# A Chaotic Compressive Sensing Based Data Transmission Method for Sensors within BBNs
## 7 Security Analysis
### Chaotic Compressive Sensing Security Analysis  
In a classical cryptosystem, assume the plaintext is \(p\), the ciphertext is \(c\), if \(\mathrm{P}(c)=\mathrm{P}(c|p)\), then the cryptosystem is considered to be secure [46]. For modern cryptosystems, they are often designed to be computationally secure. Namely, the cryptosystem could not be broken by existing sophisticated tools within polynomial time. Specifically, if an encryption scheme is sensitive to initial conditions and has tremendous secret key space, then we regard such an encryption scheme as a secure encryption scheme [47].  
Chaotic systems are famous for sensitive dependence on initial conditions. When this characteristic is applied to encryption, it behaves as an entirely different decryption result, even if the secret key changes only slightly.  
Figure 6 shows that, after a slight change is applied on the initial value of the chaotic sequence, that is \(z_{0}^{{}^{\prime}}\), the experimental reconstruction results greatly change.  
In the proposed method, there are four parameters participating in generating chaotic measurement matrices. The secret key A, which must be used by either restricted-authorized receivers or full-authorized receivers, is determined by chaotic parameter \(b\), initial value \(z_{0}^{{}^{\prime}}\), initial position \(n_{0}\), and sampling distance \(d\). Accordingly, we define \(K_{b},K_{z},K_{u},K_{d}\), and the key space \(K\) is,  
\[K=K_{b}\times K_{z}\times K_{n}\times K_{d}. \tag{24}\]  
Suppose we use a 32-bit processor in which the data precision of a double-precision floating point type is 16 significant digits after the decimal point, and suppose \(K_{n}\) and \(K_{d}\) are 100 and 10, respectively, then we can calculate from Equation (24) that \(K\approx 10^{35}\) (See Table 4). Such a number could be enlarged by adjusting the value ranges of the parameters. For instance, if we change the value range of \(K_{d}\) from [1; 10] to [1; 100], then  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Compression Rate 0.7**} \\ \hline
**Concealing** & **Channel r** & **Channel g** & **Channel b** & **Average** \\
**Section** & & & & \\ \hline none & 31.455124 & 31.512282 & 31.615987 & 31.527797 \\
**5** & 31.319548 & 31.435341 & 31.660291 & 31.471726 \\
**15** & 31.22286 & 31.284546 & 31.357155 & 31.288187 \\
**25** & 29.367135 & 29.356614 & 29.023638 & 29.249129 \\
**35** & 20.061426 & 19.881269 & 18.988067 & 19.643587 \\
**40** & 14.417344 & 14.175233 & 13.895409 & 14.162662 \\
**50** & 8.000954 & 8.06832 & 8.150955 & 8.073409 \\ \hline \hline  & \multicolumn{4}{c}{**Compression Rate 0.8**} \\ \hline
**Image** & **Channel r** & **Channel g** & **Channel b** & **Average** \\ \hline none & 32.278187 & 32.253436 & 32.396164 & 32.309262 \\
**5** & 32.221204 & 32.178524 & 32.447941 & 32.282556 \\
**15** & 32.018806 & 32.040822 & 32.220331 & 32.093319 \\
**25** & 30.019168p & 30.058696 & 29.816343 &  
the key space will be increased 10 times. In addition, increasing the precision of numbers that participate in operations could also be helpful to enlarge the key space of the proposed method. However, such an operation may increase the running time and complexity of the proposed method. Therefore, there should be a compromise when setting the range and number precision of parameters, according to the security and efficiency requirements.