Research On Chaotification Of Dynamic Systems And Its Application In Image Encryption

Chaos as one of the brilliant scientific miracles of physics in 20th century has attracted extensive attention from scientists all over the world. Chaos anti-control (chaotification) is an important research topic in chaos theory, the anti-control of continuous-time system has achieved some relevant results, but it has not yet formed a systematic theory with universal-ity, and the design of n-dimensional hyperchaotic systems with multiple positive Lyapunov exponents has been an important problem for research. Meanwhile, chaotic system has the characteristics of high sensitive to the initial conditions, boundedness and randomness, which makes the chaotic system has a significant advantage in the design of secure communication and information encryption, and it is very suitable for the design and application of stream cipher, so how to design chaos-based cryptographic algorithm is one of important research directions of information security. To this end, this paper further studies the method on chaos anti-control of dynamical systems and the application of chaos-based information encryption,the main works are given as follows:1. Two improved algorithms based on QR orthogonal decomposition and SVD orthog-onal decomposition approaches are investigated. The Lyapunov exponents are calculated by an eigenvalue method for discrete-time dynamical systems, in theory, the more accurate cal-culations of Lyapunov exponent can be obtained with the increment of iterations according to eigenvalue method, and the limits also exist. However, due to the finite precision of computer and other reasons, the results will be numeric overflow, unrecognized, or inaccurate, which can be stated as follows: (1) The iterations k cannot be too large, otherwise, the simulation result will appear as an error message of NaN or Inf; (2) With the increment of iterations k, all Lyapunov exponents will get close to the largest Lyapunov exponent; (3) If the iterations k are too small, then the results are also inaccurate. Finally, some examples are given to illustrate the feasibility and effectiveness of the improved algorithms.2. Theoretical analysis and design method of some anti-controlled higher-dimensional hyperchaotic systems are investigated. First,some theoretical analyses for Lyapunov-exponent generating algorithms are given, and then the relationship between the number of positive Lya-punov exponents and the number of eigenvalues with positive real parts of the Jacobi matrix in the controlled system is qualitatively described and analyzed, that is, the more the number of eigenvalues with positive real parts in the controlled system are configured, the more the number of positive Lyapunov exponents in the controlled system can be generated. Therefore,one can resolve the positive Lyapunov exponents allocation problem by purposefully design-ing the number of positive real parts of the corresponding eigenvalues. Finally, two examples of such anti-controlled higher-dimensional hyperchaotic systems are given for demonstration.3. A chaotification approach based on an average eigenvalue criterion is proposed to design n-dimensional hyperchaotic systems with n-2 positive Lyapunov exponents (i.e. n-dimensional non-degenerate hyperchaotic systems). The approach consists of four steps: (1)a globally bounded controlled system is designed based on an asymptotically stable nominal system with a uniformly bounded controller; (2) a closed-loop pole assignment technique is utilized to ensure that the numbers of eigenvalues with positive real parts of the controlled system be equal to n-1 and n-2, respectively, at two saddle-focus equilibrium points; (3) the number of average eigenvalues with positive real parts is ensured to be equal to n - 2 for the controlled system over a given control period; (4) the smallest value of the positive real parts of the average eigenvalues is ensured to be greater than a given threshold value. Finally, this paper is closed with some typical examples which illustrate the feasibility and performance of the proposed design methodology.4. The method of constructing higher-dimensional non-degenerate hyperchaotic systems with multiple controllers is further investigated. First, a dissipative linear system is designed after a similarity transformation. Second,the master controller is added to the linear system,one needs to find the best control positions and parameters of master controller such that the number of average eigenvalues with positive real parts over one period time of the master con-troller is equal to n - 2, and the smallest value of average eigenvalue with the positive real parts satisfies a given condition of threshold. Third, the non-master controller is added to the controlled system, which the control positions are selected arbitrarily and the parameters are given in advance, so one gets a globally bounded controlled system. Meanwhile, two exam-ples of constructing higher-dimensional non-degenerate hyperchaotic systems with multiple controllers are given for demonstration.5. A new three-dimensional dynamical system is designed to be chaotic by the chaos anti-control method. Then, by the theory of topological horseshoe, it is proved that there exists a invariant compact set in the Poincare mapping section of the new anti-control system such that it is semi-conjugate to a 2-shift dynamics with computer-assisted. Furthermore, an image encryption scheme is designed based on the chaotic system. First, the image pixel positions are scrambled by the pseudo-random sequences generated from the chaotic system. Second,the pixel values are encrypted for multiple rounds by the chaotic stream cipher, so it achieves double chaotic encryption. Finally, the statistical analyses are given to show the feasibility and security of the proposed encryption scheme.6. A color image encryption scheme is investigated based on fractional-order hyperchaot-ic systems. Firstly, a plain image, which is known to users in advance, is chosen as a secret key to confuse the original image. Then, the confused image is encrypted by the sequences generated from the fractional-order hyperchaotic systems. With this encryption method,one can get an encrypted image that is fully scrambled and diffused, and this encryption scheme enhances the security and effectiveness. Experiments show that the algorithm is suitable for image encryption, and some statistical tests are provided to show the high security in the end.