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Based on the robust chaos theorem of S-unimodal maps, this paper studies a kind of cubic polynomial discrete maps (CPDMs) and sets up a novel theorem. This theorem gives general conditions for the occurrence of robust chaos in the CPDMs. By using the theorem, we construct a CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed point arithmetic, we investigate the cycle lengths of the CPDM and a Logistic map. The results show that the maximum cycle lengths of chaotic sequences with length generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach , the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches , there is the situation without cycle for cubic polynomial chaotic sequences with length . By using the CPDM and Logistic map, we design four chaos-based pseudorandom number generators (CPRNGs): CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. The randomness of two key streams consisting of bits is tested, respectively, generated by the four CPRNGs. The result suggests that CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.

Robust Chaos of Cubic Polynomial Discrete Maps with Application to Pseudorandom Number Generators