A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains

Abstract

Based on 1-D fractional Fourier transform, we proposed an image encryption algorithm in order to hide two images simultaneously. When the fractional order is closed to 1, most energy in frequency domain is centralized in the center part of spectrum. The image can be recovered acceptable by using a half of spectrum, which locates in the middle part at x-direction or y-direction. Cutting operation is employed in order to combine two spectra. Double random phase encoding is employed for image encryption. The corresponding numerical simulations are performed to demonstrate the validity and efficiency of the algorithm.

Introduction

In the recent years, many optical image encryption algorithms [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] based on some tools or models, such as double random phase encoding, digital holography, phase retrieval, intensity random coding and virtual optics, have been proposed. A classical one of them is random phase encoding [1], which has been proved that the security is attractive in the aspect against some attacks [21], [22]. Moreover, this algorithm has been developed into the domains of fractional Fourier transform (FRFT) [23] and Fresnel transform [24]. Recently, the FRFT and its improved versions have been used in the field of image encryption [16], [17], [18], [19], [20]. In fact, the former algorithms are researched for encrypting single image with some random elements. However, the problem that how to hide more than one image was ignored in a long time.

Multiple image encryption (MIE) is a new concept and interesting in the field of optical information security. There multiple 2-D data, such as a series of pictures of an object or many different images, is encrypted with certain relations, which can be often stored or transmitted together. Situ and Zhang first proposed an algorithm of MIE by using wavelength multiplexing [25]. Because the multiple original images are directly added into one image, the qualities of the corresponding decrypted images in the algorithms [25] are not perfect due to the cross-talk effects between images. Four kinds of double image encryption schemes [18], [26], [27], [28] were reported, which belong to the algorithm of MIE. One of them can be expanded to encrypt more than two images [26]. However, the phase retrieval algorithm in the paper [26] requires complex computational load.

In this paper, we propose a new kind of double image encryption by use of cutting spectrum in the FRFT domains. The center part of spectrum includes main information of an image, which is considered for describing and recording the image approximatively. The higher frequency of FRFT is cut down and replaced by zero in the algorithm. Although this operation sacrifices the detail information of image, the visual difference between original image and recovered image is very small. The part spectra of two images are scrabbled up a new spectrum, which is encrypted with double random phase encoding technique. All transforms are limited in the format of one-dimension, which was used for optical image encryption [29].

The rest of the paper is organized in the following sequence. In Section 2, the algorithm of double image encryption is described and analyzed in detail. In Section 3, numerical result is given to demonstrate the validity of the algorithm. Some concluding remarks are summarized in the final section.