Mosaic is a popular approach to provide privacy of data and image. However, the existing demosaicing techniques cannot accomplish efficient perfect-reconstruction. If the receiver wants to recover the original image, the extra transmission of the original subimage to be mosaicked is necessary, which consumes much channel resource and is therefore inefficient. In this paper, we propose a novel efficient recoverable cryptographic mosaic technique by permutations. A mosaic, or a privacy-protected subimage, can be constructed through either of the three permutations (Busch’s, Wu’s, and Sun’s/Minmax). These three permutations are designed to maximize the objective function as the sum of the absolute row/column index-differences. This objective is related to the sum of the pixel-to-pixel cross-correlation by our pertinent theoretical study. To measure the effectiveness of the image-mosaicing methods, we propose two image-discrepancy measures, namely summed cross-correlation (SCC) and Kullback-Leibler divergence of discrete cosine transform (DCT-KLD). Compared to the big majority of random permutations for image-mosaicing, our proposed three permutation methods can achieve much better performances in terms of SCC. Nevertheless, the advantage of the three proposed permutation methods over random permutations is not obvious according to DCT-KLD.