Image encryption based on the finite field cosine transform

Highlights

• A novel image encryption scheme is proposed.

• The technique involves two steps, where the finite field cosine transform is applied to blocks of an image.

• The proposed scheme provides benefits related to computational complexity and encoding of the ciphered-images.

Abstract

In this paper, a novel image encryption scheme is proposed. The technique involves two steps, where the finite field cosine transform is recursively applied to blocks of a given image. In the first step, the image blocks to be transformed result from the regular partition of subimages of the original image. The transformed subimages are regrouped and an intermediate image is constructed. In the second step, a secret-key determines the positions of the intermediate image blocks to be transformed. Besides complying with the main properties required by image encryption methods, the proposed scheme provides benefits related to computational complexity and encoding of the ciphered-images.

Introduction

The ever-increasing multimedia communication traffic has demanded high transmission rates and security in distribution and storage of data. In manifold applications, such as medical imaging systems, military image database and television, schemes devoted to reliability and security of image and video are particularly important [1], [2], [3]. This scenario has attracted the attention of companies and academia and led to the development of techniques with different purposes and based on several principles.

This paper encompasses image encryption, where visual and statistical aspects of an image are modified in order to obtain a noisy ciphered-image. In [4], for example, image encryption is done by employing chaotic sequences. In [5] and [6], image encryption schemes based on local random phase encoding in fractional Fourier transform and gyrator transform domains, respectively, are proposed; in such schemes, random phase encoding is iteratively applied to different regions of an input image. Other encryption techniques use fractional Fourier transforms, discrete cosine transform, Arnold transform, jigsaw transform and discrete fractional random transforms in the intensity-hue-saturation space [7], [8], [9], [10], [11]. Furthermore, image encryption can be implemented by the means of optical systems [12], [13], [14], [15]. Although a method for image encryption has specific theoretical foundations, it should be resistant against statistical analysis, differential attacks and other strategies that can be used by an adversary [1].

In this work, an image encryption technique based on the finite field cosine transform (FFCT) is introduced. The FFCT was first defined in [16] and corresponds to a finite field version of the discrete cosine transform (DCT). It exhibits interesting properties which are valuable for cryptographic purposes. In particular, it is possible to obtain FFCT matrices with large periods, compared to the periods of other transforms [17]. This means that the FFCT of an image block can be recursively computed a large number of times before returning to the original block. Such a property was previously explored in [18], where a simple method for uniformizing histograms of digital images was introduced (encryption was not performed in that case).

The image encryption scheme proposed in this paper consists of two steps. In the first step, 64 subimages are generated from the original image by a systematic pixel selection procedure. Each subimage is regularly divided into non-superposed 8×8 blocks, which are submitted to the recursive application of the FFCT. An intermediate image is produced by regrouping the transformed subimages. In the second step, a permutation is used as a key to determine the position of new superposed8×8 blocks of the intermediate image. Such blocks are then sequentially processed by the FFCT also in a recursive manner. The decryption scheme consists simply in inverting the order of the operations applied to the image in the encryption process.

The most attractive property of our approach is that only modular arithmetic is necessary to process the images. This means that arithmetic operations can be done with less complexity, compared to those performed using floating-point computations [19], [20]. Since rounding is not needed, round-off errors are avoided and the decrypted image is rigorously equal to the original one. Moreover, due to the recursive application of the FFCT, the ciphered-image can be stored using the same encoding scheme of the original image. Such benefits are not present in image encryption schemes based on transforms defined over real (or complex) numbers [7]. Additionally, the steps which compose our approach do not depend on any iterative strategy to achieve specific security constraints. This contrasts with most image encryption methods based on pixel scrambling and chaotic sequences, which may require several rounds of iterations to give acceptable results [9], [21].

This paper is organized as follows. After this introduction, the theory of the finite field cosine transform is reviewed in Section 2. We briefly discuss some important characteristics of the FFCT and give the transform matrix to be used in the proposed application. In Section 3, we introduce our image encryption scheme and explain the steps necessary to its implementation. In Section 4, we present computer simulations used to illustrate and evaluate the performance of our approach. In particular, the security of the method is analyzed by the use of some well-known metrics. Finally, in Section 5, the concluding remarks of the paper are presented.