Application of residue number system (RNS) to image processing using orthogonal transformation

Several techniques for image encryption have been proposed over the years with significant consideration of basic cryptographic goals such as authentication, integrity and confidentiality. Recently, another method for encrypting image data using an Orthogonal transform namely Walsh Hadamard transform on residual number system have been proposed. In this paper, we basically analyse this approach and propose an efficient method for this type of encryption scheme. We modify the transform algorithm of the previous technique by implementing the Fast Walsh Hadamard transform algorithm. The fast Walsh Hadamard transform algorithm has been proved to be the efficient algorithm to compute the Walsh Hadamard transform with computational complexity of O(N log N). We emphasise that the naive implementation of the Walsh Hadamard transform yields a computational complexity of O(N²). In this paper we demonstrate the efficiency of our approach using 32 chosen eigenvalues in the key generation algorithm. The eigenvalues are derived from a reference image. Throughout the processes on the image data, we use modular arithmetic to ensure that computations with the resulting RNS become very efficient. Moreover, the approach considers image in a divided matrix domain and finally combines all independent cryptographic operations as encryption is a one-to-one mapping. This deals with the possibility of having any pixel value ill-stored or wrongly received at the receiver end, without affecting the decryption process. However, the final recovered image will differ by a negligible amount.