## 3 Our Proposed Scheme
### Encryption Scheme
The true keys of our cryptosystem are two parameters \(p\in(0,0.5)\) and two initial values \(x\in(0,1)\) of two PWLCM systems, and four initial values \(x_{0}\in(-40,40)\), \(y_{0}\in(-40,40)\), \(z_{0}\in(1,81)\), \(w_{0}\in(-250,250)\) of the HCLS system. We suppose the change step of each initial value and parameter is \(10^{-15}\), the key space of our proposed cryptosystem can be calculated as \(S=(0.5\times 10^{15})^{2}\times(1\times 10^{15})^{2}\times(80\times 10^{15}) ^{3}\times(500\times 10^{15})=6.4\times 10^{127}\approx 2^{424}\). The key space is required to be at least \(2^{100}\) to protect from a brute-force attack [16], and our key space is large enough for security. ### Differential Attack

The sensitivity of plain image is a very important index to an image cryptosystem, and it can help cryptosystem to resist the differential attack which is a kind of chosen-plaintext attack. The differential attack usually changes the value of one pixel in the plain image, and compares the difference between two corresponding cipher images. There are two features to evaluate the sensitivity of the plain image: number of pixels change rate (NPCR) and unified average changing intensity (UACI) [8; 27]. The NPCR and UACI are calculated by Equations (3) and (4), respectively. Figure 3: The block diagram of the decryption scheme. Figure 4: Encryption simulation. (**a–e**) The plain images of ‘Lena’, ‘Baboon’, ‘Pepper’, ‘Airplane’ and ‘Lake’; (**f–j**) the corresponding cipher images; (**k–o**) the recovery images from the decryption. \[\left\{ \begin{array}{ll}\text{NPCR}&=\frac{\sum_{i=1}^{M}\sum_{j=1}^{N}D(ij)}{M \times N}\times 100\%\\ D(i,j)&=\left\{\begin{array}{ll}0,&C_{1}(i,j)=C_{2}(i,j)\\ 1,&C_{1}(i,j)\
eq C_{2}(i,j)\end{array}\right.,\\ \end{array} \right. \tag{3}\]

\[\text{UACI}=\frac{1}{M\times N}\left(\sum_{i=1}^{M}\sum_{j=1}^{N}\frac{|C_{1} (i,j)-C_{2}(i,j)|}{255}\right)\times 100\%\quad, \tag{4}\]

where \(C_{1}(i,j)\) and \(C_{2}(i,j)\) are the cipher images which correspond two one-pixel different images. \(M\) and \(N\) are the height and width of those images, respectively. The NPCR calculates the percentage of pixels which are different values at the same position between \(C_{1}(i,j)\) and \(C_{2}(i,j)\). The UACI is a quantitative assessment of the average distance between two pixels at the same position in \(C_{1}(i,j)\) and \(C_{2}(i,j)\). To evaluate the NPCR and UACI results, the critical value is given by [PERSON] et al. [28]. For a significance level \(\alpha\), a critical value of NPCR is given by:

\[\mathcal{N}_{\alpha}^{*}=\frac{Q-\Phi^{-1}(\alpha)\sqrt{Q/H}}{Q+1}, \tag{5}\]

where \(H\) represents the total number of pixels in an image, \(Q\) represents the largest allowed pixel value in the image. An image encryption scheme can be considered to pass the NPCR if the obtained NPCR is larger than \(\mathcal{N}_{\alpha}^{*}\). The critical interval \((\mathcal{U}_{\alpha}^{*-},\mathcal{U}_{\alpha}^{*+})\) of UACI is given by:

\[\left\{ \begin{array}{ll}u_{\alpha}^{*-}=\mu_{u}-\Phi^{-1}\left(\frac{\alpha }{2}\right)\sigma_{u},\\ u_{\alpha}^{*+}=\mu_{u}+\Phi^{-1}\left(\frac{\alpha}{2}\right)\sigma_{u}, \end{array} \right. \tag{6}\]

where

\[\mu_{u}=\frac{Q+2}{3Q+3} \tag{7}\]

and

\[\sigma_{u}=\sqrt{\frac{(Q+2)\left(Q^{2}+2Q+3\right)}{18(Q+1)^{2}QH}}. \tag{8}\]

If the UACI value falls into interval \((\mathcal{U}_{\alpha}^{*-},\mathcal{U}_{\alpha}^{*+})\), it means that the two test images have much difference. When the size of test image is \(512\times 512\) and the significance level \(\alpha=0.05\), the critical value of NPCR is \(\mathcal{N}_{0.05}^{*}=99.5893\%\) and the critical interval of UACI is \((\mathcal{U}_{0.05}^{*-},\mathcal{U}_{0.05}^{*+})=(33.3730\%,\,33.5541\%)\). For this test, we randomly changed one pixels' value from plain images, and calculated the NPCR and UACI. We test NPCR and UACI 100 times, and the test results are shown in Table 5. The expectation of NPCR and UACI between two random images are given in Equations (9) and (10), respectively.