Discrete Wavelet Transform and Data Expansion Reduction in Homomorphic Encrypted Domain

Signal processing in the encrypted domain is a new technology with the goal of protecting valuable signals from insecure signal processing. In this paper, we propose a method for implementing discrete wavelet transform (DWT) and multiresolution analysis (MRA) in homomorphic encrypted domain. We first suggest a framework for performing DWT and inverse DWT (IDWT) in the encrypted domain, then conduct an analysis of data expansion and quantization errors under the framework. To solve the problem of data expansion, which may be very important in practical applications, we present a method for reducing data expansion in the case that both DWT and IDWT are performed. With the proposed method, multilevel DWT/IDWT can be performed with less data expansion in homomorphic encrypted domain. We propose a new signal processing procedure, where the multiplicative inverse method is employed as the last step to limit the data expansion. Taking a 2-D Haar wavelet transform as an example, we conduct a few experiments to demonstrate the advantages of our method in secure image processing. We also provide computational complexity analyses and comparisons. To the best of our knowledge, there has been no report on the implementation of DWT and MRA in the encrypted domain.