Solutions to the Byzantine Agreement Problem Problem here, a protocol for Byzantine Agreement. Problem here is, if you have to write a protocol for the Byzantine problem, that protocol must guarantee that all non-faulty processes agree on the same value if the source processor is non-faulty, right? Okay, there is a chance that the source processor may be faulty, and the problem of agreement arises. It must ensure that non-faulty processors agree on the same value, which is initiated or assumed by the source—a common single value initiated by the source processor, okay? While processors exchange values over messages, a faulty processor may cause confusion. A faulty processor will be called a confused node, as it may send conflicting values. Since we are allowing faulty processors in the system, right? There is a chance of malicious behavior, leading to a high probability of confusion by sending conflicting values. So, the Byzantine Agreement Problem is also referred to as the Byzantine Generals Problem. Now, coming to the problem solutions. Here, yes, the first solution says upper bound on the number of faulty processors, right? The number of multiprocessors allowed in the system should not exceed a limit in order to reach consensus, right? So, for example, N is the total number of processors. M here is the total number of faulty processors. Now, , what is ? It is 9. , so M should be either 0, 1, or 2, which is less than 3, right? So, we have a maximum limit for M as 2. The maximum bound is (N - 1) / 3. If there are 10 processors in the system, there could be a maximum of 2 faulty processors, always taking the maximum limit for multiprocessors as (N - 1) / 3. Impossibility result: randomly, Byzantine agreement cannot be reached among three processes, right? So, this agreement cannot be reached among three processes, right? Where you have three processors… Assume that there are two values only being communicated. All three processors are communicating only two values, either 0 or 1. Now, as per the Byzantine agreement, P₀ is the source process and initiates the initial value. Two possibilities: 1. Non-faulty P₀ – If P₀ is non-faulty, then all the non-faulty processors can agree upon any common value. 2. P₀ is faulty – Then, it is irrelevant what value the multiprocessor agrees on. Whether they agree or not, it’s not at all a problem. Okay, okay, okay… For reference, I have given one example here. Now, Case 1 says P₀, which is the source processor, is non-faulty. Okay, let’s assume P₁ and P₂ are faulty for now. But P₁ is sending the initial value, while P₀ is the source processor here and takes the initial value as 1. So, 1 is broadcast, and P₁ computes. P₁ is not broadcasting anything. Again, in the Byzantine problem, only the source process should broadcast. Here, P₁ should agree, right? And P₂ will forward whatever P₀ sends, because P₀ is a non-faulty processor. Either P₂’s value is agreed upon, or a single same value (1) satisfies the case. Now, in Case 2, if P₀ is faulty, even the source processor is faulty, then P₀ sends 1 to P₁ but sends 0 to P₂. In this case, right? So, this is how the impossibility result works, right? We cannot take up a maximum decision. There are three processors, with one faulty processor, then it is impossible to come up with consensus.