Considering the prospects of public key embedding (PKE) mechanism in active forensics on the integrity or identity of ciphertext for distributed deep learning security, two reversible data hiding in encrypted domain (RDH-ED) algorithms with PKE mechanism are proposed, in which all the elements of the embedding function shall be open to the public, while the extraction function could be performed only by legitimate users. The first algorithm is difference expansion in single bit encrypted domain (DE-SBED), which is optimized from the homomorphic embedding framework based on the bit operations of DE in spatial domain. DE-SBED is suitable for the ciphertext of images encrypted from any single bit encryption and learning with errors (LWE) encryption is selected in this paper. Pixel value ordering is introduced to reduce the distortion of decryption and improve the embedding rates (ER). To apply to more flexible applications, public key recoding on encryption redundancy (PKR-ER) algorithm is proposed. Public embedding key is constructed by recoding on the redundancy from the probabilistic decryption of LWE. It is suitable for any plaintext regardless of the type of medium or the content. By setting different quantization rules for recoding, decryption and extraction functions are separable. No distortion exists in the directly decrypted results of the marked ciphertext and ER could reach over 1.0 bits per bit of plaintext. Correctness and security of the algorithms are proved theoretically by deducing the probability distributions of ciphertext and quantization variable. Experimental results demonstrate the performances in correctness, one-way attribute of security and efficiency of the algorithms.
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