## 6 Efficiency and Security Analysis
### Security
#### 6.2.1 External Attack  
Before we lay out the formal analysis regarding the possible actions undertaken by [PERSON] in her effort to breach the security of the GHZ\({}_{3}\)MQPEC protocol and obtain some useful information, it is essential to recall two standard detection techniques that are among the first line of defense against malicious adversaries. 1. **Eavesdropping detection. ** A typical scheme for the detection of a possible eavesdropper is to have the source responsible for the generation and distribution of the entangled particles, the \(|GHZ_{3}\rangle\) triplets in our case, also produce decoy photons randomly chosen from one of the \(|0\rangle,|1\rangle,|+\rangle,|-\rangle\) states, which are randomly embedded into the transmission sequences. [PERSON], by meddling with the transmission sequences, and unaware of the positions of the decoys, will invariably cause errors that are easily detectable by the legitimate players upon a typical verification check. This technique, which is referred to as the _decoy technique_, has been extensively studied in the literature [20; 26; 36; 41; 43; 44; 47; 69]. 2. **Entanglement validation. ** The effective execution of any protocol based on entanglement relies fundamentally on the presence of entanglement itself. In the absence of assured entanglement, the operational integrity of the protocol is jeopardized. The process of validating entanglement may yield two possible results. Should entanglement be confirmed, the protocol is then able to advance and successfully fulfill its designated purpose. The inability to confirm entanglement suggests a lack of the essential entanglement itself. This situation may arise due to either the presence of noisy quantum channels or intentional disruption by an adversary. No matter what the underlying reason might be, the only appropriate course of action is to cease the current execution and restart the entire process, ensuring that corrective measures are put in place. Given its importance, the validation of entanglement has been extensively examined by numerous researchers. The GHZ\({}_{3}\)MQPEC protocol follows the advanced techniques described in previous studies, and in particular [77; 78; 79; 80; 81; 82]. We shall now examine the most prominent tactics that the external eavesdropper [PERSON] might utilize. 1. **Measure and Resend. ** [PERSON] intercepts the \(|GHZ_{3}\rangle\) qubits as they are being transmitted from Trent to [PERSON], and [PERSON]\({}_{i}\), \(0\leq i\leq n-1\), measures them and then returns them to the intended recipients in this kind of attack. However, there is no information encoded in the \(|GHZ_{3}\rangle\) triplets. Therefore, [PERSON] will be unable to find any information by doing this. Additionally, the legitimate players will be able to identify a malicious presence with ease if they use the decoy technique. As a result, the protocol is immune to attacks of this kind. 2. **Intercept and Resend. ** [PERSON]'s approach in this scenario involves intercepting the \(|GHZ_{3}\rangle\) particles while they are being transmitted from Trent to [PERSON] and [PERSON]\({}_{i}\), where \(0\leq i\leq n-1\). Due to the no-cloning theorem, she is unable to create copies of these particles. Consequently, she opts to prepare and send new qubits to the designated recipients. [PERSON] does not learn anything by measuring the \(|GHZ_{3}\rangle\) triplets because, as we have already mentioned, they carry no information at all during the protocol's transmission phase. Additionally, the legitimate players can deduce the existence of an eavesdropper thanks to the decoy technique. So, the GHZ\({}_{3}\)MQPEC protocol is also safe from such an attack. 3. **Entangle and Measure. ** [PERSON] starts this kind of attack by intercepting the qubits of the \(|GHZ_{3}\rangle\) triplets during their transmission.