A Novel 3 Dimensional Chaotic System and Design of Pseudorandom Number Generator

In this paper, a novel 3-dimensional discrete chaotic map is introduced. The calculated Lyapunov exponents of the map are 0.0705, 0.0150 and -0.0743. Numerical simulations show that the dynamic behaviors of the chaotic map have chaotic attractor characteristics. Based on the chaotic map and a chaos generalized synchronization (GS) theorem, a 6-dimensional chaotic GS system is constructed. Using the chaotic GS system and a transformation T form ℝ³ to binary set {0, 1} designs two chaos-based pseudorandom number generators (CPRNGs). Using FIPS 140-2 test suite and SP 800-22 test suit tests the randomness of the CPRNGs and a RC4 PRNG. The 100 outputs of the CPRNGs and the RC4 PRNG all passed the FIPS 140-2 criteria. The sub-test of NIST SP 800-22 which has maximum failure passing rate for the CPRNGs and the RC4 PNG are the Random Excursions Test. The failure percentages of this test to the CPRNG3 and the RC4 PRNG are 48%, 36% and 43%, respectively. The mean p-values of SP 800-22 tests of the three PRNGs do not have significant differences. Numerical simulations show that for the perturbations of the keys of the CPRNG which are larger than 10^-14, the key streams have an average 99.61% different codes which are different from the codes generated by unperturbed keys. The results imply that the key streams of the CPRNG have sound pseudorandomness.