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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mn mathvariant="normal">1000</mml:mn></mml:mrow></mml:math> chaotic sequences with length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn mathvariant="normal">3</mml:mn><mml:mo>×</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msup></mml:math> generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msup><mml:mo>~</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">13</mml:mn></mml:mrow></mml:msup></mml:math>, the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">14</mml:mn></mml:mrow></mml:msup></mml:math>, there is the situation without cycle for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mn mathvariant="normal">1000</mml:mn></mml:mrow></mml:math> cubic polynomial chaotic sequences with length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mn mathvariant="normal">3</mml:mn><mml:mo>×</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msup></mml:math>. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mn mathvariant="normal">1000</mml:mn></mml:mrow></mml:math> key streams consisting of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mn mathvariant="normal">20000</mml:mn></mml:mrow></mml:math> 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.