Pseudorandom number generators have been widely used in Monte Carlo methods,communication systems, cryptography and so on. For cryptographic applications,pseudorandom number generators are required to generate sequences which havegood statistical properties, long period and unpredictability. A Dicksongenerator is a nonlinear congruential generator whose recurrence function isthe Dickson polynomial. Aly and Winterhof obtained a lower bound on the linearcomplexity profile of a Dickson generator. Moreover Vasiga and Shallit studiedthe state diagram given by the Dickson polynomial of degree two. However, theydo not specify sets of initial values which generate a long period sequence. Inthis paper, we show conditions for parameters and initial values to generatelong period sequences, and asymptotic properties for periods by numericalexperiments. We specify sets of initial values which generate a long periodsequence. For suitable parameters, every element of this set occurs exactlyonce as a component of generating sequence in one period. In order to obtainsets of initial values, we consider a logistic generator proposed by Miyazaki,Araki, Uehara and Nogami, which is obtained from a Dickson generator of degreetwo with a linear transformation. Moreover, we remark on the linear complexityprofile of the logistic generator. The sets of initial values are described byvalues of the Legendre symbol. The main idea is to introduce a structure of ahyperbola to the sets of initial values. Our results ensure that generatingsequences of Dickson generator of degree two have long period. As aconsequence, the Dickson generator of degree two has some good properties forcryptographic applications.
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