# Estimating Interception Density in the BB84 Protocol: A Study with a Noisy Quantum Simulator
## 3 Background
### Background on BB84  
QKD protocols represent a method of secure communication that implements a cryptographic protocol involving quantum mechanical components. The first QKD protocol was presented in 1984 by [PERSON] and [PERSON], from which the BB84 protocol name derives: [11]. This protocol is provably secure when assuming a perfect implementation and two conditions: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Classical Bit** & **+ Basis** & \(X\) **Basis** \\ \hline
0 & \(\ket{\uparrow}\) & \(\ket{\
earrow}\) \\
1 & \(\ket{\rightarrow}\) & \(\ket{\
warrow}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of mapping classical bits to quantum states. 1. Information can only be gained at the expense of disturbing the signal. If the no-cloning theorem is valid, the act of measurement required to obtain information about the quantum state disturbs the original state, preventing exact replication. 2. The use, in parallel, of an authenticated public classical channel. BB84 or other analogous protocols permit the continued utilization of established symmetric encryption algorithms, such as Advanced Encryption Standard (AES) [32], since they are not as significantly affected by the advent of quantum computers, as instead, it affects public key algorithms, including Diffie-Hellman [33]. BB84 Scheme  
The BB84 algorithm begins with [PERSON] randomly selecting a set of \(n\) bits. Considering the polarization encoding, for each bit, it is necessary to establish the basis and the quantum states, using for example the mapping shown in Table 1. In this manner, [PERSON] obtains \(n\) qubits that are subsequently transmitted to [PERSON] through the quantum channel. Before reaching [PERSON], the eavesdropper [PERSON] may attempt to intercept the qubits, decode them, and then, according to the read value, forward new qubits to [PERSON]. Due to the no-cloning theorem, [PERSON] is unable to copy the qubits unless she takes a measurement that inevitably disturbs the original state of that qubit half of the time if she chooses the wrong basis. Consequently, the only option for [PERSON] is to randomly select the basis, measure and decode the intercepted qubit, regenerate the new one using the selected basis, and then forward it to [PERSON]. Similarly, [PERSON] is unable to ascertain in advance the set of bases used by [PERSON] to generate the qubits. Thus, [PERSON] can randomly select the set of \(n\) bases to measure the received qubits and decode the related bits. After the decoding of the \(n\) bits, [PERSON] publicly announces he has received [PERSON]'s signals. [PERSON] responds by sharing, on the public classical channel, the set of \(n\) bases used in the previous transmission. At this point, this operation is safe because the shared bases refer to the previous signal. Similarly, [PERSON] communicates to [PERSON] over the public channel the \(n\) bases he has randomly selected to decode the signal. Both [PERSON] and [PERSON] now compare the received \(n\) bases with those they have used, and they discard the bits corresponding to mismatches. The remaining bits measured with the same bases are instead retained. Assuming that the number of such bits is \(k\), [PERSON] selects a certain fraction of them (in our subsequent simulation analysis, for reasons of simplicity, we will assume the first \(k/2\) bits). She then discloses her selection to [PERSON] over the public channel. Subsequently, both [PERSON] and [PERSON] publicly announce the values of the selected bits and verify whether a predetermined number of them are identical. If this verification fails, the issue may be attributed to [PERSON], random noise, or a combination of both. In such cases, the algorithm must be restarted. Otherwise, the algorithm concludes, and both parties can proceed to generate a secret shared key with the remaining bits, as detailed in [34]. It is worth mentioning that _information reconciliation_ and _privacy amplification_ techniques [2] can be employed to create the secret shared key.