# An Adaptive and Secure Holographic Image Watermarking Scheme
## 5 Experiments and Results
### Test for the Effectiveness of Our Watermarking Scheme
#### 5.2.3 Key Sensitivity  
The sensitivity of the reconstructed image of the extracted CGH to slight alterations of the secret keys \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\), is investigated. Figure 7a-i show the decrypted reconstructed images with wrong keys \(X(1)=0.352+10^{-15}\), \(Y(1)=0.865+10^{-15}\), \(SZ=0.752+10^{-15}\), \(\alpha=0.998+10^{-14}\), \(\beta=3-10^{-14}\), \(\gamma=4-10^{-15}\), \(t1=3528+1\), \(t2=7832+1\) and \(t3=6832+2\), respectively. The \(N\)Cs of the decrypted CGH with the above wrong permutation keys and the \(BER\)s of the corresponding reconstructions are presented in Table 6. Please note that in the above experiments, the other keys remain correct while a key is varied in decryption. As illustrated in Figure 7a-f, we cannot obtain any information from the decrypted reconstructions visually when the absolute values of deviations of \(SX(1)\),\(SY(1)\), \(SZ\) and \(\gamma\) are up to \(10^{-15}\) and those of \(\alpha\) and \(\beta\) are up to \(10^{-14}\). In addition, we know from Figure 7g-h that if the parameters \(t_{1}\) and \(t_{2}\) are less 1 or more 1 than the right value, the decoded images are noise-like images. Similarly, the decrypted reconstructed image shown in Figure 6i cannot be recognized when the key \(t_{3}\) are less 2 or more 2 than the correct value. So, the keys \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\) are highly sensitive to the proposed method. Now we evaluate the key space of the proposed encrypted hologram. In light of the description of the proposed scheme, we know that the key space of the cryptosystem consists of the parameters \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\). Let \(KS_{1}\), \(KS_{2}\), \(KS_{3}\), \(KS_{4}\), \(KS_{5}\), \(KS_{6}\), \(KS_{7}\), \(KS_{8}\) and \(KS_{9}\) be the key spaces of the secret keys \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\), respectively. From Table 6, the parameters \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\) and \(\gamma\) maintain 15, 15, 15, 14, 14 and 15 digits after the decimal point respectively. So \(KS_{1}\)\(\times\)\(KS_{2}\)\(\times\)\(KS_{3}\)\(\times\)\(KS_{4}\)\(\times\)\(KS_{5}\)\(\times\)\(KS_{6}\)\(=\)\(10^{88}\). Since \(1\leq t1\leq T+PQ\), \(1\leq t2\leq T+PQ\) and \(1\leq t3\leq T\)\(\times\)\(KS_{7}\)\(\times\)\(KS_{8}\)\(=\)\((T+PQ)^{2}\times T\). Since \(T\) and \(PQ\) are 10000 and \((64\times 64)^{2}\) in the experiments, \(S_{7}\)\(\times\)\(S_{8}\)\(\times\)\(S_{9}\)\(\approx\)\(2\times 10^{12}\). Hence, the total key space of the encryption system is \(KS_{1}\)\(\times\)\(KS_{2}\)\(\times\)\(KS_{3}\)\(\times\)\(KS_{4}\)\(\times\)\(KS_{5}\)\(\times\)\(KS_{6}\)\(\times\)\(KS_{7}\)\(\times\)\(KS_{8}\)\(\times\)\(KS_{9}\)\(\approx\)\(10^{88}\)\(\times\)\(2\times 10^{12}\)\(\approx\)\(2^{333}\). It is clear that the key space of the proposed cryptosystem is far larger than \(2^{100}\) and enormous enough to resist a brute force attack [29]. Therefore, by use of the encrypted hologram, the security level of the proposed watermarking scheme can be improved. Figure 6: The corresponding reconstructed images of the CGH extracted from the distorted watermarked images in Figure 4. (**a**) Gaussian low-pass filtering; (**b**) average filtering; (**c**) median filtering; (**d**) occlusion; (**e**) JPEG; (**f**) unsharp; (**g**) burring; (**h**) Gaussian noise; (**i**) salt & pepper noise; (**j**) brighten; (**k**) darken; (**l**) rescaling; (**m**) rotation; (**n**) painting. Figure 7: The reconstructed images of the CGHs extracted from watermarked Elaine.