This document presents two high capacity reversible data hiding methods for encrypted images called CPE-MHCRDH and EPE-MHCRDH. The CPE-MHCRDH method corrects prediction errors in the most significant bits of pixels before encryption, allowing two bits of secret data to be embedded per pixel. The EPE-MHCRDH method directly encrypts the original image and embeds the locations of prediction errors after encryption. Both methods compress the data-hidden encrypted image losslessly using LZ77 compression. Experimental results show the proposed methods achieve better reconstructed image quality and higher embedding capacity than previous reversible data hiding in encrypted image methods.