Traditionally, chaotic systems are built on the domain of infinite precision in mathematics. However, the quantization is inevitable for any digital devices, which causes dynamical degradation. To cope with this problem, many methods were proposed, such as perturbing chaotic states and cascading multiple chaotic systems. This paper aims at developing a novel methodology to design the higher-dimensional digital chaotic systems (HDDCS) in the domain of finite precision. The proposed system is based on the chaos generation strategy controlled by random sequences. It is proven to satisfy the Devaney's definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The application of HDDCS in image encryption is demonstrated via FPGA platform. As each operation of HDDCS is executed in the same fixed precision, no quantization loss occurs. Therefore, it provides a perfect solution to the dynamical degradation of digital chaos.

This paper analyzes the security of an image encryption algorithm based on a two-dimensional hyperchaotic map. This encryption algorithm generated chaotic sequences through a combination of two one-dimensional chaotic maps and used them as the permutation and diffusion key. Then, the image was encrypted by using the structure of row–column permutation, forward-diffusion, and backward-diffusion. The proposer claimed that the above algorithm was secure. However, it was found through cryptanalysis that the algorithm cannot withstand the chosen plaintext attack. Although the forward-diffusion and backward-diffusion of the original algorithm use two different diffusion keys and there is a ciphertext feedback mechanism, the analysis of the diffusion by iterative optimization showed that it can be equivalent to global diffusion. In addition, the generation of chaotic sequences in the encryption process is independent of the plaintext image, so the equivalent diffusion and permutation key stream can be obtained by adjusting the individual pixel values of the chosen plaintexts. Aiming at the security loopholes in the encryption algorithm, the theoretical and experimental results are presented to support the efficiency of the proposed attack and suggestions for improvement are given. Finally, compared with the performance analysis of the existing cracking algorithm, our cryptanalysis greatly improved the cracking efficiency without increasing the complexity of the data.

Due to the equivalent keys revealed by a chosen-plaintext attack or a chosen-ciphertext attack, most of the existing chaotic image encryption schemes are demonstrated to be insecure. In order to improve security performance, some scholars have recently proposed the plaintext-related chaotic image encryption scheme. Although the equivalent effect of a one-time pad is achieved, an additional secure channel is required to transmit the hash values or other parameters related to the plaintext before the ciphertext can be decrypted at the receiving end. Its main drawback is that an absolutely secure channel is needed to transmit the information related to the plaintext, which is not feasible in practical applications. To further solve this problem, this paper proposes a chaotic image encryption scheme based on global dynamic selection of a multi-parallel structure. First, a chaotic sequence is employed to dynamically select DNA encoding rules. Secondly, the permutation with a multi-parallel structure is performed on the DNA-encoded matrix, and the DNA decoding rules are dynamically selected according to another chaotic sequence. Finally, the diffusion rules obtained by the ciphertext feedback mechanism are introduced to determine the dynamic diffusion. Compared with the existing local dynamic encryption schemes, the main advantage of this scheme is that it can realize global dynamic selection, so as to ensure that there is no equivalent key, and it can resist the chosen-ciphertext attack or chosen-plaintext attack and does not need an additional secure channel to transmit parameters related to plaintext, which is practical. A theoretical analysis and numerical experiments demonstrate the feasibility of the method.

This paper aims to refine and expand the theoretical and application framework of higher-dimensional digital chaotic system (HDDCS). Topological mixing for HDDCS is strictly proved theoretically at first. Topological mixing implies Devaney's definition of chaos in a compact space, but not vice versa. Therefore, the proof of topological mixing promotes the theoretical research of HDDCS. Then, a general design method for constructing HDDCS via loop-state contraction algorithm is given. The construction of the iterative function uncontrolled by random sequences (hereafter called iterative function) is the starting point of this research. On this basis, this paper put forward a general design method to solve the construction problem of HDDCS, and several examples illustrate the effectiveness and feasibility of this method. The adjacency matrix corresponding to the designed HDDCS is used to construct the chaotic Echo State Network (ESN) for predicting Mackey-Glass time series. Compared with other ESNs, the chaotic ESN has better prediction performance and is able to accurately predict a much longer period of time.