# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 4 The \(n\) Player EPRQDBA Protocol
### The \(n\) Player Agreement Phase
The probability that [PERSON] places _all_ the \(\approx\frac{m}{4}\) bits \(0\) correctly in the \(\approx\frac{m}{2}\) uncertain pairs is

\[P(\text{ [PERSON] places all }0\text{ correctly })\approx\frac{1}{\binom{m/2}{m /4}}\, \tag{2}\]

which tends to zero as \(m\) increases. Put it another way, the cardinality of the symmetric difference \(\mathbb{P}_{0,1}(0_{B})\vartriangle\mathbb{P}_{0,1}(\mathds{1}_{C})\) will be significantly greater than \(0\) from a statistical point of view. Hence, when [PERSON] uses the Algorithm 2 during \((\mathbf{Round}_{3})\) to verify the validity of the command vector \(\mathds{1}_{C}\) sent by [PERSON], he will detect inconsistencies and stick to his preliminary decision. Ergo, [PERSON] follows [PERSON]'s order. The case where [PERSON] conveys the command \(1\) is virtually identical. Finally, the situation where [PERSON] and [PERSON] are loyal and [PERSON] is a traitor, is completely symmetrical. **Proposition A2** (Traitor [PERSON]). : _If [PERSON] and [PERSON] are loyal and [PERSON] is a traitor, the \(3\) player EPRQDBA protocol will enable [PERSON] and [PERSON] to reach agreement, in the sense that they will either follow the same order or abort._Proof. : Let us now consider the case where the commanding general [PERSON] is a traitor, but both [PERSON] and [PERSON] are loyal. We distinguish the following cases. * [PERSON] sends a consistent command vector to [PERSON] ([PERSON]) and an inconsistent command vector to [PERSON] ([PERSON]). In this case, [PERSON] ([PERSON]) will stick to his initial decision, but [PERSON] ([PERSON]) will use the CheckWBV Algorithm 3 to verify [PERSON]'s ([PERSON]'s) command vector. Then, according to \(\mathbf{Rule}_{3,5}\) of the 3 player EPRQDBA Protocol 1, [PERSON] ([PERSON]) will change his decision to that of [PERSON] ([PERSON]). * [PERSON] sends to [PERSON] and [PERSON] different orders with consistent command vectors. Let us assume, without loss of generality, that [PERSON] sends the order 0 together with a consistent command vector \(\mathbf{0}_{B}\) to [PERSON] and the order 1 together with a consistent command vector \(\mathbf{1}_{C}\) to [PERSON]. [PERSON] knows the positions of almost all 00 and 01 pairs in \(\mathbf{a}\). If [PERSON] had forged even a single 10 or 11 pair, claiming to be either 00 or 01, then [PERSON], when using the CheckAlice Algorithm 1, he would have immediately detected the inconsistency. Therefore, [PERSON] knows the sets \[\mathbb{P}_{0,0}(\mathbf{0}_{B}) =\{p_{1},p_{2},\ldots,p_{t_{1}}\}\,\ \text{where}\ t_{1}\approx\frac{m}{4}\,\ \ \text{and}\] (11) \[\mathbb{P}_{0,1}(\mathbf{0}_{B}) =\{p^{\prime}_{1},p^{\prime}_{2},\ldots,p^{\prime}_{t_{2}}\}\,\ \text{where}\ t_{2}\approx\frac{m}{4}\,\] (12) of the positions of the 00 and 01 pairs in \(\mathbf{a}\). Symmetrically, [PERSON] knows the positions of almost all 01 and 11 pairs in \(\mathbf{a}\):

\[\mathbb{P}_{0,1}(\mathds{1}_{C}) =\{q_{1},q_{2},\ldots,q_{t_{3}}\}\,\ \text{where}\ t_{3}\approx\frac{m}{4}\,\ \ \text{and} \tag{13}\] \[\mathbb{P}_{1,1}(\mathds{1}_{C}) =\{q^{\prime}_{1},q^{\prime}_{2},\ldots,q^{\prime}_{t_{4}}\}\,\ \text{where}\ t_{4}\approx\frac{m}{4}. \tag{14}\]

According to the rules of the protocol, [PERSON] will sent to [PERSON] the command vector \(\mathbf{0}_{B}\), and, simultaneously, [PERSON] will send to [PERSON] the command vector \(\mathds{1}_{C}\). Both will use the CheckWCV Algorithm 2 to verify the consistency of the others' command vector. Then, according to \(\mathbf{Rule}_{3,3}\), both [PERSON] and [PERSON] will abort. * [PERSON] sends to both [PERSON] and [PERSON] inconsistent command vectors. Then, according to \(\mathbf{Rule}_{3,1}\) of the protocol, both [PERSON] and [PERSON] agree to abort. Ergo, both [PERSON] and [PERSON] will either follow the same order or abort. **Proposition A3** (\(n\) Player Loyal [PERSON]). : _If [PERSON] is loyal, then the \(n\) player EPRQDBA protocol will enable all loyal lieutenant generals to follow [PERSON]'s order._

Proof.