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Designing a discrete chaotic system via fractal transformation has become a new method for engineering applications. This method generates new discrete chaotic system through external mechanisms, instead of the traditional way of internal mechanisms. The way of building novel discrete chaotic system is enriched by fractal and mathematical operation. Taking one-dimensional ICMIC map and two-dimensional Hénon map as the seed maps, dynamics of the generated chaotic map is analyzed by bifurcations, complexity and spectrum distribution characteristics. The results show that the new discrete chaotic map has the advantages in complexity and distribution in the parameter space. Finally, the digital circuit of fractal chaotic system is implemented based on DSP technique. The feasibility of the circuit is verified. Therefore, it has good application prospects in secure communication.

Based on a stream encryption scheme with avalanche effect (SESAE), a stream encryption scheme with both key avalanche effect and plaintext avalanche effect (SESKPAE) is introduced. Using this scheme and an ideal $2^d$-word ($d$-segment) pseudorandom number generator (PRNG), a plaintext can be encrypted such that each bit of the ciphertext block has a change with the probable probability of $(2^d-1)/2^d$ when any word of the key is changed or any bit of the plaintext is changed. To that end, a novel four-dimensional discrete chaotic system (4DDCS) is proposed. Combining the 4DDCS with a generalized synchronization (GS) theorem, a novel eight-dimensional discrete GS chaotic system (8DDGSCS) is constructed. Using the 8DDGSCS, a $2^{16}$-word chaotic pseudorandom number generator (CPRNG) is designed. The keyspace of the $2^{16}$-word CPRNG is larger than $2^{1195}$. Then, the FIPS 140-2 test suit/generalized FIPS 140-2 test suit is used to test the randomness of the 1000-key streams consisting of 20000 bits generated by the $2^{16}$-word CPRNG, the RC4 algorithm PRNG and the ZUC algorithm PRNG, respectively. The test results show that for the three PRNGs, there are 100%/98.9%, 99.9%/98.8%, 100%/97.9% key streams passing the tests, respectively. Furthermore, the SP800-22 test suite is used to test the randomness of four 100-key streams consisting of 1000000 bits generated by four PRNGs, respectively. The numerical results show that the randomness performances of the $2^{16}$-word CPRNG is promising, showing that there are no significant correlations between the key streams and the perturbed key streams generated via the $2^{16}$-word CPRNG. Finally, using the $2^{16}$-word CPRNG and the SESKPAE to encrypt two gray-scale images, test results demonstrate that the $2^{16}$-word CPRNG is able to generate both key avalanche effect and plaintext avalanche effect, which are similar to those generated via an ideal CPRNG, and performs better than other comparable schemes.

The chaotic system is widely used in chaotic cryptosystem and chaotic secure communication. In this paper, a universal method for designing the discrete chaotic system with any desired number of positive Lyapunov exponents is proposed to meet the needs of hyperchaotic systems in chaotic cryptosystem and chaotic secure communication, and three examples of eight-dimensional discrete system with chaotic attractors, eight-dimensional discrete system with fixed point attractors and eight-dimensional discrete system with periodic attractors are given to illustrate how the proposed methods control the Lyapunov exponents. Compared to the previous methods, the positive Lyapunov exponents are used to reconstruct a hyperchaotic system.

Numerous image encryption schemes are hindered by weak correlations with plaintex images, inadequate image reconstruction quality, and inefficient transmission and storage. To address these problems, this paper proposes an image compression–encryption algorithm that combines a three-dimensional discrete chaotic system, Josephus circle and compression perception. First, a new three-dimensional discrete hyperchaotic system is constructed, utilizing a hash function to generate the initial value of the chaotic system, through which the measurement matrix and the sequences required for the encryption process are constructed. An encryption algorithm based on chaotic sequence and Josephus circle is designed. Finally, the proposed compressed encryption algorithm is simulated and tested, and the results of experimental tests indicate that the proposed algorithm provides robust security and high-quality reconstruction performance.

Based on a generalized synchronization (GS) theorem of discrete chaotic systems, this paper introduces an 8-dimensional discrete chaotic GS system. Simulations show that the trajectories of the GS system have significant characteristics of chaotic attractor. Using the GS system designs a 2¹⁶ number string chaotic pseudorandom number generator (CPRNG). The FIPS 140-2 test suit/generalized FIPS 140-2 test suit are used to test the randomness of three 1000-key streams consisting of 20000 bits generated by the CPRNG , the RC4 algorithm and the ZUC algorithm, respectively. The test results confirm that there are 100%/98.6%, 99.9%/98.8%, 100%/97.9% key streams passing the tests, respectively. Numerical simulations suggest that there are no significant correlations between the key stream and the perturbed key stream. The key space of the CPRNG is larger than 2¹¹⁹⁵. Using the CPRNG and a stream encryption scheme with avalanche effect (SESAE) encrypts a text. The results suggest that the CPRNG is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs.