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Author: Admin | 2025-04-28
U an honest party has a chain \(\mathcal {C}\) of length \(\ell\), then that party broadcasts \(\mathcal {C}\) at a round earlier than u. It follows that every honest party will receive \(\mathcal {C}\) by round \(u-1+\Delta =v\).For the inductive step, assume the inductive hypothesis for \(v-1\) and consider two cases. First, consider \(X^{\prime }_{v-\Delta }=0\), in which case we have \begin{equation*} \ell +\sum _{u\le r\le v-\Delta }X^{\prime }_r=\ell +\sum _{u\le r\lt v-\Delta }X^{\prime }_r=\ell ^{\prime } . \end{equation*} By the inductive hypothesis every honest party has received a chain of length at least \(\ell ^{\prime }\) by round \(v-1\).For the second case, \(X^{\prime }_{v-\Delta }=1\). By the inductive hypothesis, by round \(v-\Delta\), every honest party has adopted a chain of length at least \begin{equation*} \ell ^{\prime } =\ell +\sum _{u\le r\le v-2\Delta }X^{\prime }_r =\ell +\sum _{u\le r\lt v-\Delta }X^{\prime }_r , \end{equation*} where the second equality holds because \(X^{\prime }_{v-\Delta }=1\) implies \(X^{\prime }_r=0\) for all \(v-2\Delta \lt r\lt v-\Delta\). It follows that every honest party queried the oracle with a chain of length at least \(\ell ^{\prime }\) at round \(v-\Delta\). Hence, all honest parties successful at round \(v-\Delta\) broadcast a chain of length at least \(\ell ^{\prime }+X_{v-\Delta }\). This chain will be received by every honest party by round v. Since \(X^{\prime }_{v-\Delta }=1\), using the expression for \(\ell ^{\prime }\) displayed above, we have \begin{equation*} \ell ^{\prime }+X_{v-\Delta }=\ell ^{\prime }+X^{\prime }_{v-\Delta }=\ell +\sum _{u\le r\le v-\Delta }X^{\prime }_r \end{equation*} and this completes the case and the proof. □Definition 7.2 (Typical Execution in the Bounded-delay Model).An execution is \((\epsilon ,\lambda ,\Delta)\)-typical, with \(\epsilon \in (0,1)\), \(\lambda \ge 2/f\), and integer \(\Delta\), if, for any set S of at least \(\lambda\) consecutive rounds, the following hold.(a)\((1-\epsilon)\mathbb {E}[X^{\prime }(S)]\lt X^{\prime }(S)\), \(X(S)\lt (1+\epsilon)\mathbb {E}[X(S)]\) and \((1-\epsilon)\mathbb {E}[Y^{\prime }(S)]\lt Y^{\prime }(S)\).(b)\(Z(S)\lt \mathbb {E}[Z(S)]+\epsilon \mathbb {E}[X^{\prime }(S)]\).(c)No insertions, no copies, no guesses, and no predictions occurred.Theorem 7.3.An execution is typical with probability at least \begin{equation*} 1-4L^2e^{-\Omega (\epsilon ^2\lambda f^2(1-f)^{4\Delta -2})}-3Q^22^{-\kappa }-[(n-t)L]^22^{-\nu } . \end{equation*} Proof.Note that \(Y^{\prime }_i\) and \(Y^{\prime }_j\) are not independent anymore when \(|i-j|\lt 2\Delta\) and the standard Chernoff bound does not apply. (Similarly for \(X^{\prime }_i\) and \(X^{\prime }_j\).) However, \(Y^{\prime }(S)\), as a function of the honest queries in S, is 2-Lipschitz (see Definition A.2). This is because each query in a round i affects \(Y^{\prime }_j\) only if \(|i-j|\lt \Delta\) and there
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