## 2 Preliminary Concepts  
### \(|GHZ_{3}\rangle\) States  
Numerous quantum protocols can be characterized as interactions among familiar fictional characters, typically identified as [PERSON], [PERSON], [PERSON], etc., and in this particular case, [PERSON] and [PERSON] besides [PERSON]. The players are engaged in efforts to exchange information in a manner that is both private and secure. Generally, secure communication is achieved through a combination of classical authenticated channels and quantum channels. The transmission of confidential information typically occurs via the quantum channel, employing a variety of techniques, followed by the exchange of messages through a classical public channel to complete the protocol. During this process, an adversary, commonly called [PERSON], may attempt to intercept this communication and extract any available information. In such scenarios, the primary advantage of quantum cryptography over classical methods lies in the protection afforded to the communicating parties during the information transmission through the quantum channel, which is grounded in fundamental principles of quantum mechanics, including the no-cloning theorem [62] and the concept of entanglement monogamy, among others. Entanglement is the cornerstone of numerous quantum protocols, including the one discussed in this paper, and could potentially become the standard approach to quantum cryptography in the future thanks to its wide range of uses across the entire domain. It is a key concept in quantum mechanics and can be represented mathematically as the sum of two or more product states. GHZ states are a prominent class of maximally entangled states, involving three or more qubits. At its core, quantum entanglement suggests that quantum particles can become linked together, and the state of one particle can instantly affect the state of another, regardless of the distance between them. For those interested in exploring this topic further, standard textbooks like [63; 64; 65] are a great resource for a more in-depth explanation. The Hadamard and CNOT gates, two common quantum gates, can be used to create \(|GHZ_{n}\rangle\) states on modern quantum computers, as demonstrated by the power of the most recent IBM quantum computers [2; 3]. \(n\) qubits, \(n-1\) CNOT gates, and one Hadamard gate applied to the first qubit are needed to implement such a circuit. To build effective GHZ circuits, we follow [61], which provides a workable approach that requires only \(\lg n\) stepsto prepare the \(|GHZ_{n}\rangle\) state. Our new protocol employs \(|GHZ_{3}\rangle\) triplets. A \(|GHZ_{3}\rangle\) state is formally expressed by the following formula. To make clear that \(|GHZ_{3}\rangle\) entanglement requires three separate entities, we use the subscripts A, S, and T to indicate which qubit belongs to [PERSON], [PERSON], and [PERSON]. Following this convention, qubits denoted by \(|\cdot\rangle_{A}\) belong to [PERSON], those denoted by \(|\cdot\rangle_{S}\) belong to [PERSON], and those denoted by \(|\cdot\rangle_{T}\) belong to [PERSON]:  
\[|GHZ_{3}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{T}|0\rangle_{S}|0\rangle_{A}+|1 \rangle_{T}|1\rangle_{S}|1\rangle_{A})\;. \tag{1}\]  
In view of the scope of the proposed protocol, it is necessary to use \(m\)\(|GHZ_{3}\rangle\) triplets because a single \(|GHZ_{3}\rangle\) triplet will not suffice. The mathematical description of the state of a composite system containing \(m\)\(|GHZ_{3}\rangle\) triplets is shown below (see also [14; 66]):  
\[|GHZ_{3}\rangle^{\otimes m}=\frac{1}{\sqrt{2^{m}}}\sum_{\mathbf{x}\in\mathbb{ B}^{m}}|\mathbf{x}\rangle_{T}|\mathbf{x}\rangle_{S}|\mathbf{x}\rangle_{A}\;. \tag{2}\]  
In writing Formula (2), the following notation is used:  
* \(\mathbb{B}\) is the binary set \(\{0,1\}\). * To differentiate syntactically the bit vector \(\mathbf{x}\in\mathbb{B}^{m}\) from the bit \(x\in\mathbb{B}\), the bit vector \(\mathbf{x}\) is written in boldface.