## 6 Security Analysis  
In order to assess the security of the implemented setup, we had performed the analysis of sensitivity of encryption key to distortions. For the analysis, we used MCDDC with 128 \(\times\) 128 elements (see Figure 8a) encrypted (see Figure 10a) using the key with NAE 0.001, which contains 19 nonzero elements (see Figure 9a).  
We relocated single nonzero element and made an attempt at decrypting an encrypted image from Figure 10a using the distorted key. Decryption resulted in 2501 errors (24% errors); decrypted MCDDC was decoded; and only 1 (0.6%) symbol out of 170 was decoded incorrectly. The relocation of two nonzero elements resulted in 3024 errors (29% errors) and led to 18 (11%) incorrectly decoded symbols. The relocation of three nonzero elements resulted in 3548 errors (34% errors) and 43 (25%) incorrectly decoded symbols and made encoded text unreadable.  
As described in [49], if one possesses the information about the exact number of nonzero elements \(k\), then the number of possible key combinations is:  
\[C=\frac{(N_{x}N_{y})!}{k!(N_{x}N_{y}-k)!} \tag{1}\]  
For a key with 128 \(\times\) 128 elements, 19 out of which are not zero, this number is equal to 9.6 \(\times\) 10\({}^{62}\). It corresponds to a one-dimensional binary key with a 209 bit length.  
Next, let us evaluate the number of acceptable decryption keys [52] which allow one to successfully decode encrypted data. In the case of MCDDC, which can partially decode text even when the damage threshold is exceeded, we consider decoding successful when the decoded text can still be read. If \(r\) nonzero elements out of total \(k\) can be randomly  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Key NAE**} & \multicolumn{2}{c}{**MCDDC Size**} \\ \cline{2-3}  & **128 \(\times\) 128** & **254 \(\times\) 254** \\ \hline
0.0005 & 1925 (18.4\%) & 2392 (14.8\%) \\
0.001 & 1944 (18.5\%) & 2230 (13.8\%) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numbers of incorrectly decrypted pixels and error rates.  
Figure 11: Quantized decrypted images (**a,c**) and error maps (**b,d**) for MCDDCs with 128\(\times\)128 (**a,b**) and 254 \(\times\) 254 (**c,d**) elements.  
relocated and the resulting key allows for the successful decryption of an encrypted data, then the number of acceptable decryption keys can be found as the number of possible variants of choosing of any of \((k-r)\) correct nonzero elements multiplied by the number of possible variants of positioning of \"unnecessary\" \(r\) nonzero elements:  
\[S=\frac{k!(N_{x}N_{y})!}{\left(k-r\right)!(r!)^{2}(N_{x}N_{y}-r)!} \tag{2}\]  
For our case of a key with 128 \(\times\) 128 elements, 19 out of which are nonzero and 2 can be relocated, this corresponds to 2.3 \(\times\) 10\({}^{10}\) combinations or 34 bit length of one-dimensional binary key.  
To find one acceptable key, the number of combinations that should be tried can be found as:  
\[K=\frac{C}{2S}=\frac{(k-r)!(r!)^{2}(N_{x}N_{y}-r)!}{2\cdot\left(k!\right)^{2}(N _{x}N_{y}-k)!} \tag{3}\]  
which in our case makes \(4.2\times 10^{52}\) combinations. This corresponds to a one-dimensional binary key with a 175 bit length. It is a sufficiently large key to effectively withstand brute-force attacks (the minimum key length for that is 100 bit [4]).