From f5e0d90fb4665a9afa6302f5af522760a94bb2e9 Mon Sep 17 00:00:00 2001 From: Amaury JOLY Date: Thu, 15 Jan 2026 09:30:46 +0100 Subject: [PATCH] update --- Recherche/BFT-ARBover/6_BFT_ARB/index.tex | 141 ++++++++++++---------- Recherche/BFT-ARBover/main.pdf | Bin 473706 -> 473019 bytes 2 files changed, 75 insertions(+), 66 deletions(-) diff --git a/Recherche/BFT-ARBover/6_BFT_ARB/index.tex b/Recherche/BFT-ARBover/6_BFT_ARB/index.tex index 4cf662c..19a74a6 100644 --- a/Recherche/BFT-ARBover/6_BFT_ARB/index.tex +++ b/Recherche/BFT-ARBover/6_BFT_ARB/index.tex @@ -36,119 +36,128 @@ There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that : \paragraph{Termination.} Every operation $\BFTAPPEND(x)$, $\BFTPROVE(x)$, and $\BFTREAD()$ invoked by a correct process always returns. -\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. +% \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. -\paragraph{PROVE Validity.} If the invocation of a $op = \BFTPROVE(x)$ by a correct process $p$ is not valid, then: -\begin{itemize} - \item $p \not\in \Pi_V$; \textbf{or} - \item At least $t+1$ valid $\BFTAPPEND(x)$ appears before $op$ in $\Seq$. -\end{itemize} -Otherwise, the operation is valid. +\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 $|C| \leq t$ \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. \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. -\subsubsection{t-BFT-GE} +\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 $\BFTPROVE(x)$. -We consider a t-Byzantine Fault Tolerant Group Election Object (t-$\BFTGE[r]$) per round $r \in \mathcal{R}$ with the following properties. +\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 $p_i$. -There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$. +% \subsubsection{t-BFT-GE} -\paragraph{Termination.} Every operation $\BFTVOTE(i, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$ invoked by a correct process always returns. +% We consider a t-Byzantine Fault Tolerant Group Election Object (t-$\BFTGE[r]$) per round $r \in \mathcal{R}$ with the following properties. -\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 $W_r$. +% There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$. -\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. +% \paragraph{Termination.} Every operation $\BFTVOTE(i, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r)$ invoked by a correct process always returns. -\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 $\BFTRESULT(r)$. +% \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 $W_r$. + +% \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. + +% \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 $\BFTRESULT(r)$. \subsection{DL $\Rightarrow$ t-BFT-DL} \begin{lemma} - For any fixed value $t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object. + For any fixed value $t$ such that $3t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object. \end{lemma} \begin{proof} - Fix $t < |M|$. Let + Fix $3t < |M|$. Let \[ - \mathcal{T} = \{\, T \subseteq M \mid |T| = t \,\}. + \mathcal{U} = \{\, U \subseteq M \mid |U| = |M| - t \,\}. \] - For each $T \in \mathcal{T}$, we instantiate one DenyList object $DL_T$ whose authorization sets are + For each $U \in \mathcal{U}$, we instantiate one DenyList object $DL_U$ whose authorization sets are \[ - \Pi_M(DL_T) = S_T = M \setminus T + \Pi_M(DL_T) = S_T = U \qquad\text{and}\qquad \Pi_V(DL_T) = V. \] Let \[ - K = \{\, DL_T \mid T \in \mathcal{T} \,\}, + K = \{\, DL_U \mid U \in \mathcal{U} \,\}, \qquad\text{so that}\qquad - |K| = \binom{|M|}{t}. + |U| = \binom{|M|}{|M| - t}. \] - \begin{algorithmic} - \State $K \gets \{DL_T : T \subseteq M, |T|=t\}$ + \begin{algorithmic}[1] + % \State $K \gets \{DL_T : T \subseteq M, |T|=t\}$ - \Function{BFTAPPEND}{x} - \If{$p_i \notin M$} - \State \Return \textbf{false} - \EndIf - \For{\textbf{each } $DL_T \in K$} - \State $DL_T.\APPEND(x)$ - \EndFor - \State \Return \textbf{true} - \EndFunction + \renewcommand{\algletter}{DL} + \begin{algorithm}[H] + \caption{\BFTAPPEND} + \Function{$\BFTAPPEND$}{x} + \For{\textbf{each } $DL_U \in K$ such that $p_i \in U$} + \State $DL_U.\APPEND(x)$ + \EndFor + \EndFunction + \end{algorithm} - \Function{BFTPROVE}{x} - \If{$p_i \notin V$} - \State \Return $\bot$ - \EndIf - \State $state \gets false$ - \For{\textbf{each } $DL_T \in K$} - \State $state \gets state \textbf{ OR } DL_T.\PROVE(x)$ - \EndFor - \State \Return $state$ - \EndFunction + % \renewcommand{\algletter}{B} + \begin{algorithm}[H] + \caption{\BFTPROVE} + \Function{$\BFTPROVE$}{x} + \State $state \gets false$ + \For{\textbf{each } $DL_U \in K$} + \State $state \gets state \textbf{ OR } DL_U.\PROVE(x)$ + \EndFor + \State \Return $state$ + \EndFunction + \end{algorithm} - \Function{BFTREAD}{} - \State $results \gets \emptyset$ - \For{\textbf{each } $DL_T \in K$} - \State $results \gets results \cup DL_T.\READ()$ - \EndFor - \State \Return $results$ - \EndFunction + % \renewcommand{\algletter}{C} + \begin{algorithm}[H] + \caption{\BFTREAD} + \Function{$\BFTREAD$}{$\bot$} + \State $results \gets \emptyset$ + \For{\textbf{each } $DL_U \in K$} + \State $results \gets results \cup DL_U.\READ()$ + \EndFor + \State \Return $results$ + \EndFunction + \end{algorithm} \end{algorithmic} - \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 $|A|\ge t+1$. + % \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 $|A|\ge t+1$. \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 $\Seq$. - \smallskip - \noindent\textbf{Case (i): $i\notin V$.} - 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. - - \smallskip - \noindent\textbf{Case (ii): $|A|\ge t+1$ and $i\in V$.} Fix any $T\in\mathcal{T}$. By BFT-APPEND Validity, $A\cap S_T\neq\emptyset$. Pick $j\in A\cap S_T$. Since $j\in S_T$, the call $\BFTAPPEND^{(j)}(x)$ triggers $DL_T.\APPEND(x)$, and because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, this induces a valid $DL_T.\APPEND(x)$ that appears before the induced $DL_T.\PROVE(x)$. By \textbf{PROVE Validity} of $\DL$, the induced $DL_T.\PROVE(x)$ is invalid. As this holds for every $T\in\mathcal{T}$, there is \emph{no} component $DL_T$ where $\PROVE(x)$ is valid, so $op$ is invalid. - - \smallskip - \noindent\textbf{Case (iii): $|A|\le t$ and $i\in V$.} - By BFT-APPEND Validity, there exists $T^\star\in\mathcal{T}$ such that $A\cap S_{T^\star}=\emptyset$, i.e., $A\subseteq T^\star$. For any $j\in A$, we have $j\notin S_{T^\star}$, so $\BFTAPPEND^{(j)}(x)$ does \emph{not} call $DL_{T^\star}.\APPEND(x)$. Hence no valid $DL_{T^\star}.\APPEND(x)$ appears before the induced $DL_{T^\star}.\PROVE(x)$. Since also $i\in V=\Pi_V(DL_{T^\star})$, by \textbf{PROVE Validity} of $\DL$ the induced $DL_{T^\star}.\PROVE(x)$ is valid. Therefore, there exists a component with a valid $\PROVE(x)$, so by the lifting convention $op$ is valid. - + \begin{itemize} + % \item \textbf{Case (i): $i\notin V$.} + % 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. + + \item \textbf{Case (i): $|A|\ge t+1$.} + 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. + + \item \textbf{Case (ii): $|A|\le t$.} + 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. + \end{itemize} + \smallskip Combining the cases yields the claimed characterization of invalidity. \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 $\Seq$. - By BFT-PROVE Validity, since $i\in V$, 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 $|A|\ge t+1$. + 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 $|A|\ge t+1$. - Fix any $T\in\mathcal{T}$ and $DL_T$ with $\Pi_M(DL_T)=S_T=M\setminus T$. By BFT-APPEND Validity, we have $A\cap S_T\neq\emptyset$. Pick $j\in A\cap S_T$. Since $j\in S_T$, the call $\BFTAPPEND^{(j)}(x)$ triggers a call $DL_T.\APPEND(x)$. Moreover, because $\BFTAPPEND^{(j)}(x)\prec op$ in $\Seq$, the induced $DL_T.\APPEND(x)$ appears before the induced $DL_T.\PROVE(x)$ of $op$ in the projection $\Seq_T$. + 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 $\Seq_U$. - Hence, in $\Seq_T$, there exists a \emph{valid} $DL_T.\APPEND(x)$ that appears before the $DL_T.\PROVE(x)$ induced by $op$. By \textbf{PROVE Validity} the base $\DL$ object, the induced $DL_T.\PROVE(x)$ is therefore \emph{invalid} in $\Seq_T$. + 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 $\Seq_U$. - Now let $op'=\BFTPROVE(x)$ be any invocation such that $op\prec op'$ in $\Seq$. Fix again any $T\in\mathcal{T}$. Hence, the $DL_T.\PROVE(x)$ induced by $op'$ appears after the $DL_T.\PROVE(x)$ induced by $op$ in $\Seq_T$. Since the induced $DL_T.\PROVE(x)$ of $op$ is invalid, by \textbf{PROVE Anti-Flickering} of $\DL$, \emph{every} subsequent $DL_T.\PROVE(x)$ in $\Seq_T$ is invalid. + 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. - As this holds for every $T\in\mathcal{T}$, there is no component $DL_T$ in which the induced $\PROVE(x)$ of $op'$ is valid. + 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. + + \paragraph{Local consistency.} + + \paragraph{Liveness.} + \end{proof} diff --git a/Recherche/BFT-ARBover/main.pdf b/Recherche/BFT-ARBover/main.pdf index 14f59721a0c795c7e4a99a2a190e81ded36a1306..77479a854d3dfaaecfecdd8ebac91f1f966e8a4a 100644 GIT binary patch delta 154000 zcmZs?Q;;r9v@F=RZQI&y+qP}nUu(B_+qP}nwr$(I|G6^{6BFm9qUt4|Rzzm5%({5O z>%7H7Ay*U?r)Os1fFYk-8eW58X5y?zc}0|v{rjpw4FBr2YD&d<%zaXLweUKq8}?hW z#~%HYy)c^BPCbkktBNg?=jngH{R>p&yvPq@W^d}^>TG6Y_dhoWV`~^@)>NT82-HRs zq-X@Gl7U+V;^4P-n!|Nk`F4KTTTLL`YGx+m4YaM$&fmNz9FBF8>3VgfIg!6Rh~QI 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