**Table 2.**

*Image encryption scheme.*

The decryption process of the proposed algorithm is the reverse process of the encryption algorithm. A flowchart of the decryption process is shown in **Figure 13**.

The detailed decryption process includes the following steps.

Input: Plain image; Initial conditions for the chaotic system; Control parameter of the chaotic system; Average pixel value of the plain image

Output: Decrypted image

Step 1: Generate the pseudorandom sequence via the initial conditions and the average pixel values of the plain image

Step 2: Sort the pseudorandom sequence for row and column recovery.

*Chaotic Systems with Hyperbolic Sine Nonlinearity DOI: http://dx.doi.org/10.5772/intechopen.94518*

#### **Figure 13.**

*A flowchart of the decryption scheme.*

Input: Ciphered image En\_Img, Initial conditions for the chaotic system, control parameter for the chaotic system, Avg\_pixel\_value of Org\_Img Output: Plain Image Org\_Img

```
[m,n] size(En_Img);
  x(1) x(1) + Avg_pixel_value
  y(1) y(1)
  z(1) z(1)
  u(1) u(1)
  s(1) u(1)*10^4 – floor(u(1)*10^4)
  For i=1:1:m*n % Generate a pseudorandom sequence that will
  % be used for decryption
  [dx, dy, dz, du] Runge-Kutta (x(i), y(i), z(i), u(i))
  x(i+1) x(i) +dx
  y(i+1) y(i) +dy
  z(i+1) z(i) +dz
  u(i+1) u(i) +du
    s(i+1) u(i+1)*10^4 – floor(u(i+1)*10^4)
  End
  S_index Sort(s)
  For i=1:m % Row-wise permutation recovery
   For j=1:n
   De_per_row (i,j) Sort (En_Img, S_index)
   End
  End
 For i=1:n % Column-wise permutation recovery
  For j=1:m
  De_per_col (i,j) Sort (De_per_row, S_index)
  End
 End
Count=1 % Count flag
For i=1:m % Diffusion recovery
  For j=i:n
  diff(Count) mod (s(Count)*10^14, 256) % transform s, which could be used for XOR
  Org_Img (i,j)=bitxor(De_per_col (i,j), diff (Count)); % Bitwise exclusive OR
    Count= Count+1;
  End
 End
```
**Table 3.** *Image decryption scheme.* Step 3: Shift the pixel positions by row

Step 4: Shift the pixel positions by column

Step 5: Transform the pseudorandom sequence and recover the pixel values of the image via XOR

To provide a better understanding of this scheme, the pseudo-code is provided in **Table 3**

The testing results of encryption and decryption are shown in **Figure 14**.

In this system, all the initial conditions and control parameters can be considered as secret keys. Because the basin of attraction of each initial condition is greater than 1, it could have more than 1015∗<sup>4</sup> =10<sup>60</sup> choices via a resolution of 1015, in terms of a numeric calculation. Moreover, if a range of control parameters are considered for the key space, the key space of this system would far exceed 1090. Such a large key space provides sufficient security against brute-force attacks.

#### **Figure 14.**

*The testing results of encryption and decryption: (a) is the plain image of cameraman; (b) is the encrypted image of cameraman; (c) is the decrypted image of cameraman; (d) is the plain image of breast CT image; (e) is the encrypted image of breast CT image; (f) is the decrypted image of breast CT image; (g) is the plain image of thorax CT image; (h) is the encrypted image of thorax CT image; (i) is the decrypted image of thorax CT image.*

*Chaotic Systems with Hyperbolic Sine Nonlinearity DOI: http://dx.doi.org/10.5772/intechopen.94518*

Correlation coefficients of adjacent pixels in the plain and encrypted image are shown in **Table 4**.

The NPCR and UACI score of CT image are 99.5804% and 33.3227%.

From the above security analysis, the proposed scheme can provide high security for cryptographic applications.

## **4.3. Spread spectrum communication**

Chaotic systems can also use for spread spectrum communication propose. Different chaos shift keying (DCSK) technology employs nonperiodic and wideband chaotic signals as carriers so as to achieve the effect of spectrum spreading in the process of digital modulation. **Figure 15** shows the scheme of modulation for DCSK.

In this scheme, every bit has two time slots. The first time slot is used for transmission of a chaotic sequence for the reference signal. The second time slot is used for transmission of another chaotic sequence for the reference signal which has the same length as the first time slot. If the information bit is +1, then the information signal is exactly the same as the reference signal. If the information signal bit is �1, then the information signal is the negative of the reference signal. For bits *bk*, the signal at time k is:

$$\mathbf{s}\_{i} = \begin{cases} \mathbf{x}\_{i} & \mathbf{2}k\boldsymbol{\beta} < i \le (2k+1)\boldsymbol{\beta} \\\ b\_{k}\mathbf{x}\_{i-\boldsymbol{\beta}} & (2k+1)\boldsymbol{\beta} < i \le 2(k+1)\boldsymbol{\beta} \end{cases} \tag{10}$$

Where *β* is the number of sampling points. The spreading factor (SF) in the DCSK system is *SF* ¼ 2*β* .


#### **Table 4.**

*Correlation coefficients of adjacent pixels in the plain and encrypted image.*

**Figure 15.** *Scheme of DCSK modulation.*

For demodulation as shown in **Figure 16**, the receiver calculates the correlation between the received signal *ri* and the signal *ri*�*<sup>β</sup>*, which is *ri* delayed by *β* . After a time k, the output of the correlator is:

$$Z\_k = \sum\_{2(k+1)\beta}^{i=(2k+1)\beta+1} r\_i r\_{i-\beta} \tag{11}$$

Thus, the information bit *bk* can be restored by the sign of the decision variable:

$$
\hat{b}\_k = \text{sgn}\left[Z\_k\right] \tag{12}
$$

The obtained BER performance under additive white Gaussian noise (AWGN) channels for spreading factor 2*β* ¼ 200 is shown in **Figure 17**. From the comparison results, DCSK can have a lower BER when using this system as a carrier signal in the presence of noise.

**Figure 16.** *Scheme of the DCSK demodulation.*

**Figure 17.**

*Comparison of the bit error rate for a Chebyshev sequence and the hyperbolic sine system with DCSK.*
