## 3 Our Proposed Scheme
### Encryption Scheme
\[\mathrm{H(s)}=\sum_{i=0}^{2^{K}-1}P\left(s_{i}\right)\log_{2}\frac{1}{P\left( s_{i}\right)}, \tag{15}\]

where \(K\) is the bit depth of the test image, e.g., \(K=8\) for an 8-bit gray image, and \(P(s_{i})\) means the probability of \(s_{i}\). In the ideal case, the information entropy of 8-bit gray image is \(H(s)=8\) bits. The entropy test results are shown in Table 9. In our test results, entropies of cipher images are very close to the ideal value 8. Thus, our cryptosystem has good diffusion performance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cipher Images** & **NPCR(\%)** & \(\mathbf{\mathcal{N}_{0.05}^{*}}\) & **UACI(\%)** & \(\mathbf{\mathcal{U}_{0.05}^{*-}}=\mathbf{33.3730\%}\) \\  & & **99.5893\%** & & \(\mathbf{\mathcal{U}_{0.05}^{*+}}=\mathbf{33.5541\%}\)** \\ \hline C1 and C2 & 99.6185 & Pass & 33.4405 & Pass \\ C1 and C3 & 99.6197 & Pass & 33.3950 & Pass \\ C1 and C4 & 99.6174 & Pass & 33.4428 & Pass \\ C1 and C5 & 99.5956 & Pass & 33.4201 & Pass \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Quantitative analysis of key sensitivity. #### 4.5.2 Local Shannon Entropy

Global Shannon entropy reflects the total randomness of an image, and it has certain limitations in some occasions. Therefore, local Shannon entropy (LSE) was proposed by [PERSON] et al. [29]. To measure local entropy, we first randomly select \(k\) non-overlapping image blocks \(B_{1},B_{2},\cdots,B_{k}\) with \(T_{B}\) pixels from image \(I\), then the LSE is defined by:

\[\overline{\mathrm{H}_{k,T_{B}}}(I)=\sum_{i=1}^{k}\frac{H\left(B_{i}\right)}{k}, \tag{16}\]

where \(H(B_{i})\) is the Shannon entropy of image block \(B_{i}\), and can by calculated by Equation (15). In this test, we select parameters \((k,T_{B})=(30,\,1936)\). Under these parameters, the ideal value of LSE is \(7.902469317\). When we set significance \(\alpha=0.05\), we consider the tests passed when the test LSE values fall into the interval (\(7.901901305,\,7.903037329\)). The LSE test results are shown in Table 10.

### Computation and Complexity Analysis

Due to DNA computing having advantages when processing in parallel, our scheme is applicable to encrypt a large amount of image data in the actual DNA computing environment. In this section, we only discuss the time-consuming part of cryptography for computing floating point numbers. We use PWLCM to control randomly DNA encoding and to generate key sequence, it needs \(\Theta(5\times M\times N)\) iterations of computing floating point number. And we use HCLS to control permutation, it needs \(\Theta(M\times N)\) iterations of computing floating point number. The Matlab programming to simulate DNA calculation is low efficiency, under the hardware and software environment mentioned at beginning of Section 4, the encryption of \(512\times 512\) image is total cost \(3.233\) s. Although MATLAB simulation results have long running time, the proposed algorithm is still applicable to the actual DNA computing scene. After all, our purpose of this paper is to design a secure image encryption scheme in a real DNA computing scene, and simulation is only a means to verify the correctness and security of the proposed image cryptosystem. ### Security Comparison

In this part, we make a comparison with other works on some common security analysis indicators, the comparison results are shown in Table 11, and shown that our proposed scheme has very excellent security performance. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & **Lena** & **Baboon** & **Pepper** & **Airplane** & **Lake** \\ \hline Local Shannon Entropy & 7.902361 & 7.902319 & 7.902437 & 7.902321 & 7.902892 \\ (\(7.901901305,7.903037329\)) & Pass & Pass & Pass & Pass & Pass \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The local Shannon entropy test. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Image** & **Lena** & **Baboon** & **Pepper** & **Airplane** & **Lake** \\ \hline Entropy & 7.999258 & 7.999334 & 7.999414 & 7.999328 & 7.999282 \\ \hline \hline \end{tabular}
\end{table}