Impossibility result. Whereas the case t = 0 is obviously are optimal for both, broadcast with extended validity and broadcast with extended consistency, it still needs to be proven that the bound t + 2t+ < n is optimal. Note that this impossibility result even holds for the ordinary variants without consistency detection, or validity detection, respectively. The proof proceeds along the lines of the impossibility proof in [FLM86] that broadcast is impossible if t ≥ n/3. ∈ { } ≥
Impossibility result. What exactly makes Consensus hard to solve? When the only failures considered are process crashes, this problem has relatively sim- ple solutions in synchronous distributed systems. However, solving consensus in a purely asynchronous distributed system prone to crash failures is far from being a trivial task. Let us consider a simple scenario described in Figure 2.1: a system consisting of three processes that communicate by message passing. A very simple protocol has the following behavior: each process broadcasts its own value and gathers the messages from the other two pro- cesses. Each process then determines the minimum value among the gathered values, and decides on it. Such a protocol, called the strawman protocol, would work only if all processes were correct, albeit limited in speed by the slowest process or link. decision = min{v1, v2, v3} p2 p1 v1 v2 v1 v2 v3 v3 p3 vd = v1 vd = v1 vd = v1 Figure 2.1: Strawman Protocol (best case scenario: all processes are correct). However, in the presence of a single failure, the processes might gather different sets of values and the minimum among them might not be the same for each process. If processes use timeouts, then each of them might use different timeout on different sets of proposals which would lead to processes deciding on different values, therefore violating the Agree- ment property. Thus, each process should wait until it has received a value from each other process. But if only one process is faulty and crashes, every other correct process would then wait forever and may never decide on a value (see Figure 2.2). p2 p1 v1 v2 v1 v2 v3 v3 p3 vd = v1 vd = v2 Figure 2.2: Strawman Protocol (when a crash failure occurs). Solving Consensus in a synchronous system In a synchronous setting, a straightforward implementation of the strawman protocol relies on a timeout mechanism. In such a context, there exist timing bounds on execution steps and communication latency, a straightforward implementation of the strawman protocol relies on a simple timeout mechanism. It is possible to tune the timeouts to ensure that any suspected process is really crashed. Processes can proceed in simultaneous steps and failures can be detected in the following way. First, a process waits for a certain reply from another process, for a given period of time. If the timeout expires and no reply is received, the process is considered to be crashed. One of the fundamental results in distributed computing is also a negative one. ...
Impossibility result. In this section, we show that there is no Approximate Agreement protocol that is 𝑡𝑠 -secure under a synchronous network, and 𝑡𝑎 -secure under an asynchronous network, for 2 · 𝑡𝑠 + 𝑡𝑎 ≥ 𝑛. This shows that our protocol Π𝐴𝐴 achieves the optimal corruption threshold.
Impossibility result. We justify our reliance on a trusted dealer by showing that some form of setup is necessary for (non-amortized) subquadratic BA tol- erating Θ(n) corrupted parties. Moreover, this holds even when secret channels and erasures are available.
