From d5a865dd0a47432338da87bd7049ebfa69771e47 Mon Sep 17 00:00:00 2001 From: Amaury JOLY Date: Tue, 20 Jan 2026 10:28:46 +0100 Subject: [PATCH] fin (?) de la spec de BFT-DL + debut de preuve de l'algo pour BFT ARB --- Recherche/BFT-ARBover/6_BFT_ARB/index.tex | 226 +++++++++++++--------- Recherche/BFT-ARBover/main.pdf | Bin 472970 -> 479191 bytes 2 files changed, 130 insertions(+), 96 deletions(-) diff --git a/Recherche/BFT-ARBover/6_BFT_ARB/index.tex b/Recherche/BFT-ARBover/6_BFT_ARB/index.tex index de43a2d..4e6aeed 100644 --- a/Recherche/BFT-ARBover/6_BFT_ARB/index.tex +++ b/Recherche/BFT-ARBover/6_BFT_ARB/index.tex @@ -48,62 +48,65 @@ There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that : \subsection{DL $\Rightarrow$ t-BFT-DL} -\begin{lemma} - For any fixed value $t$ such that $3t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object. + +Fix $3t < |M|$. Let +\[ +\mathcal{U} = \{\, U \subseteq M \mid |U| = |M| - t \,\}. +\] +For each $U \in \mathcal{U}$, we instantiate one DenyList object $DL_U$ whose authorization sets are +\[ +\Pi_M(DL_T) = S_T = U +\qquad\text{and}\qquad +\Pi_V(DL_T) = V. +\] + +\[ +|\mathcal{U}| = \binom{|M|}{|M| - t}. +\] + +\begin{algorithmic}[1] + + % \renewcommand{\algletter}{} + \begin{algorithm}[H] + \caption{t-BFT-DL implementation using multiple DL objects} + + % \caption{\BFTAPPEND} + \Function{$\BFTAPPEND$}{x} + \For{\textbf{each } $U \in \mathcal{U}$ st $i \in U$} + \State $DL_U.\APPEND(x)$ + \EndFor + \EndFunction + + \vspace{1em} + + % \caption{\BFTPROVE} + \Function{$\BFTPROVE$}{x} + \State $state \gets false$ + \For{\textbf{each } $U \in \mathcal{U}$} + \State $state \gets state \textbf{ OR } DL_U.\PROVE(x)$ + \EndFor + \State \Return $state$ + \EndFunction + + \vspace{1em} + + % \caption{\BFTREAD} + \Function{$\BFTREAD$}{$\bot$} + \State $results \gets \emptyset$ + \For{\textbf{each } $U \in \mathcal{U}$} + \State $results \gets results \cup DL_U.\READ()$ + \EndFor + \State \Return $results$ + \EndFunction + \end{algorithm} +\end{algorithmic} + +\begin{lemma}[BFT-PROVE Validity] + The invocation of $op = \BFTPROVE(x)$ by a correct process is invalid iff there exist at least $t+1$ distinct processes in $M$ that invoked a valid $\BFTAPPEND(x)$ before $op$ in $\Seq$. \end{lemma} \begin{proof} - Fix $3t < |M|$. Let - \[ - \mathcal{U} = \{\, U \subseteq M \mid |U| = |M| - t \,\}. - \] - For each $U \in \mathcal{U}$, we instantiate one DenyList object $DL_U$ whose authorization sets are - \[ - \Pi_M(DL_T) = S_T = U - \qquad\text{and}\qquad - \Pi_V(DL_T) = V. - \] - - \[ - |\mathcal{U}| = \binom{|M|}{|M| - t}. - \] - - \begin{algorithmic}[1] - - \renewcommand{\algletter}{DL} - \begin{algorithm}[H] - \caption{\BFTAPPEND} - \Function{$\BFTAPPEND$}{x} - \For{\textbf{each } $U \in \mathcal{U}$ st $i \in U$} - \State $DL_U.\APPEND(x)$ - \EndFor - \EndFunction - \end{algorithm} - - \begin{algorithm}[H] - \caption{\BFTPROVE} - \Function{$\BFTPROVE$}{x} - \State $state \gets false$ - \For{\textbf{each } $U \in \mathcal{U}$} - \State $state \gets state \textbf{ OR } DL_U.\PROVE(x)$ - \EndFor - \State \Return $state$ - \EndFunction - \end{algorithm} - - \begin{algorithm}[H] - \caption{\BFTREAD} - \Function{$\BFTREAD$}{$\bot$} - \State $results \gets \emptyset$ - \For{\textbf{each } $U \in \mathcal{U}$} - \State $results \gets results \cup DL_U.\READ()$ - \EndFor - \State \Return $results$ - \EndFunction - \end{algorithm} - \end{algorithmic} - - \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 $\BFTAPPEND(x)$ before $op$ in $\Seq$. + 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 $\BFTAPPEND(x)$ before $op$ in $\Seq$. \begin{itemize} \item \textbf{Case (i): $|A|\ge t+1$.} @@ -115,8 +118,14 @@ There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that : \smallskip Combining the cases yields the claimed characterization of invalidity. +\end{proof} - \paragraph{BFT-PROVE Anti-Flickering.} Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i$ that is \emph{invalid} in $\Seq$. +\begin{lemma}[BFT-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. +\end{lemma} + +\begin{proof} + Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i$ that is \emph{invalid} in $\Seq$. 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 $U\in\mathcal{U}$. 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 $\Seq_U$. @@ -126,51 +135,57 @@ There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that : 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 $U\in\mathcal{U}$, there is no component $DL_U$ in which the induced $\PROVE(x)$ of $op'$ is valid. - - \paragraph{READ Liveness.} Let $R$ the result of a $READ()$ operation submit by any correct process. $(i, \PROVEtrace(x)) \in R$ implie that $\exists U^\star \in \mathcal{U}$ such that $(i, \PROVEtrace(x)) \in R^{U^\star}$ with $R^{U^\star}$ the result of $DL_{U^\star}.\READ()$. By \textbf{READ Validity} $(i, \PROVEtrace(x)) \in R^{U^\star}$ implie that there exist a valid $DL_{U^\star}.\PROVE^{(i)}(x)$. The for loop in the $\BFTPROVE(x)$ implementation return true iff there at least one valid $DL_{U}.\PROVE^{(i)}(x)$ for any $U \in \mathcal{U}$. +\end{proof} + +\begin{lemma}[READ Liveness] + Let $op = \BFTREAD()$ invoke by a correct process such that $R$ is the result of $op$. For all $(i, \PROVEtrace(x)) \in R$ there exist a valid invocation of $\BFTPROVE(x)$ by $p_i$. +\end{lemma} + +\begin{proof} + Let $R$ the result of a $READ()$ operation submit by any correct process. $(i, \PROVEtrace(x)) \in R$ implie that $\exists U^\star \in \mathcal{U}$ such that $(i, \PROVEtrace(x)) \in R^{U^\star}$ with $R^{U^\star}$ the result of $DL_{U^\star}.\READ()$. By \textbf{READ Validity} $(i, \PROVEtrace(x)) \in R^{U^\star}$ implie that there exist a valid $DL_{U^\star}.\PROVE^{(i)}(x)$. The for loop in the $\BFTPROVE(x)$ implementation return true iff there at least one valid $DL_{U}.\PROVE^{(i)}(x)$ for any $U \in \mathcal{U}$. Hence because there exist a $U^\star$ such that $DL_{U^\star}.\PROVE^{(i)}(x)$, there exist a valid $\BFTPROVE^{(i)}(x)$. $(i, \PROVEtrace(x)) \in R \implies \exists \BFTPROVE^{(i)}(x)$ - - \paragraph{READ Anti-Flickering.} - - \paragraph{READ Safety.} - \end{proof} - -\subsection{t-BFT-DL $\Rightarrow$ t-BFT-GE} - -\begin{lemma} - For any fixed value $r \in S$, multiple BFT-DenyList Object can be used to implement a BFT-Group Election Object. +\begin{lemma}[READ Anti-Flickering] + Let $op_1, op_2$ two $\BFTREAD()$ operations that returns respectively $R_1, R_2$. Iff $op_1 \prec op_2$ then $R_2 \subseteq R_1$. Otherwise $R_1 \subseteq R_2$. \end{lemma} \begin{proof} - \begin{algorithmic} - \State $Y[i]$ \Comment{Is a set of $n$ $\BFTDL$ with $\Pi_M = \Pi_V = \Pi$} + Let $R_1, R_2$ respectively the output of two $\BFTREAD()$ operations $op_1, op_2$ such that $op_1 \prec op_2$. + By the implementation of $\BFTREAD$, $R_k = \bigcup_{U \in \mathcal{U}} R_k^U$ where $R_k^U$ is the result of $DL_U.\READ()$ during $op_k$. - \vspace{1em} + Because $op_1 \prec op_2$ for any $U \in \mathcal{U}$, the $DL_U.\READ()$ induced by $op_1$ happen before the $DL_U.\READ()$ induced by $op_2$. Hence we have for all $U, R_2^U \subseteq R_1^U$. + Therefore + \[ + \bigcup_U R_2^U \subseteq \bigcup_U R_1^U \implies + R_2 \subseteq R_1 + \] +\end{proof} - \Function{BFTVOTE}{j, r} +\begin{lemma}[READ Safety] + Let $op_1, op_2$ respectively a valid $\BFTPROVE(x)$ operation submited by the process $p_i$ and a $\BFTREAD()$ operation submited by any correct process such that $op_1 \prec op_2$. Let $R$ the result of $op_2$ then $R \ni (i, \PROVEtrace(x))$ +\end{lemma} - \EndFunction - \vspace{1em} +\begin{proof} + Let $op_1 = \BFTPROVE^{(i)}(x)$ be a valid operation by a correct process $p_i$ and $op_2 = \BFTREAD()$ be any $\BFTREAD()$ operation such that $op_1 \prec op_2$ in $\Seq$. + By BFT-PROVE Validity, there exist at most $t$ distinct processes in $M$ that invoked a valid $\BFTAPPEND(x)$ before $op_1$ in $\Seq$. Let $A\subseteq M$ denote that set, with $|A|\le t$. - \Function{BFTCOMMIT}{r} - \EndFunction - \vspace{1em} - \Function{BFTRESULT}{r} - \State $Z \gets \emptyset$ - \For{\textbf{each } $j \in \Pi$} - \If{$|\{(\_, \PROVEtrace(\_, r)) \in Y[j].\BFTREAD(r)\}| \geq n-f$} - \State $P \gets \BFTREAD()$ - \State \Return $\{j : (j, \PROVEtrace), \}$ - \EndIf - \EndFor - \EndFunction - \vspace{1em} - \end{algorithmic} + 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 $DL_{U^\star}.\PROVE(x)$ of $op_1$. Since also $i\in \Pi_V(DL_{U^\star})$, by \textbf{PROVE Validity} of $\DL$ the induced $DL_{U^\star}.\PROVE^{(i)}(x)$ is valid. + + Now, because $op_1 \prec op_2$ in $\Seq$, the induced $DL_{U^\star}.\PROVE^{(i)}(x)$ appears before the induced $DL_{U^\star}.\READ()$ of $op_2$ in $\Seq_{U^\star}$. By \textbf{READ Safety} of $\DL$, the result $R^{U^\star}$ of the induced $DL_{U^\star}.\READ()$ contains $(i, \PROVEtrace(x))$. + + Finally, by the implementation of $\BFTREAD()$, we have $R = \bigcup_{U \in \mathcal{U}} R^U$, so $(i, \PROVEtrace(x)) \in R$. +\end{proof} + +\begin{theorem} + 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{theorem} + +\begin{proof} + Follows directly from the previous lemmas. \end{proof} \subsection{Algorithm} @@ -182,9 +197,9 @@ Each process $p_i$ maintains the following local variables: \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 $W_r \gets \bot, \forall r$ \State $\resolved[r] \gets \bot, \forall r$ + \State $Y[j]$ \Comment{Set of $n$ \BFTDL{}} \end{algorithmic} \renewcommand{\algletter}{A} @@ -192,15 +207,16 @@ Each process $p_i$ maintains the following local variables: \caption{ABroadcast$(m)$} \begin{algorithmic}[1] \Function{ABroadcast}{$m$} - \State $r \gets \current$ - \State $S \gets (\received \cup \{m\}$) + % \State $r \gets \current$ + \State $S \gets (\received \cup \{m\})$ \For{\textbf{each}\ $r \in \{\current, \current +1, \dots\}$} \State $\RBcast(i, PROP, S, r)$ - \State \textbf{wait} until $|W_r| \geq n - f$ where $W_r = \BFTRESULT[r]$ - \State $\BFTCOMMIT(r)$ + \State \textbf{wait} until $|W_r| \geq n - f$ where $W_r = \bigcup_{j \in \Pi} Y[j].\BFTREAD()$ + \State $\forall j \in W_r, Y[j].\BFTAPPEND(r)$ + \State $\RBcast(i, COMMIT, r)$ \State \textbf{wait} until $|\resolved[r]| \geq n - f$ - \State $W_r \gets \BFTRESULT[r]$ - \If{$i \in W_r \vee (\exists j, r': j \in W_r \wedge \prop[r'][j] \ni m)$} + \State $W_r \gets \bigcup_{j \in \Pi} Y[j].\BFTREAD()$ + \If{$i \in W_r \vee (\exists j, r': j \in W_{r'} \wedge \prop[r'][j] \ni m)$} \State \textbf{break} \EndIf \EndFor @@ -217,7 +233,7 @@ Each process $p_i$ maintains the following local variables: \If{$|\resolved[r]| < n - f$} \State \Return $\bot$ \EndIf - \State $W_r \gets \BFTRESULT[r]$ + \State $W_r \gets \bigcup_{j \in \Pi} Y[j].\BFTREAD()$ \If{$\exists j \in W_r,\ \prop[r][j] = \bot$} \State \Return $\bot$ \EndIf @@ -239,13 +255,31 @@ Each process $p_i$ maintains the following local variables: \Function{Rreceived}{j, PROP, S, r} \State $\received \gets \received \cup \{S\}$ \State $\prop[r][j] \gets S$ - \State $\BFTVOTE(j, r)$ + \State $Y[j].\BFTPROVE(r)$ \EndFunction \vspace{1em} \Function{Rreceived}{j, COMMIT, r} - \State $\received[r] \cup \{j\}$ + \State $\resolved[r] \cup \{j\}$ \EndFunction \end{algorithmic} \end{algorithm} + +\begin{definition}[BFT Closed round for $i$] + Given $Seq^{(i)}$ the linearization of the $\BFTDL$ $Y[i]$, a round $r \in \mathcal{R}$ is \emph{closed} in $\Seq$ iff there exist at least $n - f$ distinct processes $j \in \Pi$ such that $\BFTAPPEND^{(j)}(r)$ appears in $\Seq^{(i)}$. +\end{definition} + +\begin{definition}[BFT Closed round] + A round $r \in \mathcal{R}$ is \emph{closed} iff for all process $p_i$, $r$ is closed in $\Seq^{(i)}$. +\end{definition} + +\subsection{Proof of correctness} + +\begin{lemma}[BFT Stable round closure] + % If a round $r$ is closed by a correct process, +\end{lemma} + +\begin{theorem} + The algorithm implements a BFT Atomic Reliable Broadcast. +\end{theorem} diff --git a/Recherche/BFT-ARBover/main.pdf b/Recherche/BFT-ARBover/main.pdf index 4afc073bd19138f23ea976fdf6aa859a68b2d027..f582a8f108a989c46413ca4bd4b676bcef8c8516 100644 GIT binary patch delta 210138 zcmV)FK)=6=?;O{k9grshHFvF_fH`W_j%W3C^+Rh=A+!w_rqL=|EAz3 zjQ*X{$Dp|>Zos6~Vhmb^{8^M)@2hh>X ze>uWUr~M7If3qn+P$BQ<3t?UHqP|?+ZL4)nu+Q(GOVRD>{X!~Hytod}jOjjOQ3j`i zaHmosPhP)z_2N7^h^0u> zk?6CO;mLC;h*arJDu5R;zK1mtMtzgjz-?%4wi=bUna}vK{vAKd?HHm73L0fH?c*J` zdwu?FXiGE$5U}Bd%8Ygj)r6!(f8~1IOp20#MMFs@=E-tRvbzPy>=# zSg;c{AUP+NpQ-`LEurcZ4XkfHV3eEX4xC<8M#SG$RhP?6^=EnIk%(m|WWn>u=nXDH 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