \forall r$
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\end{algorithmic}
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\renewcommand{\algletter}{A}
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\begin{algorithm}[H]
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\caption{ABroadcast$(m)$}
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\begin{algorithmic}[1]
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\Function{ABroadcast}{$m$}
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\State $r \gets \current$
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\State $S \gets (\received \cup \{m\}$)
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\For{\textbf{each}\ $r \in \{\current, \current +1, \dots\}$}
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\State $\RBcast(i, PROP, S, r)$
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\State \textbf{wait} until $|W_r| \geq n - f$ where $W_r = \BFTRESULT[r]$
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\State $\BFTCOMMIT(r)$
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\State \textbf{wait} until $|\resolved[r]| \geq n - f$
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\State $W_r \gets \BFTRESULT[r]$
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\If{$i \in W_r \vee (\exists j, r': j \in W_r \wedge \prop[r'][j] \ni m)$}
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\State \textbf{break}
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\EndIf
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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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\renewcommand{\algletter}{B}
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\begin{algorithm}[H]
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\caption{ADeliver$(m)$}
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\begin{algorithmic}[1]
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\Function{ADeliver}{m}
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\State $r \gets \current$
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\If{$|\resolved[r]| < n - f$}
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\State \Return $\bot$
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\EndIf
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\State $W_r \gets \BFTRESULT[r]$
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\If{$\exists j \in W_r,\ \prop[r][j] = \bot$}
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\State \Return $\bot$
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\EndIf
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\State $M_r \gets \bigcup_{j \in W_r} \prop[r][j]$
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\State $m \gets \ordered(M_r \setminus \delivered)[0]$ \Comment{Set $m$ as the smaller message not already delivered}
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\State $\delivered \leftarrow \delivered \cup \{m\}$
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\If{$M_r \setminus \delivered = \emptyset$} \Comment{Check if all messages from round $r$ have been delivered}
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\State $\current \leftarrow \current + 1$
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\EndIf
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\State \textbf{return} $m$
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\EndFunction
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\end{algorithmic}
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\end{algorithm}
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\renewcommand{\algletter}{C}
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\begin{algorithm}[H]
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\caption{RB handlers}
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\begin{algorithmic}[1]
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\Function{Rreceived}{j, PROP, S, r}
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\State $\received \gets \received \cup \{S\}$
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\State $\prop[r][j] \gets S$
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\State $\BFTVOTE(j, r)$
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\EndFunction
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\vspace{1em}
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\Function{Rreceived}{j, COMMIT, r}
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\State $\received[r] \cup \{j\}$
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\EndFunction
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\end{algorithmic}
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\end{algorithm}
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% \subsection{Example execution}
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% \begin{figure}[H]
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% \centering
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% \input{diagrams/classic_seq.tex}
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% \caption{Expected Executions of P1 willing to send a message at round r}
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% \end{figure}
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