Cryptoanalysis on optical image encryption systems based on the vector decomposition technique in the Fourier domain

In this paper, the security of optical cryptosystems based on the vector decomposition technique in the Fourier domain is analyzed. Compared to the conventional cryptosystem based on the equal modulus decomposition (EMD) technique, an additional EMD structure is introduced in the cascaded EMD-based cryptosystem; hence, the mask including the phase information of the Fourier spectrum is further encoded in the second EMD structure to enhance the security level. However, it is shown that the number of the private keys has not been increased in the cascaded EMD-based cryptosystem, which makes it possible to crack the cascaded EMD-based cryptosystem. Therefore, a chosen-plaintext attack (CPA) and a special attack with an arbitrarily given private key are proposed to retrieve information from encoded images obtained by the cascaded EMD-based cryptosystem. In addition, the security of the cryptosystem based on the random modulus decomposition (RMD) technique is also analyzed. Compared to the EMD-based cryptosystem in which the Fourier spectrum is decomposed into two vectors with equal moduli, the security level of the cryptosystem has been improved by using the RMD technique to decompose the spectrum into vectors with unequal moduli to decrease the number of the amplitude constraints. However, it is found that the arbitrarily given ciphertext provides the attackers enough information to retrieve the precise information of the plaintext without any knowledge of the private keys. A special attack is proposed to crack the RMD-based cryptosystem. This is the first time to report that these two cryptosystems based on the vector decomposition technique are attacked successfully. Numerical simulation is conducted to validate the feasibility and effectiveness of the proposed attacks.