# Quantum Authentication Evolution: Novel Approaches for Securing Quantum Key Distribution
## 5 QKD Authentication Model Setup and Details
### Quantum Communication Metrics: QBER, Maximum Corrected Errors, and Overhead  
In our analysis, we introduce the concept of the probability \(\rho\) representing a random flip induced by the noise within the quantum channel. This probability aligns closely with the QBER, serving as an estimation of errors within the quantum channel. For the sake of simplicity, we treat QBER and \(\rho\) as interchangeable, assuming their equivalence for easier notation. The QBER holds significance in our study and serves as a crucial variable. To ground our investigation, we reference the experimentally estimated QBER in an authenticated QKD implementation [23]. Our examination of the QBER extends to an error threshold of 1.1%, the maximum value observed in experimental settings. Additionally, we delve into assessing the authentication cost concerning the time required for protocol execution, a metric we term as overhead. In certain instances, we break down the authentication process into its signature and verification components. This division allows for a more detailed comparison of the performance of each algorithm relative to the others. In such cases, we explicitly distinguish between mono-signatures/verifications, contributing to a nuanced analysis of algorithmic efficiency. An additional parameter under consideration is the count of corrected errors. To manage computational resources effectively, we limit the correction process to address up to three errors. This decision is based on practical considerations, as a brute force search for four errors proves to be time-consuming. Details on the time required for worst-case error correction scenarios are presented in Table 3. Whenever the actual number of errors surpasses the predetermined limit for correction, the entire key associated with that exchange is discarded. Consequently, the exchange is aborted, and the protocol is restarted. The frequency of such aborts becomes another parameter of interest in our study, shedding light on the robustness and reliability of the key distribution process. The failure time is computed for the error correction process. Table 3 presents the maximum (averaged) time that the script incurs in conducting a brute force search to address errors. This metric serves as an indicator of the computational overhead and time investment involved in rectifying errors during the error correction phase. The count of hashes performed, presented in each cell, is derived from the combinatorial formula, which is used to determine the total number of unique hash operations conducted during the specified computational processes. This calculation offers insights into the computational complexity and hashing requirements associated with different scenarios and parameters. \[nC_{r}=\frac{n!}{r!(n-r)!} \tag{4}\]  
The formula utilized for calculating the number of hashes performed is expressed as \(nC_{r}\), where \"\(n\)\" represents the total number of bits under consideration (128, 192, or 256), and \"\(r\)\" corresponds to the maximum number of corrected bits. This formula provides a systematic approach to quantify the hashing operations required for various configurations, facilitating a detailed analysis of the computational demands involved. In this context, it is important to note that the cumulative count of hashes includes those from the preceding rounds. For instance, in the scenario of a 128-bit key with a brute force search allowing up to one error, the process involves hashing the original string (1 hash) and subsequently hashing the 128 possible combinations corresponding to flipping only one bit (128 hashes). In total, this results in 129 hashes. This principle applies consistently across the different cases under consideration. Having thoroughly examined the introduced parameters, our focus shifted to investigating how the quantity of exchanged keys influences the performance of the algorithms. This exploration entailed studying a practical QKD setup characterized by a continuous photon stream.