# Advancement of the DRPE Encryption Algorithm for Phase CGHs by Random Pixel Shuffling
## 2 Preliminaries
### Image Encryption Methods  
Image encryption is based on converting the intensity matrices for each channel into others with indecipherable patterns for unauthorized access. There are multiple proposals to achieve this: chaotic systems [17] offer high security, computational efficiency, and simplicity of implementation: they have features as ergodicity, unpredictability, randomness, and extreme sensitivity to initial conditions and key accuracy, but they show some weaknesses related with high numerical precision or security [18]. There is intensive academic work to address these limitations [19, 20, 21].  
Currently, there are also proposals based on quantum computation [22, 23] that provide robust security against quantum attacks, but face challenges due to the lack of mature quantum infrastructure and its heavy reliance on specialized hardware. Operations based on DNA computing [24, 25] offer parallel processing and the ability to handle large amounts of data, but come at the cost of complexity of implementation and the need for specialized laboratory infrastructure.  
Image encryption using neural networks [26, 27, 28] allows the generation of complex (non-linear) keys and is resistant to classical attacks (such as pattern analysis). They are systems that require a high computational cost in the training process, and this must be directed to specific applications.  
Optical encryption using optical transformations [4, 29, 30, 31] leverages properties such as diffraction and Fourier transforms to securely encode images, offering high-speed processing and resistance to traditional computational attacks, but is limited by the need for specialized hardware and environmental sensitivity.  
CGH Encryption Methods: DRPE Encryption  
A widely used optical encryption method for image applications is Double Random Phase Encoding (DRPE) [32]. In DRPE, the input image is represented by the amplitude of light, which is modulated by a random phase before being transformed by a Fouriertransform. A second random phase mask is then applied in the frequency domain as an encryption key. The resulting complex amplitude is considered the encrypted information. An evolution of this approach is optical encryption using digital holography [33]. This technique can also be modeled using algebraic procedures, such as vector-matrix multiplications [34].  
The double-phase encryption technique has been successfully tested using both static and dynamic keys. However, this method presents certain vulnerabilities to attacks, as recognized in the literature. If the process is purely linear, there is a possibility of breaking the encryption key [35]. To enhance DRPE, various improvements have been proposed. One approach involves modifying pixels using chaotic functions [36], while another leverages deep learning techniques [37].  
Traditionally, the phase masks proposed for DRPE have aimed to ensure uniformity in both the amplitude of the hologram and the phase distribution of the encrypted CGH to make unauthorized decryption more challenging. Various photosensitive materials (e.g., photorefractive media) have been used for their implementation [38]. Currently, with advancements in spatial light modulator (SLM) technology, diffractive elements computed digitally (such as CGHs) and phase masks can be directly applied to a light beam in the laboratory. This enables both the encrypted data and the encryption keys to be dynamic.