# Multi-Stage Quantum Secure Direct Communication Using Secure Shared Authentication Key
## 5 Example of the Proposed Protocol
### Message Sharing Procedure  
In this section, the implementation of SSAK protocol over free space optics using passive optical components is discussed. Figure 3 shows the setup of the implementation. [PERSON] will have three half-wave plates (HWPs) in her possession while [PERSON] will have two HWPs. A linearly polarized laser is used as the photon source in our implementation [28]. At the beginning of the protocol, [PERSON] generates a state with a \(0^{\circ}\) linear polarization using a \(0^{\circ}\) polarizer. The polarization procedure is to encrypt the classical bits using the photons to produce the quantum bit (qubit). After the classical bits are encrypted to the photons, she will apply a transformation using HWP-1 at the angle \(\theta\) and apply her second HWP-2 at angle \(\theta_{A}\), which results in the superposition state, \(|\psi\rangle\). The superposition states carry the secret message that will be shared between [PERSON] and [PERSON] in quantum manners [27]. The rotation of HWP operation is represented as  
\[M_{HWP}(\theta)=\left[ \begin{array}{cccc}1&0&0&0\\ 0&cos(4\theta)&sin(4\theta)&0\\ 0&sin(4\theta)&-cos(4\theta)&0\\ 0&0&0&-1\end{array} \right] \tag{6}\]  
where  
\[0^{\circ}<\theta<45^{\circ} \tag{7}\]  
It is noted that HWP-1 at the angle \(\theta\) is the authentication key set from \(0^{\circ}\) to \(45^{\circ}\), coupled with information \(X\) to make the protocol more secure. Alice then applies the transformation associated with the encoded bit using HWP-3 that will be set at an angle \(\theta_{X}=0^{\circ}\) in the case of bit 0 being sent or \(\theta_{X}=45^{\circ}\) if bit 1 is being sent. \[ \begin{array}{c}\textit{Message }X\Rightarrow\textit{bits}\in\{0,1\}^{n} \\ \{0,1\}^{n}\Rightarrow\left\{\begin{array}{c}|0\rangle,\left|S_{out}=M_{HWP }(0^{\circ})\times S_{in}\\ |1\rangle,\left|S_{out}=M_{HWP}(45^{\circ})\times S_{in}\end{array}\right. \right.\end{array}  \tag{8}\]  
where \(S_{in}\) is the Stokes parameter's sequence of the input light that is illustrated as  
\[S_{in}=\left[ \begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{c}1\\ 1\\ 0\\ 0\end{array} \right] \tag{9}\]  
Figure 3: Implementation of secure shared authentication key protocol over Free Space Optic (FSO). while \(S_{out}\) is the Stokes parameter's sequence of the output light  
\[S_{out}=\left[ \begin{array}{c}S^{\prime}{}_{0}\\ S^{\prime}{}_{1}\\ S^{\prime}{}_{2}\\ S^{\prime}{}_{3}\end{array} \right] \tag{10}\]  
\[S_{out}=M_{HWP}\times S_{in} \tag{11}\]  
The light polarization angle after passing through the HWP can be measured [28; 29] by  
\[\theta=\cos^{-1}\frac{S_{in}[1:3].S_{out}[1:3]}{\parallel S_{in}[1:3] \parallel\times\parallel S_{out}[1:3]\parallel} \tag{12}\]  
where \". \" is the multiplication operator and \(\parallel\cdot\parallel\) is the norm operator. [PERSON] will send the optical beam containing the message to [PERSON], who will first pass it through his first HWP (HWP-4). Then he will remove his transformation using a HWP-5 sets at the angle \(\theta_{A}\). [PERSON] will receive a beam polarizer at either \(0^{\circ}\) or \(90^{\circ}\) and pass it through a polarization beam splitter. The detectors will detect whether it is bit 0 or bit 1. A basic description of the proposed protocol is discussed here. First, the message that is translated into the binary code is encrypted into a single particle called a quantum bit or qubit. The quantum bit is then transmitted from [PERSON] to [PERSON] using the proposed protocol as follows. 1. Suppose that [PERSON] prepares a photon. [PERSON] wants to send the first qubit, \(|0\rangle\), the qubit is then encrypted to the photons using the first agreed angle from the first procedure, \(\theta_{A}=30^{\circ}\). [PERSON] prepares photons based on Equation (9) and encrypts it using the rotation of HWP based on Equation (6). \[