Collision in double-image encryption scheme based on spatial encoding and phase-truncation Fourier transforms

In this paper, the security strength of a double-image cryptosystem using spatial encoding and phase-truncation Fourier transforms (PTFTs) is evaluated. Unlike the conventional PTFT-based cryptosystem, where two random phase masks (RPMs) are used as public keys to provide enough phase constrains in the estimation, in the improved cryptosystem, the RPM generated by a random amplitude mask (RAM) is treated as an unknown parameter. Due to this fixed RAM, the number of constraints in the estimation decreases to achieve high robustness against potential iterative attacks. Moreover, instead of two phase-only masks (POMs), here the two POMs and the RAM are utilized as the private keys in the improved cryptosystem; thus, the key space of the double-image cryptosystem has been enlarged. However, we noticed that the RAM used to encode plaintexts spatially and to generate the phase encryption key is independent of the plaintexts. This could be recovered by a known pair of plaintexts and the ciphertext. Once the information of the RAM is retrieved, the phase key RPM can also be produced making the cryptosystem vulnerable. Based on this finding, new hybrid algorithms, including a known-plaintext attack and a known key attack are proposed to crack the enhanced PTFT-based cryptosystem. The information of the plaintexts can be retrieved from one POM using the proposed algorithms without any knowledge of another POM and the corresponding ciphertext. Numerical simulations have been carried out to validate the information disclosure problem still exists in the double-image cryptosystem based on spatial encoding and PTFTs.