## 4 Sequential Signature Transmission Scheme for Sensor Data  
To ensure the integrity of sensor data and the time and sequence logic and save the cost of nodes, based on the one-time signature scheme proposed by [PERSON] (2018), we propose a time-based sequential logic signature scheme for sensors to send the authentication of the monitoring data to the remote state estimation system. In our asymmetric key signature scheme, the remote state estimation system is responsible for generating the private key and public key and distributing the public key to the sensors through a secure channel; only the sensor generates the signature, and the remote state estimation system verifies the signature. In the signature scheme proposed in this paper, \(sk_{1},sk_{2},\ldots,sk_{n}\) is \(n\) different random \(l\) bit strings with a fixed length, \(H()\) is an encrypted hash function using algorithms such as SHA1, SHA256 and SHA384 used to generate private key \(sk=(H(sk_{1}),H(sk_{2})\), \(\ldots\), \(H(sk_{n})\) ); \(f()\) is a one-way function, used to generate the corresponding public key \(pk=(pk_{1},pk_{2},\ldots,\ pk_{n})\). \(data_{k+T}\) is the local state estimation data of the sensor in the time period \(k+T\), \(S_{k+T}\) is the current cycle time of the sensor end, and \(R_{k+T}\) is the current cycle time of the remote state estimation end. To prevent sequential logic errors of the sensor data caused by information attacks such as replay and selective forwarding, we intended to sign the sensor data \(data_{k+T}\) at the current cycle time \(S_{k+T}\) of the sensor, and only transmit the data and signature sent in the current time cycle \(k+T\) in the network; when the remote state estimation system verifies the signature, it uses its current cycle time \(R_{k+T}\) to sign and verify the data \(data_{k+T}\). That is, the remote state estimation system uses the private key \(sk\) to sign the sensor identity \(id_{S}\), the data \(data_{k+T}\) sent in the time period \(k+T\), and the corresponding time \(S_{k+T}\), and obtains \(h=H(id_{S}||data_{k+T}||S_{k+T})\); to maintain the time logic, the time stamp \(T_{\Delta}\) of the data \(data_{k+T}\) is generated at the same time; when the receiving end executor verifies the signature, it first checks whether the timestamp meets \(T_{0}\leq T_{\Delta}+T_{\mathcal{S}}\); if so, the public key \(pk\) is used for signature verification, otherwise the command message is discarded. The specific protocol is shown in Algorithm 1. Figure 2: Network attack model. ```
Key generation of the remote state estimation system (R): Input: The parameter \(n\) represents the number of strings, \(l\) represents the length of the string, and \(k\) represents the number of substrings. 1. Randomly generate \(n\) strings of \(l\) bit \(sk_{1},sk_{2},\ldots,sk_{n}\), with different lengths; 2. Generate a hash string : \(H(sk_{1}),H(sk_{2})\), \(\ldots,H(sk_{n})\); 3. Calculate the public key \(pk_{i}=f(H(sk_{i});1\leq i\leq n\); Output : public key \(pk=(k,pk_{1},pk_{2},\ldots,pk_{n})\) and private key \(sk=(k,H(sk_{1}),H(sk_{2})\), \(\ldots,H(sk_{n}))\). Signature generation of sensor node (SN): Input : the integer value of the data \(data_{k+T}\). sent in the time period \(k+T\), and the private key set \(\mathrm{sk}=(sk_{1},sk_{2},\ldots,sk_{n})\). 1. Calculate \(h=H(id_{sk}||data_{k+T}||S_{k+T})\); 2. Split \(h\) into \(m\) substrings \(h_{1},h_{2},\ldots,h_{k}\) with a length of log\({}_{2}\)\(n\) bit; 3. Express \(h_{j}\) as an integer \(i_{j}(1\leq j\leq k)\); Output : data \(data_{k+T}\) sent in the time period \(T\), signature \((s_{i1},s_{i2},\ldots,s_{i\bar{k}})\) of \(data_{k+T}\) Where \(s_{i}=H(sk_{i})\); timestamp is \(T_{\Delta}\); Signature verification of the remote state estimation system (R): Input : data \(data_{k+T}\) sent in time period \(k+T\), signature set \((s_{i}^{\prime},s_{2}^{\prime},\ldots,s_{k}^{\prime})\); time stamp \(T_{\Delta}\); 1.