# Advancement of the DRPE Encryption Algorithm for Phase CGHs by Random Pixel Shuffling
## 5 Conclusions  
Encoding information using CGH and random phases (DRPE) has proven to be a robust technique for information encryption in the past. However, the time required to find the decryption key with brute force attacks decreases as the available computing power increases.  
We present an evolution of DRPE that adds the random permutation of pixels. This proposal allows the use of a single key to generate the CGH that serves as ciphertext or the use of two independent keys: one for the permutation and another for the synthesis of the random phase mask.  
The entire process can be simulated on a computer or demonstrated in a laboratory. The permutation possibilities add a new layer of complexity against attacks.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**CGH Size (in Pixels)** & **t(s.)** \\ \hline \(1920\times 1080\) & 1600.37 \\ \(1024\times 1024\) & 809.08 \\ \(512\times 512\) & 204.27 \\ \(256\times 256\) & 56.06 \\ \(64\times 64\) & 16.27 \\ \hline \hline \end{tabular}
\end{table}
Table 3: CGH computation times in seconds vs. CGH resolution using double precision for coding amplitude and phase.  
Figure 7: (**a**): Pearson Correlation Coefficient (PCC) as a function of the percentage of shuffled pixels (orange line) or when only the phases are modified (light blue). The behavior of both graphs is identical. PCC when both are used (dark blue line). (**b**): Distribution of grayscale levels (phases) of the encrypted CGH in Figure 3(a).  
We have analyzed the performance of this proposal through a visual assessment. We have significantly increased the size of the key space compared with the DRPE proposals, while preserving the robustness of the information by storing it using CGH. In this context, variables such as pixel correlation and image entropy show substantial improvement in the field of encryption.  
From a weakness perspective, our proposal offers a key size larger than those in previous approaches; however, it remains compatible with using some of these proposals for key generation. This presents an area for future work, particularly with encryption algorithms based on chaotic systems. Additionally, another potential direction involves applying this process to color CGH, which would essentially triple the information size.  
Encoding the information in CGH makes the generated ciphertext robust to localized modifications, the random phase modification makes the phase histogram distribution more resilient to statistical attacks, and the shuffling of already-encrypted pixels significantly increases the combinatorial possibilities to explore, rendering a brute force attack unfeasible.  
Conceptualization, A.B. and F.J.S.; methodology, A.B.; software, A.B.; validation, A.B. and F.J.S.; formal analysis, A.B. and F.J.S.; investigation, A.B. and F.J.S.; resources, A.B. and F.J.S.; data curation, A.B. and F.J.S.; writing--original draft preparation, A.B.; writing--review and editing, F.J.S.; visualization, A.B. and F.J.S.; supervision, A.B. and F.J.S.; project administration, A.B.; funding acquisition, A.B. All authors have read and agreed to the published version of the manuscript.  
This initiative is carried out within the framework of the EU-funded \"Plan de de Recuperacion, Transformacion y Resiliencia\" (Next Generation), Instituto Nacional de Ciyenseguridad (INCIBE) (Spain).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors would like to thank [PERSON] (trainee student of the Department of Computer Science at the University of Zaragoza) for his contribution in the design and synthesis of Figure 2a.  
The authors declare no conflicts of interest.