With the steady advancement of cloud computing, data privacy protection has become a crucial concern during the processes of storing and processing data in the cloud. Reversible Data Hiding in Encrypted Images (RDHEI) can recover the image losslessly and ensure the transmission of secret information, thus becoming a research hotspot. Aiming to improve embedding efficiency and enhance image security, this paper puts forward an efficient RDHEI method based on Block and Bit Plane Recomposing (BBPR) scheme (BBPR-RDHEI). Content owners reserve embedding room by using the BBPR scheme. For high bit planes, they employ Prediction Error and Bitplane Coding (PEBC). For low bit planes, they carry out bit plane recomposing. After that, they encrypt the image using a four-dimensional chaotic system to acquire encrypted image. Data hiders encrypt additional data at first and then embed it into the encrypted image, generating the marked encrypted image. Receivers can perform image recovery or data extraction operations based on the keys they hold. Numerous experimental outcomes manifest that the embedding performance of our BBPR-RDHEI method outperforms the existing advanced schemes. Meanwhile, the security level of the image is enhanced as well.