Asymmetric Optical Cryptosystem with Secret-Key Sharing Based on Coherent Superposition and Normalized Decomposition

Electrical Engineering Electronic Information Engineering Optics Information Science Physics
optical encryption multiple image encryption secret image sharing phase-only mask normalized decomposition

Asymmetric Optical Cryptosystem Based on Coherent Superposition and Normalized Decomposition

Background Introduction

With the growing demand for information security, optical image encryption technology has attracted significant attention over the past three decades. This technology leverages various degrees of freedom of light (such as amplitude, phase, wavelength, polarization, etc.) to achieve high-speed parallel processing, offering unique advantages for image encryption. However, traditional optical encryption methods have some limitations, such as the “silhouette problem” (where partial original information may leak during decryption), increased storage requirements for complex-valued ciphertext images, and crosstalk noise issues in multiple-image encryption (MIE).

To overcome these limitations, Mohamed G. Abdelfattah et al. proposed an asymmetric optical cryptosystem based on coherent superposition and normalized decomposition. The study aims to address the following key issues: 1. Silhouette Problem: Eliminate potential leakage of the original image silhouette during decryption by introducing a Chaotic Random Amplitude Mask (CRAM). 2. Secret Sharing Capability: By decomposing the spectrum of each image into multiple Phase-Only Masks (POMs), one part serves as the shared ciphertext image while the rest act as independent keys, enabling a secret-sharing mechanism. 3. Unlimited Encryption Capacity: Support simultaneous encryption of a large number of images without degrading decryption quality. 4. Robustness and Security: Enhance system resistance to Gaussian noise, statistical attacks, and chosen-plaintext attacks.

Paper Source

This paper was authored by Mohamed G. Abdelfattah, Salem F. Hegazy, and Salah S. A. Obayya, affiliated with the Department of Electronics and Communications Engineering at Mansoura University, the National Institute of Laser Enhanced Sciences at Cairo University, and the Center for Photonics and Smart Materials at Zewail City of Science and Technology in Egypt. It was published in the journal Optical and Quantum Electronics in 2025, article number 57:158, with the DOI 10.1007/s11082-025-08061-y.

Research Details

a) Research Workflow

1. Spectrum Decomposition Algorithm Design

The core of the research is a novel algorithm called “M-POM Normalized Decomposition.” This algorithm decomposes the spectrum of each image into a set of m Phase-Only Masks (POMs). The specific steps are as follows: 1. Obtain Image Spectrum: Use Fourier Transform (FT) to convert the input image into its frequency-domain representation. 2. Normalize Amplitude: Ensure that the spectral amplitude range is within [0, m] to meet subsequent decomposition conditions. 3. Randomly Generate (m-2) POMs: Randomly generate (m-2) phase masks according to constraint conditions and calculate their superposition results. 4. Solve Remaining Two POMs: Determine the last two unknown phase angles through geometric analysis and cosine law formulas. 5. Output Results: Finally, obtain m POMs, where one serves as the shared ciphertext image and the remaining (m-1) act as independent keys.

2. Multiple-Image Encryption (MIE) Scheme

The study further applies the above algorithm to multiple-image encryption, with the following workflow: 1. Preprocessing Stage: Process each RGB channel of the color images separately using Random Phase Masks (RPMs) for encoding. 2. CRAM Modulation: Multiply the spectral amplitude by a Chaotic Random Amplitude Mask (CRAM) to eliminate the silhouette problem. 3. Spectral Decomposition: Apply the M-POM normalized decomposition algorithm to decompose the spectrum of each image into m POMs. 4. Key Distribution: Use one POM as the shared ciphertext image and distribute the remaining (m-1) POMs as independent keys to authorized users. 5. Decryption Process: Restore the original image through inverse Fourier transform and correct key combination.

3. Optical Experimental Setup

The study designed a compact optical system to verify the feasibility of the encryption and decryption processes: - Encryption System: Use Spatial Light Modulators (SLMs) to display input images and RPMs, capturing the spectrum through a Fourier lens. - Decryption System: Employ a Mach-Zehnder Interferometer (MZI) to achieve coherent superposition, recording the decryption results via a CCD camera.

b) Main Results

1. Encryption Performance

  • Resolution of Silhouette Problem: When any single POM is missing or incorrectly used, the correlation coefficient (CC) of the decrypted image falls below 0.008, indicating that the original image cannot be correctly recovered. This demonstrates that the silhouette problem has been effectively resolved.
  • Unlimited Encryption Capacity: The study tested the encryption of 100 different images, and all decrypted images had a CC of 1, showing that the system has unlimited encryption capacity.

2. Security Analysis

  • Key Sensitivity: Even a phase deviation of just 0.02 radians in a single POM reduces the CC of the decrypted image to below 0.015, demonstrating extremely high key sensitivity.
  • Noise Resistance: Under added Gaussian noise, the system maintains high decryption quality. For instance, with a noise strength factor of 0.01, the CC of the decrypted images remains above 0.97.
  • Resistance to Occlusion Attacks: Even when the ciphertext image is occluded by 15%, the system can partially recover the original image, though the CC significantly decreases.

3. Statistical Analysis

  • Histogram Analysis: The histogram of the ciphertext image is entirely different from that of the original image, and the ciphertext histograms of different images are similar, indicating that the ciphertext image contains no information about the original image.
  • Adjacent Pixel Correlation: The correlation between adjacent pixels in the ciphertext image is nearly zero, whereas it is relatively high in the original image, further verifying the system’s security.

c) Conclusion

The asymmetric optical cryptosystem proposed in this study offers the following advantages: 1. Resolves Silhouette Problem: CRAM modulation completely eliminates potential leaks of the original image silhouette during decryption. 2. Supports Secret Sharing: (m-1) POMs can be distributed to authorized users, enhancing access security. 3. Unlimited Encryption Capacity: Capable of encrypting a large number of images simultaneously without degrading decryption quality. 4. High Robustness and Security: Demonstrates strong resistance to Gaussian noise, statistical attacks, and chosen-plaintext attacks.

d) Research Highlights

  1. Novel Normalized Decomposition Algorithm: The M-POM normalized decomposition algorithm not only simplifies the encryption process but also generates real-valued ciphertext images, reducing storage requirements.
  2. Secret Sharing Mechanism: Distributing (m-1) POMs as independent keys to authorized users realizes secure secret sharing.
  3. Strong Attack Resistance: The system is highly sensitive to keys and exhibits good robustness against noise and occlusion attacks.
  4. Unlimited Encryption Capacity: Supports simultaneous encryption of a large number of images, breaking the limitations of traditional MIE methods.

e) Other Valuable Information

The study also explores the system’s potential applications, such as in military communications, medical image protection, and digital rights management. Additionally, the authors note that future work should further optimize the system’s robustness against high levels of occlusion.

Research Significance and Value

This study proposes an asymmetric optical cryptosystem based on coherent superposition and normalized decomposition, bringing significant breakthroughs to the field of multiple-image encryption. Its innovative normalized decomposition algorithm and secret-sharing mechanism not only enhance the system’s security and efficiency but also resolve the silhouette problem and increased storage demands present in traditional methods. The research findings indicate that the system is of great theoretical and practical significance, providing new directions for the development of optical image encryption technology.