Randomized version of Dolev-Reischuk. As might be expected, any BA protocol (possibly randomized) must perform at least $Omega (f^2) $ of communication in the presence of a highly adaptive adversary capable of performing a “posteriori distance”, $f$ specifying the number of damaged nodes. In our previous contributions to the lower limit, we discussed limit values for the contradictory threshold (i) DLS in partial synchronization and (ii) FLM in the absence of a reliable configuration. In this article, we discuss another classic impossibility outcome on the limits of the complexity of communication on Byzantine radio. The search for the construction of an evolutionary Byzantine agreement involves many challenges. In this article, we highlight the lower limit of Dolev and Reischuk of 1982. At a very high level, the dolev and resichuk limit says that if you send few messages < $< (f/2)^$2, then an honest party will not receive any message! The party that does not receive a message has no way of reaching an agreement with the rest. We use a trivial argument of indivisibility and create a world where a $p$ node receives no message. Therefore, nodes $p$ cannot tell the difference between a world where the specified sender sends 0 and a world where it sends 1. If the protocol has a communication complexity $leq (f/2)^2$, there must be a node $p in V$ to receive messages $leq f/2$. In world 2, the opponent does everything as in world 1, except (i) he does not damage $p$ and (ii) he damages all the nodes in $U$ that have sent messages at $ $p (this may include the specified sender). However, their solution had exponential communication complexity (in $n$, the number of parties).
An obvious question is then to find the slightest complexity of communication that can be achieved. Dolev and Resichuk showed that the barrier to square communication complexity cannot be crossed by deterministic protocols. Please join the Simons Foundation and our generous member organizations to support arXiv during our fundraising campaign from September 23-27. 100% of your contribution will fund improvements and new initiatives for the benefit of arXiv`s global scientific community. Both the individuals and organizations that collaborate with arXivLabs have accepted and accepted our values of openness, community, excellence and user privacy. arXiv upholds these values and only works with partners who comply with them. In World 1, the attacker damages a set $V$ $f/$2, which does not contain the specified sender. Suppose $U$ is the remaining nodes. All parties with $$V behave like honest nodes except (i) they ignore the first $f/$2 messages sent to them and (ii) they don`t send messages to each other. .