Some Polynomial Chaotic Maps Without Equilibria and an Application to Image Encryption with Avalanche Effects

This study uses seven four-dimensional four-variable polynomial chaotic maps without equilibria in combination with generalized chaos synchronization (GCS) theorem to construct eight-dimensional bidirectional discrete generalized chaos synchronization (8DBDGCS) systems without equilibria. By combining the 8DBDGCS system with the GCS theorem, a 12-dimensional GCS system is designed. Numerical simulation verifies the chaotic dynamics of the 12-dimensional GCS system, which is used to design a 2¹⁶-word chaotic pseudorandom number generator (CPRNG). The SP-8002 test suite is used to test the randomness of four 100-key streams consisting of 1 000 000 bits generated respectively by the CPRNG, a six-dimensional GCS-based CPRNG, the RC4 algorithm and the ZUC algorithm. The results show that the randomness performances of the two CPRNGs are promising, suggesting that there are no significant correlations between the key stream and the perturbed key streams generated via the 2¹⁶-word CPRNG. In addition, theoretically the key space of the CPRNG is larger than 2¹¹⁹⁵. The CPRNG is used with an avalanche-encryption scheme to encrypt an RGB balloon image, demonstrating that the CPRNG is able to generate the avalanche effects which are similar to those generated via ideal 2¹⁶-word CPRNGs.