## 4 Sequential Signature Transmission Scheme for Sensor Data
Check whether the timestamp meets \(T_{0}\leq T_{\Delta}+T_{\bar{\zeta}}\), where \(T_{0}\) is the current timestamp value, and \(T_{\bar{\zeta}}\) is the threshold value; If the conditions are met, verify the signature further, otherwise discard the data; 2. Calculate \(h=H(id_{\bar{\zeta}}||data_{k+T}||R_{k+T})\); 3. Split \(h\) into \(k\) substrings \(h_{1},h_{2},\ldots,h_{k}\) with a length of log\({}_{2}\)\(n\) bit 4. Express \(\mathrm{hj}\) as an integer \(i_{j}(1\leq j\leq k)\); Output : For each \(j,\ (1\leq j\leq k)\), if \(f\Big{(}s_{j}\Big{)}=pk_{ij}\), it is accepted; otherwise it is rejected. ```  
**Algorithm 1**Sequential signature algorithm  
In the scheme we propose, the sensor end and the remote state evaluation system end use their own current cycle time \(S_{k+T}\) and \(R_{k+T}\) to sign and verify, respectively, which requires a synchronized clock between the sensor and the remote state evaluation system. Time deviation, or the natural delay between the time of signature generation and the time of verification, may cause failure of data verification. However, time-sensitive networks such as the industrial control system have higher synchronization accuracy and stable time synchronization; for example, the IEEE 1588 Precision Clock Synchronization Protocol of network measurement and control system can achieve a synchronization accuracy higher than microseconds, so as to fully meet the signature and verification requirements proposed in this paper. Since \(data_{k+T}\) in \(id_{S}||data_{k+T}||S_{k+T}\) or \(id_{S}||data_{k+T}||R_{k+T}\) changes with each data message from the sensor node to the remote state estimation system, \(S_{k+T}\) and \(R_{k+T}\) change with the time cycle; these two parameters determine the value of \(id_{S}||data_{k+T}||S_{k+T}\) or \(id_{S}||data_{k+T}||R_{k+T}\) within the cycle time. \(S_{k+T}\) and \(R_{k+T}\) never reappear and are not used repeatedly:\(sk_{i}\) and \(pk_{i}\) are used only once discarded. In the authentication process where the sensor sends monitoring data to the remote state estimation system, the remote state estimation system provides a set of four public keys for two sensor nodes (such as \(SN_{1}\) and \(SN_{2}\)). After signing the message, the sensor nodes \(SN_{1}\) and \(SN_{2}\) provide the signatures \(s_{1}\) and \(s_{2}\) for the remote state estimation system, and then the remote state estimation system verifies the signature. The signatures \(s_{1}\) and \(s_{2}\) are never reused by any sensor node by the remote state estimation system, even if every remote state estimation system has some unused signatures. Once the sensor nodes have used all the signatures generated, the remote state estimation system provides each sensor node with a set of new public keys. Let us understand signature generation and verification through an application example. Assuming that in the sensor node's communication with the remote state estimation system: \(n=160\), \(l=4\), and \(k=32\). Initially, the remote state estimation system generates a private key \(sk=(H(sk_{1}),\ldots,H(sk_{32}))\) and a public key \(pk=(pk_{1};\ldots,\ pk_{32})\), and sends the public keys set \(pk\) to the sensor node through a secure channel. The sensor node calculates the hash value \(h=H(id_{S}||data_{k+T}||S_{k+T})\), then splits the hash value \(h\) into four substrings \(h_{i}\) (\(1\leq i\leq 4\)), namely \(h_{1},h_{2},h_{3},h_{4}\), and expresses each hash value \(h_{i}\) as an integer \(i_{j}\) (\(1\leq j\leq 4\)). Here, the security level provided by the hash function is (\(l\times log_{2}\)\(k\)), which is 20 bits. It generates four hash substrings, each of which is 5 bits long.