A Novel Approach for Designing Dynamical S-Boxes Using Hyperchaotic System

A Novel Approach for Designing Dynamical S-Boxes Using Hyperchaotic System

Jun Peng (Chongqing University of Science and Technology, China), Du Zhang (California State University, Sacramento, USA) and Xiaofeng Liao (Chongqing University, China)
DOI: 10.4018/jcini.2012010105


In the information security field, the substitution boxes (S-boxes) have been extensively used in many cryptographic systems. This paper presents a novel approach for generating dynamically cryptographically S-boxes using a four-dimensional hyperchaotic Lorenz system. Within the algorithm, the initial condition is employed to drive the hyper-chaotic system to generate a chaotic sequence which is used to construct a chaotic key-dependent S-box. With different system initial conditions, many of distinct S-boxes can be obtained dynamically. Some cryptographic properties for a good S-box such as bijection, nonlinearity, SAC (Strict Avalanche Criterion), BIC (Bit Independence Criterion), and differential approximation probability are found to hold in the obtained S-boxes. The analytic results indicated that all the criteria for designing strong S-boxes can be achieved. The comparison of the proposed method for generating S-boxes with other chaos-based schemes indicates that our S-boxes have a better performance with respect to some properties. Finally, the authors give an example of a digital image encryption algorithm using their S-box and the results of image statistical analysis show that the algorithm has the desirable cryptographic properties.
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In the information security field, the substitution boxes (S-boxes) have been extensively used in most of cryptographic systems, such as DES, IDEA and AES. S-boxes are core component of the DES-like cryptosystems and the only nonlinear component of these ciphers. The security of a cryptographic system primarily depends on the cryptographic strength of the S-box (Feng, 2000). In block ciphers, they are typically used to obscure the relationship between the secret key and the ciphertext's Shannon property of confusion. In many cases, the S-boxes are carefully chosen to provide the cryptosystem with abilities of resisting cryptanalysis. Therefore, the construction of cryptographically strong S-boxes is an important task. Mathematically, an S-box is a nonlinear mapping (or substitution) from to , where and represent the vector spaces of and tuples of elements from , respectively.

As chaotic sequences have many significant properties favorable to the information security, such as random-like and extreme sensitivity to the initial condition and control parameters, the study on using chaos theory in information security has attracted great attentions, such as cryptographic ciphers (Yang, 1997; Fridrich, 1998; Kocarev, 2001), image encryption systems (Chen, 2004; Rhouma, 2009), and chaos-based hash functions (Xiao, 2008; Peng, 2008; Li, 2012). Recent research shows that it is a promising direction to use the distinct properties of chaos to design S-boxes, and there are many methodologies and design criteria for constructing chaos-based S-boxes in the literature. Jakimoski and Kocarev presented a map from chaotic Logistic map, in which map actually can be viewed as an S-box (Jakimoski, 2001). Tang and Liao (2005) and Tang, Liao, and Chen (2005) proposed a new method for obtaining cryptographically strong dynamic S-boxes based on the iterating discretized chaotic map. Later, Chen et al. presented a scheme to construct S-box by employing a three-dimensional chaotic Baker map, which has more intensive chaotic characters than the two-dimensional one (Chen, 2007). Recently, Wang et al. proposed a method for designing S-box based on chaotic neural network (Wang, 2010). After that, Wang et al. studied a novel method to design S-box using chaotic map and genetic algorithm (Wang, 2012), where the problem of constructing S-box was transformed to a Traveling Salesman Problem. Obviously, these research achievements are very useful to the later studies on the construction algorithms of S-boxes using chaotic map.

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