Proving culpability Sample Clauses

Proving culpability.Β How do disagreeing processes decide which processes were malicious? When a process decides in round π‘Ÿ , it sends its certificate to all the other processes. Any process that decides a different value in a round > π‘Ÿ can prove the culpability of at least 𝑛 3 Byzantine processes by comparing this certificate to its logged ledgers. (It can then broadcast the proper logged ledgers to ensure that everyone can identify the malicious processes.) βˆ’ We will say that a certificate (e.g., from 𝑝1) and a ledger (e.g., from 𝑝2) conflict if they are constructed in the same round π‘Ÿ , but for different values 𝑣 and 𝑀. That is, both the certificate and the ledger attest to (𝑛 𝑑0) ECHO messages from round π‘Ÿ sent to 𝑝1 and 𝑝2 (respectively) that contain only value 𝑣 and only value 𝑀, respectively. Since every two sets of size (𝑛 βˆ’ 𝑑0) intersect in at least (𝑛 βˆ’ 2 Β· 𝑑0) locations,
Sample 1
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Proving culpability.Β How do disagreeing processes decide which processes were malicious? When a process decides in round π‘Ÿ , it sends its Polygraph certificate to all the other processes. Any process that decides a different value in a round > π‘Ÿ can prove the culpability of at least βŒˆπ‘›/3βŒ‰ Byzantine processes by comparing this certificate to its logged ledgers. (It can then broadcast the proper logged ledgers to ensure that everyone can identify the malicious processes.) We will say that a certificate (e.g., from 𝑝1) and a ledger (e.g., from 𝑝2) conflict if they are constructed in the same round π‘Ÿ , but for different values 𝑣 and 𝑀. That is, both the certificate and the ledger attest to (𝑛 βˆ’ 𝑑0) ECHO messages from round π‘Ÿ sent to 𝑝1 and 𝑝2 (respectively) that contain only value 𝑣 and only value 𝑀, respectively. Since every two sets of size (𝑛 βˆ’ 𝑑0) intersect in at least (𝑑0 + 1) locations, this conflict identifies (𝑛 βˆ’ 𝑑0) processes that sent different Phase 2 messages in round π‘Ÿ to 𝑝1 and 𝑝2 and hence they are malicious. We now discuss how to find conflicting certificates and ledgers. Assume that process 𝑝𝑖 decides value 𝑣 in round π‘Ÿ , and that process 𝑝 𝑗 decides a different value 𝑀 in a round > π‘Ÿ . (Recall that 𝑣 is the only possible value that can be decided in round π‘Ÿ .) There are two cases to consider, depending on whether 𝑝 𝑗 decides in round π‘Ÿ + 1 or later. β€’ Round π‘Ÿ +1: If 𝑝 𝑗 decides in round π‘Ÿ +1, then value𝑀 was the only candidate value after Phase 2. This implies that 𝑀 was received by some bv-broadcast message. Since π‘Ÿ > 1, we know that the message must have contained a valid ledger β„“ from round π‘Ÿ for value 𝑀 β‰  𝑣. This ledger β„“ conflicts with the decision certificate of 𝑝𝑖 . β€’ Round β‰₯ π‘Ÿ + 2: Since 𝑝 𝑗 decides 𝑀 β‰  𝑣, it does not decide 𝑣 in round π‘Ÿ + 2. This means that 𝑝 𝑗 has 𝑀 as a candidate value, which implies that 𝑝 𝑗 received 𝑀 in a bv-broadcast. Since π‘Ÿ > 1, we know that the message must have contained a valid ledger β„“ from round π‘Ÿ + 1 for value 𝑀 β‰  𝑣. This ledger β„“ consists of a copy of a ledger from round π‘Ÿ for value 𝑀 which conflicts with the decision certificate of 𝑝𝑖 . In either case, if 𝑝 𝑗 does not decide 𝑣, then, by looking at the messages received in round π‘Ÿ + 1 and π‘Ÿ + 2, it can identify a ledger that conflicts with the decision certificate of 𝑝𝑖 and hence can prove the culpability of at least 𝑑0 + 1 malicious processes.
Sample 1

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