Common use of Proof sketch Clause in Contracts

Proof sketch. To derive a contradiction, assume that there exists an algorithm A that solves Byzantine agree- ment with ℓ ≤ t. In the argument below, we consider only executions of A with some fixed set of ℓ Byzantine processes, chosen so that each of the ℓ identifiers is held by one Byzan- tine process. We consider configurations of the the algorithm A at the end of a synchronous round. Such a configuration can be completely specified by the state of each process. A config- uration C is 0-valent if, starting from C, the only possible decision value that correct processes can have is 0; it is 1- valent if, starting from C, the only possible decision value that correct processes can have is 1. C is univalent if it is either 0-valent or 1-valent; C is multivalent if it is not univalent. The following lemma encapsulates a Byzantine agent’s abil- ity to influence the decision value. ′ Lemma 17 Let C and C be two configurations of A such that the state of only one correct process is different in C accepted message. More precisely, this multiplicity is greater than the number of correct processes that sent the message and does not exceed the number of correct processes by more than the actual number of Byzantine processes in the exe- cution. Furthermore, all correct processes agree eventually on the multiplicity of each message. This authenticated broadcast with multiplicity is used to ensure the agreement property. As ℓ > t, at least one iden- tifier is assigned only to correct processes. This property is used to ensure the termination property of the agreement algorithm.