Cryptanalyzing a chaos-based image encryption algorithm using alternate structure

Abstract

Recently, a chaos-based image encryption algorithm with an alternate structure (IEAS) was proposed. This paper applies the differential cryptanalysis on the IEAS and finds that some of its properties favor the differential attack which can recover an equivalent secret key with only a few number of chosen plain-images. Detailed procedures for cryptanalyzing IEAS with a lower round number are presented. Both theoretical analysis and experimental results are provided to show the vulnerability of IEAS against differential attack. In addition, some other security defects of IEAS, including insensitivity with respect to changes of plain-images and insufficient size of the key space, are also pointed out and verified.

Introduction

The security of multimedia data such as images and videos is considered more and more important as the transmission of this type of data becomes more and more frequently via the internet. However, the differences between this type of data and texts, such as bulk data size and high redundancy among neighboring raw pixels, make most traditional encryption algorithms like DES (Data Encryption Standard) ineffective for multimedia data. In addition, multimedia encryption has other special requirements, for example fast encryption and easy cascading with other functional blocks in the system. Therefore, designing specific multimedia encryption algorithm is a desirable but challenging task.

Chaos theory was developed in depth in the 1960s. The most essential characteristic of chaos is the so-called “butterfly effect”, i.e., states of a chaotic system are very sensitive to the changes of its initial conditions and control parameters. This characteristic is similar to the confusion and diffusion properties of a cryptosystem used for measuring sensitivity of encryption results with respect to changes of the secret key and the plaintext. The subtle similarity inspired researchers to design various secure encryption algorithms by combining chaos and cryptography.

Roughly speaking, the role of chaos in designing digital encryption schemes can be categorized into the following three classes:

• creating position permutation matrices (Fridrich, 1998, Chen et al., 2004, Wang et al., 2005, Solak et al., 2010, Li and Lo, 2011);

• generating a pseudo-random bit sequence, which is then used to control combination and composition of some basic arithmetical operations like modulo addition and exclusive OR operation (Li and Zheng, 2002, Yi et al., 2002, Li et al., 2008, Li et al., 2009, Li et al., 2010, Rhouma and Belghith, 2008, Alvarez and Li, 2009, Yoon et al., 2011);

• producing ciphertext directly when the element values of the plaintext are converted to initial conditions and control parameters of a chaotic map (Pisarchik et al., 2006, Arroyo et al., 2008, Solak and Cokal, 2011).

Meanwhile, some research work show that security of chaos-based cryptography is questionable (Li et al., 2004, Masuda et al., 2006, Alvarez and Li, 2006, Zhou and Au, 2011) and that some specific properties of chaotic system can be used to support the corresponding attack (Chen et al., 2012). Due to the simple syntax of uncompressed image and easy extension of image encryption schemes for other multimedia data, a great number of chaos-based encryption schemes consider image data as the encryption object. In the past decade, hundreds of chaos-based image encryption schemes have been proposed, the most famous schemes are the ones proposed in Fridrich (1998) and Chen et al. (2004).

In Zhang et al. (2007), a new image encryption algorithm with an alternate structure (IEAS) based on the generalized cat map and OCML (One-way Coupled Map Lattice) was proposed, where the two maps are used for realizing position permutation/diffusion and value substitution, respectively. Essentially, the structure of IEAS belongs to a Feistel network, i.e., an iterative block cipher where the output of the current round is determined by that of the previous one. This paper focuses on the security analysis of IEAS. It is found that, when the integer parameter is even, some properties of IEAS favor a differential attack which can recover an equivalent secret key with a small number of chosen plain-images. The procedure of the differential attack is presented in detail when the round number of IEAS is less than or equal to four. In addition, the cryptanalysis finds some other security defects of IEAS, like insensitivity with respect to changes of plain-images and an insufficient key space.

The rest of this paper is organized as follows. The next section introduces the image encryption algorithm under investigation, i.e., IEAS. Section 3 presents a comprehensive cryptanalysis on the algorithm with some experimental results demonstrated. The last section concludes the paper.