Verifiable Chebyshev maps-based chaotic encryption schemes with outsourcing computations in the cloud/fog scenarios

Based on cloud servers' powerful storage and computing resources, users can store mass encrypted data in the cloud and outsource complex encryption computations to the cloud servers. Since cloud servers cannot be completely trusted, then data privacy and integrity are concerned about hot issues. We focus on the following problems in outsourced encryptions: how to protect the data privacy and how to check the integrity of data and the correctness of cloud server's outsourcing computations. In this paper, we at first propose a verifiable chaotic encryption based on Chebyshev polynomials. The scheme supports verifiable function for data integrity. To further improve the efficiency of the scheme, a corresponding outsourced encryption scheme is constructed, where the heavy overhead evaluations of Chebyshev polynomials are transferred from the user side to the cloud server. The outsourced encryption also provides the checkability for data integrity and correctness of cloud computations. The scheme is suitable for mobile users with limited computing resources. Moreover, the newly proposed scheme no longer depends upon the simple heuristic analysis. It achieves the indistinguishability under chosen-ciphertext attacks (IND-CCA) in the standard model based on the Chebyshev-based Decisional Diffie-Hellman (CDDH) assumption. Thus, we answer a long-term open problem for building a chaotic encryption scheme with provable security in the sense of the IND-CCA.