Cascaded double-phase encoding (CDPE) is an optical cryptosystem, and it comprises two phase-only masks (ciphertext mask and key mask). Among optical cryptosystems, CDPE is of great importance due to its superiority in security. Its ciphertext is a phase-only mask whose content cannot be directly read out by the intensity-sensitive device such as the charge-coupled device (CCD) or human eyes. Although there is already published research on CDPE, few of them focus on simultaneous compression and encryption. In this paper, we propose a novel iterative encryption algorithm (IEA) to achieve double-image encryption in CDPE, which employs the position of the key mask as a controllable parameter. Compared with that of the traditional CDPE, the encryption capacity of the proposed algorithm has been substantially improved. The proposed algorithm opens up a new way for simultaneous compression and encryption in CDPE, and it may offer new inspiration for the design of other cryptosystems.

The optical architecture for decryption in this paper comprises two phase-only masks (ciphertext mask and key mask). Parallel monochromatic light is employed for illumination. The positions of the ciphertext mask and the output plane are fixed during decryption. Two positions along the axis are specified for the key mask. When the key mask locates respectively at these positions, two distinct plaintexts can be individually generated at the output plane. Essentially, the decryption employs two different optical architectures which differ only in the position of the key mask. According to the decryption principle, an IEA is proposed to encrypt the two plaintexts into the ciphertext mask. The IEA requires parallel iteration in the two optical architectures. For each architecture, the light wave virtually illuminates the scheme and finally reaches the output plane after being modulated by the two phase-only masks. At the output plane, the amplitude of the wavefront is replaced with the plaintext. The renewed wavefront at the output plane then propagates back to the input plane and forms a complex amplitude. The two complex amplitudes from the two architectures are superposed and then phase-reserved to obtain a new estimation of the ciphertext mask. The first iteration completes after the update of the ciphertext mask, and then the second iteration begins. The iteration will continue until the decrypted plaintexts at the output plane sufficiently approximate the original ones.

First, we validate the effectiveness of the proposed algorithm with binary images in the simulation context of MATLAB R2016a. The proposed IEA shows excellent convergence, and it terminates after 238 iterations. Both the subjective and objective metrics indicate the high quality of decrypted plaintexts (Fig. 3), which verifies the effectiveness of the proposed algorithm. Second, we analyze the key space created by each of the secret keys, including the wavelength, axial distances, and key mask. The key space of the proposed algorithm is as large as 2¹⁰⁴⁸⁵⁶, which is robust enough to resist brute-force attacks. Third, we investigate the condition for successful multiplexing. The results show that a minimum position interval of 2 mm of the key mask is required (Fig. 7), and an interval exceeding this value will cause obvious cross-talk noise in the decrypted images. Fourth, we verify the proposed algorithm with grayscale images and successfully extend it to multiple-image encryption (Fig. 8 and Fig. 9). The corresponding results show that the quality of the decrypted images decays with the image number for multiplexing. Therefore, there must be a compromise between the encryption capacity and the quality of decryption. Fifth, we test the robustness of the proposed algorithm against noise attacks, and the results show that the ciphertext can still ensure high-quality decryption in spite of severe contamination (Fig. 10). Sixth, we analyze the robustness of the proposed algorithm to cryptoanalysis. It is found that a chosen-plaintext attack (CPA) based on the G-S algorithm fails to crack the proposed algorithm (Fig. 11). Seventh, we further demonstrate the effectiveness of the proposed algorithm with experimental results (Fig. 13 and Fig. 14), which highly agree with the simulated ones.

In this paper, a double-image encryption method based on position-multiplexing in the CDPE system is proposed. A new IEA is presented to encrypt two plaintext images into a single phase-only mask (ciphertext mask). Compared with that of the traditional CDPE, the encryption capacity of this method is doubled. For decryption, the key mask is placed respectively at two preset axial positions, and two different plaintext images can be individually obtained at the same output plane. In addition, for successful position multiplexing to avoid crosstalk, the distance between the two axial positions of the key mask must be greater than a certain value. The security analysis shows that the proposed algorithm has a huge key space that is enough to resist brute-force attacks. Furthermore, the G-S algorithm is adopted to provide a CPA to the proposed algorithm, and the results show that the proposed algorithm is robust to the CPA. In addition, the feasibility of extending the proposed algorithm to multiple-image encryption is proven, which indicates that the encryption efficiency of CDPE systems can be further enhanced.