# Quantum Authentication Evolution: Novel Approaches for Securing Quantum Key Distribution
## 5 QKD Authentication Model Setup and Details
### Mono-Authentication
The QBER is defined as  
\[QBER=\frac{\left(\[PERSON]^{B0}_{chkAB}-\[PERSON]^{A0}_{chkAB}\right)+\left(\[PERSON]^{B1}_{ chkAB}-\Psi^{A1}_{chkAB}\right)}{n_{r}} \tag{1}\]  
where  
\(\Psi^{B0}_{chkAB}\) is the number of 0 bits in \(\[PERSON]^{B}_{chkAB}\)  
\(\Psi^{B1}_{chkAB}\) is the number of 1 bit in \(\[PERSON]^{B}_{chkAB}\)  
\(\Psi^{A0}_{chkAB}\) is the number of 0 bit in \(\[PERSON]^{A}_{chkAB}\)  
\(\Psi^{A1}_{chkAB}\) is the number of 1 bit in \(\Psi^{A}_{chkAB}\)  
The length of the string containing the revealed bits is denoted as \(n_{r}\), where \(n_{r}=len(\Psi^{i}_{chkAB}\big{|}i=A\ or\ B)\) equals the length of the string.  
At [PERSON]'s error correction and privacy amplification stage, the primary objective is to securely exchange a secret key between [PERSON] and [PERSON]. The secret keys derived from previous steps are denoted as \(\Psi^{A}_{kAB}\) and \(\Psi^{B}_{kAB}\). Due to inherent noise in the communication channel, these keys may not match perfectly \(\left(\Psi^{A}_{kAB}\
eq\Psi^{B}_{kAB}\right)\). Our goal is to synchronize these keys to ensure they are identical. It is important to note that the original BB84 protocol did not account for noise in the communication channel, which led to the later introduction of error correction mechanisms. These mechanisms were added to rectify discrepancies and ensure the successful and secure exchange of the key between [PERSON] and [PERSON].  
Once it is confirmed that no eavesdropper has tampered with the quantum communication, the participants begin the error correction process. The goal is to align their respective strings, achieving the highest level of mutual information between [PERSON] and [PERSON]. [PERSON] starts the error correction process by creating a hash of her secret key, \(\Psi^{A}_{kAB}\), using the function \(F\) as:  
\[F:\ F^{\prime}\ \gets F\Big{(}\Psi^{A}_{kAB}\Big{)} \tag{2}\]  
\(F^{\prime}\) denotes the outcome of a secure hashing procedure, guaranteeing consistent results for specific inputs while preventing the original input from being inferred. This deterministic hashing is vital for synchronizing keys between [PERSON] and [PERSON] during error correction. After error correction, the subsequent step is privacy amplification, a pivotal process aimed at bolstering key confidentiality. In mathematical terms, this entails minimizing the mutual information between [PERSON] and [PERSON] to the fullest extent possible. To accomplish this, [PERSON] introduces a random permutation, denoted as \(P\in\mathcal{P}\), where \(\mathcal{P}\) represents the set of permutations. [PERSON] applies this permutation \(\mathcal{P}\) to her secret key and then hashes the result using the same hash function utilized in the prior error correction phase. The resultant value, denoted as \(\Psi_{skAB}^{A}=F\big{(}P\big{(}\Psi_{kAB}^{A}\big{)}\big{)}\), signifies the ultimate secret key following privacy amplification. This method ensures the continued security of the key. The overall procedure unfolds as:  
\[[PERSON]:\quad P\in\mathcal{P}\overset{F,P}{\rightarrow}[PERSON]:\quad\Psi_{skAB}^{A} :=F\Big{(}P\Big{(}\Psi_{kAB^{\prime}}^{B}\Big{)}\Big{)} \tag{3}\]  
At this point, [PERSON] completes her part in the protocol. In the last phase, she communicates crucial details to [PERSON] essential for error correction and privacy amplification. This involves sharing information such as the hash of the private key (\(F^{\prime}\)), the random permutation (\(P\)), and specifying the hash function she employed (\(F\)). Together, these elements equip [PERSON] with the required parameters to execute the final protocol steps, guaranteeing the synchronization and security of the resultant secret key. In the final phase of the key exchange, [PERSON] initiates error correction and privacy amplification upon receiving the transmitted data from [PERSON]. He begins by utilizing the hash (\(F^{\prime}\)) provided by [PERSON] to conduct error correction. [PERSON] computes the hash of his own key (\(F^{\prime\prime}=F\big{(}\Psi_{kAB}^{B}\big{)}\)).