Advancing multi-party quantum key agreement protocols: the power of a two-layer approach

Abstract

Quantum key agreement holds great promise for enabling secure and efficient data transfer in the era of advanced technologies. This paper proposes a novel approach to address the key generation problem by introducing a two-layer multiparty quantum key agreement (TMQKA) protocol based on non-maximally entangled states. The TMQKA protocol generates two layers of keys among multiple users by employing the properties of non-maximally entangled states and local operations. The first layer of keys is established among all users to achieve secure multiparty communication. The second layer of keys is between two neighboring users to ensure point-to-point communication between them. In addition, the analysis demonstrates that the proposed TMQKA protocol satisfies the requirements of correctness, security, and fairness. The unique two-layer structure of TMQKA makes it have a high efficiency. Our work lays the foundation for the establishment of two-layer quantum networks in the future.

Appendix A: The efficiency calculations

In the following, we consider that the MQKA protocol combined with QKD accomplishes the same task. It is well known that BB84 is a relatively efficient and practical QKD scheme. The protocols in Refs. [13, 19, 24, 25] are some of the typical and efficient MQKA protocols. These MQKA protocols utilize different quantum resources such as single photons, Bell states, and six-particle entangled states. We will use the MQKA protocol and BB84 from Ref. [19] as an example to calculate the efficiency of the method.

1.1 A.1 The combined method

The goal of the users is to obtain the two-layer keys. All users obtain a 0.5l-bit first layer key by executing an MQKA protocol. Any two neighboring users obtain a l-bit second layer key based on the BB84 protocol.

The first layer key. All users execute MQKA in Ref. [19] to get a 0.5l-bit first layer key. Based on Ref. [19], n users need to consume $n(l+\delta )$ qubits in this process.

The second layer key. The method of negotiating a second layer key between two neighboring users based on QKD is as follows.

(1) Any two neighboring users $U_{i}$ and $U_{i{\tilde{\oplus }}1}$ generate two l-bit secret keys $R_i^{\prime }=\{r^{\prime }_{i,1},r^{\prime }_{i,2},...,r^{\prime }_{i,l}\}$ and $R_{i{\tilde{\oplus }}1}^{\prime }=\{r^{\prime }_{i{\tilde{\oplus }}1,1},r^{\prime }_{i{\tilde{\oplus }}1,2},...,r^{\prime }_{i{\tilde{\oplus }}1,l}\}$, respectively. $U_{i}$ prepares a particle sequence $A_{i}=\{a_{i,1},a_{i,2},...,a_{i,l}\}$. Here, if $r^{\prime }_{i,j}=0~(1)$, $a_{i,j}$ is in state $|0\rangle ~(|1\rangle )$. Similarly, $U_{i{\tilde{\oplus }}1}$ prepares a particle sequence $A_{i{\tilde{\oplus }}1}$. They insert $\delta$ decoy particles into the particle sequences $A_{i}$ and $A_{i{\tilde{\oplus }}1}$, respectively, where each decoy particle is randomly in one of the four states in $\{|0\rangle , |1\rangle , |+\rangle ,|-\rangle \}$. They then send the processed particle sequences to each other.

(2) After $U_{i}$ and $U_{i{\tilde{\oplus }}1}$ receive the particle sequences, they detect the channel security. More specifically, they announce the corresponding bases of the decoy particles and measure them, respectively. Then, they announce the positions and the measurement results and compare these results with the initial states.

(3) If there are no errors, they measure the remaining particles in $\{|0\rangle , |1\rangle \}$ basis. $U_{i}$ and $U_{i{\tilde{\oplus }}1}$ obtain the keys $R_{i{\tilde{\oplus }}1}^{\prime }$ and $R_i^{\prime }$, respectively.

(4) $U_{i}$ and $U_{i{\tilde{\oplus }}1}$ get a l-bit second layer key by computing $R_i=R_i^{\prime } \oplus R_{i{\tilde{\oplus }}1}^{\prime }$. The second layer key is influenced equally by two neighboring users.

Table 2 Efficiency calculation of BB84 protocol combined with MQKA protocol

In total, they consume $2(l+\delta )$ qubits. In addition, every two neighboring users need to establish a l-bit second layer key. In total, all users need to consume $2n(l+\delta )$ qubits.

1.2 A.2 The efficiency calculations

Based on the above analysis, let’s calculate the efficiency of this method. In total, it needs to consume $2n(l+\delta )+n(l+\delta )=3n(l+\delta )$ qubits, i.e., $q=3n(l+\delta )$. At the end of the protocol, each user gets a 0.5l-bit second layer key and two l-bit second layer keys, i.e., $b=2.5l$. Therefore, we have

$$\begin{aligned} \begin{aligned} \eta _1=\frac{2.5l}{3n(l+\delta )}=\frac{2.5}{3n(1+\nu )}. \end{aligned} \end{aligned}$$

(A1)

Based on the same calculation method and Refs. [13, 19, 24, 25], we can calculate the efficiencies of some typical MQKA protocols combined with QKD protocols to accomplish the same tasks as TMQKA, as shown in Table 2.