# A Novel CDMA-Based Secret Message Multiplexing Scheme by Image Data Hiding +
## 4 Experimental Results and Discussion
### Robustness against Noise Attacks  
Noise can be treated as a random intensity variation present in an image during the process of image acquisition and transmission. It can significantly deteriorate the image quality. In this subsection, to test the robustness of the proposed scheme, two kinds of noise attacks, Gaussian noise attack and the salt-and-pepper noise attack, are considered. The marked Lena images with different values of \(N\) are taken as the test images. For the Gaussian noise attack, based on the 8-bit grayscale image values, three different values of \(\sigma\) (noise standard deviation) are chosen, which are 2.55, 8.06, and 18.03, respectively. The corresponding noise variance values with pixel values normalized to the range [0, 1] are 0.0001, 0.001, and 0.005, respectively. Figure 6 displays the tainted Lena images with number of users \(N\) equal to 16. The degradation in image quality is unobvious \begin{table}
\begin{tabular}{c c c c c c} \hline
**Images** & \(N\) **= 2** & \(N\) **= 4** & \(N\) **= 8** & \(N\) **= 16** & \(N\) **= 32** \\ \hline Lena & 45.13 & 39.91 & 34.38 & 28.65 & 22.66 \\ Airplane & 45.14 & 39.95 & 34.26 & 28.86 & 22.16 \\ Baboon & 45.11 & 39.90 & 34.32 & 28.66 & 22.37 \\ Peppers & 45.13 & 39.91 & 34.30 & 28.30 & 22.25 \\ \hline \end{tabular}
\end{table} 
Table 2: PSNR values (dB) versus number of users \(N\) for four test images. Figure 4: The four grayscale test images. (**a**) Lena; (**b**) Baboon; (**c**) Airplane; (**d**) Peppers. Besides PSNR, to further evaluate the robustness of the proposed scheme, the bit-error rate (BER) between the extracted secret message and the embedded secret message is calculated as as:  
\[\text{BER}=\frac{\text{Error bits}}{\text{Total number of embedded bits}}. \tag{11}\]  
At the secret message extraction stage, the inner product operation of an \(N\)-digit CDMA codeword group and a secret key is performed. For a noise-free marked image, with a correct secret key, the result value will be \(N\) or \(-N\). When the data is corrupted by Gaussian noise of variance \(\sigma^{2}\), the inner product operation will result in a Gaussian random variable with mean \(N\) or \(-N\) and variance \(N\sigma^{2}\) if the effect of modulo operation is ignored. The error occurs when the noise component is less than \(-N\) corresponding to mean \(N\) or the noise component is greater than \(N\) corresponding to mean \(-N\). By symmetry of the Gaussian distribution, the probabilities are the same in either case. Hence, the error probability can be calculated as:  
\[P_{b}=\int_{N}^{\infty}\frac{1}{\sqrt{2\pi N\sigma^{2}}}\exp\biggl{(}\frac{-x^ {2}}{2N\sigma^{2}}\biggr{)}dx. \tag{12}\]  
with variable substitution, Equation (12) can be rewritten as:  
\[P_{b}=\int_{\frac{N}{\sigma}}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\biggl{(}\frac {-u^{2}}{2}\biggr{)}du, \tag{13}\]  
where the integrand is the standard Gaussian distribution function with mean 0 and variance 1. When the number of users \(N\) grows, the longer secret key will lead to a larger inner product value. From Equations (12) and (13), we can observe that a larger \(N\) helps to reduce the error probability. This is the benefit of CDMA multiplexing. The BER values of the extracted secret from the Lena image with different values of \(N\) under Gaussian noise are listed in Table 4 and the corresponding expected BER values calculated according to Equation (12) are listed in Table 5. For comparison, those values are also plotted in Figure 8. It is observed that when the number of users increases, BER will decrease and  
Figure 7: PSNR values versus number of users \(N\) for the Lena image under Gaussian noise. get closer to its expected value, especially for moderate values of \(\sigma\) such as 2.55 and 8.06. The reason is that for small value of \(N\) or larger value of \(\sigma\), modulo wrap-around of the tainted pixel value occurs more frequently, causing the random variable to deviate from the Gaussian distribution.