## 3 Preliminaries
### Proposed Approach  
Our proposed architecture can be mainly divided into three phases: **local training**, **server aggregation**, and **model update**. Initially, the clients decide on the number \(t\) of servers. Table 1 lists all the symbols we use in the model and their respective meanings. **Local training. **\(\mathrm{C_{i}}\) trains a model and obtains new model gradients \(M_{i}\). When the training reaches the specified epoch, the clients execute \(\mathbf{Share_{t-n}(M)}\) to generate \(q\) groups, with each group containing \(t\) shares, as shown in Algorithm 1 and Figure 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline Svr & server \\ C & client \\ \(n\) & number of servers \\ \(t\) & number of selected threshold \\ \(q\) & number of bits converted from server number, \(q=\log_{t}(n-1)\) \\ \(z\) & number of clients \\ \(M_{i}\) & the gradient of the \(i\)-th client’s model \\ \(m_{i,j,k}\) & the \(k\)-th share of the \(j\)-th secret shared by the \(i\)-th client \\ \(l_{i}\) & the amount of data from the \(i\)th client \\ \(L\) & the total data amount, \(L=\sum l_{i}\) \\ \(h\) & the different bit numbers of the selected server \\ \(w\) & the weighted average obtained by servers \\ \(num\) & the set of numbers of the selected \(t\) servers \\ \(S\) & the set of the number of all servers \\ \(r_{i}\) & the random number for adjusting gradient of model \(i\)-th layer \\ \(p\) & the number of layers in the model \\ \(lv\) & the membership level \\ \(T_{\mathrm{G}}\) & the Gradient Adjustment Table generated by the level server \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 2: \(\mathrm{C_{i}}\) splits the gradients after local training and sends the gradients and data volume to \(\mathrm{Svr_{s}}\). ```
0: all servers' number length \(q\), \(M_{i}\) is the gradients obtained after training for \(\text{C}_{\text{i}}\), \(t\) is the number of servers that the clients choose to eventually aggregate. for\(j\gets 0\) to \(q-2\)do for\(k\gets 0\) to \(t-1\)do \(m_{i,j,k}=\) a random integer in the range \(F\) endfor \(m_{i,j,t-1}=M_{i}-\sum_{k=0}^{t-2}m_{i,j,k}\) endfor
```  
**Algorithm 1** Splitting gradients after local training  
For example, \(\text{C}_{\text{i}}\) receives the new gradients \(M_{i}\), and executes \(\mathbf{Share_{t-n}(M_{i})}\) to obtain \((m_{i,0,0},m_{i,0,1},\cdots,m_{i,0,t-1})\), \((m_{i,1,0},m_{i,1,1},\cdots,m_{i,1,t-1})\), \(\cdots\), \((m_{i,q-1,0},m_{i,q-1,1},\cdots,m_{i,q-1,t-1})\), among which  
\[ \begin{split} M_{i}&=m_{i,0,0}+m_{i,0,1}+\cdots+m_ {i,0,t-1}\\ &=m_{i,1,0}+m_{i,1,1}+\cdots+m_{i,1,t-1}\\ &\vdots\\ &=m_{i,q-1,0}+m_{i,q-1,1}+\cdots+m_{i,q-1,t-1}\end{split}  \tag{1}\]  
After dividing the gradients into multiple groups of shares \(m_{i,j,k}\), the clients need to send these shares to the servers. Next, we introduce the rules for which share the clients send to which server, as shown in Algorithm 2 and Figure 2. The clients decide which server to send the share \(m_{i,j,k}\) to based on the second parameter \(j\) and third parameter \(k\). When \(\text{C}_{\text{i}}\) wants to send shares to \(\text{Svr}_{\text{s}}\), \(\text{C}_{\text{i}}\) sends \(m_{i,0,s_{0}},\ldots,m_{i,q-1,s_{q-1}}\), a total of \(q\) shares, and data volumes \(l_{i}\) to \(\text{Svr}_{\text{s}}\). ```
0: \((m_{i,0,0},\ldots,m_{i,0,t-1}),\ldots,(m_{i,q-1,0},\cdots,m_{i,q-1,t-1})\) is the \(q\) group of gradient shares of \(\text{C}_{\text{i}}\), \(s_{0}\ldots s_{q-1}\) is the identifier of \(\text{Svr}_{\text{s}}\), \(l_{i}\) is the data volume of \(\text{C}_{\text{i}}\). for\(j\gets 0\) to \(q-1\)do \(\text{C}_{\text{i}}\) sends \(m_{i,j,j_{i}}\) to \(\text{Svr}_{\text{s}}\) endfor \(\text{C}_{\text{i}}\) sends \(l_{i}\) to \(\text{Svr}_{\text{s}}\)
```  
**Algorithm 2** Sending shares and data volume to servers  
This is in contrast to Fedshare [8], where each client sends one share and one data size value to each server. In our proposed architecture, each server receives \(q\) shares and a data size value from each client.