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I started reading PBFT and I could not understand: why do we need 2F prepare certificates?

The client needs only F + 1 replies, not 2F, so why can't the same rule be applicable in the prepare phase of the protocol?

Same question in the commit phase, but this time it is even weirder: we need at least 2F + 1, again why not only F + 1, I just know that the quorum intersection must be more than 2F + 1 but we only need F + 1 replies from that quorum intersection as far as I know.

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For a detailed answer, please read Lamport et al. paper.

For a short answer, I'll explain it downwards. To coming readers, please note that we are discussing the PBFT paper.

Assume that we have only three processes (named p0, p1 and p2) and a client c, where p2 is a faulty process. That said, we have 2F+1 processes.

In the Prepare phase, p1 receives two votes one from p0 (an honest vote) and one from p2 (a faulty vote). Which decision should p1 write into the database? You can simply notice that no-one (neither p0 nor p2) is overweighing, which leads to an inconsistency. That's why we need 3F+1 (and accordingly, each process requires 2F+1 replies) processes to achieve consistency.

Roughly speaking: XF+1 replicas means XF replies.

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