Message complexity. GWTS executes a possibly infinite sequence of decisions. Thus, we restrict our message complexity analysis to the number of messages needed for each decision. The messages are counted per proposer, we include messages created by correct acceptors in response to proposer actions. Each proposer decides exactly once for each algorithm round. Therefore, we count messages from start to end of a generic round. A proposer has to reliably broadcast its batch (line 15 -cost O(n2)), it has to broadcast its proposal (line 25 - cost O(n)), then, in the worst case, it refines its proposal at most f times (see Lemma 10 -line 33 - cost O(n)), however each ack from a correct acceptor has to be reliably broadcast (line 10 - cost O(n2)). The total cost is therefore upper-bounded by O(f · n2).
Message complexity. O · O The Byzantine reliable broadcast used at Line 8 costs (n2) messages [14], this cost dominates the other algorithm operations: in the 2f + 5 delays needed to reach the decision at most (f n) messages are generated.
Message complexity. The Byzantine Reliable broadcast used at Line 9 costs (n2) messages [14], this cost dominates the other algorithm operations: in the 2f + 5 delays needed to reach the decision at most O(f · n) messages are generated. if Decided Accepted set state = proposing r = r ⊆ ∧ ∧ 39: DECIDE(Accepted set) 40: Decided = Accepted set 41: state = newround 42: function SAFE(m)
Message complexity. The asymptotical complexity of our algorithm is the same of WTS once the cost of the Byzantine Reliable Broadcast has been removed, that is O(f · n).
Message complexity. The message complexity fol- lows the same analysis of Section VI-E, the removal of the Byzantine reliable broadcast leads to (f n) messages per decision.
Message complexity. In the dispersal phase, which in- curs 𝑛2 messages, where each node sends a total of O( ) O( ) O( ) 𝑛 messages. In the Election & Recast phase, beyond a single common coin invocation, the recast phase requires one all-to-all multicast, incurring 𝑛2 messages. In the MBA phase, there is only one MBA instance. Moreover, the Election & Recast phase and the MBA phase are expected to be repeated two times. To summarize, the overall message complexity of the HMVBA protocol is 𝑛2 . O( + ) •
Message complexity. In the filter phase, incurring 𝑛 messages, each node sends a total of 𝑛 messages. In the ABA phase, beyond common coin invocation, it needs to ex- change 𝑛2 messages. The output phase occurs without any message exchange. To summarize, the overall message complexity of the MBA protocol is O(𝑛2).
