  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 have
good statistical properties, long period and unpredictability. A Dickson
generator is a nonlinear congruential generator whose recurrence function is
the Dickson polynomial. Aly and Winterhof obtained a lower bound on the linear
complexity profile of a Dickson generator. Moreover Vasiga and Shallit studied
the state diagram given by the Dickson polynomial of degree two. However, they
do not specify sets of initial values which generate a long period sequence. In
this paper, we show conditions for parameters and initial values to generate
long period sequences, and asymptotic properties for periods by numerical
experiments. We specify sets of initial values which generate a long period
sequence. For suitable parameters, every element of this set occurs exactly
once as a component of generating sequence in one period. In order to obtain
sets of initial values, we consider a logistic generator proposed by Miyazaki,
Araki, Uehara and Nogami, which is obtained from a Dickson generator of degree
two with a linear transformation. Moreover, we remark on the linear complexity
profile of the logistic generator. The sets of initial values are described by
values of the Legendre symbol. The main idea is to introduce a structure of a
hyperbola to the sets of initial values. Our results ensure that generating
sequences of Dickson generator of degree two have long period. As a
consequence, the Dickson generator of degree two has some good properties for
cryptographic applications.
