In this paper, we will discuss properties of different computational methods of the logistic map over integers. We will give four computational methods of the map in order to implement the map on a computer with limited memories, and then will evaluate average processing times and lengths of sequences generated by these methods. By these results, we will provide a constructive use of these four computational methods of the logistic map over integers for a design of the pseudorandom number generator.

Several methods for finding a basis conversion matrix between two different bases in an extension field have been proposed. Among them, the one based on Gauss period normal basis (GNB) is on average the most efficient. However, since it needs to construct a certain tower field , some inefficient cases in which the towering degree becomes large have been reported. This paper first determines that such inefficient cases are caused by the GNB condition. In order to overcome this inefficiency, we propose a method that does not use any GNB in the target extension field , but instead uses a certain polynomial basis in derived by a certain small cyclic group in . This causes re- laxation of the condition for the towering degree . In addition, our experimental results show that the proposed method substantially accelerates the computation time for finding a basis conversion matrix.

We are investigating the logistic map over integers and its pseudorandom number generator. Focusing on differences between the logistic maps over floating-point numbers and over integers, we herein show that the integer format is a suitable representation for implementing the logistic map on computers with limited memories.

On the studies of the logistic map, many reports are presented because of the simple structure of the equation. Calculations with finite precision are usually used for implementations of the map for computers. Then a rounding is needed to these calculations. There are five major methods to implement rounding, but there was not any paper on the rounding of the map. In the present paper, we get some experimental results that the properties of sequences change greatly by roundings, and give some theoretical analyses of them. Then we will give principal reasons that the map with a floor function for rounding generates long aperiodic subsequences.

Self-dual normal basis is efficient for the arithmetic operations in extension field and especially trace calculation. However, self-dual normal bases do not exist in when characteristic p is odd and degree m is even. This paper proposes a method to construct an efficient normal basis for trace calculation when extension degree is even. In this paper, we call it pseudo self-dual normal basis.