## 2 Problem Formulation  
Given a system consisting of \(m\) mutually independent binary observations, consider the \(l\)-th observing system, \(l=1,2,\ldots,m\):  
\[y_{l,k}=a_{1}u_{l,1}+a_{2}u_{l,2}+\cdots+a_{n}u_{l,n}+d_{l,k}=\phi_{l}^{T} \theta+d_{l,k}, \tag{1}\]  
where \(\theta=[a_{1},a_{2},\ldots,a_{n}]^{T}\) is the system parameter; \(\phi_{l}=[u_{l,1},u_{l,2},\ldots,\)\(u_{l,n}]^{T}\) is the input parameter for each observation system, which is a fixed value that does not change by moment \(k\); and \(d_{l,k}\) is the system noise.  
\(y_{l,k}\) is the system output of the \(l\)-th observing system at the \(k\)-th moment, measured by a binary observation sensor with threshold \(C_{l}\), which is expressed as:  
\[s_{l,k}^{0}=I_{\{y_{l,k}\leq C_{l}\}}=\left\{\begin{array}{ll}1,&y_{l,k}\leq C _{l};\\ 0,&\text{otherwise}.\end{array}\right. \tag{2}\]  
As shown in Figure 1, \(s_{l,k}^{0}\) denotes the data that were transmitted to the estimation center through the \(l\)-th network channel at the moment of \(k\), but were subjected to a cyberattack during transmission, resulting in data tampering. The data received by the estimation center are denoted as \(s_{l,k}\), which is related to \(s_{l,k}^{0}\):  
\[\left\{\begin{array}{ll}\Pr\{s_{l,k}=0|s_{l,k}^{0}=1\}=p_{l};\\ \Pr\{s_{l,k}=1|s_{l,k}^{0}=0\}=q_{l}.\end{array}\right. \tag{3}\]The data tampering attack strategy of the \(l\)-th observation system described in the above equation is abbreviated as \((p_{l},q_{l})\).  
In the given system, the identification algorithm of the unknown parameter \(\theta\) needs to be constructed by the defender, while the attacker aims to disrupt the effectiveness of the identification algorithms under multiple binary observations. Since there are multiple communication networks, so the attacker can adopt different attack strategies in different networks, the attacker's set of tampering strategies is noted as  
\[\Omega=\{(p_{l},q_{l}),\;l=1,2,\ldots,m\}.\]  
In this system, the following questions are given and addressed in this paper:  
1. From the perspective of the attacker, when the system parameters are known, how to adjust the attack strategy so that the system can achieve the maximum attack effect with the minimum energy under the energy constraint.
2. How to construct the attacker's identification algorithm to accomplish the parameter estimation when the system parameters are unknown.
3. How to implement the optimal data tampering attacks while meeting the real time.  
**Assumption 1**.: _The system noise \(\{d_{l,k}\}\) is a sequence of independent and identically distributed Gaussian random variables whose probability distribution function and probability density function are denoted as \(F(\cdot)\) and \(f(\cdot)\), respectively._  
**Remark 1**.: _To solve the case that the attacker does not have the knowledge of the threshold and the system parameters, one can view the distribution and system parameters jointly as uncertainties, and extract the information about the noise distribution from the designed input and relevant output data of the system with a modified algorithm._  
**Remark 2**.: _The system in (1) is a quantized linear system model with multiple observing systems, which has a simplified structure but is easy to extend. In the subsequent research, the nonlinearization module can be used to combine and connect, which can well portray the complex nonlinear system. By adding a static nonlinear function with this system, it can be combined into a Hammerstein system or Wiener system._  
Figure 1: System structure.