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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 2, FEBRUARY 2010

Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps Kwok-Wo Wong, Senior Member, IEEE, Qiuzhen Lin, and Jianyong Chen

Abstract—Based on the observation that iterating a skew tent map reversely is equivalent to arithmetic coding, a simultaneous compression and encryption scheme is proposed in which the chaotic map model for arithmetic coding is determined by a secret key and keeps changing. Moreover, the compressed sequence is masked by a pseudorandom keystream generated by another chaotic map. This two-level protection enhances its security level, which results in high key and plaintext sensitivities. The compression performance of our scheme is comparable with arithmetic coding and approaches Shannon’s entropy limit. Index Terms—Arithmetic coding, chaotic map, simultaneous compression and encryption.

I. I NTRODUCTION

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RADITIONALLY, source coding and encryption are performed one after another to reduce the data volume while maintaining information secrecy. A typical example is the compression of a private photo using JPEG format and then the encryption of the compressed file using the Advanced Encryption Standard. However, there is an increasing interest in simultaneous compression and encryption [1]–[7]. This can be achieved by either embedding compression into encryption algorithms or adding cryptographic sense in compression schemes. An attempt using the first approach was reported in [1]. The allocation of plaintext symbols in the dynamic lookup table used in a chaos-based cryptographic scheme is determined by the plaintext statistics. The resultant ciphertext is shorter than the plaintext, and so compression is achieved while performing encryption. There were more reports based on the second approach, particularly the introduction of cryptographic sense in entropy coding methods. However, most of these schemes are found insecure or inefficient. The multiple Huffman table approach [4] was cryptanalyzed in [8] and [9]. The arithmetic coding method using key-based interval splitting [5] suffers from known-plaintext attack [9]. The secure arithmetic coding [6] was broken in [10]. Randomized arithmetic coding [7] is not cryptanalyzed but is considered inefficient when compared with the traditional compress-then-encrypt approach [9].

In recent years, some approaches [11]–[13] utilizing chaotic systems for source coding, particularly arithmetic coding, have been suggested. However, they did not investigate the relationship between these two areas, but just employed chaotic systems as pseudorandom bitstream generators. Moreover, a design fault of the chaos-based adaptive arithmetic coding scheme [11] has been recently found, and a modified approach was suggested [14]. Unfortunately, the modified version is also vulnerable to chosen-plaintext attack [14]. The relationship between arithmetic coding and chaotic maps was studied in [15] and [16]. In [15], arithmetic coding is found equivalent to finding the best initial condition for iterating a chaotic map to generate a symbolic sequence corresponding to the source message. In [16], source coding and chaotic systems are related to each other by treating messages emitted by independent and identically distributed sources as symbol sequences of a chaotic nonlinear dynamical system known as the generalized Luroth series (GLS). It is proven that GLS achieves Shannon’s entropy bound and is a generalization of arithmetic coding [16]. Inspired by [15] and [16], here, we propose a scheme for the simultaneous compression and encryption of message sequences with multiple symbols. Compression is achieved by iterating a multisegment piecewise linear chaotic map, whereas encryption is realized by changing the chaotic map model continuously using a secret key, without affecting the essence of arithmetic coding. The use of a chaotic map for this purpose is better than existing schemes based on traditional arithmetic coding [5]–[7]. This is because the position and the direction of the linear segments in the map are governed by the secret key. To further enhance the security, the compressed sequence is masked by a pseudorandom keystream generated by another chaotic map, as suggested in [10]. The rest of this brief is organized as follows. In Section II, the concept of arithmetic coding using a skew tent map is illustrated. The proposed simultaneous arithmetic coding and encryption scheme is described in Section III. The performance of our scheme is reported in Section IV, whereas conclusions are drawn in Section V. II. A RITHMETIC C ODING U SING S KEW T ENT M AP

Manuscript received September 16, 2009. First published February 8, 2010; current version published February 26, 2010. This work was supported by a grant from Research Grants Council of the Hong Kong Special Administrative Region, China (Project CityU 122308). This paper was recommended by Associate Editor Y. Horio. K.-W. Wong is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong (e-mail: itkwwong@ cityu.edu.hk). Q. Lin and J. Chen are with the Department of Computer Science and Technology, Shenzhen University, Shenzhen 518060, China. Digital Object Identifier 10.1109/TCSII.2010.2040315

Here, the concept of arithmetic coding using a skew tent map is illustrated. The map is defined as [17]  x/p, 0≤x