  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.
