Search

Classical one-dimensional chaotic map has many ideal characteristics which is quite suitable for many different kinds of scientific fields, especially cryptography. In this paper, we propose an idea of constructing high-dimensional (HD) cyclic symmetric chaotic maps by using one-dimensional (1D) chaotic map. Two constructed 3D cyclic symmetric chaotic maps are taken as the examples, named three-dimensional cyclic symmetric logistic map (3D-CSLM) and three-dimensional cyclic symmetric Chebyshev map (3D-CSCM), respectively. Numerical experiments show that the new maps possesses better dynamical performances, and their parameters have a wider range, compared with the original map. Furthermore, to verify its effect in image encryption, a novel image encryption algorithm based on 3D-CSLM and DNA coding is proposed. DNA method for image encryption can improve the efficiency of permutation and diffusion. Firstly, the algorithm uses 3D-CSLM to generate chaotic sequences for DNA operation rule selection and pixel permutation. Then through the DNA XOR operation to achieve diffusion, and finally form an encrypted image. Several simulation tests results indicate that the proposal has a promising security performance and strong anti-attack ability.

Chaotic systems are widely used in various fields, but under the finite precision device, chaotic systems would fall into a cycle and subsequently the performance degrade. Thus, the suppression method of the dynamic degradation of digital chaos is receiving increasing attention. This paper proposes a new improvement model to suppress the dynamical degradation under finite computing accuracy equipment. By using the difference between two maps of the same type but with different initial values, and the state feedback function to improve the performance of the digital chaotic map and extend the time before the chaotic map enters the cycle. Take the 1D Logistic map and x-dimensional of Baker map as examples to prove the effectiveness of the improvement model. Then we proposed a new key selection method, in what part of information of the image would be selected by using a chaotic map to generate a special value. The special value would be used as part of the key. Based this method, a new image encryption algorithm was proposed. The information entropy of the image encrypted by our encryption algorithm is 7.9972, the NPCR and UACI are 0.996095 and 0.334635, respectively, what both are very close to ideal values. The experimental simulation results show that the image encryption scheme exhibits good performances and high security and effectively resists various attacks.

In a finite precision computing environment, the trajectories of all chaotic sequences enter a cycle leading to degradation of their dynamics. In this paper a new perturbation feedback hybrid control method to reduce the influence of finite precision. A traditional logistic map is introduced as a pseudo-random sequence generator for time-varying perturbation to perturb the coefficients of chaotic map and make them iteratively changed in the chaotic region. The nonlinear feedback mechanism has high complexity. Numerical analysis results show that the perturbation-feedback hybrid control method can effectively attenuate the dynamic degradation of digital chaotic systems. Further, we propose a simple encryption algorithm based on the perturbation-feedback hybrid control method and apply it to image encryption. The NPCR and UACI of our encryption method are 0.99609 and 0.33464, respectively and the information entropy is as high as 7.9976. All the numerical experiments results prove that the proposed algorithm is highly secure, resistant to multiple attacks, and is more competitive than other encryption algorithms.

A chaotic map, which is realized on finite precision device, such as computer, will suffer dynamical degradation. Such chaotic maps cannot be regarded as rigorous chaos anymore, since their chaotic characteristics are degraded, and naturally, these kinds of chaotic maps are not secure enough for cryptographic use. Therefore, in this paper, a novel perturbation method is proposed to reduce the dynamical degradation of digital chaotic maps. Once the state is repeated during the iteration, the parameter and state are both perturbed to make the state jump out from a cycle. This method is convenient to implement without any external sources and can be used for different kinds of digital chaotic maps. The most widely used logistic map is used as an example to prove the effectiveness of this method. Several numerical experiments are provided to prove the effectiveness of this method. Under the same precision, the number of iterations when entering a cycle and the period of the improved map are greater than those of the original one. The complexity analysis shows that the improved map can get an ideal complexity level under a lower precision. All these results prove that this perturbed method can greatly improve the dynamical characteristics of original chaotic map and is competitive with other remedies. Furthermore, we improve this method by using a variable perturbation, where the perturbation is affected according to the number of iteration steps. Numerical experiments further prove that this improved perturbation method has a better performance in suppressing dynamical degradation.

To improve the security level of image encryption based on chaos theory, a novel bit-level transform method which could enhance the dynamical performance of chaotic maps is proposed and a new image encryption algorithm based on a random irregular blocking method is also designed in this paper. Firstly, in the bit-level transform method, three chaotic maps are combined to generate a novel chaotic sequence. From a series of performance analyses results, the generated sequence not only successfully passes NIST tests, but also performs better than the original sequence, with a larger chaotic parametrial range, high complexity, etc. Then, an image encryption algorithm based on the newly generated chaotic sequence is proposed, in which a random irregular blocking method is adopted. Random irregular blocking could solve the problem of insecurity in the fixed blocking method. With a color Lena image as an encryption example, we analyze the security of the encryption algorithm by conducting a sufficiently large number of experiments using the encrypted image. For the encrypted Lena image, the average values of information entropy, NPCR, and UACI are 7.9931, 0.9961, and 0.3352, respectively. The experimental results and security analyses results provide evidence that the proposed encryption algorithm has good performance and can resist various typical attacks.

Chaos maps are widely used in image encryption systems due to their intrinsic advantages such as extreme sensitivity to initial values., ergodicity and pseudo-randomness. 1D Logistic map has attracted the attention of researchers due to its simple structure and easy implementation, but also because of this, the map is easy to be affected by finite precision, resulting in dynamic degradation, at low precision, the sequence generated by this map not only enters a period quickly, but also has a shorter period. Thus, taking 1D Logistic map as an example, we proposed a method to suppress the dynamic degradation of digital chaotic systems by using parameter variables and state variables to influence each other, and using sine function as feedback function to destroy the state space. The simulation results show that the improved logistic mapping with the proposed method has better randomness and higher complexity than the original logistic mapping. To prove the practicability and applicability of the improved chaotic map, we design a new image encryption algorithm, which is suitable for both color image and grayscale image. The numerical results indicate that the proposed algorithm has high encryption efficiency, good resistance to various attacks and certain competitiveness with other encryption algorithms.

Digital chaotic maps are not secure enough for cryptographic applications due to their dynamical degradation. In order to improve their dynamics, in this paper, a novel method with time-delay linear feedback and parameter perturbation is proposed. The delayed state variable is used to construct the linear feedback function and parameter perturbation function. This method is universal for all different digital chaotic maps. Here, two examples are presented: one is 1D logistic map and the other is 2D Baker map. To show the effectiveness of this method, we take some numerical experiments, including trajectory and phase space analysis, correlation analysis, period analysis, and complexity analysis. All the numerical results prove that the method can greatly improve the dynamics of digital chaotic maps and is quite competitive with other proposed methods. Furthermore, a simple pseudorandom bit generator (PRBG) based on digital Baker map is proposed to show its potential application. The proposed PRBG is completely constructed by the digital chaotic map, without any other complex operations. Several numerical results indicate that this PRBG has good randomness and high complexity level.

The eigenvalue is one of the important cryptographic complexity measures for sequences. However, the eigenvalue can only evaluate sequences with finite symbols&mdash, it is not applicable for real number sequences. Recently, chaos-based cryptography has received widespread attention for its perfect dynamical characteristics. However, dynamical complexity does not completely equate to cryptographic complexity. The security of the chaos-based cryptographic algorithm is not fully guaranteed unless it can be proven or measured by cryptographic standards. Therefore, in this paper, we extended the eigenvalue complexity measure from the finite field to the real number field to make it applicable for the complexity measurement of real number sequences. The probability distribution, expectation, and variance of the eigenvalue of real number sequences are discussed both theoretically and experimentally. With the extension of eigenvalue, we can evaluate the cryptographic complexity of real number sequences, which have a great advantage for cryptographic usage, especially for chaos-based cryptography.

Synchronization of complex dynamical network has been a hot topic for its practical applications. In this paper, the pinning impulsive cluster synchronization of complex dynamical network is investigated. Based on the theories of Lyapunov stability and impulsive differential equations, our pinning impulsive control strategy is proved to be correct and effective for realizing cluster synchronization once the sufficient condition given in the theorem is satisfied, and is also universal for different dynamical networks. Moreover, some numerical simulations are performed to show that our control strategy is quite effective and convenient.