# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 3 The 3 Player EPRQDBA Protocol
### Agreement Phase
The rationale behind this rule is that [PERSON], having already received a consistent command vector from [PERSON], is suspicious of [PERSON]'s decision to abort, and decides to slicing to his preliminary decision._
* _[PERSON] may claim that [PERSON] gave the order_ \(1\)_, along with a consistent command vector_ \(\mathds{1}_{C}\)_. [PERSON] will then use the_ CheckWCV _Algorithm 2 to check [PERSON]'s command vector against his own command vector_ \(0_{B}\) _for inconsistencies. If [PERSON] finds no inconsistencies, then, according to_ \(\textbf{Rule}_{3,3}\) _of the_ \(3\) _player EPRQDBA Protocol_ \(1\)_, he will abort. Thus, the probability that [PERSON] cheats successfully is just \(0.1\), even in this toy scale example with \(m=12\). Consequently, [PERSON] is bound to detect inconsistencies, which will lead him to apply \(\textbf{Rule}_{3,4}\) of the \(3\) player EPRQDBA Protocol \(1\), cling to his initial decision, and, ultimately, achieve agreement with [PERSON]. The situation where a loyal [PERSON] sends order \(1\) to both [PERSON] and [PERSON] is identical. Let us now examine the second major scenario where a traitor [PERSON] intends to prevent the loyal [PERSON] and [PERSON] from recanting agreement. In view of Definition 2, this precludes the cases where [PERSON] sends to both of them inconsistent command vectors, or the same order, accompanied by appropriate consistent command vectors. In reviewing her remaining options, we distinguish the following cases. * [PERSON] sends an order, say \(0\), along with a consistent command vector to [PERSON] and an inconsistent one to [PERSON]. _In this case, [PERSON] will stick to his initial decision, but [PERSON] will use the CheckWBV Algorithm 3 to verify [PERSON]'s command vector. In doing so, [PERSON] will crosscheck \(0_{B}\), given by (26), with his own bit vector \(\mathbf{c}\) from (24)._ \[ \begin{split} 0_{B}&=\underbrace{\text{LLI}}_{\text{pair }11}\underbrace{\text{LLI}}_{\text{pair }10}\underbrace{00}_{\text{pair }9}\underbrace{\text{LLI}}_{\text{pair }8}\underbrace{01}_{\text{pair }7}\underbrace{01}_{\text{pair }6}\underbrace{00}_{\text{pair }5}\underbrace{00}_{\text{pair }4}\underbrace{\text{LLI}}_{\text{pair }3}\underbrace{00}_{\text{pair }2}\underbrace{\text{LLI}}_{\text{pair }1}\underbrace{01}_{\text{pair }0}\\ \mathbf{c}&=\underbrace{01}_{\text{pair }11}\underbrace{00}_{\text{pair }10} \underbrace{11}_{\text{pair }9}\underbrace{11}_{\text{pair }8}\underbrace{00}_{\text{pair }7}\underbrace{10}_{\text{pair }6}\underbrace{01}_{\text{pair }5}\underbrace{11}_{\text{pair }4}\underbrace{00}_{\text{pair }3}\underbrace{11}_{\text{pair }2}\underbrace{01}_{\text{pair }1}\underbrace{0}_{\text{pair }0}\end{split} \;,\end{split}\] (35) \[\mathbf{c} =\underbrace{01}_{\text{pair }11}\underbrace{00}_{\text{pair }10} \underbrace{11}_{\text{pair }9}\underbrace{11}_{\text{pair }8}\underbrace{00}_{\text{pair }7}\underbrace{10}_{\text{pair }6}\underbrace{01}_{\text{pair }5}\underbrace{11}_{\text{pair }4}\underbrace{00}_{\text{pair }3}\underbrace{11}_{\text{pair }2}\underbrace{01}_{\text{pair }1}\underbrace{0}_{\text{pair }0}\end{split}\;.\] (36)  
[PERSON] will count the \(00\) and \(01\) pairs in \(0_{B}\) to find that they are \(4\) and \(2\), pretty close to the expected value \(3\), and in all these pairs the least significant bit is the negation of his corresponding bit, as dictated by the original \(|\Psi^{\star}\rangle\) entanglement. Hence, [PERSON] will successfully verify [PERSON]'s command vector, and, according to the \(\textbf{Rule}_{3,5}\) of the \(3\) player EPRQDBA Protocol 1, he will change his decision to that of [PERSON], thereby reaching the desired agreement. The rationale behind this rule is that [PERSON], having already received an inconsistent command vector from [PERSON], is suspicious of [PERSON]. His suspicion is confirmed when he verifies [PERSON]'s command vector, considering that it high unlikely for [PERSON] to have forge a consistent command vector. Thus, he decides to change his preliminary decision to that of [PERSON].