Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1496329, 14 pages

http://dx.doi.org/10.1155/2016/1496329

## A Chaos Robustness Criterion for 2D Piecewise Smooth Map with Applications in Pseudorandom Number Generator and Image Encryption with Avalanche Effect

^{1}Schools of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China^{2}School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received 28 August 2015; Revised 3 January 2016; Accepted 24 January 2016

Academic Editor: Jonathan N. Blakely

Copyright © 2016 Dandan Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study proposes a chaos robustness criterion for a kind of 2D piecewise smooth maps (2DPSMs). Using the chaos robustness criterion, one can easily determine the robust chaos parameter regions for some 2DPSMs. Combining 2DPSM with a generalized synchronization (GS) theorem, this study introduces a novel 6-dimensional discrete GS chaotic system. Based on the system, a 2^{16}-word chaotic pseudorandom number generator (CPRNG) is designed. The key space of the CPRNG is larger than 2^{996}. Using the FIPS 140-2 test suit/generalized FIPS 140-2 test suit tests the randomness of the 1000 key streams consists of 20,000 bits generated by the CPRNG, the RC4 algorithm, and the ZUC algorithm, respectively. The numerical results show that the three algorithms do not have significant differences. The CPRNG and a stream encryption scheme with avalanche effect (SESAE) are used to encrypt an image. The results demonstrate that the CPRNG is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs. The SESAE with one-time-pad scheme makes any attackers have to use brute attacks to break our cryptographic system.

#### 1. Introduction

The dynamic behaviors of chaotic systems have some specific features, such as their extreme sensitivity to the variables of initial conditions and system parameters, pseudorandom property, and ergodic and topological transitivity. Particularly, the property of sensitive dependence on initial conditions and parameters and robustness are suitably used in information security field [1–4].

Piecewise smooth dynamical systems (PSDSs) can exhibit complex dynamic phenomena, including chaos. PSDSs are particularly relevant in many areas of engineering and applied science. As early as in the last seventies, Feigin published his pioneering work on the analysis of C-bifurcations in -dimensional PWS systems (e.g., see [5–7]), which proposed the classification of the piecewise linear normal form for two- and three-dimensional piecewise smooth continuous maps. It makes it possible to follow closely the process of emergence of complex structures due to parameter variation.

In 1999, Banerjee and Grebogi [8] redeveloped the classification proposed by Feigin, putting his earlier results in the context of modern bifurcation analysis. Banerjee and Grebogi investigated the various types of border collision bifurcations that can occur in piecewise smooth maps by deriving a piecewise affine approximation of the map in the neighborhood of the border. In di Bernardo et al.’s book [9], the authors offer a very good survey of the rapidly developing area of the dynamics of nonsmooth systems and many beautiful examples of chaotic dynamics induced by nonsmooth phenomena.

Practical applications in chaos-based cryptography require the corresponding chaotic dynamical systems to be robust with respect to system parameters. In [10], Banerjee et al. have shown that such robust chaos can occur in piecewise smooth maps and obtained the conditions of existence of robust chaos. In [8], Banerjee and Grebogi have researched two-dimensional piecewise smooth maps and proposed the corresponding robust chaos theorems.

Since Matthews first proposed a chaotic encryption algorithm [11], there are increasing researches of chaotic encryption technology [12–21]. In [14], a fast chaos-based image encryption system with stream cipher structure is proposed. The major core of the encryption system is a pseudorandom key stream generator based on a cascade of chaotic maps, serving the purpose of sequence generation and random mixing. In [18], a novel image encryption scheme was presented, which uses a chaotic random bits generator. The chaotic random bits generator is based on the coexistence of two different synchronization phenomena. In [19], a novel stream encryption scheme with avalanche effect (SESAE) was introduced. Using the scheme and an ideal pseudorandom number generator to generate a -word key stream, one can encrypt a plaintext such that by using any key stream generated from a different seed to decrypt the ciphertext, the decrypted plaintext will become an avalanche-like text which has consecutive one’s with a high probability.

Based on one theorem proposed by Banerjee and Grebogi, this paper introduces a chaos robustness criterion for a kind of 2-dimensional piecewise smooth maps (2DPSMs) and constructs a 2DPSM with robust chaos feature. Combing the chaos generalized synchronization (GS) theorem with the 2DPSM, this paper proposes a 6-dimensional chaotic generalized synchronization system (6DCGSS) and designs a 2^{16}-word chaotic pseudorandom number generator (CPRNG). At last, using the CPRNG and the SESAE encrypts an RGB image Panda and shows the performance of the CPRNG.

The rest of this paper is organized as follows. Section 2 proposes the chaos robustness criterion for the 2DPSMs and constructs a novel 2DPSM with robust chaos feature. Section 3 introduces the definition and theorem for GS and presents a novel 6DCGSS. Section 4 designs a 2^{16}-word CPRNG and makes the statistic tests for the CPRNG. Section 5 makes an image encryption experiment with avalanche effect. Section 6 performs security analysis on the proposed image encryption scheme. Finally, some concluding remarks are presented in Section 7.

#### 2. The 2-Dimensional Piecewise Robust Chaotic Map

##### 2.1. The Robust Chaos of Normal Form

In [22], Nusse and Yorke have proved that, using some coordinate transformations, any 2-dimensional piecewise smooth map (2DPSM) can be reduced to the normal form in some small neighborhood of the fixed point of the 2DPSM. The normal form is defined as follows: where is a parameter and and are the traces and determinants of the corresponding matrices of the linearized map in the two subregions and given by

Banerjee et al. have proposed the robust chaos theorem on the normal form as follows [8, 10].

Theorem 1 (see [8, 10]). *If where and are the eigenvalues of coefficient matrix, then the 2DPSM has a bifurcation from no attractor to a chaotic attractor. The chaotic attractor for is robust.*

Formulas (3)–(5) give the criteria of the chaotic attractor appearing in 2DPSM (1). However, it will be difficult to determine the robust chaos regions for the system parameters.

Based on Theorem 1, this study proposes the following theorem which provides parameters inequalities to determine easily the robust chaos regions for the system parameters.

Theorem 2. *Let . Denote If the following inequalities hold, then conditions (3)–(5) hold. That is, 2DPSM (1) has a chaotic attractor.*

*Proof. *First, inequalities (6)–(8) are equivalent to conditions (3)-(4). Second, we show that inequality (9) implies that condition (5) holds.

The eigenvalues of coefficient matrix are shown as follows: Let ; then Substituting (10) into (5) gives Denote Substituting (13) into (12) gives because inequality (9) holds. In summary, this completes the proof.

*Remark 3. *Compared with inequalities (3)–(5), inequalities (6)–(9) more easily determine the robust chaos regions for the system parameters.

For any given nonnegative real numbers , one can determine the chaos regions on parameters from inequalities (6)–(9). For example, choosing , the robust chaos regions of 2DPSM are shown in Figure 1. The position of the red dot in Figure 1 is located in the robust chaos region surrounding the two planes.