Long Period Sequences Generated by the Logistic Map over Finite Fields with Control Parameter Four

Recently, binary sequences generated by chaotic maps have been widely studied. In particular, the logistic map is used as one of the chaotic map. However, if the logistic map is implemented by using finite precision computer arithmetic, rounding is required. In order to avoid rounding, Miyazaki, Araki, Uehara and Nogami proposed the logistic map over finite fields, and show some properties of sequences generated by the logistic map over finite fields. In this paper, we show some properties of periods of sequences generated by the logistic map over finite fields with control parameter four. In particular, we show conditions for parameters and initial values to have a long period, and asymptotic properties for periods by numerical experiments. Conditions for initial values are described by values of the Legendre symbol. The main idea is to introduce a structure of a hyperbola to certain sets of initial values. It follows that periods of sequences generated by the logistic map over finite fields on the sets of initial values are induced by periods of sequences generated by the square map on the parameter spaces of the hyperbola.