To reduce prediction error and improve the embedding rate of secret messages, a new reversible data hiding algorithm for encrypted images (RDHEI) is designed using adaptive total variation and cross-cyclic shift called RDHEIAC. The adaptive total variation is first used to generate the prediction error image. Then, the bit-plane rearrangement based on Hilbert scanning is performed to make room for data hiding. Thirdly, an improved run-length encoding is applied to compress the error image. Subsequently, a cross-cyclic shift operation and a diffusion operation based on chaotic maps are used to produce encrypted images. Finally, secret messages are inserted into the encrypted image using bit substitution. Experimental results indicate that the proposed RDHEI scheme is privacy secure and has high embedding capacity and image fidelity. Moreover, information extraction and image restoration are separable. Compared with previous RDHEI algorithms, the proposed RDHEI algorithm has a 47.07% higher embedding rate.