Byzantine Agreement with Faulty Majority using Bounded Broadcast
  Byzantine Agreement introduced in [Pease, Shostak, Lamport, 80] is a widely used building block of reliable distributed protocols. It simulates broadcast despite the presence of faulty parties within the network, traditionally using only private unicast links. Under such conditions, Byzantine Agreement requires more than 2/3 of the parties to be compliant. [Fitzi, Maurer, 00], constructed a Byzantine Agreement protocol for any compliant majority based on an additional primitive allowing transmission to any two parties simultaneously. They proposed a problem of generalizing these results to wider channels and fewer compliant parties. We prove that 2f < kh condition is necessary and sufficient for implementing broadcast with h compliant and f faulty parties using k-cast channels. 
Jeffrey Considine, Leonid A. Levin, David Metcalf