## 2 The Proposed Quantum Secret Sharing Protocol and Its Security
_Step 4.3._ The dealer randomly choose another group of raw data and requests all dishonest players to disclose their corresponding Gaussian random numbers. _Step 4.4._ The dealer can displace measurement results of the group in step 4.3 utilizing \(x_{M}=x_{r}-\sum_{i=2}^{n}\sqrt{T_{i}}x_{i}\) and \(p_{M}=p_{r}-\sum_{i=2}^{n}\sqrt{T_{i}}p_{i}\). Therefore, a two-party CVQKD between the dealer and [PERSON]\({}_{1}\) is established. Then they can estimate a lower bound of secret key rate (SKR) \(R_{1}\) with the standard post-processing procedures in the GMTS QKD [12; 36]. After that, all the players abandon the disclosed data. _Step 4.5._ Repeat steps 4.2-4.4 \(n\) times. In each round, the dealer chooses a different player as the honest player and at final obtains \(n\) secret key rates \(\{R_{i}\}_{i=1}^{n}\). _Step 4.6._ Finally, the dealer employs the minimum of \(\{R_{i}\}_{i=1}^{n}\) as the SKR of the THz-CVQSS protocol and obtains the final SKR from undisclosed data by taking advantage of the reverse reconciliation method. The dealer can use the final shared key to implement a QSS protocol. Through cooperation, \(n\) players can recover the shared key. However, any group of fewer \(n\) players cannot recover the shared key correctly, since only an exponentially small amount of information about the shared key can be obtained by them. Next, we will analyze the security of the protocol. In a word, the proposed QSS protocol actually establishes \(n\) independent point-to-point CVQKD links in each round of quantum transmission. As assumed in steps 4.2-4.5, there is a honest player ([PERSON]\({}_{i}\)) and the remaining \(n-1\) dishonest players in each CVQKD. Note that this assumption is the worst case, since if all the players are dishonest, then the QSS is meaningless. In this most pessimistic case, the QSS is actually reduced to a standard CVQKD model, that is, there are two legitimate players, i.e., the sender, [PERSON]\({}_{i}\) and the receiver, [PERSON] (the dealer). Now we need to analyze whether the remaining \(n-1\) players (and potential eavesdroppers in the channel) can cooperate to recover the key shared by [PERSON]\({}_{i}\) and [PERSON]. As mentioned in step 4.3, the dealer requests all the players except [PERSON]\({}_{i}\) to publish their corresponding Gaussian random numbers, which makes [PERSON]\({}_{i}\) own the complete information of all the players, while the remaining \(n-1\) players cannot infer the information about the shared key between [PERSON]\({}_{i}\) and [PERSON] in this CVQKD link. As a result, whether there are \(n-1\) dishonest players or not, [PERSON] and [PERSON]\({}_{i}\) can share a secure key. As for the eavesdroppers in the channel, we can consider their quantum attack in each individual CVQKD link. This allows us to apply the standard security proof of GMTS QKD. Some previous work [11; 12] has proved the security of GMTS QKD. Thus, we can use the existing security proof for GMTS QKD to evaluate the SKR of THz-CVQSS. In addition, since the players inject the locally prepared thermal Gaussian states into the circulating spatiotemporal mode, the detection signals of the eavesdropper cannot reach the modulators within the secure stations of the honest players. In other words, the honest players can prevent the eavesdropper from accessing the preparation process of the signal states, so that the protocol is immune to Trojan horse attacks.