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This study proposes a quantum logistic image encryption algorithm based on the Rivest–Shamir–Adleman (RSA) and secure hash (SHA-3) algorithms. First, the asymmetric encryption algorithm RSA is used to randomly generate key pairs with private key and public key. A fixed matrix is then produced to confuse the plain image by additive modular operation and get preprocessed image. Second, the preprocessed image is computed by the hash function SHA-3 to obtain the plain message that is then stored secretly. Using RSA algorithm, the cipher message can be obtained corresponding to the plain message. After combining both the plain and cipher messages, the initial conditions of the quantum logistic map are calculated using a new mathematical model. Random keystream can then be generated from the quantum logistic map. Third, the keystream is further employed to cyclically confuse row and column pixels in the preprocessed image with different directions. Then, diffusion operations to odd row and odd column are carried out followed by even row and column, respectively. Finally, the cipher image can be reconstructed by both inverse cyclic confusion and inverse diffusion to each row and column. The test results show that the algorithm has strong plaintext sensitivity and key sensitivity as well as an enhanced ability to resist various cryptographic attacks. Therefore, the proposed algorithm is suitable for secure communication of network image in conjunction with a public cryptosystem.