Asynchronous Byzantine Agreement. Algorithm 3.21 Asynchronous Byzantine Agreement (Ben-Or, for f < n/10) 1: xi 0, 1 a input bit 2: r = 1 a round 3: decided = false 4: Broadcast propose(xi,r) 5: repeat
Asynchronous Byzantine Agreement. 33 If the correct nodes have different (binary) input values, the validity condition becomes trivial as any result is fine. − − What about agreement? Let u be the first node to decide on value x (in Line 8). Due to asynchrony another node v received messages from a different subset of the nodes, however, at most f senders may be different. Taking into account that byzantine nodes may lie (send different propose messages to different nodes), f additional propose messages received by v may differ from those received by u. Since node u had at least n 2f propose messages with value x, node v has at least n 4f propose messages with value x. Hence every correct node will propose x in the next round, and then decide on x. So we only need to worry about termination: We have already seen that as soon as one correct node terminates (Line 8) everybody terminates in the next round. So what are the chances that some node u terminates in Line 8? Well, we can hope that all correct nodes randomly propose the same value (in Line 12). Maybe there are some nodes not choosing at random (entering Line 10 instead of 12), but according to Lemma 3.22 they will all propose the same. − Thus, at worst all n f correct nodes need to randomly choose the same bit, which happens with probability 2−(n−f)+1. If so, all correct nodes will send the same propose message, and the algorithm terminates. So the expected running time is exponential in the number of nodes n.
Asynchronous Byzantine Agreement. Definition 1 (ABA). Let π be any asynchronous protocol in which each process has a binary input. We say that π is an almost-surely terminating, t-resilient ABA protocol if the following properties hold for every t-adversary and every input:
Asynchronous Byzantine Agreement. De nition 1 Let be an asynchronous protocol for which each one of the n players has binary input. We say that is a (1 )-completing, t-resilient Byzantine Agreement protocol if the following requirements hold, for every t-adversary and every input vector for the players.
Asynchronous Byzantine Agreement. The most basic form of asynchronous Byzantine consensus is asynchronous binary agreement (ABA), where each node has a binary value as input and will agree on a binary value. The validity of ABA is defined in an unanimous manner, i.e., if all honest nodes input the same binary value, they will agree on this value. As there are only two candidate values, the validity of ABA implies the so-called strong validity, i.
Asynchronous Byzantine Agreement. ( ABA)). An asynchronous Byzantine agreement protocol among 𝑛 parties is one in which each party receives an input and outputs a value 𝑣 and has the following guarantees: • • •
Asynchronous Byzantine Agreement
