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
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Abstract

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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Introduction

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 jcini.2012010105.m01 S-box is a nonlinear mapping (or substitution) from jcini.2012010105.m02 to jcini.2012010105.m03, where jcini.2012010105.m04 and jcini.2012010105.m05 represent the vector spaces of jcini.2012010105.m06 and jcini.2012010105.m07 tuples of elements from jcini.2012010105.m08, 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 jcini.2012010105.m09jcini.2012010105.m10jcini.2012010105.m11jcini.2012010105.m12jcini.2012010105.m13 from chaotic Logistic map, in which map jcini.2012010105.m14 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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