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@@ -99,9 +99,9 @@ These will typically only hold with high probability. In the asynchronous case,
\subsection{Our approach}
To come up with a solution to the blockchain Byzantine finality gadget problem, we will typically look at various Byzantine agreement protocols and use those to find protocols for the multi-valued Byzantine finality gadget problem. Protocols for that with appropriate properties can used to find protocols for the blockchain Byzantine finality gadget problem by considering running them in parallel at every block number. If the one block protocol has the right properties then they will agree on blocks consistently so if we finalise a block then we also finalise its ancestors and we can come up with a succinct protocol.
To discover up with a solution to the blockchain Byzantine finality gadget problem, we will typically look at various Byzantine agreement protocols and use those to find protocols for the multi-valued Byzantine finality gadget problem. Agreement protocols with appropriate properties can used to find protocols for the blockchain Byzantine finality gadget problem by considering running them in parallel at every block number. If the one block protocol has the right properties then they will agree on blocks consistently, so if we finalise a block then we also finalise its ancestors and we can come up with a succinct protocol.
For example, suppose we have a one block protocol that calls for a vote on blocks which requires a participant to observe a supermajority, say votes from $2/3$ of voters, for some block (or else the participant observes that the vote is undecided). Now imagine running this vote in parallel for every block number and have any honest voter vote for blocks from a particular chain. Byzantine voters may vote more than once, but if we count a vote for a block as a vote for each ancestor of the block in the vote for the instance of the one block protocol with its number, then Byzantine voters must also vote for chains, though they can vote for multiple chains. If we do this, then we see that if a block has a supermajority in a vote, then so does all its ancestors in their votes. Thus the blocks with a supermajority form a chain. Furthermore, if only $1/3$ of voters equivocate then if a participant sees a subset of the votes for chains, then they must see a prefix of the chain of blocks that all the votes have supermajorities for. Intuitively, the protocol can agree on the prefix that $2/3$ of voters agree on using this.
For example, suppose we have a one block protocol that calls for a vote on blocks which requires a participant to observe a supermajority, say votes from $2/3$ of voters, for some block, or else the participant observes that the vote is undecided. Now imagine running this vote in parallel for every block number and have any honest voter vote for blocks from a particular chain. Byzantine voters may vote more than once, but if we count a vote for a block as a vote for each ancestor of the block in the vote for the instance of the one block protocol with its number, then Byzantine voters must also vote for chains, though they can vote for multiple chains. If we do this, then we see that if a block has a supermajority in a vote, then so does all its ancestors in their votes. Thus the blocks with a supermajority form a chain. Furthermore, if only $1/3$ of voters equivocate then if a participant sees a subset of the votes for chains, then they must see a prefix of the chain of blocks for which all the votes have supermajorities. Intuitively, the protocol can agree on the prefix that $2/3$ of voters agree on using this.
To ensure safety, each participant maintains an estimate $E_r$ of the last block that could have been finalised in a round $r$. This has the property that in future rounds it overestimates the block that could have been finalised so that in round $r$, the chain with head $E_{r-1}$ contains all blocks that could have been finalised. Any honest voter only votes in round $r$ for chains containing their estimate $E_{r-1}$ and this guarantees that any block that could have been finalised in round $r-1$ will be finalised in round $r$.