## 6 Efficiency and Security Analysis
### Security
#### 6.2.2 Participants' Attack  
In this subsection, we examine the scenario in which one of the players in this game seeks to acquire information regarding the wealth of one of the millionaires. It is useful to categorize the following situations based on the dishonest participant attempting to obtain unauthorized information:  
**[PERSON]\({}_{i}\)**: Let us assume that [PERSON]\({}_{i}\), \(0\leq i\leq n-1\), intercepts the \(|GHZ_{3}\rangle\) qubits during their transmission from Trent to [PERSON] and another Alice\({}_{j}\), \(0\leq i\
eq j\leq n-1\). [PERSON]\({}_{i}\) has the option to measure them and send them back to their intended recipients or transmit new, previously prepared qubits to their intended recipients. Taking into consideration the fact that the \(|GHZ_{3}\rangle\) triplets contain no information at all, we see that, in both cases, [PERSON]\({}_{i}\) will fail to gain any information. Moreover, the decoy technique will allow the legitimate players to infer that someone has tampered with the transmitted sequences. Another scenario for [PERSON]\({}_{i}\) is to intercept the qubits from the \(|GHZ_{3}\rangle\) triplets during their transmission from [PERSON] to [PERSON] and [PERSON]\({}_{j}\), \(i\
eq j\), and, without measuring them, entangle them with her ancilla state. Then, [PERSON]\({}_{i}\) forwards the relevant qubits to [PERSON] and [PERSON]\({}_{j}\) and waits for the quantum part of the protocol to finish. At this point, she measures her qubits in the hope of gaining useful information. The critical flaw in [PERSON]\({}_{i}\)'s strategy is that her actions result in having \(|GHZ_{4}\rangle\) quadruples evenly distributed among [PERSON], [PERSON], [PERSON]\({}_{j}\), and [PERSON]\({}_{i}\) instead of having \(|GHZ_{3}\rangle\) triplets evenly distributed among [PERSON], [PERSON], and [PERSON]\({}_{j}\). Ultimately, [PERSON]\({}_{i}\) will fail to compute the correct modulo 2 sum as the classical portion of the protocol unfolds, the reason being that [PERSON]\({}_{i}\) will not know [PERSON]'s private bit vector \(\mathbf{y}_{2}\). Recall that [PERSON] never divulges \(\mathbf{y}_{2}\) or the modulo 2 sum \(\mathbf{y}_{2}\oplus\mathbf{y}_{1}\oplus\mathbf{y}_{0}\) to any other in-game player or outside eavesdropper, as ordained by (\(\mathbf{A}_{4}\)) and (\(\mathbf{D}_{3}\)). Once again, [PERSON]\({}_{i}\) fails to gain any information. Therefore, the GHZ\({}_{3}\)MQPEC protocol is immune to attacks by [PERSON]\({}_{i}\), \(0\leq i\leq n-1\). **[PERSON]**: Let us assume that [PERSON] intercepts the \(|GHZ_{3}\rangle\) particles en route from Trent to Alice\({}_{i}\). [PERSON] has the option to measure these particles and subsequently return them to [PERSON]\({}_{i}\), or alternatively, she may send new qubits that she has prepared in advance to [PERSON]\({}_{i}\). Since the \(|GHZ_{3}\rangle\) triplets carry no information, [PERSON] will not gain any information in any of the earlier cases. Additionally, the other legitimate players will be able to identify the tampering thanks to the use of the decoy technique. Another possibility for [PERSON] is to intercept the qubits from the \(|GHZ_{3}\rangle\) triplets during their transmission from [PERSON] to [PERSON]\({}_{i}\) and entangle them with her ancilla state, without measuring them. [PERSON] sends corresponding qubits to [PERSON]\({}_{i}\) and waits for the protocol to finish. Then, she measures her qubits, expecting to uncover some information. The problem with this strategy is that [PERSON]'s tampering has resulted in having \(|GHZ_{4}\rangle\) quadruples evenly distributed among [PERSON], [PERSON], and [PERSON]\({}_{i}\), instead of having \(|GHZ_{3}\rangle\) triplets. Ultimately, [PERSON] will fail to compute the correct modulo 2 sum as the classical portion of the protocol unfolds because she will not know [PERSON]'s private bit vector \(\mathbf{y}_{2}\).