A reversible data hiding scheme based on double image encryptions and spatial correlation comparisons is proposed, which improves the performance of two previous methods when dealing with flat cover images. A grayscale cover image is first encrypted. The four LSBs of each block pixel in the resulting encrypted image are encrypted once more to embed a message bit, thereby solving the problem of being unable to handle flat images encountered by the previous methods which flip three LSBs of each involved pixel at this stage. Data extraction and image recovery are achieved by a scheme of checking gray-value differences between blocks, which utilizes spatial correlations existing in nature images. Moreover, a side-match scheme that makes use of the spatial correlations of both recovered and unrecovered blocks is proposed to decrease the bit extraction error rate, in contrast with the two previous methods that consider only recovered blocks. Experimental results showing the feasibility of the proposed method are also included.