A novel hybrid image encryption–compression scheme by combining chaos theory and number theory

Compression and encryption are often performed together for image sharing and/or storage. The order in which the two operations are carried out affects the overall efficiency of digital image services. For example, the encrypted data has less or no compressibility. On the other hand, it is challenging to ensure reasonable security without downgrading the compression performance. Therefore, incorporating one requirement into another is an interesting approach. In this study, we propose a novel hybrid image encryption and compression scheme that allows compression in the encryption domain. The encryption is based on Chaos theory and is carried out in two steps, i.e., permutation and substitution. The lossless compression is performed on the shuffled image and then the compressed bitstream is grouped into 8-bit elements for substitution stage. The lossless nature of the proposed method makes it suitable for medical image compression and encryption applications. The experimental results shows that the proposed method achieves the necessary level of security and preserves the compression efficiency of a lossless algorithm. In addition, to improve the performance of the entropy encoder of the compression algorithm, we propose a data-to-symbol mapping method based on number theory to represent adjacent pixel values as a block. With such representation, the compression saving is improved on average from 5.76% to 15.45% for UCID dataset.