# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 5 A QKD Protocol Based on a Shared Collaboration among Three Parties
### Considerations for Complexity and Number of Resources in the Procedure  
Secure communications remain safe against attacks, particularly by those performed by quantum computers. They require effective and strong protocols of QKD. In any case, even when the use of a minimum of resources has been mainly pursued in the original contributions to those protocols, extreme security only requires a sizable, but a finite number of resources and signals. The BB84 protocol is the earliest QKD protocol [11]. It has inspired a variety of other similar slender protocols as the B92 protocol [12] but reducing the communication efficiency and the practicability. Still, the six-state protocol is a more secure extension of the BB84 protocol, despite raising the upper limit of the QBER. Otherwise, limited to the current single-photon source technology, it is not possible to obtain ideal single photons, instead of using multi-photon sources [13] as in the decoy-state [51], the most widely implemented QKD scheme. Such protocols anyway introduce higher complexity compared to the more theoretical QKD protocols being proposed. As well, single-photon QKD systems commonly include polarization and phase encoding, thus introducing a higher number of resources to the theoretical ideal approaches. Thus, the BB84 protocol uses the fewest resources to produce each bit of the key, a single ideal qubit properly prepared on a certain basis and two signals, one travelling through a single quantum channel and a classical one to share the basis used by both main parties, after the receiver measures the incoming qubit. Compared with the procedure being presented, the current protocol uses five qubits partially entangled by pairs. This number, as it was seen, increases to nine qubits for feasible implementations: two for performing the \(C^{C}\mathcal{U}_{\mathbb{I}_{4}}\) and two more for performing the \(C_{SWAP_{4}}\) among the faraway qubits \(2,4\). In addition, two more classical signals should be added in each case (see Appendices A and B). A more elaborated infrastructure is expected in terms of quantum gates and entanglement control. Despite this, the asymmetric processing combined with the shared assembling via teleportation allow for a strategy to lower the eavesdropping rate of success to a great extent. In any case, the number of resources grows linearly concerning the key block length. More complex proposals have been considered in real scenarios to avoid environmental factors lowering the efficiency of quantum cryptography [52]. Other complex deployments currently consider more specialized quantum and optical resources in QKD protocols [53]. Thus, entangled resources and shared multipartite schemes as in the current proposal should be considered to set more secure procedures, particularly in the small block length regime [54]. Complexity is not equivalent to impossibility, until now, first approaches to QKD have arisen in parallel with our technological scope. Nevertheless, while more control is reached on quantum systems, more new audacious proposals are being tried combining full quantum resources to reach more outstanding outcomes. Considering the outcomes for the success of [PERSON], \(\overline{P}_{E_{H}}=0.073\) for \(\theta=0\) (basis \(H\)) and \(\overline{P}_{E_{D}}=0.164\) for \(\theta=\frac{\pi}{2}\) (basis \(D\)), we note she has a higher probability to guess outcomes in the last basis. The last outcomes are not the complete picture because still [PERSON] should decide which \(\theta\) will use (basis) and also the pair \(j,k\) to impose the adequate prescriptions. Thus, the global average success probability for [PERSON] in a key of length \(N=n_{H}+n_{D}\) (when [PERSON] decides to use \(n_{H}\) times the basis \(H\) and \(n_{D}\) times the basis \(D\)) is:  
\[\overline{P}_{E_{Total}} = (\frac{n_{H}}{N}\overline{P}_{E_{H}}+\frac{n_{D}}{N}\overline{P} _{E_{D}})^{N}=(\overline{P}_{E_{H}}+f_{D}(\overline{P}_{E_{D}}-\overline{P}_{ E_{H}}))^{N} \tag{31}\]  
where \(f_{D}\equiv\frac{n_{D}}{N}\).